### Definition

In e-NVH, saturation force harmonics refer for **Maxwell force harmonics linked to magnetic saturation** combined with slotting effects. Saturation tends to flatten the airgap flux, resulting in a modification of the harmonic content of Maxwell stress. In particular airgap permeance harmonics of wavenumbers 2p appear.

More precisely, as any force harmonic is due to the product of two flux density harmonics B1 and B2, these saturation harmonics can be written as

`F_{sat}=B_{1}B_{2}=P_{1}F_{1}P_{2}F_{2}=P_{1}F_{0}P_{2}F_{0}`

where

- P1 is a saturation permeance harmonic
- P2 is a slotting permeance harmonics
- F1=F2=F0 is the fundamental magnetomotive force

As the noise and vibration frequencies are the same as the exciting force frequencies, **saturation harmonics** can also refer to some specific harmonics of the acoustic noise spectrum or vibration velocity spectrum.

Saturation harmonics should disappear when the machine runs in linear conditions (low magnetic excitation).

### Application to e-NVH

In **induction machines** at no-load, slotting harmonics can be more precisely derived as
`F_{sat}=P_{a}F_{0s}P_{sr}F_{0s}`
where

- Pa stands for saturated permeance harmonic
- Psr stands for combined rotor and stator slot permeance harmonic
- F0s is the fundamental stator magnetomotive force

One can show that the resulting force wavenumbers r and frequencies f are given for the first saturated permeance harmonic by

`r_{sat}=k_{r}Z_{r}-k_{s}Z_{s} \pm 0,4p`

`f_{sat}=f_{s}((1-s)k_{r}Z_{r}/p \pm 0,4)`
where

- ks (resp. kr) is the rank of stator (resp. rotor) permeance harmonic (0 for DC permeance, 1 for highest magnitude permeance harmonic)
- fs is the fundamental electrical frequency
- Zs (resp Zr) is the stator (resp. rotor) slot number
- s is the fundamental slip

Saturation harmonics can therefore interfere with some slotting harmonics. In spectrograms saturation harmonics appear as straight lines crossing the origin:

**Illustration of main electromagnetic noise and vibration lines in spectrograms (case of an induction motor)**

### Noise control action

Reducing the saturation levels allows to cancel the magnetic force excitation due to saturation, it can be obtained by a change of the control law, modification of the magnetic circuit geometry or change of magnetic sheet material. Besides, changing the slot/ople combination has also an influence on these specific excitations as they combine both saturation and slotting effects.

### Application to MANATEE

MANATEE software electromagnetic models can account for saturation effects using non linear finite element calculations or calculating the saturated permeance wave based on semi analytical techniques.