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),
FA1 x,t n qI 2K Kc q1 y1 cos xcos 1t (4.9) with Kq1 sin /6/ qsin /6q 1;K y1 sin (4.10)
Kq1 is known as the zone (or spread) factor and Ky1 the chording factor. For q = 1, Kq1 = 1 and for full pitch coils, y/τ = 1, Ky1 = 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 = 2p1qnc (4.11)
With (4.11), Equation (4.9) becomes
FA1(x,t)= π2p1 W I 2K K1q1 y1 cos πτ xcosω1t (4.12)
For three phases we obtain
F x,t1( ) = F1m cosπτ x − ω1t (4.13)
= 3W I 2K K1 q1 y1 (ampereturns per pole) (4.14)
with F1m πp1
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
A1m is the maximum value of the current sheet and is also identified as current loading. The current loading is a design parameter (constant) A1m≈ 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 ⋅
p1ν
⋅KBI cosνπτ x − ω1t − ν −( 1)23π − KBII cosνπτ x + ω1t − ν +( 1)23π
with |
(4.17) |
|
Kqν = qsinsinνπνπ/6/6q ;K yν = sinνπ2τy |
(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 νd = 3K + 1, KBI = 1 and KBII = 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, KBI = 0, and KBII = 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 Kqν and the chording factor Kyν is called the stator winding factor Kwν.
K wν = KqνK 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 Kyν leads to
sinνπ2τy = 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).
yτ = 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 Ky1. 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 5th 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 Ns = 24, number of poles 2p1 = 4, y/τ = 5/6, two-layer winding; slot area Aslot = 100 mm2, total slot fill factor Kfill = 0.5, current density jCo = 5 A/mm2, number of turns per coil nc = 25. Let us calculate
a.) The rated current (RMS value), wire gauge
b.) The pole pitch τ
c.) Kq1 and Ky1, Kw1
d.) The amplitude of the mmf F1m and of the current sheet A1m
a.) Kq7, Ky7 and F7m (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 Ic = Aslot ⋅Kfill ⋅JCo =100⋅0.5 5⋅ = 250Aturns As nc = 25; I = 250/(2⋅25) = 5A (RMS). The wire gauge dCo is:
dCo = π4 JCoI = π4 55 =1.128 mm
The pole pitch τ is
From (4.10)
Kq1 K y1 sin ⋅ = ; Kw1 = K Kq1 y1 = 0.9659⋅0.966 = 0.933
The mmf fundamental amplitude, (from 4.14), is
W1 = 2p1qnc = 2 2 2⋅ ⋅ ⋅25 = 200turns/ phase
From (4.16) the current sheet (loading) A1m is,
From (4.18),
Kw7 = −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 F1m = 628Aturns/pole.
It may be shown that for 1200 phase belts [10], the distribution (spread) factor Kqν is
For the same case q = 2 and ν = 1, we find Kq1 = 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.