Block Diagram Reduction Techniques

Block Diagram Reduction
Techniques
Anjali A Jagtap
Department of Electronics and Telecommunication
International Institute of Information Technology, I²IT
www.isquareit.edu.in

International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057
Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - info@isquareit.edu.in
Introduction:
•A control system is generally represented by multiple blocks connected
from input to output.
•It’s a pictorial relationship between input and output of the system.
•Each block represents individual gain of sub system of complex control
system.
•Mapping systems input to its output or response of the system can be
obtained by a mathematical model known as transfer function .
•For a Linear Time Invariant (LTI) system, transfer function is defined as a
ratio of Laplace transform of output function to Laplace transform of input.
•To reduce system complexity, set of rules called as Block Diagram
Reduction Rules/Techniques are used.

Block diagram basics:
• Input : Single input or multiple input applied to system for processing.
• Output: A signal received with some process applied on the input.
• Summing block/point : Multiple inputs are added or subtracted based on
sign assigned.
•Takeoff point: A point from which signal is fed to input side or output side.
• Block : Indicate function of a sub system/particular system of complex
system.
Forward gain block: block in forward path with defined gain.
Feedback gain block: block in feedback path with defined gain.
• Forward path: Path (direction) from input to output.
• Feedback path: Backward path from output to input.

Input
R(S)
Summing Block/Point
Output
C(S)
G1(S)
G2(S)
Blocks
(Forward gain block)
H(S)
Block
(Feedback gain block)
Forward Path
Take off point
Forward Path
Feedback Path
Note: Input, output, gains of the system are represented in Laplace transform.

Rules:
1. Blocks in series :
If two or more blocks are in series i.e. in between two block if there is no
other component such as summing point and takeoff point, then their
equivalent is block with gain as product of gains of blocks in series.
G1(S) G2(S)R(S) C(S)
G1(S)*G2(S)
R(S) C(S)

2. Blocks in parallel :
If two or more blocks are in parallel i.e. arrow direction from the block is in
the same direction and towards summing point, then their equivalent is block
with gain as addition and/or subtraction of gains of blocks in parallel.
+
-
R(S) C(S)
X(S)
G1(S)
G2(S)
Note: sings are mentioned on the arrow near summing point for each block.
R(S)*G1(S) – X(S)*G2(S)
C(S)
X(S)*G2(S)
R(S)*G1(S)
=X(S)*G2(S) – X(S)*G2(S)

3. Associative law:
If two summing point are in series i.e. in between two summing point if there
is no other component such as block and/or takeoff point, then the signal
associated with summing points can be interchanged with signs.
R(S)
X(S) Y(S)
C(S)
R(S)
Y(S) X(S)
C(S)

4. Shifting summing point before block:
Summing point can be shifted before block. Consider the case below and
check its equivalent.
+
-
+
-
C(S) = [R(S)*G1(S)]-Y(S)R(S)
G1(S)
Y(S)
R(S)*G1(S)
C(S)
= {R(S)+Y(S)*[1/G1(S)]}*G1(S)
= R(S)*G1(S) – Y(S)*(1/G1(S))*G1(S)
= R(S)*G1(S) – Y(S)
1/G1(S)
R(S)
G1(S)
Y(S) Y(S)*[1/G1(S)]
R(S)+{Y(S)*[1/G1(S)]}

5. Shifting summing point after block:
Summing point can be shifted after block. Consider the case below and check
its equivalent.
+
-
+
-
Y(S)*G1(S)
R(S) - Y(S)
=[R(S) – Y(S)]*G1(S)
Y(S)
R(S)
G1(S)
C(S)
R(S)
G1(S)
Y(S) G1(S)
C(S)
R(S)*G1(S)
=R(S)*G1(S) – Y(S)*G1(S)
=[R(S) – Y(S)]*G1(S)

6. Shifting takeoff point before block:
Takeoff point can be shifted before block. Consider the case below and check
its equivalent.
R(S)*G1(S)
C(S)R(S)
G1(S)
Y(S)
= R(S)*G1(S)
= R(S)*G1(S)
R(S)
G1(S)
Y(S)
G1(S)
C(S) = R(S)*G1(S)
= R(S)*G1(S)

7. Shifting takeoff point after block:
Takeoff point can be shifted after block. Consider the case below and check its
equivalent.
R(S)
G1(S)
Y(S)
C(S)= R(S)*G1(S)
= R(S)
R(S)
G1(S)
Y(S)
C(S)
1/G1(S)
= R(S)*G1(S)
= R(S)*G1(S)*[1/G1(S)]
=R(S)
= R(S)*G1(S)

8. Shifting summing point before takeoff point:
Consider the case below and check its equivalent.
+
-
+
-
+
= [R(S) - X(S)]
= R(S)
= R(S) - X(S)
R(S) – X(S)
R(S)
Y(S)
C(S)
X(S)
R(S) - X(S)
=R(S) – X(S) +X(S)
=R(S)
R(S)
Y(S)
C(S)
X(S)

9. Shifting summing point after takeoff point:
Consider the case below and check its equivalent.
+
-
+
+ -
-
R(S)
Y(S)
C(S)
X(S)
R(S) - X(S)
= R(S) – X(S)
=R(S) – X(S)
R(S)
Y(S)
C(S)
X(S)=R(S) – X(S)
= R(S) – X(S)

10. Minor Feedback loop:
Consider the case below, it’s a minor feedback loop with G(S) as forward or
open loop gain of system, H(S) as feedback gain with R(S) as input and C(S)
as output of system. Let B(S) is total feedback signal at input of summing
point and E(S) is error signal at the output of summing point.
+
-
C(S) = E(S)*G(S)
= C(S)*H(S)
C(S)
R(S) G(S)
H(S)B(S)
E(S) = R(S) - B(S)
C(S) = [R(S) – B(S)]*G(S)
= R(S)*G(S) – B(S)*G(S)
C(S) = R(S)*G(S) – C(S)*H(S)*G(S)
C(S) + C(S)*H(S)*G(S) = R(S)*G(S)
C(S) [1 + H(S)*G(S)] = R(S)*G(S)
C(S) = R(S) * G(S)
[1 + H(S)*G(S)]
C(S) = G(S)
R(S) [1 + H(S)*G(S)]
Note: If feedback input sign is positive then the transfer function C(S)/R(S) = G(S)/[1 – G(S)*H(S)]
R(S) C(S)
G(S)
[1 + H(S)*G(S)]
Transfer function = C(S) = G(S)
R(S) [1 + H(S)*G(S)]

Thank you
For any queries
Contact : anjalij@isquareit.edu.in
Website: www.isquareit.edu.in

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