# Computation of the Scattering Matrix of a Transmission Line

let's consider this transmission line (with losses) in which a and b denote respectively the reflected and the incident waves at each port:

This exercise asks to find its scattering parameters normalized with respect to an arbitrary impedance Zc1, which is different from Zc.

It starts with the computation of S12 and S22: in this case, according to the Scattering Parameters' definitions, we have to analyze the network in case a1 = 0 (i.e. port 1 adapted) and with a source applied to port 2. The exercise so analyzes this situation:

What I do not understand is this: why is port 1 closed on Zc1 and not to Zc? The exercise asks to find S-matrix normalized with respect to Zc1, but if we close port 1 on Zc1, it will be non true that a1 = 0, from my point of view. In fact, there will be physically reflection, since there is an impedance mismatch between the load Zc1 and the transmission line which has characteristic impedance Zc. How can this procedure be correct?

why is port 1 closed on Zc1 and not to Zc?

Because in the problem statement you specifically said you wanted, "scattering parameters normalized with respect to an arbitrary impedance Zc1, which is different from Zc".

To test (or simulate) the scattering parameters of a network normalized to an impedance $$\Z_{c1}\$$, you terminate it with the impedance $$\Z_{c1}\$$.

if we close port 1 on Zc1, it will be non true that a1 = 0, from my point of view. In fact, there will be physically reflection, since there is an impedance mismatch between the load Zc1 and the transmission line which has characteristic impedance Zc.

You are correct there will be a reflection there. But this reflection is only within the transmission line being tested (the DUT), so it is not considered as part of the wave $$\a_1\$$.

Imagine you replaced the termination $$\Z_{c1}\$$ with a length of transmission line of characteristic impedance $$\Z_{c1}\$$, and terminated that with the impedance $$\Z_{c1}\$$. As far as the behavior of your DUT is concerned, this would not change anything. But it would be clear that there is no forward travelling wave $$\a_1\$$ in this transmission line. The reflection is entirely within the DUT, not in the termination.

• So, with this choice, a and b are not exactly the direct and reverse waves of the transmission line, correct? Because in that situation we consider a1 = 0 but there is a reverse wave on the transmission line because of the mismatch. So is it correct to say that a and b coincide with direct and reverse waves only if the normalization impedance is the same of the characteristic impedance? Dec 1, 2019 at 18:53
• They're the forward and reverse waves in the feedlines, not the forward and reverse waves in the DUT. Consider that these waves should be defined for any kind of DUT, not just for transmission line DUTs. Dec 1, 2019 at 19:00
• Perfect, thank you very much Dec 1, 2019 at 19:00