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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Signal Flow graphs


Introduction

• Block diagram is a pictorial representation of the sequence of events in a process

• For complicated systems difficult to reduce block diagrams

• Signal flow graph – a graphical representation of the relationship between variables – Provides relationship between system variables

without the need for reduction


Signal Flow Graph (SFG)

Block Diagram to signal flow graph


Signal Flow Graph (SFG)

• A signal-flow graph is a diagram consisting of

nodes that are connected by several directed

branches and is a graph representation of a set of

linear relation.• Only valid for linear system


Basic elements of SFG

• Branch: A unidirectional path segment

• Nodes: The input and output points or

junctions



Input Node Output NodeMixed Node


• Path: A branch or a continuous sequence of branches

that can be traversed from one node to anther node.


1. Forward Path2. Feedback path


• Loop: A closed path that originates and terminates on the same node, and along the path no node is met twice.



• Non-touching loops: If two loops do not have a common node.



Touching loops: Two touching loops share one or more common nodes.



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




1. Forward Path Gain: P1 = G1G2G3

P2 = G4

P3 = G1G2(-1)


2. Feedback Path Gain: -G1H1


In this case there is one loop or Feed back path



3. Non-touching loop gain: Two Non-touching loops gain: G2H2H3

G2H2H7G7

G7H7H3

Three Non-touching loops gain: G2H2H3H7G7



)(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.


Block Diagram to SFG


G4(s)


E(s) x1 (s) x2 (s) Y(s)

-H1(s)

G1(s) G2(s) G3(s)

-1

R(s)

1


Mason’s gain formula

The linear dependence (Gain) T i j between input variable x i and output variable x j is given by the following formula:

k ijkijk

ij

PT


Mason’s Gain Rule

Mason’s gain rule is as follows: the transfer function of a system with signal-input, signal-output flow graphs is

332211)(

pppsT

Δ=1-(sum of all loop gains)+(sum of products of gains of all combinations if 2 non-touching loops)- (sum of products of gains of all combinations if 3 non-touching loops)+…

A path is any succession of branches, from input to output, in the direction of the arrows, that does not pass any node more than once.

A loop is any closed succession of branches in the direction of the arrows that does not pass any node more than once.


Ex 1: Signal-Flow Graph Models


P 1 = G1G2G3G4 P 2 = G5G6G7G8



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 2: SFG


Ex 2: SFG

Forward Paths

P1 = G1G2G3G4G5G6

P2 = G1G2G7G6

P3 = G1G2G3G4G8


L 3 = -G 8 H 1

LOOPS

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

Ex 2: SFG


L 4 = - G2 G 7 H2

L5 = -G 4 H 4

L1= -G 5 G 6 H 1

LOOPS

Ex 2: SFG


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

Ex 2: SFGLOOPS


Ex 2: SFG

Pair of 2 non-touching loops


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


Simultaneous equations to SFG

𝑦 2=𝑡 21 𝑦1+𝑡 23 𝑦3𝑦 3=𝑡 32 𝑦2+𝑡33 𝑦 3+𝑡31 𝑦1𝑦 4=𝑡43 𝑦 3+𝑡42 𝑦2

𝑦 5=𝑡 54 𝑦4𝑦 6=𝑡 65 𝑦5+𝑡 64 𝑦4


t21t 23

t31

t32 t33

𝑦 2=𝑡 21 𝑦1+𝑡 23 𝑦3

𝑦 3=𝑡 32 𝑦2+𝑡33 𝑦 3+𝑡31 𝑦1

𝑦 4=𝑡43 𝑦 3+𝑡42 𝑦2



𝑦 5=𝑡 54 𝑦4

𝑦 6=𝑡 65 𝑦5+𝑡 64 𝑦4



After joining all SFG

Csl8 2 f15

Engineering

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