Question

MATLAB question. Please answer all the questions and also upload the code by MATLAB. Thanks. Down vote if no code provided.

For the circuit shown above, at the moment t = 0, the switch is closed, find w(t) for 120, No energy is stored in the capacitor and inductor at moment t-0 1. Write the dynamic model for RLC circuit after t> 0? a. Show all vour work and calculations b. Write down the characteristic equation of the transfer function in an open loop c. Write down the characteristic equation of the transfer function in a closed loop with feedback H(s)- 1/S d. What form of step response is the open loop system (overdamped, damped. ctc) Use the values on step 3 2. Identify the initial conditions Write an m-file that does the following: If Vs = 12V, R = 280Ω, L=10mH, C = 0.9μF, use matlab solve the roots of characteristic equation, find the expression of vc(t), choose proper time range to plot vc. What type of damping is it? 4. What is the difference in response with both open and closed loop systems. Explain the differences 5. Apply a PID controller on the closed loop system and write a MatLab scrip that will allow you to change the values of P, I, and D coefficients and plot the effects of this values on the step response of the system. Show all posible forms obtained when changing the values of the PID controller. When is the system damped, overdamped, etc Chose values of P starting from 0 to any value you want. Do the same for I and D and show all posible conbinations and results. Document your results 6. Plot and explain all the results

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