block diagrams and signal flows
TRANSCRIPT
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2 Block Diagrams
(1)
Solution: By eliminating the feed-back paths, we get
Combining the blocks in series, we get
Eliminating the feed back path, we get
+
- - -
+
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+C(S)
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3Control Systems
(2)
Solution: Shifting the take-off beyond the block , we get
Combining and eliminating (feed back loop), we get
Eliminating the feed back path , we get
Combining all the three blocks, we get
R(S)- -
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C(S)
R(S)- -
-
C(S)
R(S)- -
C(S)
R(S) C(S)
R(S)-
C(S)
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4 Block Diagrams
(3)
Solution: Re-arranging the block diagram, we get
Eliminating loop & combining, we get
Eliminating feed back loop
Eliminating feed back loop , we get
C(S) R(S)- - -
C(S) R(S)- -
R(S) C(S)
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C(S) R(S)
C(S) R(S)-- -
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Signal Flow GraphsBy: Sheshadr i.G.S.
CIT, Gubbi.
For complicated systems, Block diagram reduction method becomes tedious & time consuming. Analternate method is that signal flow graphs developed by S.J. Mason. In these graphs, each node representsa system variable & each branch connected between two nodes acts as Signal Multiplier. The direction of
signal flow is indicated by an arrow.
1. A node is a point representing a variable.2. A transmittance is a gain between two nodes.
A branch is a line joining two nodes. The signal travels along a branch.
It is a node which has only out going signals.
It is a node which is having only incoming signals.
It is a node which has both incoming & outgoing branches (signals).
It is the traversal of connected branches in the direction of branch arrows. Such that no nodeis traversed more than once.
It is a closed path.
It is the product of the branch transmittances of a loop.
Loops are Non-Touching, if they do not possess any common node.
It is a path from i/p node to the o/p node which doesnt cross any node more than
once.
It is the product of branch transmittances of a forward path.
The relation between the i/p variable & the o/p variable of a signal flow graphs is given by the net gain between the i/p & the o/p nodes and is known as Overall gain of the system.
Mas ons gain formula for the determination of overall system gain is given by,
Where, Path gain of forward path.
Determinant of the graph.
The value of the for that part of the graph not touching the forward path.
T Overall gain of the system.
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2 Signal Flow Graphs
Solution:
Masons gain formula is,
No. of forward paths:
No. of individual loops:
No. of three non-touching loops = 0.
Solution:
Masons gain formula is,
Forward Paths:
Gain Products of all possible combinations of two non-touching loops:
Contd......
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No. of individual loops: Two Non-touching loops:
Where is i/p variable & is o/p variable.
Solution:
No. of forward paths:
Individual loops: Two non-touching loops:
Three non-touching loops = 0
Masons gain formula is,
Contd......
R(S)
C(S)
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4 Signal Flow Graphs
Solution: Re-arranging the summing points,
Signal flow graphs:
No. of forward paths:
No. of individual loops:
Solution: Shifting the take-off point ahead of the block . The BD reduces to,
asons gain formula is,
R(S) C(S
R(S)
C(S)
R(S) C(S)
Contd......
R(S) C(S)
C(S
R(S)
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Signal flow graph:
No. of forward paths:
No. of individual loops:
Solution:Shifting beyond , we get
C(S) R(S)
C(S) R(S)
R(S)
C(S
R(S)
C(S)
R(S) C(S)
R(S) C(S)
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6 Signal Flow Graphs
Eliminating feed back loop , we get
Eliminating feed back loop , we get
Eliminating the another feed back loop , we get
Signal flow graph:
R(S)
C(S)
R(S)
C(S)
R(S)
C(S)
R(S)
C(S)
C(S) R(S)
R(S) C(S)
Contd......
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No. of forward paths:
No. of individual loops:
Solution: No. of forward paths:
No. of individual loops: Two non-touching loops:
Solution:Shifting the take off point of beyond block & Simplifying for the blocks , we get
R(S) C(S)
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Eliminating loop, we get
Solution: No. of forward paths:
Individual loops:
Two non-touching loops = 0
Contd......
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Solution:
No. of forward paths:
No. of individual loops:
Three non-touching loops:
Four non-touching loops = 0
Two non-touching loops:
Contd......
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10 Signal Flow Graphs
Solution: No. of forward paths:
No. of individual loops:
Three non-touching loops = 0
Solution: No. of forward paths:
Two non-touching loops:
Two non-touching loops:
o. of individual loops:
Contd......
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Solution: No. of forward paths:
No. of individual loops:
Solution:
No. of forward paths:
Two non-touching loops:
Three non-touching loops = 0
o. of individual loops:
Two non-touching loops:
Contd......
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12 Signal Flow Graphs
Three non-touching loops = 0
Solution:Same block diagram can be re-arranged as shown below.
Shifting the take-off points beyond we get
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Substituting x value in the block diagram. The block diagram becomes,
Signal flow graph:
No. of forward paths:
No. of individual loops: Two non-touching loops = 0
Contd......
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14 Signal Flow Graphs
Solution:Same Block Diagram can be written as,
Substituting the value of x
Signal flow Graph:
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No. of forward paths:
No. of individual loops: Two non-touching loops = 0
Solution:(i) Let then we can find
No. of forward paths:
No. of individual loops: Two non-touching loops:
Three non-touching loops = 0
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(ii) Let Determine
No. of forward paths:
No. of individual loops:
remains same.
(iii) Let Determine
No. of forward paths:
(iv) Let Determine(i.e.,Response at 2 when source 2 is acting). (figure is in next page)
No. of forward paths:
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Hence,