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Homework Help: Impedance matching derivation (next best after complex conjugate method)

  1. Aug 28, 2012 #1
    (This is not a question I was given to solve, it is a question about the course notes.)

    1. The problem statement, all variables and given/known data
    In impedance matching, what is the next best method after the complex conjugate method?
    If the source has V_s and Z_s, what should Z_L be?
    V_s, Z_s, and Z_L are in series.

    2. Relevant equations
    The best impedance match is when Z_s = (Z_L)*
    And this was derived from maximum power transfer, or
    P = (I_rms)^2 * R_L = 1/2 * (|V_s|/( |Z_s + Z_L| )) = ...

    3. The attempt at a solution
    The notes say
    Z_L = R = |Z_s|

    This says the next best choice is having Z_L be a resistance equal to the magnitude of Z_s.

    However, from the math I did, this assumes that there cannot be a reactance for Z_L.
    Why this assumption?

    Edit: Why is this the "second-best method"? Why is there the assumption that there is no load reactance?
    (I do not have any criteria either. It is from a handout. I will ask the professor after the 1st day of class, I guess.)
    Last edited: Aug 28, 2012
  2. jcsd
  3. Aug 28, 2012 #2

    rude man

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    By defintion, |Z_s| is a real quantity, i.e. a resistance, not a complex impedance.

    As to what constitutes "second-best" matching impedance I have no idea. No criterion(a) was/were offered.
  4. Aug 28, 2012 #3
    I am aware that this method yields a real number.

    My question is, why is this the second best matching impedance.

    No criteria were offered to me either...

    Anyone with real-world experience?
  5. Aug 29, 2012 #4

    rude man

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    OK I think I know.

    What is meant is: if a complex load is not permitted, i.e. if it has to be resistive, then the highest power transfer is when Z_L = R = |Z_s|. And that is correct.

    (Has to be that: otherwise you could just say, OK, let Z_L = R_s - jX_s + ε where Z_s = R_s + jX_s and ε is an arbitrarily small resistance!)

    So: can you proceed with that info? Hint: what is |i|, the current? Then what is power P_L, then how do you determine optimum R_L for max P_L?
  6. Aug 29, 2012 #5
    I'm sorry, but that is not what I meant.

    I want to know Why is a complex load not permitted in the second best method? Why is this the second best method?

    I've already done the math showing that this R_L corresponds to maximizing P_L when Z_L is only a resistance, not a reactance.
  7. Aug 29, 2012 #6

    rude man

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    There used to be a commercial on TV about a hair dye product that went: "Only your hairdresser knows for sure".

    Only your instructor knows for sure.
  8. Aug 29, 2012 #7
    So the bad news: The professor does not remember any more why this is so. :( He said it was just best practices.

    And the better news: He says it may have something to to do with minimizing the reflection coefficient.

  9. Aug 30, 2012 #8

    rude man

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    Can't be that either. You'd get zero reflection coefficient if Z_L = Z_s. So if Z_s is complex, so is Z_L.
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