1. Chapter 4. Microwave Network Analysis
It is much easier to apply the simple and intuitive idea of circuit analysis to a microwave problem than it is to solve Maxwell’s equations for the same problem.
Maxwell’s equations for a given problem is complete, it gives the E & H fields at all points in space.
Usually we are interested in only the V & I at a set of terminals, the power flow through a device, or some other type of “global” quantity.
A field analysis using Maxwell’s equations for problems would be hopelessly difficult.

2. 4.1 Impedance and Equivalent Voltages and Currents
The voltage of the + conductor relative to the – conductor
After having defined and determined a voltage, current, and characteristic impedance, we can proceed to apply the circuit theory for transmission lines to characterize this line as a circuit element.

3. Figure 4.1 (p. 163) Electric and magnetic field lines for an arbitrary two-conductor TEM line.

4. Figure 4.2 (p. 163) Electric field lines for the TE10 mode of a rectangular waveguide.

5. There is no “correct” voltage in the sense of being unique.
There are many ways to define equivalent voltage, current, and impedance for waveguides.
V&I are defined only for a particular waveguide mode.
The equivalent V&I should be defined so that their product gives the power flow of the mode.
V/I for a single traveling wave should be equal to Z0 of the line. This impedance may be chosen arbitrarily, but is usually selected as equal to the wave impedance of the line.

6. For an arbitrarily waveguide mode, the transverse fields
where e and h are the transverse field variations of the mode. Since Et & Ht are related by Zw,
Defining equivalent voltage and current waves as

7. The complex power flow for the incident wave
Since we want this power to be (1/2)V+I+*,
where the surface integration is over the cross section of the waveguide.
If it is desired to have Z0 = Zw,

8. For higher order modes,
Ex 4.1

9. The Concept of Impedance
Various types of impedance
Intrinsic impedance ( ) of the medium: depends on the material parameters of the medium, and is equal to the wave impedance for plane waves.
Wave impedance ( ): a characteristic of the particular type of wave. TEM, TM and TE waves each have different wave impedances which may depend on the type of the line or guide, the material, and the operating frequency.
Characteristic impedance ( ): the ratio of V/I for a traveling wave on a transmission line. Z0 for TEM wave is unique. TE and TM waves are not unique.

10. Ex 4.2
Figure 4.3 (p. 167) Geometry of a partially filled waveguide and its transmission line equivalent for Example 4.2.

11. Figure 4.4 (p. 168) An arbitrary one-port network.

12. The complex power delivered to this network is:
where Pl is real and represents the average power dissipated by the network.
If we define real transverse modal fields, e and h, over the terminal plane of the network such that
with a normalization

13. The input impedance is
If the network is lossless, then Pl = 0 and R = 0. Then Zin is purely imaginary, with a reactance

14. Even and Odd Properties of Z(ω) and Γ(ω)
Consider the driving point impedance, Z(ω), at the input port of an electrical network.  V(ω) = I(ω) Z(ω).
Since v(t) must be real v(t) = v*(t),
 Re{V(ω)} is even in ω, Im{V(ω)} is odd in ω. I(ω) holds the same as V(ω).

15. The reflection coefficient at the input port

16. 4.2 Impedance and Admittance Matrices
At the nth terminal plane, the total voltage and current is as seen from (4.8) when z = 0.
The impedance matrix
Similarly,
where

17. Figure 4.5 (p. 169) An arbitrary N-port microwave network.

18. Zij can be defined as
In words, Zij can be found by driving port j with the current Ij, open-circuiting all other ports (so Ik=0 for k≠j), and measuring the open-circuit voltage at port i.
Zii: input impedance seen looking into port i when all other ports are open.
Zij: transfer impedance between ports i and j when all other ports are open.
Similarly,

19. Reciprocal Networks
Let Fig. 4.5 to be reciprocal (no active device, ferrites, or plasmas), with short circuits placed at all terminal planes except those of ports 1 and 2.
Let Ea, Ha and Eb, Hb be the fields anywhere in the network due to 2 independent sources, a and b, located somewhere in the network.
From the reciprocity theorem,

20. The fields due to sources a and b at the terminal planes t1 and t2: (4.31)
where e1, h1 and e2, h2 are the transverse modal fields of port 1 and 2. Therefore,
where S1, S2 are the cross-sectional areas at the terminal planes of ports 1 and 2.
Comparing (4.31) to (4.6), C1 = C2 = 1 for each port, so that from (4.10).

21. This leads to
For 2 port,
Generally,

22. Lossless Networks Consider a reciprocal lossless N-port network.
If the network is lossless, Re{Pav} = 0.
Since the Ins are independent, only nth current is taken.

23. Take Im and In only 
Since (InIm*+ ImIn*) is purely real, Re{Zmn} = 0,.
Therefore, Re{Zmn} = 0 for any m, n.
Ex 4.3

24. 4.3 The Scattering Matrix
The scattering matrix relates the voltage waves incident on the ports to those reflected from the ports.
The scattering parameters can be calculated using network analysis technique. Otherwise, they can be measured directly with a vector network analyzer.
Once the scattering matrix is known, conversion to other matrices can be performed.
Consider the N-port network in Fig. 4.5.

25. or
Sii the reflection coefficient seen looking into port i when all other ports are terminated in matched loads,
Sij the transmission coefficient from port j to port i when all other ports are terminated in matched loads.

26. Figure 4.7 (p. 175) A photograph of the Hewlett-Packard HP8510B Network Analyzer. This test instrument is used to measure the scattering parameters (magnitude and phase) of a one- or two-port microwave network from 0.05 GHz to 26.5 GHz. Built-in microprocessors provide error correction, a high degree of accuracy, and a wide choice of display formats. This analyzer can also perform a fast Fourier transform of the frequency domain data to provide a time domain response of the network under test. Courtesy of Agilent Technologies.

27. Ex.4, Evaluation of Scattering Parameters
Figure 4.8 (p. 176) A matched 3B attenuator with a 50 Ω Characteristic impedance

28. Show how [S]  [Z] or [Y]. Assume Z0n are all identical, for convenience Z0n = 1.
where
Therefore,
For a one-port network,

29. To find [Z],
Reciprocal Networks and Lossless Networks
As in Sec. 4.2, the [Z] and [Y] are symmetric for reciprocal networks, and purely imaginary for lossless networks.
From

30. If the network is reciprocal, [Z]t = [Z].
If the network is lossless, no real power delivers to the network.

31. For nonzero [V+], [S]t[S]*=[U], or [S]*={[S]t}-1.  Unitary matrix 
If i = j,
If i ≠ j,
Ex 4.5 Application of Scattering Parameters
The S parameters of a network are properties only of the network itself (assuming the network in linear), and are defined under the condition that all ports are matched.

32. A Shift in Reference Planes
Figure 4.9 (p. 181) Shifting reference planes for an N-port network.

33. [S]: the scattering matrix at zn = 0 plane.
[S']: the scattering matrix at zn = ln plane.

35. Generalized Scattering Parameters
Figure (p. 181) An N-port network with different characteristic impedances.

37. The generalized scattering matrix can be used to relate the incident and reflected waves,

38. Figure on page 183

39. 4.4 The Transmission (ABCD) Matrix
The ABCD matrix of the cascade connection of 2 or more 2-port networks can be easily found by multiplying the ABCD matrices of the individual 2-ports.

40. Ex. 4.6 Evaluation of ABCD Parameters

41. Relation to Impedance Matrix
From the Z parameters with -I2 ,

42. If the network is reciprocal, Z12=Z21, and AD-BC=1.
Equivalent Circuits for 2-port Networks
Table 4-2
A transition between a coaxial line and a microstrip line. Because of the physical discontinuity in the transition from a coaxial line to a microstrip line, electric and/or magnetic energy can be stored in the vicinity of the junction, leading to reactive effects.

43. Figure (p. 188) 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 [6].

44. Figure (p. 188) Equivalent circuits for a reciprocal two-port network. (a) T equivalent. (b) π equivalent.

45. 4.5 Signal Flow Graphs
Very useful for the features and the construction of the flow transmitted and reflected waves.
Nodes: Each port, i, of a microwave network has 2 nodes, ai and bi. Node ai is identified with a wave entering port i, while node bi is identified with a wave reflected from port i. The voltage at a node is equal to the sum of all signals entering that node.
Branches: A branch is directed path between 2 nodes, representing signal flow from one node to another. Every branch has an associated S parameter or reflection coefficient.

46. Figure (p. 189) The signal flow graph representation of a two-port network. (a) Definition of incident and reflected waves. (b) Signal flow graph.

47. Figure (p. 190) The signal flow graph representations of a one-port network and a source. (a) A one-port network and its flow graph. (b) A source and its flow graph.

48. Decomposition of Signal Flow Graphs
A signal flow graph can be reduced to a single branch between 2 nodes using the 4 basic decomposition rules below, to obtain any desired wave amplitude ratio.
Rule 1 (Series Rule): V3 = S32V2 = S32S21V1 .
Rule 2 (Parallel Rule): V2 = SaV1 + SbV1 = (Sa + Sb)V1.
Rule 3 (Self-Loop Rule): V2 = S21V1 + S22V2, V3 = S32V2.
 V3 = S32S21V1/(1-S22)
Rule 4 (Splitting Rule): V4 = S42V2 = S21S42V1.

49. Figure 4. 16 (p. 191) Decomposition rules. (a) Series rule
Figure (p. 191) Decomposition rules. (a) Series rule. (b) Parallel rule. (c) Self-loop rule. (d) Splitting rule.

50. Ex 4.7 Application of Signal Flow Graph
Figure (p. 192) A terminated two-port network.

51. Figure (p. 192) Signal flow path for the two-port network with general source and load impedances of Figure 4.17.

52. Figure 4. 19 (p. 192) Decompositions of the flow graph of Figure 4
Figure (p. 192) Decompositions of the flow graph of Figure 4.18 to find Γin = b1/a1 and Γout = b2/a2. (a) Using Rule 4 on node a2. (b) Using Rule 3 for the self-loop at node b2. (c) Using Rule 4 on node b1. (d) Using Rule 3 for the self-loop at node a1.

53. Figure 4.20 (p. 193) Block diagram of a network analyzer measurement of a two-port device.

54. Figure 4.21a (p. 194) Block diagram and signal flow graph for the Thru connection.

55. Figure 4.21b (p. 194) Block diagram and signal flow graph for the Reflect connection.

56. Figure 4.21c (p. 194) Block diagram and signal flow graph for the Line connection.

57. Figure 4.22 (p. 198) Rectangular waveguide discontinuities.

58. Figure 4. 23 (p. 199) Some common microstrip discontinuities
Figure (p. 199) Some common microstrip discontinuities. (a) Open-ended microstrip. (b) Gap in microstrip. (c) Change in width. (d) T-junction. (e) Coax-to-microstrip junction.

59. Figure 4.24 (p. 200) Geometry of an H-plane step (change in width) in rectangular waveguide.

60. Figure 4.25 (p. 203) Equivalent inductance of an H-plane asymmetric step.

61. Figure on page 204 Reference: T. C
Figure on page 204 Reference: T.C. Edwards, Foundations for Microwave Circuit Design, Wiley, 1981.

62. Figure (p. 205) An infinitely long rectangular waveguide with surface current densities at z = 0.

63. Figure (p. 206) An arbitrary electric or magnetic current source in an infinitely long waveguide.

64. Figure 4.28 (p. 208) A uniform current probe in a rectangular waveguide.

65. Figure (p. 210) Various waveguide and other transmission line configurations using aperture coupling. (a) Coupling between two waveguides wit an aperture in the common broad wall. (b) Coupling to a waveguide cavity via an aperture in a transverse wall. (c) Coupling between two microstrip lines via an aperture in the common ground plane. (d) Coupling from a waveguide to a stripline via an aperture.

66. Figure (p. 210) Illustrating the development of equivalent electric and magnetic polarization currents at an aperture in a conducting wall (a) Normal electric field at a conducting wall. (b) Electric field lines around an aperture in a conducting wall. (c) Electric field lines around electric polarization currents normal to a conducting wall. (d) Magnetic field lines near a conducting wall. (e) Magnetic field lines near an aperture in a conducting wall. (f) Magnetic field lines near magnetic polarization currents parallel to a conducting wall.

67. Figure (p. 213) Applying small-hole coupling theory and image theory to the problem of an aperture in the transverse wall of a waveguide. (a) Geometry of a circular aperture in the transverse wall of a waveguide. (b) Fields with aperture closed. (c) Fields with aperture open. (d) Fields with aperture closed and replaced with equivalent dipoles. (e) Fields radiated by equivalent dipoles for x < 0; wall removed by image theory. (f) Fields radiated by equivalent dipoles for z > 0; all removed by image theory.

68. Figure 4.32 (p. 214) Equivalent circuit of the aperture in a transverse waveguide wall.

69. Figure 4.33 (p. 214) Two parallel waveguides coupled through an aperture in a common broad wall.