Notes 15 - Signal Flow Graph Analysis

8/10/2019 Notes 15 - Signal Flow Graph Analysis

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Prof. David R. Jackson

Dept. of ECE

Notes 15

ECE 5317-6351

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., 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.

2



Signal-Flow Graph Analysis (cont.)

3

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.

S o u r c e

N e

t w o r k

L

o a d g b

g a 1

b La

Lb2

a

2b

1a

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



0 Z L

Z

La

Lb

L L Lb a

L

1

1

L

a

L L Lb a

0

0

L L

L

Z Z

Z Z

Example (Single Load)

Signal flow graphSingle load

4



g a

Th Z

g b

g a

s

0 Z

ThV

+

sb 1 1

1

g b

-

0

0 0

1

g g s Th

Th

Z V

Z

b a

Z Z

0

0

Th s

Th

Z Z

Z Z

g s g sb b a

Example (Source)

Hence

0

0

s Th

Th

Z

b V Z Z

g g s sb a b

5

where



1a

1b

1 1

11

22S

12S

21S

11S

1b

2a

2b

1a

0 Z

2b

2a

0 Z

Example (Two-Port Device)

6

1 11 1 12 2

2 21 1 22 2

b S a S a

b S a S a



1a

1b

22S

12S

21S

11S

2b

2a g a

g b sb

L s

11

1 1

La

Lb

Complete Signal-Flow Graph

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

7

S o u r c e

N e t w o r k

L o a d g b

g a1

b La

Lb2

a

2b

1a

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



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 physicalunderstanding.

c) Decomposition:

Straightforward graphical technique, requires experience, retains physical

understanding.

Solving Signal-Flow Graphs

8



1a

1b

22S

12S

21S

11S

2b

2a

L

1a

1b

Example: Direct Solution Technique

1

1

in

b

a

A two-port device is connected to a load.

9

N e t w o r k

L o a d

1b

La

Lb

2a

2b

1a



1a

1b

22S

12S

21S

11S

2b

2a

L

1a

1b

Example: Direct Solution Technique (cont.)

1

1

in

b

a

2 1 21 22 2

2 2

1 11 1 12 2

L

b a S S a

a bb S a S a

1 21 12

11

1 221

Lin

L

b S S

S a S

Solve :

10

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



1a

2a

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.

11



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.

12



1a

2a

1a

2a

21S

bS

1a

1a 2a21

S 1a

2a

2a

21 bS S

21

1

1b

LS S

1a 2a21S L

3) Self-loop

1 1 1 21 ba a a S S

1 1

21

1

1 b

a aS S


Note that we have removed the self loop.

13

1 1 2

2 1 21

ba a a S

a a S



1a

2a

3a

1a

3a

21S

32S

42S

21 42S S

4a

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


Note that we have shifted the splitting point.

14



Example


Solve for in = b1 / a1

15

Two-port device

Th Z

ThV +- L Z

in

S

1a

1b

0 Z 0

Z

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).



1a

1b

22S

12S

21S

11S

2b

2a

L s

sb

Example

16

Two-port device

The signal flow graph is constructed:



22S

12S

21S

11S

2b

L s

sb1

a

2a1b

22S

12S

21S

11S

2b

L s

sb

1a

2a

1b

Consider the following decompositions:

Example (cont.)

17

The self-loop at the end is rearranged

To put it on the outside (this is optional).



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

L s

sb1

a

2a

1b

1

22

1

1 L

LS

Example (cont.)

18

Remove self-loop

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

Rewrite self-loop



1

11 21 1 12

1

in L

bS S L S

a

Example (cont.)

Hence:

19

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

21 12

11

221

Lin

L

S S S

S

122

1

1 L L S

We then have

21 1S L

11S

2b

12 LS

s

sb

1a

1b



Example

20


Solve for b2 / b s

Two-port device

Th Z

ThV +- L

Z

in

S

1a

1b

2a

2b

sb 0 Z

0 Z

2 2 020 1 1 L L LV V b Z

Note :

(Hence, since we know b s, we could find the load voltage from b2/b s if we wish.)



Example (cont.)

Using the same steps as before, we have:

21

1

22

1

1 L

LS

21 1S L

11S

2b

12 LS

s

sb1

a

1b



21 1

S L

2b

12 LS

11 S S 2

L

2b

sb

12 LS

21 1S L

11S

2b

12 LS s

sb

1a

1b

1a

sb

s

s

1a

2 21 1 3 L S L L

2b sb

1a

21 1S L

2

2 21 1 3

2 21 1

2 21 1 121

s

L S

b L S L L

b

L S L L S L S

2

11

1

1 S

LS

Example (cont.)

22

Remove self-loop

Rewrite self-loop on the left end

3

2 21 1 12

1

1 L S

L L S L S

Remove final self-loop

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



2 21 1 2

21 12 1 2

21

21 12

1 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 S L L

S

S S S S

Example (cont.)

Hence

23

2 21

22 11 21 121 1 s L S s L

b S

b S S S S

Two-port device

Th

Z

ThV +- L Z

in

S

1a

1b

2a

2b

sb 0

Z 0

Z

Notes 15 - Signal Flow Graph Analysis

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