This document contains several examples of using Mason's Rule to calculate transfer functions from signal flow graphs. The examples demonstrate applying Mason's Rule to calculate loop gains, forward path gains, non-touching loops, and the overall transfer function. Signal flow graphs are constructed from block diagrams and the steps of Mason's Rule are systematically worked through.

1. Control System Engineering
Kuntumal Sagar M.
B.TECH (E.E)
UID-U41000000484
Email: skuntmal@yahoo.com
TOPIC
Signal Flow Graph
Examples

2. 2
Example#1: Apply Mason’s Rule to calculate the transfer function
of the system represented by following Signal Flow Graph
774663332221 HGLHGLGHLHGL  ,,,
2. Calculate all loop gains.
3. Consider two non-touching loops.
L1L3 L1L4
L2L4 L2L3
2)(pathand1)(path 8765243211 GGGGPGGGGP 
1. Calculate forward path gains for each forward path.
P1
P2

3. 4. Consider three non-touching loops.
None.
5. Calculate Δ from steps 2,3,4.
   4232413143211 LLLLLLLLLLLL 
3
 
 7733663377226622
776633221
HGGHHGGHHGHGHGHG
HGHGGHHG


Example#1: continue

4. 4
Example#1: continue
Eliminate forward path-1
 431 1 LL 
 212 1 LL 
Eliminate forward path-2
 77661 1 HGHG 
 33222 1 HGHG 

5. 

 2211 PP
sR
sY
)(
)(
5
Example#: continue
     
   773366337722662277663322
3322876577664321
1
11
HGGHHGGHHGHGHGHGHGHGGHHG
HGHGGGGGHGHGGGGG
sR
sY



)(
)(

6. Example#2
• Find the transfer function, C(s)/R(s), for the signal-flow
graph in figure below.

7. Example#2
• There is only one forward Path.
)()()()()( sGsGsGsGsGP 543211 

8. Example#2
• There are four feedback loops.

9. Example#2
• Non-touching loops taken two at a time.

10. Example#2
• Non-touching loops taken three at a time.

11. Example#2
Eliminate forward path-1

12. Example#3: Apply Mason’s Rule to calculate the transfer function
of the system represented by following Signal Flow Graph




 
  332211
3
1 PPP
P
sR
sC i
ii
)(
)(
There are three forward paths, therefore n=3.

13. Example#3: Forward Paths
722 AP 76655443321 AAAAAP 
766554423 AAAAP 

14. Example#3: Loop Gains of the Feedback Loops
23321 AAL 
34432 AAL 
45543 AAL 
56654 AAL 
67765 AAL 
776 AL 
2334427 AAAL 
6776658 AAAL 
23344557729 AAAAAL 
23344556677210 AAAAAAL 

15. Example#3: two non-touching loops
31LL
41LL
51LL
61LL
81LL
42LL
52LL
62LL
82LL
53LL
63LL
64LL
74LL
75LL 87LL

16. Example#3: Three non-touching loops
31LL
41LL
51LL
61LL
81LL
42LL
52LL
62LL
82LL
53LL
63LL
64LL
74LL
75LL 87LL

17. G1 G4G3
From Block Diagram to Signal-Flow Graph Models
Example#4
－
－
－
C(s)R(s)
G1 G2
H2
H1
G4G3
H3
E(s) X1
X2
X3
R(s) C(s)
－H2
－H1
－H3
X1 X2 X3E(s)1 G2

18. 1;
)(1
143211
14323234321


GGGGP
HGGHGGHGGGG
14323234321
4321
1)(
)(
HGGHGGHGGGG
GGGG
sR
sC
G


R(s)
－H2
1G4G3G2G11 C(s)
－H1
－H3
X1 X2 X3E(s)
From Block Diagram to Signal-Flow Graph Models
Example#4

19. G1
G2
+－
+
－
－
－
+ C(s)R(s) E(s)
Y2
Y1
X1
X2
－
1
-1
1
-1
-1
-1
-1
1
1
G1
G2
1
R(s) E(s) C(s)
X1
X2
Y2
Y1
Example#5

20. Example#5
1
-1
1
-1
-1
-1 -1
1
1
G1
G2
1
R(s) E(s) C(s)
X1
X2
Y2
Y1
7 loops:
3 ‘2 non-touching loops’ :

21. Example#5
1
-1
1
-1
-1
-1 -1
1
1
G1
G2
1
R(s) E(s) C(s)
X1
X2
Y2
Y1
212 G4G2G1Δ Then:
4 forward paths:
211 G1Δ1G1)(p 1
1Δ1G1)(G1)(p 221 2
132 G1Δ1G1p 3
1Δ1G1G1p 412 4

22. Example#5
We have
212
2112
421
2
GGG
GGGG
p
sR
sC kk




 

)(
)(

23. Example-6: Determine the transfer function C/R for the block diagram below
by signal flow graph techniques.
• The signal flow graph of the above block diagram is shown below.
• There are two forward paths. The path gains are
• The three feedback loop gains are
• No loops are non-touching, hence
• Since no loops touch the nodes of P2,
therefore
• Because the loops touch the nodes of P1,
hence
• Hence the control ratio T = C/R is

24. Example-6: Find the control ratio C/R for the system given below.
• The two forward path gains are
• The signal flow graph is shown in the figure.
• The five feedback loop gains are
• Hence the control ratio T =
• There are no non-touching loops, hence
• All feedback loops touches the two forward
paths, hence

25. Design Example#1
RsIsI
Cs
sV )()()( 111
1

RsIsV )()( 12 
)()()( sIsCsVsCsV 121 
)(sV1 )(sI1
)(sV2
Cs R
Cs

26. Design Example#2
)( 2111
2
1 XXkXsMF  221212
2
20 XkXXkXsM  )(

27. Design Example#2

28. Design Example#2

29. THANK YOU