signal flow graph
TRANSCRIPT
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Gandhinagar Institute of Technology
Active learning assignments onSignal Flow Graph
Batch -B3Elements of Electrical design
Electrical dept.(2016)
• Prepared by:-• Sodha Manthan (140120109057)• Gundigara Naitik (150123109003)• Ranpura Rutul (150123109013)
• Guided by:- • Prof. Jaimini Gohel
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Outline
• Introduction to Signal Flow Graphs– Definitions– Terminologies
• Mason’s Gain Formula– Examples
• Signal Flow Graph from Block Diagrams
• Design Examples
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Signal Flow Graph (SFG)• Alternative method to block diagram
representation, developed by Samuel Jefferson Mason.
• Advantage: the availability of a flow graph gain formula, also called Mason’s gain formula.
• A signal-flow graph consists of a network in which nodes are connected by directed branches.
• It depicts the flow of signals from one point of a system to another and gives the relationships among the signals.
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Important terminology :• Branches :- line joining two nodes is called branch.
x ya
Branch • Dummy Nodes:- A branch having one can be added at i/p as well as o/p.
Dummy Nodes
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Out put node
Input node
b
x4x3x2x1x0 h
fg
e
dc
a
Input & output node• Input node:- It is node that has only outgoing
branches.
• Output node:- It is a node that has incoming
branches.
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Forward path:-• Any path from i/p node to o/p node.
Forward path
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Loop :-
• A closed path from a node to the same node is called loop.
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Self loop:- •A feedback loop that contains of only one node is called self loop.
Self loop
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Loop gain:-
The product of all the gains forming a loop
Loop gain = A32 A23
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Path & path gain
Path:- A path is a traversal of connected branches in the direction of branch arrow.
Path gain:- The product of all branch gains while going through the forward path it is called as path gain.
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Feedback path or loop :-
• it is a path to o/p node to i/p node.
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Touching loops:-
• when the loops are having the common node that the loops are called touching loops.
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Non touching loops:-
• when the loops are not having any common
node between them that are called as non- touching loops.
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Non-touching loops for forward paths
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Chain Node :-
• it is a node that has incoming as well as outgoing branches.
Chain node
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SFG terms representation
input node (source)
b1x a
2x c
4x
d1
3x3x
Chain node Chain
node
forward path
path
loop
branch
node
transmittance input node
(source)
Output node
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Mason’s Rule (Mason, 1953)• The block diagram reduction technique requires
successive application of fundamental relationships in order to arrive at the system transfer function.
• On the other hand, Mason’s rule for reducing a signal-flow graph to a single transfer function requires the application of one formula.
• The formula was derived by S. J. Mason when he related the signal-flow graph to the simultaneous equations that can be written from the graph.
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Mason’s Rule :-
• The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is;
• Where
• n = number of forward paths.• Pi = the i th forward-path gain.• ∆ = Determinant of the system• ∆i = Determinant of the ith forward path
n
iiiP
sRsC 1
)()(
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n
iiiP
sRsC 1
)()(
∆ is called the signal flow graph determinant or characteristic function. Since ∆=0 is the system characteristic equation.
∆ = 1- (sum of all individual loop gains) + (sum of the products of the gains of all possible two loops that do not touch each other) – (sum of the products of the gains of all possible three loops that do not touch each other) + … and so forth with sums of higher number of non-touching loop gains
∆i = value of Δ for the part of the block diagram that does not touch the i-th forward path (Δi = 1 if there are no non-touching loops to the i-th path.)
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Example1: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph
2211 PPRCTherefore,
24313242121411 HGGGLHGGGLHGGL ,,
There are three feedback loops
Continue……
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∆ = 1- (sum of all individual loop gains)
There are no non-touching loops, therefore
243124211411 HGGGHGGGHGG
3211 LLL
Continue……
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∆2 = 1- (sum of all individual loop gains)+...Eliminate forward path-2
∆2 = 1
∆1 = 1- (sum of all individual loop gains)+...Eliminate forward path-1
∆1 = 1
Continue……
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From Block Diagram to Signal-Flow Graph Models
• Example2
--
-C(s)R(s)
G1 G2
H2
H1
G4G3
H3
E(s) X1 X
2
X3
R(s)
- H2
1G4G3G2G11 C(s)- H1
- H3
X1 X2 X3E(s)
Continue……
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R(s)
- H2
1G4G3G2G11 C(s)- H1
- H3
X1 X2 X3E(s)
14323234321
43211)(
)(HGGHGGHGGGG
GGGGsRsCG
1;)(1
143211
14323234321
GGGGPHGGHGGHGGGG
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