What are slotting force harmonics?

Definition

In e-NVH, slotting force harmonics refer for Maxwell force harmonics linked to slotting effects only (airgap permeance fluctuations due to rotor and stator slots), which means magnetic force harmonics independent of saturation harmonics, eccentricity or uneven airgap harmonics, as well as stator winding harmonics.

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

F_{slotting}=B_{1}B_{2}=P_{1}F_{1}P_{2}F_{2}=P_{1}F_{0}P_{2}F_{0}

where

  • P1 is a first permeance harmonic
  • P2 is a second permeance harmonics
  • F1=F2=F0 is the fundamental magnetomotive force

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

Slotting harmonics should disappear when the machine has virtually closed slots (equivalently, high permeability magnetic wedges) or is slotless (airgap winding).

A particular case of slotting harmonics is cogging torque in PMSM open-circuit conditions, corresponding to tangential magnetic force harmonics due to pole / slot interactions.

Application to e-NVH

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

  • Ps (resp. Pr) stands for stator (resp. rotor) slot permeance harmonic
  • F0s is the fundamental stator magnetomotive force

One can show that the resulting force wavenumbers r and associated frequency f are given by

r_{slotting}=k_{r}Z_{r}-k_{s}Z_{s} \pm 0,2p

f_{slotting}=f_{s}((1-s)k_{r}Z_{r}/p \pm 0,2) 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

One can show that pulsating forces (r=0) occur at mutiples of LCM(Zs,Zr,2p) times mechanical frequency.

In synchronous machines, the resulting force wavenumbers r and associated frequency f are given by

r_{slotting}=k_{s}Z_{s} \pm 2p h_{r}

f_{slotting}=\pm 2h_{r}f_{s}

One can show that pulsating forces (r=0) occur at mutiples of LCM(Zs,2p) times mechanical frequency.

In spectrograms slotting harmonics appear as straight lines crossing the origin:

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

In EV HEV NVH issues, slotting effect can therefore occur around multiples of the stator slot passing frequencies or around multiples of the rotor slot passing frequencies depending on topology.

Noise control actions

A key design parameter to control slotting effects is the slot/pole combination. Additional influential parameters include slot opening geometry, notches, magnetic wedges.

Application to MANATEE

MANATEE software electromagnetic models account for all slotting effects. Force harmonics are automatically labelled with the corresponding frequency and wavenumber expressions.

An automated analysis of slotting harmonics is available. Slotting effects can be cancelled to check if they are the root cause of acoustic noise and vibrations.

MANATEE includes the modelling of notches and magnetic wedges.

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