---
title: "Power angle characteristics of cylindrical rotor machine"
subject: "Electric Machines II"
chapter: 1
key: "semester_5/electric_machines_ii:power-angle-cylindrical-rotor"
source: https://www.bctengineer.com/q/semester_5/electric_machines_ii/power-angle-cylindrical-rotor
site: BCT Engineering
---

# Power angle characteristics of cylindrical rotor machine

Electric Machines II · Chapter 1: Three Phase Synchronous Generator

- Asked in: 2074 Asoj, 6 marks; 2070 Chaitra, 6 marks (asked 2 times)
- Video: [Power Angle Characteristics of Cylindrical Machine](https://www.youtube.com/watch?v=6N8LVn-U2mg) — Sharad Dhamal
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/electric_machines_ii/power-angle-cylindrical-rotor
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machines_ii/chapter-1

## Question

Explain power angle characteristics of cylindrical rotor machine.

## Short answer

The electrical power P of a cylindrical-rotor machine varies sinusoidally with load angle δ: P = (3EV/X_s) sin δ. Maximum power P_max = 3EV/X_s occurs at δ = 90°, the steady-state stability limit. Positive δ gives generator action, negative δ motor action; normal operation is at 20°–35°.

## Answer

The **power angle characteristic** is the curve of electrical power ($P$) against the power (load) angle $\delta$, the angle between the excitation emf $E$ and the terminal voltage $V$.

#### Derivation (armature resistance neglected)
For a cylindrical-rotor generator with $\bar E = E\angle\delta$, $\bar V = V\angle 0$:
$$
\begin{aligned}
\bar I &= \frac{E\angle\delta - V\angle 0}{jX_s} \\
\bar S &= 3\bar V\bar I^{*} \\
P &= \frac{3EV}{X_s}\sin\delta \\
Q &= \frac{3V}{X_s}\left(E\cos\delta - V\right)
\end{aligned}
$$
(per-phase values; for a motor, $\delta$ is negative and power flows into the machine).

#### The curve
```text
  P
  |  generator       Pmax
  |  region       .-"""""-.
  |            .'           '.
  |          .'               '.
  |        .'                   '.
 -+------+------------+-----------+-----> delta
 -180   .0           90          180 deg
     .'
  .'   motor region (delta < 0)
```

#### Features
1. Power varies **sinusoidally** with $\delta$.
2. **Maximum power** occurs at $\delta = 90^\circ$:
   $$
   P_{max} = \frac{3EV}{X_s}
   $$
   This is the **steady-state stability limit**. Beyond 90°, more $\delta$ gives less power, the machine cannot hold synchronism and falls out of step (pole slipping).
3. Positive $\delta$ (E leads V): **generator action**; negative $\delta$: **motor action**.
4. Normal operation is at $\delta$ = 20°–35°, leaving a margin for sudden load changes.
5. $P_{max}$ increases with excitation $E$ and with lower $X_s$; strong excitation improves stability.
6. **Synchronising power coefficient** $P_{syn} = \dfrac{dP}{d\delta} = \dfrac{3EV}{X_s}\cos\delta$ is the slope of the curve; it is largest at $\delta = 0$ and zero at $90^\circ$.

## Check yourself

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

1. The power angle δ is the angle between
   A. excitation emf E and terminal voltage V
   B. current I and voltage V
   C. stator mmf and rotor mmf in space
   D. E and current I
   Answer: A. δ is the load angle between E and V. (Key point: Power angle)

2. Neglecting Ra, the power of a cylindrical-rotor machine (per phase values, 3 phases) is
   A. P = (3EV/Xs) cos δ
   B. P = (3EV/Xs) sin δ
   C. P = (3E^2/Xs) sin δ
   D. P = (EV/Xs) tan δ
   Answer: B. From I = (E∠δ - V)/(jXs) and S = 3VI*, P = 3EV sinδ / Xs. (Key point: Power formula)

3. The maximum power and steady-state stability limit occur at
   A. δ = 0°
   B. δ = 45°
   C. δ = 90°
   D. δ = 180°
   Answer: C. sin δ is maximum at 90°; beyond it the machine loses synchronism. (Key point: Stability limit)

4. The synchronising power coefficient dP/dδ is
   A. 3EV sinδ / Xs, largest at 90°
   B. constant for all δ
   C. 3EV / Xs, zero at δ = 0
   D. 3EV cosδ / Xs, largest at δ = 0 and zero at 90°
   Answer: D. Differentiating gives cosδ dependence; slope is greatest at zero and vanishes at 90°. (Key point: Synchronising power)

## Related in Electric Machines II

- [Power angle characteristics of salient pole machine](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asalient-power-angle-derivation) — Electric Machines II, chapter 2
- [Construction of synchronous machine and rotor types](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asynchronous-machine-construction-rotors) — Electric Machines II, chapter 1, asked 2×
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- [Regulation and power angle of 1650 kVA generator](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aregulation-power-angle-1650kva) — Electric Machines II, chapter 1
- [Power angle and regulation of 10 kVA alternator](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Apower-angle-regulation-10kva) — Electric Machines II, chapter 1

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- [Initial rotor angle and critical clearing angle](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Arotor-angle-critical-clearing-2076) — Power System Analysis II, chapter 6