The 4-port Scattering matrix.

Transmission line notes.
Introductions to Scattering Parameters.
Microwave measurements.
Microwaves for satellite communications.

The properties of the S-matrix.

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.

The n input wave complex amplitudes can be written as a column n-vector. If we consider the voltage amplitudes, this can be represented as the column vector

```        V1+
V2+
V3+
.              =         [V+]
.

.
Vn+

```
And there will be a similar n-vector of outgoing voltage complex amplitudes [V-]

The S-matrix, written [S], links these two n-vectors by the relationship [V-] = [S][V+]

The scattering matrix S consists of n x n elements, each of which is a complex number.

For a lossless junction, the total output power must equal the total input power. Considering a system where the [V+] and [V-] are rms complex quantities and the characteristic impedance is normalised to unity, then the total input power is given by

```
total input power =  [V+]t [V+]*

where [V+]t is the transpose (a row vector) of [V+] and
[V+]* is the complex conjugate of the column vector [V+].

similarly

total output power = [V-]t [V-]*

Each of these products is a scalar formed from the sum of
n terms for the power in (or out) of each of the n ports.

Using our definition of the S-matrix we can write

[V-]t = ([S][V+])t

and so the condition that the total output power is equal
to the total input power, which is that

[V-]t [V-]*  =  [V+]t [V+]*

may be expressed as

[V-]t [V-]* = ([S][V+])t ([S][V+])*
= [V+]t ([S]t [S]*) [V+]*

and so the product

[S]t [S]* must equal the unit matrix [U]

[U] has elements uij which are 1 if i=j and 0 otherwise.

```

Thus, in words, the transpose of the S matrix is equal to the inverse of the complex conjugate of the S matrix. This is known as the Unitary property of the S matrix.

It is a consequence of the lossless nature of the n-port microwave circuit, and does not hold for lossy n-ports.

We note that a variation of the S-matrix by a complex multiplicative factor of modulus unity does not affect the Unitary property. Suppose we multiply every term in the S-matrix by a factor exp{-j phi}. Then the corresponding complex conjugate [S]* has every term multiplied by the complex conjugate factor exp{+j phi} and this does not affect the unitary property since exp{-j phi} exp{+j phi} = 1. This multiplicative phase factor is physically equivalent to altering the frequency, or to adding a transmission line to each port, each additional length being the same.

Four-port circuits.

A four-port network has a 4 x 4 S-matrix. It can be shown that if the diagonal elements of the S-matrix are all zero, then the 4-port has the properties of a directional coupler. This curious fact arises from the Unitary property of the S-matrix as discussed above.

If the diagonal S-matrix elements are all zero, there is no reflection from any port when the other three ports are terminated. Such a device is said to be "simultaneously matched on all ports".

The S-matrix for such a 4-port, with zero diagonal elements, has the following form. We assume the S-matrix is symmetric as the 4-port has no ferrite in it and is a reciprocal device.

```

0   s12  s13  s14
s21   0  s23  s24
s31 s32    0  s34
s41 s42  s43    0

here, sij = sji so there are at most 6 independent
complex elements, represented by at most 12 independednt
real numbers.

so the S-matrix reduces to

0   s12  s13  s14
s12   0  s23  s24
s13 s23    0  s34
s14 s24  s34    0

Now we apply the unitary property of the S-matrix.

Considering the diagonal unit elements of [U] we have, summing the
squared moduli of the column elements,

s12s12* + s13s13* + s14s14*                               = 1
s12s12*                       +s23s23* + s24s24*          = 1
s13s13*             +s23s23*           +s34s34* = 1
s14s14*            + s24s24* +s34s34* = 1

This set of equations consists of four relations between six
unknowns, so there are only two independent s-parameter
amplitudes or sizes.

Considering the off-diagonal (zero) elements of [U] we have

s12s23* + s14s34* = 0
s12s14* + s23s34* = 0
s12s13* + s24s34* = 0      and complex conjugates
s12s24* + s13s34* = 0
s13s23* + s14s24* = 0
s13s14* + s23s24* = 0

This further set of six constraints taken with the four we had
earlier means that there are a total of 10 constraints between
12 real numbers, so only two numbers are needed to describe the
matched 4-port completely.

```

The directional coupler

A directional coupler has four ports which we can draw diagramatically as

```
1              2
4              3

```
taken in clockwise order around the device. For illustration we consider a directional coupler where port 1 couples to ports 2 and 3 only but not to port 4, and where port 2 couples to ports 1 and 4 only, but not to port 3.

The absence of coupling between ports 1 and 4, and between ports 2 and 3, means that s14 = 0 and s23 = 0.

Applying these properties to our general matched 4-port relationships above, we find that

```
s12s12* + s13s13* = 1
s12s12* + s24s24* = 1

so that
s13s13* = s24s24*

and similarly
s12s12* + s24s24* = 1
s34s34* + s24s24* = 1

so that

s12s12* = s34s34*

```
Let us call the size or complex modulus of s13 the coupling strength k. Then we see that the size of s12 is sqrt(1-k^2). The sizes of s24 and s34 are then k and sqrt(1-k^2) respectively. The coupling strength k is the first of our two arbitrary adjustable parameters.

We now have only one remaining parameter that we can choose; and it has to be a phase angle. Now, since the S-matrix is only determined to within a complex multiplicative factor of modulus unity, as explained above, we can choose the phase angle of s12 to be zero degrees, so that s12 is the real number sqrt(1-k^2). This does not use up one of the constraints imposed by the unitary nature of the S-matrix. Then the relationships above allow us to determine the S-matrix for our directional coupler and we will find that we have a single phase angle which we can specify.

Looking again at our relationships for the off-diagonal (zero) elements of [U] and substituting our values above for the directional coupler we find that since s14=s23=0 there are only two remaining complex constraints..

```

s12s13* + s24s34* = 0
s12s24* + s13s34* = 0

the size of each of the terms in these two equations is
k*sqrt(1-k^2). If we call the phase angle of s13 theta,
the phase angle of s24 phi, and the phase angle of s34
psi then the equations reduce to (since the phase of s12 is 0)

exp{-j theta} + exp {j phi - j psi}   = 0
exp{-j phi}   + exp {j theta - j psi} = 0

therefore

cos{theta} + cos{phi-psi} = 0
-sin{theta} + sin{phi-psi} = 0
cos{phi} + cos{theta-psi} = 0
-sin{phi} + sin{theta-psi} = 0

and from the sin equations

theta = phi-psi
phi = theta-psi

so that psi = 0 and s34=s12 = sqrt(1-k^2) so that both are entirely
real.

we have therefore that sin{phi} = -sin{theta}
and  cos{phi} =  cos{theta}

so that theta = - phi.

```
We can therefore choose theta arbitrarily. Common angles for theta are 90 degrees (pi/2 radians), or 180 degrees (pi radians).

Copyright D.Jefferies 1997, 2004.
D.Jefferies
22nd December 2004.