# Impedance Matching: The mathematical Conditions for Matching Network

The Tortoise-Man
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:

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):

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:

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?