What is Electrical Engineering? - GUCeee.guc.edu.eg/Courses/Communications/COMM903 Microwave... · 2016-09-19 · Signal Flow Graphs • A signal-flow graph can be used only when

GUC (Dr. Hany Hammad) 9/19/2016

COMM (903) Lecture #2 1

© Dr. Hany Hammad, German University in Cairo

Lecture # 2

• Signal flow graphs:

– Definitions.

– Rules of Reduction.

– Mason’s Gain Rule.

– Signal-flow graph representation of a:

• Voltage Source.

• Passive single-port device.


Signal Flow Graphs

• A signal-flow graph is a graphical means of portraying the relationship among the variables of a set of linear algebraic equations.

• Originally introduced by S.J. Mason.

• Consider a linear network that has N input and output ports. Which is described by a set of linear algebraic equations.

j

N

j

iji IZV

1

Ni ,,2,1

• This says that the effect Vi at the ith port is a sum of gain times causes at its N ports. Vi represents the dependent variable (effect), and Ij are the independent variables (cause). Nodes or junction points of the signal-flow graph represent these variables.




Signal Flow Graphs

Nodes or Junction Points

Branches

ijZ

Coefficient or gain of a branch the connects the ith dependent node with

the jth independent node.


Signal Flow Graphs

• A signal-flow graph can be used only when the system is linear.

• A set of algebraic equations must be in the form of effects as functions of causes before its signal-flow graph can be drawn.

• A node is used to represent each variable. Normally, these are arranged from left to right, following is succession of inputs (causes) and outputs (effects) of the network.

• Nodes are connected together by branches with an arrow directed toward the dependent node.

• Signal travel along the branches only in the direction of the arrows.

• Signal Ik traveling along a branch that connects nodes Vi and Ik is multiplied by the branch gain Zik. The dependent node (effect) Vi is equal to the sum of the branch gain times the corresponding independent nodes (causes).




Example 1

2121111 aSaSb Output b1 of a system is caused by two inputs a1 and a2 as represented

by the equation.

Find its signal-flow graph.

Solution

1b2a

1a

11S

12S

2221212 aSaSb Output b2 of a system is caused by two inputs a1 and a2 as represented

by the equation.


Solution

2b 2a

1a

21S

22S

Example 2

Cause

Cause

Effect

Gain

node

branch


Example 3

The input-output characteristics of a two-port network are given by the set of linear algebraic equations.

2221212

2121111

aSaSb

aSaSb


Solution

1a

1b 2a

11S

12S

2b

22S

21S




Cascade connection of two port networks

BB

BB

SS

SS

2221

1211

AA

AA

SS

SS

2221

1211

1

11b2a

1a

AS11

AS12

2b

AS22

AS21

1b 2a

1a

BS11

BS12

2b

BS22

BS21

2221212

2121111

aSaSb

aSaSb

AA

AA

2221212

2121111

aSaSb

aSaSb

BB

BB

12

21

ba

ba

1a

1b

2a

2b1a

1b2a

2b


Example 4

The following set of linear algebraic equations represents the input-output relations of a multiport network. Find the corresponding signal-flow graphs.

Solution

2112

1X

sRX

212124

174 X

sYRXX

2213

Xs

sY

112 10 sYXY

1R 1X1 2Xs2

1

2112

1X

sRX




Example 4

212124

174 X

sYRXX

11R 1X s2

1

4

1

s

2X

4

2R

1Y

7

1


Example 4

2213

Xs

sY

1R 1X1

2X

s2

1

4

2R

1Y

7

4

1

s

1

32 s

s




Example 4

112 10 sYXY

1R1X1

2X

s2

1

4

2R

1Y7

4

1

s

1

32 s

s

10

s

2Y


Input

2RInput

Definitions (Input and Output nodes)

Input Node: Node with only outgoing branches.

Output Node: Node with only incoming branches.

1R1X1

2X

s2

1

4

1Y

7

4

1

s

1

1

2X1X

1

11 XX 22 XX

Converting any node to an output node by adding a unity

gain branch.




Definitions (Input and Output nodes)

Converting any node to an input node by rearranging the equations

2112

1X

sRX

211

2

1X

sXR

1R

1X

1 2X

s

2

1

4

2R

1Y

7

4

1

s

1


Definitions (Path)

• A continuous succession of branches traversed in the same direction is called the path.

• It is known as a forward path if it starts at an input node and ends at an output node without hitting a node more than once.

• The product of branch gains along a path is defined as the path gain.

1R

1X

12X

s

2

1

4

2R

1Y

7

4

1

s

1

Path gain between X1 and R1

Path 1 “1”

Path 2 “4/(2+S)”




Definitions (Loop)

• A loop is a path that originates and ends at the same node without encountering other nodes more than once along it traverse.

• When a branch originates and terminates at the same node, it is called a self-loop.

• The path gain of a loop is defined as the loop gain.

Self Loop

1R1X1

2X

s2

1

4

2R

1Y

7

4

1

s

1

Loop Loop Gain “-4/(2+S)”


Rules of Reduction (Rule 1)

• When there is only one incoming and one outgoing branch at a node (i.e. two branches are connected in series), it can be replaced by a direct branch with branch gain equal to the product of the two.

1X 1R 2X

s5 2

2X

s10

1X





• Two or more parallel paths connecting two nodes can be merged into a single path with a gain is equal to the sum of the original path gains.

2X

25 s

1X1X

2X

s5

2



• A self loop of gain G at a node can be eliminated by multiplying its input branches by 1/(1-G).

2X

4

1X

s2

2X

S21

4

1X

212 24 sXXX

122 42 XsXX

s

XX

21

4 12

Proof





• A node that has one output and two or more input branches can be split in such a way that each node has just one input and one output branch.

2X

1X

3X

5X

3C

4C

2C

1C

2X

1X

3X

5X

3C

4C

2C

1C

4C

4C

4X

4X

4X

X4



• This is similar to rule 4. A node that has one input and two or more output branches can be split in such a way that each node has just one input and one output branch.

2X

1X

3X

5X

3C

4C

2C

1C4X

2X

3X

5X

3C

4C

2C

1C

1C

1C

4X

4X

4X




Mason’s Gain Rule

• Ratio T of the effect (output) to that of the cause (input) can be found using Mason’s rule as follows:

332211 PPP

P

sT k

kk

Where, Pi is the gain of the ith forward path

3211 LLL

)1()1()1(

1 3211 LLL

)2()2()2(

2 3211 LLL

)3()3()3(

3 3211 LLL


Mason’s Gain Rule

1L

Stands for the sum of all second-order loop gains. 2L

Stands for the sum of all first-order loop gains.

1

1L Denotes the sum of those first-order loop gains that do not touch path P1 any node.

1

2L Denotes the sum of those second-order loop gains that do not touch path P2 any node.

2

1L Denotes the sum of those first-order loop gains that do not touch path P1 any node.

Second-order loop gain is the product of two first-order loops that do not touch at any point. Third-order loop gain is the product of three first-order loops that do not touch at any point.




Mason’s Gain Rule

∆ = 1 – (sum of all different loop gains) + (sum of products of all pairs of loop gains, for non-touching loops) – (sum of products of all triples of loop gains, for non-touching loops) + …

Pk = kth path from input to output.

∆k = The quantity ∆, but with all loops touching the kth path, Pk, removed.

332211 PPP

P

sT k

kk


Example 7

A signal-flow graph of a two-port network is given in Figure Using Mason’s rule, find its transfer function Y/R.

Solution

21

313

21

1

1111

ss

s

s

s

sP 61612 P

1

414

1

1113

ssP

1

31

sL

2

52

s

sL

4

3 1

53

11R Y

1

2ss 11 s

6

2121

23212121

1

11

LLLL

LPLLLLPP

R

Y




Signal-flow graph representation of a voltage source

sZ

sI

-

o

sE 0 sV

sa

sb

ref

S

in

SSSssS

ref

S

in

Ss IIZEIZEVVV

reflected incident

in

SS

s

ref

SS V

Z

ZEV

Z

Z

00

11

in

S

S

Ss

S

ref

S VZZ

ZZE

ZZ

ZV

0

0

0

0

00

0

00

0

0 222 Z

V

ZZ

ZZ

Z

E

ZZ

Z

Z

V in

S

S

Ss

S

ref

S

SSGS abb

S

s

GZZ

EZb

0

0

2

02Z

Vb

ref

SS

02Z

Va

in

SS

0

0

ZZ

ZZ

S

SS

o

ref

S

in

SSSs

Z

VVZEV


Signal-flow graph representation of a voltage source

sI

sZ

-

o

sE 0 sV

sa

sb

11GbSb

SaSa 1

S

Sb

SSGS abb

1Gb

Sa

S

Sb

Input

Output




Signal-flow graph representation of a passive single-port device

LI

-

LV

La

Lb

LZ

ref

L

in

LLLL

ref

L

in

LL IIZIZVVV

in

LLref

LL V

Z

ZV

Z

Z

11

00

in

LL

in

L

L

Lref

L VVZZ

ZZV

0

0

0Z

VVZV

ref

L

in

LLL

LLL ab o

in

LL

o

ref

L

Z

V

Z

V

22

1 La

Lb1

La

L

Lb

La

Lb

L


Example 8

ZL in

1b2a

1a

11S

12S

2b

22S

21S1 La

Lb1

L

Load Two-port network

1

211

1

1

1

1

L

PLP

a

bin

111 SP

122112212 11 SSSSP LL

22221 11 SSL LL

22

12212211

1

1

S

SSSS

L

LLin

22

122111

1 S

SSS

L

Lin

1

1

a

bin

Two-port Network

Impedance ZL terminates port 2 of a two-port network as shown in Figure. Draw the signal flow graph and determine the reflection coefficient at its input port using Mason’s rule

Solution

a1

b1




Example 8 (another possible solution)

1b2a

1a

11S

12S

2b

22S

21S1 La

Lb1

L

1b2a

1a

11S

12S

2b

22S

21S

L

1b2a

1a

11S

12S

2b21S

L22SL

2a

1b2a

1a

11S

12S

2b

L

2a

LS

S

22

21

1

Rule 3

Rule 1

Rule 5


Example 8 (another possible solution)

1b2a

1a

11S

12S

2b

L

2a

LS

S

22

21

1

1b

1a

11SL

L

L

S

SS

22

2112

1

1b

1a

L

L

S

SSS

22

211211

1

Rule 1

Rule 2




Example 9

ZL

VS

Two-port Network

2

2

a

bout

221 SP

211221122 11 SSSSP SS

out

11111 11 SSL SS

S

SSout

out

S

SSSS

L

PLP

a

b

11

12211122

1

211

2

2

1

1

1

1

S

Sout

S

SSS

11

122122

1

Lb

ZS

A voltage source is connected at the input port of a two-port network and the load impedance ZL terminats its output, as shown. Draw its signal-flow graph and find the output reflection coefficient out.

Solution

1b2a

1a

11S

12S

2b

22S

21S1 La

1

L

Load Two-port network

S

1

1

Gb 1

Source

Sb

Sa


Example 10

The signal-flow graph shown represents a voltage source that is terminated by a passive load. Analyze the power transfer characteristics of this circuit and establish the conditions for maximum power transfer.

Lb

La

S

1

1

Gb 1 Sb

Sa

L

Solution

Output power of the source 2

Sb

Power reflected back into the source 2

Sa

Power delivered by the source dP

22

SS ab

Power incident on the load 2

La

Power reflected from the load 222

LLL ab

VS

ZS

ZL

Sb

Sa

Lb

La

Power absorbed by the load 22222211 LSLLLLL babaP

SLSGLLSGLSGSSGS bbabbbabb

LS

GS

bb

1

??dP




Example 11

GS

S

S bba

1

dP

2

2

11

L

LS

Gd

bP

2

2

11

L

LS

GL

bP

To maximize PL we should minimize the dominator

22))(( bajbajba

To minimize the dominator the product SL must be positive and pure real

*

SL

*

oS

oS

oL

oL

ZZ

ZZ

ZZ

ZZ *

SL ZZ

2

2

2

2

21

11 S

G

L

S

GL

bbP

SSGS abb

G

LS

G

S

bb

1

1

LS

GLb

1

22

SS ab

22

11 LS

GL

LS

G bb

LP 221 LSb

SL

1

SL1

Documents

What is Electrical Engineering? - GUCeee.guc.edu.eg/Courses/Communications/COMM903 Microwave... · 2016-09-19 · Signal Flow Graphs • A signal-flow graph can be used only when