hacer la inversa y la transformada de laplace en matlab

UTILIZACIÓN DEL MATLAB PARA HALLAR LA TRANSFORMADA DE LAPLACE.

CRISTIAN SUAREZ VANEGAS CRISTIAN CAMILO GARCIA NAYIB SESIN ESPELETA

PROGRAMA DE INGENIERIA ELECTRONICA UNIVERSIDAD DE IBAGUE SEPTIEMBRE 10 DE 2012 IBAGUE TOLIMA 1

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UTILIZACIÓN DEL MATLAB PARA HALLAR LA TRANSFORMADA DE LAPLACE.

PROFESOR: MSc. Ing. RICARDO ENRIQUE TRONCOSO

LABORATORIO #3 DE CIRCUITOS CIRCUITOS IV

CRISTIAN SUAREZ VANEGAS CRISTIAN CAMILO GARCIA NAYIB SESIN ESPELETA

PROGRAMA DE INGENIERIA ELECTRONICA E LECTRONICA

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TABLA DE CONTENIDO

CONTENIDO

PAG

INTRODUCCIÓN/ ABSTRACT …………......................................... …………............................................................... ............................4 ......4 OBETIVOS................................... OBETIVOS......................................................... ............................................ ............................................ ........................................5 ..................5 ANEXO..................................... ANEXO........................................................... ............................................ ............................................ ............................................ ......................66 MARCO TEORICO……………………………………………………………………. TEORICO……………………………………………………………………..7 .7 PRACTICA……….................................. PRACTICA………............ ............................................ .......................................... .................... ...............................9

DESARROLLO DE LA PRÁCTICA.................................... PRÁCTICA........................................................... .........................................10 ..................10 CONCLUSIONES........................... CONCLUSIONES................................................. ............................................ ............................................ ....................................25 ..............25 BIBLIOGRAFIA................................... BIBLIOGRAFIA......................................................... ............................................ ............................................ ..............................26 ........26

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INTRODUCCION

Matlab es un programa interactivo de cálculo numérico y de visualización de datos basado en software de matrices, en un entorno de desarrollo totalmente integrado y orientado a proyectos que requieren un elevado cálculo numérico y visualización gráfica. En las universidades Matlab se ha convertido en una herramienta básica tanto para estudiantes, como para docentes e investigadores por su amplio amplio abanico de programas especializados llamados Toolboxes que cubren casi todas las áreas del conocimiento, como por ejemplo las utilizadas en este trabajo para resolver ecuaciones diferenciales con ayuda de la transformada de la place y los comandos laplace para hacer la transformada de la place y ilaplace para hacer la inversa de la place

ABSTRAC Matlab is an interactive program for numerical computation and data visualization software based on matrices, in a fully integrated development environment-oriented projects that require a large lar ge numerical and graphical display. In Matlab universities has become a basic tool for both students and for teachers and researchers for its wide range of specialized programs called Toolboxes covering almost all areas of knowledge, such as those used in this work to solve differential equations using the transform of the place and commands laplace transform to the place and ilaplace to the inverse of the place

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OBJETIVOS



Actualizar nuestros conocimientos frente el uso de MATLAB y sus diferentes Symbolic Math Toolbox para resolver ecuaciones diferenciales de manera fácil u rápida



Resolución de ecuaciones diferenciales con ayuda de la transformada de la place, introducción de la función de transferencia y análisis de estabilidad a partir de ella.



Solución de ecuaciones diferenciales modelos matemáticos de circuitos eléctricos..

lineales

que

podrían

ser

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ANEXO

Anexo a este trabajo se encuentran los respectivos archivos archivos .m que genera Matlab para que pueda revisar nuestros códigos de las soluciones de las ecuaciones diferenciales y ver cada una de las graficas de estas mismas, expuestas en este trabajo mas adelante. Cada archivo esta enumerado respectivamente respecti vamente con su punto correspondiente a la guía de trabajo dejada por el profesor

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MARCO TEORICO TRANSFORMADA DE LA PLACE La transformada de Laplace de una función f (t ) definida (en ecuaciones diferenciales, o en análisis matemático o en análisis funcional) para todos los números positivos t ≥ 0, es la función F (s), definida por:

siempre y cuando la integral esté definida. def inida. Cuando f (t ) no es una función, sino una distribución con una singularidad en 0, la definición es

Cuando se habla de la transformada de Laplace, generalmente se refiere a la l a versión unilateral. También existe la transformada de Laplace bilateral, que se define como sigue:

La transformada de Laplace F (s) típicamente existe para todos los números reales s > a, donde a es una constante que depende del comportamiento de crecimiento de  f (t ). ). operador de la transformada de Laplace. es llamado el operador de

TRANSFORMADA INVERSA DE LA PLACE En matemática, la transformada inversa de Laplace de una función F(s) es la función f(t) que cumple con la propiedad donde

es la transformada de Laplace.

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PRIMER TEOREMA DE TRASLACIÓN.

Este primer teorema de traslación trasla ción se conoce también con el nombre de primer teorema te orema de desplazamiento Si se considera a s una variable real, entonces la gráfica de F (s – a) es la gráfica de F(s) desplazada en el eje s por la cantidad , tal como se muestra en la figura 7.11. Para dar énfasis a esta traslación en el eje s, a veces es útil usar el simbolismo siguiente: a

Donde s  s  a significa que la transformada de Laplace F(s) de f(t) el símbolo s se remplaza por s-a siempre que aparezca.

SEGUNDO TEOREMA DE TRASLACIÓN.

Este segundo teorema de traslación se conoce también con el nombre de segundo teorema de desplazamiento En el teorema anterior se puede observar que un múltiplo exponencial de  f(t) da como resultado una traslación de la transformada F(s) en el eje s. Como una consecuencia del segundo teorema se nota que siempre que F(s) se multiplique por una función  , 0 exponencial , la transformada inversa del producto e F  ( s ) es la función f desplazada a lo largo del eje , tal como se muestra en la figura 7.16 (b) e

 as

a



as

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PRACTICA

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DESARROLLO DE LA PRÁCTICA

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    [ ]                                                                                                                  ⁄                      ⁄            ⁄                 Remplazando condiciones iniciales y dejando a un lado de la igualdad todos los elementos de Y(s) obtenemos:

Sacando factor común de y(s) a un lado de la igualdad y al otro factorizando términos semejantes obtenemos que:

Ahora despejamos Y(s) pasando el polinomio al otro lado de la igualdad y obtenemos: Ahora vamos a hallas las constantes de los 3 polinomios

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       ⁄                                 ⁄                                                             ⁄               ⁄        (     )

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close all all; ; clc; numerador1 = [2 5 1]; denominador1 denominador1 = [1 3 2 0 ]; [residuos, polos, ganancia] = residue(numerador1, denominador1) numerador2 = [0 0 1]; denominador2 denominador2 = [1 3 2 0 0]; [residuos, polos, ganancia] = residue(numerador2, denominador2) numerador3 = [2 0 1]; denominador3 denominador3 = [1 3 2 0 0]; [residuos, polos, ganancia] = residue(numerador3, residue(numerador3, denominador3)

Matlab nos muestra en pantalla los siguientes resultados residuos = -0.5000 2.0000 0.5000 residuos = -0.2500 1.0000 -0.7500

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CONCLUSIONES

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