extra note-block diagram reduction signal flow graph.doc
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EXERCISE SIGNAL FLOW GRAPH
Block Diagram Reduction & Signal Flow GraphBlock DiagramsBasic components of a block diagram for a LTIV system
Cascade or series subsystems,
Parallel Subsystems,
Feedback Form
a. Feedback control system;b. simplified model;c. equivalent transfer functionMoving blocks to create familiar forms,
Example 1
Reduce the following block diagram to form a single transfer function.
Solution,
Example 2
Reduce the following block diagram to form a single transfer function.
Solution,
1.1 Control signal
R(s) +
E(s)
Y(s)
_
B(s)
E(s) error signal
B(s) feedback signal
R(s) reference signal
Y(s) output signal
Feed forward transfer function
Feedback transfer function
Open-loop transfer function Closed-loop transfer function
E(s)
Y(s)
B(s)
E(s)
B(s)
Open-loop
Assume, gives .
LISTNUM
LISTNUM
LISTNUM
Variable difference
LISTNUM Characteristic equation
LISTNUM Signal Flow Graphs
SFG may be viewed as a simplified form of block diagram. SFG consists of arrows (represent systems) and nodes (represent signals).
Signal-flow graph components:a. system;b. signal;c. interconnection of systems and signals
Converting common block diagrams to SFG
Converting a block diagram to SFG
Signal-flow graph development:a. signal nodes;b. signal-flow graph;c. simplified signal-flow graph
Mason Gain Formula
The transfer function of a given system represented by a SFG is:
where
k = no. of paths
= the kth forward-path gain
= 1 - loop gains + non-touching loop gains 2 at a time -
non-touching loop gains 3 at a time + non-touching
loop gains 4 at a time -
k= - ( loop gain terms in that do not touch the k-th forwad-path. In
other words, k is formed by eliminating from those loop gains that do
touch the k-th forward path.SIGNAL FLOW GRAPHDefinitions:
i)Loop Gains:
G2(s)H1(s), G4(s)H2(s), etc
ii)Forward-path gains:
G1(s)G2(s)G3(s)G4(s)G5(s)G7(s), G1(s)G2(s)G3(s)G4(s)G6(s)G7(s)
iii)Non-touching Loopsiv)Non-touching Loop Gains
[G2(s)H1(s)][G4(s)H2(s)]
[G2(s)H1(s)][G4(s)G5(s)H3(s)]
[G2(s)H1(s)][G4(s)G6(s)H3(s)]
Example
Find the transfer function, C(s)/R(s), for the signal-flow graph below:
Solution:i)Firstly, identify the forward-path gains.G1(s)G2(s)G3(s)G4(s)G5(s)ii)Secondly, identify the loop gains.
G2(s)H1(s), G4(s)H2(s), G7(s)H4(s), G2(s)G3(s)G4(s)G5(s)G6(s)G7(s)G8(s)
iii)Thirdly, identify the non-touching loops taken two at a time.
Loop 1 & Loop 2
:G2(s)H1(s)G4(s)H2(s)
Loop 1 and Loop 3:G2(s)H1(s)G7(s)H4(s)
Loop 2 and Loop 3:G4(s)H2(s)G7(s)H4(s)
iv)Finally, identify the non-touching loops taken three at a time.
Loops 1, 2 and 3
:G2(s)H1(s)G4(s)H2(s)G7(s)H4(s)
Hence,
=1 [G2(s)H1(s) + G4(s)H2(s) + G7(s)H4(s) +
G2(s)G3(s)G4(s)G5(s)G6(s)G7(s)G8(s)] + [G2(s)H1(s)G4(s)H2(s) +
G2(s)H1(s)G7(s)H4(s) +G4(s)H2(s)G7(s)H4(s)]
[G2(s)H1(s)G4(s)H2(s)G7(s)H4(s)]
Then form k by eliminating from the loop gains that do not touch the kth forward-path :
1=1 - G7(s)H4(s)
Hence:
EXERCISE
1.
Find the transfer function, G(s) = C(s)/R(s) of the figure above using:1) Block diagram reduction technique2) Masons ruleSOLUTION:1) Block diagram reduction technique:Combine the parallel blocks in the forward path. Then, push 1/s to the left past the pickoff point.
Combine the parallel feedback paths and get 2s. Then, apply the feedback formula, simplify and get
2) Signal flow graph technique:
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H(s)G(s)
G(s)
H(s)s
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S
1/S
S2+1/S
+
-
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R(s)
C(s)
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