[Article] A Study of the Effect of a Magnetic Field on by King A.S.

By King A.S.

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42, harmonic waves of the MMF of each phase are shifted by the angle (ν ∓1)(2π )/3 one from each other. Harmonics ν = 3k = 3, 9, 15, . . (in general, ν = m 1 k where k = 1, 3, 5, . . and m 1 = 3) are shifted by (ν ∓ 1)(2π )/3 = (3k ∓ 1)(2π )/3 = 2π k ∓ 2π/3 or 120o . Thus, the sum of those harmonic waves is equal to zero, so that harmonics ν = 3k do not exist in three phase systems. For harmonic waves ν = 6k + 1 (in general, ν = 2m 1 k + 1) the shift angle is (ν − 1)(2π )/3 = 4kπ or 0o . The resultant wave is the arithmetic sum of waves in all three phases.

The rotor magnetic flux density waveforms are calculated on the basis of MMF waveforms and permeances of the air gap. Magnetic forces are calcualated on the basis of Maxwell stress tensor. The natural frequencies of the stator system are calculated with the aid of equations given in Chapter 5. , from the FEM structural package. Then, using the damping coeffcient as a function of frequency, amplitudes of radial velocities are calculated. The damping factor affects significantly the accuracy of computation.

40. 35 I1 . 30. 46) DK3193 noise November 3, 2005 15:28 34 Noise of Polyphase Electric Motors where ω = 2π f is the angular frequency for the fundamental harmonic, τ is the pole pitch and ν = 2m 1 k ± 1, k = 0, 1, 2, 3, 4, 5, . . 47) The “−” sign before νπ x/τ is for the forward-rotating field and the “+” sign before νπ x/τ is for the backward-rotating field. The total MMF exicted by all m 1 phases is the sum of MMFs excited by each phase. 46 F1ν (x, t) = F11 (x, t) + F12 (x, t) + · · · + F1m 1 (x, t) = 1 2 ∞ Fmν cos ωt − ν ν=1 × 1 + cos cos π x τ 1 2 (ν − 1)2π + cos (ν − 1)2π + · · · m1 m1 m1 − 1 (ν − 1)2π m1 − 1 2 ∞ Fmν sin ωt − ν ν=1 π x τ 1 2 × 0 + sin (ν − 1)2π + sin (ν − 1)2π + · · · m1 m1 m1 − 1 sin (ν − 1)2π m1 1 + 2 ∞ Fmν cos ωt + ν ν=1 π x τ 1 2 × 1 + cos (ν + 1)2π + cos (ν + 1)2π + · · · m1 m1 cos m1 − 1 (ν + 1)2π m1 + 1 2 ∞ Fmν sin ωt + ν ν=1 π x τ 1 2 × 0 + sin (ν + 1)2π + sin (ν + 1)2π + · · · m1 m1 m1 − 1 sin (ν + 1)2π .

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