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

# Effect of feedback on time constant

Control Systems · Chapter 1: Control System Background

- Asked in: 2078 Bhadra, 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/feedback-time-constant
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

Discuss the effect of feedback on time constant of a control system.

## Short answer

Negative feedback reduces the time constant by a factor of (1+K), making the response faster. For a first-order plant K/(τs+1), closed-loop time constant becomes τ/(1+K). Example: τ=1s, K=9 gives τ_cl=0.1s.

## Answer

Negative feedback **reduces the time constant** of a system, making the response faster.

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):**

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

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

So the time constant is reduced by the factor $(1+K)$ and bandwidth increases by the same factor. Gain also falls by $(1+K)$, which is recovered with an amplifier.

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 on a first-order system
   A. increases the time constant
   B. leaves the time constant unchanged
   C. makes the response oscillatory always
   D. reduces the time constant, making the response faster
   Answer: D. tau_cl = tau/(1+K). (Key point: Effect)

2. Bandwidth after unity feedback is
   A. (1+K)/tau
   B. 1/tau
   C. K/tau
   D. tau/(1+K)
   Answer: A. Bandwidth increases by the same factor (1+K). (Key point: Bandwidth)

3. The price paid for the faster response is
   A. DC gain rises to K(1+K)
   B. DC gain falls from K to K/(1+K)
   C. bandwidth falls
   D. time constant rises
   Answer: B. The reduced gain is recovered by an amplifier. (Key point: Cost)

## Related in Control Systems

- [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
- [Derivative feedback controller: block diagram and effect](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-bd-effect) — Control Systems, chapter 7, asked 2×
- [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
- [Derivative feedback in second order time response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-second-order) — Control Systems, chapter 7
- [Effect of feedback on dynamic response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dynamic-response) — Control Systems, chapter 1

## Related in other subjects

- [Short note: Feedback control system](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Afeedback-control-note) — Project Engineering, chapter 4, asked 2×
- [Elements of cost control; feedback control](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acost-control-elements-feedback-system) — Project Engineering, chapter 4
- [Thermocouple time constant and frequency response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Athermocouple-first-order-response-numerical) — Instrumentation and Measurement, chapter 4
- [Excitation system and changing excitation at constant load](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aexcitation-system-constant-load) — Electric Machines II, chapter 2