1. Notes 15 ECE 5317-6351 Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 15
Signal-Flow Graph Analysis

2. Signal-Flow Graph Analysis
This is a convenient technique to represent and analyze circuits characterized by S-parameters.
It allows one to “see” the “flow” of signals throughout a circuit.
Signals are represented by wavefunctions (i.e., ai and bi).
Signal-flow graphs are also used for a number of other engineering applications (e.g., in control theory).
Note: In the signal-flow graph, ai(0) and bi(0) are denoted as ai and bi for simplicity.

3. Signal-Flow Graph Analysis (cont.)
Construction Rules for signal-flow graphs
Each wave function (ai and bi) is a node.
S-parameters are represented by branches between nodes.
Branches are uni-directional.
A node value is equal to the sum of the branches entering it.
In this circuit there are eight nodes in the signal flow graph.

4. Example (Single Load)
Single load
Signal flow graph

5. Example (Source)
Hence
where

6. Example (Two-Port Device)

7. Complete Signal-Flow Graph
A source is connected to a two-port device, which is terminated by a load.
When cascading devices, we simply connect the signal-flow graphs together.

8. Solving Signal-Flow Graphs
a) Mason’s non-touching loop rule:
Too difficult, easy to make errors, lose physical understanding.
b) Direct solution:
Straightforward, must solve linear system of equations, lose physical understanding.
c) Decomposition:
Straightforward graphical technique, requires experience, retains physical understanding.

9. Example: Direct Solution Technique
A two-port device is connected to a load.

10. Example: Direct Solution Technique (cont.)
For a given a1, there are three equations and three unknowns (b1, a2, b2).

11. Decomposition Techniques
1) Series paths
Note that we have removed the node a2.

12. Decomposition Techniques (cont.)
2) Parallel paths
Note that we have combined the two parallel paths.

13. Decomposition Techniques (cont.)
3) Self-loop
Note that we have removed the self loop.

14. Decomposition Techniques (cont.)
4) Splitting
Note that we have shifted the splitting point.

15. Example
A source is connected to a two-port device, which is terminated by a load.
Solve for in = b1 / a1
Two-port device + -
Note: The Z0 lines are assumed to be very short, so they do not affect the calculation (other than providing a reference impedance for the S parameters).

16. Example
The signal flow graph is constructed:
Two-port device

17. Example (cont.) Consider the following decompositions:
The self-loop at the end is rearranged
To put it on the outside (this is optional).

18. Example (cont.) Next, we apply the self-loop formula to remove it.
Rewrite self-loop
Remove self-loop

19. Example (cont.)
Hence:
We then have

20. Example
A source is connected to a two-port device, which is terminated by a load.
Solve for b2 / bs
Two-port device + -
(Hence, since we know bs, we could find the load voltage from b2/bs if we wish.)

21. Example (cont.)
Using the same steps as before, we have:

22. Example (cont.) Rewrite self-loop on the left end Remove self-loop
Remove final self-loop

23. Example (cont.)
Two-port device + -
Hence

24. Example (cont.)
Alternatively, we can write down a set of linear equations:
There are 5 unknowns: bg, a1, b1, b2, a2.