Question

The frequency response of a linear, time -invarient system is given by $H\left(f\right)=\frac{5}{1+j10\pi f}.$ The step response of the system is

• A. $5(1-e^{5t})u\left(t\right)$
• B. $5(1-e^{-t/5})u\left(t\right)$
• C. $\frac{1}{5}(1-e^{-5t})u\left(t\right)$
• D. $\frac{1}{(s+5)(s+1)}$

Answer 1

The correct option is B $5(1-e^{-t/5})u\left(t\right)$

$H\left(f\right)=\frac{5}{1+j10\pi f}$
$H\left(s\right)=\frac{5}{1+5s}$
$=\frac{5}{5(s+\frac{1}{5})}=\frac{1}{s+\frac{1}{5}}$
$Stepresponse=\frac{1}{s}\frac{1}{(s+\frac{1}{5})}$
$\frac{1}{s}\times \frac{1}{(s+\frac{1}{5})}=\frac{A}{s}+\frac{B}{(s+\frac{1}{5})}$ $\Rightarrow A(s+\frac{1}{5})+Bs=1$
$whens=0,A=5$
$Whens=\frac{-1}{5},B=-5$
$Y\left(s\right)=\frac{5}{s}-\frac{5}{s+\frac{1}{5}}$
$\Rightarrow y\left(t\right)=5[1-e^{-t/5}]u\left(t\right)$