---
title: "Effect of feedback on gain and speed of response"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:feedback-gain-speed"
source: https://www.bctengineer.com/q/semester_3/control_systems/feedback-gain-speed
site: BCT Engineering
---

# Effect of feedback on gain and speed of response

Control Systems · Chapter 1: Control System Background

- Asked in: 2078 Kartik, 4 marks
- Video: [Introduction to feedback 6 - quantifying impact of feedback for 1st order systems](https://www.youtube.com/watch?v=iUQWELYjpu8) — John Rossiter
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/feedback-gain-speed
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

What is the effect of feedback on system gain and speed of response and how?

## Short answer

Negative feedback reduces overall gain by factor (1+GH), making T ≈ 1/H for large GH. It increases speed of response by reducing the time constant from τ to τ/(1+K) for a first-order plant, so the system responds (1+K) times faster.

## Answer

Negative feedback **reduces the overall gain** and **increases the speed of response**, both by the factor $(1+GH)$.

#### Effect on gain
$$
\text{Open loop: } T = G, \qquad \text{Closed loop: } T = \frac{G}{1+GH}
$$

The gain falls by $(1+GH)$. Example: $G = 100$, $H = 0.1$ gives closed-loop gain $100/11 = 9.09$. For large $GH$, $T \approx 1/H$, so the gain depends mainly on the (accurate) feedback element. The lost gain is recovered with an amplifier.

#### Effect on speed of response
Take a first-order plant $G(s) = \dfrac{K}{\tau s + 1}$.

**Open loop:** for a unit step input

$$
c(t) = K\left(1 - e^{-t/\tau}\right), \qquad \text{time constant} = \tau
$$

**Closed loop (unity feedback):**

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

$$
c(t) = \frac{K}{1+K}\left(1 - e^{-t(1+K)/\tau}\right), \qquad \tau_{cl} = \frac{\tau}{1+K}
$$

The time constant is reduced by $(1+K)$ and the bandwidth increases by the same factor, so the response is faster. Gain and speed are traded: gain $\times$ bandwidth stays constant ($K/\tau$).

Example: $\tau = 1$ s, $K = 9$. Open loop: $\tau = 1$ s, settling time $4\tau = 4$ s. Closed loop: $\tau_{cl} = 1/10 = 0.1$ s, settling time $= 0.4$ s, i.e. ten times faster.

## Check yourself

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

1. Negative feedback changes the overall gain from G to
   A. G(1+GH)
   B. G/(1+GH)
   C. G/(1-GH)
   D. GH
   Answer: B. Gain falls by factor (1+GH). (Key point: Effect on gain)

2. G = 100 and H = 0.1 give closed-loop gain
   A. 10
   B. 9.09
   C. 100
   D. 1.1
   Answer: B. 100/(1+10) = 9.09. (Key point: Numerical gain)

3. When GH is very large, T is approximately
   A. G
   B. 1/H
   C. H
   D. 0
   Answer: B. T = G/(1+GH) tends to 1/H, set by the accurate feedback element. (Key point: Large loop gain)

4. Which quantity stays constant when feedback is applied to K/(tau s+1)?
   A. Gain alone
   B. Gain x bandwidth (K/tau)
   C. Time constant
   D. Bandwidth alone
   Answer: B. Gain falls and bandwidth rises by (1+K), keeping their product constant. (Key point: Gain-speed trade-off)

## Related in Control Systems

- [Speed of response increases with gain](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aspeed-increases-with-gain) — Control Systems, chapter 1
- [Feedback effect in DC motor speed control](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dc-motor-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
- [Effect of feedback on dynamic response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dynamic-response) — 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
- [True/false: feedback makes response faster](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-feedback-faster) — Control Systems, chapter 1

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