Cascading S-Parameters in Plain English: Part 2

This is part 2 of a 5 part series: Cascading S-Parameters in Plain English

Previous << Part 1: Introduction

In this section, we attempt to simply explain what S-parameters are. We described in Part 1 that S-parameters describe the frequency dependant performance of a device. But that isn't plain English. In fact, S-parameters describe electromagnetic energy waves, either power or voltage, depending on the context.

S-parameters describe electromagnetic energy waves, either power or voltage, depending on the context.

A Rock in A Pond

Imagine dropping a rock in a pond; it creates a circle of waves propagating away from the rock. Depending on the size of the rock, the waves are a certain size, a certain distance apart, and depending on when you dropped the rock in the pond, the waves may be close to or far away from where they started.

S-parameters basically contain all 3 key characteristics, but the terminology is a little different. S-parameters typically describe wave height as amplitude, length in terms of frequency (waves per second), and position as phase. To re-iterate, S-parameters have 3 pieces of information: frequency, amplitude, and phase. And that's it!... well, almost.

S-parameters are meaningless without a reference context. Like our rock in the pond analogy, the output depends on the size of the rock. So S-parameters are defined as a ratio of the output wave relative to some input wave. In plain English, an S-parameter quantifies the ratio of an output wave over an input wave.

In plain English, an S-parameter quantifies the ratio of an output wave over an input wave.

Basic Case: A 2 Port Network

For a simple 2 port network, if we define an input wave on port 1 as a₁ and output wave on port 1 as b₁, and so on, there are 4 ratios we can quanitify: b₁/a₁, b₂/a₁, b₁/a₂, and b₂/a₂.

These are typically expressed in matrix notation per below:

This might look a little confusing, but each S parameter has a very literal plain English description, following the below pattern.

That wasn't so bad, was it? Just keep in mind that S-parameters are frequency specific, so in practice, your S-parameter file will typically contain many S-parameters; one for each frequency measured or simulated.

A More Complex Case: Balanced S-Parameters

Another case worth considering is a 4 port network. As you can see below, we can just think of a 4 port S-parameter as the big brother of the 2 port case.

However, there is also a popular digital signalling scheme which couples two ports on each side of the circuit. This essentially means we apply our input waves to two conductors at the same time, and then measure our output wave on two conductors at the same time.The challenge is that when we have a port made of two conductors, we have some decisions to make: For our input wave, we can either send a positive wave down both legs, or we can send a positive wave down one and a negative wave down the other. And then when we measure a port that has two conductors, we can either take the two signals and sum them together, or we can subtract them. These different 'modes' are called common and differential

There are the same number of terms as a 4 port network, but they are described a little differently

It turns out differential mode has some really useful properties in high speed digital signaling. These properties are beyond scope of this series, but the plain English definitions of balanced mode S-parameters can follow a very similar pattern as their simple single port brothers.

Next >> Part 3: T-Parameters in Plain English

By: Matt Wright 7/10/2020

Tags: Tech, S-parameters