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# S-parameters

Updated December 4, 2009

When you come down to it, there are really only a few things that separate a microwave engineer from a "normal" electrical engineer: knowledge of the Smith chart, S-parameters, transmission lines including waveguide, and decibels. Thankfully, these are all simple concepts and we'll help you master them right here at Microwaves101!

### History of S-parameters

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 only, 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:

Note that each S-parameter is a vector, so if actual data were presented in matrix format, a magnitude and phase angle would be presented for each Sij.

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!

If we assume that each port is terminated in impedance Z0, we can define the four S-parameters of the 2-port as:

There's a missing step to this derivation, which was pointed out by Alex (thanks!) You'll find the complete derivation on Wikipedia, we'll update this page soon.

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:

If we want to measure S11, we inject a signal at port one and measure its reflected signal. In this case, no signal is injected into port 2, so a2=0; during all laboratory S-parameter measurements, we only inject one signal at a time. If we want to measure S21, we inject a signal at port 1, and measure the resulting signal exiting port 2. For S12 we inject a signal into port 2, and measure the signal leaving port 1, and for S22 we inject a signal at port 2 and measure its reflected signal.

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

Sij(dB)=20*log[Sij(magnitude)]

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).

### Types of S-parameters

When we are talking about networks that can be described with S-parameters, we are usually talking about single-frequency networks. Receivers and mixers aren't referred to as having S-parameters, although you can certainly measure the reflection coefficients at each port and refer to these parameters as S-parameters. The trouble comes when you wish to describe the frequency-conversion properties, this is not possible using S-parameters.

Small signal S-parameters are what we are talking about 99% of the time. By small signal, we mean that the signals have only linear effects on the network, small enough so that gain compression does not take place. For passive networks, small-signal is all you have to worry about, because they act linearly at any power level.

Large signal S-parameters are more complicated. In this case, the S-matrix will vary with input signal strength. Measuring and modeling large signal S-parameters will not be described on this page (perhaps we will get into that someday)

Mixed-mode S-parameters refer to a special case of analyzing balanced circuits. We're not going to get into that either!

Pulsed S-parameters are measured on power devices so that an accurate representation is captured before the device heats up. This is a tricky measurement, and not something we're gonna tackle yet.

### Other matrices

S-parameters are just one matrix that can fully describe a network. Other matrices include ABCD parameters, Y-parameters and Z-parameters. ABCD parameters are actually used "behind the scenes" in many calculations, because they are easily cascadable. By cascadable, we mean that if you want to simulate an attenuator followed by an amplifier, the S-parameter math will drive you insane, while the ABCD math involves nothing more than multiplication. This will remain a topic for another day!

Come back soon!

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