# RC circuit with AC - finding current

• Akhilleus
In summary, the conversation discusses solving a problem involving an RC circuit with an alternating current and finding a differential equation for the current in terms of the applied voltage using KCL. The conversation also covers the use of the differential equation method and the impedance method, as well as the steps for solving the problem and verifying the solution.
Akhilleus

## 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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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)?

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

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.

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?

Last edited:
Looks good!

## 1. What is an RC circuit?

An RC circuit is a circuit that contains a resistor (R) and a capacitor (C) connected in series or in parallel. This type of circuit is commonly used in electronic devices to control the flow of current and store electrical energy.

## 2. What is the purpose of an RC circuit?

The purpose of an RC circuit is to filter or modify the current flowing in a circuit. This can include smoothing out signals, blocking certain frequencies, or delaying the flow of current.

## 3. How does an RC circuit behave with AC current?

When an RC circuit is connected to an AC power source, the capacitor will alternately charge and discharge as the polarity of the current changes. This creates a lag in the current flow, which can affect the overall behavior of the circuit.

## 4. How do you find the current in an RC circuit with AC?

To find the current in an RC circuit with AC, you will need to use Ohm's Law and the equations for capacitive reactance (Xc) and impedance (Z). First, calculate the total impedance of the circuit using Z = √(R^2 + Xc^2). Then, use Ohm's Law (I = V/Z) to find the current (I) where V is the voltage of the AC power source.

## 5. What factors can affect the current in an RC circuit with AC?

The current in an RC circuit with AC can be affected by the values of the resistor and capacitor, the frequency of the AC power source, and the phase relationship between the voltage and current. Additionally, any other components in the circuit can also affect the current flow.

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