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GATE 2026 EE – Question 13

Control Systems · Transient and Steady-state analysis of linear time invariant systems · 1 mark · Multiple choice

The Laplace transform of the step response of a system is given by

$$Y(s)=\frac{100}{s(s+100)}$$

The rise time is defined as the time taken for the response to go from 0.1 to 0.9 of its final value. The settling time is defined as the time taken for the response to reach 0.98 of its final value.

For this system, the rise time ($T_r$), settling time ($T_s$), and time constant ($T_c$), all expressed in seconds, are

  1. $T_r=0.022,\ T_s=0.04,\ T_c=0.01$
  2. $T_r=0.22,\ T_s=0.404,\ T_c=0.01$
  3. $T_r=2.2,\ T_s=4.04,\ T_c=1.01$
  4. $T_r=22,\ T_s=40.4,\ T_c=10.1$

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Correct answer: (A) $T_r=0.022,\ T_s=0.04,\ T_c=0.01$

Explanation

$y(t)=1-e^{-100t}$, a first-order response with time constant $T_c=1/100=0.01$ s. Rise time (10% to 90%) is $2.2T_c=0.022$ s. Settling time to 98% is $\ln(50)\,T_c=3.91\times0.01\approx0.04$ s.