How to interpret the theoretical harmonic force analysis?

Once MANATEE software electromagnetic and vibro-acoustic simulation is over, one should find in the console text output a table like this one:

210. Theoretical harmonic force analysis
Theoretical harmonic force analysis

This table identifies analytically the main magnetic force harmonics depending on the machine topology and identifies potential harmful force harmonics. These harmonics represent both radial and tangential force harmonics.

  • the wavenumber r is the "spatial frequency" or wavenumber (half the number of circumferential nodes along the airgap) of each magnetic force wave. For instance, the fundamental flux density has a wavenumber r=p (number of pole pairs).
  • the integers kr and ks are the ranks of the slotting permeance harmonics (ranks of the Fourier transform of the airgap permeance, the inverse of airgap reluctance), due to respectively rotor slots and stator slots. ks=1 is the principal stator slot harmonic, whereas ks=0 stands for the average airgap permeance.
  • the integers hr and hs are the ranks of the Fourier transform of the rotor and stator magnetomotive forces along the airgap (hr=0 and hs=0 corresponds to the fundamental rotor and stator mmf).
  • the variable eps equals +/-1 comes from addition or substraction of waves . For induction machines it can also be +/-2 for saturation harmonics.
  • the variable k is the proportionality factor between the force harmonic frequency and the fundamental stator frequency. For instance k=2 for the largest magnetic force wave at wavenumber r=2p and frequency f=2fs.

- *the boolean resonance indicates if there is a resonance that should theoretically occur, based on the specified minimum and maximum speed, and on the natural frequencies calculated by the structural model.

  • freso and Nreso respectively indicate the supply frequency (Hz) / speed (rpm) at which this resonance occurs.

The higher kr, ks, hr and hs are, and the lower is the force magnitude. In practice kr and ks values above 4 should not raise any acoustic issue.

The lower are kr and ks, the higher are the reluctance harmonics magnitude and the higher are vibration and noise at resonance. In this example of an induction machine with Zs=36 Zr=28 and p=3, one can notice a high force harmonic of order r=-2=Zr-Zs+2p=28-36+6 and frequency fs((1-s)Zr/p+2) that theoretically resonate with the stator ovalization mode near 1170 rpm. Another important force line (kr=ks=1) is given by r=-8=Zr-Zs, but it does not make any resonance in theory (forced excitation of the stack).

Note that for induction machines, this theoretical study does not include higher space harmonic effects, in particular the space harmonics present in the stator winding mmf.

Further guidelines to identify the root cause of noise can be found here.

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