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

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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

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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.

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Example (Single Load) Signal flow graphSingle load 4

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Example (Source) Hence 5 where

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Example (Two-Port Device) 6

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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.

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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

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

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Example: Direct Solution Technique (cont.) 10 For a given a 1, there are three equations and three unknowns ( b 1, a 2, b 2 ).

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Decomposition Techniques 1) Series paths Note that we have removed the node a 2. 11

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

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

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

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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).

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

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

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

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Example (cont.) Hence: 19 We then have

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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.)

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

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

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

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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.

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