RC circuit with AC - finding current

  • Thread starter Akhilleus
  • Start date
  • #1
9
0

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."



Homework Equations



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?
 
Last edited:

Answers and Replies

  • #2
Delphi51
Homework Helper
3,407
10
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.
 
  • #3
9
0
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)?
 
  • #4
Delphi51
Homework Helper
3,407
10
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.
 
  • #5
9
0
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
 
  • #6
Delphi51
Homework Helper
3,407
10
dV/dt = iwVo*e^(iwt)
 
  • #7
9
0
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.
 
  • #8
Delphi51
Homework Helper
3,407
10
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.
 
  • #9
9
0
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?
 
Last edited:
  • #10
Delphi51
Homework Helper
3,407
10
Looks good!
 
  • #11
9
0
Excellent! Thanks for your help.
 

Related Threads on RC circuit with AC - finding current

Replies
6
Views
1K
Replies
5
Views
2K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
11
Views
2K
Replies
9
Views
711
  • Last Post
Replies
4
Views
742
  • Last Post
Replies
6
Views
878
Replies
11
Views
2K
  • Last Post
Replies
3
Views
2K
Replies
1
Views
676
Top