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 $u\left(t\right)$ 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. $[e^{-2t}+e^{-4t}]u\left(t\right)$
• B. $[-4e^{-2t}+12e^{-4t}-e^{-t}]u\left(t\right)$
• C. $[-4e^{-2t}+12e^{-4t}]u\left(t\right)$
• D. $[-0.5e^{-2t}+1.5e^{-4t}]u\left(t\right)$

Answer 1

The correct option is C $[-4e^{-2t}+12e^{-4t}]u\left(t\right)$

Transfer function can be written as,
$T\left(s\right)=\frac{K(s+1)}{(s+2)(s+4)}$

$Input,R\left(s\right)=\frac{1}{s}$

$Output,C\left(s\right)=R\left(s\right)H\left(s\right)$

Given :
$\underset{s\to 0}{lim}sC\left(s\right)=1$

$or\underset{s\to 0}{lim}\frac{s.K(s+1)}{s(s+2)(s+4)}=1$

$or\frac{k}{8}=1$

$\Rightarrow K=8$

$H\left(s\right)=\frac{8(s+1)}{(s+2)(s+4)}$

$=-\frac{4}{s+2}+\frac{12}{s+4}$
so,
$h\left(t\right)=(-4e^{-2t}+12e^{-4t})u\left(t\right)$
Which is also required impulse response of the system