Can these S-parameters be derived?

Question

I have been trying to derive the S-parameters of a lossless T-junction of transmission lines. Here's a diagram from Google:

This is an exercise in microwave network analysis, not in design, so the purpose is to obtain analytical expressions, not a useful circuit. This is a theoretical question only, no Wilkinson dividers here, and no convenient values to get ideal matching or symmetry.

Background Assume the T-junction is fed by a source Vg with source impedance Zg feeding a line of characteristic impedance Z0, connecting to a junction of two transmission lines, with characteristic impedances Z1 and Z2, and lengths of lambda/4 (quarterwave commensurate lines). These lines are terminated in loads of ZL1 and ZL2 respectively.

Denote the input impedance to line 1 (with char. impedance Z1) by Zin1, the input impedance to line 2 by Zin2, and assume they both have their own coordinate systems, with z=0 at the loads and z=-l represents the junction. The voltage on line 1 is V1(z), the voltage on line 2 is V2(z), so that, at the junction, V1(-l) = V2(-l) = Vx, and at the loads, V1(0) is the load voltage of line 1, and V2(0) is the load voltage of line 2.

That's all the background! It is simply to show that the input reflection, S11, is $$ S_{11} = \frac{Z_{in,1}||Z_{in,2} - Zg}{Z_{in,1}||Z_{in,2} + Zg} $$ Which is the basic reflection coefficient. Calculation of S21 is not so easy. To help, we can calculate the line voltages (and currents), as (using V1 as the template, V2 is identical): $$ V_1(z) = V_{0}^1 ( e^{-j\beta z} + \Gamma_{L,1} e^{j\beta z} ) $$ Where $$ V_0^1 = \frac{V_x}{j(1 - \Gamma_{L,1})} $$ And $$ V_x = V_g \frac{Z_{in,1}||Z_{in,2}}{Z_{in,1}||Z_{in,2} + Z_g} $$ These were derived by pretty straightforward analysis, nothing too tough, and I compared them with simulation to confirm. They've held up in every way.

The Problem I wanted to obtain S21 from the line voltages and currents. I thought this would be simple, but it was not; no matter how I analyzed it, the results weren't making sense, and weren't agreeing with simulation. After much experimentation, I actually found that S21 varies with V1(0), with a correction factor that depends ONLY on the ratio of Vg to ZL1, the load resistor on line 1 (port 2 in the S-parameter context). After some experimenting, I obtained the following result: $$ S_{21} = 2 V_1(0) \sqrt{\frac{Z_g}{Z_{L,1}}} $$ And this expression agrees with simulation even if Z1 and Z2 are different, or are changing, and even if ZL1 and ZL2 are different.

I was really surprised to find this expression. It's not unheard of to have square-roots in microwave analysis, but the square root of a ratio of impedances? For all my effort, I don't know how this happens. Why in the world does this expression work? Where does this come from? If anyone can shine a light on this, I would be forever grateful.

Can this value of S21 be derived?

Answer 1

I've solved the problem, and it was slightly subtle. It's an interesting problem, so I'll write a detailed solution here for anyone that needs it. This is a lossless transmission-line T-network power divider.

To answer this question, it's important to note that because the generator and load impedances are, in general, different. If we define the s-matrix from the traveling waves, V+(z) and V-(z), but don't take the different reference impedances into account in these definitions, then our results won't agree with those of generalized scattering parameters.

In order to obtain results which are consistent, it is necessary to replace V+ and V- variables with values normalized to the reference impedance of that port, often denoted a and b (in the style of Collin). $$ a_i=\frac{V_i^+}{\sqrt{Z_{0i}}} $$ $$ b_i=\frac{V_i^-}{\sqrt{Z_{0i}}} $$ The s-parameters are now more properly given by: $$ S_{ij} = \left. \frac{a_i}{b_j}\right\vert_{V_k=0, k\neq j} = \left.\frac{V_i^+}{V_j^-}\frac{\sqrt{Z_{0j}}}{\sqrt{Z_{0i}}}\right\vert_{V_k=0, k\neq j} $$

Using previously obtained results, it was found that: $$ V_2^- = -jV_g\frac{Z_{in,p}}{Z_{in,p}+Z_g}\frac{1+\Gamma_{L1}}{1-\Gamma_{L1}} $$ $$ V_1^+ = \frac{1}{2} V_g $$ Which becomes $$ \frac{V_2^-}{V_1^+} = -j2\frac{Z_{in,p}}{Z_{in,p}+Z_g}\frac{1+\Gamma_{L1}}{1-\Gamma_{L1}} = 2V_1(0) $$ Now normalizing for the different port impedances, we get the value of S21, $$ S_{21} = 2V_1(0)\sqrt{\frac{Z_g}{Z_{L1}}} $$ Exactly what was obtained empirically in the question.