1. ENE 428 Microwave Engineering
Lecture 10 Signal Flow Graphs and Excitation of Waveguides

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. Signal flow graphs and applications
A convenient technique to represent and analyzed the transmission and reflection of waves in a microwave amplifier.
Relations between the variables can be obtained using Mason’s rule.
The flow graph technique permits expression, such as power gains and voltage gains of complex microwave amplifiers, to be derived easily.

4. Rules of signal flow graph constructions
Each variable is designated as a node.
The S parameters and reflection coefficients are represented by branches.
Branches enter dependent variable nodes and emanate from independent variable nodes. The independent variable nodes are the incident waves, and the reflected waves are dependent variable nodes.
A node is equal to the sum of the branches entering it.

5. Signal flow graph of the S parameters of a two-port network
Observe that b1 and b2 are the dependent variable nodes and a1 and a2 are the independent variable nodes.

6. Signal flow graph of a signal generator

7. Signal flow graph of a load impedance

8. Signal flow graph of a microwave amplifier
Observe that the nodes bg, ag, bL, and aL are identical to a1, b1, a2, and b2, respectively.

9. Mason’s rule
Mason’s rule is used to determine the ratio of transfer function T of a dependent to an independent variable.

10. Variables’descriptions (1)
P1, P2, (and so on)
= paths connecting the dependent and independent variables whose transfer function T is to be determined.
A path is defined as a set of consecutive, codirectional branches along which no node is encountered more than once as we move in the graph from the independent to the dependent node.
 L(1)
= the sum of all first-order loops. A first-order loop is defined as the product of the branches encountered in a round trip as we move from a node in the direction of the arrows back to that original node.

11. Variables’descriptions (2)
= the sum of all second-order loops. A second-order loop is defined as the product of any two nontouching first-order loops.
 L(3)
= the sum of all third-order loops. A third-order loop is defined as the product of any three nontouching first-order loops.
 L(4),  L(5), and so on represent fourth-, fifth-, and higher order loops.

12. Variables’descriptions (3)
 L(1)(P)
= the sum of all first-order loops that do not touch the path P between the independent and dependent variables.
 L(2)(P)
= the sum of all second-order loops that do not touch the path P between the independent and dependent variables.
 L(3)(P),  L(4)(P) and so on represent third-, fourth-, and higher order loops that do not touch the path P.

13. Ex1 Use Mason’s rule to obtain b1/bS as shown in a microwave amplifier’s signal flow graph

14. Applications of Signal flow graphs (1)
The calculation of the input reflection coefficient, IN
Observing that P1 = S11, P2 = S21L S12, L(1) = S22 L , and L(1)(1) = S22 L, we can use Mason’s rule to obtain

15. Applications of Signal flow graphs (2)
The calculation of the output reflection coefficient, OUT
Observing that P1 = S22, P2 = S21S S12, L(1) = S11 S , and L(1)(1) = S11 S, we can use Mason’s rule to obtain

16. Excitation of Waveguides
Several propagating modes can be excited in the waveguide along with evanescent modes.
Formalism for excitation of the give wg mode due to an arbitrary electric and magnetic current source will be determined.
Excitation of the wg using aperture coupling will also be briefly discussed.

17. Single mode excitation using current sheets

18. The amplitudes must be equal to satisfy BCs.
When (5) is applied to (1) and (2), we get
The discontinuity in the transverse magnetic field in (6) is equal
to the electric surface current density thus at z = 0,

19. The surface current density at z = 0 can be found.
This current will excite only the TEmn mode since
Maxwell’s equations and all boundary conditions are
satisfied.

20. The electric current that excites only the TM mode can be determined analogously.

21. Excitation of WG modes by an arbitrary electric or magnetic current source.
where n represents any possible TE or TM mode.

22. The unknown amplitude An+ can be determined using the Lorentz reciprocity theorem. (1)
where S is a closed surface enclosing the volume V, and
are the fields due to the current source .
Let the volume V be the region between the wg walls,
Let and ,
And let be the nth wg mode in the –z direction,

23. The unknown amplitude An+ can be determined using the Lorentz reciprocity theorem. (2)
Substitution (12) and (13) into (11) with and
, gives
The portion of the surface integral over the wg walls vanishes because the tangential electric field is zero. This reduces the integration to the guide cross section, S0, at the planes z1 and z2.

24. The unknown amplitude An+ can be determined using the Lorentz reciprocity theorem. (3)
Using (7)-(10) and (15), then reduces (14) to
Since the second integral vanishes, this further reduces to
where

25. The unknown amplitude An- can be determined using the same procedure.
By repeating with and , we get
These above results can be applied to any type of wg such as stripline and microstrip, where modal fields can be defined.