What are stator winding force harmonics?


In e-NVH, stator winding force harmonics (sometimes called stator mmf force harmonics, or armature field force harmonics) refer to Maxwell force harmonics linked to stator magnetomotive force space harmonics combined with slotting effects.

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



  • P1 is a first permeance harmonic
  • P2 is a second permeance harmonic
  • F1=F0 is the fundamental magnetomotive force
  • F2 is a stator magnetomotive force (mmf) space harmonic

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

Winding force harmonics should disappear when the machine has an ideal winding distribution (infinite number of stator slots per pole per phase).

Application to e-NVH

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

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

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

r_{swind}=k_{r}Z_{r}-k_{s}Z_{s} \pm p \pm p(2q_{s}h_{s}+1)

f_{swind}=f_{s}((1-s)k_{r}Z_{r}/p \pm 1 \pm (2q_{s}h_{s}+1))


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

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

r_{swind}=k_{s}Z_{s} \pm p \pm (\xi \tau q{s}h_{s}+p)

f_{swind}=f_{s}(\pm 1 + \pm  (\xi \tau q{s}h_{s}/p+1))

where \tau =GCD(Zs,p) and \xi=1 if Z_{s}/\tau is odd, \xi=2 otherwise

For integral windings \tau =p and  \xi = 2

Stator mmf harmonics can therefore interfere with some slotting harmonics. In spectrograms stator mmf 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)

Noise control actions

The key design parameter to control stator winding harmonics is the winding connection matrix itself. Increasing the number of slots per pole per phase reduces mmf space harmonics. The choice of the winding coil pitch / coil throw is also important but it has no influence on the stator mmf stepped harmonics of wavenumber Zs+/-p.

Application to MANATEE

MANATEE software electromagnetic models account for all winding space harmonics in permeance / mmf model, subdomain and finite element models. Any type of winding can be modelled.

Stator winding space harmonic effects can be cancelled to check if they are the root cause of acoustic noise and vibrations.

The harmonic content of the stator winding mmf can be analyzed using post processing such as plot_smmf_space.

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