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Región de Convergencia para la Transformada de Laplace

Module by: Richard Baraniuk. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Region of Convergence for the Laplace Transform by Richard Baraniuk

Summary: Explica como encontrar la ROC para sistemas LTI.

Con la transformada de Laplace, el plano –s representa un conjunto de señales (exponenciales complejos). Para cualquier sistema LTI, alguna de estas señales puede causar que la salida del sistema inversa, mientras otras hacen que la salida del sistema diverja (“explote”). El conjunto de las señales que causa que la salida de los sistemas converja se encuentran en la región de convergencia (ROC).este modulo discutirá como encontrar la región de convergencia para cualquier sistema LTI continuo.

Recuerde la definición de la transformada de Laplace,

Transformada de Laplace

Hs=hte(st)d t H s t h t s t
(1)
Si consideramos un exponencial complejo causal, ht=e(at)ut h t a t u t , obtenemos la siguiente ecuación,
0e(at)e(st)d t =0e((a+s)t)d t t 0 a t s t t 0 a s t
(2)
evaluando esto obtenemos,
-1s+a(limit   t e((s+a)t)1) -1 s a t s a t 1
(3)
Nota que esta ecuación ira infinito cuando limit   t e((s+a)t) t s a t vaya al infinito. Para entender por que pasa esto, tomaremos un paso mas al usar s=σ+jω s σ ω para realizar ecuaciones como
limit   t e(jωt)e((σ+a)t) t ω t σ a t
(4)
Al reconocer que e(jωt) ω t es senosoidal, se vuelve aparente e(σat) σ a t va determinar si la ecuación explota o no. lo que encontramos es que si σ+a σ a es positivo, el exponencial va a tener una potencia negativa, lo que va a causar que esto se vaya a cero cuando tt vaya a infinito. Pero si σ+a σ a es negativa o cero, el exponencial no tendrá una potencia negativa, lo que prevendrá que vaya a cero y el sistema no va a converger. Lo que todo esto nos dice es que para una señal causar, tenemos convergencia cuando

Condición para Convergencia

Res>a s a
(5)

Aunque no pasaremos por este proceso otra vez paras señales anticausales, podríamos hacerlo. Al hacerlo, nos daríamos cuenta que la condición necesaria para convergencia es cuando

Condición Necesaria para Convergencia Anti-causal

Res<a s a
(6)

Entendiendo el ROC Gráficamente

Talvez la mejor manera del ver la región de convergencia es el ver el plano –S lo que observamos es que para un solo polo, la región de convergencia se encuentra a la derecha de las señales causales y a la izquierda de las señales anticausales.

Figura 1
(a) La ROC para una señal causal. (b) La ROC para una señal anti-causal.
Figura 1(a) (laplaceroc1.png)Figura 1(b) (laplaceroc2.png)

Después de reconocer esto, la pregunta necesaria es esta: ¿Que hacemos cuando tenemos polos múltiples? La respuesta más simple es que tenemos que tomar la intersección de todas las regiones de convergencias para cada respectivo polo.

Ejemplo 1

Encuentre Hs H s y diga la región de convergencia para ht=e(at)ut+e(bt)ut h t a t u t b t u t

Al separa esto en dos términos obtenemos que las funciones de transferencia y la respectivas regiones de convergencia de

H 1 s=1s+a  ,   Res>a    s s a H 1 s 1 s a
(7)
y
H 2 s=-1s+b  ,   Res<b    s s b H 2 s -1 s b
(8)
Combinando esto obtenemos la región de convergencia de b>Res>a b s a . Si a>b a b , podemos representar esto gráficamente. Si no, no abra una región de convergencia.

Figura 2: Región de convergencia de ht h t si a>b a b .
Figura 2 (laplaceroc3.png)

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