Air Gap and Carter Factor

To be able to calculate the magnetic voltage manually over an air gap, the geometry of the air gap has to be simplified. Often in an electrical machine, the surfaces of both the stator and the rotor are split with slots. The flux density always decreases at the slot opening (Figure 3.4), and therefore it is not easy to define the average flux density of the slot pitch between the stator and the rotor.

However, in 1901 F.W. Carter provided a solution to the problem of manual calculation (Carter, 1901). On average, according to Carter’s principle, the air gap seems to be longer than its physical measure. The length of the physical air gapδincreases with the Carter factor kC. The first correction is carried out by assuming the rotor to be smooth. We obtain

δes=k_Csδ. (3.6)

The Carter factor k_Cs is based on the dimensions in Figure 3.5. When determining the Carter factor, the real flux density curve is replaced with a rectangular function so that the flux remains constant under the teeth and is zero at the slot opening; in other words, the shaded

τu

(a) (b)

slot pitch Bδ

τu

Figure 3.4 (a) Flux diagram under a stator slot along one slot pitch, and (b) the behaviour of the air-gap flux densityB_δalong a slot pitch. At the slot opening, there is a local minimum of flux density. The flux density on the right side of the slot is slightly higher than on the left side, since a small current is flowing in the slot towards the observer

b_e

b₁ B(α)

Bav

–0.5 0 0.5

δ 1.0

0

τu

B_min/Bav av

av

B B

av max

C B

k = B

s

2 Q α 2B₀/B_av

S₁ S₁

S₂

Figure 3.5 Distribution of air-gap flux densityB_δ(α) in a distance of one slot pitchτu.αis the angle revolving around the periphery of the machine.beis the equivalent slot opening

areasS1+S1in Figure 3.5 are equal toS2. The equivalent slot openingbe, in which the flux

The Carter factor is also the ratio of the maximum flux densityB_max to the average flux densityB_av

k_C= Bmax

Bav . (3.9)

The variation of the flux density assuming no eddy currents damping the flux variation is β= B0

When both the stator and rotor surfaces are provided with slots, we calculatekCsfirst by assuming the rotor surface to be smooth. The calculations are repeated by applying the cal-culated air gapδesand the slot pitch of the rotorτr, and by assuming the stator surface to be smooth. We then obtainkCr. Finally, the total factor is

k_C,tot≈k_Cs·k_Cr, (3.11)

which gives us the equivalent air gapδe

δe≈k_C,totδ≈kCrδes. (3.12)

The influence of slots in the average permeance of the air gap is taken into account by replacing the real air gap by a longer equivalent air gapδe. The result obtained by applying the above equations is not quite accurate, yet usually sufficient in practice. The most accurate result is obtained by solving the field diagram of the air gap with the finite element method.

In this method, a dense element network is employed, and an accurate field solution is found as illustrated in Figure 3.4. If the rotor surface lets eddy currents run, the slot-opening-caused

flux density dips are damped by the values suggested by the Carter factor. In such a case, eddy currents may create remarkable amounts of losses on the rotor surfaces.

Example3.1: An induction motor has an air gapδ =0.8 mm. The stator slot opening is b₁ =3 mm, the rotor slots are closed and the stator slot pitch is 10 mm. The rotor magnetic circuit is manufactured from high-quality electrical steel with low eddy current losses. Calculate the Carter-factor-corrected air gap of the machine. How deep is the flux density dip at a slot opening if the rotor eddy currents do not affect the dip (note that the possible squirrel cage is designed so that the slot harmonic may not create large opposing eddy currents)? How much three-phase stator current is needed to magnetize the air gap to 0.9 T fundamental peak flux density? The number of stator turns in series isNs=100, the number of pole pairs isp=2, and the number of slots per pole and phase isq=3. The winding is a full-pitch one.

Solution:

The depth of the flux density dip on the rotor surface is 2B0:

2B0=2βBmax=2βkCBav=2·0.179·1.148Bav=0.41Bav. The equivalent air gap isδe≈kCsδ=1.148·0.8 mm=0.918 mm.

At 0.9 T the peak value of the air gap field strength is Hˆ_δ = 0.9 T

4π·10⁻⁷V s/A m=716 kA/m.

The magnetic voltage of the air gap is

Uˆ_m,δe=Hˆ_δδe=716 kA/m·0.000 918 m=657 A.

According to Equation (2.15), the amplitude of the stator current linkage is Θˆs1= m

To calculate the amplitude, we need the fundamental winding factor for a machine with q=3. For a full-pitch winding withq=3, we have the electrical slot angle

αu= 360^◦

2mq = 360^◦

2·3·3 =20^◦.

We have six voltage phasors per pole pair: three positive and three negative phasors. When the phasors of negative coil sides are turned 180^◦, we have two phasors at an angle of−20^◦, two phasors at an angle of 0^◦ and two phasors at+20^◦. The fundamental winding factor will hence be

k_ws1=2 cos (−20^◦)+2 cos (0^◦)+2 cos (+20^◦)

6 =0.960.

Since this is a full-pitch winding the same result is found by calculating the distribution factor according to Equation (2.23)

kds1=kws1= sinqsαus

2 qssinαus

2

= sin 320 2 3 sin20

2

=0.960.

We may now calculate the stator current needed to magnetize the air gap Ism= Θˆs1πp

mk_ws1N_s√

2 = 657·π·2 3·0.960·100√

2A=10.1 A.

If an analytic equation is required to describe the flux distribution in case of an air gap slotted on only one side, an equivalent approximation introduced by Heller and Hamata (1977) can be employed. This equation yields a flux density distribution in the case of a smooth rotor in a distance of one slot. If the origin is set at the centre of the stator slot, the Heller and Hamata equations are written for a stator (see Figure 3.5)

B(α)= 1−β−βcos π 0.8α0

α

Bmax, when 0< α <0.8α0

and

B(α)=Bmaxelsewhere,when 0.8α0< α < αd. (3.13) Hereα0=2b₁/Dandαd=2π/Q_s=2τu/D.

Drops in the flux density caused by stator slots create losses on the rotor surface. Corre-spondingly, the rotor slots have the same effect on the stator surface. These losses can be reduced by partially or completely closing the slots, by reshaping the slot edges so that the drop in the flux density is eliminated, or by using a semi-magnetic slot wedge, Figure 3.6.

semi-magnetic slot wedge

Rotor surface along a stator slot pitch B

semi-magnetic slot wedge

special slot opening

no w edge

rotor stator slot

(a) (b) (c)

Figure 3.6 (a) Slot opening of a closed stator slot of Figure 3.4 has been filled with a semi-magnetic filling (µr=5). The flux drop on the rotor surface, caused by the slot, is remarkably reduced when the machine is running with a small current at no load. (b) Simultaneously, the losses on the rotor surface are reduced and the efficiency is improved. The edges of the stator slot can also be shaped according to (c), which yields the best flux density distribution. (b) The curve at the top