# What is LCM (Least Common Multiple)?

### Definition

LCM(a,b) is the Least Common Multiple between two integers a and b. In electrical machines it is usually applied to calculate LCM(Zs,2p), the least common multiple between slot number (Zs) and pole numbers (2p).

To calculate the LCM(Zs,2p) one can multiply progressively the number of slots by k=1, 2, 3 etc until kZs can be divided by 2p, the LCM is then kZs. Alternatively one can use the Matlab/Octave command lcm(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 LCM(Zs,2p)=Zs. The concept of LCM is therefore more interesing for fractional slot windings.

### Application to e-NVH

In synchronous machines (permanent magnet or wound rotor synchronous machines), some radial and tangential pulsating magnetic forces (wavenumber r=0) occur at multiples of LCM(Zs,2p) times the rotational frequency due to pole/slot interactions. This can be extended to distributed winding squirrel cage induction machines where pulsating harmonic forces occur at LCM(Zs,Zr,2p)=LCM(Zs,Zr) times rotational frequency.

Increasing the LCM is a well known technique to reduce cogging torque in synchronous machines. Increasing LCM might lower the GCD which also sizes the magnitude of cogging torque harmonics. In open circuit condition, radial and tangential force harmonics are correlated which explains why minimizing cogging torque might reduce electromagnetically-excited noise and vibrations. However, the LCM itself does not account for potential resonances between structural modes and magnetic excitations.

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