---
title: "Effect of adding a zero"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:effect-add-zero"
source: https://www.bctengineer.com/q/semester_3/control_systems/effect-add-zero
site: BCT Engineering
---

# Effect of adding a zero

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2075 Asoj, 4 marks
- Video: [How Transfer Function Zeros Affect Transient Response – Quick Concepts in Control 2](https://www.youtube.com/watch?v=U6UnMxbUKPw) — Ben Dickinson
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/effect-add-zero
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

Discuss effect of addition of a zero to a system.

## Short answer

Adding a zero to the open loop shifts root locus left, improves stability and damping. In the closed loop, it increases overshoot but gives faster rise and peak times due to the derivative term; a zero far from dominant poles has little effect.

## Answer

Adding a zero to a system, either in the open-loop $G(s)H(s)$ or in the closed-loop transfer function, generally makes the response **faster** and changes the stability.

#### Zero added to the open-loop transfer function
- The root locus is pulled to the **left**, away from the imaginary axis.
- Relative stability improves. A system that was unstable for high $K$ can become stable for all $K$. Example: $K/[s^2(s+a)]$ is always unstable, but $K(s+b)/[s^2(s+a)]$ with $b < a$ is stable for all $K > 0$.
- Damping increases, so overshoot reduces. This is the basis of PD control and lead compensation.

#### Zero added to the closed-loop transfer function
If $T(s)$ becomes $T(s)(1 + s/z)$, the new output is
$$
c_{new}(t) = c(t) + \frac{1}{z}\frac{dc(t)}{dt}
$$
- The derivative term adds to the response while it is rising, so **rise time and peak time decrease**.
- **Peak overshoot increases**. The closer the zero is to the origin (small $z$), the larger the overshoot.
- **Bandwidth increases**, so the system lets more high-frequency noise through.
- A zero far to the left (about 5 times further than the dominant poles) has little effect.

```text
 c(t)       with zero (faster, more overshoot)
  |       _/\_
  |     /  _--\___________
  |    / /                original
  |   //
  +--------------------------- t
```

| Parameter | Effect of adding zero |
|---|---|
| Rise time | decreases |
| Overshoot (CL zero) | increases |
| Root locus | shifts left |
| Stability | improves |
| Bandwidth | increases |

## Check yourself

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

1. What happens to the root locus when a zero is added to the open loop transfer function?
   A. It is pulled to the left, improving relative stability
   B. It is pushed to the right
   C. It is unchanged
   D. It becomes unbounded in the right half-plane
   Answer: A. Open loop zero attracts branches to the left, increasing damping, as in PD or lead control. (Key point: Open loop zero and root locus)

2. What is the effect of a zero added to the closed loop transfer function?
   A. Overshoot decreases and rise time increases
   B. Peak overshoot increases and rise time decreases
   C. Only steady state error changes
   D. Overshoot and rise time both increase
   Answer: B. The new response is c(t) + (1/z)dc/dt, which adds a derivative term making the rise faster and the peak higher. (Key point: Closed loop zero and overshoot)

3. When does a closed loop zero have little effect?
   A. When it is very close to the origin
   B. When it is on the imaginary axis
   C. When it is far to the left, about 5 times further than the dominant poles
   D. When it is to the right of the poles
   Answer: C. A zero close to the origin has a larger overshoot effect; a distant one is negligible. (Key point: Position of the zero)

4. How does adding a zero affect bandwidth?
   A. Bandwidth increases, passing more high-frequency noise
   B. Bandwidth decreases
   C. Bandwidth is unchanged
   D. Bandwidth becomes zero
   Answer: A. A faster response means a wider bandwidth and more noise passes through. (Key point: Bandwidth)

## Close variants set in other exams

- [Effect of adding poles and zeros](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aeffect-add-pole-zero) — Control Systems, chapter 3, asked 2×

## Related in Control Systems

- [Root locus: poles −2±j3, zero at origin](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-2j3-zero-0-critical) — Control Systems, chapter 5
- [Root locus with poles 0, −1 and zero −3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-poles-0-1-zero-3) — Control Systems, chapter 5
- [Nyquist criterion from given pole-zero and Nyquist plot](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-criteria-from-plot) — Control Systems, chapter 6

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