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I have a doubt on a step usually performed in matching network design. Consider a load ZL: a method for getting impedance matching (for instance with an amplifier) is that of inserting a transmission line in order to modify its impedance, like in the following picture:

enter image description here

This operation can be seen also in terms of reflection coefficient. Precisely, I found this relation:

enter image description here

where ΓL is the reflection coefficient of the Load. My question is a bout a generic case in which ZL may be also another transmission line, a port etc

where does this relationship come from? Are there any assumption (for instance that the transmission line must have the same characteristic impedance of ZL - in case is a transmission line - , or it must be a no loss line etc) for it?

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  • \$\begingroup\$ You might be interested in this section of the transmission line article on Wikipedia. \$\endgroup\$ – The Photon Jun 29 at 14:24
  • \$\begingroup\$ Perfect, thank you very much. \$\endgroup\$ – Kinka-Byo Jun 29 at 14:43
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where does this relationship come from?

It comes from the fact that if you have a forward travelling wave at \$z=-l_1\$, then it will travel forward a distance \$l_1\$, then reflect off the load, then travel in reverse distance \$l_1\$ again before you see the reverse travelling wave at your measurement point.

Are there any assumption (for instance that the transmission line must have the same characteristic impedance of ZL - in case is a transmission line - , or it must be a no loss line etc) for it?

As shown in the diagram, the transmission line must have characteristic impedance \$Z_0\$ (it must be matched to the system impedance), and propagation constant \$\beta\$.

If it was matched to the load (it had characteristic impedance numerically equal to \$Z_L\$), then the reflection coefficient at \$z=-l_1\$ would just be \$\Gamma_L\$, since a transmission line terminated with a matched load just "looks like" a lumped impedance equal to its characteristic impedance.

You are correct that this formula only applies to a lossless line. If it were a lossy line, the magnitude of the reflection constant at \$z=-l_1\$ would be less than \$|\Gamma_L|\$.

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  • \$\begingroup\$ I am sorry, a last question: why does the line has to be of the same characteristic impedance of the system, to satisfy the previous equation? \$\endgroup\$ – Kinka-Byo Jun 29 at 16:58

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