BLOCK DIAGRAM

REDUCTION
G2

+ +
G1 + G3 + G4 G6 Y
X
-

G5
Components
 Signals
R(s) C(s)

 System Blocks
G H

 Summing points/junctions
+ +
+ +
-
-

 Pick-off points
R(s)
R(s) R(s)
Rules and Forms
 Cascade systems

R(s) C1(s) C(s)

G1(s) G2(s)

R(s) C(s)
G1(s)G2(s)
Rules and Forms
 Systems in parallel

R(s) C1(s) + C(s)

G1(s)
+
C2(s)
G2(s)

R(s) C(s)
G1(s) + G2(s)
Rules and Forms
 Feedback loop

R(s) + J2(s) C(s)

G(s)
+

H(s)
J1(s)

R(s) G(s)
C(s)
±
1 G(s)H(s)
Rules and Forms
 Moving a summing block beyond systems

R1(s) + R(s) C(s)

G(s)
+
R2(s)

R1(s) + C(s)
G(s)
+
R2(s)
G(s)
Rules and Forms
 Moving a summing block ahead systems

R1(s) R3(s) + C(s)

G(s)
+
R2(s)

R1(s) + C(s)
G(s)
+
R2(s) 1
G(s)
Rules and Forms
 Moving a pick-off point beyond a system

R(s) C1(s)
G(s)

C2(s)

R(s) C1(s)
G(s)

1 C2(s)
G(s)
Rules and Forms
 Moving a pick-off point ahead a system

R(s) C1(s)
G(s)

C2(s)

R(s) C1(s)
G(s)

C2(s)
G(s)
Examples: Block Diagram Reduction via Familiar
Forms
Reduce the block diagram below to a single transfer function

R(s) + + + C(s)
G1(s) G2(s) G3(s)
- + -

H1(s)

H2(s)

H3(s)
 Examples
 Block Diagram Reduction by Moving Blocks

G2

+ +
G1 + G3 + G4 G6 Y
X
-

G5
Going Further
 Multiple Input Systems

R2(s)

R1(s) + + + C(s)
G1(s) G2(s) G3(s)
-
Going Further
 Multiple Output Systems

C1(s)

R1(s) + C(s)
G1(s) G2(s) G3(s)
-
SIGNAL FLOW GRAPHS
u7

u5

X u1 u2 u3 u4 Y

u6
Components
 Nodes u7

u5
X
u1 u2 u3 u4 Y

u6

 Branches

G2 +1
+1 G3 G4 G6

G1 +1 +1

-1 G5
Rules and Forms
 Cascade systems

G1(s) G2(s)
R(s) C(s)
u

G1(s) G2(s)
R(s) C(s)
Rules and Forms
 Systems in parallel
G1(s)

R(s) C(s)
G2(s)

G1(s) + G2(s)
R(s) C(s)
Rules and Forms
 Feedback

+1 G(s)
R(s) C(s)

+1 H(s)

G(s)
±
1 G(s)H(s)
R(s) C(s)
 Rules and Forms
 Shifting transmittance: Starting and Termination Points

G1(s) G2(s)
R(s) C(s)

G3(s)

G1(s) G2(s) G1(s) G2(s)

R(s) C(s) R(s) C(s)

G3(s) G3(s)
G2(s) G1(s)
 Rules and Forms
 Shifting transmittance: Starting or Termination Points

G1(s) G2(s)
R(s) C(s)

G3(s)

G1(s) G2(s) G1(s) G2(s)

R(s) C(s) R(s) C(s)

G3(s) G2(s) G3(s)

G1(s)
 Rules and Forms
 Y Transformation
G2(s) C1(s)
G1(s)
R(s)

G3(s) C2(s)

G2(s) C1(s) G1(s) G2(s)

C1(s)
G1(s)
R(s) R(s)

G1(s)G3(s) C2(s) G1(s) G3(s) C2(s)

Rules and Forms
 Star-to-Mesh Transformation

x2 x2

ab bc
b
a c
x1 x3 x1 x3

d
ad dc

x4 x4
Rules and Forms
 General Reduction Rule of Multiple-loop Single-path Signal Flow Diagrams

P 1n
𝐺𝑒 =
Δ

P1n= 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑝𝑎𝑡h 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑠𝑜𝑢𝑟𝑐𝑒𝑛𝑜𝑑𝑒 𝑥1 𝑎𝑛𝑑 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡𝑛𝑜𝑑𝑒 𝑥 𝑗

Example
h

x2 x3 x4 x5
x1 x6
a b c d e

f g
Rules and Forms
 General Reduction Rule of Multiple-loop Multi-path Signal Flow Diagrams:
Mason’s Rule

Σ (𝑃 ¿ ¿𝑘 Δ 𝑘)
𝐺𝑒 = ¿
Δ

𝑃 𝑘=𝑎 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑝𝑎𝑡h 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑠𝑜𝑢𝑟𝑐𝑒𝑛𝑜𝑑𝑒 𝑥 1 𝑎𝑛𝑑 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑛𝑜𝑑𝑒 𝑥 𝑗

Δ 𝑘=𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡 𝑜𝑓 𝑡h𝑒 𝑜𝑡h𝑒𝑟 𝑝𝑎𝑟𝑡𝑠 𝑜𝑓 𝑡h𝑒 𝑔𝑟𝑎𝑝h 𝑡h𝑎𝑡 𝑑𝑜𝑒𝑠𝑛𝑜𝑡 𝑡𝑜𝑢𝑐h 𝑃 𝑘
Example

e f

x2 x3
x1 x4
a b c

d
Case Study
 Applied block diagram and Signal flow graph