AIRGAP FIELD SPACE HARMONICS, PARASITIC TORQUES, RADIAL FORCES, AND NOISE

The airgap field distribution in induction machines is influenced by stator and rotor mmf distributions and by magnetic saturation in the stator and rotor teeth and yokes (back cores).

Previous chapters introduced the mmf harmonics but were restricted to the fundamentals. Slot openings were considered but only in a global way, through an apparent increase of airgap by the Carter coefficient.

We first considered magnetic saturation of the main flux path through its influence on the airgap flux density fundamental. Later on a more advanced model was introduced (AIM) to calculate the airgap flux density harmonics due to magnetic saturation of main flux path (especially the third harmonic). However, as shown later in this chapter, slot leakage saturation, rotor static, and dynamic eccentricity together with slot openings and mmf step harmonics produce a multitude of airgap flux density space harmonics. Their consequences are parasitic torque, radial uncompensated forces, and harmonics core and winding losses. The harmonic losses will be treated in the next chapter.

In what follows we will use gradually complex analytical tools to reveal various airgap flux density harmonics and their parasitic torques and forces. Such treatment is very intuitive but is merely qualitative and leads to rules for a good design. Only FEM–2D and 3D–could depict the extraordinary involved nature of airgap flux distribution in IMs under various factors of influence, to a good precision, but at the expense of much larger computing time and in an intuitiveless way. For refined investigation, FEM is, however, “the way”.

10.1. STATOR MMF PRODUCED AIRGAP FLUX HARMONICS

As already shown in Chapter 4 (Equation 4.27), the stator mmf per-polestepped waveform may be decomposed in harmonics as

F x,t1( )= 3W Iπ1 1p1 2 Kw1 cosπτ x −ω1t+ K5w5 cos 5τπ x +ω1t+ (10.1)

+ K7w7 cos 7τπ x −ω1t+ K11w11 cos11τπ x +ω1t+ K13w13 cos13τπ x −ω1t

where K_wν is the winding factor for the ν^th harmonic,

νπ

K sin ^νπy (10.2)

6 ; K _yν = sin _wν = K_qνK _yν; K_qν = qsin _νπ 2_τ

6q

In the absence of slotting, but allowing for it globally through Carter’s coefficient and for magnetic circuit saturation by an equivalent saturation factor

K_sν , the airgap field distribution is

Bg1(θ,t)= 3µπ0p gKW I1 1 1 c 2 1+KKw1s1 cos(θ − ω1t)+ 5 1( K+w5Ks5 )cos 5( θ + ω1t)+

+ 7 1( K+w7Ks7 )cos 7( θ − ω1t)+ 11(1K+w11Ks11)cos(11θ + ω1t)+ (10.3)

⁺

13(1^K+^w13K )cos(13θ − ω₁t ;) θ = π_τ x

_s13

In general, the magnetic field path length in iron is shorter as the harmonics order gets higher (or its wavelength gets smaller). K_sν is expected to decrease with ν increasing.

Also, as already shown in Chapter 4, but easy to check through (10.2) for all harmonics of the order ν,

ν = C₁ ^Np1^s ±1 (10.4)

the distribution factor is the same as for the fundamental.

For three-phase symmetrical windings (with integer q slots/pole/phase), even order harmonics are zero and multiples of three harmonics are zero for star connection of phases. So, in fact,

ν = 6C₁ ±1 (10.5)

As shown in Chapter 4 (Equations 4.17 – 4.19) harmonics of 5th, 11th, 17th, … order travel backwards and those of 7th, 13th, 19th, … order travel forwards – see Equation (10.1).

The synchronous speed of these harmonics ω_ν is

ω =_ν ^ddt^θν = ^ων¹ (10.6)

In a similar way, we may calculate the mmf and the airgap field flux density of a cage winding.

10.2. AIRGAP FIELD OF A SQUIRREL CAGE WINDING

A symmetric (healthy) squirrel cage winding may be replaced by an equivalent multiphase winding with N_r phases, ½ turns/phase and unity winding factor. In this case, its airgap flux density is

Bg2 (θ,t)= Nπrp gKµ10 1I c2 ∑µ=∞1 cos(µθµ(1m+ωK1tsν− ϕ) 12ν) = µgK0F t2 c( ) (10.7)

The harmonics order which produces nonzero mmf amplitudes follows from the applications of expressions of band factors K_BI and K_BII for m = N_r:

µ = C₂ ^Np1^r ±1 (10.8)

Now we have to consider that, in reality, both stator and rotor mmfs contribute to the magnetic field in the airgap and, if saturation occurs, superposition of effects is not allowed. So either a single saturation coefficient is used (say K_sν = K_s1 for the fundamental) or saturation is neglected (K_sν = 0).

We have already shown in Chapter 9 that the rotor slot skewing leads to variation of airgap flux density along the axial direction due to uncompensated skewing rotor mmf. While we investigated this latter aspect for the fundamental of mmf, it also applies for the harmonics. Such remarks show that the above analytical results should be considered merely as qualitative.

10.3. AIRGAP CONDUCTANCE HARMONICS

Let us first remember that even the step harmonics of the mmf are due to the placement of windings in infinitely thin slots. However, the slot openings introduce a kind of variation of airgap with position. Consequently, the airgap conductance, considered as only the inverse of the airgap, is

g( )θ = g + ∆ θ + ∆₁( ) ₂ (θ − θ_r )= _f1¹( )_θ + _f2 (_{θ − θ}¹ r ₎ − g (10.11)

∆₁(θ) and ∆₂(θ − θ_r) represent the influence of stator and rotor slot openings alone on the airgap function.

As f₁(θ) and f₂(θ − θ_r) are periodic functions whose period is the stator (rotor) slot pitch, they may be decomposed in harmonics:

f a a cos N

(10.12) f b b cos N

Now, if we use the conformal transformation for airgap field distribution in presence of infinitely deep, separate slots–essentially Carter’s method–we obtain [1]

= βg Fν btos,rs,r  (10.13)

a ,bν ν

  νbos,r 2 

 b 

Fν tos,rs,r  = ν π1 4 0.5 +  ts,r νbos,r 2 sin 1.6 πνtbs,ros,r  (10.14)



 0.18− 2 ts,r 

β is [2]

Table 10.1 β(b_os,r/g)

Equations (10.13 – 10.14) are valid for ν = 1, 2, … . On the other hand, as expected,

a 0 ≈ K g¹_c1 b0 ≈ K¹_c2g (10.15) where K_c1 and K_c2 are Carter’s coefficients for the stator and rotor slotting, respectively, acting separately.

Finally, with a good approximation, the inversed airgap function 1/g(θ,θ_r) is λg (θ θ, r )= g(θ θ1, r ) ≈ g1 K Kc11 c2 − Kac21 cos Ns pθ1 − Kb1c1 cos Nr (θ − θp1 r ) +

+ a b1 1cos (Ns p+1 Nr )θ − Np1r θr  + cos (Ns p−1 Nr )θ + Np1r θr __ (10.16)

 

As expected, the average value of λ_g(θ,θ_r) is

(λg (θ θ, r ))average = K Kc11 c2g = K g1c (10.17)

θ, θ_r – electrical angles.

We should notice that the inversed airgap function (or airgap conductance) λ_g has harmonics related directly to the number of stator and rotor slots and their geometry.