How Do You Determine Transient Response Voltages in a Series RLC Circuit?

[lvjudge]

Homework Statement

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

Homework Equations

Since this circuit turns out to be overdamped:
I(t)=A₁e^s1*t+A₂e^s2*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)

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

 

[LightHero]

I'm not sure what I am doing wrong, but the values are coming out much higher than expected. Any help is greatly appreciated.

 

[Mene]

I would first like to commend the effort put into solving this problem. However, there are a few areas where I believe some mistakes may have been made. Firstly, the equation for the current in a series RLC circuit is not I(t)=A1es1*t+A2es2*t, but rather I(t)=A1e^(s1t)+A2e^(s2t). This is because the current in a series circuit is the sum of the currents through each element, and each of these currents follows an exponential decay or growth function. Additionally, the value of I(0) is not -2/Z, but rather -2/R. This is because the voltage across the resistor is equal to the voltage of the source (since they are in series) and therefore the current through the resistor is simply V/R. Finally, the value of A2 should be positive, as it represents the initial current through the inductor. With these adjustments, the equations for the current and voltage across each element should be as follows:

I(t)=-0.00004e^(-alpha+sqrt((alpha^2)-(omega_0^2))t)+0.00002e^(-alpha-sqrt((alpha^2)-(omega_0^2))t)

Vr(t)=I(t)*R

Vc(t)=I(t)*X_C

Vl(t)=I(t)*X_L

I hope this helps in solving the problem and provides a more accurate analysis of the series RLC circuit.