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LC circuit

Discussion in 'Electronics Homework Help' started by Daedalus, Nov 11, 2018.

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  1. Daedalus

    Daedalus

    3
    0
    Nov 11, 2018
    I have the LC circuit of the figure:

    circuit.png

    At initial time C2 has a charge Q2, and C1 has Q1=0. I am asked to find the voltage across C2, and the current at any other time.

    So what I did was first to consider the equivalent circuit, with

    eq1.gif

    Then I have:

    eq3.gif

    So I have, using the initial condition:

    eq4.gif

    Then, to find the voltage across C2, I've used (which I'm not sure if this right):

    q(t)=Q1(t)+Q2(t), the total charge should be the sum of charges in both capacitors. And then I used that the charge should be related to the voltage as: Q1=C1V, Q2=C2V, where I am assuming that the voltage drop is the same in both capacitors (I'm not sure if that is correct). Then I have:

    eq5.gif

    So, is this correct?

    Or should I consider, instead of what I did:

    CodeCogsEqn.gif

    If I use the last equation, I get:

    CodeCogsEqn (1).gif

    I think this last result is correct, because I am not doing any ad hoc assumption. But at the same time, I find a different current that the one I obtain for the equivalent circuit. Shouldn't the current be the same across the capacitor than for the equivalent circuit?
     

    Attached Files:

    • eq2.gif
      eq2.gif
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    Last edited: Nov 11, 2018
  2. Harald Kapp

    Harald Kapp Moderator Moderator

    10,025
    2,138
    Nov 17, 2011
    Assuming switch S is closed at t=0 ?
     
  3. Daedalus

    Daedalus

    3
    0
    Nov 11, 2018
    Yes, sorry. I might have an error sign in the last equations.
     
  4. Ratch

    Ratch

    1,075
    327
    Mar 10, 2013
    Tsk, tsk, you really are lost in the wilderness. You cannot combine the two capacitors because only one cap has a charge imbalance on it. How is that going to affect the "equivalent" capacitor? So, one cap has a voltaqe across it and the other cap does not. Do you know what that voltage is? Find the current in the loop, then you can easily calculate the voltage across any of the circuit elements. Since you are asked to solve the equations symbolically, you are in for a long derivation. I am going to use Laplace transforms to simplify the path to the solution. By the way, did you check and proof the quality of your embedded images after the first one in your post? All I see is garbage.

    The loop expressed in Laplace terms is:

    Daedalus1.JPG

    Solving for Is we obtain:

    Daedalus2.JPG

    Finding the inverse we get a real time current of :

    Daedalus3.JPG

    Normalizing the circuit parameters we can simplify and plot the current.

    Daedalus4.JPG

    Daedalus5.JPG

    Daedalus6.JPG

    From the simplified equation we can see the amplitude is 1/sqrt(2) = 0.707 and the period is 2 Pi/sqrt(2) = 4.44 seconds. You can expect a sine wave for the current because there is both capacitance and inductance in the circuit. The wave will last forever because no resistance is present. Now that you know the current, can you find the voltages across each circuit element?

    Ratch
     
    Daedalus and Harald Kapp like this.
  5. Daedalus

    Daedalus

    3
    0
    Nov 11, 2018
    Hi. Thanks a lot. I did check the images, they look fine in my computer after I logged in (I'm in a different place than when I wrote the post, in both systems I use ubuntu and mozilla ff, however the forums should enable the use of latex!). I didn't expected to be so complicated! it was supposed to be an exercise for students learning a very short course in electromagnetism! I will try to avoid laplace transforms, because the student don't have that backgrounds (first year at university level).

    Thanks again!
     
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