# What will be the time response expression for a standard first order system having unit step function $\frac{1}{s}$ as the input

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What will be the time response expression for a standard first order system having unit step function $\frac{1}{s}$ as the input
1. $t - T + Te^\frac{-t}{ T}$
2. $1 - {{\rm{e}}^{ - \frac{{\rm{t}}}{{\rm{T}}}}}$
3. 1
4. $\frac{1}{T} e^{\frac{-t}{ T}}$

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Correct Answer - Option 2 : $1 - {{\rm{e}}^{ - \frac{{\rm{t}}}{{\rm{T}}}}}$

Let the standard first-order open-loop transfer function is:

${\rm{G}}\left( {\rm{s}} \right) = \frac{1}{{{\rm{sT}}}}$

then,

The closed-loop transfer function is:

${\rm{H}}\left( {\rm{s}} \right) = \frac{1}{{1 + {\rm{sT}}}}$ (assume unity negative feedback)

${\rm{Y}}\left( {\rm{s}} \right){\rm{\;}} = {\rm{\;X}}\left( {\rm{s}} \right).{\rm{\;H}}\left( {\rm{s}} \right)$

If input = unit step

${\rm{\;X}}\left( {\rm{s}} \right) = \frac{1}{{\rm{s}}}$

$\begin{array}{l} {\rm{Y}}\left( {\rm{s}} \right) = \frac{1}{{1 + {\rm{sT\;}}}}.\frac{1}{{\rm{s}}} \\= \frac{1}{{\rm{s}}} + \frac{{ - {\rm{T}}}}{{\left( {1 + {\rm{sT}}} \right)}} \\= \frac{1}{{\rm{s}}} - \frac{1}{{\frac{1}{{\rm{T}}} + {\rm{s}}}}\\ {\rm{y}}\left( {\rm{t}} \right) = 1 - {{\rm{e}}^{ - \frac{{\rm{t}}}{{\rm{T}}}}} \end{array}$

Hence, option 2 is correct.