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

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.

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.

Example (Single Load) Single load Signal flow graph

Example (Source) Hence where

Example (Two-Port Device)

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.

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.

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

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

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

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

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

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

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).

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

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

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

Example (cont.) Hence: We then have

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.)

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

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

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

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