Experimento 3 MATLAB Analisis Frecuencia

PONTIFICIA UNIVERSIDAD JAVERIANA
FACULTAD DE INGENIERIA - DEPARTAMENTO DE ELECTRONICA

4096 CIRCUITOS EN FRECUENCIA
EXPERIMENTO 3: MATLAB ANALISIS EN EL DOMINIO DE LA FRECUENCIA
Objetivos:
1. Computar, comparar y evaluar métodos de solución (CDIO 2.1.5.1/4/5)
2. Inferir el comportamiento del circuito a partir de representaciones entrada – salida

(CDIO 2.1.3.4)
3. Describir propiedades funcionales y de comportamiento (intencionales y no

intencionales) que surgen de un sistema (CDIO 2.3.2.2)
Alcance:
 Emplear MATLAB para analizar las funciones de sistema.

 Emplear MATLAB para obtener la respuesta de sistemas lineales e invariantes
empleando funciones de sistema.
Funciones de matemática simbólica (Notas 1,2 y 3)
Experimento 3 1
Ejemplo 1.
a. Encontrar la Transformada de Laplace de:
ft = exp(a*t)*cos(w*t)
b. Encontrar la transformada inversa
Model Objects Represent Linear Systems (Nota 4)
In Control System Toolbox™ software, you represent linear systems as model objects. Model
objects are specialized data containers that encapsulate model data and other attributes in
a structured way. Model objects allow you to manipulate linear systems as single entities
rather than keeping track of multiple data vectors, matrices, or cell arrays.
Model objects can represent single-input, single-output (SISO) systems or multiple-input,

multiple-output (MIMO) systems. You can represent both continuous- and discrete-time
linear systems.
The main families of model objects are:
 Numeric Models —Basic representation of linear systems with fixed numerical

coefficients. This family also includes identified models that have coefficients
estimated with System Identification Toolbox software.
 Generalized Models —Representations that combine numeric coefficients with
tunable or uncertain coefficients. Generalized models support tasks such as
parameter studies or compensator tuning using Robust Control Toolbox tuning
commands.
About Model Data
Experimento 3 2
The data encapsulated in your model object depends on the model type you use. For
example:
 Transfer functions store the numerator and denominator coefficients.

 State-space models store the A, B, C, and D matrices that describe the dynamics
of the system.
 PID controller models store the proportional, integral, and derivative gains.
Other model attributes stored as model data include time units, names for the model inputs
or outputs, and time delays.
Types of Model Objects
The following diagram illustrates the relationships between the types of model objects in
Control System Toolbox™, Robust Control Toolbox™, and System Identification Toolbox™
software. Model types that begin with id require System Identification Toolbox software.
Model types that begin with u require Robust Control Toolbox software. All other model types
are available with Control System Toolbox software.
Numeric Models
Experimento 3 3
Numeric Linear Time Invariant (LTI) Models
Numeric LTI models are the basic numeric representation of linear systems or components
of linear systems. Use numeric LTI models for modeling dynamic components, such as
transfer functions or state-space models, whose coefficients are fixed, numeric values. You
can use numeric LTI models for linear analysis or control design tasks.
The following table summarizes the available types of numeric LTI models
Transfer Function Representations
Control System Toolbox software supports transfer functions that are continuous-time or discrete-
time, and SISO or MIMO. You can also have time delays in your transfer function representation.
Transfer Function Representations
• Control System Toolbox software supports transfer functions that are

continuous-time or discrete-time, and SISO or MIMO
• The tf model object represents transfer functions in polynomial form.
tf : Create transfer function model, convert to transfer function model
Syntax
tf
sys = tf(num,den)
sys = tf(num,den,Ts)
sys = tf(M)
sys = tf(num,den,ltisys)
tfsys = tf(sys)
tfsys = tf(sys, 'measured')
tfsys = tf(sys, 'noise')
tfsys = tf(sys, 'augmented')
The zpk model object represents transfer functions in factorized form.
zpk : Create zero-pole-gain model; convert to zero-pole-gain model
Experimento 3 4
Syntax
sys = zpk(z,p,k)
sys = zpk(z,p,k,Ts)
sys = zpk(M)
sys = zpk(z,p,k,ltisys)
s = zpk('s')
z = zpk('z',Ts)
zsys = zpk(sys)
zsys = zpk(sys, 'measured')
zsys = zpk(sys, 'noise')
zsys = zpk(sys, 'augmented'
pzplot: Pole-zero map of dynamic system model with plot customization options
Syntax:
h = pzplot(sys)
pzplot(sys1,sys2,...)
pzplot(AX,...)
pzplot(..., plotoptions)
tfdata: access transfer function data.
zpkdata: Access pole – zero gain data
Experimento 3 5
SISO Model Creation
Transfer Function Model Using Numerator and Denominator Coefficients

This example shows how to create continuous-time single-input, single-output (SISO) transfer functions from
their numerator and denominator coefficients using tf.
Create the transfer function :
num = [1 0];
den = [1 3 2];
G = tf(num,den);
num and den are the numerator and denominator polynomial coefficients in descending powers of s. For
example, den = [1 3 2]represents the denominator polynomial s2 + 3s + 2.
G is a tf model object, which is a data container for representing transfer functions in polynomial form.
Tip Alternatively, you can specify the transfer function G(s) as an expression in s:
Create a transfer function model for the variable s.
s = tf('s');
1. Specify G(s) as a ratio of polynomials in s.
G = s/(s^2 + 3*s + 2);
Transfer Function Model Using Zeros, Poles, and Gain

This example shows how to create single-input, single-output (SISO) transfer functions in factored form
using zpk.
Experimento 3 6
Create the factored transfer function
:
Z = [0];
P = [-1-1i -1+1i -2];
K = 5;
G = zpk(Z,P,K);
Z and P are the zeros and poles (the roots of the numerator and denominator, respectively). K is the gain of
the factored form. For example,G(s) has a real pole at s = –2 and a pair of complex poles at s = –1 ± i. The
vector P = [-1-1i -1+1i -2] specifies these pole locations.
G is a zpk model object, which is a data container for representing transfer functions in zero-pole-gain
(factorized) form.
Ejemplo 2.
Para la función:
50 s
----------------------------------------------
s^4 + 6 s^3 + 50 s^2 + 150 s + 625
Encontrar:
a. Polos y ceros.
b. Diagrama de polos y ceros.
c. Expansión en fracciones parciales.
Experimento 3 7
Ejemplo 3.
Crear la función de transferencia factorizada:
Experimento 3 8
Ejemplo 4
Crear la función de transferencia y graficar el diagrama de polos y ceros
𝟎, 𝟖
𝑮(𝒔) =
(𝟑𝒔 + 𝟏)(𝟏𝟎𝒔 + 𝟏)(𝟑𝟎𝒔 + 𝟏)
Experimento 3 9
Problema 1
Para el circuito de la figura 1:

𝑉𝑠
a. Evaluar la impedancia de entrada 𝑍(𝑠) = 𝐼
b. Obtener el diagrama de polos y ceros.
Problema 2
Para la función de transferencia dada:
1
𝐻(𝑠) =
𝑠4 + 2.6131𝑠 3 + 3.41412𝑠 2 + 2.6131𝑠 + 1
a. Obtener el diagrama de polos y ceros empleando MATLAB, en 2-d.

b. ¿Cuál es la respuesta impulso h(t)?. Graficar
Problema 3
La impedancia de entrada de un circuito RLC tiene el diagrama de polos y ceros:
Experimento 3 10
6
x 10 Pole-Zero Map
3
System: zin System: zin

2 Zero : -2.78e+05 + 2.57e+06i Pole : -0 + 2.58e+06i
Damping: 0.108 Damping: 0
Overshoot (%): 71.2 Overshoot (%): 100
Frequency (rad/s): 2.58e+06 Frequency (rad/s): 2.58e+06
Imaginary Axis (seconds -1)
-1
System: zin
Zero : -2.78e+05 - 2.57e+06i
Damping: 0.108
-2 Overshoot (%): 71.2
Frequency (rad/s): 2.58e+06
-3
-3 -2.5 -2 -1.5 -1 -0.5 0
-1 5
Real Axis (seconds ) x 10
La Impedancia de entrada evaluada para s = 0 es:
Z(0) = 1,8 kΩ
a. Construya un circuito que cumpla dichas condiciones.

b. Cuánto valen R, L y C. Ajuste a valores comerciales.
c. Si se tiene en cuenta la tolerancia de los componentes, ¿cuál es la máxima
variación en la ubicación de los polos?
d. Verifique su respuesta empleando MATLAB.
Notas
1. The MathWorks, Inc . MATLAB® R2011b Getting Started Guide. – COPYRIGHT

1984–2011 by The MathWorks, Inc.
2. Dr. Bouzid Aliane. Getting started with Matlab. Fifth Edition Copyright © 1996,
1997, 1998, 1999, and 2001. Disponible en:
http://doctord.dyndns.org/courses/Topics/Matlab/index.htm
3. The MathWorks, Inc. Symbolic Math Toolbox™ User’s Guide © COPYRIGHT 1993–
2009 by The MathWorks, Inc.
http://www.mathworks.com/help/symbolic/index.html
4. The MathWorks, Inc. Control System Toolbox™ Getting Started Guide ©
COPYRIGHT 2000–2010 by The MathWorks, Inc.
Experimento 3 11
5. Realización experimento: 16 de agosto 2 a 4 PM. Salón 611 de Ingeniería.

6. Entrega reporte: Semana 30 de agosto antes de las 5 PM
7. No es necesario incluir los ejemplos en el informe del experimento
8. Para los problemas se debe guardar el archivo .m empleado, y adjuntarlo en el
informe de laboratorio.
9. COPIA DETECTADA DURANTE LA CALIFICACIÓN SE SANCIONARÁ CON LA
ANULACIÓN DEL EXPERIMENTO Y LA CORRESPONDIENTE SANCIÓN
ESTABLECIDA EN EL REGLAMENTO.
Revisiones:
Revisión 1 Agosto 2012 CCB

Revisión 3 Febrero 2015 CCB
Revisión 5 Febrero 2016 CCB
Revisión 6 Agosto 2017 JBG
Experimento 3 12

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PONTIFICIA UNIVERSIDAD JAVERIANA

FACULTAD DE INGENIERIA - DEPARTAMENTO DE ELECTRONICA

EXPERIMENTO 3: MATLAB ANALISIS EN EL DOMINIO DE LA FRECUENCIA

1. Computar, comparar y evaluar métodos de solución (CDIO 2.1.5.1/4/5)

2. Inferir el comportamiento del circuito a partir de representaciones entrada – salida

3. Describir propiedades funcionales y de comportamiento (intencionales y no

 Emplear MATLAB para analizar las funciones de sistema.

Funciones de matemática simbólica (Notas 1,2 y 3)

a. Encontrar la Transformada de Laplace de:

b. Encontrar la transformada inversa

Model Objects Represent Linear Systems (Nota 4)

Model objects can represent single-input, single-output (SISO) systems or multiple-input,

The main families of model objects are:

 Numeric Models —Basic representation of linear systems with fixed numerical

About Model Data

 Transfer functions store the numerator and denominator coefficients.

Types of Model Objects

Numeric Linear Time Invariant (LTI) Models

Transfer Function Representations

Transfer Function Representations

• Control System Toolbox software supports transfer functions that are

tf : Create transfer function model, convert to transfer function model

The zpk model object represents transfer functions in factorized form.

zpk : Create zero-pole-gain model; convert to zero-pole-gain model

tfdata: access transfer function data.

zpkdata: Access pole – zero gain data

SISO Model Creation

Transfer Function Model Using Numerator and Denominator Coefficients

Create the transfer function :

G = s/(s^2 + 3*s + 2);

Transfer Function Model Using Zeros, Poles, and Gain

Create the factored transfer function

Crear la función de transferencia factorizada:

Crear la función de transferencia y graficar el diagrama de polos y ceros

Para el circuito de la figura 1:

Para la función de transferencia dada:

a. Obtener el diagrama de polos y ceros empleando MATLAB, en 2-d.

La impedancia de entrada de un circuito RLC tiene el diagrama de polos y ceros:

System: zin System: zin

La Impedancia de entrada evaluada para s = 0 es:

a. Construya un circuito que cumpla dichas condiciones.

1. The MathWorks, Inc . MATLAB® R2011b Getting Started Guide. – COPYRIGHT

5. Realización experimento: 16 de agosto 2 a 4 PM. Salón 611 de Ingeniería.

Revisión 1 Agosto 2012 CCB

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