For block diagram shown in figure C(s)/R(s) is given by

\frac{G_{1} G_{2} G_{3}}{1 + H_{2} G_{2} G_{3} + H_{1} G_{1} G_{2}}

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\frac{G_{1} G_{2} G_{3}}{1 + G_{2} G_{3} G_{3} H_{1} H_{2}}

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\frac{G_{1} G_{2} G_{3}}{1 + G_{1} G_{2} G_{3} H_{1} + G_{1} G_{2} G_{3} H_{1}}

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\frac{G_{1} G_{2} G_{3}}{1 + G_{1} G_{1} G_{3} H_{1}}

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Solution

The correct option is A $\frac{G_{1} G_{2} G_{3}}{1 + H_{2} G_{2} G_{3} + H_{1} G_{1} G_{2}}$
Number of forward paths
$= 1 = (G_{1} G_{2} G_{3})$

Number of loops
$= 2 = (- G_{1} G_{2} H_{1}, - G_{2} G_{3} H_{2})$

Number of non-touching loops = 0

$∴ T . F . = \frac{C (s)}{R (s)} = \frac{G_{1} G_{2} G_{3}}{1 + G_{1} G_{2} H_{1} + G_{2} G_{3} H_{2}}$

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