A 68 nF capacitor has a parasitic parallel resistance Rp = 1 MΩ.
If a voltage V = Vmsin(ωt) is applied, as shown in the diagram, find the frequencies at which:
1. The current amplitude through the parasitic resistance is 1% of the current amplitude
through the capacitor.
2. The amplitudes of the currents through the resistor and the capacitor are equal.
I=C(dv/dt) Through capacitor
I=V/R Through Resistance.
The Attempt at a Solution
I want to try to solve the second part of the question first because it would provide be a more fundamental solution.
In any case, I've equated the current through the capacitor with the current through the parasitic resistor:
Cdv/dt = V/Rp
dv/dt = (w)(Vm)(Cos)(wt)
Substituting in we get:
(C)(w)(Vm)(Cos)(wt) = Vmsin(wt)/(Rp).
(Rp)(C)(w)(Vm)(Cos)(wt) = Vmsin(wt) <-- just moving the resistance to the other side
At t=0. We know that sin waves have 0 amplitude, and cos waves have 1 amplitude. So setting t=0.
(Rp)(C)(w) = 0
Or in another words w=0, and frequency also =0.
Clearly that isn't correct though. What am I missing.
At the stage
Cdv/dt = V/Rp
I've tried integrating both sides with respect to dv and dt. which produces:
(C)(Rp)V = -(1/W)(V)(Cos(wt))
-(C)(Rp)(W) = 1
And we can solve for W
and thus frequency.
that produces the right answer, but I can't see why the first approach produces the answer f=0. I think the problem is to do with how I've used/understood calculus in the first attempt. Can anyone shed any light?