# Scattering parameters.

## What are scattering parameters?

"Scattering" is an idea taken from billiards, or pool. One takes a cue ball and fires it up the table at a collection of other balls. After the impact, the energy and momentum in the cue ball is divided between all the balls involved in the impact. The cue ball "scatters" the stationary target balls and in turn is deflected or "scattered" by them.

In a microwave circuit, the equivalent to the energy and momentum of the cue ball is the amplitude and phase of the incoming wave on a transmission line. (A rather loose analogy, this). This incoming wave is "scattered" by the circuit and its energy is partitioned between all the possible outgoing waves on all the other transmission lines connected to the circuit. The scattering parameters are fixed properties of the (linear) circuit which describe how the energy couples between each pair of ports or transmission lines connected to the circuit.

Formally, s-parameters can be defined for any collection of linear electronic components, whether or not the wave view of the power flow in the circuit is necessary. They are algebraically related to the impedance parameters (z-parameters), also to the admittance parameters (y-parameters) and to a notional characteristic impedance of the transmission lines.

A visual demonstration of the meaning of scattering may be given by throwing a piece of chalk at a blackboard....

## Definitions.

An n-port microwave network has n arms into which power can be fed and from which power can be taken. In general, power can get from any arm (as input) to any other arm (as output). There are thus n incoming waves and n outgoing waves. We also observe that power can be reflected by a port, so the input power to a single port can partition between all the ports of the network to form outgoing waves.

Associated with each port is the notion of a "reference plane" at which the wave amplitude and phase is defined. Usually the reference plane associated with a certain port is at the same place with respect to incoming and outgoing waves.

The n incoming wave complex amplitudes are usually designated by the n complex quantities an, and the n outgoing wave complex quantities are designated by the n complex quantities bn. The incoming wave quantities are assembled into an n-vector A and the outgoing wave quantities into an n-vector B. The outgoing waves are expressed in terms of the incoming waves by the matrix equation B = SA where S is an n by n square matrix of complex numbers called the "scattering matrix". It completely determines the behaviour of the network. In general, the elements of this matrix, which are termed "s-parameters", are all frequency-dependent.

```
For example, the matrix equations for a 2-port are

b1 = s11 a1 + s12 a2

b2 = s21 a1 + s22 a2

And the matrix equations for a 3-port are

b1 = s11 a1 + s12 a2 + s13 a3

b2 = s21 a1 + s22 a2 + s23 a3

b3 = s31 a1 + s32 a2 + s33 a3

```

The wave amplitudes an and bn are obtained from the port current and voltages by the relations a = (V + ZoI)/(2 sqrt(2Zo)) and b = (V - ZoI)/(2 sqrt(2Zo)). Here, a refers to an if V is Vn and I In for the nth port. Note the sqrt(2) reduces the peak value to an rms value, and the sqrt(Zo) makes the amplitude normalised with respect to power, so that the incoming power = aa* and the outgoing power is bb*.

A one-port scattering parameter s is merely the reflection coefficient gamma, and as we have seen we can relate gamma to the load impedance zL = ZL/Zo by the formula gamma = (zL-1)/(zL+1).

Similarly, given an n by n "Z-matrix" for an n-port network, we obtain the S matrix from the formula S = (Z-I)(Z+I)^-1, by post-multiplying the matrix (Z-I) by the inverse of the matrix (Z+I). Here, I is the n by n unit matrix. The matrix of z parameters (which has n squared elements) is the inverse of the matrix of y parameters.

## 2-port scattering matrix.

In the case of a microwave network having two ports only, an input and an output, the s-matrix has four s-parameters, designated

```	    s11     s12

s21     s22

```

These four complex quantites actually contain eight separate numbers; the real and imaginary parts, or the modulus and the phase angle, of each of the four complex scattering parameters.

Let us consider the physical meaning of these s-parameters. If the output port 2 is terminated, that is, the transmission line is connected to a matched load impedance giving rise to no reflections, then there is no input wave on port 2. The input wave on port 1 (a1) gives rise to a reflected wave at port 1 (s11a1) and a transmitted wave at port 2 which is absorbed in the termination on 2. The transmitted wave size is (s21a1).

If the network has no loss and no gain, the output power must equal the input power and so in this case |s11|^2 + |s21|^2 must equal unity.

We see therefore that the sizes of S11 and S21 determine how the input power splits between the possible output paths.

NOTE s21 relates power OUT of 2 to power IN to 1, not vice versa as it is easy to think at first sight.

Clearly, if our 2-port microwave network represents a good amplifier, we need s11 rather small and s21 quite big, let us say 10 for a 20dB amplifier.

In general, the s-parameters tell us how much power "comes back" or "comes out" when we "throw power at" a network. They also contain phase shift information.

## Reciprocity

Reciprocity has to do with the symmetry of the s-matrix. A reciprocal s-matrix has symmetry about the leading diagonal. Many networks are reciprocal. In the case of a 2-port network, that means that s21 = s12 and interchanging the input and output ports does not change the transmission properties. A transmission line section is an example of a reciprocal 2-port. A dual directional coupler is an example of a reciprocal 4-port. In general for a reciprocal n-port sij = sji.

Amplifiers are non-reciprocal; they have to be, otherwise they would be unstable. Ferrite devices are deliberately non-reciprocal; they are used to construct isolators, phase shifters, circulators, and power combiners.

## Examples of scattering matrices.

### One-port S-matrix

This is a matrix consisting of a single element, the scattering parameter or reflection coefficient. You may think of it as a 1 by 1 matrix; one row and one column.

• A matched transmission line, s11 = 0

• A short circuit, at the short, s11 = 1 angle -180 degrees

• The input to a 0.2 lambda line feeding a short cicuit, s11 = 1 angle -324 degrees

• Your turn again. The input to a transmission line of length 20.35 lambda connected to a normalised load 2+j1?

### Two port S-matrices

These are 2 by 2 matrices having the following s parameters; s11 s12 s21 s22.

• A 0.1 lambda length of transmission line

```           s11 = 0             s12 = 1 angle -36 degrees
s21 = 1 angle -36   s22 = 0
```

• A 10dB amplifier, matched on input and ouput ports

```           s11 = 0                   s12 = small
s21 = 3.16 angle -theta   s22 = 0
```

• An isolator having 1dB forward loss, 21 dB backward loss, matched on ports 1 and 2

```           s11 = 0                   s12 = 0.0891 some angle
s21 = 0.891 some angle    s22 = 0

```

• Your turn. A 9 cm length of waveguide of cross sectional dimensions 2.5 cms by 1.8 cms at a frequency of 8GHz

• Your turn again. The waveguide above has a capacitative iris placed mid way along it. Assume that 30% of the power incident on the iris gets through it, and 70% is reflected. Hint. represent the normalised load admittance of the iris as y = 1 + js and calculate s for 70% power reflection. That will give you the phase shift on reflection as well.

### Three port S-matrices

These are 3 by 3 matrices having the following s parameters

```            s11  s12  s13
s21  s22  s23
s31  s32  s33
```
• Your turn. Write down the s matrix for a perfect circulator with -70 degrees of phase shifts between successive ports.

• Your turn again. A coaxial cable is connected in a Y arrangement with each arm 12.3 wavelengths long. At the junction, the cable arms are all connected in parallel. Write down its S matrix and comment on this method of splitting power from a TV down lead to serve two television sets.

## Stability.

If a 1-port network has reflection gain, its s-parameter has size or modulus greater than unity. More power is reflected than is incident. The power usually comes from a dc power supply; Gunn diodes can be used as amplifiers in combination with circulators which separate the incoming and outgoing waves. Suppose the reflection gain from our 1-port is s11, having modulus bigger than unity. If the 1-port is connected to a transmission line with a load impedance having reflection coefficient g1, then oscillations may well occur if g1s11 is bigger than unity. The round trip gain must be unity or greater at an integer number of (2 pi) radians phase shift along the path. This is called the "Barkhausen criterion" for oscillations. Clearly if we have a Gunn source matched to a matched transmission line, no oscillations will occur because g1 will be zero.

If an amplifier has either s11 or s22 greater than unity then it is quite likely to oscillate or go unstable for some values of source or load impedance. If an amplifier (large s21) has s12 which is not negligibly small, and if the output and input are mismatched, round trip gain may be greater than unity giving rise to oscillation. If the input line has a generator mismatch with reflection coefficient g1, and the load impedance on port 2 is mismatched with reflection coefficient g2, potential instability happens if g1g2s12s21 is greater than unity.

D.Jefferies email
6th June 2007