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The slot-leakage inductance of a single coil is given by an equation of the form

where N is the number of turns, Lstk is the stack length, and P1 and P2 are the permeance coefficients

for the two slots in which lie the two coilsides. The permeance coefficient for each slot permits the inductance to be calculated as though all N conductors linked the same flux. In practice they do not: conductors towards the bottom of the slot link more flux than those towards the top. Therefore in calculating the permeance coefficient the distribution of flux within the slot must be taken into account.

When the slot can be considered to be made up of sections, that are segments of circles or trapezoids, Fig. 3.11, analytical expressions for P can be derived if it is assumed that the flux crosses the slot in the

x-direction (that is, B = (Bx,0,0). The contribution )P of any section depends on the variation Bx(y) through the depth of the slot, and on the MMF of all sections lying below that section.

Consider an isolated section as in Fig. 3.11 (b) or (c) with an elementary flux tube dN. The MMF driving this flux element is JA(y), where J is the average current-density in the wound part of the slot: i.e., J =

NI/A_w, where A_w is the total wound area. Then dN = µ₀JA(y)L_stkdy/x = [µ₀L_stkNI/A_w] A(y)dy/x. The flux

dN is linked by the fraction A(y)/Aw of the total turns N, so the contribution dR to the total flux-linkage

is equal to dR = dN.N A(y)/Aw

= [µ

where dP is the contribution of the flux element to P.

The contribution )P of any whole section is obtained by integrating dP over the height h of the section. If there is another current-carrying section of area Ubelow the current section, then we must integrate

dP = (1/Aw2) [U + A(y)]2dy/x over h.

The feasibility of building up P in this way depends on the integrability of the expression for dP. Simple slots can be treated algebraically, using a few sections, but slots with more complicated shapes may need to be divided into a large number of layers, each of which is calculated with eqn. (3.23). As a simple example, )P is calculated for a slot bottom that is a circular segment spanning an angle 2$, Fig. 3.11(c). Since the section is at the bottom, U = 0 and dP = (1/Aw2) [A(y)]2dy/x. It is convenient to

integrate with respect to 2 rather than y, so we write y = r(1 – cos 2); dy = r sin 2 d2; and x = 2r sin 2.

A(y) is the sector area r2_{(22 – sin 22)/2, and A}

w is also given by this formula with 2 = $. Making all these

substitutions and performing the integration with respect to 2 from 0 to $, we get

)P ' $[4$2/3 % 1/2 % 2cos2$] & (5/4)sin2$

)P ' 2k A_w2 B 2 _ln w1 w₀ % h 2(w1%w0){ B % k 4(w 2 1 % w 2 0)} (3.22) )P ' 1 A_w2 U(U % a) % a2 3 h w. (3.23) )P ' 1 3 h w (3.24) )P ' h w₁ ! w₀ ln w₁ w₀ (3.26) )P ' 2h w₀ % w₁ (3.27) )P ' 4h 3w₁ % w₀ ; w1 < w0 (3.28)

When $ = B/2, the slot-bottom is semicircular and )P = 0.1424. With $ = B we get the "classical" value for the slot permeance coefficient of a round slot, 0.6231.

Now consider the trapezoidal section, Fig. 3.11 (b). The area A(y) is written in terms of x as k(x2_–w

02)

where k = h/2(w₁ – w₀) and x = w₀ + (w₁ – w₀)y/h, so that when dP is integrated with respect to x from

x = w0 to x = w1, we get the following expression (with B = U – kw02):

When the trapezium has parallel sides w₁ = w₀ = w, so k 6 4 and if a = hw, )P simplifies to

For a rectangular section at the bottom of the slot, U = 0; and if this is the only section there are no conductors above it, so Aw = a and

which is the well known formula for a rectangular slot. Another special case arises at the bottom of a slot if w0 = 0; then the section is triangular and

Empty sections: For a section that is empty of conductor we must integrate dP = (1/Aw2) U2dy/x over

h. For a trapezoidal section this gives

and if w1 and w0 are nearly equal this becomes

which is commonly quoted in textbooks. Veinott in his VICA-31 program for slot constants uses a modified form

in which w₁ is equal to the slot opening and w₀ is the width at the bottom of the slot wedge. By giving three times more weight tow1 than to w0, he increases the value of )P and makes an allowance for

fringing in a section of the slot where it is generally most significant. (See table 3.4).

Finally, the contribution of the slot opening region is given by eqn. (3.26) with w1 = w0 = w equal to the

slot opening, and h equal to the depth of the tooth-tip: i.e., )P = h/w if there is no conductor in the slot opening. If there is conductor in the slot-opening, eqn. (3.22) is used; it gives a slightly lower result.

Closed slots: For slots closed at the top there is no formula for )P that gives a finite result, because this theory assumes infinitely permeable iron. Closed slots, and even slots with significant saturation of the tooth tips, require a different treatment and their effective permeance depends on the slot current.

)P ' 1 A_w2

h3_w 1

Fig. 3.12 Typical finite-element flux plots. The permeance coefficients are a few percent higher than those calculated on the assumption that the flux crosses the slot in parallel tubes.

Comparison with finite-element calculations

The analysis assumes that the flux crosses the slot in the x-direction with no fringing. In practice fringing increases the permeance, and finite-element studies of all the standard example slots in PC-IMD indicate that the analytical P is typically 10% low. Fig. 3.12 shows a typical flux-plot from this study, and the table summarizes the results. The permeance coefficient is calculated from the expression

E/µ0I2, where E is the energy in J/m of axial length.

Bar type FE PC-IMD

1 1.98 1.8 2 2.01 1.88 3 2.36 2.2 4 2.95 2.73 5 2.41 2.3 6 2.72 2.57 7 4.44 3.63 8 3.78 3.41 9 2.03 1.85 10 2.57 2.3 Open custom 2.42 2.16 Rectangular slot 2.166 2.167 Table 3.4

Comparison with VICA-31

(Type 13 ER=0.5, DR=1.0, CR=0.455, BR=4.0, A1R=3.0)

VICA-31 FE PC-IMD

Slot opening region empty Slot opening fullPC-IMD Kostenko & PiotrovskyClassical formula

2.85 2.87 2.71 2.58 2.2 TABLE 3.5 Stator slots Rectangular Round-bottomed FE PC-IMD FE PC-IMD 2.03 2.03 1.97 2.02 TABLE 3.6 Deep-bar effect

For the deep-bar effect (skin effect in rotor conductors), PC-IMD has two alternative methods: one is the classical method for a rectangular slot, and the other is an integration of the complex diffusion equation throughout the slot, using a layered model developed for SPEED by Prof. I. Boldea. This is similar to the analysis above, except that the integration of contributions from the layers in the slots is complex, to account for the change in phase of the current density throughout the conductor.

2 _{If the amp-conductor distribution is C cos 2 then the MMF distribution is the space-integral !C sin 2. The axis of the MMF}

distribution is also the axis of the corresponding space vector.

3 _{PC-IMD's phasor diagram shows only the current I}

2N (labelled as I2).

Fig. 3.13 Sine-distributed ampere- conductor distribution

Fig. 3.14 Distribution of ampere-conductors.