The document provides examples of block diagram reduction techniques. It begins with 12 examples showing the step-by-step process of reducing block diagrams to obtain transfer functions. It then provides block diagram examples of an armature controlled DC motor and a liquid level system. The examples illustrate techniques for moving pickoff points, eliminating feedback loops, and applying rules to reduce block diagrams to single transfer functions.

1. Control System Engineering
Kuntumal Sagar M.
B.TECH (E.E)
UID-U41000000484
Email: skuntmal@yahoo.com
TOPIC
BLOCK DIAGRAM EXAMPLES

2. Example 9
Find the transfer function of the following block diagrams
2G 3G1G
4G
1H
2H
)(sY)(sR

3. 1. Moving pickoff point A ahead of block 2G
2. Eliminate loop I & simplify
324 GGG 
B
1G
2H
)(sY
4G
2G
1H
AB
3G
2G
)(sR
I
Solution:

4. 3. Moving pickoff point B behind block
324 GGG 
1G
B
)(sR
21GH 2H
)(sY
)/(1 324 GGG 
II
1G
B
)(sR C
324 GGG 
2H
)(sY
21GH
4G
2G A
3G 324 GGG 

5. 4. Eliminate loop III
)(sR
)(1
)(
3242121
3241
GGGHHGG
GGGG

 )(sY
)()(1
)(
)(
)(
)(
32413242121
3241
GGGGGGGHHGG
GGGG
sR
sY
sT



)(sR
1G
C
324
12
GGG
HG

)(sY
324 GGG 
2H
C
)(1 3242
324
GGGH
GGG


Using rule 6

6. 2G1G
1H 2H
)(sR )(sY
3H
Example 10
Find the transfer function of the following block diagrams

7. Solution:
1. Eliminate loop I
2. Moving pickoff point A behind block
22
2
1 HG
G

1G
1H
)(sR )(sY
3H
BA
22
2
1 HG
G

2
221
G
HG
1G
1H
)(sR )(sY
3H
2G
2H
BA
II
I
22
2
1 HG
G

Not a feedback loop
)
1
(
2
22
13
G
HG
HH



8. 3. Eliminate loop II
)(sR )(sY
22
21
1 HG
GG

2
221
3
)1(
G
HGH
H


21211132122
21
1)(
)(
)(
HHGGHGHGGHG
GG
sR
sY
sT


Using rule 6

9. 2G 4G1G
4H
2H
3H
)(sY)(sR
3G
1H
Example 11
Find the transfer function of the following block diagrams

10. Solution:
2G 4G1G
4H
)(sY
3G
1H
2H
)(sR
A B
3H
4
1
G
4
1
G
I
1. Moving pickoff point A behind block
4G
4
3
G
H
4
2
G
H

11. 2. Eliminate loop I and Simplify
II
III
443
432
1 HGG
GGG
1G
)(sY
1H
B
4
2
G
H
)(sR
4
3
G
H
II
332443
432
1 HGGHGG
GGG

III
4
142
G
HGH 
Not feedbackfeedback

12. )(sR )(sY
4
142
G
HGH 
332443
4321
1 HGGHGG
GGGG

3. Eliminate loop II & IIII
143212321443332
4321
1)(
)(
)(
HGGGGHGGGHGGHGG
GGGG
sR
sY
sT


Using rule 6

13. 3G1G
1H
2H
)(sR )(sY
4G
2G A
B
Example 12
Find the transfer function of the following block diagrams

14. Solution:
1. Moving pickoff point A behind block
3G
I
1H
3
1
G
)(sY
1G
1H
2H
)(sR
4G
2G
A B
3
1
G
3G

15. 2. Eliminate loop I & Simplify
3G
1H
2G B
3
1
G
2H
32GG B
2
3
1
H
G
H

1G
)(sR )(sY
4G
3
1
G
H
23212
32
1 HGGHG
GG

II

16. )(sR )(sY
12123212
321
1 HGGHGGHG
GGG

3. Eliminate loop II
12123212
321
4
1)(
)(
)(
HGGHGGHG
GGG
G
sR
sY
sT


4G

17. Example-13: Simplify the Block Diagram.

18. Example-13: Continue.

19. Example-14: Reduce the Block Diagram.

20. Example-14: Continue.

21. Example-15: Reduce the Block Diagram. (from Nise: page-242)

22. Example-15: Continue.

23. Example-16: Reduce the system to a single transfer function.
(from Nise:page-243).

24. Example-17: Simplify the block diagram then obtain the close-
loop transfer function C(S)/R(S). (from Ogata: Page-47)

25. Example-18: Multiple Input System. Determine the output C
due to inputs R and U using the Superposition Method.

26. Example-18: Continue.

27. Example-18: Continue.

28. Example-19: Multiple-Input System. Determine the output C
due to inputs R, U1 and U2 using the Superposition Method.

29. Example-19: Continue.

30. Example-19: Continue.

31. Example-20: Multi-Input Multi-Output System. Determine C1
and C2 due to R1 and R2.

32. Example-20: Continue.

33. Example-20: Continue.
When R1 = 0,
When R2 = 0,

34. Block Diagram of Armature Controlled D.C Motor
Va
ia
T
Ra La
J

c
eb
 
  (s)IK(s)cJs
(s)V(s)K(s)IRsL
ama
abaaa





35. Block Diagram of Armature Controlled D.C Motor
  (s)E(s)K(s)IRsL abaaa  

36. Block Diagram of Armature Controlled D.C Motor
  (s)IK(s)cJs ama 

37. Block Diagram of Armature Controlled D.C Motor

38. Block Diagram of liquid level system
1
1
1 qq
dt
dh
C 
1
21
1
R
hh
q


21
2
2 qq
dt
dh
C 
2
2
2
R
h
q 

39. Block Diagram of liquid level system
)()()( sQsQssHC 111 
1
21
1
R
sHsH
sQ
)()(
)(


2
2
2
R
sH
sQ
)(
)(  )()()( sQsQssHC 2122 
1
1
1 qq
dt
dh
C 
1
21
1
R
hh
q


21
2
2 qq
dt
dh
C 2
2
2
R
h
q 
L
L
L
L

40. Block Diagram of liquid level system
)()()( sQsQssHC 111 
1
21
1
R
sHsH
sQ
)()(
)(


2
2
2
R
sH
sQ
)(
)(  )()()( sQsQssHC 2122 

41. Block Diagram of liquid level system