1. ENE 429 Antenna and Transmission Lines Theory
Lecture 7 Matching network and
Scattering parameters RS

2. Review Input impedance for finite length line Quarter wavelength line
Half wavelength line
Smith chart
A graphical tool to solve transmission line problems
Use for measuring reflection coefficient, VSWR, input impedance, load impedance, the locations of Vmax and Vmin
Impedance matching

3. Parallel and Series Connections (1)
Parallel connection of R and L elements

4. Parallel and Series Connections (2)
Parallel connection of R and C elements

5. Parallel and Series Connections (3)
Series connection of R and L elements

6. Parallel and Series Connections (4)
Series connection of R and C elements

7. Ex1 Adding a series inductor L (zL = j0.8) to an impedance z = 0.3-j0.3.

8. Ex2 Adding a shunt inductor L (yL = -j2. 4) to an admittance y = 1
Ex2 Adding a shunt inductor L (yL = -j2.4) to an admittance y = 1.6+j1.6.

9. L matching networks

10. Forbidden regions
Sometimes a specific matching network cannot be used to accomplish a given match.

11. 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.

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

13. z Parameters v1 = z11i1 + z12i2 v2 = z21i1 + z22i2
or in the matrix form

14. y, h, and ABCD parameters y parameters These two-port h parameters
representations
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.

15. Two-port connected in series

16. Two-port connected in shunt

17. Two-port connected in cascade fashion

18. 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.

19. Introduction of 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 incident 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