[hsdd] High-Speed Digital Design Newsletter
- From: "Dr. Howard Johnson" <howiej@xxxxxxxxxx>
- To: <hsdd@xxxxxxxxxxxxx>
- Date: Wed, 19 Feb 2003 14:26:48 -0800
SCATTERING PARAMETERS
HIGH-SPEED DIGITAL DESIGN ? online newsletter ?
Vol. 6 Issue 03
______________________________________________________
Just as the snow begins to slowly turn my ranch to mud,
I'll be hitting the road to visit clients in the
beautiful Southwest. On March 24-25, I'll be in Dallas
conducting a High-Speed Digital Design seminar at the
Omni Richardson. See www.sigcon.com for more details.
______________________________________________________
Have you ever finished a big project--and then the
next morning wished you could have added just one
more feature? Well, thanks to the miracle of the
internet, I get to keep writing all the articles I
want to supplement the information in my latest
book, High-Speed Signal Propagation. This article on
Scattering Parameters (S-parameters) relates the
transmission matrices described in that book to the
S-parameter matrices provided by a network analyzer.
______________________________________________________
Nicer versions of the figures and equations, which
are too big to send through this newsletter
reflector, are posted in the web version of this
article at www.sigcon.com, under "archives".
______________________________________________________
SCATTERING PARAMETERS
DR. HOWARD JOHNSON
A scattering matrix (S-parameter matrix) is one way
to describe the operation of a linear, time-
invariant two-port circuit. A two-port network is
defined as any linear device where a signal goes in
one side and comes out the other.
The S-parameter matrix is rapidly becoming very
popular as a way to characterize connectors and
cables for high-speed applications above 1 Gb/s.
The measurement setup associated with S-parameters
is as follows (Figure 1).
in1 --->| DUT |<--- in2
out1 <---| DUT |---> out2
Figure 1 -- A two-port S-parameter matrix records the
reflection coefficients and transmission gain for
signals coming and going on both sides of the DUT.
From the test equipment, two cables having
characteristic impedance Z0 lead to the left and
right sides, respectively, of the device under test
(DUT).
Using the first (left-side) cable, a sinusoidal
signal (in1) of unit amplitude is injected into the
DUT. The test equipment records the amplitude and
phase of the signal (out1) reflected back onto the
first cable from the DUT, and also the amplitude and
phase of the signal (out2) conveyed through the DUT
to the second cable on the other side.
Using the second (right-side) cable, another
sinusoidal signal (in2) of unit amplitude is
injected into the DUT. The test equipment records
the amplitude and phase of the signal (out2)
reflected from the right side of the DUT, and the
amplitude and phase of the signal (out1) conveyed
through the DUT to the other (left) side.
The complete S-parameter matrix is a combination of
these four basic measurements.
The left column of the S-parameter matrix is
defined when injecting a signal from the left:
s11 = out1/in1 (L-side reflection)
s21 = out2/in1 (L->R transmission)
The right column of the matrix is defined when
injecting a signal from the right:
s12 = out1/in2 (R->L transmission)
s22 = out2/in2 (R-side reflection)
The four elements of the s-parameter matrix then may
be used to compute the signals out1 and out2
emanating from the two-port device when stimulated
by input signals in1 and in2:
out1 = s11*in1 + s12*in2
out2 = s21*in2 + s22*in2
The four elements of an S-parameter matrix may be
reported as complex numbers (with real and imaginary
parts) or in logarithmic units (as dB magnitude and
phase).
[Note that the procedure above provides a model for
the calculation of circuit performance only at one
single frequency. The entire procedure is usually
performed on a dense grid of frequencies spanning
the range of interest, such that the parameters s11,
etc., are all functions of frequency.]
PROVIDED that the reflection coefficients s11 and
s22 are relatively small, you may estimate the
effect of cascading several two-port networks by
merely multiplying the s21 coefficients of the
individual components (or, if they are in
logarithmic units, by adding the dB values of the
s21 coefficients). Such a calculation determines, TO
FIRST ORDER, the magnitude and phase of a signal
that propagates straight through the cascade
proceeding from left to right through each
component. This is the beauty of S-parameter
analysis, and one key reason it is used in the
design of highly cascaded systems like radio
receivers and chains of linear amplifiers.
Unfortunately, the overall transfer function of a
highly cascaded system equals the product of the s21
terms ONLY when the reflections are negligeable. If
the reflections are significant, the gain does NOT
equal the product of s21 terms.
If you want to model the reflections, a more
sophisticated analysis is required. That's the
purpose of the Transmission Matrix, also called the
Transfer Matrix, or A-parameters.
The measurement setup associated with transmission
parameters is defined as follows (Figure 2).
V1, I1 --->| DUT |---> I2, V2
Figure 2 -- The transmission parameters directly relate
input voltages and currents to output voltages and
currents.
On the left side of the DUT define the voltage V1
and current I1 entering the DUT. On the right side
of the DUT define the voltage V2 and current I2
LEAVING the DUT.
The voltage and current on the left are related to
the voltage and current on the right:
V1 = a11*V2 + a12*I2
I1 = a21*V2 + a22*I2
The four transmission parameters are measured by
first stimulating the circuit on the left side (V1
and I1) while holding the right side open-circuited.
This condition ensures I2=0, under which condition
you may easily determine a11 = V1/V2 and a21 =
I1/V2.
Then the right side is shorted to ground, ensuring
V2=0, while you measure a12 = V1/I2 and a22 = I1/I2.
The measurements are repeated on a dense grid of
frequencies spanning the range of interest, such
that the parameters a11, etc., all become functions
of frequency.
One difficulty associated with transmission
parameters is that in a practical high-frequency
circuit it is extrememly difficult to obtain the
perfect open-circuit and short-circuit conditions
required to make good measurements. For example, the
open-circuit measurement is always corrupted by
parasitic capacitance shunting port 2. The short-
circuit measurement is corrupted by parasitic
inductance in series with port 2.
The S-parameter method circumvents the open/short
difficulties by always connecting both ports to
transmission lines with stable, well-defined
impedances. To the extent that such a test setup
better represents the actual working conditions of
your circuit, the S-parameter method generates a
more accurate model. Note, however, that the S-
parameter method requires that the measurements be
made at a PARTICULAR impedance Z0, and that this
impedance, for best results, should match closely
the impedance under which your circuit element will
actually operate.
To make clear the importance of the test-circuit
impedance, consider that a good 50-ohm connector, if
implemented with proper vias, and when measured with
Z0=50 ohms may exhibit very small reflection
coefficients s11 and s22. The same connector, if
measured with a test impedance of Z0=75 ohms might
produce terrible reflection coefficients. The values
of the S-parameter matrix are thus quite sensitive
to the impedances used during the test. The subject
of correcting an S-parameter matrix to compute the
result you would have gotten at a different level of
test impedance is considered at the end of this
article.
The transmission parameters are the parameters best
suited for calculating the performance of cascaded
systems, taking into account reflections.
Given a set of system components represented by
transmission matrices A, B,...Z, the transmission
matrix representing the cascaded combination of all
three pieces is constructed by simply forming the
matrix-multiplication product of all the pieces:
(AB...Z). The resulting combined transmission matrix
properly represents all the transmission gains and
all the internal reflections associated with the
combination of system components.
The transmission-matrix description excels at
modeling systems that incorporate multiple cascaded
sections with noticeable reflections, such as a chip
I/O driver followed by a chip package, a pcb trace,
a connector, another pcb trace, another chip
package, and a parasitic load at the receiver.
Supposing that you have an S-parameter description
of a connector, and an S-parameter description of a
backplane, how do you combine these pieces to
produce an S-parameter description of the whole
system?
This is done by first converting each S-parameter
description to a transmission-matrix description and
then multiplying together the transmission matrices
corresponding to the system components you intend to
cascade. The result is a transmission-matrix
description of the whole system.
To facilitate your work, here are the conversions
from S-matrix format to transmission-matrix (A-
matrix) format. These formulas assume that Z0 is
purely real and the same on both sides of the S-
parameter test setup. If that isn't true, more
complicated formulas apply (see for example, Chan
Chan and Chan, "Analysis of Linear Networks and
Systems", Addison Wesley, 1972, Lib. Congress Cat.
Card No. 70-156589).
Given S, find A
a11 = [1 + s11 - s22 - (s11*s22 - s12*s21)]/2*s21
a21 = [1 - s11 - s22 + (s11*s22 - s12*s21)]/2*s21*Z0
a12 = Z0*[1 + s11 + s22 + (s11*s22 - s12*s21)]/2*s21
a22 = [1 - s11 + s22 - (s11*s22 - s12*s21)]/2s21
Given A, find S
D = a12 + Z0*(a11 + a22) + Z0*Z0*a21
s11 = [a12 + Z0*(a11 - a22) - Z0*Z0*a21]/D
s21 = 2*Z0/D
s12 = 2*Z0*(a11*a22 - a12*a21)/D
s22 = [a12 - Z0*(a11 - a22) - Z0*Z0*a21]/D
[Note that the conversion calculcations are done at
each frequency on your dense grid. This results in a
lot of computation, but that's why we have
computers.]
Given the source and load impedances in the network
you may extract the transmission gain of the system
(including the effects of internal reflections)
directly from the combined tranmission matrix (see
Appendix C of High-Speed Signal Propagation).
Alternately, if you have access to an S-parameter
simulator you can convert the combined transmission
matrix back to a single S-parameter matrix
representing the whole system, and then use it.
You'll get the same answer either way.
Now let's look at one of the very interesting uses
of the transmission matrix: scaling the length of a
transmission line.
Sometimes it happes that you have measured the S-
parameters of a piece of cable, or a backplane
trace, and wish to have an S-parameter description
of a longer (or shorter) length of the same cable.
The solution to this problem is to first convert the
S-parameter description to a transmission matrix,
then scale the length, and finally convert back to S-
parameters.
To see how length-scaling works with transmission
parameters, first consider the simple case of
doubling the length of the transmission medium. This
effectively cascades two identical sections of
transmission line. If the tranmission-matrix for a
single section of line is A, then the result for a
double-length section would be A*A (using matrix
multiplication). Further integer-length extensions
are provided by compounding higher and higher powers
of the matrix A, such that the complete model for a
section n-times longer than the basic unit section
equals A raised to the power of n.
An even more interesting fact (which I don't have
space to prove here) is that the power-of-A formula
works even for non-integer n, and also for
fractional n. Therefore if you want to model a
section of transmission line whole length is, say,
.316 times the length of your basic unit section you
simply form the matrix A raised to the power of .316
If you haven't previously encountered the idea of
raising a matrix to a fractional power, let me tell
you a little about it. The concept is that you first
decompose the matrix using an eigenvalue
decomposition. Routines for eigenvalue decomposition
exist in all the major mathematical-spreadsheet
packages (MatLab, Mathematica, and my favorite,
MathCad). The eigenvalue decomposition produces
three matrices, U, D, and V, which when multiplied
together (UDV) equal your original matrix A. Without
going into a lot of detail, the matrix D is always a
diagonal matrix. The diagonal elements of D are
called the eigenvalues of A (see any book on linear
algebra, for example Gilbert Strang, "Linear Algebra
and Its Applications", Academic Press, 1976, ISBN 0-
12-673650-2). To form the matrix A raised to the
power of n what you do is first form a new matrix E
by raising each element of the diagonal matrix D to
the power of n, and then use E to form a new matrix
(UEV), which is your answer. Convert (UEV) from
transmission-matrix format back to S-parameter
format and you now have a length-scaled S-parameter
model of the whole system.
Let me advise you that such manipulations work best
on systems, such as long uniform transmission lines,
that incorporate few if any internal reflections.
It's OK if you have reflections at the front and
back ends of the line, as those will properly scale
in time, but if you have internal reflections (such
as vias) internal to your unit-length standard
measurement then when you scale that measurement to,
say, half length the equations will produce some
kind of system with different internal reflections,
such that when you cascade two half-length systems
in series you will get the original reflection
pattern observed during the test. That answer is
mathematically correct but it probably
isn't what you want. The original unit-length S-
parameter setup should incorporate only the main
body of the transmission structure, not any
imperfections along the way. Any mismatches at the
ends of the transmission structure are properly
handled by this technique.
My final subject has to do with correcting the value
of Z0 used during S-parameter characterization.
The conversion from S-parameter to transmission-
matrix format depends in each direction upon an
assumed value for the characteristic impedance Z0 of
the test setup. If you have made S-parameter
measurements at one level of impedance and wish to
see what S-parameter matrix you would have gotten
with a different level of impedance Z1, then convert
from S-parameters to transmission-parameters using
Z0, and then go back the other way using Z1.
I hope these brief comments are helpful to you.
______________________________________________________
Join us at our upcoming seminar in Dallas, TX, Mar.
24-25, 2003. A full schedule of cities and dates
appears at: www.sigcon.com.
WIN A FREE BOOK! If you recommend my seminar to any of
your colleagues and they sign up, I will send you one
free autographed copy of my new book, "High-Speed Signal
Propagation" when it is published in March. Be sure
your colleague mentions your name when they are
registering. Thanks for your support.
Check out our newsletter archives at www.sigcon.com,
under "Archives".
High-Speed Signal Propagation: More Black Magic is
now available for presale at www.barnesandnoble.com.
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