Impedance Matching: The mathematical Conditions for Matching Network

  • #1
The Tortoise-Man
57
1
Assume we have a network consisting of a source with impedance ##Z_S## and load with impedance ##Z_L## and
we want to perform impedance matching on them in order to obtain the maximum power transfer:

Zs Zl MATCH.png


Note that in practice there may occur sitations where causes more harm than profit (see eg Baluncore's example here: https://www.physicsforums.com/threa...g-when-the-transmitter-line-and-load.1009889/ ; see post #2)


But the motivation of this question has pure conceptional nature; ie I'm not asking here IF it should be done but HOW it should be correctly done, if one has to do it.

In general, ##Z_S \neq Z_L^*##. So principally, what one do in order to match impedances, one places a matching network between source and load (the components which the matching network contains depend on concrete problem):

Zs Zl Matching Network.png


Indeed, the matching network can consist of resistive components, it might be a L- or T-network and and and... ) and one tries to adjust the parameters of the components of the matching network to satisfy certain matching conditions; see below.

Now having implemented the matching network in our circuit in order to impose the right mathematical conditions we introduce following two auxilary impedances: the input impedance ##Z_{in}## and output impedance ##Z_{out}## defined as follows:


Zs Zl Matching ZIN.png


Zs Zl Matching ZOUT.png



My question is: In order to obtain the impedance matching
for maximal power transfer which conditions should be satisfied?

##Z^*_s= Z_{in}##, ##Z_{out}= Z^*_L## or both simultaneously?

Or even more interesting question is: Assume we succeed in adaping the components within the matching network such that ##Z^*_s= Z_{in}## holds. Is then ##Z_{out}= Z^*_L##
automatically satisfied?

So the question is basically about if it really neccassary to adapt the components within the matching network such that they should satisfy both conditions ##Z^*_s= Z_{in}## AND ##Z_{out}= Z^*_L##, or is one of these conditions really redundant in the sense that it suffice to adjust the matching box only to satisfy ONE of them and the second is the satisfied automatically?
 

Answers and Replies

  • #2
berkeman
Mentor
64,124
15,327
Thread closed temporarily as a potential repost of previous threads. Hopefully I can deal with this before the weekend.

Sorry, you are right, presumably sending a PM to you about how to continue the discussion there would have been a better option. I will keep that in mind.
 

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