tag:blogger.com,1999:blog-31089126149147617502024-03-27T07:37:04.488+01:00Reflexiones de un IngenieroAquí se puede encontrar cualquier cosa que se pase por mi mente y crea interesante compartir (electrónica, ingeniería, coches, fotografía...)JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.comBlogger36125tag:blogger.com,1999:blog-3108912614914761750.post-10839049065290360062021-02-06T04:07:00.003+01:002021-02-06T13:39:18.257+01:00Reparar fallo SRS Mercedes W639 Vito-Viano en cinturón de seguridad<p></p><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7zLg_TjFfLArNKVBXXXX-vo7KHr93cCTNJuXheXIL0MQXPm2HV8Ph35H3VORUtMRaUOudbNCgkjr-pGQ7bD8zVBAtnnFjk_NLTTeLsSDYeCcuDIsjCwz0QtaNgf6y1F7dUP1BMoYi_hw/s1024/Mercedes-Viano-W639-SRS-failure-dashboard.jpg" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img alt="Vito-Viano w639 dashboard SRS failure" border="0" data-original-height="550" data-original-width="1024" height="173" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7zLg_TjFfLArNKVBXXXX-vo7KHr93cCTNJuXheXIL0MQXPm2HV8Ph35H3VORUtMRaUOudbNCgkjr-pGQ7bD8zVBAtnnFjk_NLTTeLsSDYeCcuDIsjCwz0QtaNgf6y1F7dUP1BMoYi_hw/w320-h173/Mercedes-Viano-W639-SRS-failure-dashboard.jpg" title="Cuadro Mercedes Vito-Viano w639 con fallo SRS" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig.1 - Cuadro Mercedes Vito-Viano </td></tr></tbody></table> <b>Resumen</b><p></p><p style="text-align: justify;">En este artículo aprenderemos a identificar y reparar uno de los modos de fallo asociados al sistema SRS (Sistema de Retención de Seguridad) de Mercedes en el modelo de furgoneta W639 Vito/Viano. En concreto, trataremos el fallo asociado a la presencia y cierre de la hebilla del cinturón de seguridad.</p>
<b><div><b><br /></b></div><div><b><br /></b></div>Abstract</b><span style="font-size: small;"><span style="font-weight: normal;"> </span></span><div><p style="text-align: justify;">In this article we will learn how to identify and repair one of the failure modes associated with the Mercedes SRS (Supplemental Restraint Systems) system on the W639 Vito / Viano van model. Specifically, we will deal with the failure associated with the presence indentification and closure of the seat belt buckle.</p><a name='more'></a><p style="text-align: center;">
----oooOOOooo----</p><p style="text-align: justify;">El sistema de retención de seguridad (SRS) permite proteger a los ocupantes en distintos tipos de accidente. Son muchas las fuentes de fallo que pueden originar en el cuadro de mando el fallo SRS. Sin embargo, hoy vamos a centrarnos en el fallo que corresponde a los cierres de la hebilla del cinturón de seguridad.</p><p style="text-align: justify;">El fallo que se describe se presentó en una furgoneta Viano W639. El cuadro de mando diagnosticaba un fallo en el sistema SRS y no proporcionaba aislamiento al componente afectado. Para el aislamiento nos tuvimos que apoyar en una máquina de diagnosis OBDII. En particular se usó el escaner <a href="https://www.icarsoft.com/Product/s-300-iCarsoft_MB_V2.html" target="_blank">Icarsoft MB V2.0</a></p><p style="text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrIwgbknu97wiQuhR7ofj-0rYc3jF0L6BRk5ssoSXxlxAn6Bcn_R_S93gY0tXDTL38EE6PqSqfXFhiG_5IBBqs_Dno5R9yW6G_XN33ej6zw7wPXD4VPuwy9VKi97qVaTKbJ0dANsTk3kM/s2048/Mercedes-W639-SRS-seat-belt-buckle-failure-.jpg" style="margin-left: auto; margin-right: auto;"><img alt="Icarsoft MB V2.0 showing failure 9023 in Mercedes Viano W639" border="0" data-original-height="1054" data-original-width="2048" height="206" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrIwgbknu97wiQuhR7ofj-0rYc3jF0L6BRk5ssoSXxlxAn6Bcn_R_S93gY0tXDTL38EE6PqSqfXFhiG_5IBBqs_Dno5R9yW6G_XN33ej6zw7wPXD4VPuwy9VKi97qVaTKbJ0dANsTk3kM/w400-h206/Mercedes-W639-SRS-seat-belt-buckle-failure-.jpg" title="Fallo SRS 9023 Mercedes Vito-Viano" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 2 -Fallo SRS 9023 Mercedes Vito-Viano</td></tr></tbody></table><span style="text-align: justify;"><br /></span></div><div><span style="text-align: justify;">El diagnóstico fue el fallo 9023 correspondiente al interruptor del sistema de retención del cierre del cinturón delantero derecho (S68/2). Como nota adicional, el cierre correspondiente al asiento delantero izquierdo es el S68/1</span></div><div><p style="text-align: justify;"><span style="text-align: left;">Se procede a desmontar el sistema de retención (6398600069) ya que se comprobó mediante un polímetro que las conexiones estaban cortadas.</span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip7yq86_1H41BCbicBtsOBOvtVLgS0IBNRLPvg_EZ29mk8uj3fvNa9LhuoqkrfAnyRI6eLzvairglGGNVS8PkufjBnrIrLw1Qek86fFZVG-2Vuc3NYdoCnkZMXA80hgeDahqU2w4bwJqM/s1920/Mercedes-W639-seat-belt-buckle-switch-SRS-fail.jpg" style="margin-left: auto; margin-right: auto;"><img alt="Mercedes W639 SRS Seat Belt Buckle opened PN:6398600069" border="0" data-original-height="1080" data-original-width="1920" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip7yq86_1H41BCbicBtsOBOvtVLgS0IBNRLPvg_EZ29mk8uj3fvNa9LhuoqkrfAnyRI6eLzvairglGGNVS8PkufjBnrIrLw1Qek86fFZVG-2Vuc3NYdoCnkZMXA80hgeDahqU2w4bwJqM/w400-h225/Mercedes-W639-seat-belt-buckle-switch-SRS-fail.jpg" title="Cierre cinturón Mercedes W639" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig. 3 - Cierre del cinturón</td></tr></tbody></table><div style="text-align: left;"><br /></div><div style="text-align: justify;"><span style="text-align: left;">En el interior del cierre se encuentran dos resistencias y un interruptor que obedecen al esquema de la figura 4.</span></div></div><div style="text-align: justify;"><span style="text-align: left;"><br /></span></div><div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2tMdGlmWnNmiiFOL6B3bhyH5l0NCrpWkpOaBum587rIMiWgQEYO_MXP65Hwl2q5mjfqGZt3LGKlWrKfeEI5ig5dqLEEH8ad8amYbaaFBAluZOuJiohuhzbr5d9HYGaseRWhgyZLE68Z0/s432/Mercedes+W639+SRS+Seat+Belt+Buckle+Schematic.png" style="margin-left: auto; margin-right: auto;"><img alt="Mercedes W639 SRS Seat Belt Buckle resistors Schematic" border="0" data-original-height="432" data-original-width="419" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2tMdGlmWnNmiiFOL6B3bhyH5l0NCrpWkpOaBum587rIMiWgQEYO_MXP65Hwl2q5mjfqGZt3LGKlWrKfeEI5ig5dqLEEH8ad8amYbaaFBAluZOuJiohuhzbr5d9HYGaseRWhgyZLE68Z0/w388-h400/Mercedes+W639+SRS+Seat+Belt+Buckle+Schematic.png" title="Mercedes W639 SRS Seat Belt Buckle resistors Schematic" width="388" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fig.4 - Esquema eléctrico cierre cinturón seguridad</td></tr></tbody></table><div style="text-align: center;"><br /></div><div style="text-align: center;"><div style="text-align: justify;">La resistencia que se ve en los terminales del conector es "medida" por la unidad de SRS. La unidad SRS espera los siguientes valores:</div><div style="text-align: justify;"><ul><li>Cinturón desabrochado: 400 ohms</li><li>Cinturón abrochado: 100 ohms</li></ul><div>Cualquier otra combinación originará fallo de SRS</div><div><br /></div><div>En el <a href="https://youtu.be/Vb1gT6qqVKs" target="_blank">siguiente video</a> tenemos la explicación del diagnóstico y reparación de una forma mucho más detallada.</div></div><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/Vb1gT6qqVKs" width="320" youtube-src-id="Vb1gT6qqVKs"></iframe></div><br /></div><div class="separator" style="clear: both; text-align: left;">Para cualquier duda, no dudéis en escribirme en el apartado comentarios.</div></div></div>JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com11España40.463667000000008 -3.7492212.153433163821163 -38.90547 68.773900836178854 31.40703tag:blogger.com,1999:blog-3108912614914761750.post-58733941863627097302018-02-16T01:15:00.000+01:002018-02-16T01:15:06.492+01:00Consumo Mercedes Viano W639 3.0 CDI 204 CV<b>Resumen</b><br />
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<div style="text-align: justify;">
En esta entrada se analizan en pruebas reales los consumos de gasóleo de la furgoneta Mercedes Viano chasis W639, motor V6 3.0 CDI de 204 CV.</div>
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<div style="text-align: justify;">
<span style="text-align: center;">Cuando hablamos de los consumos de combustible de este vehículo, existen grandes discrepancias entre la información que reza en foros de usuarios y las publicaciones oficiales. Por ello, me decidí a realizar ensayos lo más objetivos posibles caracterizando el escenario del test.</span></div>
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<span style="text-align: center;"><b> </b></span></div>
<div style="text-align: left;">
<b>TEST 1</b><br />
<b> </b></div>
<div style="text-align: justify;">
<span style="text-align: center;">Condiciones de test</span></div>
<ul>
<li style="text-align: justify;"><span style="text-align: center;">Circulación por autovía generalmente llana a velocidad sostenida con el control de crucero a 100 Km/h</span></li>
<li style="text-align: justify;"><span style="text-align: center;">Furgoneta sin carga.</span></li>
<li style="text-align: justify;"><span style="text-align: center;">Ocupantes: Solo el conductor.</span><span style="text-align: center;"> </span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpQ43uPrpjQ2aFvy59Z6PiLgkV3japCkTJ1c_rvZkDVZWxUx0zjr0LI0sWAgmAMhIQmn9_nMc4yB5wKFLlz1UqP4jhwPHy6m4fArC5zdFuk2FxwCGKrsgCHpiWehVdGfR75NJ0C9gFyNk/s1600/Consumo-viano-w369-v6-30-cdi-100-kmh.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Consumo gasóleo mercedes Viano W639 V6 3.0 CDI a 100 Km/h" border="0" data-original-height="558" data-original-width="1024" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpQ43uPrpjQ2aFvy59Z6PiLgkV3japCkTJ1c_rvZkDVZWxUx0zjr0LI0sWAgmAMhIQmn9_nMc4yB5wKFLlz1UqP4jhwPHy6m4fArC5zdFuk2FxwCGKrsgCHpiWehVdGfR75NJ0C9gFyNk/s400/Consumo-viano-w369-v6-30-cdi-100-kmh.jpg" title="Cuadro instrumentos mercedes Viano W639" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1: consumo a 100 Km/h</td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<span style="text-align: center;">Como se puede observar en la figura 1, el consumo real a <b>100 Km/h</b> es de <b>7.5 l/100 Km</b> </span><br />
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<span style="text-align: center;"> <b>TEST 2 </b></span><br />
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<div style="text-align: justify;">
<span style="text-align: center;">Condiciones de test</span></div>
<ul>
<li style="text-align: justify;"><span style="text-align: center;"><span style="text-align: center;">Circulación por autovía generalmente llana a velocidad sostenida con el control de crucero a 120 Km/h</span></span></li>
<li style="text-align: justify;"><span style="text-align: center;"><span style="text-align: center;">Furgoneta con equipaje de maletas pesadas y lo típico de cuando uno regresa de Asturias de ver a la familia (patatas, manzanas, berzas, calabacines, calabazas y media huerta) </span></span></li>
<li style="text-align: justify;"><span style="text-align: center;"><span style="text-align: center;">Ocupantes: 5 personas a bordo. </span></span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img alt="Consumo gasóleo mercedes Viano W639 V6 3.0 CDI a 120 Km/h" border="0" data-original-height="577" data-original-width="1024" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEido-IYxzhtVNyS9IkicAkHuuE8bYlTYcI_ya8O7kGtDbKQ164oUySV74UU73nUfzm6CNpjyul5Jy_i_HwDE_k_z1vIm8ee4fPSYMq-AfAML9HZTD0b950STkdHsSVGBNB5kEdTndJ5jlk/s400/Consumo-viano-w369-v6-30-cdi-120-kmh.jpg" style="margin-left: auto; margin-right: auto;" title="Cuadro instrumentos mercedes Viano W639" width="400" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 2: Consumo a 120 Km/h</td></tr>
</tbody></table>
<span style="text-align: center;"> </span><span style="text-align: center;">Como se puede observar en la figura 2, el consumo real a<b> 120 Km/h</b> es de <b>9.4 l/100 Km</b> </span><br />
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<div style="text-align: justify;">
<span style="text-align: center;"><b>NOTA</b>: Es reseñable que el velocímetro analógico marca ligeramente más que el digital el cual se usa para los cálculos de velocidad en el ordenador de a bordo. Tras contrastar con la velocidad GPS, se comprueba que el velocímetro analógico mide en exceso y el digital va acorde al GPS.</span></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com1Madrid, España40.4167754 -3.703790199999957640.0300434 -4.3492371999999575 40.8035074 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-57074671207000683102016-09-27T23:59:00.001+02:002016-10-03T16:59:19.680+02:00La injusticia detrás de la "justa" distribución normal en la gestión del desempeño<div style="text-align: justify;">
<b>Resumen</b><br />
<b><br /></b></div>
<div style="text-align: justify;">
Cada año toca la difícil y subestimada tarea de revisión del desempeño de los trabajadores en la empresa. Cada empleado es evaluado por sus superiores y encasillado en un "slot" que, según los modelos clásicos, nos llevan a una distribución normal. Pero, <b>¿Sigue el desempeño de la plantilla una distribución normal?</b></div>
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Actualmente existe la creencia en muchas empresas que el rendimiento de los trabajadores se puede ajustar a una distribución normal (curva de Gauss). Estudios recientes demuestran que este “dogma de fe” está seriamente comprometido y, que la distribución de Gauss no refleja la realidad de una empresa. Pese a ello, a día de hoy<b> muchas empresas continúan haciendo "normal" el talento original de la individualidad del capital humano</b>.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxaRBuO_8LcL39LQj_eZB0vAGf6BWrwnYpqNw-R7AOXg2gN470o49syYuGNamqvrXkDNhAYoOtNCae_r3Q3hYNSSzuNrp74kjzeE2ItV6EadlQdc4t1Tkd8pKB69K6uPUKWkreZI670iE/s1600/Dilbert+distribuci%25C3%25B3n+forzada.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxaRBuO_8LcL39LQj_eZB0vAGf6BWrwnYpqNw-R7AOXg2gN470o49syYuGNamqvrXkDNhAYoOtNCae_r3Q3hYNSSzuNrp74kjzeE2ItV6EadlQdc4t1Tkd8pKB69K6uPUKWkreZI670iE/s640/Dilbert+distribuci%25C3%25B3n+forzada.jpg" width="640" /></a></div>
A continuación se expondrán dos puntos de vista de un mismo problema, la gestión del desempeño. </div>
<ul>
<li style="text-align: justify;"><b>Teoría clásica: </b>Se asume como irrefutable la distribución normal y se ajusta el desempeño de los trabajadores para que quede bajo la curva.</li>
<li style="text-align: justify;"><b>Nuevas tendencias:</b> Se evalúa el desempeño real de los trabajadores y se realiza un contraste de hipótesis para ver qué distribución estadística es la más acertada al desempeño de la plantilla.</li>
</ul>
<div style="text-align: justify;">
<b>1. Teoría clásica para la gestión del desempeño (distribución normal)</b></div>
<div style="text-align: justify;">
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<div style="text-align: justify;">
Desde principios del siglo pasado, la estadística evolucionó de manera considerable y con ello la aplicación al mundo de la empresa. En estos inicios de la industria donde abundaba la mano de obra no cualificada, con el apoyo de la teoría de los grandes números y el teorema central del límite, se estableció que la gestión del desempeño de los trabajadores se ajustaba a una distribución normal y actualmente, este criterio continúa vigente en muchas empresas.<br />
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Siguiendo esta corriente, si durante el proceso de evaluación se observa que el desempeño de los trabajadores no se ajusta a una normal, RRHH atribuye el problema a una serie de factores subjetivos como una evaluación benevolente, severa o errores de halo. Ante esa situación, se modifican los resultados y, bajo ningún pretexto se cuestiona si realmente el desempeño se puede ajustar mejor a otro modelo estadístico. Estudios recientes demuestran que hoy en día, esta asunción es muy discutible aunque muchas empresas (y no precisamente pequeñas) aún la siguen siguen aplicando.</div>
<div style="text-align: justify;">
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<div style="text-align: justify;">
Acorde al modelo de la distribución normal, el desempeño de los trabajadores se puede ajustar en torno a la media de desempeño de toda la plantilla. Según éste modelo, la plantilla se distribuye según: </div>
<ul>
<li style="text-align: justify;"><b>Rango 1:</b> Un 2.28 % tiene un rendimiento mediocre. Correspondiente al intervalo [-inf ,-2 sigma]</li>
<li style="text-align: justify;"><b>Rango 2:</b> Un 13.59 % tiene un rendimiento inferior a la media. Correspondiente al intervalo [-2 sigma , -1 sigma]</li>
<li style="text-align: justify;"><b>Rango 3:</b> Un 68.26 % tiene un rendimiento en la media. Correspondiente al intervalo [-1 sigma , 1 sigma]</li>
<li style="text-align: justify;"><b>Rango 4: </b>Un 13.59 % tiene un rendimiento superior a la media. Corresponde al rango [1 sigma , 2 sigma]</li>
<li style="text-align: justify;"><b>Rango 5:</b> Un 2.28 % tiene un rendimiento excelente. Correspondiente al intervalo [2 sigma , Inf] </li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGW5lOwQsZjPaRKQZz7HFFxdA_1_-3V1jPFZ6jPnck0X6rbBt_QCmxAcrjfkBAw7IOjjjGgOmY-K4ORZOeOmBqdJ9vvqFQvgODhKXF92uw6K1pJIxQy94CRb_MHMeSPUW0izt96J473B8/s1600/Distribuci%25C3%25B3n-normal-desempe%25C3%25B1o.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGW5lOwQsZjPaRKQZz7HFFxdA_1_-3V1jPFZ6jPnck0X6rbBt_QCmxAcrjfkBAw7IOjjjGgOmY-K4ORZOeOmBqdJ9vvqFQvgODhKXF92uw6K1pJIxQy94CRb_MHMeSPUW0izt96J473B8/s400/Distribuci%25C3%25B3n-normal-desempe%25C3%25B1o.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.1 - Distribución normal del desempeño</td></tr>
</tbody></table>
<div style="text-align: justify;">
<div>
Cabe destacar que a ambos lados de la media y en iguales proporciones, se encuentran los trabajadores con un rendimiento superior a la media (lado derecho) y los trabajadores con un rendimiento inferior a la media (lado izquierdo). </div>
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<div>
<b>1.1-<span class="Apple-tab-span" style="white-space: pre;"> </span>Ventajas</b></div>
<div>
<ul>
<li>No es necesario un análisis previo del desempeño de la plantilla. Se supone que el desempeño de esta sigue <b>una distribución normal facilitando los cálculos estadísticos</b> y haciendo la vida fácil al departamento de RRHH.</li>
</ul>
<ul>
<li>Si el desempeño sigue una curva normal, entonces <b>es bastante fácil prever las compensaciones basadas en el rendimiento</b> ya que la distribución de los trabajadores está claramente predeterminada independientemente de la realidad. Es decir, se puede calcular la partida presupuestaria para incremento salarial independientemente de la evaluación real del talento.</li>
</ul>
</div>
<div>
<b>1.2-<span class="Apple-tab-span" style="white-space: pre;"> </span>Inconvenientes</b></div>
<div>
<ul>
<li>Acorde a la distribución normal, <b>estamos recortando el número de empleados etiquetados con rendimiento excelente</b>. Muchas empresas establecen criterios a la hora de evaluar. Si se usa una escala del 1 al 5, dicen... “no puede haber más del 3 % de la plantilla en el rango excelente (rango 5) y, forzosamente tiene que haber un 3 % de la misma con en el rango mediocre (rango 1)</li>
</ul>
<ul>
<li><b>Se fuerza a que el 16 % de la plantilla esté en el rango inferior a la media</b>, los perdedores del grupo. Pero… ¿Qué pasa si el equipo de trabajo es el mejor que podemos tener? Aun así, aunque tengamos a los mejores trabajadores que puede haber en el mercado con nosotros, estamos forzando a que un 16 % de ellos estén catalogados como “poco útiles” originando frustraciones, malestar y muy posiblemente fugas de conocimiento de la empresa ya que a esos empleados no se les retribuirá en consecuencia.</li>
</ul>
<ul>
<li><b>La distribución normal debe incluir a una población completa</b>, sin sesgar, lo que no ocurre en las empresas, ya que el departamento de RRHH se encarga de hacer la selección del personal en la contratación buscando a los mejores perfiles. Estos profesionales fuerzan el sesgo de la distribución promocionando el talento y desincentivando a su vez la mediocridad. Buscan la élite entre la multitud de candidatos y, por lo tanto, la población analizada no está distribuida de manera aleatoria tal y como Dios nos trajo al mundo, violando así la hipótesis de partida de la estadística. </li>
</ul>
</div>
<div>
<ul>
<li><b>Distribución forzada: </b>¿y si el desempeño no sigue una distribución normal? ¿Estamos falseando la realidad? La respuesta es un rotundo SÍ. <b>La distribución normal obliga a las empresas a hacer una distribución forzada</b> de sus recursos para poder construir la curva. Los empleados son encasillados en “slots” y todos los "slots" deben estar rellenos con un número predefinido independientemente del desempeño de la plantilla. <b>Se asume que el desempeño sigue una distribución normal</b> como un acto de fe.</li>
</ul>
</div>
<div>
<ul>
<li><b>Siempre hay perdedores:</b> imaginemos ahora que a los trabajadores que están en el rango 1 (los perdedores) los despedimos y, no hay nuevas contrataciones en la empresa. Seis meses después, se procede a hacer una nueva evaluación acorde a la distribución normal (véase figura 2) Claramente, se ve que la media de la plantilla ha mejorado en rendimiento, pero de nuevo vuelve a haber “slots” para perdedores (que antes no lo eran) y el número de trabajadores en rango 5 (excelentes) desciende originando la frustración del personal que realmente aporta a la empresa. Sorprendente ejercicio ¿verdad?</li>
</ul>
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6sUzE_PdtrNPgCVE5AFRSrekL_L51r9xr6lJQRvwu3KjN0pdvG6rO9-LiBAoOEBtoKUfvpxQqVcJ-0R3V78AbQsDGwP2ky7jLhGga1XO-0QAPm6xrC3Y9v8F_SsUhxoF8HRWbbKjwA3k/s1600/Distribuci%25C3%25B3n-normal-desempe%25C3%25B1o-y-despidos.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="La distribución normal del desempeño y los perdedores" border="0" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6sUzE_PdtrNPgCVE5AFRSrekL_L51r9xr6lJQRvwu3KjN0pdvG6rO9-LiBAoOEBtoKUfvpxQqVcJ-0R3V78AbQsDGwP2ky7jLhGga1XO-0QAPm6xrC3Y9v8F_SsUhxoF8HRWbbKjwA3k/s400/Distribuci%25C3%25B3n-normal-desempe%25C3%25B1o-y-despidos.jpg" title="La distribución normal del desempeño y los perdedores" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.2- Distribución normal del desempeño y los perdedores</td></tr>
</tbody></table>
<div style="text-align: center;">
</div>
<div>
<div>
<b>1.3 La distribución forzada y la frustración en la plantilla</b></div>
<div>
<b><br /></b></div>
<div>
Los <b>empleados </b>evaluados bajo una <b>distribución forzada</b> a la normal podrían estar afectados por los siguientes puntos:</div>
<div>
<ul>
<li><b>Búsqueda del protagonismo individual </b>y disminución de la colaboración en el grupo. ¿Por qué colaborar cuando compito contra mi compañero?</li>
</ul>
<ul>
<li>El personal en rango 4 (siendo 5 el máximo sobre 5), muchas veces son personal de alto potencial que se sienten <b>frustrados </b>porque, aun mereciéndolo, ese año no han podido (o no les ha correspondido) el 5 por temas de “cupo”.</li>
</ul>
<ul>
<li>El personal en rango 3 muchas veces son personas que, sabiendo que no van a poder ser 4 o 5, no aspiran a ello y, simplemente <b>se quedan en su zona de confort</b>, no aspirando a mejora personal, ni por supuesto, a su mejor desempeño dentro de la empresa.</li>
</ul>
<ul>
<li>En términos económicos, el grueso de la compensación generalmente se reparte entre ese 68 % de la media y, se limitan las posibilidades de dar mayores salarios a las personas que realmente contribuyen más a la empresa….¡¡ Cuidado….ser justos no significa ser iguales!!</li>
</ul>
<ul>
<li><b>No fomenta la mejora continua</b>, ya que el número 5 está muy limitado, con lo que en muchos casos ¿Para qué competir si ya sé a quién se lo van a dar?</li>
</ul>
</div>
<div>
Resumiendo y con todo lo expuesto, ¿es la distribución normal una herramienta justa de evaluación del talento? O por el contrario su uso genera frustraciones, desmotivación en personas de valía y, como consecuencia, posibles fugas de talento en la empresa.</div>
<div>
<br /></div>
<div>
<b>2.<span class="Apple-tab-span" style="white-space: pre;"> </span>Nuevos enfoques para la gestión del desempeño (Distribución de Pareto)</b></div>
<div>
<b><br /></b></div>
<div>
Tras lo expuesto en el punto anterior, ¿sigue el mundo real del desempeño una distribución normal? La respuesta es claramente NO. </div>
<div>
<br /></div>
<div>
Según <b>un estudio realizado entre los años 2011 y 2012 por Ernest O’Boyle Jr. y Herman Aguinis </b>(198 muestras repartidas en 633,263 investigadores, artistas, políticos y atletas profesionales), el desempeño sigue una distribución exponencial que se asemeja a la de Pareto. Basándose en las muestras del estudio, el 94 % de las ocasiones se encontró que el desempeño de los grupos analizados no seguía una distribución normal. En su lugar, la distribución se ajustaba a una de Pareto. Para el estudio, Ernest O’Boyle Jr. y Herman Aguinis utilizaron <b>técnicas estadísticas de contraste de hipótesis</b> para comparar la muestra de población con las distribuciones normal y de Pareto observando casi un perfecto ajuste a esta última y total disonancia con la "normal"</div>
<div style="text-align: center;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYKurL3ztc-x2nzj91cIUjTKJ_fIwrpR-Cim_xEzF8dDKBKH02Ehfma9m76tkUAMENsCAyd0AVGmXhyTVRcOskYKJNGVtzOr-QiS_Dp0a15sWmg5OFxfhguqodnTYiotR4USs0Wc66Q3g/s1600/distribuci%25C3%25B3n-pareto-desempe%25C3%25B1o.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Distribución de Pareto del desempeño" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYKurL3ztc-x2nzj91cIUjTKJ_fIwrpR-Cim_xEzF8dDKBKH02Ehfma9m76tkUAMENsCAyd0AVGmXhyTVRcOskYKJNGVtzOr-QiS_Dp0a15sWmg5OFxfhguqodnTYiotR4USs0Wc66Q3g/s400/distribuci%25C3%25B3n-pareto-desempe%25C3%25B1o.jpg" title="Distribución de Pareto del desempeño" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.3 - Distribución de Pareto del desempeño</td></tr>
</tbody></table>
</div>
<div>
<br /></div>
<div>
En la realidad de cualquier empresa suele haber:</div>
<div>
<ul>
<li>Un porcentaje mínimo que son extraordinarios y que realmente marcan la diferencia.</li>
</ul>
<ul>
<li>Otro porcentaje muy cerca del anterior que son los que ayudan a marcar esa diferencia (entre ambos quizá suman el 20%)</li>
</ul>
<ul>
<li>Resto de personas que con distinto rendimiento (nada de simetrías), se distribuye a lo largo de una curva “Long Tail”.</li>
</ul>
</div>
<div>
En esta situación, las evaluaciones se forzarían únicamente para homogeneizar los criterios de evaluación y no para buscar la simetría de una distribución. <b>Lo que se buscaría con ella sería:</b></div>
<div>
<ul>
<li>Realmente <b>identificar </b>a ese porcentaje crítico que son los que con su trabajo y <b>talento </b>marcan la diferencia en la empresa.</li>
</ul>
<ul>
<li>Poder hacer una <b>distribución salarial más acorde</b> a esa criticidad, de forma que la mayoría de la retribución no esté repartida en los “average” sino en los que realmente hacen que la empresa sea exitosa.</li>
</ul>
<ul>
<li>Tener una <b>distribución </b>mucho más ajustada a lo que es la <b>realidad </b>de la empresa, y no a algo puramente estadístico e “igualitario”</li>
</ul>
<ul>
<li><b>Reducir las frustraciones</b> y los inconvenientes ocasionados que se producen al utilizar distribución normal.</li>
</ul>
</div>
<div>
<b>3.- Conclusiones</b></div>
<div>
<ul>
<li>Cada empresa debería <b>cuestionar la verdadera distribución estadística del desempeño</b> de su plantilla. Para ello, en lugar de asumir a pie juntillas la validez de la distribución normal, se debería hacer un estudio estadístico del desempeño apoyado en un contraste de hipótesis contra una distribución de Pareto o similar.</li>
</ul>
<ul>
<li><b>El salario</b> de un trabajador no debe estar limitado por el tipo de trabajo que desarrolla sino por <b>el valor que aporta a la empresa</b>.</li>
</ul>
<ul>
<li>Actualmente, los <b>departamentos de RRHH de grandes compañías</b> como Google o Microsoft se han dado cuenta que la verdadera distribución del desempeño de sus trabajadores no se podía forzar a una distribución normal sin cometer injusticias. Por ello, recientemente han adoptado <b>distribuciones </b>más ajustadas a la realidad como la de <b>Pareto</b>.</li>
</ul>
</div>
<div>
Como dijo el cabeza de <b>Recursos Humanos de Google Laszlo Bock</b> en su libro Work rules (2015):</div>
<div>
<i>“you should pay employees what they're worth, not for the job they do. It's hard work to have pay ranges where someone can make two or even 10 times more than someone else," he writes. "But it's much harder to watch your highest-potential and best people walk out the door. It makes you wonder which companies are really paying unfairly: the ones where the best people make far more than average, or the ones where everyone is paid the same." </i><br />
<br />
<i>"Se debería pagar a los empleados por lo que valen y no por el trabajo que hacen. Es un trabajo duro tener que pagar salarios donde alguien puede ganar dos o 10 veces más que otro" escribe. "Pero es mucho más duro ver como el alto potencial y la mejor gente se va por la puerta. Esto te hace pensar sobre qué empresas realmente están pagando injustamente: aquellas donde la mejor gente gana más que el promedio, o en las que todo el mundo gana lo mismo"</i></div>
<div>
<b><br /></b></div>
<div>
<b>4.-Referencias</b></div>
<div style="text-align: left;">
[1]- "<a href="https://www.amazon.com/Work-Rules-Insights-Inside-Transform/dp/1455554790" target="_blank">Work Rules</a>" by <a href="https://www.linkedin.com/in/laszlobock/es" target="_blank">Laszlo Bock</a>, april 2015</div>
<div style="text-align: left;">
[2]- "<a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1744-6570.2011.01239.x/abstract" target="_blank">The Best & the Rest: Revisiting the Norm of Normality of Individual Performance</a>", Ernest O’Boyle Jr., Herman Aguinis, Publisher: Personnel Psychology, 2012, 65, 79-119 </div>
</div>
</div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com1Madrid, España40.4167754 -3.703790199999957640.0300434 -4.3492371999999575 40.8035074 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-89690660962753747702016-07-07T01:48:00.002+02:002016-07-07T01:48:19.827+02:00Puntos Inspección tipo II BMW serie 3-E46-2.0D-320D<b>Resumen</b><br />
En el presente artículo se describe el procedimiento de inspección tipo II de BMW para la serie 3 E46 con motorización 2.0D, 320D, años 2001-2007.<br />
<br />
<b>Abstract</b><br />
The aim of this paper is to describe the BMW inspection type II for 3 series E46, 2.0D engines, 320D, years 2001-2007<br />
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<div style="text-align: center;">
<br /></div>
<b>1.- MATERIAL CONSUMIBLE NECESARIO</b><br />
<b><br /></b>
<div style="text-align: left;">
</div>
<div style="text-align: center;">
<b>Aceite motor </b></div>
<div style="text-align: center;">
SAE 0W30, 0W40, 5W30, 5W40 </div>
<div style="text-align: center;">
Especificación BMW LL04</div>
<div style="text-align: center;">
cantidad 5.5 litros<br />
<b><b>Filtro de aire del motor</b></b></div>
<b>
</b><b><div style="text-align: center;">
<b>Filtro de aceite del motor</b></div>
</b><b><div style="text-align: center;">
<b>Filtro de polen</b></div>
</b><b><div style="text-align: center;">
<b>Filtro principal diesel</b><br />
<b>Válvula ventilación del cárter</b> </div>
</b><div style="text-align: center;">
(Sólo motores M47TU1/2 09.02-06.06 cada tercer cambio de aceite)</div>
<br />
<div style="text-align: left;">
<b><br /></b></div>
<div style="text-align: left;">
<b>2.- PUNTOS DE INSPECCIÓN</b></div>
<div style="text-align: left;">
<br /></div>
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background-color: #104E8B;
color: #FFF;
font-weight: bold;
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</style>
<br />
<table class="tableizer-table">
<thead>
<tr class="tableizer-firstrow"><th>Vehículo en el suelo</th><th></th><th></th></tr>
</thead><tbody>
<tr><td>Descripción</td><td>Acción</td><td>Estado</td></tr>
<tr><td>Servofreno</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Tubos servofreno/ válvula de retención</td><td>Comprobar</td><td></td></tr>
<tr><td>Recorrido del pedal de freno</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Recorrido del freno de mano</td><td>Comprobar/ajustar</td><td></td></tr>
<tr><td>Soportes/cinturones de seguridad</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Sistema de aviso de ABS</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Códigos de diagnosis de averías</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Instrumentos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Interruptores/mandos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Ventilador calefacción</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Bocina</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Luces interiores</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Rquetas limpiaparabrisas delantero</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Lavaparabrisas delantero</td><td>Comprobar/ajustar</td><td></td></tr>
<tr><td>Iluminación instrumentos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Intermitentes de dirección/luces de emergencia</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Sistema de iluminación</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Sistema de nivelación de faros</td><td>Comprobar</td><td></td></tr>
<tr><td>Faros</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Carrocería/pintura</td><td>Comprobar estado</td><td></td></tr>
</tbody></table>
<br />
<br />
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table.tableizer-table {
font-size: 12px;
border: 1px solid #CCC;
font-family: Arial, Helvetica, sans-serif;
}
.tableizer-table td {
padding: 4px;
margin: 3px;
border: 1px solid #CCC;
}
.tableizer-table th {
background-color: #104E8B;
color: #FFF;
font-weight: bold;
}
</style>
<br />
<table class="tableizer-table">
<thead>
<tr class="tableizer-firstrow"><th>Vehículo totalmente elevado</th><th></th><th></th></tr>
</thead><tbody>
<tr><td>Descripción</td><td>Acción</td><td>Estado</td></tr>
<tr><td>Fugas de aceite del motor</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Aceite del motor</td><td>Vaciar/llenar</td><td></td></tr>
<tr><td>Filtro de aceite del motor</td><td>Sustituir</td><td></td></tr>
<tr><td>Sistema hidráulico del embrague</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Fugas de aceite de la caja de cambios</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Juntas de la dirección</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Caja/cremallera dirección</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Tubos dirección asistida</td><td>Comprobar</td><td></td></tr>
<tr><td>Acoplamientos columna de la dirección</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Fuelles/retenes/juntas de la suspensión</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Fuelles/retenes/juntas del arbol de transmisión</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Fugas de aceite del diferencial trasero</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Muelles</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Amortiguadores/soportes</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Sistema de escape/soportes</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Corrosión tubería de frenos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Fugas del sistema de combustible</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Estado de bajos/hermeticidad</td><td>Comprobar/informar</td><td></td></tr>
</tbody></table>
<br />
<br />
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table.tableizer-table {
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font-family: Arial, Helvetica, sans-serif;
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padding: 4px;
margin: 3px;
border: 1px solid #CCC;
}
.tableizer-table th {
background-color: #104E8B;
color: #FFF;
font-weight: bold;
}
</style>
<br />
<table class="tableizer-table">
<thead>
<tr class="tableizer-firstrow"><th>Vehículo medio alzado</th><th></th><th></th></tr>
</thead><tbody>
<tr><td>Descripción</td><td>Acción</td><td>Estado</td></tr>
<tr><td>Cojinetes de rueda delanteros</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Pastillas de frenos delanteras</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Discos de frenos delanteros</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Pastillas de frenos traseras</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Discos de frenos traseros</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Varillaje/cables freno de mano</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Sistema hidráulico de frenos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Líquido hidráulico de frenos</td><td>Sustituir</td><td></td></tr>
<tr><td>Estado de los neumáticos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Presión de los neumáticos</td><td>Comprobar/ajustar</td><td></td></tr>
</tbody></table>
<br />
<br />
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padding: 4px;
margin: 3px;
border: 1px solid #CCC;
}
.tableizer-table th {
background-color: #104E8B;
color: #FFF;
font-weight: bold;
}
</style>
<br />
<table class="tableizer-table">
<thead>
<tr class="tableizer-firstrow"><th>Operaciones en el compartimento motor</th><th></th><th></th></tr>
</thead><tbody>
<tr><td>Descripción</td><td>Acción</td><td>Estado</td></tr>
<tr><td>Bornes de la batería</td><td>Limpiar/engrasar</td><td></td></tr>
<tr><td>Nivel de electrolito de la batería</td><td>Comprobar/reponer</td><td></td></tr>
<tr><td>Estado de la batería/gravedad específica</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Depósito lavaparabrisas</td><td>Comprobar/reponer</td><td></td></tr>
<tr><td>Líquido de la dirección asistida</td><td>Comprobar/reponer</td><td></td></tr>
<tr><td>Grado de concentración anticongelante/nivel de refrigerante</td><td>Comprobar/ajustar</td><td></td></tr>
<tr><td>Tubos de refrigerante</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Filtro de aire</td><td>Sustituir</td><td></td></tr>
<tr><td>Filtro de polen</td><td>Sustituir</td><td></td></tr>
<tr><td>Filtro principal diesel</td><td>Sustituir</td><td></td></tr>
<tr><td>Válvula de ventilación de cárter (Sólo motores M47TU1/2 09.02-06.06 cada tercer cambio de aceite)</td><td>Sustituir</td><td></td></tr>
<tr><td>Indicador de intervalos de revisión</td><td>Reajustar</td><td></td></tr>
</tbody></table>
<br />
<br />
<!-- AQUI INSERTAMOS EL INICIO DE UNA TABLA -->
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table.tableizer-table {
font-size: 12px;
border: 1px solid #CCC;
font-family: Arial, Helvetica, sans-serif;
}
.tableizer-table td {
padding: 4px;
margin: 3px;
border: 1px solid #CCC;
}
.tableizer-table th {
background-color: #104E8B;
color: #FFF;
font-weight: bold;
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<br />
<table class="tableizer-table">
<thead>
<tr class="tableizer-firstrow"><th>Prueba en carretera</th><th></th><th></th></tr>
</thead><tbody>
<tr><td>Descripción</td><td>Acción</td><td>Estado</td></tr>
<tr><td>Juego de la dirección</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Funcionamiento del embrague</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Funcionamiento de la suspensión</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Rendimiento del motor</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Transmisión</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Frenos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Funcionamiento ABS</td><td>Comprobar/informar</td><td></td></tr>
<tr><td>Instrumentos</td><td>Comprobar/informar</td><td></td></tr>
<tr><td><b>Tras la prueba en carretera</b></td><td></td><td></td></tr>
<tr><td>Motor</td><td>Comprobar si hay fugas</td><td></td></tr>
<tr><td>Estado del vehículo</td><td>Comprobar limpieza</td><td></td></tr>
</tbody></table>
<br />
<!-- AQUI FINALIZA LA TABLA --><b>3.- REFERENCIAS</b><br />
[1]- <a href="https://www.autodata-group.com/es/" target="_blank">Autodata</a>JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com1Madrid, España40.4167754 -3.703790199999957640.0300434 -4.3492371999999575 40.8035074 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-16294467013546204982016-03-24T00:27:00.000+01:002016-03-24T00:27:12.283+01:00Cómo funciona un alternador de automóvil y su aplicación para medir rpm<b>Resumen</b><br />
En el éste artículo se presenta el principio de funcionamiento de un alternador de automóvil y su aplicación para medir las revoluciones de un motor.<br />
<b><br /></b><br />
<b>Abstract</b><br />
The aim of this entry is to explain the car's alternator operating principle and how can be used for measuring the rotation speed motor.<br />
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En éste video tutorial se tratan los siguientes puntos:<br />
<br />
<ul>
<li>Principio básico de funcionamiento.</li>
<li>Conexionado del alternador en el automóvil.</li>
<li>Principio de la pre-excitación y auto-excitación del alternador.</li>
<li>Regulación de la excitación de la corriente del rotor con el regulador.</li>
<li>Rectificación trifásica-continua.</li>
<li>Aplicación para medir las revoluciones.</li>
<li>Video práctico sobre un VW Golf II</li>
</ul>
<div>
<br /></div>
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<br />JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com3tag:blogger.com,1999:blog-3108912614914761750.post-91608311922003371842016-01-20T00:42:00.000+01:002016-02-18T01:40:46.454+01:00FSU (erizo) BMW 320, e46 cómo funciona y diagnóstico de la avería<div class="separator" style="clear: both; text-align: center;">
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<b>Resumen</b></div>
<div style="text-align: justify;">
Existe un fallo común en los BMW E46 (y otros modelos) donde el ventilador interior del habitáculo deja de funcionar causado por un fallo de la unidad FSU popularmente conocida como erizo<span id="transmark"></span>. En el presente video podemos ver los principios de funcionamiento de ésta unidad, el diagnóstico y aislamiento del fallo.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>Abstract</b></div>
<div style="text-align: justify;">
It is a common failure in the BMW E46 (and other models), the vehicle interior fan breaks down due to a failure in the FSU unit. In this video, we will see the operating principles of this unit and its diagnostics and fault isolation.</div>
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<div style="text-align: justify;">
Si alguna vez habéis tenido <b>problemas con el ventilador de climatización</b> en los BMW E46 (y otros modelos), lo más probable es que se deba a un <b>fallo de la FSU</b> (Final Stage Unit) también llamada FSR (Final Stage Resistor) y popularmente "erizo" por su semejanza con el animal.<br />
<br /></div>
<div style="text-align: justify;">
En el video encontrareis básicamente la siguiente estructura:</div>
<br />
<ol>
<li style="text-align: justify;">Explicación teórica para una mejor comprensión del problema.</li>
<li style="text-align: justify;">Análisis de los modos de fallo de la FSU y cómo aislarlo.</li>
<li style="text-align: justify;">Explicación de la acción de control que actua sobre la FSU en un vídeo práctico.</li>
</ol>
<br />
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JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com96Madrid, Madrid, España40.4167754 -3.703790199999957640.0300434 -4.3492371999999575 40.8035074 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-53414149152844347562015-11-12T01:44:00.001+01:002015-11-12T01:44:58.167+01:00Optoma HD25-LV lumens and contrast ratio test review<div style="text-align: justify;">
The present study evaluates a production unit of the Optoma HD25-LV projector. Illuminance and contrast ratio measurements were performed on this system according to IEC 61947-1:2002.<br />
<br />
<div style="text-align: start;">
<b>PERFORMANCE SUMMARY:</b></div>
<div style="text-align: start;">
The projector excluded the screen produces light output (illuminance) of 2037 <span style="background-color: white;">ANSI l</span>umens at 1080p and contrast ratio of 331:1</div>
</div>
<a name='more'></a><br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvT1M9TGOyPQviYgDxBQkhKfDR6_qUGQDhG4CN-xFB_prnhVRNPMJ9DNNYdS2kQ40n5Ylvck0sRtrq1Ndib2RuKdKekfrjSFVgxlc3uh1-k5-RFsuNk8Dvm0RFjG3hBGAXeH-wZAgavRM/s1600/optoma-hd25-lv.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvT1M9TGOyPQviYgDxBQkhKfDR6_qUGQDhG4CN-xFB_prnhVRNPMJ9DNNYdS2kQ40n5Ylvck0sRtrq1Ndib2RuKdKekfrjSFVgxlc3uh1-k5-RFsuNk8Dvm0RFjG3hBGAXeH-wZAgavRM/s320/optoma-hd25-lv.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1 - Optoma HD25-LV</td></tr>
</tbody></table>
<div style="text-align: justify;">
</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
I provide below a description and details of the setup procedures used to prepare the proyector for measurements according to ANSI . </div>
<div style="text-align: justify;">
</div>
<ul>
<li>Section 2 presents the test requirements.</li>
<li>Section 3 shows the measurement setup.</li>
<li>Section 4 provides the instruments to be used.</li>
<li>Sections 5 and 6 results and conclusions.</li>
</ul>
<br />
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>1. GENERAL PRODUCT DETAILS</b></div>
<div style="text-align: justify;">
</div>
<ul>
<li style="text-align: justify;">The Optoma HD25-LV delivers outstanding high-definition 1080p video and graphics.</li>
<li style="text-align: justify;">According to manufacturer the HD25-LV has 3500 ANSI lumens and 20,000:1 contrast ratio.</li>
</ul>
<div>
<b>2. TEST REQUIREMENTS</b><br />
Projector and measurement test setups were according to<b> </b><span style="text-align: justify;">IEC 61947-1.</span><br />
<span style="text-align: justify;">Pattern used:</span></div>
<div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0vESNUklJZTQx2ph6vpFwYXfnblcNqfjuuSjaz91RF9WX0rRnr0o1AwOzBjFkqadzxFsdaS3NAzGXTRTfH2XNpLEsZGJOIdRmQXcgIF5IER1L4nZhEf-w_W4mnHf5_B9NWB7OS2K01D0/s1600/ANSI-screen-locations-for-measuring-light-output-in-lumens.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="ANSI screen locations for measurement the light output in lummens" border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0vESNUklJZTQx2ph6vpFwYXfnblcNqfjuuSjaz91RF9WX0rRnr0o1AwOzBjFkqadzxFsdaS3NAzGXTRTfH2XNpLEsZGJOIdRmQXcgIF5IER1L4nZhEf-w_W4mnHf5_B9NWB7OS2K01D0/s320/ANSI-screen-locations-for-measuring-light-output-in-lumens.png" title="ANSI screen locations for measurement the light output in lummens" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 2 - White test pattern specification</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: left;">
For test purposes readers can download the figures 3 and 4 ready for 1080 in aspect ratio 16/9 to test their systems.</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK2lE-Z-uurbBg6uu_XUA1CeEI7N67zA3LRtaesR5tKgne93XiHWSUQ2Yp-qgJoRBMVYkrPc9ul44h7Xdd9SdnmkE3HbB93dySQMl5Tbo4i8lFj6dX4Og8Xx5U0IgFl7Lr0MHiRjXW4RU/s1600/White+test+IEC+2002.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Test pattern for ANSI light measurements" border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK2lE-Z-uurbBg6uu_XUA1CeEI7N67zA3LRtaesR5tKgne93XiHWSUQ2Yp-qgJoRBMVYkrPc9ul44h7Xdd9SdnmkE3HbB93dySQMl5Tbo4i8lFj6dX4Og8Xx5U0IgFl7Lr0MHiRjXW4RU/s320/White+test+IEC+2002.jpg" title="Test pattern for ANSI light measurements" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 3 - White test pattern</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdV1-3VWR_EX_ZCtNjKsIrxCnEPoyJa2GdDelQfCtVos9K_AxfE3JY5NYECnWdyyV7daZM1T-wiko6CwZyqdaM2Wey11dPkag1SoFkwiAXB1MxhKNPuv5V03gGxRzHvUMH4HTH0ZzkAEY/s1600/Contrast+measurement+IEC+2002.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Contrast pattern for ANSI light measurements" border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdV1-3VWR_EX_ZCtNjKsIrxCnEPoyJa2GdDelQfCtVos9K_AxfE3JY5NYECnWdyyV7daZM1T-wiko6CwZyqdaM2Wey11dPkag1SoFkwiAXB1MxhKNPuv5V03gGxRzHvUMH4HTH0ZzkAEY/s320/Contrast+measurement+IEC+2002.jpg" title="Contrast pattern for ANSI light measurements" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 4 - Contrast pattern</td></tr>
</tbody></table>
<br /></div>
<div>
<b>3. TEST SETUP</b></div>
<div>
The projection system was setup according to the figure 5.</div>
<div style="text-align: center;">
<ul>
<li style="text-align: justify;">Projection distance (D): 3.40 meters approximated.</li>
<li style="text-align: justify;">Screen width (W): 2.245 m</li>
<li style="text-align: justify;">Screen height (H): 1.265 m</li>
<li style="text-align: justify;">Area (W*H): 2.84 m2</li>
</ul>
</div>
<div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizXPmDuTBIXJ52G808iGODJf52cB0Md9vmO0J4obEnAAz3IvILtO_D4Kj7yNia8JCh0hxu4u_lHwtsQEbpPwRonpV8VifkccnP9EL_wSOd2kpdgbLCo1xKxNsQIynrdJIbikW_OFM7VFY/s1600/Test-setup.png" imageanchor="1" style="margin-left: auto; margin-right: auto; text-align: center;"><img alt="Test setup for testing optoma HD25-LV according to IEC 61947-1" border="0" height="377" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizXPmDuTBIXJ52G808iGODJf52cB0Md9vmO0J4obEnAAz3IvILtO_D4Kj7yNia8JCh0hxu4u_lHwtsQEbpPwRonpV8VifkccnP9EL_wSOd2kpdgbLCo1xKxNsQIynrdJIbikW_OFM7VFY/s640/Test-setup.png" title="Test setup for testing optoma HD25-LV according to IEC 61947-1" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 5 - Test setup</td></tr>
</tbody></table>
</div>
<div>
<div style="text-align: justify;">
Brightness and contrast controls were adjusted to achieve visually <b>distinguishable steps</b> of a gray scale pattern (figure 6). Lamp mode was configured to maximum power. Other adjustments were made as indicated in figures from 7 to 10.</div>
</div>
<div>
<div style="text-align: justify;">
<br /></div>
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvVIlIk_Jr8LaqOaxdSlTH4TujI2DjcOTcbiBZiubxIzBEPOjunhPMk5awjknFkJEz3x37Ywy_7YGaEi2E7WJ5QpA_8-iJ6ecV_QHvOtP6FEKfnQhChR4L7BPu-aN5D0sj2UJ0VYf3jMM/s1600/Gray-Scale-Step-8.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Gray scale pattern 1080 according to IEC 61947-1" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvVIlIk_Jr8LaqOaxdSlTH4TujI2DjcOTcbiBZiubxIzBEPOjunhPMk5awjknFkJEz3x37Ywy_7YGaEi2E7WJ5QpA_8-iJ6ecV_QHvOtP6FEKfnQhChR4L7BPu-aN5D0sj2UJ0VYf3jMM/s400/Gray-Scale-Step-8.png" title="Gray scale pattern 1080 according to IEC 61947-1" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 6 - Gray scale step 8</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
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<div>
<div style="text-align: justify;">
<span style="text-align: start;">Software projector configuration was as indicated in the following figures.</span><br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIwguI0_E8gDDtzrCwZD6IpK9utktZMb_Nrktressz7QRApLIbqG_DbRJQVHUtlXm7s17Bg6yGzvaxqqLmY-wjg9GH_gNpXoLrKxHVWPpb7sW1wqZ7GnjBepSE8l3RE5mmcv9atKNbTKw/s1600/Optoma_HD25-LV_configuration_03.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Display configuration OPTOMA HD25-LV" border="0" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIwguI0_E8gDDtzrCwZD6IpK9utktZMb_Nrktressz7QRApLIbqG_DbRJQVHUtlXm7s17Bg6yGzvaxqqLmY-wjg9GH_gNpXoLrKxHVWPpb7sW1wqZ7GnjBepSE8l3RE5mmcv9atKNbTKw/s320/Optoma_HD25-LV_configuration_03.jpg" title="Display configuration OPTOMA HD25-LV" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 7 - Display configuration</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWkbpS0rhxxqNZncB-VyEcl8puai13430pxcee-nyHrCKpDfdlbgx-L-RZVFRSJ0T7t5zeCTUHWaPPF3a7wMwoH3-mX8Xz7jchD29eq3u-nSZ-s1WqG0n7mC2KLYV_qQzR8gnVyUkpvk4/s1600/Optoma_HD25-LV_configuration_02.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Image configuration OPTOMA HD25-LV" border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWkbpS0rhxxqNZncB-VyEcl8puai13430pxcee-nyHrCKpDfdlbgx-L-RZVFRSJ0T7t5zeCTUHWaPPF3a7wMwoH3-mX8Xz7jchD29eq3u-nSZ-s1WqG0n7mC2KLYV_qQzR8gnVyUkpvk4/s320/Optoma_HD25-LV_configuration_02.jpg" title="Image configuration OPTOMA HD25-LV" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 12.8px;">Fig. 8 - Image configuration</span></td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgI__VwnbU3PBBYlVSzqYUiFESh2-H0-jqh-KHSM0de20chOR9JqH_PiDI9drSfsK4vuqjxB03Fcu_cH2JmXpkI8HNSqS21I-krTi_PMAl5-Fep7QJAdZj84rMDgMaPXZVgludJXHNSyQo/s1600/Optoma_HD25-LV_configuration_04.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Advance image configuration OPTOMA HD25-LV" border="0" height="290" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgI__VwnbU3PBBYlVSzqYUiFESh2-H0-jqh-KHSM0de20chOR9JqH_PiDI9drSfsK4vuqjxB03Fcu_cH2JmXpkI8HNSqS21I-krTi_PMAl5-Fep7QJAdZj84rMDgMaPXZVgludJXHNSyQo/s320/Optoma_HD25-LV_configuration_04.jpg" title="Advance image configuration OPTOMA HD25-LV" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 12.8px;">Fig. 9 - Advance image configuration</span></td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIW8L8RTp6EBTsoUwhLTeIhcg8pNt-DLGbbgY-xhrlZwa3yRfTj4-WBvenrGrPLQHz8xDx9G3eeY8I-NaqffaE4NCHGyBmRQss0raleQ-Hpq1xKWoHLTOG26HRWEUCNTOqZxkTrm0FgQE/s1600/Optoma_HD25-LV_configuration_03.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Color space configuration OPTOMA HD25-LV" border="0" height="258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIW8L8RTp6EBTsoUwhLTeIhcg8pNt-DLGbbgY-xhrlZwa3yRfTj4-WBvenrGrPLQHz8xDx9G3eeY8I-NaqffaE4NCHGyBmRQss0raleQ-Hpq1xKWoHLTOG26HRWEUCNTOqZxkTrm0FgQE/s320/Optoma_HD25-LV_configuration_03.jpg" title="Color space configuration OPTOMA HD25-LV" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 12.8px;">Fig. 10 - Color space configuration</span></td></tr>
</tbody></table>
<span style="text-align: start;"><br /></span>The screen illuminance were measured with a light meter. The sensor was placed on the plane of the focused image at the centre of each of nine points indicated in the figure 11. In this case, illumination sensor was placed on axis with the projector light ray in order to <b>correct the cosine effect </b>and maximize the flux entry into the sensor as indicated in figure 11.</div>
</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZbBn5LDEBrRuAmcsYTNC1ONVMZ49pqNLvbPA211gc2SX_qSMW60jnlXa69luJL0OZv6kDqEA_2St2wqM9H7hi42E_HVxW3fp_y2Sklb4n6r36UA4O41QbLGioraGco2nZXGxC2AptP-Y/s1600/Illumination-sensor-alignment.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Lux meter alignment" border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZbBn5LDEBrRuAmcsYTNC1ONVMZ49pqNLvbPA211gc2SX_qSMW60jnlXa69luJL0OZv6kDqEA_2St2wqM9H7hi42E_HVxW3fp_y2Sklb4n6r36UA4O41QbLGioraGco2nZXGxC2AptP-Y/s200/Illumination-sensor-alignment.jpg" title="Lux meter alignment" width="191" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.11 - Sensor alignment</td></tr>
</tbody></table>
<b><br /></b>
<b>4. INSTRUMENTATION</b></div>
<div>
Instruments used in these test include:</div>
<div>
<ul>
<li>Illumination meter DX-200</li>
</ul>
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4JV0i9kG7iTNmgWsTeMGqCqzap2xJ6y37lbktPUqPmVMHJ6lQEvcW9oIpZITPq7XtT5vQ70mhWmpQnUG_AgbsEjjV8ANvrtrw6YZbPd6FyFsZbEETR-9V_aBfvtk3MBfUGq-FloA8QeA/s1600/Illumination-meter-DX-200.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Illumination meter" border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4JV0i9kG7iTNmgWsTeMGqCqzap2xJ6y37lbktPUqPmVMHJ6lQEvcW9oIpZITPq7XtT5vQ70mhWmpQnUG_AgbsEjjV8ANvrtrw6YZbPd6FyFsZbEETR-9V_aBfvtk3MBfUGq-FloA8QeA/s320/Illumination-meter-DX-200.jpg" title="Illumination meter" width="242" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 12 - Illumination meter</td></tr>
</tbody></table>
<div>
<br /></div>
<div>
<b>5. DATA AND RESULTS</b><br />
<b><br /></b>
<div style="text-align: justify;">
<b><b>5.1 ILLUMINANCE</b></b></div>
<br />
<div style="text-align: justify;">
The average of the nine (points 1 to 9 in figure <span style="background-color: white;">3</span>) readings in lux (lumens per square metre) was multiplied by the number of square metres covered by the image at the plane of the meter readings. The result was taken as the light output of the projector, in lumens.<br />
<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghYmnEYzC_kfzQaFzk7wc6BZGO_G0X7-MVbIbGKQuk7Y9tJbbsBKMW9kAQZV-b_tuzdkb654XU4zCsLUslHmd4HGdgzvkB3GQQGOqmW7URvik5xVPpOmtXNgMjouQMaEOzOXJAR0xUr0c/s1600/Optoma_HD25-LV_illuminance_test_IEC_2002.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Optoma HD25-LV illuminance test IEC 2002" border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghYmnEYzC_kfzQaFzk7wc6BZGO_G0X7-MVbIbGKQuk7Y9tJbbsBKMW9kAQZV-b_tuzdkb654XU4zCsLUslHmd4HGdgzvkB3GQQGOqmW7URvik5xVPpOmtXNgMjouQMaEOzOXJAR0xUr0c/s320/Optoma_HD25-LV_illuminance_test_IEC_2002.PNG" title="Optoma HD25-LV illuminance test IEC 2002" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 13 - Optoma HD25-LV illuminance test IEC 2002</td></tr>
</tbody></table>
<br />
Based on the measurements taken, illuminance measured according to the ANSI procedure yielded a 9-point <b>average of <span style="background-color: white;">2037 </span>ANSI lumens</b> with a variation of <b><span style="background-color: white;">13.6 </span>% greater than average, and <span style="background-color: white;">-41.3 </span>% less than average</b> for the thirteen ANSI screen locations shown in figure 13.<br />
<br />
Note: It has to be emphasize that at the time of this test, the life of the lamp was 160 hours.<br />
<b><br /></b>
<b>5.2 CONTRAST RATIO</b></div>
<div style="text-align: justify;">
Contrast ratio is the ratio of higher illuminance to lower illuminance. Illuminance measurements in lux was made at the centre of each of the bright (white) rectangles and the dark (black) rectangles. The average illuminance value of the bright rectangles was divided by the average illuminance value of the dark rectangles. The contrast ratio was then expressed as this ratio:1 (e.g. bright rectangles with an average value of 15 lx and dark rectangles with an average value of 0,10 lx provide a contrast ratio of 150:1).<br />
<br />
Illuminance was measured in both horizontal and vertical directions at 16 screen positions for white and black rectangular targets simultaneously displayed in a 4 x 4 chess board configuration as specified by ANSI IT7.215.<br />
<br /></div>
</div>
<div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPj_Jec1RDzyMiEKXSUU30QwEj2ODB2ZCzwejIBtGWufnrdh9xNQw0lJHaoFVVU9QjziKwhqTJAE-gpCP4y3n56igh_LAyX2XsDvtfW4lP4k3eDGf-xQHyaH1YFUYFILm6p9irQ5N2U4o/s1600/Optoma_HD25-LV_contrast_ratio_test_IEC_2002.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Optoma HD25-LV contrast ration test IEC 2002" border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPj_Jec1RDzyMiEKXSUU30QwEj2ODB2ZCzwejIBtGWufnrdh9xNQw0lJHaoFVVU9QjziKwhqTJAE-gpCP4y3n56igh_LAyX2XsDvtfW4lP4k3eDGf-xQHyaH1YFUYFILm6p9irQ5N2U4o/s320/Optoma_HD25-LV_contrast_ratio_test_IEC_2002.PNG" title="Optoma HD25-LV contrast ration test IEC 2002" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 12.8px;">Fig. 14 - Optoma HD25-LV contrast ratio test IEC 2002</span></td></tr>
</tbody></table>
<span style="text-align: justify;">Based on the measurements taken (figure 14), <b>Contras Ratio</b> computed from measurements made under dark room conditions yielded a <b>CR = 331:1</b></span><br />
<span style="text-align: justify;"><br /></span></div>
<div>
<b>6. CONCLUSIONS</b><br />
<br />
<ul>
<li style="text-align: justify;">The collected data reveals a <b>clear difference between the manufacturer data and the test results</b>. </li>
<li style="text-align: justify;">There is a clear discrepancy between the manufacturer <b>illuminance </b>(3000 lumens) and the data obtained (<b>2037 lumens</b>).</li>
<li style="text-align: justify;">The same clear discrepancy is observed between the manufacturer <b>contrast ratio</b> (20000:1) and the data obtained (<b>331:1</b>). However, it has to be empathized that for this measurement, dynamic black was setted to off.</li>
</ul>
<div style="text-align: justify;">
Seeing those data, we can scold Optoma for their advanced “optimism” (or total lack of reality) but the bottom line is that roughly 2000 measured lumens is a lot of lumens, and makes this a projector that can tackle many family rooms and living rooms while providing a good picture.</div>
</div>
<div style="text-align: justify;">
<br /></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0Madrid, Madrid, España40.4167754 -3.703790199999957640.0300434 -4.3492371999999575 40.8035074 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-63574515407352686202015-08-17T00:24:00.000+02:002015-08-17T00:24:28.315+02:00Estado actual sobre la homologación del título de Ingeniería Industrial (II)<div style="text-align: justify;">
Después de mucho pelear para obtener la equiparación del título de Ingeniería industrial en planes pre-Bolonia al nivel europeo, por fin los Ingenieros Industriales podemos obtener la correspondencia.</div>
<div style="text-align: justify;">
</div>
<a name='more'></a><br />
<div style="text-align: justify;">
Con fecha 12/8/2015, el B.O.E. ha publicado el acuerdo del Consejo de Ministros según el cuál se establece la correspondencia del título Universitario Oficial de <b>Ingeniero Industrial al nivel 3 del Marco Español de Cualificación de Estudios Superiores (MECES) y EQF-7 a nivel europeo</b>.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
En consecuencia a partir de ahora ya es posible obtener el certificado de correspondencia. Basta seguir los pasos que se indican en el siguiente <a href="http://www.coiias.es/Contenidos/Ficha.aspx?IdMenu=b4f8f15e-1224-43a5-b2f7-a0acce2b94cc&Idioma=es-ES" target="_blank">enlace</a>, o bien accediendo directamente a la página del <a href="http://www.mecd.gob.es/servicios-al-ciudadano-mecd/catalogo-servicios/gestion-titulos/estudios-universitarios/titulos-espanoles/correspondencias-titulos-meces.html" target="_blank">Ministerio de Educación Cultura y Deporte</a>.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Tras realizar todo el trámite, se obtendrá un certificado como el que se indica en la figura 1.</div>
<div style="text-align: justify;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjode3cbQgYFJjI5MF_Ts7Ust8d7eOcdhmeLmtJzNoMJhI3sP9TAdMl-L2sQH-iVM0nsVkm-7M_6e4bt8-Mj81ZJC6gXJwCejkU402NwKV2zLGkG6qm_HdgPnCZ73b4KpEgQtpq4tz-mas/s1600/Certificado-Ingeniero-Industrial-MECES-3-Y-EQF-7.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Certificado Ingeniero Industrial MECES-3 y EQF-7" border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjode3cbQgYFJjI5MF_Ts7Ust8d7eOcdhmeLmtJzNoMJhI3sP9TAdMl-L2sQH-iVM0nsVkm-7M_6e4bt8-Mj81ZJC6gXJwCejkU402NwKV2zLGkG6qm_HdgPnCZ73b4KpEgQtpq4tz-mas/s640/Certificado-Ingeniero-Industrial-MECES-3-Y-EQF-7.png" title="Certificado Ingeniero Industrial MECES-3 y EQF-7" width="451" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1 - Certificado Ingeniero Industrial MECES-3 y EQF-7</td></tr>
</tbody></table>
<div style="text-align: justify;">
<b>REFERENCIAS</b></div>
<div style="text-align: justify;">
<b>[1] - </b><a href="http://www.coiias.es/" target="_blank">Colegio Oficial de Ingenieros Industriales del Principado de Asturias</a></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0España40.463667000000008 -3.749220000000036614.545167000000006 -45.057814000000036 66.38216700000001 37.559373999999963tag:blogger.com,1999:blog-3108912614914761750.post-82867104226024579052015-08-14T23:40:00.002+02:002015-08-22T23:26:10.093+02:00Arduino Mega 2560 - Test de velocidad<div class="separator" style="clear: both; text-align: justify;">
<b>Resumen</b></div>
<div class="separator" style="clear: both; text-align: justify;">
En esta entrada vamos a comparar la velocidad de salida de una línea discreta del Arduino Mega 2560 usando el código de Arduino frente a la programación directa por registros.</div>
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<b><br /></b></div>
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<b>Abstract</b></div>
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In this entry we will compare the discrete output speed using Arduino Mega 2560 code versus specific register programming.</div>
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-------ooooOOOOoooo-------</div>
<br />
<div style="text-align: justify;">
Una de los factores que ha hecho tremendamente popular al Arduino y ha acercado el HW al usuario medio es la capacidad para programarlo sin ser un experto en programación y microcontroladores.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Escribir código usando <b>las librerías que proporciona Arduino</b> es un gran avance ya que nos proporciona un nivel de abstracción respecto al HW. Sin embargo, todo ése código que nos hace la vida fácil<b> tiene un coste en tiempo de ejecución</b>. Dependiendo del destino que le demos a nuestro diseño, ése tiempo puede ser asumible o no.</div>
<br />
<div style="text-align: justify;">
Veamos un par de ejemplos comparando la programación con librerías Arduino con la programación directa por registros.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>1. PROGRAMACIÓN CON LIBRERÍAS ARDUINO</b></div>
<div style="text-align: justify;">
Para éste ejemplo, vamos a mover lo más rápido posible el pin 12 del Arduino usando las librerías Arduino.</div>
<div>
<br /></div>
<pre><span style="color: #00979c;">void</span> <span style="color: #5e6d03;">setup</span><span style="color: black;">(</span><span style="color: black;">)</span> <span style="color: black;">{</span>
<span style="color: #d35400;">pinMode</span><span style="color: black;">(</span><span style="color: black;">12</span><span style="color: #434f54;">,</span> <span style="color: #00979c;">OUTPUT</span><span style="color: black;">)</span><span style="color: black;">;</span>
<span style="color: black;">}</span>
<span style="color: #00979c;">void</span> <span style="color: #5e6d03;">loop</span><span style="color: black;">(</span><span style="color: black;">)</span> <span style="color: black;">{</span>
<span style="color: #d35400;">digitalWrite</span><span style="color: black;">(</span><span style="color: black;">12</span><span style="color: #434f54;">,</span> <span style="color: #00979c;">HIGH</span><span style="color: black;">)</span><span style="color: black;">;</span>
<span style="color: #d35400;">digitalWrite</span><span style="color: black;">(</span><span style="color: black;">12</span><span style="color: #434f54;">,</span> <span style="color: #00979c;">LOW</span><span style="color: black;">)</span><span style="color: black;">;</span>
<span style="color: black;">}</span></pre>
<pre><span style="color: black;">
</span></pre>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_VC-1Ust3_ZXPWdPDaT_sBTlwruJ1yHSgLkbP8E7ABxvNfoyfDOk1N2iUPjACcDSMftEMm3g6XNjePtGv3O5y-PdvMMLXq0afXAO9-iyAOMRAOGNQQ0CdS_r-ND5dq8JzmjpigIaVPzQ/s1600/arduino-mega-2560-test-de-velocidad-00.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Arduino Mega 2560 Test de Velocidad" border="0" height="251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_VC-1Ust3_ZXPWdPDaT_sBTlwruJ1yHSgLkbP8E7ABxvNfoyfDOk1N2iUPjACcDSMftEMm3g6XNjePtGv3O5y-PdvMMLXq0afXAO9-iyAOMRAOGNQQ0CdS_r-ND5dq8JzmjpigIaVPzQ/s400/arduino-mega-2560-test-de-velocidad-00.png" title="Arduino Mega 2560 Test de Velocidad" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.1 - Velocidad usando librerías Arduino</td></tr>
</tbody></table>
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<span style="background-color: white; font-size: 14.8500003814697px; line-height: 20.7900009155273px;"><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;"><br /></span></span></div>
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</div>
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</div>
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Como se aprecia en la figura 1, el resultado es una velocidad de conmutación media de 64 KHz. Se observa que esporádicamente el ciclo es inestable alterando a la frecuencia.</div>
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<br /></div>
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<b>2. PROGRAMACIÓN CON DIRECTA CON REGISTROS</b></div>
<div style="text-align: justify;">
Para éste ejemplo, vamos a mover lo más rápido posible el pin 12 del Arduino programando directamente los registros del microcontrolador.<br />
<br /></div>
<!-- INSERCIÓN DE CODIGO ARDUINO-->
<br />
<pre><span style="color: black;">#include</span> <span style="color: #434f54;"><</span><span style="color: black;">avr</span><span style="color: #434f54;">/</span><span style="color: black;">io</span><span style="color: #434f54;">.</span><span style="color: black;">h</span><span style="color: #434f54;">></span> <span style="color: #7e7e7e;">/* Defines pins, ports, etc */</span>
<span style="color: #00979c;">int</span> <span style="color: black;">main</span><span style="color: black;">(</span><span style="color: #00979c;">void</span><span style="color: black;">)</span> <span style="color: black;">{</span>
<span style="color: #434f54;">// -------- Inits --------- //</span>
<span style="color: #434f54;">//Arduino mega 2560 pin 12 enable as output</span>
<span style="color: black;">DDRB</span> <span style="color: #434f54;">|</span><span style="color: #434f54;">=</span> <span style="color: black;">(</span><span style="color: black;">1</span> <span style="color: #434f54;"><<</span> <span style="color: black;">PB6</span><span style="color: black;">)</span><span style="color: black;">;</span> <span style="color: #7e7e7e;">/* Data Direction Register B: writing a one to the bit</span><span style="color: #7e7e7e;">enables output. */</span>
<span style="color: #434f54;">// ------ Event loop ------ //</span>
<span style="color: #5e6d03;">while</span> <span style="color: black;">(</span><span style="color: black;">1</span><span style="color: black;">)</span> <span style="color: black;">{</span>
<span style="color: black;">PORTB</span> <span style="color: #434f54;">=</span> <span style="color: black;">0b01000000</span><span style="color: black;">;</span> <span style="color: #7e7e7e;">/* Turn on pin 12 (PB6 pin) */</span>
<span style="color: black;">PORTB</span> <span style="color: #434f54;">=</span> <span style="color: black;">0b00000000</span><span style="color: black;">;</span> <span style="color: #7e7e7e;">/* Turn off all B pins, including LED */</span>
<span style="color: black;">}</span> <span style="color: #7e7e7e;">/* End event loop */</span>
<span style="color: #5e6d03;">return</span> <span style="color: black;">(</span><span style="color: black;">0</span><span style="color: black;">)</span><span style="color: black;">;</span> <span style="color: #7e7e7e;">/* This line is never reached */</span>
<span style="color: black;">}</span></pre>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWCHsBUbXFe-TKlLKCfEShv_BJPJU_68GVgEaziVx5-RDoLJZoa_JGTab77R2NR_gjaWhR9HeP5NGJQmbGt-WkM2_hEksmOIubpni0EC3z5si040rzC1s8MrSzZD4n-pCk0fYjxS24r9Y/s1600/arduino-mega-2560-test-de-velocidad-01.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Arduino Mega 2560 speed test" border="0" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWCHsBUbXFe-TKlLKCfEShv_BJPJU_68GVgEaziVx5-RDoLJZoa_JGTab77R2NR_gjaWhR9HeP5NGJQmbGt-WkM2_hEksmOIubpni0EC3z5si040rzC1s8MrSzZD4n-pCk0fYjxS24r9Y/s400/arduino-mega-2560-test-de-velocidad-01.png" title="Arduino Mega 2560 speed test" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 2 - Velocidad con programación por registros</td></tr>
</tbody></table>
<div style="text-align: justify;">
En la figura 2 se puede apreciar una velocidad significativamente superior 4 MHz. La frecuencia de generación es estable y no se producen inestabilidades como el mostrado en la figura 1.</div>
<div style="text-align: justify;">
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<div style="text-align: justify;">
<b>3. CONCLUSIONES</b></div>
<div style="text-align: justify;">
<ul>
<li>La velocidad de funcionamiento del diseño definirá la forma de programar el Arduino.</li>
<li>En éste caso particular, <b>programar directamente por registros multiplica la velocidad x63 </b>respecto al uso de la librería de Arduino.</li>
<li>En el ejemplo se ha movido una sola línea. Mover el byte entero (8 líneas) a la vez, no consume más tiempo mientras que con las librerías de Arduino se multiplicaría por 8. <b>Si moviésemos el byte entero</b> supondría un incremente de velocidad de 63*8 = <b>504 veces más rápido</b>.</li>
</ul>
</div>
<div style="text-align: justify;">
<br /></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com2España40.463667000000008 -3.749220000000036614.545167000000006 -45.057814000000036 66.38216700000001 37.559373999999963tag:blogger.com,1999:blog-3108912614914761750.post-34285036866035665912015-07-26T02:57:00.001+02:002015-07-26T02:57:56.744+02:00Cambio latiguillos de freno VW Golf II 1.6TD<div style="text-align: justify;">
<b>Resumen</b></div>
<div style="text-align: justify;">
En esta entrada se describe el cambio de los latiguillos de freno delanteros de un Volkswagen Golf serie II, 1.6 GTD.</div>
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<br /></div>
<div style="text-align: justify;">
<b>Abstract</b></div>
<div style="text-align: justify;">
The aim of this entry is to describe the procedure for changing the flexible brake hoses in VW Golf II, 1.6 GTD series model.</div>
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<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVvshY2TWw-c8llDbvU2269YzoHmExfOgOupSFsIhm3deyRGaEH6hxjUowgfivXpyMeIKACrSkRPolSAvyKdcnwyK4FgjguQH-52qMfLpPkGwFpd2jOi2gzUDioHfPXQsTlVLJvuTqjuk/s1600/fuga-liguido-frenos-VW-Golf-II.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img alt="fuga de liquido de freno VW golf II" border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVvshY2TWw-c8llDbvU2269YzoHmExfOgOupSFsIhm3deyRGaEH6hxjUowgfivXpyMeIKACrSkRPolSAvyKdcnwyK4FgjguQH-52qMfLpPkGwFpd2jOi2gzUDioHfPXQsTlVLJvuTqjuk/s200/fuga-liguido-frenos-VW-Golf-II.jpg" title="fuga de liquido de freno VW golf II" width="135" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.1 - Inspección visual de fugas</td></tr>
</tbody></table>
El sistema hidráulico de frenado del VW Golf II se compone de tuberías rígidas de metal y de latiguillos flexibles. Por lo general, las tuberías de metal no van a presentar problemas con los años (salvo que reciban algún golpe o se oxiden), pero los latiguillos de freno presentan una vida útil bastante más corta y agravada dependiendo del de las condiciones ambientales que tengamos.<br />
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<b>En éste artículo nos centraremos en cómo cambiar los latiguillos de freno.</b><br />
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<b>1. COMPROBACIÓN VISUAL DEL SISTEMA DE FRENADO</b></div>
<div style="text-align: justify;">
Periódicamente se procederá a una inspección visual de todo el conjunto hidráulico. </div>
<ul>
<li style="text-align: justify;">En las <b>tuberías de metal </b>buscaremos indicios de oxidación, golpes o falta de sujeción. </li>
</ul>
<ul>
<li style="text-align: justify;">En los <b>latiguillos</b>, basta con doblarlos con la mano y observar si presenta indicio de grietas en la goma. Si la respuesta es afirmativa, se procederá a su cambio como mantenimiento preventivo. <b>Es muy recomendable cambiar los dos latiguillos simultáneamente</b>. Es decir, si uno de ellos presenta grietas y el otro aparentemente está bien, por precaución se cambiarán los dos.</li>
</ul>
<div style="text-align: justify;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsfCHtOvEbrasgPA97i3ulCSXMu15DcZDwg8QgRRwMdh273jXeCnzEeA1gfEV25yg25OoaUDnQNy1foduuHkFhScZ973oLeu8k8QVz_snWfdhFrNStNmElNVlldr0vp9bdfLlyBZA0mhU/s1600/test-latiguillo-freno.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="test de doblado de latiguillo de freno" border="0" height="232" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsfCHtOvEbrasgPA97i3ulCSXMu15DcZDwg8QgRRwMdh273jXeCnzEeA1gfEV25yg25OoaUDnQNy1foduuHkFhScZ973oLeu8k8QVz_snWfdhFrNStNmElNVlldr0vp9bdfLlyBZA0mhU/s320/test-latiguillo-freno.jpg" title="test de doblado de latiguillo de freno" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.2 - Test de doblado </td></tr>
</tbody></table>
<b>SINTOMAS DE PERDIDA DE LIQUIDO DE FRENO</b></div>
<ul style="text-align: justify;">
<li>Cuando con el tiempo se observa que <b>desciende el nivel de líquido de freno en el tanque</b>. </li>
<li>Cuando se pisa el pedal de freno a tope, se observa que poco a poco <b>perdemos pedal</b>.
Es decir, el pedal de freno se desplaza lentamente hasta el final de su
recorrido. Nota: Éste síntoma se podría confundir con un fallo en la
bomba del servofreno.</li>
<li>Cuando se observan <b>fugas y goteos</b> en el suelo (caso extremo como el de la figura 1).</li>
</ul>
<br />
<b>2. MATERIAL CONSUMIBLE NECESARIO</b><br />
<ul>
<li>Latiguillos de freno:</li>
<ul>
<li>Longitud [mm] : 340</li>
<li>Rosca 1 : M 10X1 (lado pinza de freno).</li>
<li>Rosca 2 : F 10X1 (lado tubería rígida). </li>
</ul>
<li> Líquido de freno: especificación DOT 4.</li>
<li>Tarro de cristal.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYPp30Sb-_Lpgkje-Xe3rTVz356Fl5mqG8jTYvhdy5AXSmbQhaMEM0wwLw5Asi3emlsbpUjzQaAp2Qh9ei7j9qYokZJ4Z4QJToNoLo__ik95yrzC1Avs3ztSLvjUyTOjBwc36URr5fxq4/s1600/Latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Latiguillo de freno delantero VW Golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYPp30Sb-_Lpgkje-Xe3rTVz356Fl5mqG8jTYvhdy5AXSmbQhaMEM0wwLw5Asi3emlsbpUjzQaAp2Qh9ei7j9qYokZJ4Z4QJToNoLo__ik95yrzC1Avs3ztSLvjUyTOjBwc36URr5fxq4/s400/Latiguillo-freno-VW-golf-II.jpg" title="Latiguillo de freno delantero VW Golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.3 - Latiguillo de freno VW Golf II </td></tr>
</tbody></table>
<b>3. HERRAMIENTA NECESARIA</b><br />
<br />
[1].- <a href="http://www.irimo.es/product.php?id_product=145" target="_blank">Llave para racores</a> nº 11<br />
[2].- <a href="http://www.irimo.es/product.php?id_product=91" target="_blank">Llave fija</a> de dos bocas 14x15<b></b><br />
[3].- Llave de vaso nº 17<br />
[4].- Llave fij<span style="background-color: white;">a nº 7</span><br />
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<b>4. LATIGUILLOS DE FRENO- DESMONTAJE Y MONTAJE</b><br />
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<div style="text-align: center;">
<b>PRECAUCIÓN</b></div>
<div style="text-align: center;">
<b>Cuidado no derramar líquido de frenos sobre la pintura de la carrocería. El líquido de frenos ataca la pintura creando un efecto decapante. En caso de derrame, aclarar con agua fría.</b></div>
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<b>4.1 DESMONTAJE</b></div>
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<ul>
<li>Montar el coche sobre caballetes o en un elevador y quitar las ruedas delanteras.</li>
</ul>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCmaRJeH00grmJpHWms76Nz599gpRS1615jZ07w-ZLFuRiQKLV-e51uBdU_9cxFsoWvx9ZaiWM7N-c5DaRJMbp_lkgKj-wL2RvC1nBE-hQBvPM3CkTHH15Nq3kESI-T5xbcZ6ahj7Pzy8/s1600/01-aflojar-rueda-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="rueda VW golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCmaRJeH00grmJpHWms76Nz599gpRS1615jZ07w-ZLFuRiQKLV-e51uBdU_9cxFsoWvx9ZaiWM7N-c5DaRJMbp_lkgKj-wL2RvC1nBE-hQBvPM3CkTHH15Nq3kESI-T5xbcZ6ahj7Pzy8/s400/01-aflojar-rueda-VW-golf-II.jpg" title="rueda VW golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.4 - Extracción rueda delantera</td></tr>
</tbody></table>
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<ul>
<li>Una vez liberada la rueda, se puede observar el latiguillo de freno como se indica en la figura 5.</li>
</ul>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIKJc3-y5l13erpxaY3unna8OFqU-kXdYlovgOfOe67RxsFuxit6OrtFTEfy69pE2MdEiZU0Ba-NRlMLx7jTAfp4pOOqinoFdpx5TBcVSIhQtyee7eRZKSpU9o60-6fFwPn7SxGDURwws/s1600/02-disco-freno-latiguillo-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Identificación latiguillo freno vw Golf II" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIKJc3-y5l13erpxaY3unna8OFqU-kXdYlovgOfOe67RxsFuxit6OrtFTEfy69pE2MdEiZU0Ba-NRlMLx7jTAfp4pOOqinoFdpx5TBcVSIhQtyee7eRZKSpU9o60-6fFwPn7SxGDURwws/s400/02-disco-freno-latiguillo-vw-golf-II.jpg" title="Identificación latiguillo freno vw Golf II" width="260" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.5 - Identificación del latiguillo</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: justify;">
</div>
<ul>
<li>El latiguillo de freno une la tubería rígida (figura 6) con la pinza de freno (figura 7)</li>
</ul>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4em3rJFE0w1RHuhnr5JluIJv8GxZJKM1ocPLC9brt3ULnzGPXZ-shEvT_LcStX3ODuBSKGv2mm56YknZCgku7A3ykhB50Ixs49npclPR7CddN-3wkFUtexuwY-KZJJYICf5a7CQC_RLU/s1600/03-Partes-del-sistema-de-frenado-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="conexión latiguillo freno con tubería rígida vw golf II" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4em3rJFE0w1RHuhnr5JluIJv8GxZJKM1ocPLC9brt3ULnzGPXZ-shEvT_LcStX3ODuBSKGv2mm56YknZCgku7A3ykhB50Ixs49npclPR7CddN-3wkFUtexuwY-KZJJYICf5a7CQC_RLU/s400/03-Partes-del-sistema-de-frenado-vw-golf-II.jpg" title="conexión latiguillo freno con tubería rígida vw golf II" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.6 - Conexión latiguillo con tubería rígida</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJ3m4tZ1aKv2UQ6lkSSU9VSmmswSElHPcv7ait4_DsjOQx5ACcmgGPkju9DNTjibpU9zg2w-vtwpxwaIlI7WIFFQVTYkTO69g_R-aO7JkeKxVHM2NW1eSzra5afg-JOLvCWtWevgmTBlc/s1600/04-barro-en-pinza-freno-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="conexión latiguillo freno vw Golf II con finza de freno" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJ3m4tZ1aKv2UQ6lkSSU9VSmmswSElHPcv7ait4_DsjOQx5ACcmgGPkju9DNTjibpU9zg2w-vtwpxwaIlI7WIFFQVTYkTO69g_R-aO7JkeKxVHM2NW1eSzra5afg-JOLvCWtWevgmTBlc/s400/04-barro-en-pinza-freno-vw-golf-II.jpg" title="conexión latiguillo freno vw Golf II con finza de freno" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.7 - Conexión latiguillo con pinza de freno</td></tr>
</tbody></table>
<br />
<ul>
<li style="text-align: justify;">Antes de liberar el latiguillo, es conveniente limpiar concienzudamente las uniones de cualquier tipo de impurezas. En el caso que se muestra, hay una gran cantidad de barro reseco acumulado por el paso de los años.</li>
</ul>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0Ks17hmAWUhwX9-7eVGXOq39VASO6iswZUKSvR-eVsLbLW2s6SeaoP6wDrlsk8rYXNCyEEvsRq4O3Nwc-rG7mXliS99TTU9GSjBnDFQsLLZAEwiqxFeAzHoNMxcwJ7N0a0HZ_122Xu24/s1600/05-sustituir-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="limpieza superficial latiguillo freno vw golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0Ks17hmAWUhwX9-7eVGXOq39VASO6iswZUKSvR-eVsLbLW2s6SeaoP6wDrlsk8rYXNCyEEvsRq4O3Nwc-rG7mXliS99TTU9GSjBnDFQsLLZAEwiqxFeAzHoNMxcwJ7N0a0HZ_122Xu24/s400/05-sustituir-latiguillo-freno-VW-golf-II.jpg" title="limpieza superficial latiguillo freno vw golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.8 - Limpieza superficial (1/2)</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtduxuZ_1JCItrVzC4Q03FlOks0M6vXy3JPqTFtsVsz3ppPqyPQauFC7XW1bzsPM3hcLwelT-yHczObR1UtlhTCQjQtGcZ20DxKzvYw-Vr7S1nEuYordcYVS44V28m9dZ66IsE9WDz9E8/s1600/06-Limpieza-sustituci%25C3%25B3n-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="limpieza superficial latiguillo freno vw golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtduxuZ_1JCItrVzC4Q03FlOks0M6vXy3JPqTFtsVsz3ppPqyPQauFC7XW1bzsPM3hcLwelT-yHczObR1UtlhTCQjQtGcZ20DxKzvYw-Vr7S1nEuYordcYVS44V28m9dZ66IsE9WDz9E8/s400/06-Limpieza-sustituci%25C3%25B3n-latiguillo-freno-VW-golf-II.jpg" title="limpieza superficial latiguillo freno vw golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.9 - Limpieza superficial (2/2)</td></tr>
</tbody></table>
<br />
<ul>
<li style="text-align: justify;">Una vez limpias la uniones, se procede a aflojar en primer lugar la tubería rígida. <b>Es muy importante que se use una llave especial para racor </b>(ref: [1]). De esta forma, no se deformará la tuerca al intentar aflojar o apretar.</li>
<li style="text-align: justify;">En mi caso particular, la tuerca se encontraba agarrada por el paso de los años. Tuve que emplear en primer lugar WD-40 para intentar aflojarla y como me resultó imposible, al final tuve que combinarlo con la aplicación de calor.</li>
</ul>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEir61Tcw4XotV_stMrwzNrEW3jqKemGHUEoxc9VXaqGzLLPcNR5Q3FMaxsLxZAYZWS-ILzeur6kmhJrRe62zCPpS0zrlO7lDFTThXObVGsor5oL5GKswn4uAE-9xM9T58j9fHNFrYarczw/s1600/07-WD40-aflojar-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="aplicación lubricante WD-40 latiguillo freno VW Golf II" border="0" height="271" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEir61Tcw4XotV_stMrwzNrEW3jqKemGHUEoxc9VXaqGzLLPcNR5Q3FMaxsLxZAYZWS-ILzeur6kmhJrRe62zCPpS0zrlO7lDFTThXObVGsor5oL5GKswn4uAE-9xM9T58j9fHNFrYarczw/s400/07-WD40-aflojar-latiguillo-freno-VW-golf-II.jpg" title="aplicación lubricante WD-40 latiguillo freno VW Golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.10 - Aplicación de lubricante WD-40</td></tr>
</tbody></table>
<br />
<ul>
<li style="text-align: justify;">La aplicación de calor hace que se dilaten los materiales y combinado con el WD-40 es una combinación ganadora. El único inconveniente es que el latiguillo se estropea, pero como es para tirar... no hay ningún problema.</li>
</ul>
<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuKqFGZysskKc_olyca1RhcBJqfd9KZ0UVJRmjkiuzQVwMObpGPTwzsAhTYrA0S4whWBNgHEct0our8zcakUlMy_64EHa-j57C2_DL6TpcTHJ55_9U_0b09s49iF_Jt6hDBvDwIhwpC0s/s1600/08-aflojar-latiguillo-freno-con-calor-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="aplicación de calor para aflojar latiguillo freno VW Golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuKqFGZysskKc_olyca1RhcBJqfd9KZ0UVJRmjkiuzQVwMObpGPTwzsAhTYrA0S4whWBNgHEct0our8zcakUlMy_64EHa-j57C2_DL6TpcTHJ55_9U_0b09s49iF_Jt6hDBvDwIhwpC0s/s400/08-aflojar-latiguillo-freno-con-calor-VW-golf-II.jpg" title="aplicación de calor para aflojar latiguillo freno VW Golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.11 - Aplicación de calor </td></tr>
</tbody></table>
<br />
<ul>
<li>Tras la aplicación de calor, por fin pude liberar el racor de la tubería rígida.</li>
</ul>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIbt-mijoik6tKnbaM4He4PnRjToZTJA0CqwrtkikaXjZID7xWrkohiWHlbwIz4Xn5KUeGXYwuN2F-_1jD7-j5K1PQeZE8yAp6LdrZUz53sSmjCeoBB_PXKUOS_I_0EvKRkzh97Y7rs68/s1600/09-aflojar-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Aflojar tuerca tubería rígida latiguillo freno VW Golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIbt-mijoik6tKnbaM4He4PnRjToZTJA0CqwrtkikaXjZID7xWrkohiWHlbwIz4Xn5KUeGXYwuN2F-_1jD7-j5K1PQeZE8yAp6LdrZUz53sSmjCeoBB_PXKUOS_I_0EvKRkzh97Y7rs68/s400/09-aflojar-latiguillo-freno-VW-golf-II.jpg" title="Aflojar tuerca tubería rígida latiguillo freno VW Golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.12 - Aflojar tuerca tubería rígida (1/2)</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcYTvTO_moDjiLTkvRPBl0Kvo-t5fKj2BVLvw24XxmbPdmDLEoiJCVQU00ODlOw_pGpNpA6_LKpJjVtf2iOGVa6vWkh6IXeBlAGOSEq7z9fXXN5uUeQ6l35j6M27riPYPa93nHvxRbPZQ/s1600/11-aflojar-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Aflojar tuerca tubería rígida latiguillo freno VW Golf II" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcYTvTO_moDjiLTkvRPBl0Kvo-t5fKj2BVLvw24XxmbPdmDLEoiJCVQU00ODlOw_pGpNpA6_LKpJjVtf2iOGVa6vWkh6IXeBlAGOSEq7z9fXXN5uUeQ6l35j6M27riPYPa93nHvxRbPZQ/s400/11-aflojar-latiguillo-freno-VW-golf-II.jpg" title="Aflojar tuerca tubería rígida latiguillo freno VW Golf II" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.13 - <span style="font-size: 12.8000001907349px;">Aflojar tuerca tubería rígida (2/2)</span></td></tr>
</tbody></table>
<br />
<ul>
<li>Una vez liberado el latiguillo de la tubería rígida, se procederá a aflojarlo de la pinza de freno y ya queda liberado.</li>
</ul>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2VaYaB4zChuhnaX87O424L8qAekzIBtdqWcm_K_HoSwwPrbeFCFeMbylDdYMRqWGojy8C7lNw3DnsqlMpIzg3mqSIfyg2XHJ2VIDaAn9K4ihH1oWMyAGjb231KbKRZzJMbspTTJhnfZc/s1600/10-aflojar-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Liberar latiguillo de pinza de freno VW Golf II" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2VaYaB4zChuhnaX87O424L8qAekzIBtdqWcm_K_HoSwwPrbeFCFeMbylDdYMRqWGojy8C7lNw3DnsqlMpIzg3mqSIfyg2XHJ2VIDaAn9K4ihH1oWMyAGjb231KbKRZzJMbspTTJhnfZc/s400/10-aflojar-latiguillo-freno-VW-golf-II.jpg" title="Liberar latiguillo de pinza de freno VW Golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.14 - Liberar latiguillo de pinza de freno</td></tr>
</tbody></table>
<b>4.2 MONTAJE</b><br />
<br />
<ul>
<li>Realizar los pasos inversos al desmontaje. </li>
<li>En primer lugar se abocará el latiguillo a la pinza de freno.</li>
</ul>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbbUNAkaAPJOYfDsZyzdGCwtlA9v1dSpXlPtmOVZDldBmdRgU9Q-NsJzTMiRgYivH7rJzz9b_EE8AYk8kE1zxIoelHhEF3-XDzv451skP3tvZ4lUHVbVG5lnU1ULBiAmmPgqiqYr1Xer4/s1600/12-aflojar-latiguillo-freno-VW-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="conexión latiguillo freno VW Golf II a pinza de freno" border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbbUNAkaAPJOYfDsZyzdGCwtlA9v1dSpXlPtmOVZDldBmdRgU9Q-NsJzTMiRgYivH7rJzz9b_EE8AYk8kE1zxIoelHhEF3-XDzv451skP3tvZ4lUHVbVG5lnU1ULBiAmmPgqiqYr1Xer4/s400/12-aflojar-latiguillo-freno-VW-golf-II.jpg" title="conexión latiguillo freno VW Golf II a pinza de freno" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.15 - Conexión nuevo latiguillo (1/2)</td></tr>
</tbody></table>
<br />
<ul>
<li>Abocar el extremo superior del latiguillo a la tubería rígida.</li>
</ul>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfApJrUDoVJHN7duL0Ozb96k1ANRla8o5fNToXgML-gHmuCQPOJKpTP7sxl7MlEmfjxIHqi0xmtPFq8WON2ukUyN3TZEAjDV2d3NS_XpuOcdqAlBySWe6J1Zrn829hiqci5iJ2odp7Dv8/s1600/13-instalar-latiguillo-freno-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="conexión latiguillo freno VW Golf II a tubería rígida" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfApJrUDoVJHN7duL0Ozb96k1ANRla8o5fNToXgML-gHmuCQPOJKpTP7sxl7MlEmfjxIHqi0xmtPFq8WON2ukUyN3TZEAjDV2d3NS_XpuOcdqAlBySWe6J1Zrn829hiqci5iJ2odp7Dv8/s400/13-instalar-latiguillo-freno-vw-golf-II.jpg" title="conexión latiguillo freno VW Golf II a tubería rígida" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.16 - <span style="font-size: 12.8000001907349px;">Conexión nuevo latiguillo (2/2)</span></td></tr>
</tbody></table>
<br />
<ul>
<li>Una vez abocado todo el conjunto, apretar en primer lugar la parte de la pinza de freno.</li>
</ul>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTcyKBaHOf-DW93y9VtJPzr24-nIuAD1rm5Te5tcoV__LpDjDhwj4oEahxNlLpAFkq4hYCX74U8Sp5aE3s5mUyIyKoLz_i0J1BEj1UNqWhDt-AvEgyjtSqeHlRSZNvOev6scmbZAp8s0I/s1600/14-instalar-latiguillo-freno-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="apriete latiguillo freno VW Golf II a pinza de freno" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTcyKBaHOf-DW93y9VtJPzr24-nIuAD1rm5Te5tcoV__LpDjDhwj4oEahxNlLpAFkq4hYCX74U8Sp5aE3s5mUyIyKoLz_i0J1BEj1UNqWhDt-AvEgyjtSqeHlRSZNvOev6scmbZAp8s0I/s400/14-instalar-latiguillo-freno-vw-golf-II.jpg" title="apriete latiguillo freno VW Golf II a pinza de freno" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.17 - Apriete nuevo latiguillo a pinza freno</td></tr>
</tbody></table>
<br />
<ul>
<li style="text-align: justify;">Por último apretar la parte del racor de la tubería rígida. No es necesario apretar de una manera desproporcionada. Los asentamientos del latiguillo son cónicos y basta con apretarlos a mano sin aplicar un par excesivo para garantizar la estanqueidad del conjunto. </li>
</ul>
<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilAiSOnKTCWl9ziZnln8SIvK564NYMBfwVPxpQjmdvyiLukO0ez9P_AC0k4ViLmqvMnc5Z0bS77xepIc40CiW9BwBLrSKYlNQ12LCV8e0wbZh59L0ZyQ7scMnIowLnIF0d8JJfscfXJug/s1600/15-instalar-latiguillo-freno-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="apriete latiguillo freno VW Golf II a tubería rígida" border="0" height="253" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilAiSOnKTCWl9ziZnln8SIvK564NYMBfwVPxpQjmdvyiLukO0ez9P_AC0k4ViLmqvMnc5Z0bS77xepIc40CiW9BwBLrSKYlNQ12LCV8e0wbZh59L0ZyQ7scMnIowLnIF0d8JJfscfXJug/s400/15-instalar-latiguillo-freno-vw-golf-II.jpg" title="apriete latiguillo freno VW Golf II a tubería rígida" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.18 - Apriete nuevo latiguillo a tubería rígida</td></tr>
</tbody></table>
<br />
<ul>
<li>Por último, una vez ensamblado el conjunto, no debemos olvidar colocar la grapa de goma que guía el latiguillo al amortiguador. </li>
</ul>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9oVcEV6ioCN0m6zw3kW104JSb4pNmOKw3AGkhZLksCCm7_9l960YxuVoiE1PuzxwpzvE6tnuEQNUmZunc_0gVHYRgllz3i5gczL647wYmTrhnWbj3huvqMypDhLTCgLHqUCFP7tpHQZ4/s1600/16-instalar-latiguillo-freno-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="vista ensamblaje latiguillo de freno VW Golf II" border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9oVcEV6ioCN0m6zw3kW104JSb4pNmOKw3AGkhZLksCCm7_9l960YxuVoiE1PuzxwpzvE6tnuEQNUmZunc_0gVHYRgllz3i5gczL647wYmTrhnWbj3huvqMypDhLTCgLHqUCFP7tpHQZ4/s400/16-instalar-latiguillo-freno-vw-golf-II.jpg" title="vista ensamblaje latiguillo de freno VW Golf II" width="270" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.19 - Vista conjunto latiguillo ensamblado</td></tr>
</tbody></table>
<br />
<b>5. PURGADO DE FRENOS. MÉTODO DE DOS HOMBRES</b> <br />
<br />
<div style="text-align: justify;">
Tras los cambios de los latiguillos, se debe extraer los restos de aire contenidos en las tuberías. Para ello, se procede al purgado del sistema de frenos.</div>
<div style="text-align: justify;">
Este procedimiento necesita de dos operarios. Uno se situará en el lugar del conductor para accionar el pedal de freno y el otro actuará sobre los purgadores.</div>
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUlTWXoD17rlIof4FmVjT8dKv3T3lB2KXa0fx-RHOKWqtPefU2CmpWxkszRCpLy_PpZW25zv2_KNE0fz7mTth8GrLnC-KdKYOK-tofGNmworFgU93XJn7_TNt5Jycz_WXDJ9ExMMtgbzE/s1600/17-purgado-frenos-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Purgado de frenos delanteros VW Golf II" border="0" height="331" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUlTWXoD17rlIof4FmVjT8dKv3T3lB2KXa0fx-RHOKWqtPefU2CmpWxkszRCpLy_PpZW25zv2_KNE0fz7mTth8GrLnC-KdKYOK-tofGNmworFgU93XJn7_TNt5Jycz_WXDJ9ExMMtgbzE/s400/17-purgado-frenos-vw-golf-II.jpg" title="Purgado de frenos delanteros VW Golf II" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.20 - Purgado latiguillo</td></tr>
</tbody></table>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<br />
<ul>
<li>Localizar un tarro vacío y un trozo de tubería que se ajuste al purgador de la pinza de freno.</li>
<li>Limpiar alrededor del purgador y conectarle el tubo de purgado.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzw1k49bA1YPzuAKjY6CS8T4SxkkIgwqlTnirUg-XXeQzh_b_uzXJIl9MbptyKaHozymZNXvUReANA4a-VfYVoT1j_zSZCcc7IYXAwWPEa2gFwQ9_tlqZ7a13wuwu7oonwXNKOLrp22q0/s1600/18-purgado-frenos-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Conexión tubo drenaje al purgador VW Golf II" border="0" height="232" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzw1k49bA1YPzuAKjY6CS8T4SxkkIgwqlTnirUg-XXeQzh_b_uzXJIl9MbptyKaHozymZNXvUReANA4a-VfYVoT1j_zSZCcc7IYXAwWPEa2gFwQ9_tlqZ7a13wuwu7oonwXNKOLrp22q0/s320/18-purgado-frenos-vw-golf-II.jpg" title="Conexión tubo drenaje al purgador VW Golf II" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.21 - Conexión tubo drenaje</td></tr>
</tbody></table>
<br />
<br />
<ul>
<li style="text-align: justify;">Comprobar el nivel del depósito del líquido de frenos. Asegurarse que esté lleno. Si es necesario, accionar varias veces el pedal de freno para bajar el líquido de frenos.</li>
</ul>
<ul>
<li style="text-align: justify;">Sumergir la parte sin conectar de la tubería en el tarro. Este tendrá en el fondo líquido de frenos. El tarro se posicionará a unos 30 cm sobre el purgador para evitar la re-entrada de burbujas de aire a través del purgador durante el proceso.</li>
</ul>
<ul>
<li style="text-align: justify;">Abrir el purgador media vuelta y presionar el pedal de freno lentamente hasta el fondo. Observar la salida de aire mediante el burbujeo en el tarro. Apretar el purgador antes de soltar el pedal de freno. Repetir el proceso hasta que deje de observarse burbujeo en el tarro.</li>
</ul>
<ul>
<li style="text-align: justify;">Rellenar el depósito de líquido de freno. Asegurarse que en ningún momento del proceso se queda por debajo del nivel mínimo ya que de lo contrario el aire podría entrar en el sistema de frenado y se debería empezar de nuevo todo el proceso.</li>
</ul>
<ul>
<li style="text-align: justify;">Repetir la operación en la diagonal opuesta al freno. Si es necesario purgar todas las ruedas, la secuencia será:</li>
<ul>
<li style="text-align: justify;">rueda trasera del lado derecho.</li>
<li style="text-align: justify;">rueda trasera lado izquierdo.</li>
<li style="text-align: justify;">rueda delantera lado derecho.</li>
<li style="text-align: justify;">rueda delantera lado izquierdo.</li>
</ul>
</ul>
<ul>
<li style="text-align: justify;">Comprobar que tras el purgado, al presionar el pedal de freno, este se mantiene firme, duro y no presenta síntomas de efecto muelle que indicarían la presencia de aire en el circuito.</li>
</ul>
<ul>
<li style="text-align: justify;">Al finalizar el proceso, desechar el líquido de freno contenido en el tarro. No utilizarlo para rellenar el depósito ya que podría contener impurezas.</li>
</ul>
<ul>
<li style="text-align: justify;">Proceder a poner el tapón protector sobre el purgador para eviar la entrada de impurezas.</li>
</ul>
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWXb7rDEi52J9PDLrBkcyD6IiS9JkpBSXgw4cek6WDdCQrqyfabF3kQT6As6JqAjoM6IBFex9jgI7B9B0DxPY6xf93ugHdkNQX6HUGptDEBgpQtozP42x0ouqSeG9_ypeHsg_WXPclBOc/s1600/19-purgado-frenos-vw-golf-II.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="tapón protector del purgador VW Golf II" border="0" height="232" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWXb7rDEi52J9PDLrBkcyD6IiS9JkpBSXgw4cek6WDdCQrqyfabF3kQT6As6JqAjoM6IBFex9jgI7B9B0DxPY6xf93ugHdkNQX6HUGptDEBgpQtozP42x0ouqSeG9_ypeHsg_WXPclBOc/s320/19-purgado-frenos-vw-golf-II.jpg" title="tapón protector del purgador VW Golf II" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.22 - Tapón protector</td></tr>
</tbody></table>
<br />JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com1Madrid, Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-66912382278102964732015-07-13T02:34:00.000+02:002015-07-13T02:34:32.572+02:00Mis latinos favoritos<div style="text-align: justify;">
<b>Resumen</b></div>
<div style="text-align: justify;">
Lista de Spotify con mi música latina favorita, salsa, bachata, merengue, son cubano y mucho más.</div>
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<a name='more'></a><div style="text-align: justify;">
Dicen que los ingenieros somos un poco "raritos" en cuanto a habilidades sociales se refiere. Quizás la estructura mental de nuestro cerebro se extiende, invadiendo sin compasión áreas sociales como la habilidad de comunicación con el sexo contrario, o hablar con los amigos de temas no cuantificables con números y fórmulas. </div>
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<br /></div>
<div style="text-align: justify;">
Dicen que si añadimos la combinación ingeniero y baile, hace que sea más probable encontrar un lince ibérico que un espécimen de ésas características (¡vaya! ¡ya estoy cuantificando!)</div>
<div style="text-align: justify;">
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<div style="text-align: justify;">
Para romper falsos mitos y leyendas urbanas, aquí os dejo mi lista de música latina favorita que tengo elaborada en Spotify, que he disfrutado y disfrutaré bailando mientras no tenga desgaste de cadera :)</div>
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<div style="text-align: center;">
<iframe allowtransparency="true" frameborder="0" height="380" src="https://embed.spotify.com/?uri=spotify:user:josemanuel.vega:playlist:2TG93xRDzOfkRfUmrYFh01" width="300"></iframe><br />
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<div style="text-align: justify;">
Espero que la disfrutéis tanto como yo, y si os gusta, no dudéis en seguirla. <br />
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<iframe allowtransparency="true" frameborder="0" height="25" scrolling="no" src="https://embed.spotify.com/follow/1/?uri=spotify:user:josemanuel.vega:playlist:2TG93xRDzOfkRfUmrYFh01&size=basic&theme=light" style="border-style: none; border-width: initial; overflow: hidden; text-align: center;" width="200"></iframe></div>
</div>
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Todos a bailar!!<br />
<br />JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0España40.463667000000008 -3.749220000000036614.545176500000007 -45.057814000000036 66.3821575 37.559373999999963tag:blogger.com,1999:blog-3108912614914761750.post-81135338525004799192015-06-04T15:41:00.001+02:002015-06-04T15:41:13.851+02:00Número F, tiempo de exposición e Irradiancia<b>Resumen</b><br />
<div style="text-align: justify;">
En esta entrada se describe la relación entre el número F, el tiempo de exposición y la irradiancia percibida por un sensor de imagen.<b></b></div>
<b><br /></b>
<b>Abstract</b><br />
In this entry is described the relationship among f-number, exposure time and irradiance.<br />
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<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDeUA-9Xa204MA85qKuaOQ7yf6HKTZ7255xm7LfypZ9oBWgmSjXnkLMjah5H8rROd-JJdUDMW_OX_Eg1oMg9tFD_V6kmDuVphGeod4LL0_Idg0UUSwWnF8O6066LWqNRWnIetkk044pcQ/s1600/camara-werlisa-color.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img alt="Cámara fotos Werlisa color" border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDeUA-9Xa204MA85qKuaOQ7yf6HKTZ7255xm7LfypZ9oBWgmSjXnkLMjah5H8rROd-JJdUDMW_OX_Eg1oMg9tFD_V6kmDuVphGeod4LL0_Idg0UUSwWnF8O6066LWqNRWnIetkk044pcQ/s1600/camara-werlisa-color.jpg" title="Cámara fotos Werlisa color" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1 - Cámara fotos Werlisa color</td></tr>
</tbody></table>
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Cuando era niño, mi padre tenía una cámara de fotos WERLISA color. Recuerdo su lente repleta de ajustes, llena de números incomprensibles y que ninguno de la familia sabía manejar. El único interfaz hombre-máquina era una burda tablilla que teníamos pegada en el interior de su funda de cuero. Allí se indicaban diferentes escenarios de luz (lluvia, sol, nublado, flash) relacionados con la distancia de enfoque, ISO y diafragma, uff, demasiadas variables y muy poca información. Al final las fotos se hacían con la técnica de ensayo y error. ¡Lástima que había que esperar al revelado 6 meses después para conocer el resultado de unos ajustes ya olvidados!</div>
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<br />
Uno de los grandes misterios para el fotógrafo novel es entender la interrelación entre el número F, longitud focal, tiempo de exposición e ISO.<br />
<br />
Todos los conceptos que se expondrán a continuación son aplicables tanto al espectro visible como al infrarrojo usado en cámaras térmicas FLIR, aunque por popularidad, se hará referencia a las cámaras de visible.<br />
<br />
Durante esta entrada y para una mejor comprensión de los conceptos expuestos, vamos a dejar la variable ISO fija y estudiar la relación entre el número F y el tiempo de exposición apoyándonos en conceptos físicos de radiometría.<br />
<br />
<b>1. DEFINICIONES </b><br />
<br />
<b>Número F (F#):</b> se define
como la relación entre la longitud focal de la lente (fl) y el diámetro de la apertura
(D). También se la conoce por apertura relativa o f/stop. Se trata de un
número adimensional.<br />
<div style="text-align: justify;">
$$f\#=\frac{fl}{D}\ \ (1)$$</div>
Por
ejemplo, una lente de focal 50 mm con una apertura (diámetro) de 35.7
mm proporciona un F# de 50/35.7= 1.4. Los fotógrafos normalmente hacen
referencia a esta lente como aquella que tiene f/stop de f/1.4<br />
<br />
<b>Tiempo de exposición Ts [s]:</b> es el tiempo necesario que tenemos
que tener expuesto el detector (o película) para formar la imagen. En
fotografía se suele medir en fracciones de segundo. Por ejemplo 1/30 s <br />
<br />
<b>Ángulo sólido</b><br />
En geometría plana, un ángulo plano mide la apertura entre dos lineas. La medida del ángulo en radianes, se define entonces como<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgg3CrL7nL_u2eV3bDDelvkXldh0NiwdsGzACIUUE3k3AGTPSoGm966afzT226U59jQUdxkXcMZnu7JKl97fhq2Ocni6CXBgmhYfHpNGpvlvlHORVCYREh8bM9akzOPGbtLAyxiBqaVB50/s1600/angulo-plano.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="ángulo plano" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgg3CrL7nL_u2eV3bDDelvkXldh0NiwdsGzACIUUE3k3AGTPSoGm966afzT226U59jQUdxkXcMZnu7JKl97fhq2Ocni6CXBgmhYfHpNGpvlvlHORVCYREh8bM9akzOPGbtLAyxiBqaVB50/s1600/angulo-plano.png" title="ángulo plano" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 2 - Ángulo plano</td></tr>
</tbody></table>
$$\delta \theta\left [ rad \right ] =\frac{\delta s}{\delta r}\ \ (2)$$<br />
Donde theta representa el ángulo medido, s la longitud del arco de circunferencia trazado, y r al radio de la circunferencia utilizada para trazar el arco. La medida del ángulo es adimendional.<br />
<br />
Si ahora migramos ésta idea a 3 dimensiones e integramos, obtendremos por analogía el concepto de ángulo sólido.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFQLEyua1AcKTOg6re2Lf-gCOIJfFfBHaWukN8bqMMslvHbgqs1tWUsom444_J9fRYthfzpGDDc7FPvlO8i0US0iEs241panY2dJ2k7AZP-fn3gLsaY1xwfx_R-6_ERMTm6Qm6ebAk234/s1600/angulo-solido.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="ángulo sólido" border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFQLEyua1AcKTOg6re2Lf-gCOIJfFfBHaWukN8bqMMslvHbgqs1tWUsom444_J9fRYthfzpGDDc7FPvlO8i0US0iEs241panY2dJ2k7AZP-fn3gLsaY1xwfx_R-6_ERMTm6Qm6ebAk234/s200/angulo-solido.jpg" title="ángulo sólido" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 3 - Ángulo sólido</td></tr>
</tbody></table>
$$\delta \Omega \left [ sr \right ] =\frac{ \delta A}{ r^{2}}\ \ (3)$$<br />
<br />
<b>Radiancia</b><br />
Una de las unidades básicas de radiometría es la <b>Radiancia </b>(Le). A partir de ella, se derivan otras muchas que iremos viendo. Se define radiancia como la cantidad de flujo [W] radiados dentro de un cono de ángulo sólido [sr] desde una fuente cuyo área se mide en metros cuadrados.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj9ROk_kzvmvY7gry-WgW0w8LFnhFVg0UT1F5geiXOcBZ5NEBZPPJC0rwnSiAKThPC1esTa36jr62yMsVpykCiDxFWQKAHD6WHbQMir9rgbhyJwPekLDN3OnFQK7VDz1r3fQaFubJ81Nw/s1600/radiancia-y-angulo-solido.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="radiancia de una superficie" border="0" height="210" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj9ROk_kzvmvY7gry-WgW0w8LFnhFVg0UT1F5geiXOcBZ5NEBZPPJC0rwnSiAKThPC1esTa36jr62yMsVpykCiDxFWQKAHD6WHbQMir9rgbhyJwPekLDN3OnFQK7VDz1r3fQaFubJ81Nw/s400/radiancia-y-angulo-solido.png" title="radiancia de una superficie" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 4 - Radiancia de una superficie</td></tr>
</tbody></table>
Para una superficie perpendicular "As" al cono radiado se puede expresar la radiancia como:<br />
$$L_{e}\left [ \frac{W}{m^{2}\cdot sr} \right ]=\frac{\delta \phi }{\delta A_{s}\cdot \delta \Omega }\ \ (4)$$<br />
<b>Para un sistema óptico sin pérdidas, la radiancia permanece invariable</b> a lo largo de todo el camino óptico.<br />
<br />
<b>2. FLUJO RECIBIDO POR EL DETECTOR </b><br />
<b><br /></b>
En la realidad, el flujo radiante de la fuente se atenúa a medida que atraviesa la atmósfera hasta que llega a la lente. Éste flujo resultante que incide sobre la apertura del objetivo, de nuevo se vuelve a atenuar a medida que va atravesando las lentes internas del mismo hasta que se enfoca finalmente sobre el detector. El flujo resultante en función del tiempo de exposición, se convierte en energía y esa energía se transforma en el detector en voltaje que está directamente relacionado con la luminosidad de la imagen.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3_09QFiybjgC8l6AMk4f2uU3__HOOfL_XoXNCG3KyBZyp_xtWSiEZ10TTF92ez0w2UmFHj9Pse50C4Nz6g4X5kOditRtbqKAi2GBYd3Ec2locFVlh56WWo4rmGvqTWWJ7ty1okUhOa9M/s1600/image-system-viewing-a-source.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="image system viewing a source" border="0" height="251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3_09QFiybjgC8l6AMk4f2uU3__HOOfL_XoXNCG3KyBZyp_xtWSiEZ10TTF92ez0w2UmFHj9Pse50C4Nz6g4X5kOditRtbqKAi2GBYd3Ec2locFVlh56WWo4rmGvqTWWJ7ty1okUhOa9M/s640/image-system-viewing-a-source.png" title="image system viewing a source" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 5 - Imagen captada de una fuente radiante</td></tr>
</tbody></table>
Supongamos ahora una fuente radiante de área As a una distancia R1 de la cámara. El flujo radiante incidente sobre la apertura del objetivo (de área A0) se puede cuantificar como:<br />
$$\phi_{lente}=\iint_{ }^{ }L_{e}\cdot \delta A_{s}\cdot \delta \Omega\ \ (5) $$<br />
Integrando sobre el ángulo sólido abarcado por la apertura de la lente y haciendo simplificaciones por el pequeño ángulo abarcado por el ángulo sólido: <br />
$$\phi_{lente}=L_{e}\frac{A_{o}}{R_{1}^{2}}\cdot A_{s}\cdot T_{atm}\ \ (6) $$<br />
$$R_{1}:distancia\ \ a \ \ la\ \ fuente\\<br />
A_{o}:área\ \ de\ \ apertura\ \ de\ \ la\ \ lente\\ <br />
T_{atm}: transmitancia\ \ de\ \ la\ \ atmósfera$$<br />
El flujo radiante que alcanza el plano de la imagen Ai es:<br />
$$\phi_{imagen}=\phi_{lente}\cdot T_{sys}\ \ (7) $$<br />
$$T_{sys}:transmitancia\ \ de\ \ la\ \ lente$$ <br />
Expresándolo en los términos anteriores:<br />
$$\phi_{imagen}=L_{e}\frac{A_{o}}{R_{1}^{2}}\cdot A_{s}\cdot T_{atm}\cdot T_{sys}\ \ (8) $$<br />
Si la imagen proyectada Ai es mucho mayor que el área del detector Ad (Ai>>Ad), el flujo incidente sobre el detector se puede expresar como una relación proporcional a las áreas de ambas:<br />
$$\phi_{detector}=\phi_{imagen}\cdot \frac{A_{d}}{A_{i}}\ \ (9) $$ <br />
Teniendo en cuenta las ecuaciones 8 y 9:<br />
$$\phi_{detector}=L_{e}\frac{A_{o}}{R_{1}^{2}}\cdot A_{s}\cdot \frac{A_{d}}{A_{i<br />
.}}\cdot T_{atm}\cdot T_{sys}\ \ (10) $$<br />
Por simetría del flujo sobre la lente y usando aproximación paraxial para los ángulos, se puede escribir:<br />
$$\tfrac{A_{s}}{R_{1}^{2}}=\tfrac{A_{i}}{R_{2}^{2}}\ \ (11)$$<br />
es decir<br />
$$A_{s}=A_{i}\frac{R_{1}^{2}}{R_{2}^{2}}\ \ (12)$$<br />
Sustituyendo en la ecuación <span style="background-color: white;">10 te</span>nemos:<br />
$$\phi _{detector}=\frac{L_{e}\cdot A_{o}\cdot A_{d}}{R_{2}^{2}}\cdot T_{atm}\cdot T_{sys}\ \ (13)$$<br />
Asumiendo una apertura circular de la lente, el área de la apertura será:<br />
$$A_{o}=\pi \frac{D^{2}}{4}\ \ (14)$$<br />
$$D:diámetro\ \ de\ \ la\ \ lente$$<br />
Por otro lado, R1 y R2 están relacionados con la longitud focal (fl) por la ecuación 15<span style="background-color: white;"> co</span>mo se indica en la figura 6:<span id="transmark"></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaCenCgkhPi80qwRUG0m5IPg3C4zC62i7FFPN-viMof4TjESi3pv4FLVxYt7pq1Mpsbl4kvgg-dueIFynxDn732TGwxXwY1ZgnWvpN3oJ0h-PT1fF8f4cKXyYC0ipi2s0_wJyPUuZN0pc/s1600/thin-lens-trace-ray.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="image formation with thin lens" border="0" height="203" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaCenCgkhPi80qwRUG0m5IPg3C4zC62i7FFPN-viMof4TjESi3pv4FLVxYt7pq1Mpsbl4kvgg-dueIFynxDn732TGwxXwY1ZgnWvpN3oJ0h-PT1fF8f4cKXyYC0ipi2s0_wJyPUuZN0pc/s640/thin-lens-trace-ray.png" title="image formation with thin lens" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 6 - Formación de imagen por aproximación con lente delgada</td></tr>
</tbody></table>
$$\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{1}{fl}\ \ (15)$$<br />
Pero cuando R1 es mucho mayor que R2 (R1>>R2) o tiende a infinito, tenemos:<br />
$$fl=R_{2}\ \ (16)$$<br />
y podemos expresar el número F (F#) de la forma:<br />
$$F\#=\frac{R_{2}}{D}\ \ (17)$$ <br />
Sustituyendo las ecuaciones 14, 16 y 17 en 13, el flujo captado por el detector será:<br />
$$\phi _{detector}\left [ W \right ]=\frac{\pi }{4}\frac{L_{e}\cdot A_{d}}{f\#^{2}}\cdot T_{atm}\cdot T_{sys}\ \ (18)$$<br />
<br />
<b>3. ENERGÍA RADIANTE SOBRE EL DETECTOR </b><br />
<br />
El flujo captado por el detector multiplicado por el tiempo de exposición de la imagen se convierte en energía que a su vez se transforma en voltaje gracias a la electrónica del propio sensor. Este voltaje estará directamente relacionado con la luminosidad de la imagen.<br />
$$Q_{e}\left [ J \right ]=\frac{\pi }{4}\frac{L_{e}\cdot A_{d}}{f\#^{2}}\cdot T_{s}\cdot T_{atm}\cdot T_{sys}\ \ (19)$$<br />
<b>3.1 ANÁLISIS DE LA ECUACIÓN ENERGÍA RADIANTE SOBRE EL DETECTOR</b><br />
Según la ecuación 19 para mantener el mismo nivel de luminosidad en la imagen podemos actuar sobre varios factores. Vamos a ver cómo afecta cada uno de los parámetros sin variar los demás.</div>
<div style="text-align: justify;">
<ul>
<li><b>Le</b>: cambiar la radiancia al doble (el doble de luz en la escena) <b>supone el doble de energía en el detector</b>. Ésto permitiría por ejemplo disparar con la mitad del tiempo de exposición. Ésta variable muchas veces no es manipulable por el fotógrafo (por ejemplo luz ambiente). Modificar Le supondría añadir focos o el uso de flash en la escena.</li>
<li><b>Ad</b>: <b>cuanto mayor sea el tamaño del pixel del sensor de la cámara, mayor energía recibirá</b>. Una cámara digital será más sensible cuanto mayor sea su tamaño de pixel y lo hará de forma proporcional. Ésto nos permitirá fotografiar escenas con poca luz sin necesidad de usar grandes tiempos de exposición o ISO alto. Éste es uno de los motivos del gran tamaño de las cámaras profesionales.</li>
<li><b>Ts</b>: cambiar el tiempo de exposición nos <b>permitirá modificar la energía percibida por el detector</b>. Al doble de tiempo de exposición, doble de energía acumulada.</li>
<li><b>Tatm</b>: sobre la transmitancia atmosférica poco podemos hacer. <b>Es la atenuación que sufre la energía al atravesar la atmósfera</b> y depende de la distancia y de las condiciones ambientales. La transmitancia siempre será menor o igual que 1 siendo el ideal 1. Dias nublados tendrán mala transmitancia y dias despejados se acercarán al ideal. También, a mayor distancia, menor transmitancia.</li>
<li><b>Tsys</b>: es la transmitancia del sistema de lentes que se usa. Sobre ésta variable, el fotógrafo sí puede actuar usando objetivos de calidad. <b>Los buenos objetivos presentarán transmitancias altas</b>.</li>
<li><b>f#:</b> desde el punto de vista de energía (sin tener en cuentra otros factores como profundidad de campo), interesa usar numeros f# pequeños<b>.</b> <b>Cuanto más pequeño es f# más luminoso es el objetivo</b>. Además ésta variable actuá al cuadrado y pequeñas variaciones influyen drásticamente en la energía percibida. Teniendo en cuenta la ecuación 17, cuanto más grande sea la apertura de la lente, mejor.</li>
</ul>
<div>
<br /></div>
<b>3.2 INFLUENCIA DEL NÚMERO F</b><br />
<br />
De la ecuación 1<span style="background-color: white;">8, par</span><span style="background-color: white;">a una radiancia Le fija, se puede justificar que la </span>irradiancia recibida por el detector (energía por unidad de superficie del detector), es inversamente proporcional al cuadrado el número F# y directamente proporcional al tiempo de exposición.<br />
$$E_{e}\left [ \frac{W}{m^{2}} \right ]=\propto\frac{T_{s}}{f_{\#}^{2}} \ \ (20)$$<br />
Si se mantiene invariable el tiempo de exposición y sólo modificamos f#, se puede establecer la siguiente tabla de proporcionalidades que se encuentra en muchos libros de óptica.<b> </b><br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIJgzLOdw_cplAMYUewQhQd0raxD8CRXtEGyCNAGhte0DxXuk5sFxUG6oPgyzqYAD8cbxc9JYEwTssJr0Sur4_h_3DOwRgqMxOHY7kzgJeJ-8B6eyC9ajf5GUyjkQ53acV31B-l9a8U_o/s1600/Relative-image-irradiance-as-function-of-f-number.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Irradiancia en función del número F" border="0" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIJgzLOdw_cplAMYUewQhQd0raxD8CRXtEGyCNAGhte0DxXuk5sFxUG6oPgyzqYAD8cbxc9JYEwTssJr0Sur4_h_3DOwRgqMxOHY7kzgJeJ-8B6eyC9ajf5GUyjkQ53acV31B-l9a8U_o/s1600/Relative-image-irradiance-as-function-of-f-number.PNG" title="Irradiancia en función del número F" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 7 - Irradiancia percibida en función del número F</td></tr>
</tbody></table>
Por ejemplo, si la irradiancia percibida a F/1 es E0, cuando cambiamos a F/1.4 la irradiancia se reduce a la mitad.<br />
<br />
Supongamos la siguiente hipótesis de partida donde la irradiancia es 100% a F/1 con un tiempo de exposición de 1 segundo. Expresado la irradiancia en función del número F y el tiempo de exposición se puede ver en la<span style="background-color: white;"> figura 8 la </span>interrelación entre ambas variables.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-9xmqbn91Xgnsw3MjS-g3pldD9sLiKeOUxMmPiObu6TuMj9VKUzgbsIzBP1vG-ZVaLqTjOwU7E07LW0aHwO2Gvt4IArCQNDDQDfMOhytb39ffYAmhizMyXhyphenhyphenEL9sciUXpAJc_0GqLjHs/s1600/Irradiancia-en-funci%25C3%25B3n-del-n%25C3%25BAmero-F-y-tiempo-exposici%25C3%25B3n.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Irradiancia-en-función-del-número-F-y-tiempo-exposición" border="0" height="441" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-9xmqbn91Xgnsw3MjS-g3pldD9sLiKeOUxMmPiObu6TuMj9VKUzgbsIzBP1vG-ZVaLqTjOwU7E07LW0aHwO2Gvt4IArCQNDDQDfMOhytb39ffYAmhizMyXhyphenhyphenEL9sciUXpAJc_0GqLjHs/s640/Irradiancia-en-funci%25C3%25B3n-del-n%25C3%25BAmero-F-y-tiempo-exposici%25C3%25B3n.png" title="Irradiancia-en-función-del-número-F-y-tiempo-exposición" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 8 - Irradiancia Ee = f(F#,Ts)</td></tr>
</tbody></table>
En la figura 9 se muestran las curvas de irradiancia constante en función del número F y del tiempo de exposición.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilRL8Q9qlg_358ehFRHod1Hy0hj6yp2wNLYrEyX_sYWfe2tEkHtwxo4g3Uww8ZCpmH0khVuEe4uNFL-aWr5juRg9HcattelJD3nP5ayIJrMPLu2hHd5kz-ueboREwWtW93x4xkNnhHOSg/s1600/Curvas-irradiancia-constante-en-funci%25C3%25B3n-del-n%25C3%25BAmero-F-tiempo-exposici%25C3%25B3n.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Curvas-irradiancia-constante-en-función-del-número-F-tiempo-exposición" border="0" height="453" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilRL8Q9qlg_358ehFRHod1Hy0hj6yp2wNLYrEyX_sYWfe2tEkHtwxo4g3Uww8ZCpmH0khVuEe4uNFL-aWr5juRg9HcattelJD3nP5ayIJrMPLu2hHd5kz-ueboREwWtW93x4xkNnhHOSg/s640/Curvas-irradiancia-constante-en-funci%25C3%25B3n-del-n%25C3%25BAmero-F-tiempo-exposici%25C3%25B3n.png" title="Curvas-irradiancia-constante-en-función-del-número-F-tiempo-exposición" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 9 - Curvas de irradiancia constante</td></tr>
</tbody></table>
<b>3.3 ¿CÓMO AFECTA LA ISO?</b><br />
<br />
Hasta ahora, todo lo expuesto es manteniendo la ISO fija a un determinado valor. veamos su influencia sobre la imagen.<br />
<br />
Básicamente, <b>aumentar ISO significa aumentar la ganancia electrónica </b>en la conversión de energía-voltaje. Es decir, con la misma energía percibida por el detector, aumentar la ISO daría la sensación de una percepción de mayor energía (mayor luminosidad en la imagen). Sería el equivalente a desplazar la figura 8 en el eje z (irradiancia) hacia arriba.<br />
<br />
De la misma manera, <b>bajar la ISO significaría reducir la ganancia electrónica</b> en la conversión energía-voltaje. Sería el equivalente a desplazar la figura 8 en el eje z (irradiancia) hacia abajo.<br />
<br /></div>
<div style="text-align: justify;">
<b>REFERENCIAS</b><br />
[1] - Alexander D. Ryer (1997). The light measurement handbook. ISBN 0-9658356-9-3<br />
[2] - <a href="http://everyspec.com/NATO/NATO-STANAG/STANAG_4347_13663/" target="_blank">NATO STANAG 4347 </a><br />
[3] - Родионов С.А. Основы оптики. Конспект лекций. - СПБ: СПБ: ГИТМО (ТУ), 2000. - 167 с.<br />
[4] - Leno S. Pedrotti. <a href="http://spie.org/Documents/Publications/00%20STEP%20Module%2003.pdf" target="_blank">Basic Physical Optics</a>. Open Access SPIE Publications.<br />
<br />
<br />
<br /></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0Madrid, Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-41022594305144270432015-03-21T03:05:00.000+01:002015-08-25T00:15:30.822+02:00Estado actual sobre la homologación del título de Ingeniería Industrial<div style="text-align: justify;">
<b><span style="color: red;">ACTUALIZADO EN ÉSTE <a href="http://maladiets.blogspot.com.es/2015/08/estado-actual-sobre-la-homologacion-del-titulo-ingeniero-industrial-parte-II.html" target="_blank">ENLACE</a></span></b><br />
<br />
Muchos somos los ingenieros que hemos cursado <b>estudios de Ingeniería Industrial en planes pre-Bolonia</b> y ahora mismo nos estamos preguntando sobre el futuro de nuestra titulación y la equiparación del título universitario en el mundo globalizado. </div>
<div style="text-align: justify;">
<a name='more'></a><br />
Haciendo un símil con la termodinámica y teniendo en cuenta el segundo principio, en nuestros estudios hemos empleado una gran cantidad de energía que mucho me temo que en esta adaptación de títulos a Bolonia parte de ella se convierta en entropía inevitablemente. </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
He trasladado esta inquietud al <b>Colegio de Ingenieros Industriales del Principado de Asturias</b> y muy amablemente me ha contestado el <b>Decano Sr. Fdez. Rico</b> con el estado actual (a fecha 21/03/2015) de dicha tramitación.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Por la importancia que supone para todos nosotros, transcribo literalmente el correo recibido:</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<i>"Este proceso está en marcha, si bien con frentes distintos.</i></div>
<div style="text-align: justify;">
<i><br /></i></div>
<div style="text-align: justify;">
<i>En este momento la tramitación que se está haciendo, de acuerdo al R.D. 967/2014 de 21 de Noviembre en cuanto a la Homologación y declaración de equivalencia es la equiparación al MECES (Marco Español de Cualificaciones para la Educación Superior) del título de Ingeniero Industrial, que se correspondería con el nivel 3.</i></div>
<div style="text-align: justify;">
<i><br /></i></div>
<div style="text-align: justify;">
<i>La tramitación citada requiere un informe de ANECA (Agencia española de evaluación de la calidad y acreditación del sistema universitario), de la Comisión de Ciencia y Tecnología, que pasaría al Ministerio, luego a información pública y finalmente al Consejo de Ministros.</i></div>
<div style="text-align: justify;">
<i><br /></i></div>
<div style="text-align: justify;">
<i>Este proceso podría finalizar en Junio. Actualmente está en fase Aneca.</i></div>
<div style="text-align: justify;">
<i>Dicha equiparación MECES no implica ser Máster Ingeniero Industrial, pero seguramente se arbitrará un mecanismo de homologación, que todos pretendemos.</i></div>
<div style="text-align: justify;">
<i><br /></i></div>
<div style="text-align: justify;">
<i>Un aspecto pendiente es también la equiparación EQF (European Qualifications Framework), Marco Europeo, en el que el Gobierno Español tiene que trasladar a Bruselas que nivel tiene la titulación. Los Colegios estamos presionando para que nuestro nivel sea EQF7, equivalente al Master, (EQF8 es Doctor)."</i></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com2Oviedo, Asturias, España43.3619145 -5.849388699999963143.315743 -5.9300696999999634 43.408086 -5.7687076999999629tag:blogger.com,1999:blog-3108912614914761750.post-18343414286616817062014-11-22T01:46:00.001+01:002015-01-16T00:16:45.434+01:00USAF-1951 target resolution to angular resolution calculator<div class="separator" style="clear: both; text-align: left;">
</div>
<div class="separator" style="clear: both; text-align: left;">
</div>
<div class="separator" style="clear: both; text-align: justify;">
<b>Abstract</b></div>
<div class="separator" style="clear: both; text-align: justify;">
An easy way for testing camera resolution is to use the USAF-1951 target pattern also called "USAF 1951 Resolving Power Test Target". The aim of this entry is provide an excel calculator to convert from target resolution to angular resolution (cycles per millimetre to cycles per milliradian). </div>
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Although nowadays there are more precise methods for testing camera
resolution, the use of the old USAF-1951 pattern is widely used when
testing infrared and visible cameras due to its simplicity and
straightforward interpretation.</div>
<div class="separator" style="clear: both; text-align: justify;">
<br /></div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD6JhN8Q2eoaElcV7jgoIgczpqT1KuzUfsoufHMCMwzEldlWnwnqPWg4ocvBHGlDeHGe6J4yqiYsRiuo2R9LkkbRa_FFejC8PdS92Jv-pNbW2Bc2YWdErdWSW68yGEuFrNI-tnyiVg17k/s1600/USAF-1951-group-pattern-element-description.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img alt="USAF-1951 group pattern and element description" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD6JhN8Q2eoaElcV7jgoIgczpqT1KuzUfsoufHMCMwzEldlWnwnqPWg4ocvBHGlDeHGe6J4yqiYsRiuo2R9LkkbRa_FFejC8PdS92Jv-pNbW2Bc2YWdErdWSW68yGEuFrNI-tnyiVg17k/s1600/USAF-1951-group-pattern-element-description.png" height="240" title="USAF-1951 group pattern and element description" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">USAF-1951 pattern explanation</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjafxG9NpjppzYP2T2E_fDCdrTxsjGwBr7tdQbXJxgie2OFFSWb_J1OEGf5xcd76TNYdgpStmjPwjliguEZDeaBAb12A38YWPN6MstCo3whW_XKRkKpuDl_KVHG_EifGTtkmGiyhQMX9kw/s1600/Usaf-1951-resolution-test-pattern.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="USAF-1951 Test Pattern" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjafxG9NpjppzYP2T2E_fDCdrTxsjGwBr7tdQbXJxgie2OFFSWb_J1OEGf5xcd76TNYdgpStmjPwjliguEZDeaBAb12A38YWPN6MstCo3whW_XKRkKpuDl_KVHG_EifGTtkmGiyhQMX9kw/s1600/Usaf-1951-resolution-test-pattern.png" height="200" title="USAF-1951 Test Pattern" width="188" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">USAF-1951 Test Pattern</td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
<div class="" style="clear: both; text-align: justify;">
<br /></div>
<div class="" style="clear: both; text-align: justify;">
The USAF-1951 is defined in the military standard MIL-STD-150. The resolution of a particular element in line pairs per millimetre (or cycles per millimetre) can be computed using the following formula.</div>
</div>
$$R=2^{K+\frac{N-1}{6}}\\<br />
R: cycles\ \ per\ \ millimetre\\<br />
K: group\ \ number\\<br />
N: Element\ \ number$$<br />
<div class="separator" style="clear: both; text-align: justify;">
<br /></div>
<div class="separator" style="clear: both; text-align: justify;">
There are, however, a few cases when is desirable to <b>get the resolution in cy/mrad</b>, for example when testing against a collimator and we need to know the camera angular resolution under these units. </div>
For these cases, the excel file <a href="https://drive.google.com/file/d/0B92QFmnTIcgvYWV4Vmk3ZHAyT1U/view?usp=sharing" target="_blank">USAF 1951 conversion cy-mm to cy-mrad</a> fits like a glove.<br />
<br />
<br />
<b>REFERENCES</b><br />
<b>[1] </b><a href="http://www.assistdocs.com/search/document_details.cfm?ident_number=35531&StartRow=201&PaginatorPageNumber=5&doc_id=STD&status_all=ON&search_method=BASIC" target="_blank">MIL-STD-150A Photographic Lenses</a><b></b><br />
<b>[2] </b><a href="http://www.aig-imaging.com/mm5/PDF/USAF%201951%20Test%20Target%20T-20_v1-04.pdf" target="_blank">USAF 1951 Chart Standard Layout Product Specifications </a><b></b><br />
<b><br /></b>
<b><br /></b>
<b></b><br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0California, EE. UU.36.778261 -119.4179323999999823.886873 -140.07222939999997 49.669649 -98.763635399999984tag:blogger.com,1999:blog-3108912614914761750.post-14269237040087092332014-08-28T02:32:00.000+02:002015-03-22T02:37:49.985+01:00Testing camera lens. Where is infinity?<div class="separator" style="clear: both; text-align: left;">
<b>Abstract</b></div>
<div class="separator" style="clear: both; text-align: justify;">
The best way for Testing cameras focused at infinite requires complex instrumentation for simulating "the infinite" by means of an optical collimator. If we have enough space, the cheaper and cost effective way for simulating the "infinite" is to place a target far away from the camera but... where we can considerer is "infinite"?</div>
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</div>
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According to
MIL-STD-150A paragraph 5.1.1.3, when testing a lens at "infinite" focus
by means of a target at infinite distance, the distance shall be
considered approximately infinite when it is greater than D measured in
feet.</div>
<div class="separator" style="clear: both; text-align: justify;">
<br /></div>
<div class="separator" style="clear: both; text-align: justify;">
$$D[ft]\geq 400\cdot EFL_{i}\cdot ED_{i}\ \ (1)$$</div>
<div class="separator" style="clear: both; text-align: justify;">
Where </div>
<div style="text-align: justify;">
$$D[ft]: Minimum\ \ distance\ \ in\ \ feets\\<br />EFL_{i}[inch]: Lens\ \ Effective\ \ Focal\ \ Length\\<br />ED_{i}[inch]: Lens\ \ Effective\ \ aperture$$<br />
<br /></div>
<div style="text-align: justify;">
The f-number f# is given by the following dimensionless equation:</div>
<div style="text-align: justify;">
$$f\#=\frac{EFL_{i}}{ED_{i}}\ \ (2)$$ </div>
<div style="text-align: justify;">
Putting the equation (1) in an user-friendly format, we will substitute the equation (2) into the equation (1) and converting to the SI, we can write the minimum distance "D" to considerer infinite as:</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
$$D[m]\geq 0.216\frac{EFL^{2}}{f\#}\ \ (3)$$</div>
<div style="text-align: justify;">
Where<br />
$$EFL[mm]:Lens\ \ Effective\ \ Focal\ \ Length\ \ in\ \ mm$$</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>EXAMPLE</b></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>Problem</b>: We have a lens NIKON 16-85mm working at 50 mm and f#5.6. Calculate the minimum distance to place one target for considering it focused at infinite.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>Solution:</b></div>
<div style="text-align: justify;">
Applying the equation (3)</div>
<div style="text-align: justify;">
$$D[m]\geq 0.216\frac{50^{2}}{5.6}\simeq 96\ \ m$$</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>REFERENCES </b></div>
<div style="text-align: justify;">
[1] - Military Standard Photographic Lenses MIL-STD-150A</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0California, EE. UU.36.778261 -119.4179323999999823.886873 -140.07222939999997 49.669649 -98.763635399999984tag:blogger.com,1999:blog-3108912614914761750.post-50541055557741638022014-06-25T19:10:00.003+02:002014-06-27T11:22:54.584+02:00Calculadora de resistencias serie y paralelo<b>Resumen</b><br />
Calculadora que usa combinaciones serie y paralelo de dos resistencias estandar ( E12 (10%), E24 (5%) and E96 (1%) ), para obtener el valor deseado.<br />
<a name='more'></a><div style="text-align: center;">
<br />
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<div style="text-align: justify;">
<br />
Muchas veces cuando diseñamos necesitamos un valor no estandar de resistencia. Para lograrlo, podemos asociar resistencias en serie o en paralelo hasta conseguir el valor deseado. Normalmente con un par de resistencias en serie nos solemos acercar bastante, pero también las podríamos agrupar en paralelo y es aquí donde ésta calculadora nos hace la vida más fácil. </div>
<br />
<body>
<script type="text/javascript">
//<![CDATA[
/* Copyright (C) 2001-2010 Claudio Girardi
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
*/
/* well, this is not the simplest way to do it but, as they say, */
/* If ain't broke, don't fix it!" */
function select_series() {
var sel_val = document.getElementById('rslist').options[document.getElementById('rslist').selectedIndex].value;
switch(sel_val) {
case "E12":
Rbase = new Array(1, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2);
break;
case "E24":
Rbase = new Array(1, 1.1, 1.2, 1.3, 1.5, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 3.0, 3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1);
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break;
default:
}
R = new Array;
for (var mult = 0; mult <= 6; mult++) {
for (var idx = 0; idx <Rbase.length; idx++) {
/* need to round to compensate for pow() errors; allow max two decimals, needed for E96 */
R[idx + mult * Rbase.length] = Math.round(Rbase[idx] * Math.pow(10, mult) * 100) / 100;
}
}
n_max = R.length - 1; /* maximum valid index */
/* compute the conductances array, lowest conductance first to have an */
/* array sorted in ascending order */
G = new Array;
for (idx = 0; idx <= n_max; idx++) {
G[idx] = 1.0 / R[n_max - idx];
}
out_r1 = new Array;
out_r2 = new Array;
out_op = new Array;
out_rres = new Array;
out_tol = new Array;
CalcRes();
}
function FindIndex(vect, value) {
var index_min = 0;
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var index = Math.floor( (index_min + index_max) / 2);
i = 0;
while (((index_max - index_min) > 1) && (i < 500)) {
if (vect[index] == value) { break; }
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index = Math.floor( (index_min + index_max) / 2);
i++;
}
if (index < n_max) {
tol1 = Math.abs(vect[index] / value - 1.0);
tol2 = Math.abs(vect[index + 1] / value - 1.0);
if (tol1 < tol2)
return index;
else
return (index + 1);
} else
return index;
}
function CalcRes() {
var rd = document.getElementById('rd').value;
var r1, r2, r1_idx, rres, rres_tol, best_tol, out_idx, op;
var out_prres, out_vrres;
var i, j, iter = 0; /* number of iterations */
document.getElementById('texta').value = ""
<!-- document.getElementById('debuga').value = "" -->
/* compute assuming resistors in series */
/* locate nearest approximation with standard resistor values */
r1_idx = FindIndex(R, rd);
r1 = R[r1_idx];
/* other resistor */
/* r2 = Number.POSITIVE_INFINITY */
r2 = 0;
rres = r1;
rres_tol = (rres - rd) / rd; /* relative tolerance */
best_tol = rres_tol;
out_idx = 0;
out_r1[out_idx] = r1;
out_r2[out_idx] = r2;
out_op[out_idx] = "+";
out_rres[out_idx] = rres;
out_tol[out_idx++] = rres_tol;
for (; R[r1_idx] >= rd / 2.0; r1_idx--) {
iter++;
r1 = R[r1_idx];
r2d = rd - r1; // this is the value needed
<!-- document.getElementById('debuga').value += '+ ' + rd + ' ' + r1 + ' ' + r2d + ' ' + 1/rd + ' ' + 1/r1 + ' ' + 1/r2d + '\n' -->
if (r2d < 0) { continue } // might happen...
r2_idx = FindIndex(R, r2d);
r2 = R[r2_idx]; // get the nearest standard value
rres = r1 + r2; // compute the resulting composition
rres_tol = rres / rd - 1.0; // and its tolerance
if (Math.abs(rres_tol) < Math.abs(best_tol)) {
//best_tol = rres_tol;
out_r1[out_idx] = r1;
out_r2[out_idx] = r2;
out_op[out_idx] = "+";
out_rres[out_idx] = rres;
out_tol[out_idx++] = rres_tol;
}
}
rd = 1.0 / rd;
/* compute assuming resistors in parallel */
r1_idx = FindIndex(G, rd);
for (; G[r1_idx] >= rd / 2.1; r1_idx--) {
iter++;
r1 = G[r1_idx];
r2d = rd - r1; // this is the value needed
<!-- document.getElementById('debuga').value += '|| ' + rd + ' ' + r1 + ' ' + r2d + ' ' + 1/rd + ' ' + 1/r1 + ' ' + 1/r2d + '\n' -->
if (r2d < 0) { continue } // might happen...
r2_idx = FindIndex(G, r2d);
r2 = G[r2_idx]; // get the nearest standard value
rres = r1 + r2; // compute the resulting composition
rres_tol = rd / rres - 1.0; // and its tolerance
if (Math.abs(rres_tol) < Math.abs(best_tol)) {
//best_tol = rres_tol;
// use values from R array to avoid rounding errors
// which will lead to something like 6800.0000001...
out_r1[out_idx] = R[n_max - r1_idx] // 1.0 / r1;
out_r2[out_idx] = R[n_max - r2_idx] // 1.0 / r2;
out_op[out_idx] = "||";
out_rres[out_idx] = 1.0 / rres;
out_tol[out_idx++] = rres_tol;
}
}
// sort the results
for (i = 1; i < out_idx; i++) {
r1 = out_r1[i];
r2 = out_r2[i];
op = out_op[i];
rres = out_rres[i];
rres_tol = out_tol[i];
for (j = i - 1; (j >= 0) &&
Math.abs(out_tol[j]) > Math.abs(rres_tol); j--) {
out_r1[j + 1] = out_r1[j];
out_r2[j + 1] = out_r2[j];
out_op[j + 1] = out_op[j];
out_rres[j + 1] = out_rres[j];
out_tol[j + 1] = out_tol[j];
}
out_r1[j + 1] = r1;
out_r2[j + 1] = r2;
out_op[j + 1] = op;
out_rres[j + 1] = rres;
out_tol[j + 1] = rres_tol;
}
for (r1_idx = 0; r1_idx < out_idx; r1_idx++) {
out_prres = (Math.round(out_rres[r1_idx] * 1000)) / 1000 ;
out_vrres = out_prres.toString();
if(out_vrres.length < 8) out_vrres = out_vrres + "\t"
document.getElementById('texta').value +=
out_r1[r1_idx] + "\t" +
out_op[r1_idx] + "\t" +
out_r2[r1_idx] + "\t=\t" +
/* leave three decimal digits maximum */
out_vrres + "\t(" +
(Math.round(out_tol[r1_idx] * 100000)) / 1000 + " %)\n"
}
// put the focus on the input field
//document.getElementById('rd').focus();
}
//]]>
</script>
<div id="wrap">
<div id="contentwide">
<form action="#" method="post">
<h2>
Entrada de datos</h2>
<pre>
Serie de resistencias a usar en los cálculos: <select id="rslist" onchange="select_series();">
<option value="E12">E12 (10 %) series</option>
<option value="E24">E24 (5 %) series</option>
<option value="E96">E96 (1 %) series</option>
</select>
Resistencia deseada, R : <input id="rd" maxlength="10" size="10" type="text" value="1234" /> ohm
<input onclick="CalcRes()" type="button" value="Calcular" />
</pre>
<h2>
VALORES CALCULADOS</h2>
<pre>
<textarea cols="70" id="texta" rows="10"></textarea>
</pre>
<!--
<pre>
<textarea id="debuga" rows="20" cols="120"></textarea>
</pre>
-->
</form>
<script type="text/javascript">
//<![CDATA[
select_series()
//]]>
</script>
</div>
</div>
</body>
<b> </b><br />
<b>REFERENCIAS</b><br />
[1] Página personal de <a href="http://www.qsl.net/in3otd/parallr.html" target="_blank">Claudio Girardi IN3OTD</a><b></b>JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com2tag:blogger.com,1999:blog-3108912614914761750.post-83724277374447748402014-06-06T23:22:00.001+02:002014-06-07T11:13:37.384+02:00Relación entre ancho de banda y tiempo de subida de una señal<div style="text-align: justify;">
<b>Resumen</b></div>
<div style="text-align: justify;">
0.35 es una de las reglas de oro que relaciona el ancho de banda de una señal con su tiempo de subida. Pero... ¿De dónde sale éste número mágico?</div>
<div style="text-align: justify;">
<br />
<b>Abstract </b></div>
<div style="text-align: justify;">
0.35 is one of the rule of thumb connecting the bandwidth and the rise time. But... Where does this number come from? </div>
<a name='more'></a><br />
<div style="text-align: center;">
----oooOOOooo----</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghud3sGby0_JaLt-4xkzOntDP1rueI8ed0mMvDSwi_Kt_CQ6hGt6Id5FidHdKCKjpjJ05HgfVLI3HI2Easgz7iJOAOPffxCdgLhM4R7MiWc-9YqpLX0gs5orHARWBew2AIJdpQCUcthxo/s1600/rise-time-versus-bandwidth.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="rise time versus bandwidth" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghud3sGby0_JaLt-4xkzOntDP1rueI8ed0mMvDSwi_Kt_CQ6hGt6Id5FidHdKCKjpjJ05HgfVLI3HI2Easgz7iJOAOPffxCdgLhM4R7MiWc-9YqpLX0gs5orHARWBew2AIJdpQCUcthxo/s1600/rise-time-versus-bandwidth.png" height="180" title="rise time versus bandwidth" width="200" /></a></div>
<br />
Muchas veces el ingeniero se enfrenta a la terrible decisión de seleccionar un osciloscopio para su sistema de medida. Entre muchas de las preguntas que se cuestiona, quizás la más básica es la selección del ancho de banda del mismo, para reproducir la señal medida sin errores. La respuesta parece obvia. El osciloscopio tendrá al menos un ancho de banda igual o mayor que el de la señal a medir. Pero... <b>¿Cómo cuantificar el ancho de banda de una señal?</b></div>
<div style="text-align: justify;">
<b><br /></b></div>
<div style="text-align: justify;">
<br />
<b> El ancho de banda ocupado por una señal se puede caracterizar conociendo su tiempo se subida</b>. Para el análisis teórico partiremos de la siguiente hipótesis:</div>
<ol style="text-align: justify;">
<li>La señal sigue la respuesta de un sistema de primer orden al estímulo escalón unitario. Para el desarrollo tomaremos como referencia un sistema de ganancia unidad.</li>
</ol>
<div style="text-align: justify;">
$$y(s)=\frac{1}{1+\tau_{p}S}\ \ (1)$$<br />
<div style="text-align: justify;">
Éste sistema presenta un polo en </div>
<div style="text-align: justify;">
$$f_{p}[Hz]=\frac{1}{2\pi\tau_{p}}\ \ (2)$$</div>
<div style="text-align: justify;">
$$\tau_{p}=\frac{1}{2\pi f_{p}}\ \ (3)$$ </div>
</div>
<div style="text-align: justify;">
Por lo tanto, su respuesta temporal será:</div>
<div style="text-align: justify;">
$$y(t)=1-e^{\frac{t}{\tau_{p}}}\ \ (4)$$<br />
Si despejamos el tiempo de la ecuación (4) tenemos<br />
$$t=-\tau _{p}\cdot ln(1-y(t))\ \ (5)$$</div>
<div style="text-align: justify;">
Se define tiempo de subida (Tr: rise time) como el tiempo necesario para que la señal pase del 10% al 90% de su amplitud. Con éste criterio definimos:</div>
<div style="text-align: justify;">
$$T_{r}=t_{2}-t_{1}\ \ (6) \\ <br />
t_{1}: Tiempo\ \ al\ \ 10\%\ \ amplitud\ \ (7)\\<br />
t_{2}: Tiempo\ \ al\ \ 90\%\ \ amplitud\ \ (8)$$</div>
Con las consideraciones (7) y (8) sustituyendo (5) en la ecuación (6) tenemos<br />
$$T_{r}=-\tau _{p}\cdot ln(1-0.9)+\tau _{p}\cdot ln(1-0.1)\\<br />
T_{r}=\tau _{p}\cdot ln(9)\ \ (9)$$<br />
Sustituyendo la ecuación (3) en (9) podemos escribir<br />
$$f_{p}=\frac{ln(9)}{2\pi T_{r}}\ \ (10)$$<br />
$$f_{p}=\frac{0.3496}{T_{r}}\simeq \frac{0.35}{T_{r}}\ \ (11)$$<br />
<div style="text-align: justify;">
<div style="text-align: center;">
----oooOOOooo----<b> </b></div>
<br />
<b>EJEMPLO</b></div>
<div style="text-align: justify;">
<b>Pregunta: </b>Se desea medir una señal cuadrada que presenta un tiempo de subida de 30 ns. ¿Qué ancho de banda mínimo necesita el osciloscopio para reproducir fielmente la señal?</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>Respuesta:</b></div>
<div style="text-align: justify;">
Para averiguar el ancho de banda de la señal, basta con aplicar la ecuación (11)</div>
<div style="text-align: justify;">
$$\frac{0.35}{20\ \ ns}=17.5\ \ MHz$$ </div>
<div style="text-align: justify;">
La señal a medir tiene un ancho de banda de 17.5 MHz. El osciloscopio junto con la sonda deberá tener como mínimo un ancho de banda mayor. Para evitar notar el propio tiempo de subida del osciloscopio, elegiremos un osciloscopio con un ancho de banda al menos 5 veces superior.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>NOTA:</b></div>
<div style="text-align: justify;">
El tiempo se subida que se verá en la pantalla del osciloscopio será la raiz cuadrada de la suma de cuadrados de todos los tiempos de subida implicados (señal, sonda, osciloscopio). Es decir:<br />
$$T_{r_{total}}=\sqrt{T_{r_{signal}}^2+T_{r_{sonda}}^2+T_{r_{osciloscopio}}^2}\ \ (12)$$</div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com1Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-87270172135568845332014-05-12T03:02:00.000+02:002014-06-24T23:39:58.541+02:00How to design an oscilloscope passive probe (part II) <div style="text-align: justify;">
<span style="background-color: white;"><b>Abstract</b></span></div>
<div style="text-align: justify;">
<span style="background-color: white;">Following the previous post <i><a href="http://maladiets.blogspot.com.es/2012/09/oscilloscope-passive-probe-design-01.html" target="_blank">how to design passive oscilloscope probe (part I)</a> , </i>In this paper we will study from the theoretical point of view, the
basic principle of a passive oscilloscope probe, calculate its input
impedance, tips to use, and finally establish the basis for DIY homemade 10:1 passive probe.</span><br />
<a name='more'></a></div>
<div style="text-align: center;">
<span style="background-color: white;">---- oooOOOooo ----</span><br />
<div style="text-align: left;">
<span style="background-color: white; text-align: justify;"><b><br /></b></span></div>
<div style="text-align: left;">
<span style="background-color: white; text-align: justify;"><b>5. HOW WORKS AN OSCILLOSCOPE PASSIVE PROBE ?</b></span><br />
<span style="background-color: white; text-align: justify;"><b><br /></b></span></div>
<div style="text-align: left;">
<span style="background-color: white; text-align: justify;">The passive probe is the most known and used of al general all purpose probes. As the name says, "passive" suggests that it is made of passive components such as resistors, cables, capacitors, and indeed it is.</span></div>
</div>
<div>
<div style="text-align: justify;">
<div style="text-align: justify;">
<span style="background-color: white;"><br /></span></div>
</div>
<div style="text-align: justify;">
<span style="background-color: white;">In Part I of this article, we have seen how the oscilloscope impedance and cables capacitance affect to the measure up to the point of being able to make it unrecognisable (it was an uncompensated probe). Given that the basic principle of any measuring equipment is to measure the signal without distorting it, we will raise the theoretical equations governing the operation of a passive probe compensation that allow us to maintain the signal integrity during the measurement. </span></div>
<div style="text-align: justify;">
<span style="background-color: white;"><br /></span></div>
<div style="text-align: justify;">
<span style="background-color: white;"> The purpose of this section is to analyse from the theoretical point of view the most basic oscilloscope passive probe. The proposed scheme is valid for general purpose medium/low oscilloscope probes. For high-end passive probes, the scheme is a bit more complicated due to there are more components inside for low and high frequency compensation doing the theoretical explanation a little more confusing leaving out of the scope of this article. </span></div>
<div style="text-align: justify;">
<span style="background-color: white;"><br /></span></div>
<div style="text-align: justify;">
<span style="background-color: white;">As a starting point for the mathematical approach, we assume that <b>the probe cable is</b> sufficiently short compared with the wavelength of the signal to be measured. This hypothesis will also greatly facilitate us the analysis in most cases. In this way, we will only consider the coaxial cable capacitance as <b>a "short line" </b>and we will not consider the mathematical model of the cable as a transmission line.</span></div>
<div style="text-align: justify;">
<span style="background-color: white;"><br /></span></div>
<div style="text-align: justify;">
<span style="background-color: white;">As we have seen in the first part, if we add a piece of coaxial cable to the oscilloscope input, forms a low pass filter that depends on the impedance measured. Fortunately, compensate the influence of that pole is relatively easy. Just remembering a little of mathematics we learned during the studies and adding a zero in the measurement circuit using the components Rp (9 MOhm) and Cp (variable) (see Figure 1).</span></div>
<div style="text-align: justify;">
<span style="background-color: white;"><br /></span></div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtC8XK2ZxytbAzBSv7TpAKrO8ot_2FJUaEUJb-Kn0tNi0pdnclnwycXSHcalNymV1DURRl7kcNLMTn9yJr4u45wPHvCMhnP3KtVQNst4P5vp1c-vhbv56F-_GfweRfONuqTekxQYXWFCk/s1600/Oscilloscope-pasive-probe-diagram.png" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe schematic" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtC8XK2ZxytbAzBSv7TpAKrO8ot_2FJUaEUJb-Kn0tNi0pdnclnwycXSHcalNymV1DURRl7kcNLMTn9yJr4u45wPHvCMhnP3KtVQNst4P5vp1c-vhbv56F-_GfweRfONuqTekxQYXWFCk/s1600/Oscilloscope-pasive-probe-diagram.png" height="249" style="border: none; position: relative;" title="passive probe schematic" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Figure 1 - Oscilloscope passive probe <span style="color: #333333; line-height: 20px;">Schematic</span><span style="color: #333333; line-height: 20px;"> </span></span></td></tr>
</tbody></table>
<br />
<span style="background-color: white;"><span style="font-size: small;"></span></span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><br /></span>
<span class="" style="background-color: white;">Let be Ct the total cable capacity over the oscilloscope input</span><br />
<span style="background-color: white;"><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-1-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 7.76em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 6.198em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.589em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-3" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.188em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">C<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; 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vertical-align: 0px; width: 0.057em;"></span></span><span class="texatom" id="MathJax-Span-10" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-11" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-12" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.38em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.828em 1000.03em 4.151em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -3.984em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-13" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-14" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-15" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-16" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-17" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="mi" id="MathJax-Span-18" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="texatom" id="MathJax-Span-19" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-20" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-21" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.38em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.828em 1000.03em 4.151em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -3.984em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-22" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-23" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-24" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-25" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">c</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.215em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -0.266em; width: 0px;"></span></span></nobr></span></div>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh61DXZBllVjpspTcNlxw5pO0iVptZbgq2HKAQUIWS0CuCbkE8OaQ3zQPclBM6JGytKIIAEXq7MHDQ31ydyUt9FV62zHA_dnjLt-a5YBvsC7ziTsNrX1r41VhlOIlVrQSVoGJArk2HHyIM/s1600/Equivalent-oscilloscope-pasive-probe-diagram.png" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe equivalent circuit" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh61DXZBllVjpspTcNlxw5pO0iVptZbgq2HKAQUIWS0CuCbkE8OaQ3zQPclBM6JGytKIIAEXq7MHDQ31ydyUt9FV62zHA_dnjLt-a5YBvsC7ziTsNrX1r41VhlOIlVrQSVoGJArk2HHyIM/s1600/Equivalent-oscilloscope-pasive-probe-diagram.png" height="212" style="border: none; position: relative;" title="Oscilloscope probe equivalent circuit" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Figure 2 - Equivalent circuit</span></td></tr>
</tbody></table>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">And the relationship between the input and the output of the probe</span><br />
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-2-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-26" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 14.925em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 11.908em;"><span style="border: 0px; clip: rect(0.649em 1000.03em 3.397em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-27" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-28" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.242em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-29" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">Z<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-30" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-31" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-32" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-33" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-34" style="border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 6.521em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.42em -0.428em); left: 60.515625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.738em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-35" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-36" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-37" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-38" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-39" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.696em -0.374em); left: 60.515625px; margin: 0px 0px 0px -3.176em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-40" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-41" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-42" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-43" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-44" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-45" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-46" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-47" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-48" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="msubsup" id="MathJax-Span-49" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-50" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-51" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-52" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-53" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-54" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="mi" id="MathJax-Span-55" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 6.521em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-56" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-57" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-58" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-59" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-60" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-61" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-62" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.168em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -1.276em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><br />
with</span><br />
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-3-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-63" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 10.723em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 8.568em;"><span style="border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-64" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-65" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.242em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-66" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">Z<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-67" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-68" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-69" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-70" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="msubsup" id="MathJax-Span-71" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-72" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-73" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-74" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-75" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-76" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-77" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-78" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">/ </span></span></span></span></span><span class="texatom" id="MathJax-Span-79" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-80" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-81" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">/ </span></span></span></span></span><span class="msubsup" id="MathJax-Span-82" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-83" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-84" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-85" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-86" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-87" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-88" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-89" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-90" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-91" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-92" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">2 </span></span></span><span class="mo" id="MathJax-Span-93" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.418em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -0.401em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><br />
<span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-4-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-94" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 10.508em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 8.407em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-95" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-96" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.242em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-97" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-98" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-99" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-100" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">2</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-101" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="msubsup" id="MathJax-Span-102" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-103" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-104" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-105" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-106" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-107" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-108" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-109" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">/ </span></span></span></span></span><span class="texatom" id="MathJax-Span-110" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-111" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-112" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">/ </span></span></span></span></span><span class="msubsup" id="MathJax-Span-113" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-114" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-115" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-116" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-117" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-118" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-119" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-120" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-121" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-122" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( 3</span></span></span><span class="mn" id="MathJax-Span-123" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;"> </span></span></span><span class="mo" id="MathJax-Span-124" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.418em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.401em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><br />
Developing terms the impedances Z1 and Z2 are obtained as</span><br />
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-5-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-125" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 14.925em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 11.908em;"><span style="border: 0px; clip: rect(0.649em 1000.03em 3.397em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-126" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-127" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.242em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-128" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">Z<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-129" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-130" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-131" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-132" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-133" style="border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 6.521em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.42em -0.428em); left: 60.515625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.738em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-134" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-135" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-136" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-137" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-138" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.696em -0.374em); left: 60.515625px; margin: 0px 0px 0px -3.176em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-139" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-140" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-141" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-142" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-143" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-144" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-145" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-146" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-147" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="msubsup" id="MathJax-Span-148" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-149" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-150" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-151" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-152" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-153" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="mi" id="MathJax-Span-154" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 6.521em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-155" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-156" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-157" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-158" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-159" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-160" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">4 </span></span></span><span class="mo" id="MathJax-Span-161" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.168em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -1.276em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><br />
<span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-6-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-162" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 14.71em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 11.747em;"><span style="-webkit-transition: none; border: 0px; clip: rect(0.757em 1000.03em 3.289em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-163" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-164" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.242em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-165" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-166" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-167" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-168" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">2</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-169" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-170" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 6.36em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 59.015625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.684em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-171" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-172" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-173" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-174" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-175" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.589em -0.374em); left: 59.015625px; margin: 0px 0px 0px -3.122em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-176" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-177" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-178" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-179" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-180" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-181" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-182" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-183" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-184" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="msubsup" id="MathJax-Span-185" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-186" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-187" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-188" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-189" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-190" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="mi" id="MathJax-Span-191" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 6.36em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-192" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-193" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-194" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-195" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-196" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( 5 </span></span></span><span class="mo" id="MathJax-Span-198" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 2.899em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.141em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><br />
Replacing the equations (4), (5) in (1) and simplifying, we obtain the probe transfer function:</span><br />
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-7-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-199" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 25.807em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 20.636em;"><span style="border: 0px; clip: rect(0.595em 1000.03em 3.451em -0.374em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-200" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-201" style="border: 0px; display: inline; margin: 0px; padding: 0px 0.11em; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.296em;"><span style="border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 12.015625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.684em; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-202" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.188em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-203" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">U<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.11em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-204" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-205" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-206" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 12.015625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -3.284em; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-207" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-208" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">U</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-209" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-210" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-211" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 1.296em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-212" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-213" style="border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 14.979em;"><span style="border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 139.015625px; margin: 0px 0px 0px -3.445em; padding: 0px; position: absolute; top: -3.014em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-214" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-215" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-216" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-217" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-218" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-219" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-220" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-221" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-222" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-223" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-224" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-225" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-226" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-227" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-228" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-229" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-230" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-231" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-232" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-233" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-234" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 139.015625px; margin: 0px 0px 0px -7.432em; padding: 0px; position: absolute; top: -1.56em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-235" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-236" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-237" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-238" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-239" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-240" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-241" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-242" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-243" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-244" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-245" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-246" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-247" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-248" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-249" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-250" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-251" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-252" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-253" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-254" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-255" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">) </span></span></span><span class="mo" id="MathJax-Span-256" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-257" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-258" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-259" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-260" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-261" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-262" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-263" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-264" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-265" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.296em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-266" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;">C</span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-267" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-268" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-269" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-270" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-271" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-272" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-273" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-274" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-275" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-276" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 14.979em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-277" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-278" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-279" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-280" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-281" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-282" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">6 </span></span></span><span class="mo" id="MathJax-Span-283" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.303em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -1.343em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><br />The transfer function shows there are a zero and a pole, in principle without relation between them, but if we can enforce the relationship (7), we can simplify the transfer function (8)</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-8-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-284" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 12.501em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 9.969em;"><span style="border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-285" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-286" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.296em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-287" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">C<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-288" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-289" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-290" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-291" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="msubsup" id="MathJax-Span-292" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-293" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-294" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-295" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-296" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-297" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="msubsup" id="MathJax-Span-298" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-299" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-300" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-301" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-302" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-303" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="msubsup" id="MathJax-Span-304" style="border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-305" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-306" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-307" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-308" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-309" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-310" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-311" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-312" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-313" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-314" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">7 </span></span></span><span class="mo" id="MathJax-Span-315" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.418em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -0.401em; width: 0px;"></span></span></nobr></span></div>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-9-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-316" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 15.195em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 12.124em;"><span style="-webkit-transition: none; border: 0px; clip: rect(0.757em 1000.03em 3.397em -0.374em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-317" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mfrac" id="MathJax-Span-318" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 12.015625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.684em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-319" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-320" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">U</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-321" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-322" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-323" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 12.015625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -3.284em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-324" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-325" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">U</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-326" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-327" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-328" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 1.296em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-329" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-330" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.72em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 34.515625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.684em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-331" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-332" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-333" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-334" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-335" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.696em -0.428em); left: 34.515625px; margin: 0px 0px 0px -1.775em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-336" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-337" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-338" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-339" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-340" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-341" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-342" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-343" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-344" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-345" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-346" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-347" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 3.72em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-348" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-349" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.457em 1000.03em 2.427em -0.374em); left: 11.015625px; margin: 0px 0px 0px -0.267em; padding: 0px; position: absolute; top: -2.96em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-350" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.457em 1000.03em 2.427em -0.374em); left: 11.015625px; margin: 0px 0px 0px -0.536em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-351" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">10</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 1.188em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-352" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-353" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-354" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-355" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-356" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-357" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">8 </span></span></span><span class="mo" id="MathJax-Span-358" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.034em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.276em; width: 0px;"></span></span></nobr></span></div>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">Therefore, in order to fulfill the attenuation ratio 10:1 is mandatory that the adjustable capacity meets the equation (7). Clearing Cp from (7) we obtain the condition to be fulfilled by the trimmer capacitor.</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-10-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-359" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 11.801em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 9.43em;"><span style="-webkit-transition: none; border: 0px; clip: rect(2.158em 1000.03em 5.444em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -4.038em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-360" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-361" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-362" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-363" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-364" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-365" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-366" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mrow" id="MathJax-Span-367" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-368" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.003em;"><span style="-webkit-transition: none; border: 0px; font-family: MathJax_Size4; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span></span><span class="mfrac" id="MathJax-Span-369" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.349em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 12.515625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.684em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-370" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-371" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-372" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-373" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-374" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.42em -0.428em); left: 12.515625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -3.284em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-375" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-376" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-377" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-378" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-379" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 1.349em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-380" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0.003em;"><span style="-webkit-transition: none; border: 0px; font-family: MathJax_Size4; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">) </span></span></span></span></span><span class="msubsup" id="MathJax-Span-381" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-382" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-383" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-384" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-385" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-386" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-387" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-388" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-389" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-390" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-391" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">9 </span></span></span><span class="mo" id="MathJax-Span-392" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 4.043em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.909em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.613em; width: 0px;"></span></span></nobr></span></div>
</div>
<span style="background-color: white;"><br style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;" /><b style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;">6. WHEN USING THE ATTENUATION X10?</b></span><br />
<span style="background-color: white;"><br />
</span><br />
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">X10 probe attenuation will be used when:</span></div>
<ol style="text-align: justify;">
<li style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b style="font-style: italic;">We want to enjoy the full bandwidth of the oscilloscope without limitations.</b> In General, we can say that the bandwidth is limited by the component that has lower bandwidth (probe or oscilloscope).</span></li>
<li style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><i><b>The circuit impedance is near to the impedance of the oscilloscope</b></i> . In this case, the insertion of the probe causes the formation of a resistive divider between the Thevenin equivalent impedance of the circuit and the oscilloscope input impedance reducing the signal level. The signal showed in the oscilloscope is not displayed as it is. Also, we may have problems due to excessive current drain from the signal.</span></li>
<li style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><i><b>The signal to measure includes high frequency components or abrupt edges that have to be accurately measured</b></i> . In the previous post we have seen how the fact of measuring a signal without a suitable probe reduces the bandwidth of the oscilloscope. In order to measure accurately rise/fall times, we have to ensure a bandwidth of at least<span class="MathJax_Preview" style="color: #888888;"></span></span><div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 618px;">
<span class="MathJax" id="MathJax-Element-11-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-393" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 10.669em;"><span style="-webkit-transition: none; border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 8.514em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.026em 1000.03em 3.558em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.529em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-394" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-395" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">B </span></span></span><span class="mi" id="MathJax-Span-396" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">W</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span class="mo" id="MathJax-Span-397" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_AMS; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-398" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.996em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.457em 1000.03em 2.427em -0.428em); left: 18.515625px; margin: 0px 0px 0px -0.913em; padding: 0px; position: absolute; top: -2.96em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-399" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">0.35</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(3.181em 1000.03em 4.312em -0.482em); left: 18.515625px; margin: 0px 0px 0px -0.374em; padding: 0px; position: absolute; top: -3.284em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-400" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.757em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.457em 1000.03em 2.427em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-401" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.38em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-402" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-403" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-404" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">r</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 1.996em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-405" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-406" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-407" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-408" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-409" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-410" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">10 </span></span></span><span class="mo" id="MathJax-Span-411" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.535em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 2.899em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.141em; width: 0px;"></span></span></nobr></span></div>
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span></span><div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 618px;">
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overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-423" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-424" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">a </span></span></span><span class="mi" id="MathJax-Span-425" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; 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font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">m </span></span></span><span class="mi" id="MathJax-Span-435" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.283em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.333em; width: 0px;"></span></span></nobr></span></div>
</li>
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;"><span style="background-color: white; font-size: 15px; line-height: 20px;"><i style="font-weight: bold;">The oscilloscope input capacitance affects the circuit under test. </i>Oscilloscope probe parallel coupling with poorly compensated probe can cause that resonant circuits as used in RF, modified their operation. For example the RF oscillator frequency could change simply by the fact of measuring its signal.</span></span></li>
<li style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b><i>The signal voltage exceeds the oscilloscope input range.</i></b> The range of the input stage of an oscilloscope is limited to a few volts <b>.</b> If we have to measure higher voltages, coupling X10 means we could measure signals 10 times larger than the oscilloscope input range.</span></li>
</ol>
<div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">For the following cases, if desired, it is not necessary to use the x10 oscilloscope probe option. Instead of it may alternatively use "wire" or other connection when:</span></div>
<ol>
<li style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><i><b style="background-color: white;">We do not mind sacrificing the bandwidth of the oscilloscope.</b></i></li>
<li style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="background-color: white;"><i><b>The circuit impedance to measure is much smaller than the input impedance of the oscilloscope.</b></i> In this case, the coupling impedance of the oscilloscope to the circuit being measured causes a slight distortion in the measured signal that will be smaller the lower the impedance of the circuit being measured and in many cases negligible.</span></li>
<li style="margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;"><span style="font-size: 15px; line-height: 20px;"><b><i style="background-color: white;">The signal frequency components are low or it is not interesting measuring the signal edges accurately.</i></b></span></span></li>
</ol>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;">
<span style="background-color: white;"><br />
</span></div>
<div style="text-align: justify;">
<b><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;"><span style="background-color: white; font-size: 15px; line-height: 20px;">7. HOW AFFECTS THE INSERTION PROBE TO THE MEASURE?</span></span></b></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">As mentioned, the simple act of measuring, alters the original signal. Our purpose will be change as less as possible the circuit under test.</span><br />
<span style="background-color: white;"><br /></span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">All Previous analysis was from oscilloscope point of view. We wanted to know how the oscilloscope sees the signal sensed by the probe. But now, we must ask ourselves the following question. <b>From the point of view of the signal, how the measuring equipment is seen?</b> . Among the factors to consider when connecting the oscilloscope probe to the circuit under test, we have:</span></div>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em; text-align: justify;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><i><b>The parasitic capacitance due to "ground wire"</b></i> and probe. "This impedance is in the order of a few pF and it will not be considered in this analysis.</span></li>
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><i><b>The equivalent impedance formed by the probe and "seen" by the signal</b></i> . In this case will be Z1 + Z2.</span></li>
</ul>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">For analysis, we will calculate the input impedance showed by the oscilloscope and the probe.</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNla-PD41ieFaDxIYiSySaXuSRX4Xiw4FmEwo4-YJLMaCMWOoj2GU06hKLZ1bGoSSZPXAgn5f0sbOVWuYQChfq6WeoQQU6677vUneoSBtRu2IcQCSNp0o-pJJyYHgVGxRQZtpr8t5LoY/s1600/Input-impedance-oscilloscope-probe.png" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope input impedance passive probe " border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNla-PD41ieFaDxIYiSySaXuSRX4Xiw4FmEwo4-YJLMaCMWOoj2GU06hKLZ1bGoSSZPXAgn5f0sbOVWuYQChfq6WeoQQU6677vUneoSBtRu2IcQCSNp0o-pJJyYHgVGxRQZtpr8t5LoY/s1600/Input-impedance-oscilloscope-probe.png" height="256" style="border: none; position: relative;" title="Input impedance probe" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Figure 3 - O<span style="color: #333333; line-height: 20px;">scilloscope plus probe i</span><span style="color: #333333; line-height: 20px;">nput impedance </span></span></td></tr>
</tbody></table>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">The signal "sees" the probe as an impedance:</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
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padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-440" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-441" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-442" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-443" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-444" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-445" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="msubsup" id="MathJax-Span-446" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.242em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-447" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-448" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-449" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-450" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-451" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-452" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.242em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-453" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-454" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-455" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-456" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; font-size: 13px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">2</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-457" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-458" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-459" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-460" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-461" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-462" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">11 </span></span></span><span class="mo" id="MathJax-Span-463" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.418em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.401em; width: 0px;"></span></span></nobr></span></div>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">Considering equations (2) and (3) we can write</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-14-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-464" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 24.514em;"><span style="-webkit-transition: none; border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 19.612em;"><span style="-webkit-transition: none; border: 0px; clip: rect(0.649em 1000.03em 3.451em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-465" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-466" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.888em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-467" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-468" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-469" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-470" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-471" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-472" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-473" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-474" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 5.875em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.42em -0.428em); left: 54.515625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.738em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-475" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-476" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-477" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-478" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-479" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.374em); left: 54.515625px; margin: 0px 0px 0px -2.906em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-480" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-481" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-482" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-483" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-484" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.296em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-485" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-486" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-487" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-488" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-489" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-490" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-491" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-492" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-493" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-494" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-495" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 5.875em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-496" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="mfrac" id="MathJax-Span-497" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.326em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 5.713em;"><span style="-webkit-transition: none; border: 0px; clip: rect(3.127em 1000.03em 4.312em -0.428em); left: 53.015625px; margin: 0px 0px 0px -0.59em; padding: 0px; position: absolute; top: -4.684em; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-498" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-499" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-500" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-501" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-502" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 3.989em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.374em); left: 53.015625px; margin: 0px 0px 0px -2.799em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-503" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-504" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-505" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-506" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-507" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-508" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-509" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-510" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-511" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-512" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-513" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-514" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-515" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-516" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-517" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-518" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 5.713em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-519" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-520" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-521" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-522" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-523" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-524" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">12 </span></span></span><span class="mo" id="MathJax-Span-525" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.236em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.343em; width: 0px;"></span></span></nobr></span></div>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">And if clearing the condition (7), the input impedance can be expressed as</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
<span class="MathJax" id="MathJax-Element-15-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-526" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 15.356em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 12.286em;"><span style="-webkit-transition: none; border: 0px; clip: rect(0.649em 1000.03em 3.397em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-527" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-528" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.888em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-529" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-530" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-531" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-532" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-533" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-534" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-535" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-536" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 5.713em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.696em -0.428em); left: 53.015625px; margin: 0px 0px 0px -1.775em; padding: 0px; position: absolute; top: -3.014em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-537" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-538" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-539" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-540" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-541" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-542" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-543" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-544" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-545" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-546" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-547" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-548" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.374em); left: 53.015625px; margin: 0px 0px 0px -2.799em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-549" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-550" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-551" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-552" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-553" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-554" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-555" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-556" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-557" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-558" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-559" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-560" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-561" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-562" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-563" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-564" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 5.713em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-565" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-566" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-567" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-568" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-569" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-570" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">13 </span></span></span><span class="mo" id="MathJax-Span-571" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.168em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.276em; width: 0px;"></span></span></nobr></span></div>
</div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">The impedance (13) corresponding to the oscilloscope and probe is connected in parallel with the signal to be measured, thereby altering the circuit. Depending on the output impedance of the signal, the fact connect the probe may cause the measure to be invalid . The lower the output impedance of the signal, the better the measure.</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><br />
From equation (13) follows:</span></div>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em; text-align: justify;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;">At low frequencies, the impedance will be about 10 MOhms.<span class="MathJax_Preview" style="color: #888888;"></span></span><div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 583.75px;">
<span class="MathJax" id="MathJax-Element-16-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-572" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 27.585em;"><span style="-webkit-transition: none; border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 22.036em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-573" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-574" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.888em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-575" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-576" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-577" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-578" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-579" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-580" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-581" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="msubsup" id="MathJax-Span-582" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-583" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-584" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-585" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-586" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-587" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-588" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-589" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-590" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-591" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-592" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-593" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mn" id="MathJax-Span-594" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">10 </span></span></span><span class="mi" id="MathJax-Span-595" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">M</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span class="mi" id="MathJax-Span-596" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Ω </span></span></span><span class="mtext" id="MathJax-Span-597" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-598" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">; </span></span></span><span class="mtext" id="MathJax-Span-599" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mi" id="MathJax-Span-600" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mi" id="MathJax-Span-601" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">i </span></span></span><span class="mtext" id="MathJax-Span-602" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mn" id="MathJax-Span-603" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mo" id="MathJax-Span-604" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">> </span></span><span style="-webkit-transition: none; border: 0px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">> </span></span></span></span><span class="msubsup" id="MathJax-Span-605" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-606" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-607" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-608" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-609" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-610" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-611" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-612" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-613" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-614" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-615" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mtext" id="MathJax-Span-616" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-617" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-618" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-619" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-620" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-621" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">14 </span></span></span><span class="mo" id="MathJax-Span-622" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 1.418em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.401em; width: 0px;"></span></span></nobr></span></div>
</li>
</ul>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em; text-align: justify;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;">At high frequency, the impedance is:<span class="MathJax_Preview" style="color: #888888;"></span></span><div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 583.75px;">
<span class="MathJax" id="MathJax-Element-17-Frame" style="border: 0px; direction: ltr; display: inline; float: none; line-height: normal; margin: 0px; padding: 0px; text-align: left; white-space: nowrap; word-spacing: normal; word-wrap: normal;"><nobr style="-webkit-transition: none; border: 0px; margin: 0px; max-height: none; max-width: none; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-623" style="-webkit-transition: none; background-color: white; border: 0px; display: inline-block; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 24.245em;"><span style="border: 0px; display: inline-block; font-size: 19px; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 19.397em;"><span style="-webkit-transition: none; border: 0px; clip: rect(0.649em 1000.03em 3.289em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-624" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-625" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.888em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-626" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-627" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-628" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-629" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-630" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-631" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-632" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-633" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.72em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.696em -0.428em); left: 34.515625px; margin: 0px 0px 0px -1.775em; padding: 0px; position: absolute; top: -3.014em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-634" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-635" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-636" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-637" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-638" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-639" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-640" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="msubsup" id="MathJax-Span-641" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-642" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-643" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-644" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-645" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.589em -0.428em); left: 34.515625px; margin: 0px 0px 0px -1.506em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-646" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-647" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-648" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-649" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-650" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-651" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-652" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-653" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-654" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-655" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-656" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-657" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 3.72em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="texatom" id="MathJax-Span-658" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-659" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-660" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span></span></span><span class="texatom" id="MathJax-Span-661" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-662" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-663" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span></span></span><span class="texatom" id="MathJax-Span-664" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-665" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-666" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span></span></span><span class="texatom" id="MathJax-Span-667" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-668" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mo" id="MathJax-Span-669" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span></span></span><span class="mtext" id="MathJax-Span-670" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-671" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">; </span></span></span><span class="mtext" id="MathJax-Span-672" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; transition: none; vertical-align: 0px;"> </span><span class="mi" id="MathJax-Span-673" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mi" id="MathJax-Span-674" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">i </span></span></span><span class="mtext" id="MathJax-Span-675" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="msubsup" id="MathJax-Span-676" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-677" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-678" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-679" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-680" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-681" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-682" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-683" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-684" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-685" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mi" id="MathJax-Span-686" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">S</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mo" id="MathJax-Span-687" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">> </span></span><span style="-webkit-transition: none; border: 0px; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">></span></span></span></span><span class="mn" id="MathJax-Span-688" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;"> 1 </span></span></span><span class="mtext" id="MathJax-Span-689" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-690" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-691" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-692" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-693" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-694" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">15 </span></span></span><span class="mo" id="MathJax-Span-695" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 2.966em; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -1.141em; width: 0px;"></span></span></nobr></span></div>
</li>
</ul>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">Analyzing in frequency the impedance who the signal "sees" (Zent), we have a pole that will cause a drop of 20 dB / decade in</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><span class="MathJax_Preview" style="color: #888888;"></span><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
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display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-698" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.834em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.642em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-699" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">f<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.541em; margin: 0px; padding: 0px; position: absolute; top: -2.044em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-700" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-701" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-702" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p ole</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-706" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;"> ( </span></span></span><span class="msubsup" id="MathJax-Span-707" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.888em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-708" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-709" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-710" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-711" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-712" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-713" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-714" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">) </span></span></span><span class="mo" id="MathJax-Span-715" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-716" style="border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 3.612em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.457em 1000.03em 2.427em -0.374em); left: 33.515625px; margin: 0px 0px 0px -0.267em; padding: 0px; position: absolute; top: -2.96em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-717" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.589em -0.428em); left: 33.515625px; margin: 0px 0px 0px -1.721em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-718" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-719" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">2 </span></span></span><span class="mi" id="MathJax-Span-720" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">π</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-721" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.188em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-722" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">C</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.811em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-723" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-724" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-725" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="msubsup" id="MathJax-Span-726" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.134em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-727" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">R</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-728" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-729" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-730" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">s</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 3.612em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mtext" id="MathJax-Span-731" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.164em; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-732" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-733" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-734" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-735" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-736" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">16 </span></span></span><span class="mo" id="MathJax-Span-737" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 2.899em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -1.141em; width: 0px;"></span></span></nobr></span></div>
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<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">In general, one decade below the cutoff frequency have an impedance of 10 MOhm and one decade above the cutoff frequency will fall to 20 dB / dec.</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><br /></span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">Note how <b>the input impedance varies with frequency</b> starting from 10 MOhms and decreasing to negligible limits. If we perform a little calculation with 1.5 meters of coaxial cable (50 Ohm RG-316, capacity 96 pF / m) gives a cutoff frequency of 1460 Hz</span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;"><br />
</span><br />
<div aria-readonly="true" class="MathJax_Display" role="textbox" style="margin: 1em 0em; position: relative; text-align: center; width: 658px;">
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display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-740" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; vertical-align: 0px; width: 1.834em;"><span style="border: 0px; clip: rect(1.403em 1000.03em 2.642em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; vertical-align: 0px;"><span class="mi" id="MathJax-Span-741" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;">f<span style="border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: 0px; width: 0.057em;"></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.541em; margin: 0px; padding: 0px; position: absolute; top: -2.044em; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-742" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-743" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-744" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p ole</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-748" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="msubsup" id="MathJax-Span-749" style="border: 0px; display: inline; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.888em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.403em 1000.03em 2.427em -0.428em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -2.26em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-750" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; left: 0.757em; margin: 0px; padding: 0px; position: absolute; top: -2.098em; transition: none; vertical-align: 0px;"><span class="texatom" id="MathJax-Span-751" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-752" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-753" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">e </span></span></span><span class="mi" id="MathJax-Span-754" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">n </span></span></span><span class="mi" id="MathJax-Span-755" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-size: 13px; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">t</span></span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-756" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">) </span></span></span><span class="mo" id="MathJax-Span-757" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">= </span></span></span><span class="mfrac" id="MathJax-Span-758" style="border: 0px; display: inline; margin: 0px; padding: 0px 0.11em 0px 0.38em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 0px; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 10.508em;"><span style="-webkit-transition: none; border: 0px; clip: rect(1.457em 1000.03em 2.427em -0.374em); left: 97.515625px; margin: 0px 0px 0px -0.267em; padding: 0px; position: absolute; top: -2.96em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-759" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1</span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(1.349em 1000.03em 2.696em -0.428em); left: 97.515625px; margin: 0px 0px 0px -5.169em; padding: 0px; position: absolute; top: -1.56em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-760" style="-webkit-transition: none; border: 0px; display: inline; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-761" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">2 </span></span></span><span class="mi" id="MathJax-Span-762" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">π</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-763" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="mo" id="MathJax-Span-764" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-765" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">96 </span></span></span><span class="mo" id="MathJax-Span-766" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">+ </span></span></span><span class="mn" id="MathJax-Span-767" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">13 </span></span></span><span class="mo" id="MathJax-Span-768" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">) </span></span></span><span class="mi" id="MathJax-Span-769" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">p </span></span></span><span class="mi" id="MathJax-Span-770" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">F</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span class="mo" id="MathJax-Span-771" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">⋅ </span></span></span><span class="mn" id="MathJax-Span-772" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.218em; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1 </span></span></span><span class="mi" id="MathJax-Span-773" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">M</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.11em;"></span></span><span class="mi" id="MathJax-Span-774" style="-webkit-transition: none; border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">Ω</span></span></span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="-webkit-transition: none; border: 0px; clip: rect(0.865em 1000.03em 1.242em -0.482em); left: 0.003em; margin: 0px; padding: 0px; position: absolute; top: -1.29em; transition: none; vertical-align: 0px;"><span style="-webkit-transition: none; border-left-style: solid; border-width: 0px 0px 0px 10.508em; display: inline-block; height: 1.25px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0.003em; width: 0px;"> </span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1.08em; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-775" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;"> = </span></span></span><span class="mn" id="MathJax-Span-776" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px 0px 0px 0.272em; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">1460 </span></span></span><span class="mtext" id="MathJax-Span-777" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mi" id="MathJax-Span-778" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">H</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.057em;"></span></span><span class="mi" id="MathJax-Span-779" style="border: 0px; display: inline; font-family: MathJax_Math; font-style: italic; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">z</span></span><span style="-webkit-transition: none; border: 0px; display: inline-block; height: 1px; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0.003em;"></span></span><span class="mtext" id="MathJax-Span-780" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-781" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">! </span></span></span><span class="mo" id="MathJax-Span-782" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">! </span></span></span><span class="mo" id="MathJax-Span-783" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">! </span></span></span><span class="mtext" id="MathJax-Span-784" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-785" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-786" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mtext" id="MathJax-Span-787" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"> </span><span class="mo" id="MathJax-Span-788" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">( </span></span></span><span class="mn" id="MathJax-Span-789" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">17 </span></span></span><span class="mo" id="MathJax-Span-790" style="border: 0px; display: inline; font-family: MathJax_Main; margin: 0px; padding: 0px; position: static; vertical-align: 0px;"><span style="-webkit-transition: none; transition: none;"><span style="-webkit-transition: none; transition: none;">)</span></span></span></span><span style="border: 0px; display: inline-block; height: 2.265em; margin: 0px; padding: 0px; position: static; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-left-style: solid; border-width: 0px 0px 0px 0.003em; display: inline-block; height: 3.034em; margin: 0px; overflow: hidden; padding: 0px; position: static; vertical-align: -1.276em; width: 0px;"></span></span></nobr></span></div>
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<span style="background-color: white;"><br /></span></div>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">In this particular case, it means that in frequencies below than 1.4 kHz the input impedance will be approximately 10 MOhm and in higher frequencies begins to decay in the order of 20 dB per decade.</span></div>
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<span style="background-color: white;"><br />
For reference, next figure shows the Tektronix P2220 probe input impedance versus frequency and how the cut-off frequency is around 1.5 KHz (Figure 5, at the intersection with 45 º).</span></div>
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</span></div>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4ffA1jXtJIxmNc1F9QON52Hg0ggbKdhxkhl62yHbo6W1EOx75AKpY_WPFdvtWEzY6Lcm00WX4LFqmfuMUGxkfFF8eBZ1N1Z38e2erFHOaKh_IkS3I6F0musNpzyjlzEI9sDdmGCRjDkE/s1600/tektronix-P2220-input-impedance.png" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="Tektronix P2220 probe input impedance" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4ffA1jXtJIxmNc1F9QON52Hg0ggbKdhxkhl62yHbo6W1EOx75AKpY_WPFdvtWEzY6Lcm00WX4LFqmfuMUGxkfFF8eBZ1N1Z38e2erFHOaKh_IkS3I6F0musNpzyjlzEI9sDdmGCRjDkE/s1600/tektronix-P2220-input-impedance.png" height="326" style="border: none; position: relative;" title="P2220 Input Impedance" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Figure 4 - Probe Tektronix P2220 Input impedance </span></td></tr>
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<span style="background-color: white;"><br />Next figure shows phase change versus frequency and we can see clearly the pole influence dephasing 90 ° at high frequency.</span></div>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS3NZahEOGIUFEmCFAdD8nOORuwUMr9uou2tw5AVD7kIIFGrzt3UUHT6vMaSR6UxRg_c4TtH3lLlrBZk6gILHA3pxZM50YrDjaJ80pgpkqvDxTwfDA3NdZ-YY4Ni-asNw5cjj1TECTskU/s1600/tektronix-P2220-phase-delay.png" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="Phase diagram Tektronix P2220 probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS3NZahEOGIUFEmCFAdD8nOORuwUMr9uou2tw5AVD7kIIFGrzt3UUHT6vMaSR6UxRg_c4TtH3lLlrBZk6gILHA3pxZM50YrDjaJ80pgpkqvDxTwfDA3NdZ-YY4Ni-asNw5cjj1TECTskU/s1600/tektronix-P2220-phase-delay.png" height="273" style="border: none; position: relative;" title="Phase P2220" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Figure 5 - Phase versus frequency Tektronix P2220 probe</span></td></tr>
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<span style="background-color: white;"><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;"><br /></span><b><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;"><span style="font-size: 15px; line-height: 20px;">8. PRACTICAL EXAMPLE OF DESIGN OF A PROBE OSCILLOSCOPE</span></span></b></span></div>
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</span></div>
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<span style="background-color: white;"><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;">In this section we will implement all the theoretical part shown above. To do this, we will design a home oscilloscope 10:1 </span><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;">ratio</span><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;"> </span><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;">probe</span><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;"> </span><span style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif;"><span style="font-size: 15px; line-height: 20px;">where we will see the influence of the adjust capacitor. As I had not a variable capacitor, I took several fixed-value capacitors in parallel, so that I can be adding capacity by coupling the circuit according to the following drawing:</span></span></span></div>
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</span></div>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibSNmoNbxsiRBIPQo-k1_xUDyNu-DeOVFBS3culFfOvQ4Hq7GR4_LZu_DgkO5ITpMYGcMR3UZLcYJmW7Mfp8-r0KojzUHMgqg5UKmq1G0yAfBiwhmZtodCR3C499Z3ELUcj3bDSh08aBs/s1600/Home-made-oscilloscope-pasive-probe.png" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="homemade oscilloscope passive probe Scheme setup" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibSNmoNbxsiRBIPQo-k1_xUDyNu-DeOVFBS3culFfOvQ4Hq7GR4_LZu_DgkO5ITpMYGcMR3UZLcYJmW7Mfp8-r0KojzUHMgqg5UKmq1G0yAfBiwhmZtodCR3C499Z3ELUcj3bDSh08aBs/s1600/Home-made-oscilloscope-pasive-probe.png" height="291" style="border: none; position: relative;" title="homemade passive probe" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.6-homemade oscilloscope probe setup</span></td></tr>
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<span style="background-color: white;"><b><br /></b><b>8.1 MATERIAL REQUIRED</b></span></div>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;">Coaxial cable RG-316, length 1.20 m.</span></li>
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;">SMD capacitor values 2.7 pF, 6.8 pF, 10 pF and 22 pF.</span></li>
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;">SMD resistor value of 9 MOhm.</span></li>
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;">Oscilloscope Tektronix TDS3000 or similar.</span></li>
</ul>
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<b style="background-color: white;"><br />8.2 SETUP</b></div>
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<span style="background-color: white;"><br />
To perform the test, I used the square wave output generator from the oscilloscope that is used to compensate probes.</span></div>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="background-color: white;"> <b>TEST 1: </b> We take as reference the oscilloscope's commercial probe properly compensated. We see that the displayed waveform is completely square and that will be our goal to achieve.</span></li>
</ul>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiU2_Ebk4fu5WMCr1bYFhLkZwl-4ZBfM71pIgave1J1Hgf3DUf4gyx-hKCXbeGvgEYSGKVUumRKp2zZ_RvpXdKoEvJ7-xH8-27AahnJU_jmzyyAeWe2s5NSdF0RPAq5cYRB7php30R7_Ag/s1600/TEK00000.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe compensated in frequency" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiU2_Ebk4fu5WMCr1bYFhLkZwl-4ZBfM71pIgave1J1Hgf3DUf4gyx-hKCXbeGvgEYSGKVUumRKp2zZ_RvpXdKoEvJ7-xH8-27AahnJU_jmzyyAeWe2s5NSdF0RPAq5cYRB7php30R7_Ag/s1600/TEK00000.PNG" height="240" style="border: none; position: relative;" title="Compensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.7-Passive probe compensated</span></td></tr>
</tbody></table>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="background-color: white;"><b>TEST 2:</b> Using the DIY assembly without adding any capacity. All switches are open.</span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn-Nr_od46AUVn0v1ulAMDOuGLR2eVScyjmpfgzqbdJpwQc5jbq1PnHi0IaWMgyB3KClhvHaqQxgcMqzA8FILfZZ4LuPlJj3V1aeUJHcIoDwpaeqqXa3MVdoXrj621-rCluJ3-dEbPZEM/s1600/TEK00001.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe undercompensated" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn-Nr_od46AUVn0v1ulAMDOuGLR2eVScyjmpfgzqbdJpwQc5jbq1PnHi0IaWMgyB3KClhvHaqQxgcMqzA8FILfZZ4LuPlJj3V1aeUJHcIoDwpaeqqXa3MVdoXrj621-rCluJ3-dEbPZEM/s1600/TEK00001.PNG" height="240" style="border: none; position: relative;" title="Uncompensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.8-Probe with 0 pF compensation</span></td></tr>
</tbody></table>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="background-color: white;"><b>TEST 3:</b> DIY assembly adding 2.7 pF.</span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0bzdDhNm2RufWoC4mtG3llvVJww5aSor_G-wPJXqSiJsE-GZa_8bQa5_Y-hWsuSxjfdHUlZ9hn9xLw7l-DIN5apcjo6P_bx3QXw9sHkUCuj93znDdMRLsMTpSFuWtYRGaL_f-L3qxMvU/s1600/TEK00002.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe undercompensated" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0bzdDhNm2RufWoC4mtG3llvVJww5aSor_G-wPJXqSiJsE-GZa_8bQa5_Y-hWsuSxjfdHUlZ9hn9xLw7l-DIN5apcjo6P_bx3QXw9sHkUCuj93znDdMRLsMTpSFuWtYRGaL_f-L3qxMvU/s1600/TEK00002.PNG" height="240" style="border: none; position: relative;" title="Slightly compensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.9-probe with 2.7 pF compensation</span></td></tr>
</tbody></table>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>TEST 4:</b> DIY assembly, adding 6.8 pF.</span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_vZkvLnXwDTRq-Us8w1mJ7JBx0P6H4Hy8n5WdS53COCW_7wr1iwoNVR4L9jlH8UYZiK1I-B0uzNPCScFkvQPUY8fLHBWhJmjpezp2BFZ3xlY4JloZjqqaBhxxLsUZ87TPhiUlkuLgLT0/s1600/TEK00003.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe slightly undercompensated" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_vZkvLnXwDTRq-Us8w1mJ7JBx0P6H4Hy8n5WdS53COCW_7wr1iwoNVR4L9jlH8UYZiK1I-B0uzNPCScFkvQPUY8fLHBWhJmjpezp2BFZ3xlY4JloZjqqaBhxxLsUZ87TPhiUlkuLgLT0/s1600/TEK00003.PNG" height="240" style="border: none; position: relative;" title="Slightly compensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.10-probe with 6.8 pF compensation</span></td></tr>
</tbody></table>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>TEST 5:</b> DIY assembly adding 10 pF.</span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMyqODRO1eh4bBujxDFQpzP9c7Jz3m6__K0bfg477maW2kBvP8yVSKTQDOuRZlF-tvyAydONg6VX_tfDBKVo1UASu-u39spksx_yp5p3vL1V0J9B4rF84LXXFePLse6QgTiZKuIV0Beq0/s1600/TEK00004.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe slightly undercompensated" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMyqODRO1eh4bBujxDFQpzP9c7Jz3m6__K0bfg477maW2kBvP8yVSKTQDOuRZlF-tvyAydONg6VX_tfDBKVo1UASu-u39spksx_yp5p3vL1V0J9B4rF84LXXFePLse6QgTiZKuIV0Beq0/s1600/TEK00004.PNG" height="240" style="border: none; position: relative;" title="Slightly compensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.11-Probe with 10 pF compensation</span></td></tr>
</tbody></table>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="background-color: white;"><b>TEST 6: </b>DIY assembly adding 22 pF. Here we can see how we have passed compensating the probe. The optimum value is between 10 pF and 22 pF.</span></li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; margin-left: auto; margin-right: auto; padding: 4px; position: relative; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_H1cHS7Rwzlx7bJe4rr1GMeWtt3IUgXNbeBInzwGzivQaS4OJ-5yzq3dv1nQpES6dgs1mxiEJ0IfFT79sc0CSXPF6Axm-Aaxj90H0yV22p7YVkQau7RV35CHpn9Lo9BOIxFQ1LS0m9n8/s1600/TEK00005.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe overcompensated" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_H1cHS7Rwzlx7bJe4rr1GMeWtt3IUgXNbeBInzwGzivQaS4OJ-5yzq3dv1nQpES6dgs1mxiEJ0IfFT79sc0CSXPF6Axm-Aaxj90H0yV22p7YVkQau7RV35CHpn9Lo9BOIxFQ1LS0m9n8/s1600/TEK00005.PNG" height="240" style="border: none; position: relative;" title="Overcompensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.12-Probe with 22 pF compensation</span></td></tr>
</tbody></table>
<ul style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 1.4; list-style-image: initial; list-style-position: initial; margin: 0.5em 0px; padding: 0px 2.5em;">
<li style="border: none; margin: 0px 0px 0.25em; padding: 0px; text-align: justify;"><span style="background-color: white;"><b>TEST 7:</b> DIY adding 12 pF (10 pF / / 2.7 pF). For this value we should be near to the optimum compensation. </span></li>
</ul>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdXLYkXuJoZwjDmagTEYTfE5hd6Fhqvn456oYyxNtuuP8-yIF2rMZ9vc-0KnccwvmkjjVtJuo_MpQ_20ttuhKQdlaVjPsscoMjyy7l12Rv0e7uCKMaVWey88cbW9eKBxz0m2vHPon809s/s1600/TEK00006.PNG" imageanchor="1" style="background-color: white; color: #100077; margin-left: auto; margin-right: auto; text-decoration: none;"><img alt="oscilloscope passive probe slightly overcompensated " border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdXLYkXuJoZwjDmagTEYTfE5hd6Fhqvn456oYyxNtuuP8-yIF2rMZ9vc-0KnccwvmkjjVtJuo_MpQ_20ttuhKQdlaVjPsscoMjyy7l12Rv0e7uCKMaVWey88cbW9eKBxz0m2vHPon809s/s1600/TEK00006.PNG" height="240" style="border: none; position: relative;" title="almost compensated probe" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12px;"><span style="background-color: white;">Fig.13-Probe with 12 pF compensation</span></td></tr>
</tbody></table>
<div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<span style="background-color: white;">Due to we do not have a variable capacitor, the setup showed in test 7, is the best we can get. Note also that if we calculated theoretically the value we need (Equation 7), we obtain a value around 14 pF <b>. </b>This value is very close to the final one. </span><br />
<span style="background-color: white;"><b><br /></b>
<b>9. PART II. </b><b>CONCLUSIONS</b></span><br />
<span style="background-color: white;">
<b></b></span></div>
<ol style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px; text-align: justify;">
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>Maximizing bandwidth by adjusting the probe.</b><i><b> </b></i>To maintain as maximum as possible the bandwidth, adjust the variable capacity of the probe to compensate the oscilloscope's pole.</span></li>
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>Avoid the exchange of probes between oscilloscopes.</b> A probe adjusted for one oscilloscope may not be right for another one, since each has its own oscilloscope input impedance.</span></li>
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>Probe bandwidth.</b> When we have to buy an oscilloscope probe, we must ensure that the bandwidth is sufficient for the oscilloscope we have. The probe chosen affect the overall bandwidth.</span></li>
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>If you are going to measure high frequency signals</b> , the proposed mathematical analysis begins to be compromised together to the feasibility of passive probe. In this type of signals, it is better <b>to consider</b> RF probes based on FET also called <b>active probes</b>.<b></b></span></li>
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>If we chose to design our own probe</b> , you have to pay special attention to the selection of components. The capacitors will be searched to have flat response versus frequency and the resistors will have low inductive components in the whole bandwidth <b>.</b></span></li>
<li style="margin: 0px 0px 0.25em; padding: 0px;"><span style="background-color: white;"><b>When connecting the probe to the circuit under test, we are adding an impedance in parallel</b> , depending on the output impedance of the signal, this extra load could affect to the measurement dramatically.</span></li>
</ol>
</div>
<span style="background-color: yellow;"><br /></span>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0California, EE. UU.36.778261 -119.4179323999999823.886873 -140.07222939999997 49.669649 -98.763635399999984tag:blogger.com,1999:blog-3108912614914761750.post-627889325802078212014-02-22T22:43:00.004+01:002014-02-22T23:20:24.478+01:00Simulador de cámara reflex <b>Resumen</b><br />
<div style="text-align: justify;">
En ésta entrada encontramos un simulador de cámara reflex muy completo que permitirá a los más nóveles, mediante la metodología de ensayo y error, comprender de una manera práctica los conceptos básicos de fotografía.</div>
<a name='more'></a><br />
<br />
<iframe allowfullscreen="" frameborder="0" height="660" src="http://camerasim.com/embed/camera-simulator/" width="640"></iframe><br />
<br />
<b><br /></b>
<b>REFERENCIAS</b><br />
[1] <a href="http://camerasim.com/apps/camera-simulator/" target="_blank">Simulador de cámara reflex</a><br />
<br />JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-22944747442706254492014-01-22T22:37:00.001+01:002015-03-28T02:08:41.810+01:00¿Cuánto cuesta subir un grado la calefacción? <div style="text-align: justify;">
<b>Resumen </b><br />
En la presente entrada se evalúa de una manera técnica el coste de incrementar un grado la temperatura de la calefacción de nuestra vivienda. <b><br /><br />Abstract </b><br />
This entry evaluates from a technical point of view, the cost of increasing the home temperature one degree.<br />
<br />
<a name='more'></a><div style="text-align: center;">
----oooOOOooo----</div>
<br />
<i>"Pepe, tras recibir su última factura del gas, quedo asustado por el importe de la misma. No sabía qué parte del incremento del coste se debía a la petición de subir la temperatura de la calefacción por parte de su friolera mujer, y cual a la subida de tarifas. Buscó por Internet alguna explicación a tal fenómeno, pero al no encontrar un razonamiento objetivo, decidió preguntar a su amigo ingeniero" </i><b> </b><br />
<br />
<b>Según la Guía Práctica de la Energía del Instituto para la Diversificación y el Ahorro de la Energía (IDAE), Por cada grado que aumentemos la temperatura, se incrementa el consumo de energía aproximadamente en un 7%</b>. Como normalmente los actos de fe los reduzco a aquello que no puedo cuantificar con las matemáticas, me he propuesto demostrar de donde sale ése 7% que tanto circula por internet y que hasta ahora no he visto expresado con el rigor científico que se merece.<br />
<br /></div>
<div style="text-align: justify;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTalc5ZTtPiDb1JJRwQIavjDHo8s4C7pT1U7LVJJ9DgVCYeQcHBtTYgmhPKHIzfQdTtDuN02iAB8RHz4bM7LyU6j73jSi-WMBskS6BmS_FCptWXY1v81GIAGKSdiTO2dgr8xKTp2PREpg/s1600/home+loss+of+heat.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="perdidas de calor en el hogar" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTalc5ZTtPiDb1JJRwQIavjDHo8s4C7pT1U7LVJJ9DgVCYeQcHBtTYgmhPKHIzfQdTtDuN02iAB8RHz4bM7LyU6j73jSi-WMBskS6BmS_FCptWXY1v81GIAGKSdiTO2dgr8xKTp2PREpg/s1600/home+loss+of+heat.PNG" height="300" title="fugas de calor" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1-Pérdidas de calor en el hogar</td></tr>
</tbody></table>
</div>
<div style="text-align: justify;">
La casa ideal sería adiabática (no produce intercambio de calor a través de sus paredes con el exterior). En invierno, ésto se traduce en que no perderíamos calor al exterior, y en verano no se calentaría el interior de la casa. <b>Cualquier intercambio de calor con el exterior, se traduce en dinero que estamos perdiendo</b>, ya que en invierno tendríamos que encender la calefacción, y en verano el aire acondicionado.<br />
<br />
Como las casas distan mucho de ser ideales, la realidad es que tenemos un flujo de energía que está atravesando la estructura de la casa (techo, paredes, suelo, ventanas...) y fugándose al exterior. A ese recinto, se le denomina técnicamente <b>envolvente térmica</b>.<br />
<br />
<b>1. UN POCO DE TEORÍA </b><br />
<br />
Recapitulando un poco de teoría explicada de manera práctica, las pérdidas/ganancias de calor pueden ser debidas a tres tipos de transferencia energética:<br />
<ul>
<li><b>Conducción:</b> Sólo se produce si hay diferencia de temperatura entre dos partes de un medio conductor (p.e. paredes, ventanas, techo suelo...). Éste tipo de transferencia sigue la ley de FOURIER. </li>
</ul>
<div style="text-align: justify;">
$$\dot{Q}=k\cdot S\cdot \Delta T\ \ (1)\\<br />
\dot{Q}\ \[W]:Flujo\ \ calorico\ \ por \ \ unidad \ \ de \ \ tiempo\\<br />
k [\frac{W}{m^{2}K}]:Conductividad \ \ termica\\<br />
S[m^{2}]: Superficie \ \ de \ \ transferencia\\<br />
\Delta T [K]: Diferencia\ \ de\ \ temperatura \ \ entre\ \ caras \ \ de\ \ la\ \ pared$$</div>
<ul>
<li><b>Convección:</b> Es la que produce el viento al "rozar" las paredes del edificio. Podríamos decir que el viento "arranca" el calor de la superficie de la pared. Cuanto mayor sea su velocidad, más calor "robará" (cambia el coeficiente de película). Dentro de éstas pérdidas también está el calor que se va por la chimenea, la campana extractora de la cocina, las rejillas de ventilación. Éste tipo de transferencia sigue la ley de enfriamiento de NEWTON.</li>
</ul>
<div style="text-align: center;">
$$\dot{Q}=h_{c}\cdot S\cdot \Delta T\ \ (2) \\<br />
\dot{Q}\ \[W]:Flujo\ \ calorico\ \ por \ \ unidad \ \ de \ \ tiempo\\<br />
h_{c}[\frac{W}{m^{2}K}]:Coeficiente \ \ de \ \ pelicula \\<br />
S[m^{2}]: Superficie \ \ de \ \ transferencia\\<br />
\Delta T[K]: Diferencia\ \ de\ \ temperatura\ \ entre\ \ la\ \ superficie\ \ y\ \ el\ \ aire $$</div>
<ul>
<li><b>Radiación:</b> Éste tipo de pérdidas que se producen por el simple hecho de tener temperatura. Durante el día, el sol calienta las paredes y techo cediendo calor al interior de la casa, y durante la noche, la casa radia calor hacia el espacio exterior, que se comporta como un cuerpo negro a muy baja temperatura. Éste tipo de transferencia sigue la ley de STEFAN-BOLTZMAN. En ingeniería, con la finalidad de hacer estimaciones rápidas, si consideramos un objeto pequeño (casa) radiando hacia un medio ambiente grande y considerado la envolvente térmica de la casa como casi negra, podemos escribir:</li>
</ul>
<div style="text-align: center;">
$$\dot{Q}=\varepsilon\cdot S\cdot \sigma \cdot (T_{1}^{4}-T_{2}^{4})\ \ (3) \\<br />
\dot{Q}\ \[W]:Flujo\ \ calorico\ \ por \ \ unidad \ \ de \ \ tiempo\\<br />
\varepsilon\ :Emitancia \ \ de\ \ la\ \ casa\\<br />
S\ \ [m^{2}]: Superficie \ \ de \ \ transferencia \\<br />
(T_{1}^{4}-T_{2}^{4})\ \ [{K}^4]: Diferencia\ \ de\ \ temperatura \ \ entre\ \ pared\ \ y\ \ ambiente$$</div>
<br />
El hecho de que la transferencia de calor por radiación dependa de la cuarta potencia, vuelve complicados los cálculos en ingeniería. Cuando T1 y T2 no difieren demasiado, conviene linealizar la ecuación para obtener:<br />
<div style="text-align: center;">
$$\dot{Q}= S\cdot \ h_{r} \cdot \Delta T\ \ (4) \\<br />
\dot{Q}\ \[W]:Flujo\ \ calorico\ \ por \ \ unidad \ \ de \ \ tiempo\\<br />
h_{r} [\frac{W}{m^{2}K}]:Coeficiente\ \ de\ \ transferencia\ \ de\ \ calor\ \ por \ \ radiaccion$$</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Siendo </div>
<div style="text-align: center;">
$$h_{r}=4\varepsilon _{1}\sigma T_{media}^{3}\ \ (5)\\<br />
T_{media}=\frac{T_{1}+T_{2}}{2}\ \ (6)$$</div>
<br />
<b>2. EL ENFOQUE PRÁCTICO </b><br />
<br />
Lo primero que hay que tener en cuenta es que ENERGIA=DINERO. Un incremento de energía del 1% implica un incremento en la factura del 1% . Por lo tanto, cuando hablemos de incrementos de energía, hablaremos de incrementos de dinero en la misma proporción.<br />
<br />
Teniendo en cuenta los tres mecanismos por los que se produce la transferencia de calor, para calcular la cantidad de energía que se pierde a través de la envolvente térmica, vamos a apoyarnos en la teoría de la transferencia de calor formulada por el físico y matemático francés FOURIER junto con la analogía eléctrica del problema.<br />
<br />
El flujo de calor es proporcional a la superficie de transferencia S (envolvente térmica) y a la diferencia de temperatura entre el interior y exterior de la casa.</div>
<br />
$$\dot{Q}=K\cdot S\cdot \Delta T\ \ (7) \\<br />
\dot{Q}\ \ [W]:Flujo\ \ calorico\ \ por \ \ unidad \ \ de \ \ tiempo$$<br />
Siendo<br />
$$R_{th}=\frac{1}{KS}\ \ (8) \\<br />
R_{th}: Resistencia \ \ Termica\ \ de\ \ la\ \ Envolvente $$<br />
<div style="text-align: justify;">
<span style="text-align: justify;">En ésta ecuación tenemos una gran incógnita en general difícil de calcular. Es la resistencia térmica de la envolvente. Dicha resistencia térmica, depende de muchos factores teniendo en cuenta que estamos metiendo en ella todos los mecanismos de transferencia de calor posibles (conducción, convencción y radiacción). Pero afortunadamente para la salud mental del ingeniero, <b>en ingeniería las variables se vuelven aproximadamente constantes </b>bajo ciertas condiciones de trabajo, y ésto se consigue <b>linealizando </b>las ecuaciones sobre una temperatura de trabajo.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Bajo las premisas anteriores, podemos suponer la resistencia térmica de la envolvente constante, y vemos claramente cómo <b>el flujo calórico depende proporcionalmente de la diferencia de temperatura entre el interior de la casa y el exterior</b>. Obviamente seguimos sin conocer el valor de la resistencia térmica ("constante" de proporcionalidad), pero no la vamos a necesitar porque se va a expresar el incremento energético de subir un grado la calefacción, como una relación de potencias caloríficas entre el "antes" y el "después".</div>
<div style="text-align: justify;">
<br />
Como todo en la vida, lo que es bueno para una cosa es malo para otra, y linealizar las ecuaciones no se iba a escapar de ésta máxima. Por ello, <b>sólo vamos a poder analizar variaciones muy pequeñas de temperatura</b> sobre el punto de trabajo elegido. En éste caso, nuestra variación de temperatura será incrementar/decrementar 1ºC.<br />
<br />
Con todo lo anterior en mente, podemos escribir la ecuación linealizada en el punto de trabajo como:<br />
$$\dot{Q}=\frac{T_{int}-T_{ext}}{R_{th}}\ \ (9) \\<br />
\dot{Q}\ \ [W]:Energia\ \ perdida\ \ a\ \ traves\ \ de \ \ envolvente\ \ termica \\<br />
T_{int}: Temperatura\ \ del\ \ interior \\<br />
T_{ext}: Temperatura\ \ del\ \ exterior \\<br />
R_{th}: Resistencia\ \ Termica\ \ envolvente$$<br />
<br />
<b>3. ¿QUÉ PASA SI INCREMENTO LA TEMPERATURA UN GRADO?</b><br />
<br />
Flujo calórico antes de incrementar la temperatura interior Tint.<br />
$$\dot{Q}_{T_{int}}=\frac{T_{int}-T_{ext}}{R_{th}}\ \ (10)$$<br />
<br />
Flujo calórico después de incrementar la temperatura interior a Tint +dT. <br />
$$\dot{Q}_{(T_{int}+dT)}=\frac{T_{int}+dT-T_{ext}}{R_{th}}\ \ (11)$$<br />
<br />
Dividiendo las ecuaciones (10) y (11)<br />
$$\frac{\dot{Q}_{(T_{int}+dT)}}{\dot{Q}_{T_{int}}}=1+\frac{dT}{T_{int}-T_{ext}}\ \ (12)$$<br />
<br />
<br />
Y expresando el incremento del flujo calórico en términos relativos<br />
<div style="text-align: center;">
$$Coste\ \ Relativo\ \%=\frac{\dot{Q}_{(T_{int}+dT)}-\dot{Q}_{T_{int}}}{\dot{Q}_{T_{int}}}\cdot 100 \ \ (13)$$</div>
<br />
Es decir<br />
$$Coste\ \ Relativo\ \%=\frac{dT}{T_{int}-T{ext}}\cdot 100\ \ (14) $$</div>
<br />
<b>4. CONCLUSIONES</b><br />
<ul style="text-align: justify;">
<li>Teniendo en cuenta que la temperatura media de España en invierno es de 8.1ºC, y suponiendo una temperatura en el interior de la casa de 21ºC,<b> incrementar un grado la temperatura supone un incremento del coste alrededor del 8%</b>.</li>
</ul>
<div style="text-align: justify;">
$$Coste\ \ Relativo\ \%=\frac{1}{21^{\circ}-8.1^{\circ}}\cdot 100=7.75\%$$</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<ul style="text-align: justify;">
<li><b>Hablando en términos relativos entre facturas, y por increíble que resulte, cuanto más frío haga fuera, menos notaremos el el coste de subir un grado la calefacción.</b> Es decir, cuanto mayor sea el salto térmico entre el exterior y el interior, menos se notará en la factura tal y como se muestra en la figura 2.</li>
</ul>
<ul style="text-align: justify;">
<li>Otro <b>punto curioso</b> y a la vez lógico es <b>cuando la Text=Tint+1</b>. En éste caso incrementar un grado la calefacción supone un incremento de coste de -100%, es decir, pasamos a coste 0 por no haber flujo de energía entre el exterior y el interior.</li>
</ul>
<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZOtBza7IMhtHjszCF54mNgqGKrzp7XFns0Nm2sREWW5iB0kNd0Ygz9ZnVt6lmVLhRt9S92N8hfDOJJMQIrtXFmmQg5qIoJIwaD1it_-XyxZIFNbAse2-hzFwG1pYjxcewcl65fcaIQR8/s1600/Coste-incrementar-calefaccion-un-grado.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Coste relativo incrementar un grado la calefacción" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZOtBza7IMhtHjszCF54mNgqGKrzp7XFns0Nm2sREWW5iB0kNd0Ygz9ZnVt6lmVLhRt9S92N8hfDOJJMQIrtXFmmQg5qIoJIwaD1it_-XyxZIFNbAse2-hzFwG1pYjxcewcl65fcaIQR8/s1600/Coste-incrementar-calefaccion-un-grado.PNG" height="480" title="Coste relaitvo" width="640" /></a></td></tr>
<tr><td class="tr-caption">Fig 2.- Coste relativo incrementar 1ºC en función de la temperatura exterior</td></tr>
</tbody></table>
<ul style="text-align: justify;">
<li>Como consecuencia de lo anterior,<b> el coste relativo </b>de subir un grado la calefacción <b>es diferente según el país de estudio</b> (ya que cambia su temperatura media de invierno).</li>
</ul>
<br />
<b>REFERENCIAS</b><br />
[1] <a href="http://www.aemet.es/documentos/es/serviciosclimaticos/vigilancia_clima/resumenes_climat/estacionales/2013/Est_invierno_12_13.pdf" target="_blank">Media de temperaturas en España en invierno.</a><br />
[2] <a href="http://dl.idae.es/Publicaciones/11046%20Guia%20Practica%20Energia%203%20Ed.rev%20y%20actualizada%20A2011.pdf" target="_blank">Guía práctica de la energía (IDAE). </a><br />
<br />JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com12Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-69644849054056502542014-01-19T03:40:00.000+01:002014-01-19T03:48:12.551+01:00Restablecimiento -reset- del intervalo de servicio Audi A4 y A6 (1995-1999)<div style="text-align: justify;">
<b>Resumen</b><br />
Tras realizar
las inspecciones de mantenimiento en el vehículo, es necesario
restablecer el intervalo de servicio. El procedimiento expuesto,
describe cómo hacerlo de forma manual a través de cuadro de
instrumentos. <br />
<br />
<b>Abstract</b><br />
After carrying out
the scheduled maintenance routines on the vehicle, it is necessary to
reset the service interval on the dashboard. The following procedure
describes how to do it manually by means of the dashboard.<br />
<br />
<a name='more'></a><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEhgPi5-QEwWZU7f5TRjvXT4jRyOUP9oTEuRRS289881CUv88fue0gvdM6Q_Z83xL5t89b-Zswbfjaz3n05Oo49fzy3rSXH6GXKcM6-Y28sItCAOY6-irB_gpDCpD5ftuyJgvMm2UCWTk/s1600/Audi-A4-A6-dashboard.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Reset cuadro instrumentos Audi A4 A6" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEhgPi5-QEwWZU7f5TRjvXT4jRyOUP9oTEuRRS289881CUv88fue0gvdM6Q_Z83xL5t89b-Zswbfjaz3n05Oo49fzy3rSXH6GXKcM6-Y28sItCAOY6-irB_gpDCpD5ftuyJgvMm2UCWTk/s1600/Audi-A4-A6-dashboard.png" height="249" title="Cuadro instrumentos Audi" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1- Cuadro instrumentos Audi A4 y A6</td></tr>
</tbody></table>
<ol>
<li>Con el interruptor de encendido en la posición OFF, mantenga pulsado el botón <b>A</b> mientras gira la llave hasta la posición ON.</li>
<li>Aparecerá el mensaje "Servicio ACEITE". Si el mensaje no aparece, repita el paso 1.</li>
<li>Tire del botón <b>B</b> hasta que desaparezca el mensaje.</li>
<li>El visor debe mostrar ahora el mensaje "Service ---", que indica que se ha reiniciado el indicador SRI.<b> </b></li>
</ol>
<b>REFERENCIAS</b><br />
[1] <b> </b><a href="http://www.vdo.com/generator/www/com/en/vdo/main/products_solutions/cars/conti_sys/hidden/tab/hc_vdo_contisys_obd_phased_out.html" target="_blank">ContiSys multi-brand vehicle diagnosis</a><b><br /></b></div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-83887772432697523852014-01-19T03:23:00.002+01:002014-01-19T03:30:12.441+01:00Restablecimiento -reset- del intervalo de servicio Alfa Romeo (1994-2000)<div style="text-align: justify;">
<b>Resumen</b><br />
Tras realizar las inspecciones de mantenimiento en el vehículo, es necesario restablecer el intervalo de servicio. El procedimiento expuesto, describe cómo hacerlo de forma manual a través de cuadro de instrumentos. <br />
<br />
<b>Abstract</b><br />
After carrying out the scheduled maintenance routines on the vehicle, it is necessary to reset the service interval on the dashboard. The following procedure describes how to do it manually by means of the dashboard. </div>
<div style="text-align: justify;">
</div>
<a name='more'></a><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKdGlWwrY1t3DLpIrjnHL7-DcP0MBshyf6rW7ozivAc7p9ov5pAipKXdZHewqxhpnZlRzxTg94m4EZ1JQxoPyOtFePjb3c6VLJ4N_P3PG23FXbLwbbNxIPvZ2yoKiwBlDVZPgWn5nokpA/s1600/Alfa-Romeo-dashboard.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Cuadro instrumentos Alfa Romeo 1994-2000" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKdGlWwrY1t3DLpIrjnHL7-DcP0MBshyf6rW7ozivAc7p9ov5pAipKXdZHewqxhpnZlRzxTg94m4EZ1JQxoPyOtFePjb3c6VLJ4N_P3PG23FXbLwbbNxIPvZ2yoKiwBlDVZPgWn5nokpA/s1600/Alfa-Romeo-dashboard.png" height="270" title="cuadro Alfa Romeo" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig 1. Cuadro Alfa Romeo 1994-2000</td></tr>
</tbody></table>
<ol>
<li>Coloque el interruptor de encendido en la posición OFF.</li>
<li>Mantenga presionado el botón A.</li>
<li>Gire el interruptor de encendido hasta la posición ON.</li>
<li>Mantenga presionado el botón A durante unos 10 segundos.</li>
<li>En la pantalla aparecerá '0' y desaparecerá el icono de la llave.<b> </b></li>
</ol>
<b>Alfa Romeo 156</b><br />
<ol>
<li>Conecte el interruptor de encendido.</li>
<li>Pulse el botón [INFO] del salpicadero para entrar en el menú de las funciones de salpicadero.</li>
<li>Utilice los botones [+] y [-] del salpicadero para abrir la opción SERVICIO y pulse</li>
<li>[INFO] para seleccionar.</li>
<li>Mantenga pulsados los botones [+] y [-] durante al menos 10 segundos.</li>
<li>El 'Número de millas hasta servicio' se debe restablecer a aproximadamente 12.500 millas.</li>
<li>Utilice los botones [+] y [-] del salpicadero para abrir la opción FIN MENÚ y pulsar [MODO] para salir del menú de funciones.</li>
<li>Apague el contacto.</li>
</ol>
<b>REFERENCIAS</b><br />
[1] <b> </b><a href="http://www.vdo.com/generator/www/com/en/vdo/main/products_solutions/cars/conti_sys/hidden/tab/hc_vdo_contisys_obd_phased_out.html" target="_blank">ContiSys multi-brand vehicle diagnosis</a><b><br /></b><br />
<br />JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0tag:blogger.com,1999:blog-3108912614914761750.post-89671120597595000742013-11-22T00:44:00.000+01:002013-11-22T00:56:52.703+01:00Kung Fu en Ingeniería<div style="text-align: justify;">
Cuando era niño mi padre siempre me decía <b><i>" hijo, escucha siempre más que hables, porque cuando hablas, estarás negando la posibilidad de aprender"</i></b></div>
<div style="text-align: justify;">
<br />
<a name='more'></a></div>
<div style="text-align: justify;">
Me convertí en adulto y, en mi vida profesional como ingeniero, he conocido a personas (a los que podríamos llamar ingenieros), que hablan demasiado presumiendo de sus supuestos saberes, que son prepotentes, se muestran altivos, altaneros, arrogantes, vanidosos e incluso muestran desprecio hacia los demás. Es en esos momentos, cuando me viene a la mente un símil con la típica escena de película china de Kung Fu, donde el chino aprendiz que hace de "malo", cargado con ignorante arrogancia, provoca al humilde y callado maestro, que intenta por todos los medios evitar la confrontación.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Y os preguntaréis <i><b>¿Qué tiene que ver el Kung Fu con la ingeniería?</b></i>. Pues bien, originalmente el término Kung Fu se definió como una habilidad adquirida a través del tiempo, con esfuerzo, dedicación y continuidad. Por éste motivo, no es sólo exclusivo de las artes marciales, sino de cualquier actividad que se realice procurando hacerlo de la mejor manera posible. Si trasladamos éste concepto al mundo de la ingeniería, el estado de maestro en Kung Fu de Ingeniería, sólo será otorgado a aquel que tras una larga vida profesional de esfuerzo, tenacidad, persistencia y conocimientos adquiridos, llegue a conocerse a si mismo y a las limitaciones de sus saberes y experiencia. <b><i>Porque no hay maestro absoluto ni perfecto</i></b>, un maestro será aquel que por sus conocimientos y experiencia acumulada se encuentra en un nivel superior que le permite ver las cosas más claramente. Un ingeniero con Kung Fu será un auténtico líder en la sombra, mentor para los jóvenes ingenieros a su cargo y, que sabrá potenciar las habilidades personales de cada persona sin temor a que en un futuro se conviertan en mejores maestros que él.<br />
<br /></div>
<div style="text-align: justify;">
<div>
Por ello, podemos expresar en el lenguaje universal de las matemáticas, el estado de Kung Fu íntimamente ligado al ego de la persona como:</div>
<div>
$$Ego=\lim_{conocimiento \to \infty}=\frac{1}{Conocimiento}$$<br />
<br /></div>
Como decía el filósofo chino Lao Tse, <i><b>"El que habla no sabe y el que sabe no habla "</b></i>. El ingeniero que profundiza en las cosas, suele hablar con cautela y discreción, porque cuanto más se ahonda en los conocimientos, más consciente se es de las limitaciones del saber. Los que hablan a la ligera y hacen juicios precipitados sobre las personas o los asuntos, suelen hablar demasiado y sin mucho conocimiento.</div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-65517689083529681752013-10-04T18:29:00.000+02:002014-05-12T03:11:18.503+02:00Cómo diseñar una sonda pasiva de osciloscopio (parte II)<div style="text-align: justify;">
<b>Resumen</b></div>
<div style="text-align: justify;">
Como continuación a la entrada <i><a href="http://maladiets.blogspot.com.es/2012/09/oscilloscope-passive-probe-design-01.html" target="_blank">cómo diseñar una sonda pasiva de osciloscopio (parte I)</a>, e</i>n éste artículo vamos a estudiar desde el punto de vista teórico, el principio básico de una sonda pasiva de osciloscopio, cómo calcular su impedancia de entrada, consejos sobre el uso de la sonda y finalmente sentaremos las bases para la construcción de una sonda casera 10:1.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>Abstract </b></div>
<div style="text-align: justify;">
Following the previous post <i><a href="http://maladiets.blogspot.com.es/2012/09/oscilloscope-passive-probe-design-01.html" target="_blank">cómo diseñar una sonda pasiva de osciloscopio (parte I)</a>,</i> in this paper we will study from the theoretical point of view, the basic principle of passive oscilloscope probe, calculate its input impedance, probe tips to use, and finally establish the base for making a 10:1 homemade pasive probe.<br />
<br />
Link to English version <a href="http://maladiets.blogspot.com.es/2014/05/how-to-design-oscilloscope-passive-probe-02.html" target="_blank"><b>here</b></a>.</div>
<div style="text-align: center;">
<a name='more'></a><br />
----oooOOOooo----</div>
<br />
<b>5. ¿CÓMO FUNCIONA UNA SONDA PASIVA DE OSCILOSCOPIO?</b><br />
<br />
<div style="text-align: justify;">
La sonda pasiva es la más conocida y usada dentro de las sondas de propósito general. Como su nombre hace intuir, el nombre "pasiva" sugiere que está hecha de componentes pasivos como resistencias, cables, condensadores y efectivamente así es.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
En la parte I del artículo hemos visto cómo afecta la impedancia del osciloscopio junto con la capacidad de los cables, a la medida, hasta tal punto de poder hacerla irreconocible (se trataba de una sonda sin compensar). Teniendo en cuenta que el principio básico de cualquier equipo de medida es intentar medir la señal sin distorsionarla, vamos a plantear las ecuaciones teóricas que rigen el funcionamiento de compensación de una sonda pasiva y que nos permitirá mantener la integridad de la señal durante la medida.<br />
<br />
El propósito de éste apartado es analizar a nivel teórico la forma más básica de sonda pasiva de osciloscopio. El esquema que se propone es válido para sondas de propósito general en osciloscopios de gama media-baja. Para sondas pasivas de gama alta, el esquema se complica un poco ya que aparecen más componentes de ajuste para baja y alta frecuencia, que lo hace a nivel teórico un poco más confuso y queda fuera del alcance de éste artículo.<br />
<br />
Como punto de partida para el planteamiento matemático, vamos a suponer que <b>el cable de la sonda es</b> lo suficientemente corto comparado con la longitud de onda de la señal a medir. Ésta hipótesis de partida nos va a facilitar mucho el análisis además de ser cierta en la gran mayoría de los casos. De ésta forma, sólo vamos a considerar la capacidad del cable coaxial al tratarse de <b>una "linea corta"</b> y no consideraremos el modelo matemático del cable como una línea de transmisión.<br />
<br />
También hemos visto en la primera parte, que si solamente añadimos un
trozo de cable coaxial a la entrada del osciloscopio, se forma un filtro
pasa bajo que depende de la impedancia del circuito a medir.
Afortunadamente, compensar la influencia de ese polo es relativamente
fácil. Basta recordar un poco de las matemáticas que nos enseñaron
durante la carrera y añadir un cero en el circuito de medida mediante los
componentes Rp de 9 MOhm y Cp variable (véase figura 1).<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtC8XK2ZxytbAzBSv7TpAKrO8ot_2FJUaEUJb-Kn0tNi0pdnclnwycXSHcalNymV1DURRl7kcNLMTn9yJr4u45wPHvCMhnP3KtVQNst4P5vp1c-vhbv56F-_GfweRfONuqTekxQYXWFCk/s1600/Oscilloscope-pasive-probe-diagram.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="esquema de sonda pasiva de osciloscopio, passive probe schematic" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtC8XK2ZxytbAzBSv7TpAKrO8ot_2FJUaEUJb-Kn0tNi0pdnclnwycXSHcalNymV1DURRl7kcNLMTn9yJr4u45wPHvCMhnP3KtVQNst4P5vp1c-vhbv56F-_GfweRfONuqTekxQYXWFCk/s1600/Oscilloscope-pasive-probe-diagram.png" height="249" title="Esquema sonda pasiva" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 1- Esquema de sonda pasiva de osciloscopio</td></tr>
</tbody></table>
Asumimos como Ct la capacidad total del cable más la entrada del osciloscopio<br />
$$C_{t}=C{_{s}}+C{_{c}}$$</div>
<div style="text-align: justify;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh61DXZBllVjpspTcNlxw5pO0iVptZbgq2HKAQUIWS0CuCbkE8OaQ3zQPclBM6JGytKIIAEXq7MHDQ31ydyUt9FV62zHA_dnjLt-a5YBvsC7ziTsNrX1r41VhlOIlVrQSVoGJArk2HHyIM/s1600/Equivalent-oscilloscope-pasive-probe-diagram.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="circuito equivalente sonda pasiva de osciloscopio, passive probe equivalent circuit" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh61DXZBllVjpspTcNlxw5pO0iVptZbgq2HKAQUIWS0CuCbkE8OaQ3zQPclBM6JGytKIIAEXq7MHDQ31ydyUt9FV62zHA_dnjLt-a5YBvsC7ziTsNrX1r41VhlOIlVrQSVoGJArk2HHyIM/s1600/Equivalent-oscilloscope-pasive-probe-diagram.png" height="212" title="Circuito equivalente sonda osciloscopio" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 2- Circuito equivalente</td></tr>
</tbody></table>
</div>
<div style="text-align: justify;">
Y la relación entre la entrada y la salida de la sonda<br />
$$Z_{1}=\frac{R_{p}}{1+C_{p}\cdot R_{p}\cdot S}\ \ \ \ (1)$$<br />
con<br />
$$Z_{1}=R_{p}//C_{p}\ \ \ \ (2)$$<br />
$$Z_{2}=C_{t}//R_{s}\ \ \ \ (3)$$<br />
Desarrollando términos se obtienen las impedancias Z1 y Z2 como<br />
$$Z_{1}=\frac{R_{p}}{1+C_{p}\cdot R_{p}\cdot S}\ \ \ \ (4)$$<br />
$$Z_{2}=\frac{R_{s}}{1+C_{t}\cdot R_{s}\cdot S}\ \ \ \ (5)$$<br />
Sustituyendo las ecuaciones (4), (5) en (1) y simplificando se obtiene la ecuación de transferencia de la sonda:<br />
$$\frac{U_{s}}{U_{e}}=\frac{R_{s}(1+C_{p}R_{p}S)}{R_{p}(1+C_{t}R_{s}S)+R_{s}(1+C_{p}R_{p}S)}\ \ \ \ (6)$$<br />
De la ecuación de transferencia se observa que a priori tenemos un cero y un polo, que en principio no tienen ninguna relación entre ellos, pero si somos capaces de hacer que se cumpla la relación (7), se puede simplificar la función de transferencia a (8)</div>
<div style="text-align: justify;">
$$C_{p}\cdot R_{p}=C_{t}\cdot R_{s}\ \ \ \ (7)$$</div>
<div style="text-align: justify;">
$$\frac{U_{s}}{U_{e}}=\frac{R_{s}}{R_{s}+R_{p}}=\frac{1}{10}\ \ \ \ (8)$$ </div>
<div style="text-align: justify;">
Por lo tanto, para que se cumpla la relación de atenuación 10:1 es de obligado cumplimiento que la capacidad ajustable cumpla la ecuación (7). Despejando Cp de (7) obtenemos la condición a cumplir por el condensador de ajuste de la sonda</div>
<div style="text-align: justify;">
$$C_{p}=\left ( \frac{R_{s}}{R_{p}}\right )C_{t}\ \ \ \ (9)$$</div>
<br />
<b>6. ¿CUANDO USAR LA ATENUACIÓN X10?</b><br />
<br />
<div style="text-align: justify;">
Se usará la sonda en atenuación X10 cuando:</div>
<ol style="text-align: justify;">
<li><b style="font-style: italic;">Queremos disfrutar de todo el ancho de banda del osciloscopio sin limitaciones. </b>En general podemos decir que el ancho de banda queda limitado por el componente que tenga menor ancho de banda (sonda u osciloscopio).</li>
<li><i><b>La impedancia del circuito se aproxima a la impedancia del osciloscopio</b></i>. En éste caso, la inserción de la sonda hace que se forme un divisor resistivo entre la impedancia equivalente Thevenin del circuito y del osciloscopio causando que la señal mostrada se reduzca y no se muestre tal como es. Además podríamos tener problemas por exceso de drenaje de corriente de la señal.</li>
<li><i><b>La señal a medir incluye componentes de alta frecuencia o flancos abruptos que deben de ser medidos con exactitud</b></i>. En la entrada anterior, hemos visto como el hecho de medir una señal sin una sonda adecuada reduce el ancho de banda del osciloscopio. Para poder medir tiempos de subida con exactitud, debemos de asegurar un ancho de banda de al menos $$BW\geqslant \frac{0.35}{t_{r}}\ \ \ \ (10)$$$$t_{r}= signal\ rise\ time$$</li>
<li><i><b>La capacidad de entrada del osciloscopio influye en el circuito a medir. </b></i>El acoplamiento en paralelo de la capacidad del osciloscopio junto con una sonda mal compensada, puede hacer que circuitos resonantes como los utilizados en RF vean alterado su funcionamiento. Por ejemplo un oscilador de RF podría cambiar de frecuencia por el simple hecho de medir su señal.</li>
<li><b><i>El voltaje de la señal excede el rango de entrada del osciloscopio. </i></b>El rango de la etapa de entrada de un osciloscopio se limita a unos pocos voltios<b>. </b>Si tenemos que medir voltajes mayores, acoplar X10 significa que podríamos medir señales 10 veces mayores que el rango de entrada del osciloscopio.</li>
</ol>
<div>
<div style="text-align: justify;">
Para los siguientes casos, si se desea, no es necesario usar una sonda de osciloscopio compensada en opción X10. Se podrá usar en su defecto "cablecitos" u otro tipo de conexión cuando:</div>
<ol>
<li style="text-align: justify;"><i><b>No nos importa sacrificar el ancho de banda del osciloscopio. </b></i></li>
<li style="text-align: justify;"><i><b>La impedancia del circuito a medir es mucho menor que la impedancia de entrada del osciloscopio.</b></i> En este caso, el acoplamiento de la impedancia del osciloscopio al circuito a medir, causaría una leve distorsión de la señal medida que será menor cuanto menor sea la impedancia del circuito a medir y en muchos casos despreciable.</li>
<li style="text-align: justify;"><i><b> Las componentes en frecuencia de la señal son bajas o no tenemos interés por medir con exactitud los flancos de la señal. </b></i></li>
</ol>
<br />
<div style="text-align: justify;">
<b>7. ¿CÓMO AFECTA LA INSERCIÓN DE LA SONDA A LA MEDIDA?</b></div>
<div style="text-align: justify;">
Como hemos comentado, el simple hecho de medir, altera la señal original y nuestro propósito será alterar lo mínimo posible el circuito a medir. </div>
<div style="text-align: justify;">
<br />
Todos los análisis anteriores eran desde el punto de vista del osciloscopio. Nos interesaba saber cómo ve el osciloscopio la señal que tiene la sonda en la punta. Pero, ahora debemos formularnos la siguiente pregunta. <b>Desde el punto de vista de la señal, ¿cómo se ve el equipo de medida?</b>. Entre los factores a considerar cuando se conecta la sonda de osciloscopio sobre el circuito a medir, tenemos:</div>
<ul style="text-align: justify;">
<li><i><b>La capacidad parásita debido al "cablecito de masa" </b></i>y la punta de prueba". Ésta impedancia es del orden de unos pocos pF y no la vamos a considerar en éste análisis.</li>
<li><i><b>La impedancia equivalente formada por la sonda y "vista" por la señal</b></i>. En ésta caso será Z1+Z2</li>
</ul>
<div style="text-align: justify;">
Para el análisis, vamos a calcular la impedancia de entrada que presenta el osciloscopio más la sonda.</div>
<div style="text-align: justify;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNla-PD41ieFaDxIYiSySaXuSRX4Xiw4FmEwo4-YJLMaCMWOoj2GU06hKLZ1bGoSSZPXAgn5f0sbOVWuYQChfq6WeoQQU6677vUneoSBtRu2IcQCSNp0o-pJJyYHgVGxRQZtpr8t5LoY/s1600/Input-impedance-oscilloscope-probe.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="impedancia de entrada sonda pasiva de osciloscopio, input impedance oscilloscope passive probe " border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNla-PD41ieFaDxIYiSySaXuSRX4Xiw4FmEwo4-YJLMaCMWOoj2GU06hKLZ1bGoSSZPXAgn5f0sbOVWuYQChfq6WeoQQU6677vUneoSBtRu2IcQCSNp0o-pJJyYHgVGxRQZtpr8t5LoY/s1600/Input-impedance-oscilloscope-probe.png" height="256" title="Impedancia de entrada sonda" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 3- Impedancia de entrada osciloscopio más sonda</td></tr>
</tbody></table>
</div>
<div style="text-align: justify;">
La señal "ve" a la sonda como una impedancia</div>
<div style="text-align: justify;">
$$Z_{ent}=Z_{1}+Z_{2}\ \ \ \ (11)$$</div>
<div style="text-align: justify;">
Teniendo en cuenta las ecuaciones (2) y (3) podemos escribir</div>
<div style="text-align: justify;">
$$Z_{ent}=\frac{R_{p}}{(1+C_{p}R_{p}S)}+\frac{R_{s}}{(1+C_{t}R_{s}S)}\ \ \ \ (12)$$</div>
<div style="text-align: justify;">
Y si se cumple la condición de compensación (7), la impedancia de entrada queda expresada como</div>
<div style="text-align: justify;">
$$Z_{ent}=\frac{R_{p}+R_{s}}{(1+C_{t}R_{s}S)}\ \ \ \ (13)$$</div>
<div style="text-align: justify;">
La impedancia (13) correspondiente al osciloscopio más la sonda, aparece conectada en paralelo con la señal a medir, alterando por lo tanto al circuito. Dependiendo de la impedancia de salida de la señal, el hecho de conectar la sonda puede hacer que la medida quede invalidada.Cuanto menor sea la impedancia de salida de la señal, mejor será la medida.</div>
<div style="text-align: justify;">
<br />
De la ecuación (13) se deduce:</div>
<ul style="text-align: justify;">
<li>En baja frecuencia, la impedancia será en torno a 10 MOhms.$$Z_{ent}=R_{p}+R_{s}= 10 M\Omega \ ;\ Si\ 1>>C_{t}R_{s}S\ \ \ \ (14)$$</li>
</ul>
<ul style="text-align: justify;">
<li>En alta frecuencia, la impedancia será:$$Z_{ent}=\frac{R_{p}+R_{s}}{C_{t}R_{s}S} \ ;\ Si\ C_{t}R_{s}S>>1\ \ \ \ (15)$$</li>
</ul>
<div style="text-align: justify;">
Analizando en frecuencia la impedancia que "ve" la señal (Zent), tenemos un polo que aportará una caída de 20 dB/decada en</div>
<div style="text-align: justify;">
$$f_{polo}(Z_{ent})=\frac{1}{2\pi C_{t}R_{s}}\ \ \ \ (16)$$</div>
<div style="text-align: justify;">
En general, hasta una decada menos que la frecuencia de corte tenemos
una impedancia de 10 MOhm y a partir de una década más que la
frecuencia de corte tendremos la caída a 20 dB/dec. </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Obsérvese cómo <b>la impedancia de entrada varía con la frecuencia</b> partiendo de 10 MOhms y decreciendo hasta límites nada despreciables. Si hacemos un pequeño cálculo con 1.5 metros de cable coaxial (50 Ohm RG-316,capacidad 96 pF/m) obtenemos una frecuencia de corte de 1460 Hz.</div>
<div style="text-align: justify;">
$$f_{polo}(Z_{ent})=\frac{1}{2\pi\cdot (96+13)pF\cdot 1M\Omega }=1460\ Hz\ !!!\ \ \ \ (17)$$ </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Para éste caso particular, quiere decir que a frecuencias menores de 1.4 KHz la impedancia de entrada será prácticamente 10 MOhm y a frecuencias mayores empieza a decaer del orden de 20 dB por década.</div>
<div style="text-align: justify;">
Como referencia, en la siguiente figura tenemos la impedancia de entrada que muestra la sonda comercial Tektronix P2220 en función de la frecuencia. Se puede apreciar la frecuencia de corte en torno a 1.5 KHz (también se observa en la figura 5, en el cruce del desfase por 45º).</div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4ffA1jXtJIxmNc1F9QON52Hg0ggbKdhxkhl62yHbo6W1EOx75AKpY_WPFdvtWEzY6Lcm00WX4LFqmfuMUGxkfFF8eBZ1N1Z38e2erFHOaKh_IkS3I6F0musNpzyjlzEI9sDdmGCRjDkE/s1600/tektronix-P2220-input-impedance.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Impedancia entrada sonda tektronix P2220, input impedance probe tektronix P2220" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4ffA1jXtJIxmNc1F9QON52Hg0ggbKdhxkhl62yHbo6W1EOx75AKpY_WPFdvtWEzY6Lcm00WX4LFqmfuMUGxkfFF8eBZ1N1Z38e2erFHOaKh_IkS3I6F0musNpzyjlzEI9sDdmGCRjDkE/s1600/tektronix-P2220-input-impedance.png" height="326" title="P2220 Impedancia de entrada" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 4- Impedancia de entrada sonda Tektronix P2220 </td></tr>
</tbody></table>
<br />
De la figura de cambio de fase con la frecuencia vemos claramente la influencia del polo añadiendo un desfase de 90º en alta frecuencia.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS3NZahEOGIUFEmCFAdD8nOORuwUMr9uou2tw5AVD7kIIFGrzt3UUHT6vMaSR6UxRg_c4TtH3lLlrBZk6gILHA3pxZM50YrDjaJ80pgpkqvDxTwfDA3NdZ-YY4Ni-asNw5cjj1TECTskU/s1600/tektronix-P2220-phase-delay.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Diagrama de fase sonda Textronix P2220, Textronix probe P2220 input phase" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS3NZahEOGIUFEmCFAdD8nOORuwUMr9uou2tw5AVD7kIIFGrzt3UUHT6vMaSR6UxRg_c4TtH3lLlrBZk6gILHA3pxZM50YrDjaJ80pgpkqvDxTwfDA3NdZ-YY4Ni-asNw5cjj1TECTskU/s1600/tektronix-P2220-phase-delay.png" height="273" title="Fase P2220" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig. 5- Fase versus frecuencia sonda Tektronix P2220</td></tr>
</tbody></table>
<br />
<b>8. EJEMPLO PRACTICO DE DISEÑO DE UNA SONDA DE OSCILOSCOPIO</b><br />
<br />
<div style="text-align: justify;">
En esta sección vamos a poner en práctica toda la parte teórica mostrada anteriormente. Para ello, vamos a diseñar una sonda de osciloscopio casera de relación 10:1 donde podremos ver la influencia del condensador de ajuste. Como no disponía de un condensador variable, he puesto varios condensadores de valor fijo en paralelo, de forma que puedo ir añadiendo capacidad acoplándolos al circuito acorde al siguiente esquema:</div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibSNmoNbxsiRBIPQo-k1_xUDyNu-DeOVFBS3culFfOvQ4Hq7GR4_LZu_DgkO5ITpMYGcMR3UZLcYJmW7Mfp8-r0KojzUHMgqg5UKmq1G0yAfBiwhmZtodCR3C499Z3ELUcj3bDSh08aBs/s1600/Home-made-oscilloscope-pasive-probe.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Esquema de sonda pasiva casera de osciloscopio, homemade passive probe schematic" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibSNmoNbxsiRBIPQo-k1_xUDyNu-DeOVFBS3culFfOvQ4Hq7GR4_LZu_DgkO5ITpMYGcMR3UZLcYJmW7Mfp8-r0KojzUHMgqg5UKmq1G0yAfBiwhmZtodCR3C499Z3ELUcj3bDSh08aBs/s1600/Home-made-oscilloscope-pasive-probe.png" height="291" title="sonda pasiva casera" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.6- Sonda casera de osciloscopio</td></tr>
</tbody></table>
<b><br /></b>
<b>MATERIAL NECESARIO</b><br />
<ul>
<li>Cable coaxial RG-316, longitud 1.20 m</li>
<li>Condensadores SMD de valores 2.7 pF, 6.8 pF, 10 pF y 22 pF </li>
<li>Resistencia SMD de valor 9 MOhm. </li>
<li>Osciloscopio Tektronix TDS3000</li>
</ul>
<b><br /></b>
<b>CASO PRÁCTICO</b><br />
<br />
Para realizar la prueba, utilicé la salida de onda cuadrada del osciloscopio que se usa para compensar las sondas.<br />
<ul>
<li style="text-align: justify;"> <b>PRUEBA 1: </b>Tomamos como referencia la propia sonda del osciloscopio debidamente compensada. Vemos que la onda mostrada es totalmente cuadrada y ése será el objetivo a conseguir de nuestro montaje casero.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiU2_Ebk4fu5WMCr1bYFhLkZwl-4ZBfM71pIgave1J1Hgf3DUf4gyx-hKCXbeGvgEYSGKVUumRKp2zZ_RvpXdKoEvJ7-xH8-27AahnJU_jmzyyAeWe2s5NSdF0RPAq5cYRB7php30R7_Ag/s1600/TEK00000.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio compensada, compensated oscilloscope passive probe " border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiU2_Ebk4fu5WMCr1bYFhLkZwl-4ZBfM71pIgave1J1Hgf3DUf4gyx-hKCXbeGvgEYSGKVUumRKp2zZ_RvpXdKoEvJ7-xH8-27AahnJU_jmzyyAeWe2s5NSdF0RPAq5cYRB7php30R7_Ag/s1600/TEK00000.PNG" height="240" title="Sonda compensada" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.7- Señal vista con sonda Textronix</td></tr>
</tbody></table>
<ul>
<li style="text-align: justify;"><b>PRUEBA 2:</b> Utilizando el montaje casero, no se añade ninguna capacidad. Todos los interruptores están abiertos. </li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn-Nr_od46AUVn0v1ulAMDOuGLR2eVScyjmpfgzqbdJpwQc5jbq1PnHi0IaWMgyB3KClhvHaqQxgcMqzA8FILfZZ4LuPlJj3V1aeUJHcIoDwpaeqqXa3MVdoXrj621-rCluJ3-dEbPZEM/s1600/TEK00001.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio sin compensar, undercompensated oscilloscope passive probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn-Nr_od46AUVn0v1ulAMDOuGLR2eVScyjmpfgzqbdJpwQc5jbq1PnHi0IaWMgyB3KClhvHaqQxgcMqzA8FILfZZ4LuPlJj3V1aeUJHcIoDwpaeqqXa3MVdoXrj621-rCluJ3-dEbPZEM/s1600/TEK00001.PNG" height="240" title="Sonda sin compensar" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.8- Sonda con 0 pF de compensación</td></tr>
</tbody></table>
<ul>
<li style="text-align: justify;"><b>PRUEBA 3:</b> Utilizando el montaje casero, se añaden 2.7 pF.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0bzdDhNm2RufWoC4mtG3llvVJww5aSor_G-wPJXqSiJsE-GZa_8bQa5_Y-hWsuSxjfdHUlZ9hn9xLw7l-DIN5apcjo6P_bx3QXw9sHkUCuj93znDdMRLsMTpSFuWtYRGaL_f-L3qxMvU/s1600/TEK00002.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio poco compensada, undercompensated oscilloscope passive probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0bzdDhNm2RufWoC4mtG3llvVJww5aSor_G-wPJXqSiJsE-GZa_8bQa5_Y-hWsuSxjfdHUlZ9hn9xLw7l-DIN5apcjo6P_bx3QXw9sHkUCuj93znDdMRLsMTpSFuWtYRGaL_f-L3qxMvU/s1600/TEK00002.PNG" height="240" title="Sonda poco compensada" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.9- Sonda con 2.7 pF de compensación</td></tr>
</tbody></table>
<ul>
<li><b>PRUEBA 4:</b> Utilizando el montaje casero, se añaden 6.8 pF.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_vZkvLnXwDTRq-Us8w1mJ7JBx0P6H4Hy8n5WdS53COCW_7wr1iwoNVR4L9jlH8UYZiK1I-B0uzNPCScFkvQPUY8fLHBWhJmjpezp2BFZ3xlY4JloZjqqaBhxxLsUZ87TPhiUlkuLgLT0/s1600/TEK00003.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio poco compensada, undercompensated oscilloscope passive probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_vZkvLnXwDTRq-Us8w1mJ7JBx0P6H4Hy8n5WdS53COCW_7wr1iwoNVR4L9jlH8UYZiK1I-B0uzNPCScFkvQPUY8fLHBWhJmjpezp2BFZ3xlY4JloZjqqaBhxxLsUZ87TPhiUlkuLgLT0/s1600/TEK00003.PNG" height="240" title="Sonda poco compensada" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.10- Sonda con 6.8 pF de compensación</td></tr>
</tbody></table>
<ul>
<li><b>PRUEBA 5:</b> Utilizando el montaje casero, se añaden 10 pF.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMyqODRO1eh4bBujxDFQpzP9c7Jz3m6__K0bfg477maW2kBvP8yVSKTQDOuRZlF-tvyAydONg6VX_tfDBKVo1UASu-u39spksx_yp5p3vL1V0J9B4rF84LXXFePLse6QgTiZKuIV0Beq0/s1600/TEK00004.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio poco compensada, undercompensated oscilloscope passive probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMyqODRO1eh4bBujxDFQpzP9c7Jz3m6__K0bfg477maW2kBvP8yVSKTQDOuRZlF-tvyAydONg6VX_tfDBKVo1UASu-u39spksx_yp5p3vL1V0J9B4rF84LXXFePLse6QgTiZKuIV0Beq0/s1600/TEK00004.PNG" height="240" title="Sonda poco compensada" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.11- Sonda con 10 pF de compensación</td></tr>
</tbody></table>
<ul>
<li style="text-align: justify;"><b>PRUEBA 6:</b> Utilizando el montaje casero, se añaden 22 pF. Aquí se puede observar cómo nos hemos pasado compensando. El valor óptimo estará entre 10 pF y 22 pF.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_H1cHS7Rwzlx7bJe4rr1GMeWtt3IUgXNbeBInzwGzivQaS4OJ-5yzq3dv1nQpES6dgs1mxiEJ0IfFT79sc0CSXPF6Axm-Aaxj90H0yV22p7YVkQau7RV35CHpn9Lo9BOIxFQ1LS0m9n8/s1600/TEK00005.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio sobrecompensada, overcompensated oscilloscope passive probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_H1cHS7Rwzlx7bJe4rr1GMeWtt3IUgXNbeBInzwGzivQaS4OJ-5yzq3dv1nQpES6dgs1mxiEJ0IfFT79sc0CSXPF6Axm-Aaxj90H0yV22p7YVkQau7RV35CHpn9Lo9BOIxFQ1LS0m9n8/s1600/TEK00005.PNG" height="240" title="Sonda sobrecompensada" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.12- Sonda con 22 pF de compensación</td></tr>
</tbody></table>
<ul>
<li style="text-align: justify;"><b>PRUEBA 7:</b> Utilizando el montaje casero, se añaden 12 pF (10 pF//2.7 pF). Para éste valor estamos casi en el óptimo de compensación. </li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdXLYkXuJoZwjDmagTEYTfE5hd6Fhqvn456oYyxNtuuP8-yIF2rMZ9vc-0KnccwvmkjjVtJuo_MpQ_20ttuhKQdlaVjPsscoMjyy7l12Rv0e7uCKMaVWey88cbW9eKBxz0m2vHPon809s/s1600/TEK00006.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="sonda pasiva de osciloscopio ligeramente sobrecompensada, overcompensated oscilloscope passive probe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdXLYkXuJoZwjDmagTEYTfE5hd6Fhqvn456oYyxNtuuP8-yIF2rMZ9vc-0KnccwvmkjjVtJuo_MpQ_20ttuhKQdlaVjPsscoMjyy7l12Rv0e7uCKMaVWey88cbW9eKBxz0m2vHPon809s/s1600/TEK00006.PNG" height="240" title="sonda casi compensada" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fig.13- Sonda con 12 pF de compensación</td></tr>
</tbody></table>
<div style="text-align: justify;">
Dado que no disponemos de un condensador variable, el ajuste de la prueba 7, es lo mejor que
podemos conseguir. Nótese también que si hubiésemos calculado de modo
teórico el valor que necesitamos (ecuación 7), obtendríamos un valor en torno a los 14 pF<b>. </b>Valor muy aproximado al real.<b><br /></b><br />
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<b>9. CONCLUSIONES PARTE II</b></div>
<ol style="text-align: justify;">
<li><b>Maximizar el ancho de banda ajustando la capacidad de la sonda.</b><i><b> </b></i>Para mantener el máximo ancho de banda posible, se debe ajustar la capacidad variable de la sonda para que compense la del osciloscopio.</li>
<li><b>Cuidado con el intercambio de sondas entre osciloscopios. </b>Una sonda que está ajustada para un osciloscopio puede no estarlo para otro, ya que cada osciloscopio tiene su propia impedancia de entrada.</li>
<li><b>Ancho de banda de la sonda.</b> Cuando tengamos que comprar una sonda de osciloscopio, debemos asegurarnos que el ancho de banda es suficiente para el osciloscopio que tenemos. La sonda elegida influirá en el ancho de banda del conjunto.</li>
<li><b>Si se van a medir señales de alta frecuencia</b>, el análisis matemático propuesto empieza a estar comprometido así como la viabilidad de la sonda pasiva. Ante éste tipo de señales, es mejor <b>considerar </b>sondas de RF basadas en FET también llamadas <b>sondas activas</b>.<b></b></li>
<li><b>Si optamos por diseñar nuestra propia sonda</b>, se debe de prestar especial atención a la selección de componentes. Los condensadores se buscarán de respuesta plana y las resistencias serán de bajo valor inductivo para todo el ancho de banda<b>.</b></li>
<li><b>Cuando se conecta la sonda al circuito a medir, estamos añadiendo una impedancia en paralelo</b>, que dependiendo de la impedancia de salida de la señal y frecuencia, puede ser nada despreciable.<b> </b></li>
</ol>
</div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com11Madrid, España40.4167754 -3.703790199999957640.0300424 -4.3492371999999575 40.8035084 -3.0583431999999577tag:blogger.com,1999:blog-3108912614914761750.post-24958750878636925222013-07-27T00:36:00.001+02:002013-07-27T00:36:36.904+02:00Generador Tesla con pentodo GU-81M<div style="text-align: justify;">
<b>Resumen </b></div>
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Os presento a mi nueva amiga de juegos. La válvula GU-81M. En ésta entrada aprenderemos a construir un generador VTTC con el pentodo GU-81M</div>
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<b>Abstract</b></div>
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I present to you my new friend of games. The incredible vacuum valve GU-81M. In this paper we will learn how to built a VTTC generator using the GU-81M pentode.</div>
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----oooOOOooo---- </div>
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La verdad es que siempre me ha apasionado el mundo de las válvulas desde que era pequeño. No sé si es por la calidez con la que alumbran, el morbo del alto voltaje, o porque te dejan encandilado cuando las miras al igual que hacía el fuego en el hombre prehistórico. El caso es que mi primer contacto con las válvulas fue durante la infancia con los televisores de blanco y negro. Por aquella época jugaba a ser técnico de TV durante las vacaciones del verano sin mucho conocimiento de lo que hacía (pero resultaba divertido). Cambiaba válvulas que dejaban de "alumbrar", movía potenciómetros mientras miraba la pantalla... Fue entonces cuando aprendí que en ingeniería simplemente con la observación y la lógica se pueden resolver muchos problemas con muy pocos conocimientos. Luego llegarían otras épocas de radio aficionado donde continué jugando las válvulas y la RF, pero eso será otra historia.</div>
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Por casualidades del destino, he tenido la oportunidad de viajar a Ucrania y conseguir una de esas viejas glorias de la antigua Unión Sovietica. El <b>pentodo GU-81M</b> (гу-81м en ruso), una válvula capaz de generar 600 W de RF. Sí, ya sé. No es de las potentes, pero la elegí principalmente por su belleza externa como podéis ver en la foto, con la idea de usarla de decoración.</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img alt="Válvula Pentodo GU-81M, Pentode Vacuum Valve GU-81M" border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuxeEyZ_GooFRAECd79NfWtv234tpiX4gDO_8xymOs6CjpyheauhR7zxEpc-17jJ2CU9tN6SNJp9vtIGMRiC1oJyXIHam7yUkUZzi-ayKBtFLdySVkvteXCKTpvI1haTX1g73z0_kbOMY/s320/GU-81M.jpg" style="margin-left: auto; margin-right: auto;" title="Pentodo GU-81M" width="240" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Pentodo GU-81M</td></tr>
</tbody></table>
</div>
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De momento sólo le he encendido los filamentos para ver su hermosura y de hecho, me dan ganas de ponerla de lámpara en el salón como luz ambiental :) No obstante tengo en mente hacer algo un poco más espectacular con ella. Cuando saque algo de tiempo, me gustaría construir una bobina Tesla para asustar a las vecinas, y matar alguna que otra mosca despistada que se atreva a cruzar el espacio aéreo, jejeje. Sólo me falta el transformador de alta para terminar de "liarla parda".</div>
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Mientras la hago y no, en Internet encontré un <b>esquema eléctrico</b> para realizar una bobina Tesla con éste tubo de vacío (<b>VTTC</b>: Vacuum Tube Tesla Coil), que por su sencillez os remito. Básicamente se trata de un circuito resonante acoplado a un transformador de alta tensión. </div>
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<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjFjQetRzXHJGA4Um9NBbz_Mx1VgdymnxDDecguJzo8b7evRND8v4qOTeb98lGuZgrocy13AcH7JwgdH1Ox6teHaEvjAAj2ZqtkiqwWnLP2gaDXH-dcPLA7bBaThDv6djvYq2DCRNLpoA/s1600/VTTC+GU-81M.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="VTTC and pentode GU-81M, bobina tesla con tubo pentodo GU-81M" border="0" height="328" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjFjQetRzXHJGA4Um9NBbz_Mx1VgdymnxDDecguJzo8b7evRND8v4qOTeb98lGuZgrocy13AcH7JwgdH1Ox6teHaEvjAAj2ZqtkiqwWnLP2gaDXH-dcPLA7bBaThDv6djvYq2DCRNLpoA/s400/VTTC+GU-81M.jpg" title="VTTC and GU-81M" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">VTTC con GU-81M</td></tr>
</tbody></table>
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Y para que veáis lo espectacular que quedaría el diseño final....</div>
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<iframe allowfullscreen="" frameborder="0" height="360" src="https://www.youtube.com/embed/aH49xLd4ecU?feature=player_detailpage" width="640"></iframe>
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Espero que lo disfrutéis.</div>
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<b>REFERENCIAS</b></div>
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[1] - <a href="http://radiomaster.org/projects/tesla_coil/52-katushka-tesly-na-lampe-gu-81m.html" target="_blank">Bobina Tesla con lámpara GU-81M</a> </div>
JMVegahttp://www.blogger.com/profile/16367681871558780924noreply@blogger.com0España40.463667000000008 -3.749220000000036614.545170500000008 -45.057814000000036 66.3821635 37.559373999999963