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Computation of Thevenin Equivalent

  1. Aug 7, 2017 #1
    Hello!
    Recently I was going through some old exams and upon encountering this problem (which seemed pretty easy) I got stuck. Exams at my university are composed of individual tasks, each having three subquestions with four plausible answers respectively. Solution sheet gives results only, so there’s no way for me to check where my attempt at solution went wrong. I ask members of this forum for help, maybe someone will see where I made a mistake. Your help is very appreciated, so thank you in advance.

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    1. The problem statement, all variables and given/known data

    Task gives a circuit with following data: ## u(t) = 60 \cos{\omega t} ##, ## R = 10 \Omega ##, ##X_C = -3j \Omega ##, ## X_{L1} = 10j \Omega ## and ##X_{L2} = 20j \Omega ##.

    First question asks for absolute value of impedance ##|Z_b|##.
    Second question asks for maximum power dissipated on the load ##Z_b##.
    Third question asks for current amplitude through load ##Z_b##.

    2. Relevant equations

    My university provides a formula sheet, so for this question I believe only ##P_max = \frac{|U_{Th}|^2}{8 \cdot R_{Th}} ## is relevant here. If anyone thinks I’m missing some equation, please mention it in reply.

    3. The attempt at a solution

    So first of all, computing Thevenin resistance (first question) was pretty simple, just use $$ R_{Th} = jX_{L2} + \frac{ (R+jX_{L1}) (jX_C) }{(R+jX_{L1}) + (jX_C)} = 0.604 + 16.577j \Omega$$ and for the final touch compute absolute value of complex number ## |R_{Th}| = 16.588 \Omega ##. Solution sheet agrees with me.

    Now for second and third question I’ve been banging my head for the past 6 hours. Correct result should be that power dissipated on the load is ##P_{max} = 45 W ## and that current amplitude through the load should be ##I_0 = 12.2 A##. My very first attempt was finding current in each loop.

    Solving system of equations ## I_1 (R + jX_{L1} + jX_C) + I_2 (jX_C) = -60 ## and ##I_2 (16.588 + jX_{L2} + jX_C) + I_1(X_C) = 0 ## yields two complex currents ##I_1 = -4.088 + 2.680i ## and ##I_2 = -0.606 – 0.118i ##. So using second current (in absolute) times previously computed resistance yields ##U_Th = 10.241 U## and ##P_{max} = 0.790 W##.
    I don’t know how to solve this task, been through videos on YouTube, visited sites like Hyper Physics (which helped me compute Thevenin impedance. Could someone please help me? At least a hint or by pointing me in the right direction.

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    By the way, I really hope you’re having a lovely Monday!
     

    Attached Files:

  2. jcsd
  3. Aug 7, 2017 #2

    cnh1995

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  4. Aug 26, 2017 #3
    At first the voltage- in rms[root mean square]- has to be 60/sqrt(2) since u(t)=60*cos(wt) 60 it is maximum value Um=sqrt(2)*U.
    Second, according to the article presented by cnh1995 the maximum power will be when Xload=-Xth and Rload=Rth.
    Iload=Vth/(Zth+Zload)
    Iload=-4.95-j7.07 A
    Iload =sqrt(4.95^2+7,07^2)= 8.63 A [RMS] or
    Iload max=sqrt(2)*8,63=12.20 maxim.
    Pload max=0.604*8.63^2=45 W
     
  5. Aug 28, 2017 #4
    Thank you so much, to the both of you! I cannot express my gratitude enough! It's been bugging me for a long time... and now as I see how easy it actually is. Thanks again!
     
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