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Coupled RC circuit

  1. Dec 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Hello I want to solve the following circuit, I want to know how the voltage on R1 and R2 develop with time,
    It start at t=0 when there is no voltage in the circuit ( all V=0). Its a symbolic question, I need to show the equation.
    V1(t) = ? ,
    V2(t) = ?
    The current is a DC

    2. Relevant equations


    3. The attempt at a solution

    Well, noting much.. I just wrote it in a better way:

    VR1 = VC1
    VR1 + VRc + VC2 = 0
    VR2 = VC2
    IR1 + IC1 + VRC = Iinj
    IRc = IR2 + VC2
    [tex] V_1 = V_{C1} = V_{R1} \\ V_2 = V_{R2} = V_{C2} [/tex]

    [tex] \frac{V_1}{R_1} + \frac{dV_1}{dt}*C_1 + \frac{V_{Rc}}{R_c} = I_{inj} \\
    \frac{V_{Rc}}{R_c} = \frac{V_2}{R_2} + C_2*\frac{dV_2}{dt}[/tex]


    [tex] \frac{dV_1}{dt}*C_1 = I_{inj} -\frac{V_{Rc}}{R_c} - \frac{V_1}{R_1} \\
    \frac{dV_2}{dt}*C_2 = \frac{V_2}{R_2} -\frac{V_{Rc}}{R_c} [/tex]
    Last edited: Dec 29, 2014
  2. jcsd
  3. Dec 29, 2014 #2


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    Gold Member

    Do you know how to do nodal analysis?

    evaluate the circuit in the frequency domain using nodal analysis.

    In case you don't know nodal analysis, simply write a KCL at node V1 and V2 and solve.
  4. Dec 29, 2014 #3


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    Staff: Mentor

    You haven't defined the question sufficiently. Do you have component values or is this strictly a symbolic problem? What are the properties of the current source? Is it a DC source or an AC source? Any initial conditions?
  5. Dec 29, 2014 #4
    Hi @gneill and @donpacino thank you for the replay, I added some info about the question.
    I also tried to make some progress.
  6. Dec 29, 2014 #5


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    Gold Member

    I would start by defining the node in the upper left as V1, the node in the upper right as V2, and the bottom node as ground. so VRc=V1-V2, V1=VR1=VC1, etc
    Do a laplace transform (will allow you to solve the system with algebra) to get to the frequency domain.

    write two kcls at the upper left and right nodes using V1,V2,Iinj,R1,R2,Rc,C1,C2
    then do some algebra and do an inverse laplace transform to get to your time domain solution.

    alternatively you can solve the second order differential equation.

    just a note: I am operating under the assumption that at t=0 the current source changes values. If it does not then you can disregard the capacitors.
  7. Dec 29, 2014 #6
    thank you donpacino but I do not know how to do laplace transform...
  8. Dec 29, 2014 #7


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    Gold Member

    ok, then do what i told you, but solve the differential equation in the time domain. you will still do a KCL, but in the time domain similar to what you did above
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