# What is GCD (Greatest Common Divider)?

### Definition

GCD(a,b) is the Greatest Common Divider between two integers a and b. In electrical machines it is usually applied to calculate GCD(Zs,2p), the greatest common divider between slot number (Zs) and pole numbers (2p).

To calculate the GCD(Zs,2p) one can use the Matlab/Octave command gcd(a,b).

An important relationship between GCD and LCM is given by

If the winding is integral, Zs is already a multiple of 2p and GCD(Zs,2p)=2p. The concept of GCD is therefore more interesing for fractional slot windings.

### Application to e-NVH

The largest is the wavenumber of magnetic forces, the lowest is the magnitude of the yoke displacement, and resulting vibration and noise. It is therefore interesting to know what is the lowest wavenumber of magnetic forces in a given electrical machine.

More especially, all electrical machines have pulsating (r=0) magnetic forces, so the lowest non-zero wavenumber of magnetic forces is more interesting.

One can show that in synchronous machines (permanent magnet or wound rotor synchronous machines), this number is given in open-circuit conditions by rmin=GCD(Zs,2p). In some cases this expression is also correct in load conditions, or it becomes GCD(Zs,2p)/2. This principle can be extended to distributed winding squirrel cage induction machines where the lowest non-zero wavenumber occur at GCD(Zs,Zr,2p)=GCD(2p,Zr).

Reducing the GCD is a well known technique to reduce cogging torque magnitude in synchronous machines. One can see that it is contradictory with the maximization of GCD to reduce static vibration of the yoke and resulting acoustic noise.

### Limits of the GCD rule

A low GCD may be reached by high permeance and magnetomotive ranks, that is to say low magnitude force harmonics. Based on EOMYS experience, comparing two machines vibroacoustic behaviour based on GCD rule can be misleading. Besides, the GCD rules the Maxwell stress airgap harmonics, but once these harmonics are projected on the stator teeth a tooth sampling effect occurs (maximum wavenumber seen by the structure is Zs/2) and the GCD expression is no longer applicable. Finally, the GCD itself does not account for potential resonances between structural modes and magnetic excitations.

The GCD should not be used as a single criterion to compare the e-NVH performance of electrical machines and simulation with MANATEE software is highly recommended.

### Application to MANATEE

The GCD is automatically calculated and given in MANATEE software text output. The presence of harmonic forces can be identified using 2D Fourier transform post processing such as plot_Fr_fft2_stem.