Correct Answer - Option 3 : 4/9
DC Gain:
The DC gain is the ratio of the magnitude of the steady-state step response to the magnitude of step input.
DC Gain of a system is the gain at the steady-state which is at t tending to infinity i.e., s tending to zero.
DC gain is nothing but the error coefficients.
For type 0 system: \({K_P} = \mathop {\lim }\limits_{s \to 0} G\left( s \right)\)
For type 1 system: \({K_v} = \mathop {\lim }\limits_{s \to 0} sG\left( s \right)\)
For type 2 system: \({K_a} = \mathop {\lim }\limits_{s \to 0} {s^2}G\left( s \right)\)
Analysis:
Given:
CLTF = (S + 4) / (S2 + 7S + 13)
\(\frac{G(S)}{1+G(s) H(s)} = \frac{(s+4)}{s^2+7s+13}\)
\(\frac{G(S)}{1+G(s) H(s)-G(s)} = \frac{(s+4)}{s^2+7s+13-s-4}\)
H(s) = 1
\(G(s) = \frac{(s+4)}{s^2+6s+9}\)
DC gain = \( \mathop {\lim }\limits_{s \to 0} G\left( s \right)\)
= 4/9
