Some basic network theory

Revised May 15, 2008

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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 four-port networks. We assume that you know a little about S-parameters and maybe even some matrix algebra. Here's a clickable index to this page:

Passive versus active devices

Unilateral versus non-unilateral devices

Reciprocal versus non-reciprocal devices

Lossless networks

Matched networks

Special properties of three-port 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 non-reciprocal
• 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 non-unilateral devices

When we are discussing unilateral devices, we are talking about ideal amplifiers or other active devices. Two-port S-parameters almost always use the convention that S21 is the forward direction (the direction of gain) and S12 is the reverse-isolation direction. A unilateral device has the property that S12=0. In practice, this is impossible, but it sure makes analysis easier.

One problem with non-unilateral amplifiers is that they have poor directivity.

Properties of reciprocal and non-reciprocal 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 non-reciprocal networks are RF amplifiers and an isolators. In both cases, scattering parameter S21 is much different from S12.

A reciprocal network always has a symmetric S-parameter matrix. That means that S21=S12, S13=S31, etc. All values along the lower-left to upper right diagonal must be equal. A two-port S-parameter 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 S-parameter matrix is one simple check to see if the data is valid. Here is an example of S-parameters of a network that is either a non-reciprocal 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 S-parameter 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' sum-of-the-squares 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 sum-of-the-squares-equals-unity 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 S-parameters of a two-port test fixture and it looks like this, you can tell her for Christsakes to re-measure 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 two-port 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 (Z₀). 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=(Z-Z0)/(Z+Z0), when gamma (which in this case could be S11, S22, etc) equals zero, Z=Z0.

S-parameters are in units of volts/volt, not ohms.

A special property of three-port networks

The math behind the theory of three-port 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 three-port network to be reciprocal, lossless and matched all at the same time. You can only have two of these properties.

Reciprocal three-port 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 (loss-less) . 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 three-port junctions, particularly in power monitoring, combining and divider applications.

If a three-port 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 four-port device -- just like a magic-tee 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 three-port 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!