Lecture On

Signal Flow Graph

Submitted By:

Ms. Anupam MittalA.P., EE Deptt

SBSSTC, Ferozepur

Flow of PPT

• What is Signal Flow Graph (SFG)?• Definitions of terms used in SFG• Rules for drawing of SFG• Mason’s Gain formula• SFG from simultaneous eqns• SFG from differential eqns• Examples• Solution of a problem by Block diagram reduction technique

and SFG• SFG from a given Transfer function• Examples

What is Signal Flow Graph? SFG is a diagram which represents a set of simultaneous equations. This method was developed by S.J.Mason. This method does n’t require any reduction technique. It consists of nodes and these nodes are connected by a directed line called branches. Every branch has an arrow which represents the flow of signal. For complicated systems, when Block Diagram (BD) reduction method becomes tedious and time consuming then SFG is a good choice.

Comparison of BD and SFG

)(sR)(sG

)(sC )(sG

)(sR )(sC

block diagram: signal flow graph:

In this case at each step block diagram is to be redrawn. That’s why it is tedious method.So wastage of time and space.

Only one time SFG is to be drawn and then Mason’s gain formula is to be evaluated.So time and space is saved.

SFG

Node: It is a point representing a variable. x2 = t 12 x1 +t32 x3

X2

X1 X2

X3

t12

t32

X1

Branch : A line joining two nodes.

Input Node : Node which has only outgoing branches.

X1 is input node.

In this SFG there are 3 nodes.

Definition of terms required in SFG

Output node/ sink node: Only incoming branches.

Mixed nodes: Has both incoming and outgoing branches.

Transmittance : It is the gain between two nodes. It is generally written on the branch near the arrow.

t12

X1

t23

X3

X4

X2

t34

t43

• Path : It is the traversal of connected branches in the direction of branch arrows, such that no node is traversed more than once.• Forward path : A path which originates from the input node and terminates at the output node and along which no node is traversed more than once.• Forward Path gain : It is the product of branch transmittances of a forward path.

P 1 = G1 G2 G3 G4, P 2 = G5 G6 G7 G8

Loop : Path that originates and terminates at the same node and along which no other node is traversed more than once.

Self loop: Path that originates and terminates at the same node.

Loop gain: it is the product of branch transmittances of a loop. Non-touching loops: Loops that don’t have any common node

or branch.

L 1 = G2 H2 L 2 = H3

L3= G7 H7

Non-touching loops are L1 & L2, L1 & L3,

L2 &L3

SFG terms representation

input node (source)

b1x a

2x c

4x

d

1

3x

3x

mixed node mixed node

forward path

path

loop

branch

node

transmittanceinput node (source)

Rules for drawing of SFG from Block diagram

• All variables, summing points and take off points are represented by nodes.

• If a summing point is placed before a take off point in the direction of signal flow, in such a case the summing point and take off point shall be represented by a single node.

• If a summing point is placed after a take off point in the direction of signal flow, in such a case the summing point and take off point shall be represented by separate nodes connected by a branch having transmittance unity.

• A technique to reduce a signal-flow graph to a single transfer function requires the application of one formula.

• The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is

k = number of forward path Pk = the kth forward path gain

∆ = 1 – (Σ loop gains) + (Σ non-touching loop gains taken two at a time) – (Σ non-touching loop gains taken three at a time)+ so on .

∆ k = 1 – (loop-gain which does not touch the forward path)

Mason’s Gain Formula

Ex: SFG from BD

EX: To find T/F of the given block diagram

Identification of Forward Paths

P 1 = 1.1.G1 .G 2 . G3. 1= G1 G2 G3

P 2 = 1.1.G 2 . G 3 . 1= G 2 G3

Individual Loops

L 1 = G 1G 2 H 1 L 2 = - G 2G 3 H 2

L 3 = - G 4 H 2

L 4 = - G 1 G 4

L 5 = - G 1 G 2 G 3

Construction of SFG from simultaneous equations

t21 t 23

t31

t32 t33

After joining all SFG

SFG from Differential equations

xyyyy 253Consider the differential equation

Step 2: Consider the left hand terms (highest derivative) as dependant variable and all other terms on right hand side as independent variables.Construct the branches of signal flow graph as shown below:-

1

-5-2

-3

y

y

y y

x

(a)

Step 1: Solve the above eqn for highest order

yyyxy 253

y

x

y

y

y

1-2

-5

-31/s

1/s

1/s

Step 3: Connect the nodes of highest order derivatives to the lowest order der.node and so on. The flow of signal will be from higher node to lower node and transmittance will be 1/s as shown in fig (b)

(b)

Step 4: Reverse the sign of a branch connecting y’’’ to y’’, with condition no change in T/F fn.

Step5: Redraw the SFG as shown.

Problem: to find out loops from the given SFG

Ex: Signal-Flow Graph Models

P 1 =

P 2 =

Individual loops

L 1 = G2 H2

L 4 = G7 H7

L 3 = G6 H6

L 2= G3 H3

Pair of Non-touching loops L 1L 3 L 1L 4

L2 L3 L 2L 4

..)21(1( LiLjLkiLjLLL

P

R

Y kk

Y s( )

R s( )

G 1 G 2 G 3 G 4 1 L 3 L 4 G 5 G 6 G 7 G 8 1 L 1 L 2

1 L 1 L 2 L 3 L 4 L 1 L 3 L 1 L 4 L 2 L 3 L 2 L 4

Ex:

Forward Paths

L5 = -G 4 H 4

L1= -G 5 G 6 H 1

L 3 = -G 8 H 1

L 2 = -G2 G 3G 4G 5 H2

L 4 = - G2 G 7 H2

Loops

Loops

L 7 = - G 1G2 G 7G 6 H3

L 6 = - G 1G2 G 3G 4G 8 H3

L 8= - G 1G2 G 3G 4G 5 G 6 H3

Pair of Non-touching loops

L 4

L 5

L 3L 7

L 4

L 5L 7

L 4L 5

L 3L 4

Non-touching loops for paths

∆ 1 = 1∆ 2= -G 4 H4

∆ 3= 1

Signal-Flow Graph Models

Y s( )

R s( )

P1 P2 2 P3

P1 G1 G2 G3 G4 G5 G6 P2 G1 G2 G7 G6 P3 G1 G2 G3 G4 G8

1 L1 L2 L3 L4 L5 L6 L7 L8 L5 L7 L5 L4 L3 L4

1 3 1 2 1 L5 1 G4 H4

Block Diagram Reduction Example

R_+

_+

1G 2G 3G

1H

2H

+ +

C

R

R

R

R_+

232121

321

1 HGGHGG

GGG

C

R

321232121

321

1 GGGHGGHGG

GGG

C

Solution for same problem by using SFG

Forward Path

P 1 = G 1 G 2 G3

Loops

L 1 = G 1 G 2 H1 L 2 = - G 2 G3 H2

L 3 = - G 1 G 2 G3

P 1 = G 1 G 2 G3

L 1 = G 1 G 2 H1

L 2 = - G 2 G3 H2

L 3 = - G 1 G 2 G3

∆1 = 1

∆ = 1- (L1 + L 2 +L 3 )

T.F= (G 1 G 2 G3 )/ [1 -G 1 G 2 H1 + G 1 G 2 G3 + G 2 G3 H2 ]

SFG from given T/F

( ) 24

( ) ( 2)( 3)( 4)

C s

R s s s s

)21()2(

11

1

s

s

s

Ex:

Thanks

Example of block diagram

Step 1: Shift take off point from position before a block G4 to position after block G4

Step2 : Solve Yellow block.

Step3: Solve pink block.

Step4: Solve pink block.