RLC Circuit

· Consider the RLC circuit below.

· The Laplace Transfer function is given by

Multiplying by s/s we get

· Enter the transfer function polynomials to create a MATLAB representation of the system

numerator=[1,0,1e6];

denominator=[1,1000,1e6];

sys = tf(numerator, denominator)

· You can now use MATLAB to compute and display many characteristics of the system

pzmap(sys) % pole zer plot (roots of numerator and demoninator)

freqs(numerator,denominator) % frequency response in rads/second

impulse(sys) % impulse repsone

step(sys) % step response

· To interactively create any number of plots use the following command. Go to Edit -> Plot Configurations to create various plots and combinations of plots.

ltiview(sys)

· Now let us create a signal to process with the system. Any data array can be used, but the following command is a simple way to generate standard periodic functions. This will generate a sin wave of period 1/160 (160Hz) that goes from 0 seconds to 0.05 seconds

[x,t]=gensig('sin',1/160,.05);

· To simulate the processing of this signal, use the lsim() command as follows:

y=lsim(sys,x,t);

· Now let us display the results

figure

subplot(211)

plot(t,x)

xlabel('t (sec) ');

ylabel('x(t) (Volts) ');

title('Input');

axis([0,.05,-1,1])

subplot(212)

plot(t,y)

xlabel('t (sec) ');

ylabel('y(t) (Volts) ');

title('Output');

axis([0,.05,-1,1])

· This circuit is a band-stop filter and is designed to remove frequencies around 160Hz. Thus, the output is much smaller than the input at this frequency.

· To simulate a frequency sweep, where sinusoidal signals with a range of frequencies are applied to the system, we can use a movie.

figure % new figure

f=linspace(100,200,10); % range of frequencies to try

for k = 1 : length(f)

[x,t]=gensig('sin',1/f(k),.05); % generate the k’th sinusoid

y=lsim(sys,x,t); % simulate the output for this signal

% plot results

plot(t,x, 'b-',t,y, 'r-')

xlabel('t (sec) ');

ylabel('x(t),y(t) (Volts) ');

title('RLC Circuit Input/Output');

axis([0,.05,-1,1])

text(.01,.75,sprintf('f=%.2f',f(k)))

legend('Input', 'Output')

% take “snapshot”

M(k)=getframe;

end

% play movie

movie(M,10,5)