4.5. THE MMF HARMONICS FOR INTEGER q

Using the geometrical representation in Figure 4.5, it becomes fairly easy to decompose the resultant mmf in harmonics noticing the step-form of the distributions.

Proceeding with phase A we obtain (by some extrapolation for integer q),

F_A1 x,t n qI 2K K_{c q1 y1} cos xcos ₁t (4.9) with K_q1 sin /6/ qsin /6q 1;K _y1 sin (4.10)

K_q1 is known as the zone (or spread) factor and K_y1 the chording factor. For q = 1, Kq1 = 1 and for full pitch coils, y/τ = 1, K_y1 = 1, as expected.

To keep the winding fully symmetric y/τ≥ 2/3. This way all poles have a similar slot/phase allocation.

Assuming now that all coils per phase are in series, the number of turns per phase W₁ is

W₁ = 2p₁qn_c (4.11)

With (4.11), Equation (4.9) becomes

F_A1(x,t)= _π²_p1 W I 2K K_{1q1 y1} cos ^π_τ xcosω₁t (4.12)

For three phases we obtain

F x,t₁( ) = F_1m cos^^π_τ x − ω₁t^_ (4.13)

= 3W I 2K K^{1 q1 y1} (ampereturns per pole) (4.14)

with F_{1m πp}1

The derivative of pole mmf with respect to position x is called linear current density (or current sheet) A (in Amps/meter)

A x,t1( ) = ∂F x,t1∂(x ) = A1m sin−πτ x + ω1t (4.15)

= 3 2W I 2K K1 q1 y1 = π F1m (4.16)

A1m p1τ τ

A_1m is the maximum value of the current sheet and is also identified as current loading. The current loading is a design parameter (constant) A_1m≈ 5,000A/m to 50,000 A/m, in general, for induction machines in the power range of kilowatts to megawatts. It is limited by the temperature rise and increases with machine torque (size).

The harmonics content of the mmf is treated in a similar manner to obtain

F x,t( )= 3W I 2K1 _π qνKyν ⋅

_p1_ν



⋅KBI cosνπτ x − ω1t − ν −( 1)23π  − KBII cosνπτ x + ω1t − ν +( 1)23π



with		(4.17)
	Kqν = qsinsinνπνπ/6/6q ;K yν = sinνπ2τy 	(4.18)
	KBI = sin((ν −ν −1))ππ ;KBII = sin((ν +ν +1))ππ	(4.19)

3sin 1 /3 3sin 1 /3

Due to mmf full symmetry (with q = integer), only odd harmonics occur. For three-phase star connection, 3K harmonics may not occur as the current sum is zero and their phase shift angle is 3K⋅2π/3 = 2πK.

We are left with harmonics ν = 3K ± 1; that is ν = 5, 7, 11, 13, 17, … .

We should notice in (4.19) that for ν_d = 3K + 1, K_BI = 1 and K_BII = 0. The first term in (4.17) represents however a direct (forward) traveling wave as for a constant argument under cosinus, we do obtain

dxdt  = ω τπν1 = 2τνf1 ;ω =1 2πf1 (4.20)

On the contrary, for ν = 3K-1, K_BI = 0, and K_BII = 1. The second term in (4.17) represents a backward traveling wave. For a constant argument under cosinus, after a time derivative, we have

dxdt ν= − = − ω τπν1 = − τ2ν f1 (4.21)

3K 1

We should also notice that the traveling speed of mmf space harmonics, due to the placement of conductors in slots, is ν times smaller than that of the fundamental (ν = 1).

The space harmonics of the mmf just investigated are due both to the placement of conductors in slots and to the placement of various phases as phase belts under each pole. In our case the phase belts spread is π/3 (or one third of a pole). There are also two layer windings with 2π/3 phase belts but the π/3 (60⁰) phase belt windings are more practical.

So far the slot opening influences on the mmf stepwise distribution have not been considered. It will be discussed later in this chapter.

Notice that the product of zone (spread or distribution) factor K_qν and the chording factor K_yν is called the stator winding factor K_wν.

K _wν = K_qνK _yν (4.22)

As in most cases, only the mmf fundamental (ν = 1) is useful, reducing most harmonics and cancelling some is a good design attribute. Chording the coils to cancel K_yν leads to

sinνπ2τy  = 0; νπ2τy = n ;π yτ >32 (4.23)

As the mmf harmonic amplitude (4.17) is inversely proportional to the harmonic order, it is almost standard to reduce (cancel) the fifth harmonic (ν = 5) by making n = 2 in (4.23).

^y_τ = 5⁴ (4.23’) In reality, this ratio may not be realized with an integer q (q = 2) and thus y/τ = 5/6 or 7/9 is the practical solution which keeps the 5th mmf harmonic low. Chording the coils also reduces K_y1. For y/τ = 5/6, sin = 0.966 <1.0 but a 4% reduction in the mmf fundamental is worth the advantages of reducing the coil end connection length (lower copper losses) and a drastical reduction of 5^th mmf harmonic.

Mmf harmonics, as will be shown later in the book, produce parasitic torques, radial forces, additional core and winding losses, noise, and vibration.

Example 4.1.

Let us consider an induction machine with the following data: stator core diameter D = 0.1 m, number of stator slots N_s = 24, number of poles 2p₁ = 4, y/τ = 5/6, two-layer winding; slot area A_slot = 100 mm², total slot fill factor K_fill = 0.5, current density j_Co = 5 A/mm², number of turns per coil n_c = 25. Let us calculate

a.) The rated current (RMS value), wire gauge

b.) The pole pitch τ

c.) K_q1 and K_y1, K_w1

d.) The amplitude of the mmf F_1m and of the current sheet A_1m

a.) K_q7, K_y7 and F_7m (ν = 7)

Solution

Part of the slot is filled with insulation (conductor insulation, slot wall insulation, layer insulation) because there is some room between round wires. The total filling factor of a slot takes care of all these aspects. The mmf per slot is

2n I_c = A_slot ⋅K_fill ⋅J_Co =100⋅0.5 5⋅ = 250Aturns As n_c = 25; I = 250/(2⋅25) = 5A (RMS). The wire gauge d_Co is:

d_Co = _π⁴ JCo^I = _π^{4 5}5 =1.128 mm

The pole pitch τ is

m

From (4.10)

K_q1 K _y1 sin ⋅ = ; K_w1 = K K_{q1 y1} = 0.9659⋅0.966 = 0.933

The mmf fundamental amplitude, (from 4.14), is

W₁ = 2p₁qn_c = 2 2 2⋅ ⋅ ⋅25 = 200turns/ phase

F Aturns/ pole

From (4.16) the current sheet (loading) A_1m is,

A F .3Aturns/ m

From (4.18),

K_q7 = − ; K _y7 =sin

K_w7 = −0.2588⋅0.2588 = −0.066987

From (4.18),

F7m = 3W I 2K1 πp 71 q7K y7 = 3⋅200⋅5 2 0.π⋅2 7⋅⋅ 066987 = 6.445Aturns/ pole

This is less than 1% of the fundamental F_1m = 628Aturns/pole.

It may be shown that for 120⁰ phase belts [10], the distribution (spread) factor K_qν is

K_q (4.24)

For the same case q = 2 and ν = 1, we find K_q1 = sinπ/3 = 0.867. This is much smaller than 0.9659, the value obtained for the 60⁰ phase belt, which explains in part why the latter case is preferred in practice.

Now that we introduced ourselves to a.c. windings through two case studies, let us proceed and develop general rules to design practical a.c. windings.

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