The C/R for the signal flow graph in figure is:

\frac{G_{1} G_{2} G_{3} G_{4}}{(1 + G_{1} G_{2}) (1 + G_{3} G_{4})}

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\frac{G_{1} G_{2} G_{3} G_{4}}{(1 + G_{1} + G_{2} + G_{1} G_{2}) (1 + G_{3} + G_{4} + G_{3} G_{4})}

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\frac{G_{1} G_{2} G_{3} G_{4}}{(1 + G_{1} + G_{2}) (1 + G_{3} + G_{4})}

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\frac{G_{1} G_{2} G_{3} G_{4}}{(1 + G_{1} + G_{2} + G_{3} + G_{4})}

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Solution

The correct option is C $\frac{G_{1} G_{2} G_{3} G_{4}}{(1 + G_{1} + G_{2}) (1 + G_{3} + G_{4})}$
By using Mason gain formulae
$\frac{C}{R} = \frac{P_{k} Δ_{k}}{Δ}$

$\frac{C}{R} = \frac{G_{1} G_{2} G_{3} G_{4}}{1 - [- G_{1} - G_{2} - G_{3} - G_{4}]}$
$+ [G_{1} G_{3} + G_{1} G_{4} + G_{2} G_{3} + G_{2} G_{4}]$

$\frac{C}{R} = \frac{G_{1} G_{2} G_{3} G_{4}}{1 + G_{1} + G_{2} + G_{3} + G_{4} + G_{1} G_{3} + G_{1} G_{4} + G_{2} G_{3} + G_{2} G_{4}}$

$\frac{C}{R} = \frac{G_{1} G_{2} G_{3} G_{4}}{(1 + G_{1} + G_{2}) (1 + G_{3} + G_{4})}$

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