The root locus of a unity feedback system is shown in the figure.

The closed-loop transfer function of the system is.

\frac{C (s)}{R (s)} = \frac{K}{(s + 1) (s + 2)}

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\frac{C (s)}{R (s)} = \frac{- K}{(s + 1) (s + 2) + K}

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\frac{C (s)}{R (s)} = \frac{K}{(s + 1) (s + 2) - K}

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\frac{C (s)}{R (s)} = \frac{K}{(s + 1) (s + 2) + K}

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Solution

The correct option is C $\frac{C (s)}{R (s)} = \frac{K}{(s + 1) (s + 2) - K}$
This is converse of root locus(called complementary root locus) having no zero.

$A s, i t s G (s) H (s) = \frac{K}{(s + 1) (s + 2)}$

Its converse root locus only valid when K < 0

$S o, C . L . T . F . = \frac{G (s) H (s)}{1 - G (s) H (s)}$

$= \frac{\frac{K}{(s + 1) (s + 2)}}{1 - \frac{K}{(s + 1) (s + 2)}} = \frac{K}{(s + 1) (s + 2) - K}$

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