analisis de circuitos en ingenieria 5ed - hayt-kermerly
TRANSCRIPT
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lnductancia y capacitancia
Ahora se puede comenzar con la segunda parte del estudio de circuitos. E n este capítulo se introducen dos nuevos elementos de circuitos, cuyas relaciones voltaje-co- rriente están dadas en términos de la tasa de cambio de un voltaje o una corriente. Antes de comenzar este nuevo estudio, es conveniente hacer una pausa para echar un vistazo a lo que ya se ha estudiado del análisis de circuitos resistivos. Un poco de Introducción retrospección filosófica ayudará a coniprender el trabajo que sigue.
Después de seleccionar un sistema de unidades adecuado, se comenzó el estudio de los circuitos eléctricos definiendo corriente, voltaje y cinco elementos de circuitos sencillos. A las fuentes dependientes e independientes, tanto de corriente como de voltaje, se les llainó elenzentos activos inientras que al resistor lineal se le conoció como elenzento pasivo (aunque las definiciones de "activo" y "pasivo" son todavía ambiguas y necesitan precisarse más). Un elenzento activo se define ahora conio aquel que es capaz de proporcionar a algún dispositivo externo una potencia promedio mayor que cero, donde el proniedio se toma sobre un intervalo de duración infinita. Las fuentes ideales son elementos activos, y el amplificador operacional también lo es. Un elenzento pasivo, por consiguiente, se define como aquel que no puede suniinistrar una potencia proiiiedio que sea mayor que cero, en un intervalo de duración infinita. En esta última categoría está el resistor, y la energía que recibe generalmente se transforma en calor.
Cada uno de estos elementos se definió en términos de las restricciones puestas sobre su relación voltaje-corriente. En el caso de la fuente independiente de voltaje, por ejemplo, el voltaje en las terminales debe ser totalmente independiente de la corriente que circula a través de ellas. Luego se consideraron los circuitos conipuestos por diferentes partes. En general, sólo se usaron voltajes y corrientes constantes, pero se obtuvo alguna familiaridad con las técnicas analíticas básicas al tratar sólo al circuito resistivo; ahora se pueden considerar circuitos prácticos mucho más intere- santes que contienen inductancia y capacitancia, y en los cuales tanto las funciones de excitación conio las respuestas casi siempre varían con el tiempo.
Tanto el inductor, que es el oljjeto de estudio de esta sección y la siguiente, como el 4 capacitor, que se analiza más adelante en este capítulo, son elementos pasivos capaces de almacenar y entregar cantidades finitas de energía. A diferencia de una fuente ideal, estos elementos no pueden suniinistrar una cantidad ilimitada de energía o una potencia promedio finita en un intervalo de tiempo de duración infinita. El inductor
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- 1116 j SEGUNDA PARTE: EL CIRCUITO TRANSITORIO
Aunque el inductor y la inductancia se definirán desde un punto de vista estrictamente de circuitos, es decir, por medio de una ecuación corriente-voltaje, se tendrá una mejor comprensión de la definición si se hacen unos pocos comentarios acerca del desarrollo histórico del campo magnético. A principios del siglo m, el científico danés Oersted demostró que un conductor con corriente producía un campo magnético, haciendo ver que el movimiento de la aguja de una brújula se veía afectado en presencia de un conductor con corriente. Poco después, en Francia, Ampere hizo a l m a s mediciones cuidadosas que mostraron que este campo magnético estaba relacionado linealmente con la corriente que lo producía. El siguiente paso se dio unos veinte años después, cuando el experimentador inglés Michael Faraday y el inventor norteaiiiericano Joseph Henry descubrieron casi simultáneamente1 que un campo magnético variable podía inducir un voltaje en un circuito cercano. Ellos mostraron que este voltaje era proporcional a la tasa de cambio en el tiempo, de la corriente que producía el cainpo magnético. A la constante de proporcionalidad ahora se le llama inductancia y se denota por L, entonces,
di v=L- d t (1)
donde debe observarse que v e i son ambas funciones del tiempo. Cuando se quiera hacer hincapié en esto, se usarán los símbolos v( t ) e i(t).
1 En la figura 3- 1 se muestra el símbolo para el inductor, y se debe notar que se ha
usado la convención pasiva de signos, igual que como se hizo con el resistor. La unidad de inductancia es el henry2 (H), y la ecuación de definición muestra que el henry es sólo una expresión corta para un volt-segundo/ampere.
Figura 3-1 i i
Los signos de referencia para el voltaje y la corriente se + v - muestran en el símbolo para un inductor v = L dildt.
El inductor cuya inductancia está definida en (1) es un modelo matemático; es un elemento ideal que se puede usar para aproximar el comportamiento de un dispositivo real. Un inductor físico se puede hacer enrollando cierta cantidad de alambre en forma de bobina. Esto es nluy efectivo para aumentar la corriente que origina el campo magnético, y también para aumentar el "número" de circuitos vecinos en donde se puede inducir un voltaje de Faraday El resultado de este efecto doble es que la inductancia de una bobina es aproximadamente proporcional al cuadrado del número de vueltas con~pletas del conductor que la forma. Por ejemplo, un inductor, o "bobina", que tenga la forma de una hélice larga despaso muy pequeño, tiene una inductancia de,uWA/s, donde A es el área de la sección transversal, s es la longitud axial de la hélice, N es el número de vueltas del alambre, y p (mu) es una constante del material que hay dentro de la hélice llamada permeabilidad. Para el espacio libre muy aproximadamente para el aire), ,u = ,u, = 4n x lov7 H/m.
Los inductores físicos deben verse en un curso simultáneo de laboratorio. Los temas relativos al flujo magnético, permeabilidad, y los métodos para usar las características de la bobina física para calcular una inductancia adecuada para .el modelo matemático, se tratan en los cursos de física y en los de teoría de campos electromagnéticos.
1 Faraday ganó. 2 Una victoria vana.
i
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CAPhlJLO 3: INDUCTANCIA Y CAPAClTANClA 1 117 1
También es posible armar redes electrónicas que no contengan inductores pero que puedan proporcionar la relación u-i dada por (1) en sus terminales de entrada. Se verá un ejemplo de esto en la sección 6-8.
Ahora se analizará la ecuación (1) para deducir algunas de las características eléctricas de este modelo matemático. Esta ecuación muestra que el voltaje en un inductor es proporcional a la tasa de canibio (en el tiempo) de la corriente que pasa a través de él. En particular, muestra que no hay voltaje en un inductor que lleva una l
corriente constante, independientemente de la magnitud de esta corriente. De acuerdo con esto, se puede ver a un inductor como un "cortocircuito para cd .
Otro hecho que esta ecuación pone en evidencia está relacionado con una tasa de cambio infinita en la corriente del inductor, tal como la que causa un cambio abrupto en la corriente, de un valor finito a otro valor finito. Este cambio súbito o discontinuo en la corriente debe estar asociado con un voltaje infinito en el inductor. En otras palabras, si se desea producir un cambio abrupto en la corriente del inductor, se debe aplicar un voltaje infinito. Aunque teóricamente puede ser aceptable una función de excitación con voltaje infinito, nunca podrá llegar a ser parte de un fenómeno mostrado por un dispositivo real. Como pronto se verá, un cambio abrupto en la corriente del inductor requiere un cambio abrupto en su energía almacenada, y este cambio repentino en energía requiere una potencia infinita en ese instante; de nuevo, una potencia infinita no es parte del inundo físico real. Para evitar voltajes y potencias infinitas, no debe permitirse que la corriente en un inductor cambie bruscamente de un valor a otro.
Si se intenta poner en circuito abierto un inductor físico a través del cual circula una corriente finita, puede aparecer un arco en el interruptor. La energía almacenada se disipa al ionizar el aire que hay en la trayectoria del arco. Esto es útil en los sistemas de encendido de los automóviles, donde la corriente en la bobina se interrunipe por el distribuidor y el arco aparece en la bujía.
Por el momento no se considerarán circuitos que se abran bruscamente. Cabe señalar, sin embargo, que esta restricción se eliminará más adelante cuando se haga la hipótesis de la existencia de una función de excitación de voltaje o de una respuesta que se vuelva infinita instantáneamente.
La ecuación (1) también se puede interpretar (y resolver, si es necesario) por métodos gráficos.un ejemplo de esta técnica se basará en la figura 3-2a.
i ( t ) j (A) Figura "2
y-\ a) Forma de onda de la corriente en un in- ductor de 3 H. b) Forma de onda del voltaje
, , correspondiente v = 3 dildt.
-- ---m -- - t ( S )
-l l o 2 3
a)
v ( t ) f ( V )
3 j
b)
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-
/ 118 / SEGUNDA PARTE: EL ClRCUiiü TRANSiiORlO
Ejemplo 3-1 Dada la forma de onda de la corriente en un inductor de 3 H, como se muestra en la figura 3-2a, determine el voltaje del inductor y grafíquelo.
Solución: La corriente es cero antes de t = -1 S, aumenta linealmente hasta 1 A en el siguiente segundo, se queda en 1 A durante 2 S y después disminuye hasta cero en el siguiente segundo y permanece en cero después. Si esta corriente está en un inductor de 3 H. v si el voltaie v la corriente se asiman Dara satisfacer la
2 u .I u - a- - 1 --
convención pasiva de signos, entonces se puede usar la ecuación (1) para hallar la forma de onda del voltaje. Como la corriente es cero y constante para t e - 1 el voltaje es cero en este intervalo. Luego, la corriente comienza a aumentar a la razón lineal de 1 Ns, por lo que se produce un voltaje constante de 3 W Durante el siguiente intervalo de 2 S, la corriente es constante, por lo que el voltaje es cero. La disminución final de la corriente causa un voltaje negativo de 3 V y ninguna respuesta después de eso. En la figura 3-2b se bosqueja la forma de la onda del voltaje en la misma escala de tiempo. e
Ahora se investigará el efecto de un aumento y caída más rápidos de la corriente entre los valores de cero y 1 A.
Ejemplo 3-2 Encuentre el voltaje del inductor que resulta al aplicar la forma de onda
I de corriente mostrada en la figura 3-3a
Figura 3-3 i 1 o) El tiempo requerido por la corriente de la figura 3-2a ik--l
para cambiar desde O hasta 1 y desde 1 a O disminuye en un factor de 10. b ) Fprma de onda del voltaje resultante. 1 \ Nótese que el ancho'de los pulsos se ha exagerado un po- 1 11 a
1 f j t i , 2 1 3 - t (S)
co para que sea más claro.
1 Solución: Nótese que los intervalos requeridos para el aumento v caída dismi- nuyeron a 0.1 s. Entonces la derivada va a ser diez veces mayor en magnitud. Esta condición se muestra en las gráficas de corriente y voltaje de las figuras 3-3a y b. Es interesante notar, en las figuras 3-2b y 3-31, que el área bajo i d a onda del voltaje es 3 V-s. e
Una mayor disn~inución en la longitud de estos dos intervalos producirá una magnitud del voltaje proporcionalmente mayor, pero sólo dentro del intervalo donde la corriente aumenta o disminuye. Un cambio abrupto en la corriente causará los "picos" de voltaje infinito (cada una con un área de 3 <S) sugeridos en las figuras 3-4a y b, o desde el igualmente válido pero opuesto punto de vista, estos picos de voltaje
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