1. Lecture On Signal Flow Graph
Submitted By:
Ms. Anupam Mittal
A.P., EE Deptt
SBSSTC, Ferozepur

2. 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

3. 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.

4. Comparison of BD and SFG
block diagram:
signal flow graph:
Only one time SFG is to be drawn and then Mason’s gain formula is to be evaluated.
So time and space is saved.
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.

5. SFG

6. Definition of terms required in SFG
Node: It is a point representing a variable.
x2 = t 12 x1 +t32 x3 X1 X2 X3
t12
t32
In this SFG there are 3 nodes.
Branch : A line joining two nodes. X1 X2
Input Node : Node which has only outgoing branches.
X1 is input node.

7. Output node/ sink node: Only incoming branches
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

8. 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

9. Non-touching loops are L1 & L2, L1 & L3, L2 &L3
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 H L 2 = H3
L3= G7 H7
Non-touching loops are L1 & L2, L1 & L3, L2 &L3

10. SFG terms representation
input node (source)
input node (source)
mixed node
forward path
path
loop
branch
node
transmittance

11. 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.

12. 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)

13. Ex: SFG from BD

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

15. 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

16. Individual Loops
L 1 = G 1G 2 H 1
L 2 = - G 2G 3 H 2
L 3 = - G 4 H 2

17. L 4 = - G 1 G 4
L 5 = - G 1 G 2 G 3

18. Construction of SFG from simultaneous equations

19. t21
t 23
t31
t32
t33

21. After joining all SFG

22. SFG from Differential equations
Consider the differential equation
Step 1: Solve the above eqn for highest order
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
(a)

23. 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) 1 -2 -5 -3
1/s
(b)
Step 4: Reverse the sign of a branch connecting y’’’ to y’’, with condition no change in T/F fn.

24. Step5: Redraw the SFG as shown.

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

26. Ex: Signal-Flow Graph Models

27. P 1 =
P 2 =

28. 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 L 1L 4
L2 L L 2L 4

30. Ex:

31. Forward Paths

32. Loops L1= -G 5 G 6 H 1 L 2 = -G2 G 3G 4G 5 H2 L 3 = -G 8 H 1

33. Loops L 6 = - G 1G2 G 3G 4G 8 H3 L 7 = - G 1G2 G 7G 6 H3
L 8= - G 1G2 G 3G 4G 5 G 6 H3

34. Pair of Non-touching loops
L 4L 5
L 4
L 7
L 3
L 4
L 5L 7
L 3L 4

35. Non-touching loops for paths
∆ 1 = 1
∆ 2= -G 4 H4
∆ 3= 1

36. Signal-Flow Graph Models

37. Block Diagram Reduction Example_ +

38. R

40. R

41. R

42. _ +

44. Solution for same problem by using SFG

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

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

47. 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 ]

48. SFG from given T/F

49. Ex:

55. Thanks

56. Example of block diagram
Step 1: Shift take off point from position before a block G4 to position after block G4

57. Step2 : Solve Yellow block.
Step3: Solve pink block.
Step4: Solve pink block.