Ok guys, so this is a slightly long question, but bear with me because I think I figured out most of it. I just need help towards the end.
So I have a circuit with a current source in parallel with a capacitor and also in parallel with a resistor (see attached image). This is what the problem states:
a) Give the differential equation between t = t0 and t = t1. During this time, the switch is open and I(t) = 0. The initial value of V(t), the voltage across the resistor, is 150 V.
b) Use a Laplace transform to solve the differential equation in part a and give the time constant.
c) If V(t) were to decay to 100 V at t = t1, calculate the time constant during this interval where t1 - t0 = .500 sec.
d) Now, consider a different interval, when 0<t<t0. During this time the switch is closed and and I(t) = I0u(t). Give the differential equation of V(t) during this interval knowing that V(0) = 100 V and I0 = 5. Write the expression of V(t) in terms of the unknown resistance R.
e) Calculate R from part d knowing that V(t) increases from 100 V to 150 V within a .2 sec time interval (i.e., t0 = .2 s).
The only equations I've used so far:
1) I = C*dVc/dt
I may also need C = Q/V? (not sure)
The Attempt at a Solution
Ok so part a using the current law I think its just C*dV(t)/dt + V(t)/R = 0. This wasn't bad.
For b using the Laplace in this scenario wasn't hard, I just plugged in the initial value from part a and I got V(t) = 150e^(-t/RC). So the time constant is RC then.
For c, this where I may be skeptical, but I think what I did makes sense. I did 150 - 100 = 50 V. So I set 50 equal to the equation in part b and I plugged in .5 sec for t in the equation. So I got the time constant, RC, equal to .455.
Parts d and e are where I'm stuck. If I(t) = I1 + I2 (currents through capacitor and resistor), then I think the differential equation should be 5u(t) = CdV(t)/dt + V(t)/R. Is this the correct equation? Because when I took the Laplace I was getting answers that seemed a bit complicated (the V(0) was ruining everything). The part that I have no idea about is how to write V(t) in terms of R. Without this I have no hope of getting e. I can't find a way to write it without also having C in there. But Im supposed to get C later on so that has to be wrong. Can anyone help?? Should I be thinking about the behavior of the circuit at steady state? What about a transfer function? I guess I just dont know what concept I should be using to get R.
Edit: Woops, Im not sure what section this should go in...sorry Im new to this.
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