Insertion of transmission lines and effects on reflection coefficient

Question

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?

Answer 1

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|\$.