Question

Please use MATLAB

Vs 2. Identify the initial conditions. Write an m-file that does the following: 3. If Vs-12V, R-280Ω, L-10mH, C-0.9μF, use matlab solve the roots of characteristic equation, find the expression of vdt), choose proper time range to plot ve What type of damping is it? 4. What is the difference in response with both open and closed loop systems. Explain the differences

Answer 1

MATLAB code is provided below in bold letters with comments for an explanation with results.

clc;
close all;
clear all;

% step1:
% the dynamic transfer function model is given below.
% deffine the RLC paratemeters
R = 280;
L = 10e-3;
C = 0.9e-6;
Vs = 12;

% define the laplace variable s
s = tf('s');

% characteristic equation is
minreal(R+1/(C*s)+L*s)

result:

0.01 s^2 + 280 s + 1.111e006 =0=> s^2 + 2800s + 1.111*10^8 = 0 is the characteristic equation .

the open loop transfer function is

Transfer function:
1.111e008
-------------------------
s^2 + 28000 s + 1.111e008

Code for obtaining the closed-loop system and the transfer function

% characteristic equation of the closed loop system
% the transfer function of the closed loop system is
T = feedback(G,H)

result:

the closed loop transfer function is
1.111e008 s
-----------------------------------------
s^3 + 28000 s^2 + 1.111e008 s + 1.111e008

The characteristic equation of the closed-loop system is given by

s^3 + 28000 s^2 + 1.111e008 s + 1.111e008 = 0.

Roots of the characteristic equation:

code:

pole(G); % poles of the system are nothing but the roots of the characteristic equation.

result: the roots are given below.

1.0e+004 *

-2.3214
-0.4786

MATLAB code to plot vc(t).

clear s G H;
syms s; % symbolic variables
G = 1/(C*s)/(R+1/(C*s)+L*s);
Vs = 12/s; % input in laplace domain
Vc = Vs*G; % in laplace domain
vc = ilaplace(Vc); % output in time domain.

% define time vector to plot vc
t = 0:1e-4:0.005;
vc = subs(vc); % substitute time in vc(t)

% now plot the time vs vc(t)
figure;
plot(t,vc,'linewidth',2);grid on;xlabel('time');ylabel('Amplitude');title('vc(t)');

result:

3 4 2 0 0 2 6 4 2 0