Help with Step Response of an RC circuit

  • Engineering
  • Thread starter stau40
  • Start date
  • #1
stau40
37
0

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!
 

Answers and Replies

  • #2
berkeman
Mentor
63,234
14,180

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?
 
  • #3
stau40
37
0
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.
 
  • #4
berkeman
Mentor
63,234
14,180
But what is i(s)? The source current Is, or some current that is a function of the complex frequency s?
 
  • #5
stau40
37
0
I(s) is the source current.
 
  • #6
berkeman
Mentor
63,234
14,180
I(s) is the source current.

Ok, then you sould probably write it as Is or [tex]I_s[/tex]

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.
 
  • #7
stau40
37
0
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.
 

Attachments

  • HW1_2011-02-07_15.49.19.pdf
    34.2 KB · Views: 217
  • #8
berkeman
Mentor
63,234
14,180
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:

[tex]\frac{V_c(t)}{R} + C \frac{dV_c(t)}{dt} = I_s[/tex]

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.
 

Suggested for: Help with Step Response of an RC circuit

Replies
7
Views
906
  • Last Post
Replies
9
Views
532
  • Last Post
Replies
3
Views
279
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
16
Views
494
  • Last Post
Replies
9
Views
490
Replies
4
Views
716
Engineering RC integrator circuit
  • Last Post
Replies
8
Views
647
  • Last Post
Replies
4
Views
364
  • Last Post
Replies
3
Views
778
Top