# Analyze Series RLC Circuit

1. May 23, 2009

### lvjudge

1. The problem statement, all variables and given/known data
Analyze a series RLC circuit and determine the voltage across each element due to the transient response. The circuit is powered by a square wave voltage source with a peak-to-peak voltage of 4V and frequency of 1kHz.
R=50 kohms
L=33mH
C=1 nF

2. Relevant equations
Since this circuit turns out to be overdamped:
I(t)=A1es1*t+A2es2*t
Alpha=R/(2L)
Omega_0=1/(sqrt(L*C))
S1=-alpha+sqrt((alpha^2)-(omega_0^2))
S2=-alpha-sqrt((alpha^2)-(omega_0^2))
I=V/Z
X_L=2*pi*f*L
X_C=1/(2*pi*f*C)

3. The attempt at a solution
I have tried to solve this the best I can, but the results do not seem to come out right. I did an experiment using the circuit, so I'm comparing the calculated results to the experimental values and I am fairly confident in the experiment. I've summarized my work below, please let me know where I am going wrong.
V(0)=-2V
I(0)=-2/Z=-2/sqrt(((X_L-X_C)^2)+(R^2))=-1.20029E-5
since I(0)=A1+A2, A1=I(0)-A2
at T=0, di/dt=0
di/dt=s1A1+s2A2==>s2A2+s1(I(0)-A2)
A2=-(S1*I0))/(S2-S1)
solving for A1 and A2 give
I(t)=1.2169E-5e^(s1t)-1.65E-7e^(s2t)
Vr(t)=I(t)*R
Vc(t)=I(t)*X_C
Vl(t)=I(t)*X_L