Signal flow graph Mason’s Gain Formula

1. Topic
Signal Flow Graph

2. 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.

3. 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.

4. SFG

5. 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

6. 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

7. • 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

8. 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

9. 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
transmittance
input node (source)

10. Mason’s Gain Formula
• 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)

11. Ex: SFG from BD

12. Construction of SFG from simultaneous
equations

13. t21 t 23
t31
t32
t33

15. After joining all SFG

16. SFG from Differential equations
xyyyy  253
Consider 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 

17. y 
x
y
y
y 
1
-2
-5
-3
1/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.

18. Step5: Redraw the SFG as shown.

19. Problem: to find out loops from
the given SFG

20. Ex: Signal-Flow Graph Models

21. P 1 =
P 2 =

22. 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

23. 




..)21(1( LiLjLkiLjLLL
P
R
Y kk
Y s( )
R s( )
G1 G2 G3 G4 1 L3 L4   G5 G6 G7 G8 1 L1 L2  
1 L1 L2 L3 L4 L1 L3 L1 L4 L2 L3 L2 L4

24. Block Diagram Reduction Example
R
_+
_
+1G 2G 3G
1H
2H
+
+
C

26. R
R

27. R

28. R
_+
232121
321
1 HGGHGG
GGG

C
R
321232121
321
1 GGGHGGHGG
GGG

C

29. Solution for same problem by using
SFG

30. Forward Path
P 1 = G 1 G 2 G3

31. Loops
L 1 = G 1 G 2 H1 L 2 = - G 2 G3 H2

32. 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 ]

33. SFG from given T/F
( ) 24
( ) ( 2)( 3)( 4)
C s
R s s s s

  
)21()2(
1
1
1




 s
s
s