1. ENE 428 Microwave Engineering
Lecture 9 Scattering parameters and their properties.

2. Impedance and Admittance Matrices
Consider an arbitrary N-port network below,

3. The impedance [Z] matrix relates voltages and currents.
So we can write [V] =[Z][I]
V1 = Z11I1 + Z12I2
V2 = Z21I1 + Z22I2,
etc.

4. The admittance [Y] matrix relates currents and voltages.
So we can write [I] =[Y][V]
I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2,
etc.

5. Zij or Yij can be found by o/c or s/c at all other ports
and
Zij can be found by driving port j with the current Ij, open
-circuiting all other ports and measuring the open-circuit
Voltage at port i.
Yij can be found by driving port j with the voltage Vj, short-circuiting all other ports and measuring the short-circuit current at port i.

6. Reciprocal Network
Many practical networks are reciprocal (not containing any nonreciprocal media such as ferrites or plasmas, or active devices)
Impedance and admittance matrices are symmetric, that is
and

7. Lossless Network
If the network is lossless, then the net real power delivered to the network must be zero. Thus, Re{Pav} = 0.
Then for a reciprocal lossless N-port junction we can show that the elements of the [Z] and [Y] matrices must be pure imaginary
where m, n = port index.

8. Single- and Two-port Networks
The analysis can be done easily through simple input-output relations.
Input and output port parameters can be determined without the need to know inner structure of the system.
At low frequencies, the z, y, h, or ABCD parameters are basic network input-output parameter relations.
At high frequencies (in microwave range), scattering parameters (S parameters) are defined in terms of traveling waves and completely characterize the behavior of two-port networks.

9. Basic definitions
Assume the port-indexed current flows into the respective port and the associated voltage is recorded as indicated.

10. Ex of h and ABCD parameters for two-port network
H parameters
ABCD parameters
These two-port representations (Z, Y, H, and ABCD) are very useful at low frequencies because the parameters are
readily measured using short- and open- circuit tests at the terminals of the two-port network.

11. Two-port connected in series

12. Two-port connected in shunt

13. Two-port connected in cascade fashion

14. Disadvantages of using these parameters at RF or microwave frequency
Difficult to directly measure V and I
Difficult to achieve open circuit due to stray capacitance
Active circuits become unstable when terminated in short- and open- circuits.

15. Scattering Matrix (1)
The scattering matrix relates the voltage waves incident on the ports to those reflected from the ports
Scattering parameters can be calculated using network analysis techniques or measured directly with a network analyzer.

16. Scattering Matrix (2)
A specific element of the [S] matrix can be determined as
Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads.
Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.

17. Reciprocal networks and lossless networks
[S] matrix for a reciprocal network is symmetric,
[S]=[S]t.
[S] matrix for a lossless network is unitary that means

18. Ex1 A two-port network has this following scattering matrix
Determine if the network is reciprocal, and lossless

19. Introduction of generalized scattering parameters (S parameters)
Measure power and phase
Use matched loads
Devices are usually stable with matched loads.
S- parameters are power wave descriptors that permits us to define input-output relations of a network in terms of incident and reflected power waves

20. Introduction of the normalized notation (1)
we can write
Let’s define
and

21. Introduction of the normalized notation (2)
We can also show a(x) and b(x) in terms of V(x) and I(x) as
and

22. Normalized wave generalization
For a two-port network, we can generalize the relationship between b(x) and a(x) in terms of scattering parameters. Let port 1 has the length of l1 and port 2 has the length of l2, we can show that
or in a matrix form,
Observe that a1(l1), a2(l2), b1(l1), and b2(l2) are the values of in-
cident and reflected waves at the specific locations denoted as
port 1 and port 2.

23. The measurement of S parameters (1)
The S parameters are seen to represent reflection and transmission coefficients, the S parameters measured at the specific locations shown as port 1 and port 2 are defined in the following page.

24. The measurement of S parameters (2)
(input reflection coefficient with output
properly terminated)
(forward transmission coefficient with
output properly terminated)
(output reflection coefficient with input
properly terminated)
(reverse transmission coefficient with
input properly terminated)

25. The advantages of using S parameters
They are measured using a matched termination.
Using matched resistive terminations to measure the S parameters of a transistor results in no oscillation.

26. The chain scattering parameters or scattering transfer parameters (T parameters) (1)
The T parameters are useful in the analysis of cascade connections of two-port networks.
The relationship between S and T parameters can be developed. Namely,

27. The chain scattering parameters or scattering transfer parameters (T parameters) (2)
and
We can also write

28. Review (2)
Normalized notation of the incident a(x) and reflected waves b(x) are defined as
The relationship between the incident and reflected waves and the scattering matrix of the two-port network,

29. Shifting reference planes
S parameters are measured using traveling waves, the positions where the measurements are made are needed to be specified. The positions are called reference planes.

30. Scattering matrix of the shifting planes
At the reference planes at port 1 and port 2, we write the scattering matrix as
and at port 1’ and port 2’ as
We can show that