A second-order control system has a transfer function \(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2{\rm{\xi }}{{\rm{\omega }}_{\rm{n}}}s + \omega _n^2}}\).

For a unit step input, match List – I with List – II and select the correct answer using the code given below the Lists:

List – I

(Time Domain Specification)

List – II

(Expression)

A.

Rise time

1.

\(\frac{{\pi - {{\tan }^{ - 1}}\left( {\frac{{\sqrt {1 - {\xi ^2}} }}{\xi }} \right)}}{{{\omega _n}\sqrt {1 - {\xi ^2}} }}\)

B.

Peak time

2.

\(\frac{\pi }{{{\omega _n}\sqrt {1 - {\xi ^2}} }}\)

C.

Peak Overshoot

3.

\({e^{\left( { - \frac{{\pi \xi }}{{\sqrt {1 - {\xi ^2}} }}} \right)}}\)

D.

Settling time

4.

\(\frac{4}{{\xi {\omega _n}}}\)

Question

	List – I (Time Domain Specification)		List – II (Expression)
A.	Rise time	1.	\(\frac{{\pi - {{\tan }^{ - 1}}\left( {\frac{{\sqrt {1 - {\xi ^2}} }}{\xi }} \right)}}{{{\omega _n}\sqrt {1 - {\xi ^2}} }}\)
B.	Peak time	2.	\(\frac{\pi }{{{\omega _n}\sqrt {1 - {\xi ^2}} }}\)
C.	Peak Overshoot	3.	\({e^{\left( { - \frac{{\pi \xi }}{{\sqrt {1 - {\xi ^2}} }}} \right)}}\)
D.	Settling time	4.	\(\frac{4}{{\xi {\omega _n}}}\)