In the signal flow graph of figure given below, the gain C/R will be

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TN TRB EC 2012 Official Paper

Option 4 : 44/23

Network Theory

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- It is a technique used for finding the transfer function of a control system. A formula that determines the transfer function of a linear system by making use of the signal flow graph is known as Mason’s Gain Formula.
- It shows its significance in determining the relationship between input and output.

Suppose there are ‘N’ forward paths in a signal flow graph. The gain between the input and the output nodes of a signal flow graph is nothing but the transfer function of the system. It can be calculated by using Mason’s gain formula.

Mason’s gain formula is

\(T = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\mathop \sum \nolimits_{i = 1}^N {P_i}{{\rm{Δ }}_i}}}{{\rm{Δ }}}\)

Where,

C(s) is the output node

R(s) is the input node

T is the transfer function or gain between *R*(*s*) and *C*(*s*)

Pi is the ith forward path gain

Δ = 1−(sum of all individual loop gains) + (sum of gain products of all possible two non-touching loops) − (sum of gain products of all possible three non-touching loops) + ........

Δi is obtained from Δ by removing the loops which are touching the ith forward path.

__Calculations:__

The forward paths are as follows:

P_{1} = 5

P_{2} = 2 × 3 × 4 = 24

The loops are as follows:

L_{1} = -2, L_{2} = -3, L_{3} = -4, L_{4} = -5

The two non-touching loops are:

L_{1}L_{3} = 8

There is no three non-touching loops

By Mason’s gain formula:-

\(\frac{C}{R} = \frac{{24\; + \;5\left( {1\; + \;3} \right)}}{{1\; + \;2\; + \;3\; + \;4 \;+ \;5\; + \;8}}\)

\(= \frac{{44}}{{23}}\)