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Prof. David R. JacksonDept. of ECE

Notes 15

ECE 5317-6351 Microwave Engineering

Fall 2011

Signal-Flow Graph Analysis

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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., ai and bi).

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, 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

1) Each wave function (ai and bi) 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.

Sourc

e

Networ

k

Loadgb

ga 1b LaLb2a

2b1a

In this circuit there are eight nodes in the signal flow graph.

4

0Z LZ

La

Lb

L L Lb a

L

1

1

La

L L Lb a

0

0

LL

L

Z Z

Z Z

Example (Single Load)

Signal flow graphSingle load

5

ga

ThZ

gb

gas

0Z

ThV+

sb 1 1

1

gb

-

0

0 0

1g g s Th

Th

ZV

Zb a

Z Z

0

0

Ths

Th

Z Z

Z Z

g s g sb b a

Example (Source)

Hence

0

0s Th

Th

Zb V

Z Z

g g s sb a b

where

6

1a

1b

1 1

11

22S

12S

21S

11S

1b2a

2b1a

0Z

2b

2a

0Z

Example (Two-Port Device)

1 11 1 12 2

2 21 1 22 2

b S a S a

b S a S a

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1a

1b

22S

12S

21S

11S

2b

2aga

gbsb

Ls

11

1 1

La

Lb

Complete Signal-Flow GraphA source is connected to a two-port device, which is terminated by a load.

Sourc

e

Networ

k

Loadgb

ga 1b LaLb2a

2b1a

When cascading devices, we simply connect the signal-flow graphs together.

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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.

Solving Signal-Flow Graphs

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1a

1b

22S

12S

21S

11S

2b

2a

L

1a

1b

Example: Direct Solution Technique

1

1in

ba

A two-port device is connected to a load.

1b LaLb2a

2b1a

10

1a

1b

22S

12S

21S

11S

2b

2a

L

1a

1b

Example: Direct Solution Technique (cont.)

1

1in

ba

2 1 21 22 2

2 2

1 11 1 12 2

L

b a S S a

a b

b S a S a

1 21 1211

1 221L

inL

b S SS

a S

Solve :

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

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1a2a

3a

1a 3a

21S 32S

1

21 32S S

1

1 1

Decomposition Techniques

1) Series paths

3 21 32 1a S S a

2 21 1

3 32 2

a S a

a S a

Note that we have removed the node a2.

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1a 2a

1a 2a

aS

bS

a bS S

2) Parallel paths

2 1 1a ba S a S a

2 1a ba S S a

Decomposition Techniques (cont.)

Note that we have combined the two parallel paths.

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1a 2a

1a 2a

21S

bS

1a

1a 2a21S1a

2a

2a

21 bS S

21

1

1 b

LS S

1a 2a21SL

3) Self-loop

1 1 1 21 ba a a S S

1 121

1

1 b

a aS S

Decomposition Techniques (cont.)

Note that we have removed the self loop.

1 1 2

2 1 21

ba a a S

a a S

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1a

2a3a

1a 3a

21S 32S

42S

21 42S S4a

4a

21 32S S

4) Splitting

4 2 42

3 2 32

2 1 21a S

a a S

a S

a

a

4 21 42 1

3 21 32 1

a S S a

a S S a

Decomposition Techniques (cont.)

Note that we have shifted the splitting point.

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ExampleA source is connected to a two-port device, which is terminated by a load.

Solve for in = b1 / a1

Two-port device

ThZ

ThV +- LZ

in

S

1a1b

0Z 0Z

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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1a

1b

22S

12S

21S

11S

2b

2a

Ls

sb

Example

Two-port device

The signal flow graph is constructed:

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22S

12S

21S

11S

2b

Ls

sb1a

2a1b

22S

12S

21S

11S

2b

Ls

sb1a

2a1b

Consider the following decompositions:

Example (cont.)

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

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22 LS

12 LS

21S

11S

2b

s

sb

21 1S L

11S

2b

12 LS s

sb

1a

1b

1a

1b

22S

12S

21S

11S

2b

Ls

sb1a

2a1b

122

1

1 L

LS

Example (cont.)

Remove self-loop

Next, we apply the self-loop formula to remove it.

Rewrite self-loop

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111 21 1 12

1in L

bS S L S

a

Example (cont.)

Hence:

1 1 11 1 21 1 12 Lb a S a S L S

21 1211

221L

inL

S SS

S

122

1

1 L

LS

We then have

21 1S L

11S

2b

12 LS s

sb1a

1b

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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

ThZ

ThV +- LZ

in

S

1a

1b2a2b

sb0Z 0Z

2 2 020 1 1L L LV V b Z

Note :

(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:

122

1

1 L

LS

21 1S L

11S

2b

12 LS s

sb1a

1b

22

21 1S L

2b

12 LS 11 SS

2L

2bsb

12 LS

21 1S L

11S

2b

12 LS s

sb1a

1b1a

sb

s

s

1a

2 21 1 3L S L L

2bsb1a

21 1S L

22 21 1 3

2 21 1

2 21 1 121

s

L S

bL S L L

b

L S L

L S L S

211

1

1 S

LS

Example (cont.)

Remove self-loop

Rewrite self-loop on the left end

32 21 1 12

1

1 L S

LL S L S

Remove final self-loop

1 1 11 1 21 1 12s s s La b a S a S L S

23

2 21 1 2

21 12 1 2

21

21 121 2

21

11 22 21 12

1

1

1 1

s L s

L s

S L L s

b S L L

b S S L L

S

S SL L

S

S S S S

Example (cont.)

Hence

2 21

22 11 21 121 1s L S s L

b S

b S S S S

Two-port device

ThZ

ThV +- LZ

in

S

1a1b

2a2b

sb0Z 0Z

24

22S

12S

21S

11S

2b

Ls

sb1agb

1b 2a

1

1

1 11 1 12 2

2 21 1 22 2

2 2

g s s

g

L

b b b

a b

b S a S a

b S a S a

a b

2 21

11 22 21 121 1s S L s L

b S

b S S S S

Solve to find

Alternatively, we can write down a set of linear equations:

Example (cont.)

There are 5 unknowns: bg, a1, b1, b2, a2.