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Thread: Circuit Differential Equation

  1. #1
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    Circuit Differential Equation

    Hi all,

    My friend and I have been trying to solve this differential equation, however we did not manage to get very far! The questions states that we need to:

    Find the currents $I_1$ and $I_2$ in the circuits shown when $R = 2.5$ Ohms, $L=1$ Henry, $C= 0.04$ Farad, and $E(t) = 169sin(t)$ Volts. Assume that there are no currents in the circuits at the initial instant of time, i.e. $I_1(0) = 0$ and $I_2(0) = 0$.

    Circuit Differential Equation-fig41.png

    Am I correct in assuming that we need to use Kirchoff's second law to begin, or am I on the wrong track to begin with? And if so, would i go about finding what each of the voltages are of each component with respect to current? Sorry if that part did not make sense, I do not have much experience at all with electrical circuits at all.

    We did a question in class which was a series circuit but this parallel circuit has thrown me right out!

    Thanks in advance!

    Cotty
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  2. #2
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    Re: Circuit Differential Equation

    Your source voltage is sinusoidal so you can just treat this as an ordinary DC circuit problem using the complex impedance of your circuit elements.

    $\begin{pmatrix}Z_R + Z_L &-Z_R \\ -Z_R & Z_R+Z_C\end{pmatrix}\begin{pmatrix}I_1 \\ I_2\end{pmatrix}=\begin{pmatrix}E \\ 0\end{pmatrix}$

    You should have learned what the complex impedances are for circuit elements.

    $Z_R=R$

    $Z_C=\dfrac 1 {\jmath \omega C}$

    $Z_L=\jmath \omega L$

    You should be able to finish from here.
    Thanks from topsquark and Cotty
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  3. #3
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    Re: Circuit Differential Equation

    Hi romsek, thanks for getting back to me! Thank you for taking the time to write this out for me, however I cannot remember seeing anything like this is my classes! We have been dealing with differential equations, and from what I can gather I think the background knowledge related to circuits (for questions like this) is assumed knowledge. Are we able to set this up in the form of an ODE? We dealt with a simple series circuit before, and ended up coming out with a differential equation of:

    $\frac{d^2I}{dt^2}+\frac{R}{L}\frac{dI}{dt}+\frac{ 1}{LC} = 0$

    By using Kirchoff's Voltage Law.

    As i said before though, thank you for taking the time to type that up It seems you help me quite a bit in these forums!!
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  4. #4
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    Re: Circuit Differential Equation

    Quote Originally Posted by Cotty View Post
    Hi romsek, thanks for getting back to me! Thank you for taking the time to write this out for me, however I cannot remember seeing anything like this is my classes! We have been dealing with differential equations, and from what I can gather I think the background knowledge related to circuits (for questions like this) is assumed knowledge. Are we able to set this up in the form of an ODE? We dealt with a simple series circuit before, and ended up coming out with a differential equation of:

    $\frac{d^2I}{dt^2}+\frac{R}{L}\frac{dI}{dt}+\frac{ 1}{LC} = 0$

    By using Kirchoff's Voltage Law.

    As i said before though, thank you for taking the time to type that up It seems you help me quite a bit in these forums!!
    ok, we'll start from the beginning. You need two bits of info to attack this problem

    1) kirchoff's circuit law for voltage - i.e. the sum of the voltages around a circuit loop is 0.
    2) the relations between voltage and current for the 3 circuit elements

    Looking at (2) first we have

    a) $V(t)=I(t) R$

    b) $V(t)=\dfrac 1 C \int I(\tau)~d\tau$

    c) $V(t) = L \dfrac{dI}{dt}(t)=L I^\prime(t)$

    Now looking at (1) for the first loop we have

    $E(t)=I_1(t)R + L I_1^\prime(t) - I_2(t) R$

    for the 2nd

    $0=I_2(t) R + \dfrac 1 C \int I_2(\tau)~d\tau - I_1(t) R$

    You can differentiate eq 2 to obtain

    $0=I_2^\prime R + \dfrac 1 C I_2(t) - I_1^\prime(t)$

    see if you can take it from here.
    Thanks from Cotty
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  5. #5
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    Re: Circuit Differential Equation

    So I have had a few different attempts so far and then had a sleep on it but haven't gotten very far. Just to clarify, should I be trying to get 1 of these equations in terms of only $I_1$ or $I_2$ + their derivatives?
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