4 febrero 2019

Leyes de Kirchhoff

Entendemos por resuelto un circuito cuando se conocen todas las tensiones en bornes de todos los elementos y las corrientes que los atraviesan. Para ello la ley de Ohm, que es una ecuación de gran importancia, muchas veces es insuficiente y necesitamos ecuaciones adicionales.

Las leyes de Kirchhoff, encunciadas en 1848 por Gustav Robert Kirchhoff (en la imagen de la izquierda), nos proporcionan estas ecuaciones adicionales, pero para ello primero necesitamos definir dos conceptos nuevos: nodo y lazo (o camino cerrado).

Un nodo es un punto donde se juntan dos o más elementos de circuito. Para definir un lazo (o camino cerrado) seleccionamos un nodo arbitrariamente, trazamos un camino cerrado en el circuito que pase a través de una serie de elementos básicos seleccionados y vuelva hasta el nodo original, sin pasar a través de ningún nodo intermedio más de una vez.

La primera ley de Kirchhoff, también denominada ley de Kirchhoff de las corrientes, enuncia que la suma algebraica de todas las corrientes entrantes existentes en un nodo es igual a la suma algebraica de todas las corrientes salientes existentes en el mismo nodo. Es decir, se basa en la conservación de la carga eléctrica.

La segunda ley de Kirchhoff, también denominada ley de Kirchhoff de las tensiones, anuncia que la suma algebraica de las tensiones a lo largo de un lazo (o camino cerrado) es igual a cero.

A continuación vemos un ejemplo:

En el circuito de arriba podemos observar que los puntos A, B, C, D, E, F, G y H son nodos. Por lo tanto, en cada uno de ellos se cumple la primera ley de Kirchhoff. Para ello, primero asignamos un sentido a cada corriente, y a continuación planteamos las ecuaciones:

  • en el punto A: 0 = i_1 + i_3
    porque no entra ninguna corriente pero salen las corrientes i1 e i3
    nota: al unirse solo dos elementos, la corriente no cambia, como ocurre en los puntos C, F y H donde ya aprovechamos este resultado
  • en el punto B:i_1 + i_2 = i_4
  • en el punto C:i_2 = i_2
  • en el punto D:i_3 + i_5 = i_6
  • en el punto E: i_4 = i_5 + i_7
  • en el punto F:i_6 = i_6
  • en el punto G:i_6 + i_7 = i_2
  • en el punto H: i_2 = i_2

De modo que obtenemos el siguiente sistema de ecuaciones:

\begin{alignedat}{3} &i_1+ &i_3 = &0 \\ &i_1+ &i_2 = &i_4 \\ &i_3+ &i_5 = &i_6 \\ &i_4+ &i_5 = &i_7 \\ &i_6+ &i_7 = &i_2 \end{alignedat}

Observamos que tenemos 7 incógnitas (i1, i2, i3, i4, i5, i6 e i7) pero solo 5 ecuaciones. Por lo cual necesitamos al menos 2 ecuaciones adicionales. Por lo tanto, usaremos la segunda ley de Kirchhoff. Para ello, definimos 3 lazos en nuestro circuito:

En el circuito podemos observar 3 lazos: el lazo número 1 que se limita por los puntos A-B-E-D-A, el lazo número 2 que se limita por los puntos D-E-G-F-D y el lazo números 3 que se limita por los puntos B-C-H-G-E-B. Planteamos las ecuaciones asignando signo negativo a los incrementos de tensión y signo positivo a las caídas de tensión respecto al sentido de recorrido indicado por las flechas de los lazos (hacerlo al revés es equivalente):

\begin{alignedat}{6} &1:     &(+500Ω·i_4) + &(-1V) + &(-600Ω·i_3) = &0 \\ &2:     &(+1V) + &(+500Ω·i_7) + &(-400Ω·i_6) = &0 \\ &3:     &(+500Ω·i_4) + &(+500Ω·i_7) + &(-10V) = &0 \end{alignedat}

Resolviendo el sistema de ecuaciones obtenemos:

\begin{alignedat}{4} &i_1 = &- \frac {41} {4900} &A \backsimeq &-8.367mA \\ &i_2 = & \frac {100} {4900} &A\backsimeq &20.408mA\\ &i_3 = & \frac {41} {4900} &A\backsimeq &8.367mA\\ &i_4 = & \frac {59} {4900} &A\backsimeq &12.040mA\\ &i_5 = & \frac {20} {4900} &A\backsimeq &4.082mA\\ &i_6 = & \frac {61} {4900} &A\backsimeq &12.449mA\\ &i_7 = & \frac {39} {4900} &A\backsimeq &7.959mA \end{alignedat}

El signo negativo indica que escogimos el opuesto al sentido de la corriente eléctrica.

Bibliografía

  • NILSSON, J. W. y RIEDEL, S. A. (2005). Circuitos eléctricos. Madrid: Pearson Educación, S.A.

Etiquetas: ,

Publicado el 2019-02-04 por Nándor Suba en la categoría "Elementos de circuito

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