# RC circuit with AC - finding current

## Homework Statement

I am given an RC circuit with an alternating current. The circuit contains a capacitor and a resistor in parallel. Part (a) says "Use KCL to find a differential equation for I in terms of V." Part (b) says "For an applied voltage V = Voexp(iwt), find the current I."

ITotal = I1 + I2
Z = impedance
ZR = R
ZC = (1/wiC)
V = IZ

## The Attempt at a Solution

I understand how KCL works, but I'm not sure how to get it in a differential equation. Ignoring this and using KCL anyway, I get:

V - I1ZR = V - I1R = 0
V - I2ZC = V - I2(1/iwC) = 0
I1 = ViwC
I2 = (V/R)
ITotal = ViwC + (V/R)

Am I on the right track, or does the fact that I need a differential equation mean I need to do something completely different?

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Delphi51
Homework Helper
Looks like you are supposed to use the differential equation method (which is more general than the impedance method you started).
The current through a capacitor is related to the voltage across it by i=C*dV/dt. Apply KCL to the node where the current splits, part through R and part through C. You should have only the combined i, the common V and of course R and C in your equation. And a derivative.

I've gotten as far as:

i = C*dV/dt + V/R

Is it mathematically correct to say that V is common to both terms on the right, and therefore:

i = V(C*d/dt + 1/R) ?

If so, I would plug in the V value given in step (b):

i = Voeiwt(C*d/dt + 1/R)

How would I go about integrating this (or should I have integrated before plugging in V)?

Delphi51
Homework Helper
The first formula looks great, but you can't factor the V out of the derivative. For the (b) part you'll have to find the derivative of the V function. After that, likely you'll be able to take out a common factor because the derivative of an exponential function is the same function multiplied by some constants.

Is the i in the voltage function V = Voeiwt the current I want? If so, I've been spending MUCH longer on this than I needed to... haha

In that case, it'd just be:

V = Voeiwt

dV/di = Vo(iwt)eiwt

Delphi51
Homework Helper
dV/dt = iwVo*e^(iwt)

RIGHT that's what I meant.

Is that my answer (solving for i) or do I set that equal to I/C and solve for I?

I guess I'm getting hung up on the fact that there's an instantaneous current (i) and an average/overall current (I), and I don't know which to solve for.

Delphi51
Homework Helper
You can't solve it for i.
The answer is your i = C*dV/dt + V/R with that expression for dV/dt subbed in. Oh, and replace V with its given value. Maybe you can simplify it. Factoring out the exponential would be nice.

The average current will always be zero with AC.

Ah, that makes sense! Thank you!

Just to double check, I got for an answer:

i = Voeiwt (iwC + 1/R)

Does it matter that there's also an i in the exponent?

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Delphi51
Homework Helper
Looks good!