# Transmitted energy and transmitted wave phase in microwave structures

I have two questions about waves in large microwave structures.

So the system I'm currently working on can be simulated as lossless transmission lines put together to one long structure. Without getting into too much unnecessary details let's just say that the system is a some number of transmission lines between two infinite 50 Ohms transmission lines.

There is no point in trying to simulated the system using some matrix formalism since this kind of computation diverges. Instead I use a method called impedance loading which is presented "Microwave Engineering" by Pozar. According to a short derivation he presents if I have a transmission line of length $l$, propagation constant $\beta$ and characteristic impedance $Z_0$ that is terminated by a line of characteristic impedance $Z_L$ I can think of the two of them together as $Z_{in} = Z_0\frac{Z_L -iZ_0\tan\beta l}{Z_0 -iZ_L\tan\beta l}$. Meaning that a wave that enters the transmission line will behave as if the transmission line together with the load are one transmission line of characteristic impedance $Z_{in}$.

So, for a large structure of many transmission lines put together I can use this formula again and again until I'm left with $Z_{in}^{total}$ that describe the whole structure besides the infinite 50 Ohms transmission line at the input. Then I can use the ordinary Fresnel relations to get the amplitude reflection and transmission coefficients: $r = \frac{50 - Z_{in}^{total}}{50 + Z_{in}^{total}}$ and $t = \frac{2\cdot50}{50 + Z_{in}^{total}}$

After this long introduction I can get to my questions. It is clear from energy conservation, together with the fact that the impedance loading method takes care of all internal reflections in the structure, that the transmitted energy that gets to the output 50 Ohms transmission line is $T = 1 - R$ where $R = |r|^2$. But, somehow I get that if I take $T = \Re\frac{Z_{in}^{total}}{50}\cdot|t|^2$ then $R + T = 1$. So my first question is why does that work? It sure cannot be $\frac{Z_{in}^{total}}{50}$ instead of $\Re\frac{Z_{in}^{total}}{50}$ since this is a complex number as easily seen from the definition of $Z_{in}$. But, I would have guessed that $|\frac{Z_{in}^{total}}{50}|$ would give energy conservation and not $\Re\frac{Z_{in}^{total}}{50}$. So I would be happy to see how to get this $\Re\frac{Z_{in}^{total}}{50}$ coefficient which I guess should be there because it gives energy conservation. Any book that covers this subject I looked at so far doesn't specify how to get the transmission and reflection coefficients of the energy when either one of the two mediums impedances is complex.

Another question is as follows. The computation of $t$ gives the phase of the transmitted wave. But, this is the phase of the wave at the input point. Is there any way to get the phase of the transmitted wave at the output point? It is important because the dispersion of this structure will later affect the operation of a device I'm trying to realize.

• I mean that I'm using MATLAB to simulate my structure. Since the structure includes many transmission lines of short lengths the phase factors accumulate large numerical errors. It causes some of the values I am trying to get from the simulation to diverge. BTW the $\Re\frac{Z_{in}^{total}}{50}$ problem is solved. Jul 19, 2017 at 14:43
Your impedance transformation formula $Z_{in} = Z_0\frac{Z_L -iZ_0\tan\beta l}{Z_0 -iZ_L\tan\beta l}$ assumes a lossless transmission line. The reason why your "loaded" impedances have non-zero real part is because the loads at either end are infinite long ideal transmission lines that are in fact behave as if they were ideal resistive loads. And that is because there is nothing that reflects from the infinite distance in a finite amount of time but actually nothing is really dissipating here, therefore at every point along the structure the incident power will always equal the reflected power + transmitted power everywhere.