S-parameters refer to the scattering matrix ("S" in S-parameters refers to scattering). The concept was first popularized around the time that Kaneyuke Kurokawa of Bell Labs wrote his 1965 IEEE article Power Waves and the Scattering Matrix. Check him out in our Microwaves101 Hall of Fame! It helped that during the 1960s, Hewlett Packard introduced the first microwave network analyzers. We'll also admit that there are several papers that predate Kurokawa's from the 1950s, one good early work was written by E. M. Matthews, Jr., of Sperry Gyroscope Company, titled The Use of Scattering Matrices in Microwave Circuits. Also, Robert Collin's textbook Field Theory of Guided Waves, published 1960, has a brief discussion on the Scattering matrix. Collin's book is extensively annotated, including an author index, which reads like a Who's Who of electromagnetic theory for the first half of the twentieth century.
Introduction to S-parameters
Before we get into the math, let's define a few things you need to know about S-parameters.
The scattering matrix is a mathematical construct that quantifies how RF energy propagates through a multi-port network. The S-matrix is what allows us to accurately describe the properties of incredibly complicated networks as simple "black boxes". For an RF signal incident on one port, some fraction of the signal bounces back out of that port, some of it scatters and exits other ports (and is perhaps even amplified), and some of it disappears as heat or even electromagnetic radiation. The S-matrix for an N-port contains a N2 coefficients (S-parameters), each one representing a possible input-output path.
S-parameters are complex (magnitude and angle) because both the magnitude and phase of the input signal are changed by the network. Quite often we refer to the magnitude onl, as it is of the most interest. Who cares how the signal phase is changed by an amplifier or attenuator? You mostly care about how much gain (or loss) you get. S-parameters are defined for a given frequency and system impedance, and vary as a function of frequency for any non-ideal network.
S-parameters refer to RF "voltage out versus voltage in" in the most basic sense. S-parameters come in a matrix, with the number of rows and columns equal to the number of ports. For the S-parameter subscripts "ij", j is the port that is excited (the input port), and "i" is the output port. Thus S11 refers to the ratio of signal that reflects from port one for a signal incident on port one. Parameters along the diagonal of the S-matrix are referred to as reflection coefficients because they only refer to what happens at a single port, while off-diagonal S-parameters are referred to as transmission coefficients, because they refer to what happens from one port to another. Here are the S-matrices for one, two and three-port networks:
The input and output reflection coefficients of networks (such as S11 and S22) can be plotted on the Smith chart. Transmission coefficients (S21 and S12) are usually not plotted on the Smith chart.
Definition of S-parameters
S-parameters describe the response of an N-port network to voltage signals at each port. The first number in the subscript refers to the responding port, while the second number refers to the incident port. Thus S21 means the response at port 2 due to a signal at port 1. The most common "N-port" in microwaves are one-ports and two-ports, three-port network S-parameters are easy to model with software such as Agilent ADS, but the three-port S-parameter measurements are extremely difficult to perform with accuracy. Measure S-parameters are available from vendors for amplifiers, but we've never seen a vendor offer true three-port S-parameters for a even a simple SPDT switch (a three-port network).
Let's examine a two-port network. The incident voltage at each port is denoted by "a", while the voltage leaving a port is denoted by "b". Don't get all hung up on how two voltages can occur at the same node, think of them as traveling in opposite directions!
See how the subscript neatly follows the parameters in the ratio (S11=b1/a1, etc...)? Here's the matrix algebraic representation of 2-port S-parameters:
Did we mention that all of the a and b measurements are vectors? It isn't always necessary to keep track of the angle of the S-parameters, but vector S-parameters are a much more powerful tool than magnitude-only S-parameters, and the math is simple enough either way.
S-parameter magnitudes are presented in one of two ways, linear magnitude or decibels (dB). Because S-parameters are a voltage ratio, the formula for decibels in this case is
Remember that power ratios are expressed as 10xlog(whatever). Voltage ratios are 20xlog(whatever), because power is proportional to voltage squared.
The angle of a vector S-parameter is almost always presented in degrees (but of course, radians are possible).