Correct Answer - Option 3 : dc gain 1 and high frequency gain 0
Concept:
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:
\({K_a} = \mathop {\lim }\limits_{s \to 0} {s^2}G\left( s \right)\)
High-frequency gain:
The high-frequency gain of a system is the gain at the steady-state which is at t tending to 0 i.e., s tending to infinity.
Calculation:
Given:
\(G(s)=\frac{1}{2s+1}\)
\(DC \ Gain = \mathop {\lim }\limits_{s \to 0} G\left( s \right)\)
\(DC \ Gain = \mathop {\lim }\limits_{s \to 0} \frac{1}{2s+1}\)
DC Gain = 1
High-frequency gain is calculated as:
\( = \mathop {\lim }\limits_{s \to ∞ } G\left( s \right)\)
\( = \mathop {\lim }\limits_{s \to ∞ } \frac{1}{2s+1}\)
High frequency gain = 0
