Introduction of Signal Flow Graph (SFG)

1. Presented By : Dhruv Shah
TOPIC : Signal Flow Graph (SFG)
Control Engineering

2.  Comparison of BD and SFG
)(sR
)(sG
)(sC )(sG
)(sR )(sC
(A) Block Diagram: (B) Signal flow
Graph:

3.  Signal Flow Graph (SFG) :

4. (01) NODE: IT IS A POINT REPRESENTING A VARIABLE.
X2 = T 12 X1 +T32 X3
X
2
X
(02)Branch : A line joining two nodes.
In this SFG there are 3 nodes.
 Definition of terms required in SFG
X1 X2
X3
t12
t32

5. (03)INPUT NODE : NODE WHICH HAS ONLY
OUTGOING BRANCHES. X1 IS INPUT NODE.
(04)OUTPUT NODE/ SINK NODE: ONLY INCOMING
BRANCHES.
(05)MIXED NODES: HAS BOTH INCOMING AND
OUTGOING BRANCHES.
t12
X1
t23
X3
X4X2
t34
t43

6. (06)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.
(07)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

7. (08) Loop : Path that originates and terminates at the same
node and along which no other node is traversed more than
once.
(09)Self loop: Path that originates and terminates at the same
node.
(10)Loop gain: it is the product of branch transmittances of a
loop.
(11)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

8.  SFG terms representation:
input node (source)
b1x a 2x
c
4x
d
1
3x
4x
mixed node
mixed node
forward path
path
loop
branch
node

9. 1a
2a
3a
1a 3a
21S 32S
1
21 32S S
1
1 1

 Techniques
(1) Series paths
3 21 32 1a S S a
2 21 1
3 32 2
a S a
a S a



Note that we have removed the node a2.

10. 1a 2a
1a 2a
aS
bS
a bS S

(2) Parallel paths
2 1 1a ba S a S a 

 2 1a ba S S a 
Note that we have combined the two parallel paths.

11.  Mason’s Gain Formula:
• The transfer function, C(s)/R(s), of a system represented
by a signal-flow graph is;
Where,
n = number of forward paths.
Pi = the i th forward-path gain.
∆ = Determinant of the system
∆i = Determinant of the ith forward path

 
 
n
i
iiP
sR
sC 1
)(
)(

12. ∆ = 1- (sum of all individual loop gains) + (sum of the
products of gains of all possible Combinations of two non-
touching loops) – (sum of the products of gains of all
possible Combinations of three non-touching loops) + …
and so forth with sums of higher number of non-touching
loop gains
∆ i = 1 – (loop-gain which does not touch the forward
path)




n
i
iiP
sR
sC 1
)(
)(

13. 01. Find out all possible Forward path and
hence the
corresponding Forward path Gains.
P1 =First Forward path,
P2 =Second Forward path………….. PN
02. Find out all possible Single
(Individual)Loop.
P11 = First Single loop,
P21 = Second Single loop,
P31, P41........... Pm1
Steps

14. 03. Find out all possible Combination of two non-
touching Loop.
P12 = Two non-touching Loop.
04. Find out all possible Combination of three
non-touching Loop.
P13 = Three non-touching Loop.
05. Find out all possible Combination of four
non-touching Loop.
& so on.
P14 = four non-touching Loop.

15. 06. The value of ∆ is given by
∆ = 1 - [P11 + P21 + P31 +………..+ Pm1] +[P12 +
P22+ P32 +…………. + Pp2]
- [P13 + P23 + P33 +………..+ Pt3] +[P14
+…………………………+ Pk4]
+..............
07. Find out ∆1, ∆2, ∆3, ∆4.......... ∆n
08.Find out P1∆1, P2∆2, P3∆3,........... Pn∆n
09. Find out Transfer Function :
T.F = P1∆1 + P2∆2 + P3∆3+....../ ∆

16. Thank You !!!!!!