# Insertion of transmission lines and effects on reflection coefficient

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:

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

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?

• You might be interested in this section of the transmission line article on Wikipedia. – The Photon Jun 29 '19 at 14:24
• Perfect, thank you very much. – Kinka-Byo Jun 29 '19 at 14:43

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|\$$.

• 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? – Kinka-Byo Jun 29 '19 at 16:58