The signal flow graph of a system is given below:
The transfer function (C/R) of the system is

(G_{1} G_{2} + G_{1} G_{3}) / (1 + G_{1} G_{2} H_{2})

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(G_{1} G_{2} + G_{1} G_{3}) / (1 - G_{1} G_{2} + G_{1} H_{2} H_{2})

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(G_{1} G_{2} + G_{1} G_{3}) / (1 - G_{1} H_{1} + G_{1} G_{2} H_{2} + G_{1} G_{3} G_{2})

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(G_{1} G_{2} + G_{1} G_{3}) / (1 - G_{1} H_{1} + G_{1} G_{2} H_{2} + G_{1} G_{2} G_{3} H_{1})

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Solution

The correct option is C $(G_{1} G_{2} + G_{1} G_{3}) / (1 - G_{1} H_{1} + G_{1} G_{2} H_{2} + G_{1} G_{3} G_{2})$
Step 1 : There are tow forward paths between input and output
$P_{1} = G_{1} G_{2}$
$P_{2} = G_{1} G_{3}$

Step 2 : G_{3} There are three loops, the loops gain are
$L_{1} = G_{1} H_{1}$
$L_{2} = G_{1}, G_{3} H_{2}$
$L_{3} = G_{1} G_{2} H_{1}$

Step 3 : There are no pairs of nontouching loops.
$Δ_{1} = 1 a n d Δ_{2} = 1$

Step 4 :
$Δ = 1 - [L_{1} + L_{2} + L_{3}]$
$= 1 - G_{1} H_{1} + G_{1} G_{3} H_{2} + G_{1} G_{2} H_{2}$

According to Mason's gain formula.
$T (s) = \frac{Σ P_{k} Δ_{k}}{Δ}$

$= \frac{G_{1} G_{2} .1 + G_{1} G_{3} .1}{1 - G_{1} H_{1} + G_{1} G_{3} H_{2} + G_{1} G_{2} H_{2}}$

$∴ T (s) = \frac{G_{1} G_{2} + G_{1} G_{3}}{1 - G_{1} H_{1} + G_{1} G_{3} H_{2} + G_{1} G_{2} H_{2}}$

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