Help with Step Response of an RC circuit

[stau40]

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!

 

[berkeman]

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?

 

[stau40]

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]

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

 

[stau40]

I(s) is the source current.

 

[berkeman]

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.

 

[stau40]

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.

 

[berkeman]

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.

Your first equation is correct:

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

That is a differential equation for Vc(t). To solve it, you assume a solution for Vc(t), and then differentiate that solution to get dVc(t)/dt. Plug those back into the differential equation, and solve for any constants or unknowns.

Since you were given the solution for Vc(t), go ahead and differentiate it, and plug all of that into the differential equation to show that it is a solution.