### Definition

In e-NVH, PWM harmonics stand for **Maxwell force harmonics linked to Pulse Width Modulation** only, or more generally by any stator armature current harmonic actively produced by control (not induced by strong electromagnetic coupling). PWM magnetic force harmonics are by definition independent of slotting force harmonics, saturation force harmonics, and stator winding force harmonics.

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

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

where

- P0 is the average permeance
- F0 is the fundamental mmf (generally the rotor mmf for PMSM, the stator mmf for SCIM)
- F1 is a stator magnetomotive force harmonic due to a PWM current harmonic (generally the modulation of the fundamental mmf with a current harmonic)

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

PWM harmonics should disappear when the machine is fed with sinusoidal currents, or when the PWM switching frequency is increased significantly (in particular above 20 kHz).

### Application to e-NVH

The largest excitation frequencies depend on the PWM strategy and modulation index. For asynchronous switching frequency at fswi and symmetrical sine/triangle PWM strategy, one can show that the resulting magnetic force wavenumbers r and associated frequency f are given by

`r_{pwm}=0 or 2p`

`f_{pwm}=nf_{swi}\pm mf_{s}`

where

- n and m are integers with opposite parity

For instance, excitations at fswi+/-fs have a wavenumber r=2p, while excitations at fswi+/-3fs have a wavenumber r=0. This is illustrated in the following simplified force / vibration / noise spectrum:

**Spectrum of PWM electromagnetic excitations**

In spectrograms PWM harmonics appear as V-shaped lines starting from multiples of the switching frequency at zero speed:

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

### Noise control actions

PWM effects can be reduced playing on the switching frequency to reduce PWM current harmonics and avoid structural resonance, or by implementing spred spectrum strategies like randomization of the switching frequency. When doing so the sound quality should be checked with proper psychoacoustic indices and panel testing.

### Application to MANATEE

MANATEE software electromagnetic models can include PWM effects - any voltage or current waveform can be imposed. A coupling with Simulink is also proposed. Simulated noise can be synthesized as audio file for listening. The resulting sound file can also be imported in third-party sound quality software to run further psychoacoustic studies.