1. Microwave Network Analysis
Chapter 4
Microwave Network Analysis

2. Equivalent Voltage and Current
For non-TEM lines, the quantities of voltage, current, and impedance are nor unique, and are difficult to measured. Following considerations can provide useful result:
1) Voltage and current are defined only for a particular mode, and are defined so that the voltage is proportional to the transverse electric field, and the current is proportional to transverse magnetic field.
2) The product of equivalent voltage and current equals to the power flow of the mode.
3) The ratio of the voltage to the current for a single traveling wave should be equal to the characteristic impedance of the line. This impedance is usually selected as equal to the wave impedance of the line.

3. The Concept of Impedance
Various types of impedance:
1) =(/)1/2 =intrinsic impedance of medium. This impedance is dependent only on the material parameters of medium, and is equal to the wave impedance of plane wave.
2) Zw=Et /Ht=1/Yw =wave impedance, e.g. ZTEM, ZTM, ZTE. It may depend on the type of line or guide, the material, and the operating frequency.
3) Z0 =(L /C)1/2 =1/Y0 =characteristic impedance. It is the ratio of voltage to current. The characteristic impedance is unique definition for TEM mode but not for TM or TE modes.
The real and imaginary parts of impedance and reflection coefficient are even and odd in 0 respectively.

4. Impedance and Admittance Matrices
The terminal plane (e.g. tN) is important in providing a phase reference for the voltage and current phasors.
At the nth terminal (reference) plane, the relations are given as:
Reciprocal Networks
If the arbitrary network is reciprocal ( no active devices, ferrites, or plasmas), [Y] and [Z] are symmetric matrices .

5. Example4.1: Find the Z parameters of the two-port network? Solution
Lossless Networks
If the arbitrary network is lossless, then the net real power delivered to the network must be zero. Besides
Example4.1: Find the Z parameters of the two-port network?
Solution

6. The Scattering Matrices
The scattering parameter Sij is the transmission coefficient from j port to port i when all other ports are terminated in matched loads.
[Z] or [Y]  [S]
[S]  [Z]

7. Example4.2: Find the S parameters of the 3 dB attenuator circuit?
Solution
A matched 3B attenuator with a
50 Ω Characteristic impedance

8. Reciprocal Networks Lossless Networks
[S] is symmetric matrix
No real power delivers to network.Besides
Example4.3: Determine if the network is reciprocal, and lossless? If port 2 terminated with a matched load, what is the return loss at port 1? If port 2terminated with a short circuit, what is the return loss seen at port 1?
Solution
Since [S] is not symmetric, the network is not reciprocal.
So the network is not lossless.

9. When port 2 terminated with a matched load, =S11=0.15.
When port 2 terminated with a short circuit,

10. Summary Reciprocal Networks (symmetric)
No active elements, no anisotropic material
Lossless Networks
No resistive material, no radiation

11. Example4.4: Determine if the network is reciprocal, and lossless ?
Solution
From the result of example 4.2
A matched 3B attenuator with a
50 Ω Characteristic impedance
Since the network is reciprocal but not lossless, [S] should be symmetric but not unitary.
Since the network is reciprocal but not lossless, [Z] should be symmetric but not imaginary.

12. Example4.5: Determine if the network is reciprocal, and lossless ?
Solution

14. Problem 1:Determine if the inductance networks are reciprocal, and lossless ?
Problem 2:Determine if the capacitance networks are reciprocal, and lossless ?

15. Three Port Network Reciprocal Network Matching at all ports
Lossless Network

16. Applications A counter-clockwise circulator Power splitters
[S] is unitary and matched at all ports, but not symmetric. Therefore, circulator is lossless and matched, but not reciprocal.
Power splitters
[S] is symmetric and matched at all ports, but not unitary. Therefore, circulator is reciprocal and matched, but not lossless.

17. A Shift in Reference Planes
Twice the electric length represents that the wave travels twice over this length upon incidence and reflection.
Generalized Scattering
Parameters
If the characteristic impedances of a multi-port network are different,

18. Generalized Scattering Matrices
The scattering parameter Sij defined earlier was based on the assumption that all ports have the same characteristic impedances ( usually Z0=50). However, there are many cases where this may not apply and each port has a non-identical characteristic impedance.
A generalized scattering matrix can be applied for network with non-identical characteristic impedances, and is defined as following:

19. The Transmission (ABCD) Matrix
ABCD matrix has the advantage of cascade connection of multiple two-port networks.

21. Table 4-2 Conversions between two-port network parameters

22. Example4.6: Find the S parameters of network? Solution
For reciprocal network, [Z] is is symmetric. Hence, Z12=Z21
Example4.6: Find the S parameters of network?
Solution
From Table 4-1

23. From Table 4-2

24. The Transmission [T] Matrix
At low frequencies, ABCD matrix is defined in terms of net voltages and currents. When at high frequencies, T matrix defined in terms of incident and reflected waves will become very useful to evaluate cascade networks.

25. Equivalent Circuit for Two-Port Networks
A coax-to-microstrip transition and equivalent circuit representations. (a) Geometry of the transition. (b) Representation of the transition by a “black box.” (c) A possible equivalent circuit for the transition.

26. Equivalent circuits for some common microstrip discontinuities
Equivalent circuits for some common microstrip discontinuities. (a) Open-ended. (b) Gap. (c) Change in width. (d) T-junction.

27. (a) T equivalent (b)  equivalent
Equivalent circuits for a reciprocal two-port network.
(a) T equivalent (b)  equivalent

28. Example4.7: Find the network as equivalent T and  model at 1GHz?
Solution
From Table 4-1
From Table 4-2

29. Equivalent T model
From Table 4-2
Equivalent  model

30. Example4.8: Find the equivalent  model of microstrip-line inductor?
Solution
From Table 4-1
From Table 4-2

31. Equivalent  model

32. Example4.9: Find the equivalent T model of microstrip-line capacitor?
Solution
From Table 4-1
From Table 4-2

33. Equivalent T model

34. Problem3: Design a 6GHz attenuator ?
(Hint: -20logS21=6  S21= )
Problem4: Design a 6nH microstrip-line inductor on a 1.6mm thick FR4 substrate. The width of line is 0.25mm. Find the length (l ) and parasitic capacitance?
Problem5: Design a 2pF microstrip-line capacitor on a 1.6mm thick FR4 substrate. The width of line is 5mm. Find the length (l ) and parasitic inductance?