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**Notes 15 ECE 5317-6351 Microwave Engineering**

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

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**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.

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**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.

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Example (Single Load) Single load Signal flow graph

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Example (Source) Hence where

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**Example (Two-Port Device)**

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**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.

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**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.

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**Example: Direct Solution Technique**

A two-port device is connected to a load.

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**Example: Direct Solution Technique (cont.)**

For a given a1, there are three equations and three unknowns (b1, a2, b2).

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**Decomposition Techniques**

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

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**Decomposition Techniques (cont.)**

2) Parallel paths Note that we have combined the two parallel paths.

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**Decomposition Techniques (cont.)**

3) Self-loop Note that we have removed the self loop.

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**Decomposition Techniques (cont.)**

4) Splitting Note that we have shifted the splitting point.

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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).

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

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**Example (cont.) Consider the following decompositions:**

The self-loop at the end is rearranged To put it on the outside (this is optional).

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

Rewrite self-loop Remove self-loop

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Example (cont.) Hence: We then have

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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.)

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

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

Remove final self-loop

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

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

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