Presentation on theme: "Prof. David R. Jackson Dept. of ECE Notes 15 ECE 5317-6351 Microwave Engineering Fall 2011 Signal-Flow Graph Analysis 1."— Presentation transcript:
Prof. David R. Jackson Dept. of ECE Notes 15 ECE Microwave Engineering Fall 2011 Signal-Flow Graph Analysis 1
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., a i and b i ). Signal-flow graphs are also used for a number of other engineering applications (e.g., in control theory). Signal-Flow Graph Analysis Note: In the signal-flow graph, a i (0) and b i (0) are denoted as a i and b i for simplicity. 2
Signal-Flow Graph Analysis (cont.) 3 Construction Rules for signal-flow graphs 1)Each wave function ( a i and b i ) is a node. 2)S -parameters are represented by branches between nodes. 3)Branches are uni-directional. 4)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) Signal flow graphSingle load 4
Example (Source) Hence 5 where
Example (Two-Port Device) 6
Complete Signal-Flow Graph A source is connected to a two-port device, which is terminated by a load. 7 When cascading devices, we simply connect the signal-flow graphs together.
a) Masons 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. Solving Signal-Flow Graphs 8
Example: Direct Solution Technique A two-port device is connected to a load. 9
Example: Direct Solution Technique (cont.) 10 For a given a 1, there are three equations and three unknowns ( b 1, a 2, b 2 ).
Decomposition Techniques 1) Series paths Note that we have removed the node a 2. 11
2) Parallel paths Decomposition Techniques (cont.) Note that we have combined the two parallel paths. 12
3) Self-loop Decomposition Techniques (cont.) Note that we have removed the self loop. 13
4) Splitting Decomposition Techniques (cont.) Note that we have shifted the splitting point. 14
Example A source is connected to a two-port device, which is terminated by a load. Solve for in = b 1 / a 1 15 Two-port device + - Note: The Z 0 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 16 Two-port device The signal flow graph is constructed:
Consider the following decompositions: Example (cont.) 17 The self-loop at the end is rearranged To put it on the outside (this is optional).
Example (cont.) 18 Remove self-loop Next, we apply the self-loop formula to remove it. Rewrite self-loop
Example (cont.) Hence: 19 We then have
Example 20 A source is connected to a two-port device, which is terminated by a load. Solve for b 2 / b s Two-port device + - (Hence, since we know b s, we could find the load voltage from b 2 /b s if we wish.)
Example (cont.) Using the same steps as before, we have: 21
Example (cont.) 22 Remove self-loop Rewrite self-loop on the left end Remove final self-loop
Example (cont.) Hence 23 Two-port device + -
Alternatively, we can write down a set of linear equations: Example (cont.) 24 There are 5 unknowns: b g, a 1, b 1, b 2, a 2.