December 4, 2009
here to go to our page on the Smith chart
here to go to our page on reference planes
here to go to our page on network analyzer measurements
here to go learn about our our S-parameter Utilities spreadsheet
here to learn some basic network theory
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!
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
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.
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.
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
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).
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.
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:
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.
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.
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.
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.
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)
S-parameters refer to a special case of analyzing balanced circuits.
We're not going to get into that either!
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.
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!