---
title: "Short note: Phase-lead/phase-lag compensator"
subject: "Digital Control System"
chapter: 4
key: "semester_6/digital_control_system:short-note-lead-lag-compensator"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/short-note-lead-lag-compensator
site: BCT Engineering
---

# Short note: Phase-lead/phase-lag compensator

Digital Control System · Chapter 4: Design and Compensation of Discrete Time Control System

- Asked in: 2068 Magh, 4 marks
- Video: [What are Lead Lag Compensators? An Introduction.](https://www.youtube.com/watch?v=xLhvil5sDcU) — Brian Douglas
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/short-note-lead-lag-compensator
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-4

## Question

Write a short note on phase-lead/phase-lag compensator.

## Short answer

A phase-lead compensator has its zero closer to z=1 than its pole, adding positive phase to improve transient response and increase damping. A phase-lag compensator has its pole closer to z=1, increasing low-frequency gain to reduce steady-state error without significantly altering transient behaviour.

## Answer

A compensator is an extra block placed in the loop to change the root locus or the frequency response so that the specifications are met. Its digital form is
$$
G_D(z) = K\,\frac{z - z_c}{z - p_c}
$$

#### Phase-lead compensator
- The zero is nearer to $z = 1$ than the pole ($p_c < z_c$), so it gives **positive phase**, at most $\phi_m$ where $\sin\phi_m = \frac{1-\alpha}{1+\alpha}$ (w-plane form $\frac{1+\tau w}{1+\alpha\tau w}$, $\alpha<1$).
- It moves the root locus to the left (towards the origin of the z-plane), so it increases phase margin, damping and bandwidth.
- Result: faster response and less overshoot, but more high-frequency noise.

#### Phase-lag compensator
- The pole is nearer to $z = 1$ than the zero ($p_c > z_c$), both close to 1. It gives **negative phase** but high low-frequency gain.
- It raises the static error constants ($K_p$, $K_v$) by the factor $\frac{1-z_c}{1-p_c} > 1$ (its DC gain at $z = 1$ for $K = 1$) without much change to the transient response.
- Result: lower steady-state error and less noise, but a slower response.

| Feature | Lead | Lag |
|---|---|---|
| Phase added | Positive | Negative |
| Main effect | Transient response | Steady-state error |
| Bandwidth | Increases | Decreases |
| Noise | Amplified | Attenuated |
| Pole–zero order | Zero closer to z=1 | Pole closer to z=1 |

A **lag–lead** compensator combines the two when both transient and steady-state performance must be improved.

## Check yourself

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

1. A phase-lead compensator G_D(z) = K(z - zc)/(z - pc):
   A. Has pole nearer to z = 1 and gives negative phase
   B. Has zero nearer to z = 1 than the pole and gives positive phase
   C. Has equal zero and pole
   D. Has no effect on phase
   Answer: B. Zero closer to z = 1 than the pole gives positive phase and shifts the locus to the left. (Key point: Lead compensator phase)

2. What is the main benefit of a phase-lag compensator?
   A. Faster response
   B. More bandwidth
   C. Less overshoot through added damping
   D. Higher static error constants (about (1 - pc)/(1 - zc) times) with little change in transient
   Answer: D. Lag raises low-frequency gain, reducing steady-state error without much altering transient response. (Key point: Lag compensator effect)

3. Which pairing of effects is correct?
   A. Lead amplifies noise and increases bandwidth; lag attenuates noise and decreases bandwidth
   B. Lead decreases bandwidth; lag increases it
   C. Lead attenuates noise; lag amplifies it
   D. Both increase bandwidth
   Answer: A. Lead gives a faster response but passes more high-frequency noise; lag does the reverse. (Key point: Trade-offs)

4. When is a lag-lead compensator used?
   A. When only noise matters
   B. When both transient and steady-state performance need improvement
   C. When no compensation is needed
   D. When the plant is unstable only
   Answer: B. Lag-lead combines the benefits of both compensators. (Key point: Lag-lead)

## Related in Digital Control System

- [When to prefer lag/lead or PID controllers](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Alag-lead-vs-pid) — Digital Control System, chapter 4
- [Lead compensator for satellite position control](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Alead-compensator-satellite) — Digital Control System, chapter 4

## Related in other subjects

- [(1+aTs)/(1+Ts) as lead or lag compensator](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Alead-or-lag-1-ats) — Control Systems, chapter 7
- [Compare lag and lead compensator applications](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Alag-lead-compare) — Control Systems, chapter 7
- [Short note: Lead-lag compensator](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ashort-lead-lag) — Control Systems, chapter 7
- [Lead vs lag; lead design for K/[s²(1+0.25s)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Alead-k-s2-025s-ka10) — Control Systems, chapter 7, asked 3×