---
title: "Speed of response increases with gain"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:speed-increases-with-gain"
source: https://www.bctengineer.com/q/semester_3/control_systems/speed-increases-with-gain
site: BCT Engineering
---

# Speed of response increases with gain

Control Systems · Chapter 1: Control System Background

- Asked in: 2074 Chaitra, 3 marks
- Video: [Part 1 of 5 : Effect of Feedback on Time Constant of Control System](https://www.youtube.com/watch?v=i_T6mfi_4I0) — Learning Electronics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/speed-increases-with-gain
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

Show that the speed of response increases with the increase of the gain of the system.

## Short answer

For a first-order system, the closed-loop time constant τ_cl = τ/(1+K) decreases as gain K increases, reducing settling time and speeding response. For a second-order system, ω_n = √K grows with gain, so rise and peak times reduce, though overshoot may increase.

## Answer

Speed of response is measured by the time constant (first order) or natural frequency (second order); both improve as the loop gain $K$ rises.

#### First-order system
With unity feedback around $G(s) = \dfrac{K}{\tau s+1}$:

$$
\frac{C(s)}{R(s)} = \frac{K}{\tau s+1+K} = \frac{K/(1+K)}{\dfrac{\tau}{1+K}s+1}
$$

$$
\tau_{cl} = \frac{\tau}{1+K}, \qquad t_s \approx 4\tau_{cl} = \frac{4\tau}{1+K}
$$

As $K$ increases, $\tau_{cl}$ decreases, so the response reaches its final value faster. Example ($\tau = 1$ s): $K = 1$ gives $\tau_{cl} = 0.5$ s; $K = 9$ gives $0.1$ s.

#### Second-order system
For $G(s) = \dfrac{K}{s(s+a)}$: $\omega_n = \sqrt{K}$. Rise time and peak time $t_p = \pi/(\omega_n\sqrt{1-\zeta^2})$ decrease as $K$ rises (though overshoot increases).

Hence speed of response increases with gain.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With unity feedback around K/(tau s+1), the closed-loop time constant is
   A. tau/(1+K)
   B. tau(1+K)
   C. tau K
   D. tau/K^2
   Answer: A. C/R = K/(tau s + 1 + K). (Key point: Time constant formula)

2. As K increases, the settling time 4tau/(1+K)
   A. Increases
   B. Decreases
   C. Stays constant
   D. Becomes infinite
   Answer: B. The denominator grows, so settling time shrinks. (Key point: Trend with K)

3. With tau = 1 s, K = 9 gives closed-loop time constant
   A. 0.5 s
   B. 0.1 s
   C. 0.9 s
   D. 10 s
   Answer: B. 1/(1+9) = 0.1 s. (Key point: Numerical check)

4. For K/[s(s+a)], wn equals
   A. K
   B. sqrt(K)
   C. K^2
   D. a
   Answer: B. s^2 + as + K gives wn = sqrt(K), so speed rises with K but overshoot rises too. (Key point: Second-order system)

## Related in Control Systems

- [Effect of feedback on gain and speed of response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-gain-speed) — Control Systems, chapter 1
- [Control system and why feedback makes response faster](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-feedback-faster) — Control Systems, chapter 1
- [Characterizing control system by speed, accuracy, stability](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aspeed-accuracy-stability) — Control Systems, chapter 1
- [Disturbance sensitivity and effect of gain](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aol-cl-disturbance-gain) — Control Systems, chapter 1

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