---
title: "Pitch factor, distribution factor and EMF equation"
subject: "Electric Machines II"
chapter: 1
key: "semester_5/electric_machines_ii:pitch-distribution-factor-emf"
source: https://www.bctengineer.com/q/semester_5/electric_machines_ii/pitch-distribution-factor-emf
site: BCT Engineering
---

# Pitch factor, distribution factor and EMF equation

Electric Machines II · Chapter 1: Three Phase Synchronous Generator

- Asked in: 2083 Baisakh (new course), 3 marks; 2078 Bhadra, 4+4 marks; 2074 Asoj, 8 marks (asked 3 times)
- Video: [distribution factor and pitch factor | distribution factor in alternator | calculation | full pitch](https://www.youtube.com/watch?v=Hro1hPxN8NE) — Mind of Engineer 
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/electric_machines_ii/pitch-distribution-factor-emf
- Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machines_ii/chapter-1

## Question

Define the pitch factor and distribution factor and their significance in synchronous machine. Derive the e.m.f equation of an alternator.

## Short answer

Pitch factor kp = cos(α/2) accounts for emf reduction when coils are short-pitched, which suppresses harmonics. Distribution factor kd = sin(mβ/2)/(m sin(β/2)) arises from distributed coils, giving a more sinusoidal wave. RMS emf per phase: Eph = 4.44 kp kd f Φ T V, where T is turns per phase.

## Answer

#### Pitch factor (coil span factor), $k_p$
The **pitch factor** is the ratio of the emf induced in a short-pitched (chorded) coil to the emf that would be induced in a full-pitched coil.

If the coil is short of full pitch (180° electrical) by an angle $\alpha$, the emfs in the two coil sides are not in phase and add as phasors:
$$
k_p = \frac{\text{phasor sum}}{\text{arithmetic sum}} = \cos\frac{\alpha}{2}
$$
**Significance:** short pitching slightly reduces the emf ($k_p < 1$) but it eliminates or reduces harmonics (e.g. $\alpha = 36^\circ$ removes the 5th harmonic, since $k_{p5} = \cos(5\alpha/2) = 0$), saves copper in end connections and reduces leakage reactance.

#### Distribution (breadth) factor, $k_d$
The **distribution factor** is the ratio of the phasor sum to the arithmetic sum of the emfs of coils distributed in several slots of a phase group.

With $m$ slots per pole per phase and slot angle $\beta = \dfrac{180^\circ}{\text{slots per pole}}$:
$$
k_d = \frac{\sin(m\beta/2)}{m\sin(\beta/2)}
$$
**Significance:** distributing the winding gives a more sinusoidal emf, better use of the core and better cooling; $k_d < 1$ only slightly reduces the emf.

#### EMF equation of an alternator
Let
- $P$ = number of poles, $N_s$ = speed in rpm,
- $\Phi$ = flux per pole (Wb), $Z$ = conductors in series per phase, $T = Z/2$ turns per phase.

In one revolution, each conductor cuts flux $P\Phi$ in time $60/N_s$ seconds.
$$
\begin{aligned}
\text{Average emf per conductor} &= \frac{d\Phi}{dt} = \frac{P\Phi}{60/N_s} = \frac{P\Phi N_s}{60} \\
\text{Since } f = \frac{PN_s}{120},\quad \frac{PN_s}{60} &= 2f \\
\text{Average emf per conductor} &= 2f\Phi\ \text{V} \\
\text{Average emf per turn} &= 2 \times 2f\Phi = 4f\Phi\ \text{V} \\
\text{Average emf per phase} &= 4f\Phi T \\
\text{RMS emf per phase} &= 1.11 \times 4f\Phi T = 4.44\, f\Phi T\ \text{V}
\end{aligned}
$$
(form factor of a sine wave = 1.11). Including the winding factors for a short-pitched, distributed winding:
$$
E_{ph} = 4.44\, k_p\, k_d\, f\, \Phi\, T\ \text{volts}
$$
For a star-connected machine, line voltage $E_L = \sqrt{3}\,E_{ph}$; for delta, $E_L = E_{ph}$. The product $k_w = k_p k_d$ is called the winding factor.

## Check yourself

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

1. The pitch factor k_p of a short-pitched coil (short by angle alpha) is
   A. cos(alpha/2)
   B. sin(alpha/2)
   C. cos(alpha)
   D. 1/cos(alpha/2)
   Answer: A. k_p is phasor sum over arithmetic sum of coil-side emfs = cos(alpha/2). (Key point: Pitch factor)

2. The distribution factor k_d is the ratio of
   A. arithmetic sum to phasor sum of emfs
   B. phasor sum to arithmetic sum of emfs of coils in a phase group
   C. emf of a short-pitched coil to a full-pitched coil
   D. line voltage to phase voltage
   Answer: B. k_d = sin(m*beta/2)/(m*sin(beta/2)), phasor sum over arithmetic sum of distributed coils. (Key point: Distribution factor)

3. Average emf per conductor in the derivation is 2fΦ. Why?
   A. Each conductor cuts Φ in one second
   B. Because the form factor is 1.11
   C. Each conductor cuts flux PΦ in 60/Ns seconds, and PNs/60 = 2f
   D. Because the flux changes sinusoidally at 2f
   Answer: C. emf = PΦ/(60/Ns) = PΦNs/60, and f = PNs/120 gives 2fΦ. (Key point: EMF derivation)

4. What is the rms emf per phase of an alternator with winding factors?
   A. 4.44 k_p k_d f Φ Z
   B. 2.22 k_p k_d f Φ T
   C. 4 k_p k_d f Φ T
   D. 4.44 k_p k_d f Φ T
   Answer: D. Average 4fΦT times form factor 1.11 gives 4.44fΦT; multiplied by k_p and k_d, with T = turns per phase. (Key point: Final emf equation)

## Related in Electric Machines II

- [Synchronous generator principle and EMF equation](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Agenerator-principle-emf-equation) — Electric Machines II, chapter 1
- [Emf and regulation of 1500 kVA alternator](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aregulation-1500kva-12000v) — Electric Machines II, chapter 1, asked 2×
- [Emf and regulation of 1500 kVA 13 kV alternator](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aemf-regulation-1500kva-13kv) — Electric Machines II, chapter 1
- [Change in emf to raise power factor to unity](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Apercentage-change-emf-unity-pf) — Electric Machines II, chapter 1
- [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×
- [Synchronous reactance and regulation from test data](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Aregulation-1500kva-2300v-test) — Electric Machines II, chapter 1

## Related in other subjects

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- [1600 kVA alternator: emf, regulation, power angle](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aalt-1600kva-regulation-power-angle) — Electrical Machines, chapter 6
- [1500 kVA alternator: excitation emf, voltage at leading pf](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aalt-1500kva-6600v-excitation-emf) — Electrical Machines, chapter 6