12.1
Time-Domain Response of Second-Order Circuits:
In previous work circuits had been limited to one energy storage element, which result in first order differential equations. Now, a second independent energy storage element will be added to result in second order differential equations:
Example: a) Find the differential equation for the circuit below in terms of vc
b) Find the differential equation for the circuit below in terms of iL
Finding the Complete Solution for Second-Order Systems:
The method for determining the forced solution is the same for both first and second order circuits. New aspects in solving a second order circuit involve new forms of natural solutions that must be determined and two independent initial conditions that must be found to resolve the unknown coefficients.
Finding the Natural Solution for second-order systems
1) Find characteristic equation of homogeneous equation (via Laplace Transform)
Convert to polynomial by the following substitution:
2) Based on the roots of the characteristic equation, the natural solution will take on one of three particular forms. Roots given by:
a) If roots are real and distinct (), natural solution becomes:
b) If roots are real and repeated (), natural solution becomes:
c) If roots are complex (), natural solution becomes:
or
Find the step response for vc and iLfor the circuit below:
when
a) R=16, L=2H, C=1/24 F
b) R=10, L=1/4H, C=1/100 F
c) R=2, L=1/3H, C=1/6 F
Show:
a)
b)
c)
% This script will plot the step response of voltage and current in
% the 3 cases of the series RLC circuit example in unit 12 lecture notes
t = [0:.01:2.5]; % Create time axis (at least 5 times the largest
% time constant in length
% Compute Capacitor voltage expressions
vca = (1+0.5*exp(-6*t)-1.5*exp(-2*t));
vcb = (1-exp(-20*t)-20*t.*exp(-20*t));
vcc =(1-exp(-3*t).*(cos(3*t)+sin(3*t)));
% Compute Inductor current expression
ila = (1/24)*(3*exp(-2*t)-3*exp(-6*t));
ilb = 4*t.*exp(-20*t);
ilc = exp(-3*t).*sin(3*t);
% Plot and label voltages and currents
figure(1)
plot(t,vca)
xlabel('seconds')
ylabel('volts')
title('Part a, Capacitor Voltage')
figure(2)
plot(t,ila)
xlabel('seconds')
ylabel('amps')
title('Part a, Inductor Current')
...... continue similar statement for parts b and c