Signal Flow Graph

 Signal Flow Graph

Signal flow graph is the graphical representation of control system in which nodes representing each of system variables are connected through the direct branches. It may be regarded as the simplified version of Block Diagram. It applies only to linear system.


Basic Terminology of Signal Flow Graph


Node :-  It represent a system variable.

Branch :- Nodes are connected by line segments called branches, these branches have associated branch gain and direction, direction is represented by arrow. A signal can transmit through the branch only in the direction of arrow.


Input Node / Source node :- It is a node having outgoing branches only.


Output Node / Sink Node :- It is the node having only incoming branches.


Note :- In a signal flow graph output can be defined from any other node by extending the it with a gain of unity.


Mixed / Chain Node :- It is a node having both incoming and outgoing branches.


Path :- It is the traversal of connected branches in the direction of branch arrow such that no node is traversed more than once. A signal flow graph can have one or more than one paths.


Forward Path :- It is a path starts from input node and end at an output node.


Loop :- It is a path which originates and terminates at the same node.


Non Touching Loop :- Two or more loops are said to be non-touching if they do not have a common node.


 

Mason Gain Formula


Transfer Function =  nƩk=1 Pk k / Δ


Where,


n = number of forward paths

Pk = Path gain of the forward Kth path


Δ = 1 – {Sum of loop gains all individual loops} + {Sum of gain of product of two non-touching loops} – {Sum of gain products of three non-touching loop}...........


ΔK = It is that value of “Δ” obtained by removing all the nodes touching Kth forward path.



Calculation of transfer function by using Mason Gain formula

 

Let us take an example

Signal flow graph

In this Signal Flow Graph,


There are seven nodes these are Y1, Y2, Y3, Y4, Y5, Y6, Y7. In which Y1 is input node or source node and Y7 is output node or sink node and rest of nodes are mixed or chain nodes.



These seven are connected through the branches having forward transfer gain G1, G2, G3, G4, G5, 1 respectively and Y2, Y6 is connected through another branch have forward transfer gain G6.



Similarly Y2 and Y3 is connected through a branch having negative gain H1 and Y4 and Y6 is also connected through a branch having negative gain H2.



So according to formula first we have to find the number of forward path and product of the gains of those forward paths.


There are two forward paths


P1 = G1G2G3G4G51

P2 = G1G61


Then, we have to find product of gain of individual loops. So there are two individual loops these are


L1 =  -G2H1

L2 = -G4G5H2


Then we have to find product of gain of two non-touching loop. There are two non-touching loop these are


L1 =  -G2H1

L2 = -G4G5H2


Product of gain of these two loops is I1 = G2G4G5H1H2


Then we have to find Δk.


Δ1 =1

Δ2 = 1


So put all those value in Mason Gain formula after putting these values we get

Transfer function =

So In this way we apply the Mason Gain formula for a particular problem.



How to convert Block Diagram into Signal Flow Graph


There are some steps which we have to follow for the conversion of Block Diagram into Signal Flow Graph



Step  :- 1   Convert all the variables, summing point, take-off point, junctions into nodes


Step :- 2 Represent all the blocks of the block diagram as branches with given directions.


Step :- 3 The transfer function of the particular block represent the gain of the corresponding branch of that particular block.


Step :- 4  Connect all the nodes with branches according to the block diagram convention and direction.



Let us take an example


Block Diagram
Block diagram of feedback control system.


In the above diagram there are three summing points S1 S2 and S3 respectively, two take-off points T1 and T2  and one junction point J1.


So, according to above steps convert

S1 ――> N2

S2 ――> N4

S3 ――> N6

T1 ――> N3

T2 ――> N5

J1 ――> N1


Connects all the nodes with their respective branches and assign the gain for all the branches accordingly.


Signal flow graph















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