Cascading S-Parameters in Plain English: Part 3

This is part 3 of a 5 part series: Cascading S-Parameters in Plain English

Previous << Part 2: S-Parameters in Plain English

In this segment, we will break down T-parameters (scattering transfer parameters) and their role in cascading S-parameters. Despite this series claiming to explain the cascade of S-parameters, the truth is that we cannot actually cascade S-parameters directly. This is where T-parameters come in.

Why the Bait and Switch?

Let's get a tiny bit more technical detail on S-Parameters to help us understand why they need to be transformed into something else before we can chain them together. Below is the full matrix definition of an S-parameter

The intuition here is that we get some output wave b when we provide some input wave a. The equivalent algebraic representation is below.

Note that there appear to be 2 unknowns, a₁ and a₂. So to solve for any of the S-parameters, one unknown must be eliminated. This means that when measuring using a₁ reference input, a₂ reference input must = 0, and vice versa. The equation then becomes trivial to solve per our previous plain English definitions, as noted below

To re-iterate, S-parameters by definition require very specific control over the ports. But this breaks down when we chain multiple devices together. Suppose, for example, that we took two identical 2 port devices and chained them together by hooking up port 2 of Device 1 to port 1 of Device 2. Now it is very difficult to make sure that a₁ and only a₁ is the only active input wave coming into Device 1. When we were just measuring Device 1, we could make sure port 2 was terminated in a matched load such that nothing coming out of the device at port 2 would be reflected back into the device. But by putting Device 2 in the way, we've lost that control. We can no longer guarantee there will not be reflections back into Device 1.

T-Parameters to the Rescue

T-Parameters give us an algebraic solution to this problem. The S-parameter to T-parameter algebraic derivation is beyond the scope of this series, and certainly not something that suits the plain English format, but one general form is shown below.

Note that the a and b coefficients have been re-organized. So the plain English definition of T-parameters can be "S-parameters algebraically re-organized to put the input and output ports on different sides of the equation".

Consider two T-parameters, T₁ and T₂ below representing two converted S-parameters we would like to cascade. I have chosen ports 1 and 2 for Device 1 and ports 3 and 4 for Device 2 to avoid confusion.

Recall the idea of cascading is conceptually the same as hooking up one devices output port to the next devices input port, so our desired final result here is to have a new matrix where port 1 is the input port, port 4 is the output port, and ports 2 and 3 sort get 'absorbed' into this new circuit. The idea of hooking up ports is similar to algebraically setting the port conditions to be equal.

So if we define the port conditions of ports 2 and 3 to be equal, this leads to the very useful substitution below where T₁ and T₂ can be combined into a form in terms of ports 1 and 4.

And once we multiply the T matrices, we can convert back to an S-parameter in terms of ports 1 and 4, which is exactly what we wanted!

Earlier I mentioned our definition of T-parameters was one acceptable form. You may see different notations in different sources, because there are multiple ways to reorganize the coefficients to get the inputs and outputs separated. These are not wrong, as long as the algebraic transform is done correctly and consistently across all S-parameters. However, in the next article, I'll describe why I prefer the form here, and how that can be expanded to cover multi-port devices.

Next >> Part 4: Converting S and T parameters

By: Matt Wright 7/11/2020

Tags: Tech, S-parameters