Outline
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Block Diagrams
M. Sami Fadali
EBME Dept.
University of Nevada
What are block diagrams?
Main rules: cascade, parallel, feedback.
Interchanging: pickoff, summation.
Combining/expanding summing junctions.
Examples.
1
2
Block Diagram Notation
Block Diagrams
• Visual algebra: use block diagram
manipulation instead of algebra.
• Block: transfer function of a subsystem.
• Line: Laplace transform of a variable.
• Simplify complex systems to obtain a
single equivalent input-output transfer
function.
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4
Cascade (Series) Rule
Cascade: No Loading Assumption
What is the transfer function of the cascade?
G1 ( s ) 
5
Assumption in Cascading
G1 ( s )G2 ( s ) 
GT (s) 
1
sC1
R1 
1
sC1

1
,  R1C1
 1s  1 1
G2 ( s ) 
1
 2s 1
, 2  R2C2
6
Buffer Amplifier: No loading.
1
 1 2 s 2  ( 1   2 ) s  1
V2 (s)
1

 G1(s)G2 (s)
2
Vi (s) 12s  (1 2 )s 1C2 C1
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8
Parallel Rule
Feedback Rule: Proof
R(s)
E (s)
+
C (s)
G (s)

E ( s )  R ( s )  H ( s )C ( s )
C (s)  G(s) E (s)
H (s)
1
E (s)

R( s ) 1  H ( s )G ( s )
T (s) 
C (s)
G(s)

R( s) 1  G ( s) H ( s)
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Feedback Control System
Feedback Rule
R(s)
+

10
C (s)
Closed- loop system
G (s)
H (s)
T (s) 
C (s)
G(s)

R( s) 1  G ( s) H ( s)
feed-forward
gain (transfer
function)
E ( s )  G1 ( s )R( s )  H 2 ( s ) H1 ( s )C ( s )
C ( s )  G3 ( s )G2 ( s ) E ( s )
E (s)
G1 ( s )

R( s ) 1  H 2 ( s ) H1 ( s )G2 ( s )G3 ( s )
Negative Feedback:
plus sign
open-loop gain
(transfer function)
11
feed-forward gain
open-loop gain
G3 ( s )G2 ( s )G1 ( s )
C (s)

transfer function
R ( s ) 1  H 2 ( s ) H1 ( sEquivalent
)G3 ( s )G
12
2 (s)
Simplify Block Diagram
Effect of Pickoff
F(s)
൅
G2 ( s )G3 ( s )
1  H 2 ( s ) H1 ( s )G2 ( s )G3 ( s )
R(s)
R(s)
G1(s)
G2 ( s )G3 ( s )
1  H 2 ( s ) H1 ( s )G2 ( s )G3 ( s )
G1 ( s )G2 ( s )G3 ( s )
1  H 2 ( s ) H1 ( s )G2 ( s )G3 ( s )
൅
C (s)
CANNOT use the feedback formula since E(s) is needed.
C (s)
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Interchange Order of Blocks
and Branching
Interchange order of Blocks and
Summing Junction
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Combining/Expanding Summing
Junctions
R1(s)
+
+
-
+
+
Problem Solution: 3 Basic Rules
C(s)
R(s)
Cascade
R2(s)
R3(s)
+
G(s)

H(s)
R4(s)
R(s)
R1(s)
C(s)
Feedback
+
R3(s)
G ( s)
1  G(s) H (s)
C(s)
+
R2(s)
C(s)
-
R4(s)
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Parallel
•If necessary,
interchange.
•Apply one of the 3 rules.
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Simplify
Example 5.1
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20
Move Pickoff Point
Example
H2
H2
R
+
+
+
-
G1
+
G2
+
C
G3
R
Move
pickoff
+
+
+
-
G1
+
+
G2
1/G3
H1
H1
C
G3
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Cascade Rule
Feedback Rule
H2
R
+
G1
+
+
+
+
-
+
+
-
R
C
G2G3
G1
+
G 2G 3
1  H 2G 2G 3
H1
H1
C
1/G3
1/G3
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24
Cascade Rule
R
+
G 1G 2 G 3
1  H 2G 2G 3
+
-
Parallel Rule
+
H1
1
G1G2G3
1  H 2 G 2 G3
R +
C

1
1/G3
C
H1
G3
H1
G3
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Feedback Rule
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Example
Move
pickoff
T (s) 
C (s)
R( s)
G1G2G3
1  H 2G2G3

 G1G2G3  H1 
1 

1  

1
H
G
G
G
2 2 3 
3 

V4 ( s )  G2 ( s )V3 ( s )
V3 ( s ) 
27
1
V4 ( s )
G2 ( s )
28
Two Feedback Loops
Book: Move Block
1/G2(s)
1/G2(s)
Easier: Use feedback rule
V4 (s)
G 2 (s)

V2 (s) 1  G 2 (s)H 2 (s)
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Feedback Rule
30
Combine Summing Junctions
1/G2(s)
1/G2(s)
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32
Parallel Rule
Can we use parallel rule?
1/G2(s)
1/G2(s)
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Expand Summation
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Parallel Rule
1/G2(s)
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36
Feedback Rule
R
G1G2
H

1  G1G2  2  H1 
 G1

1
1
G2
Simplify
G3
1  G3 H 3
C
R
G1G2
1  G2 H 2  G1G2 H1
1  G2
G2
G3
1  G3 H 3
C
37
Cascade Rule
38
Motor with Feedback
E m ( s )  K e s ( s )
Ia 
R
G1 (1  G2 )G3
1  G2 H 2  G1G2 H1 1  G3 H 3 
Ea ( s )  Em ( s )
La s  Ra  R
C
B
J
1
s
Kt
1
Ia
s Js  B
 ( s )  ( s )  
E a ( s )  K a Ec ( s )
Ec ( s )  Er ( s )  RI a ( s )
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40
Block Diagram
Move Gain Block
E m ( s )  K e s ( s )
E a ( s )  K a Ec ( s )
Ia 
Ea ( s )  E m ( s )
La s  Ra  R
‫ܧ‬௥
‫ܭ‬௔
1
s
Kt
1
Ia
s Js  B
 ( s )  ( s )  
Ec ( s )  Er ( s )  RI a ( s )
‫ܭ‬௔
Combine summing junctions
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42
Feedback Loop
1
La s  Ra  R
‫ܭ‬௔
G1 ( s ) 

Feedback Loop
Ia
G1 ( s )
G2 ( s )
ܴ
1  La s  Ra  R 
1

1  K a R  La s  Ra  R  La s  Ra  R  K a R
 ( s)
G1 ( s )G2 ( s )
1
 
 Ka
Er ( s ) s 1  K eG1 ( s )G2 ( s )
1
La s  Ra  R1  K a 
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