(S)tay True, (S)tay Humble

Every once in a while, I learn that even if I think I knew a subjet to it’s fullest, there is something I missed. In this occasion, it’s scattering parameters, commonly known as S-Params. It’s not a “subject”, per-seh, barely a chapter in the unimpeachable source of Microwave Engineering (Pozar’s book). But, to my embaressement, I missed a nuance there.

Here is the first disclaimer: Not all systems have $50\Omega$ sources and $50\Omega$ terminations. Here is a more extreme proclamation – Some systems present different impedances at different ports. Probably, none of you are shocked yet. So here is how I dealt with this issue, so far.

As part of a project, a while back, I had to model arbitrary impedances connected to my Antenna array. This happens, a lot, while dealing with amplifiers. Aצplifiers are a non-linear entity and modify their behavior with respect to the load presented to them. So I took the matrix exported from the simulation software (all ports normalized to $50\Omega$ , of course) and re-normalized it to a different impedance.

How is this re-normalization done? For a thorough explanation, you may refer to Kurokawa’s article, [1]. But the transition is done by defining a quantity known as “power waves”. The incident and relected power waves at the i-th port are define as

$p^+_i=\frac{V_i + Z_i I_i}{2\sqrt{\mathrm{Re}Z_i}}$

$p^-_i=\frac{V_i - Z_i I_i}{2\sqrt{\mathrm{Re}Z^*_i}}$

where $V_i$ and $I_i$ are the respective total voltage and current in the i-th port and $Z_i$ is the port impedance. The superscript asterisk denotes a complex conjugate.

This more generalized formulation allows taking into account the source\termination impedance. Conversion of a S-Matrix with a port impedance vector, to a different set of port impedances, is given by

$\mathbf{S}'=\mathbf{A}^{-1}\left({\mathbf{S} - \mathbf{\Gamma}^+}\right)\left({\mathbf{1} - \mathbf{\Gamma S}}\right)\mathbf{A}$ ,

where $\mathbf{\Gamma}$ is a diagonal matrix, with

$\Gamma_{ii} = r_i= \frac{{Z'}_i - Z_i^*}{{Z'}_i + Z_i}$

and $\mathbf{A}$ has the diagonal elements

$A_{ii} = \left({1-r_i^*}\right)\frac{\sqrt{1 - |r_i|^2}}{\left|{1 - r_i}\right|}$

Ok, so now I have the toolset I need for using various port impedances.

Reality check

I started using an open source engine for RF simulations and tried to define a S-Param matrix. All ports have a known, arbitrary impedance. This simulation engine yields the incident and reflected voltages, $v^+$ and $v^-$ . The setup is as following:

So I said what seemed natural to me. Cut the middle man! Calculate the S-Params like this:

$S_{11} = \frac{v_1^-}{v_1^+}\,,\,\,v_2^+ = 0$

$S_{12} = \frac{v_1^-}{v_2^+}\,,\,\,v_1^+ = 0$

$S_{21} = \frac{v_2^-}{v_1^+}\,,\,\,v_1^+ = 0$

$S_{22} = \frac{v_2^-}{v_2^+}\,,\,\,v_1^+ = 0$

Well, that’s nice and all. But not true.

I need some sort of dramatic pause here, so I’ll put a separator dots.

The voltage based derivation is not valid for the arbitrary impedance case. This is valid when all the ports in the matrix are normalized to the same impedance. All incident and reflected voltages need to be converted to their respected power waves, as such.

$S_{11} = \frac{\frac{v_1^-}{\sqrt{\mathrm{Re}Z_1}}}{\frac{v_1^+}{\sqrt{\mathrm{Re} Z_1}}}\,,\,\,v_2^+ = 0$

$S_{12} = \frac{\frac{v_1^-}{\sqrt{\mathrm{Re}Z_1}}}{\frac{v_2^+}{\sqrt{\mathrm{Re} Z_2}}}\,,\,\,v_1^+ = 0$

$S_{21} = \frac{\frac{v_2^-}{\sqrt{\mathrm{Re}Z_2}}}{\frac{v_1^+}{\sqrt{\mathrm{Re} Z_1}}}\,,\,\,v_2^+ = 0$

$S_{22} = \frac{\frac{v_2^-}{\sqrt{\mathrm{Re}Z_2}}}{\frac{v_2^+}{\sqrt{\mathrm{Re} Z_2}}}\,,\,\,v_1^+ = 0$

Classic hubris. I didn’t do anything wrong before this. I just never approached this problem from the opposite direction.

We live and we learn. The other option is staying in the same place, isn’t it.

References

[1] Kurokawa, Kaneyuki. “Power waves and the scattering matrix.” IEEE transactions on microwave theory and techniques 13.2 (1965): 194-202.‏

RF With Care

(S)tay True, (S)tay Humble

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