---
title: "Capability curve of a generator"
subject: "Power Plant Design"
chapter: 4
key: "semester_8/power_plant_design:generator-capability-curve"
source: https://www.bctengineer.com/q/semester_8/power_plant_design/generator-capability-curve
site: BCT Engineering
---

# Capability curve of a generator

Power Plant Design · Chapter 4: Electric System Design of a Power Plant

- Asked in: 2081 Chaitra, 4 marks; 2078 Chaitra, 6 marks (asked 2 times)
- Video: [GENERATOR CAPABILITY CURVE : simplest way to derive](https://www.youtube.com/watch?v=KsG6qERjvC0) — Electrocean
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/power_plant_design/generator-capability-curve
- Whole chapter: https://www.bctengineer.com/md/semester_8/power_plant_design/chapter-4

## Question

What is capability curve of a generator? Describe briefly.

## Short answer

A capability curve defines the region of active and reactive power a generator can continuously sustain. Limits: armature heating (MVA circle), field heating (over-excited circle), prime mover limit (maximum P), under-excitation limits (steady-state stability and core-end heating), and minimum load. It guides loading and VAr output for voltage control.

## Answer

A **capability curve (P–Q chart)** of a synchronous generator shows the region of active power $P$ and reactive power $Q$ in which the generator can operate continuously **without exceeding thermal and stability limits**. Operators use it to set loading and VAr output.

```text
 Q (lagging, over-excited)
  ^
  |-------._    field heating limit
  |         '.
  |           \
  |            |  <- turbine limit
  |            |     (P = Pmax)
  +------------+---> P
  |            |  <- stator heating
  |           /      limit (circle S)
  |   ______.'
  |  under-excitation /
  |  stability limit
  v
 Q (leading, under-excited)
```

#### Limits that form the curve
1. **Stator (armature) current heating limit:** $P^2 + Q^2 = (V I_a)^2 = S^2$; a circle centred at the origin with radius equal to rated MVA.
2. **Field (rotor) current heating limit:** maximum excitation emf $E_{max}$ gives a circle centred at $(0, -V^2/X_s)$ with radius $E_{max}V/X_s$. This limits the lagging (over-excited) region.
3. **Prime mover (turbine) limit:** maximum turbine output, a vertical line $P = P_{max}$.
4. **Under-excitation limits:** in the leading region, the **steady-state stability limit** ($\delta$ approaching 90°, with practical margin) and **stator end-core heating** limit the VAr absorption.
5. **Minimum load limit:** some turbines (e.g. Francis) cannot run stably below about 40% load.

#### Use
- Rated point lies where the stator circle meets the field circle at rated power factor (e.g. 0.85 lagging).
- Shows how much reactive power the unit can supply or absorb at a given $P$, needed for grid voltage control and for setting AVR limiters (over-excitation and under-excitation limiters).

## Check yourself

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

1. The capability curve (P-Q chart) shows:
   A. Efficiency versus speed
   B. Region of P and Q where the generator can run continuously within thermal and stability limits
   C. Voltage versus current only
   D. Fault current decay
   Answer: B. It defines safe operating P-Q combinations. (Key point: Definition)

2. The stator heating limit is:
   A. A vertical line at P max
   B. A circle centred at (0, -V^2/Xs)
   C. A circle centred at the origin with radius equal to rated MVA
   D. A horizontal line at Q = 0
   Answer: C. P^2 + Q^2 = S^2 is a circle centred at the origin. (Key point: Stator limit)

3. The field current limit and turbine limit are respectively:
   A. Circle centred at (0, -V^2/Xs) limiting lagging region; vertical line P = Pmax
   B. Vertical line; circle at origin
   C. Both circles at origin
   D. Both horizontal lines
   Answer: A. The field limit circle bounds over-excited operation; the turbine limits maximum P. (Key point: Field limit and turbine limit)

4. In the leading (under-excited) region, VAr absorption is limited by:
   A. Field heating
   B. Steady-state stability limit and stator end-core heating
   C. Turbine overspeed
   D. Transformer tap range
   Answer: B. Delta approaching 90 degrees and end-core heating restrict the under-excited region. (Key point: Leading region limit)

## Related in Power Plant Design

- [Limits of the alternator capability curve](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Acapability-curve-limits) — Power Plant Design, chapter 4

## Related in other subjects

- [T/F: capability curve inversely proportional to length](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Atf-capability-curve-inverse-length) — Transmission and Distribution System Design, chapter 2
- [Power capability curve of alternator](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Apower-capability-curve) — Electric Machines II, chapter 1
- [Conditions for induction generator, T-S curve](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Ainduction-generator-conditions-ts-curve) — Electrical Machines, chapter 5
- [Power duration curve and primary/secondary energy](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apower-duration-curve-energy) — Hydro Power, chapter 3