Some
basic network theory
Revised July 8,
2011
Click
here to go to our main page on Sparameters
Before you learn about microwave
topics such as couplers and splitters,
you first have to suffer through some definitions which are used
in network theory, so that we can compare the functions of various
three and fourport networks. We assume that you know a little about
Sparameters and maybe even some matrix algebra. Here's a clickable
index to this page:
Passive
versus active devices
Unilateral
versus nonunilateral devices
Reciprocal
versus nonreciprocal devices
Isotropy
(separate page)
Lossless
networks
Matched
networks
Special
properties of threeport networks
Dispersion
(separate page)
Passive
devices versus active devices
A passive device contains
no source that could add energy to your signal, with one exception.
The first law thermodynamics, conservation of energy, implies that
a passive device can't oscillate. An active device is one
in which an external energy source is somehow contributing to the
magnitude of one or more responses.
The important properties of a
passive network are:
 whether it is reciprocal or
nonreciprocal
 whether it is lossy or lossless
 whether it is impedance matched
or unmatched.
What is the exception to the
"passive rule" about not adding energy? Mixers!
Here the local oscillator adds energy, but because of the way
that a mixer works, no signal gain is possible.
Unilateral
versus nonunilateral devices
When we are discussing unilateral
devices, we are talking about ideal amplifiers
or other active devices. Twoport Sparameters almost always use
the convention that S21 is the forward direction (the direction
of gain) and S12 is the reverseisolation direction. A unilateral
device has the property that S12=0. In practice, this is impossible,
but it sure makes analysis easier.
One problem with nonunilateral
amplifiers is that they have poor directivity.
Properties
of reciprocal and nonreciprocal networks
A reciprocal network is one in
which the power losses are the same between any two ports regardless
of direction of propagation (scattering parameter S21=S12, S13=S31,
etc.) A network is known to be reciprocal if it is passive and contains
only isotropic materials.
Examples of reciprocal networks include cables, attenuators,
and all passive power splitters and couplers.
Anisotropic materials
have different electrical properties (such as relative dielectric
constant) depending on which direction a signal propagates through
them. One example of an anisotropic material is the class
of materials known as ferrites,
from which circulators and isolators are made. Two classic examples
of nonreciprocal networks are RF amplifiers and isolators.
In both cases, scattering parameter S21 is much different from S12.
A reciprocal network always has
a symmetric Sparameter matrix. That means that S21=S12, S13=S31,
etc. All values along the lowerleft to upper right diagonal must
be equal. A twoport Sparameter matrix (at a single frequency)
is represented by:
If you are measuring a network
that is known to be reciprocal, checking for symmetry across the
diagonal of the Sparameter matrix is one simple check to see if
the data is valid. Here is an example of Sparameters of a network
that is either a nonreciprocal network, or your technician has
a drinking problem.
(Need to add figure)
Although the data shows the part
is well matched (S11 and S22 magnitudes are low), and low loss (S21
and S12 magnitudes are high). The magnitudes of S12 and S21 are
equal, so what is the problem? The phase angles of S12 and S21 are
significantly different. That can't be right.
Properties
of lossless networks
For a network to be lossless,
all of the power (or energy) that is incident at any one port has
to be accounted for by summing the power output at the other ports
with the power reflected at the incident port. None of the power
is converted to heat or radiated within a lossless network. Note
that an active device is not in the same category as a lossless
part, since power is added to the network through its bias connections.
Within the Sparameter matrix
of a lossless network, the sum of the squares of the magnitudes
of any row must total unity (unity is a fancy way of saying "one").
If any of the rows' sumofthesquares is less than one, there is
a lossy element within the network, or something is radiating.
Why are we looking at sum of
the squares instead of sum of the elements themselves? Because the
S matrix is express in terms of voltage, and as we said, we are
accounting for power. Power is proportional to voltage squared,
get it?
Guess what? You can never make
a lossless network. But you can come extremely close, especially
with waveguide structures. How does the sumofthesquaresequalsunity
property of lossless networks make your life better? You can use
it as a check to see if data is bullcrap. If your Grandmother hands
you Sparameters of a twoport test fixture and it looks like this,
you can tell her for Christsakes to remeasure it after she checks
the calibration:
The "test" that it
fails is that 0.8^2 + 0.7^2 =1.13, which is different from unity
(1.00). Apparently this twoport device has gain. If it really did,
you could file a patent and become fabulously wealthy. But it doesn't,
so quit dreaming and get to the bottom of your measurement problem.
A good place to start is examining your test cables for flakiness.
For some stable VNA cables at a good price, check out Storm
Products Company.
Properties
of matched networks
A matched network is one in which
all of the ports are matched to the same impedance (Z_{0}).
A desirable quality, you must agree. Looking at the scattering matrix,
this means that the diagonal elements from top left to bottom right
are all zero. Need to add a figure!
A little more explanation of
this property (still waiting for that figure...)
If a network is matched to
fifty ohms, its reflection coefficients have magnitude zero. This
means we are at the center of the Smith chart, right at Z0.
If you look at the expression
for reflection coefficient:
gamma=(ZZ0)/(Z+Z0), when
gamma (which in this case could be S11, S22, etc.) equals zero,
Z=Z0.
Sparameters are in units of
volts/volt, not ohms.
A special
property of threeport networks
The math behind the theory of
threeport circuits such certain as couplers and splitters is not
all that complicated and has a certain elegance, like a proof of
the Pythagorean Theorem. For those of you who enjoy some good matrix
algebra derivations, we refer you to Pozar's book Microwave Engineering,
which you can find on our book recommendation
page.
We are going to skip the math
and tell you the conclusion: it is impossible for a threeport network
to be reciprocal, lossless and matched all at the same time. You
can only have two of these properties.
Reciprocal threeport junctions
are characterized by the fact that a change in the terminal conditions
at one port affects the conditions at the other ports. This effect
is particularly pronounced when the junction is dissipationless
(lossless) . The reason is that the insertion of a matching network
at one port changes the impedance characteristics at the other two
ports. This lack of isolation between ports can limit the usefulness
of threeport junctions, particularly in power monitoring, combining
and divider applications.
If a threeport is lossless,
and contains no anisotropic materials, then it will have to be reciprocal.
But it cannot have all three ports matched to 50 ohms at the same
time. Is this a bummer or what? Not really. This is the reason for
the isolation resistor in a Wilkinson
power splitter. A Wilkinson is an example of a reciprocal matched
three port network. It is only lossy between ports 2 and 3, which
has no effect on its efficiency as a combiner or splitter. The resistor
isolates the output arms as well.
This further explanation came
from an antenna guy. Thanks Bob!
In "a special property
of three port networks" it is correctly stated that a three
port device can't be reciprocal, lossless and matched all at the
same time. Then a Wilkinson is referred to as a three port device
that is matched, presumably because it is not lossless. However,
a better explanation for a Wilkinson is that it is a fourport
device  just like a magictee in waveguide  except the fourth
port is terminated by a matched load. Actually, the load (isolation
resistor) could be removed from a Wilkinson power divider and
a twin lead transmission line could be brought out of the plane
of the power divider to provide the fourth port.
Lossless threeport power splitters,
referred to as reactive combiner networks, also have applications
for microwave circuits. They are sometimes used as power dividers
and combiners when the impedance of the common leg can be different
from the impedance of the split legs, and isolation between the
split legs is not important.
That's all for now!
