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Question

A linear, time - invarient, casual continuous time system has a rational transfer function with simple poles at s=2 and s=4, and one simple zero at s=1.A unit step u(t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

A
[e2t+e4t]u(t)
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B
[4e2t+12e4tet]u(t)
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C
[4e2t+12e4t]u(t)
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D
[0.5e2t+1.5e4t]u(t)
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Solution

The correct option is C [4e2t+12e4t]u(t)
Transfer function can be written as,
T(s)=K(s+1)(s+2)(s+4)

Input,R(s)=1s

Output,C(s)=R(s)H(s)

Given :
lims0sC(s)=1

or lims0s.K(s+1)s(s+2)(s+4)=1

or k8=1

K=8

H(s)=8(s+1)(s+2)(s+4)

=4s+2+12s+4
so,
h(t)=(4e2t+12e4t)u(t)
Which is also required impulse response of the system

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