Matlab and RLC analysis

Question

I'm trying to plot the response of a series RLC circuit to a step function using Matlab.

I've read a bit around it, but I can't get it to work. Help would be much appreciated.

Here is my Matlab code:

solution = dsolve('square(5, 50) / ((47*10^(-3) * 1*10^(-6))) = D2y + 220/(47*10^(-3)) * Dy + 1/(47*10^(-3) * 1*10^(-6)) * y', 'y(0) = 0', 'Dy(0) = 0', 'x');
ezplot('solution');

The values of the RLC circuit are: \$R = 220\Omega\$, \$L = 47\$mH and \$C = 1\mu\$F. The equation for the capacitor voltage (which is what I'm tring to plot) is: $$v''(t) + \frac{R}{L}v'(t) + \frac{1}{LC}v(t) = \frac{v_s}{LC}$$

\$v(t)\$ is the capacitor voltage and \$v_s\$ is the source voltage (which is a \$5\$Vpp square wave).

Answer 1

You see there's a slight overshoot, if you are familiar with "damping coefficient" (or you can look it up) you'll see that tinkering with the values of L & C will vary the overshoot (or eliminate it).

You can download the PDF instead of the images if you wish to have it on your computer:

http://bit.ly/StepResponseRLC

Any further questions, feel free.

Answer 2

I am not too sure of doing it the way using differential equations. I am more familar iwth laplace transforms.

C=1/sC
L=sL
R=R

When you have the laplace equation to describe the transfer function in the form of num/den arange by the s terms

num = [a b c]
den = [d e f]

where a,b,c and are the double zero, zero and unit and d, e, and f are the double pole, pole and unit

then using the transfer function and step commands

sys=tf(num,den);
step(sys);

for a simple R,C,C circuit the following example:

R1 = 20e3;
C1 = 235e-9;
R2 = 2e3;
C2 = 22e-9;
num = [2*R2*C1 0];
den = [C1*R1*C2*R2*2 (2*C1*R1 + C2*2*R2) 2];
sys = tf(num,den);
step(sys);

alternatively with the differential equations you could put them directly into state space form and use the step command.

Answer 3

The problem is that square() isn't an analytical function, and AFAIK Matlab doesn't have such a thing. square(t,duty) is a "conventional" Matlab function that takes a vector t and outputs a vector of the same length. The 5 that you use in square(5, 50) is actually interpreted as a single item time vector and simply resolves to the integer -1 when evaluated.

>> square(5, 50)
ans =
    -1

You might be able to trick Matlab in to working with the square() function by performing some symbolic toolbox trickery, but I'd rather suggest you rewrite your problem and use a numeric solver like ode45() to plot the response for a given time period.

Additionally, to plot an analytical function named solution you should run ezplot(solution) (without the quotes)

Answer 4

you can use rlcdemo (gui)

Analyzing the Response of an RLC Circuit This demo shows how to use the Control System Toolbox(TM) functions to analyze the time and frequency responses of common RLC circuits as a function of their physical parameters.