Winding factor

From KYNNpedia

In power engineering, winding factor <math>k_w</math> provides a way to compare of the effectiveness of different designs of stators for alternators. Winding factor is the ratio of electromotive force (EMF) produced by a stator having a short-pitch, distributed, or skewed winding, with a stator having full-pitch, concentrated, and non-skewed, windings.

For most alternators, the stator acts as the armature. Winding factor also applies to other electric machines, but this article focuses on winding factor as it applies to alternators.

Practical alternators have a short-pitched and distributed windings to reduce harmonics and maintain constant torque. Also, either the stator or rotor may be slightly skewed from the rotor's axis to reduce cogging torque. The armature winding of each phase may be distributed in a number of pole slots. Since the EMF induced in different slots are not in phase, their phasor sum is less than their numerical sum.<ref name=Shahl>Suad Ibrahim Shahl. "Introduction to AC Machines" (PDF). p. 7. Retrieved August 3, 2022.</ref> This reduction factor is called distribution factor <math>k_d</math>. The other factors that can reduce the winding factor are pitch factor <math>k_p</math> and skew factor <math>k_s</math>.

Pitch

In alternator design, pitch means angle. The shaft makes a complete rotation in 360 degrees, and is called mechanical degrees. However, the current in a conductors makes a complete cycle in 360 electrical degrees. Electrical degrees and mechanical degrees are related as follows:

<math>electrical degrees = \frac{P }{2}\cdot mechanicaldegrees</math>

where P is the number of poles.<ref name="globe">"Armature Winding". Circuit Globe. 5 January 2016. Retrieved July 29, 2022.</ref>

No matter how many poles, each pole always spans exactly 180 electrical degrees, and it is called pole pitch. Coil pitch is the number of electrical degrees spanned by the coil.<ref name=Shahl/>

Short pitch factor

A full-pitched coil is 180 electrical degrees, meaning it spans the entire pole. A short-pitched coil is less than 180 electrical degrees, meaning it does not spans the entire pole. The amount the coil is short-pitched is given by the variable <math>a</math> in electrical degrees:

<math>a = pole pitch - coil pitch</math>, and the pitch factor is:

<math>k_p = cos(\frac{a}{2})</math>.

A short pitched coil is also called chorded, in reference to the chord of a circle.<ref name=Shahl/>

Calculating winding factor

The winding factor can be calculated as
<math>k_w = k_d k_p k_s</math><ref name="Refai">Mustafa Al-Refai (2018). "Synchronous generator" (PDF). Electrical and Communications Consulting Office (ECCO). p. 20. Retrieved August 6, 2022.</ref>

where
<math>k_d</math> is the distribution factor.
<math>k_p</math> is the pole factor.
<math>k_s</math> is the skew factor resulting from the winding being skewed from the axis of the rotor.

Example

For a 3-phase 6 slot 4 pole non-overlapping winding alternator:
<math>coil pitch = \frac{2 \pi}{6} = \frac{\pi}{3} (mech) = \frac{2 \pi}{3} (elec)</math>
<math>pole pitch = \frac{2 \pi}{4} = \frac{\pi}{2} (mech) = \pi (elec)</math>

Most of 3-phase motors have winding factor values between 0.85 and 0.95.

The winding factor (along with some other factors like winding skew) can help to improve the harmonic content in the generated EMF of the machine.

References

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