📘 Lead, Lag & Lead-Lag Compensators – Complete Design Theory
Compensators are used to improve system performance. They modify transient response and steady state error. Very important in GATE design problems.
🔹 1. Why Compensator Needed?
- Reduce steady state error
- Improve stability
- Increase phase margin
- Reduce settling time
🔹 2. Lead Compensator
Glead(s) = (1 + Ts) / (1 + αTs)
Where:- 0 < α < 1
- Zero closer to origin than pole
Effect:
- Improves transient response
- Increases phase margin
- Reduces settling time
🔹 3. Lag Compensator
Glag(s) = (1 + Ts) / (1 + βTs)
Where:- β > 1
- Pole closer to origin than zero
Effect:
- Improves steady state error
- Little effect on transient response
🔹 4. Lead-Lag Compensator
Combination of lead and lag. Used when both transient and steady state improvements needed.🔹 5. Worked Example – Lead Design
Given system:
G(s) = K / [s(s+2)]
Suppose required phase margin improvement = 30°.Step 1: Calculate required phase boost
Step 2: Choose α from:
Φmax = sin⁻¹[(1−α)/(1+α)]
Step 3: Determine new gain crossover frequency
Step 4: Place zero and pole accordingly
(GATE usually asks conceptual or small numerical design steps.)🔹 6. Important Comparison
| Compensator | Transient | SSE |
|---|---|---|
| Lead | Improves | No major change |
| Lag | Minor change | Improves |
| Lead-Lag | Improves | Improves |
🔹 7. Root Locus View
- Lead → Moves locus left
- Lag → Improves low frequency gain
🔹 8. Bode Plot View
- Lead → Adds positive phase
- Lag → Adds low frequency gain
🎯 GATE Important Points
- Lead for transient improvement
- Lag for steady state improvement
- Know phase boost formula
- Design steps sometimes asked
Compensators Shape System Performance
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