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(1)

Solution: By eliminating the feed-back paths, we get

Combining the blocks in series, we get

Eliminating the feed back path, we get

+

- - -

+

-

+C(S)

-

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(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)- -

-

C(S)

R(S)- -

-

C(S)

R(S)- -

C(S)

R(S) C(S)

R(S)-

C(S)

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(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)

-

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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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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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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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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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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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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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,