How Do You Calculate iC(t) in an RLC Circuit and Plot Its Waveform?

[asdf12312]

Homework Statement

find iC(t) in the circuit and plot its waveform for t>=0.

Homework Equations

i(t)=i(infinity)+e^(-at)[D1cos(wdt)+D2sin(wdt)
D1= i(0)-i(infinity)
D2=i'(0)+a[i(0)-i(infinity)]

aux.:
w0=(LC)^(-1/2)
wd=sqrt(W0^2 - a^2)
s1= -a+sqrt(a^2 - w0^2)
s2= -a-sqrt(a^2 - w0^2)

The Attempt at a Solution

just a bit confused would this be a parallel RLC after all? doesn't seem like it, cause after switch moves it seems to be in series, except for current srouce. then I would calculate R(eq) by adding those two resistance? 0.1+0.52=0.62ohm?? i also get iL(0-)=iL(0)=4mA and iC(0)=0 because no current through open circuit?

 

[gneill]

You could change the 4mA current source and its parallel 0.1 Ω resistor into its Thevenin equivalent. Then you'd definitely have a series circuit for t > 0.

 

[asdf12312]

i used source transformation and brought the 0.1ohm on top, and got an 0.4mV source. then i divided by the R(eq) of the two 0.1ohm resistors in series to get iL(0)=iL(0-)=2mA. then Vc(0) would be the voltage across the 0.1ohm resistors, 0.2mV. since iC(0)=iL(0) then using equaiton I=C(dv/dt) i get v'C(0)= iL(0)/C= 2mA*1.8=3.6mV. also, vC(infinity)=V(s)=0.4mV.

a (for series)= R/2L= 0.62ohm/(2*0.05)=6.2 np/s
w0 = LC^(-1/2)= 6 rad/s

since a>w0 it would be overdamped. so using the formula i listed i get s1=−4.64 and s2=−7.76
A1= (v'(0)-s2[v(0)-v(infnity)])/ s1-s2 = (3.6mV-(-7.76)(0.2mV-0.4mV))/3.12= 0.66mV
A2= -[(v'(0)-s1[v(0)-v(infnity)])/ s1-s2] = (3.6mV-(-4.64)(0.2mV-0.4mV))/3.12= -0.86mVvC(t) =vC(infnity) +A1*e^(s1t)+A2*e^(s2t)
vC(t) = 0.4mV+0.66mV*e^(-4.64t) - 0.86mV*e^(-7.76t)

so far i did this right?

 

[gneill]

Yup. Looks good so far.

 

[rezeew]

I would first clarify the circuit setup with the person asking the question. It seems that there may be some confusion about the circuit's configuration. Once the circuit is clearly defined, I would use Kirchhoff's laws and the appropriate equations for the given RLC configuration (whether it is parallel or series) to calculate the values of the components and their effects on the current. I would also double check the given equations and make sure they are correct and applicable to the given circuit. Once I have all the necessary information, I would plot the waveform of iC(t) for t>=0, using a program or software such as MATLAB or LTspice. This will give a visual representation of the current through the capacitor over time. I would also make sure to label the axes and include any relevant units.