| I guess there are many different ways of looking at the issue; here's 
another. 
If I have a system comprising: Source(50 
Ohms)>>Matching-Unit>>Transmission-Line>>Doublet, and I adjust the 
matching unit so that the Source is presented with a 50 Ohm load, as far 
as the doublet is concerned looking back towards the 
TL/Matching-Unit/Source, I have created a new source with an internal 
impedance which matches the impedance of the doublet. 
I've conveniently ignored TL and tuner losses, and assumed we have a 50 
Ohm Thevenin source, none of which is true in the real world; but the 
principle still has utility in understanding what is happening. 
I produced a worked example here:
http://www.karinya.net/g3txq/tuner_match/
Steve G3TXQ
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