1. ENE 429 Antenna and Transmission Lines
Lecture 8 Scattering parameters and their properties.

2. Review (1) Two-port network
- At low frequencies, the z, y, h, or ABCD parameters are basic network input-output parameter relations. The parameters are readily measured using short- and open-circuit tests at the terminals.
- At RF or microwave frequency, these parameter are difficult to measure
- 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.
- S parameters can be easily measured using matched loads which ensure the stability of the network.

3. 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,

4. 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.

5. 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

6. Properties of scattering parameters (1)
The average power associated with incident wave on the primed i th port (i.e., at x1=0 and x2 = 0) can be expressed as
Similarly, the average reflected power is

7. Properties of scattering parameters (2)
When port 1’ is excited by the sinusoidal voltage source with source impedance Z1 = Z01 and port 2’ is matched (Z2 = Z02), we can show that
Note: PAVS represents the power available from the source

8. Properties of scattering parameters (3)
If Z1 is not equal to Z01, the net power delivered to port 1’ or to port 1 (the line is lossless, the delivered power is then equal) is then or
The power delivered to the load when Z2 = Z02 can then be derived similarly and can expressed as
Note: PAVS represents the power available from the source

9. Meaning of S parameters
represents the ratio of the power reflected from port 1 to the power available at port 1.
From
then we can write
and
S21 represents a forward voltage transmission coefficient from port 1 to port 2.

10. Equivalent network

11. Transducer power gain GT
represents the ratio of power deliver to the load Z02 to the power available from the source,
E1, TH. The ratio is known as transducer power gain GT.
represents the ratio of power
reflected from port 2 to the power available at port 2, PAVN or power available from the network.

12. A reverse voltage transmission coefficient from and a reverse GT
S12 represents a reverse voltage transmission coefficient from port 2 to port 1.
represents a reverse transducer power gain.

13. These S parameters are measured in Z0 system
These S parameters are measured in Z0 system. If Z1 and Z2 are arbitrary then the gain GT is no longer equal to

14. Ex Evaluate the S parameters in a Z0 system of (a) series impedance Z and (b) a shunt admittance Y.

16. Power waves and generalized scattering parameters
The analysis of lumped circuits (from one-port to n-port lumped circuits) in terms of a new set of waves, called power waves.
One-port network:
where RS = Re[ZS].

17. Power transmission (1) These definitions are such that the quantity is
equal to the power available from the source, and the reflected power wave bp is zero when the load impedance is conjugately matched to the source impedance (i.e. when ZL = ZS* )
Power delivered to the load
It can be shown as

18. Power transmission (2) The reflected power
A power-wave reflection coefficient P

19. Voltage and current in terms of power waves (1)
Voltage and current can be expressed as a function of incident and reflected power waves as
and
We can also define incident and reflected voltage and currents and relate them to the power waves. That is, let
where
and

20. Voltage and current in terms of power waves (2)
and where and

21. Reflection coefficients
A voltage reflection coefficient V can be defined as
A current reflection coefficient I can be defined as
When the normalizing impedance ZS is real and positive and equal to Z0 or ZS = ZS* = Z0,

22. Two-port network
Two-port network representation in terms of generalized S parameters.

23. Two-port network with E2 = 0

24. Generalized scattering parameters (Sp parameters)
Generalized scattering parameters (Sp parameters) denoted by Sp11, Sp12, Sp21, and Sp22 can be shown in terms of power waves as follows:
The input power to the two-port network can be expressed as
where is measured when ap2 = 0 or E2 = 0.

25. Power transmission and power gain
The power delivered to the load Z2 is
Therefore the transducer power gain GT is given by
If we let Z1 = Z2 = Z0, it follows that