 Sub : system response analysis using ‘MATLAB’. prepared by, chokshi abhi ( ) Guided by, Prof. Vishvjit.K.Thakar sir(Head of ECE dept.).

 “Transfer function is a property of system which give relationship between Input of system and output of system is define like this it is ratio of Laplace transform of output to the Laplace transform of input with all initial conditions are zero.  T(s)=G(s)=L{c(t)}/L{r(t)}/initial conditions=0 T(s)=G(s) =C(s)/R(s)  We can implement transfer function in MATLAB using this function T=tf([ a,b,c,… ],[a1,b1,c1,……]) Where; T=transfer function a,b,c,…=coefficients of numerator of T.F. a1,b1,c1,…=coefficient of denumerator of T.F. T=tf([num],[denum])

 EX 1.1:Imlement transfer function, using matlab  Matlab program (1) T=tf([1,2,1],[1,3,9]) Transfer function: s^2 + 2 s s^2 + 3 s + 9 (2) T=tf([1,0,2,1],[1,0,3,0,9]) Transfer function: s^3 + 2 s s^4 + 3 s^2 + 9

 From Transfer function we can get relationship between Input and output.  Transfer function is mathematical model of system which can show relationship between input and output but can not give what is a structure of system.  From Transfer function we can find poles and zeros of system which perform very important role in every response of system.  If we don’t know the output of system by applying step or impulse we can find out output response of system.  Transfer function is only applicable to Linear time invariant system not to another any type of system.  If we don’t know the transfer function of system y applying Impulse to the system then by taking Laplace transform of output we get transfer function of system.  By using Transfer function we can find out characteristic equation of system and we can find out stability of any LTI system.

 Once we have Transfer function of system then we can do any type of analysis on system but for analysis first is POLE and ZEROS of system.  If we equating numrator with Zero then roots of that equation is zeros of system at where system magnitude has zero value.  If we equating denumrator with zero so roots o that equation give poles of system at where system magnitude has infinite value.  Pole and zeros of transfer function is plotted on special Plane known specifically S-plane  T(s)=

 For Pole zero map function in matlab is pzmap(T) at where T is transfer function of system.  Ex 1.2 pzmap for  Matlab program. clc clear all close all T=tf([1,2,-3],[1,6,9]); pzmap(T)

 Time response analysis of first order system  If any system subjected to the step or impulse input so it will give according output response with time.  For giving impulse input to the system and for observing output response simultaneously by MATLAB by function “impulse(Transfer function specified variable)”.  For giving step input to the system and for observing output response simultaneously by MATLAB by function “step(Transfer function specified variable)”.  For giving ramp input to the system and for observing output response simultaneously by MATLAB by function “lsim(Transfer function specified variable)”.”But for this function we have to specify value of signal which is to be fed to the system”.  For easy to plot use “ltiview(‘plot name ’,t)”function then right click on response >characteristics>enable all ; and observe what happen.(you can find out all specification of any response automatically without calculation)

 Find impulse and step response of  Matlab program for impulse T=tf([1],[1,1]) Transfer function: 1 >>----- s + 1 >> impulse(t)  Matlab program for step input T=tf([1],[1,1]) Transfer function: s + 1 >> step(t)

 Standard form of transfer function of second order system.  Solution:  Where, , ω d =Damping frequency  ω n =natural frequency of oscillation.  ξ= Damping factor.  Step response  If 0<ξ<1 then response will be under damped;  For ξ=1 “critically damped”.  For ξ>1”over damped system”.  For ξ=0” Undamped response(oscillatory )”.

 Ex 1.3 find impulse and step response of given (under damped)system specification by matlab only.

 Ex 1.4:find Responses of given critically damped system using matlab only.

 EX 1.5:find response of given over damped system using matlab.

 EX 1.6 :Find responses of given undamped system using matlab.

 Ex.1.7:Find Root locus of given transfer function by using matlab (HINT: use function rlocus(T))

 EX1.8:Draw frequency response, phase response,nyquist,nichols,with all specification using ltiview function for given transfer function.