Closed Loop Transfer Function

Ask

The open loop DC gain of unity negative feedback system with closed loop transfer function

$\frac{s+4}{s^{2} + 7s + 13}$ is

(a)${4\over13}$

(b)${4\over9}$

(c)4

(d)13

Answer is ( b )

Concept

Solution :The closed loop transfer function is given by :${G(s)\over( 1 + G(s)H(s))}$

$= {G(s)\over 1 + G(s)H(s)}$ ... Eq(1)

$= {s+4 \over s^{2}+7s+13}$

$H(s) = 1$ for unity feedback

$\therefore$, Eq (1) reduces to ${G(s)\over(1+G(s))} = {s+4\over(s^{2}+7s+13)}$ ... Eq(2)

To simplify this, take the reciprocal of Eq (2)

${(1 + G(s))\over G(s)} = {s^{2} + 7s + 13\over s+4 }$

${1\over G(s)} = {(s^2 + 7s+13)\over(s+4)} -1$

${1 \over G(s)} ={(s^2 +7s + 13 ) - (s+4)\over(s+4)}$

${1\over G(s)} = {s^2+6s+9\over(s+4)}$

$G(s) = {(s+4)\over(s^2+6s+9)}$

For DC gain , s = 0 $\therefore, G(s) = 4/9$