# Help with Step Response of an RC circuit

• Engineering

## Homework Statement

Derive the Voltage and Current equations for Step Response of an RC circuit.

## Homework Equations

End products will be Vc(t) = i(s)R + (Vo - i(s)R)e^(-t/RC) and i(t) = (Is - (Vo/R))e^(-t/RC)

## The Attempt at a Solution

Using KCL on a hypotetical RC circuit with a current source in parallel with a resistor, which is parallel to a capacitor, I end up with i(R)+i(c)=i(s) which converts to (V(c)/R) + C(dv/dt) = i(s). After rearranging I get (V(c)/RC) + (d(v)/d(t)) = (i(s)/C) then (d(v)/d(t)) = (i(s)/C) - (V(c)/RC). I am now stumped as to how to procede. Anybody have any ideas? Thanks!

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berkeman
Mentor

## Homework Statement

Derive the Voltage and Current equations for Step Response of an RC circuit.

## Homework Equations

End products will be Vc(t) = i(s)R + (Vo - i(s)R)e^(-t/RC) and i(t) = (Is - (Vo/R))e^(-t/RC)

## The Attempt at a Solution

Using KCL on a hypotetical RC circuit with a current source in parallel with a resistor, which is parallel to a capacitor, I end up with i(R)+i(c)=i(s) which converts to (V(c)/R) + C(dv/dt) = i(s). After rearranging I get (V(c)/RC) + (d(v)/d(t)) = (i(s)/C) then (d(v)/d(t)) = (i(s)/C) - (V(c)/RC). I am now stumped as to how to procede. Anybody have any ideas? Thanks!
What's i(s)?

I don't think you need to include any source resistance in the problem (if you use a voltage source to drive the RC with the voltage step input). Does the problem tell you to use a current source as the excitation?

The problem doesn't say to use it, but our teacher started us off using a current source in the circuit rather then a voltage source so I continued to use thru the calculations.

berkeman
Mentor
But what is i(s)? The source current Is, or some current that is a function of the complex frequency s?

I(s) is the source current.

berkeman
Mentor
I(s) is the source current.
Ok, then you sould probably write it as Is or $$I_s$$

Putting parens around it like that makes it look like "I as a function of s".

Anyway, I just re-read your first post, and you are saying that this is a parallel RC? Sorry, I'm confused now. Could you maybe post a sketch?

Using KCL on a hypotetical RC circuit with a current source in parallel with a resistor, which is parallel to a capacitor,
Usually the RC circuit would be a series RC circuit, driven by a step source.

I have attached a copy of my work including a picture of the circuit. I have been trying different methods so a good part of the attached isn't relevent.

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berkeman
Mentor
I have attached a copy of my work including a picture of the circuit. I have been trying different methods so a good part of the attached isn't relevent.
Okay. That's not the traditional RC circuit, but whatever. If that's what you are asked to solve for, then ok.

$$\frac{V_c(t)}{R} + C \frac{dV_c(t)}{dt} = I_s$$