# Analogy between EM wave reflection and S-parameters

paralleltransport
It is well known that one can solve incident an reflective wave in homogeneous linear media by matching PDE boundary conditions.

In the electrical engineering community, one solves similar problem using smith chart and scattering parameters for 1-dimensional propapation of TEM modes in transmission lines.

Is there a good references that explores this analogy? The EE method of using smith chart and summing reflection coefficients using lattice diagram should be directly applicable to PDE solution to the wave equation.

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jsgruszynski
The key factor with distributed model (s-parameters, transmission lines, etc.) is that the wavelength is pretty close to physical dimensions.

Rather than "analogy" it's a "subset" relationship like that between Newtonian vs. Relativity physics - one is a proper subset model where the subset is far easier to use in practice but isn't strictly "correct". Both give the same answers until you exceed the model correctness boundary.

Broader wave equation solutions in 3D are generally required when the simplification of this starts to fail or become unwieldy. Distributed model is about simplification of human hand calculation and intuitions (it was invented when slide rules were the most advanced calculating device). But there's a point when it's not worth the trouble or it just plain gives the wrong answers.

That point is when the number of elements or wavelength constraints demand a full Maxwell's solution. You'll want to look at the underlying math of Computational Electromagnetics which is similar but scaled up to PDE solution forms and overcoming model limitations of distributed model.

One of the most common today is FDTD. There are other techniques and usually the split is on whether the problem is a "free space" or "guided propagation" situation.

None of however do NOT provide much intuition about what is simulated. You still need the basics of both Maxwell's and distributed model to do sanity checking. This is the mistake MOST people using such solvers make - they have no intuition of the problem space and blindly trust the simulation far more than they should. Numerical methods themselves have inherent flaws and errors that they always introduce.

• paralleltransport
paralleltransport
I see, thank you.

In fact you're right. I checked a calculation I made of the transmission of a plane wave across a thin metal barrier with a certain conductivity.

This is a problem in Jackson E&M.

I tried to solve it 2 ways:
- the correct way matching boundary conditions at the interface (PDE style)
- The incorrect way by summing all transmission on a lattice diagram using the famous (2n2)/(n2+n1) transmission coefficient and reflection coefficient (n2-n1)/(n2+n1)

The 2nd answer did not match the first. I don't have a good argument under what conditions should the 2 answers match however. My guess is that when the loss is low and the metal barrier is think, where the eikonal (or WKB) approximation to wave propagation is correct, the PDE solution and summing all reflections as a geometric series will match.

Gold Member
It is well known that one can solve incident an reflective wave in homogeneous linear media by matching PDE boundary conditions.

In the electrical engineering community, one solves similar problem using smith chart and scattering parameters for 1-dimensional propapation of TEM modes in transmission lines.

Is there a good references that explores this analogy? The EE method of using smith chart and summing reflection coefficients using lattice diagram should be directly applicable to PDE solution to the wave equation.
You can certainly use transmission-line modeling techniques for analyzing things like reflections of waves at multiple boundaries as well as for waveguide types of problems. However, I believe the standard formulation assumes that each segment is long enough that evanescent waves generated at one interface have died out to a negligible amplitude by the time they get to the next interface. So if you need to Include things like lossy media or frustrated total internal reflection, then you might need a slight generalization of the method. Collin mentions this in his book "field theory of guided waves" but I think leaves that generalization as an exercise for the reader. Likewise, when multiple propagation modes are important (eg multiple TE and/or TM modes) then further generalizations are required.

jason

• paralleltransport