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Maximum power transfer theorem leading to erroneous result Why?

  1. Feb 22, 2012 #1
    Hi everybody,

    Below is a circuit that I am investigating. All the components are linear, nothing fancy here.

    circ1.jpg

    My goal is to maximize the average power P dissipated in the resistor Re. To this aim, I am only allowed to play with the angular frequency ω of the voltage source and with the value of the resistor Re. All the other parameters are set to fixed values.

    First approach: from Kirschoff's law, I derive an expression for the average power P dissipated in Re. I have then to solve simultaneously dP/dRe=0 and dP/dω=0 in terms of Re and ω. The second equation does however correspond to a high order polynomial in ω (7th I think I remember) for which analytical solutions are not known.

    Second approach: I transform the initial circuit into the Thevenin equivalent shown below. I then apply the maximum power transfer theorem stating that maximum power will be dissipated in Ze when Ze=Zm*. If some conditions on the parameters other than Re and ω are fulfilled, I obtain two couples of solution in Re and ω respecting Ze=Zm*. And here comes my problem: while these couple of solutions respect dP/dRe=0 from the first approach, they are not solutions of dP/dω=0.
    circ2.jpg

    While one may suggest an error in the derivations, I checked it many times with symbolic computations software. I rather believe that the maximum power transfer theorem does not yield in the particular situation that I consider, but I do not manage in finding the reason, neither from a mathematical or physical point of view. Any help would be appreciated.
     
    Last edited: Feb 22, 2012
  2. jcsd
  3. Feb 22, 2012 #2

    Philip Wood

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    Gold Member

    The maximum power theorem deals gives the optimum load impedance (Ze) for a given power supply impedance (Zm).

    I don't think it applies here, because, by allowing ω to alter, you no longer have a given (i.e. constant) Zm. You alter both Ze and Zm at the same time.
     
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