Due to the switching effect of the inverter, the current fed into the stator winding is not an ideal sinusoidal waveform, but can be regarded as a sum of sinusoidal waveforms with different amplitudes and frequencies after Fourier analysis. The resistive losses for the k-th current harmonic Isk can be calculated by (6-12), with the resistance Rs at oper-ating temperature in °C, which is calculated by (6-13).
s, 0
R is the resistance (in ) measured at temperature 0 in °C.
2 Cu, DC, k 3 sk s
P I R (6-12)
0
0
s s,
0
1 235
R R
(6-13)
6. E-Machine Design
95
Rs is called DC resistance which is measured for a DC current feeding. If the current is alternating, additional eddy current will be induced due to the slot stray flux, leading to additional copper losses.
According to [52], the copper losses PCu,AC,k consist of the resistive losses PCu,DC,k calcu-lated by (6-12) and the additional eddy current losses due to skin and proximity effect PCu,add,pr and due to the circulating current PCu,add,circ, as described by
Cu, AC, k Cu, DC, k Cu, add, pr Cu, add, circ
P P P P . (6-14)
The skin and proximity effect refers to the unevenly distributed current density on the cross section of each single copper wire in the slot, regardless of the circuit connections of these wires. The effect depends on the slot shape, slot stray flux, wire shapes, etc.
The designed E-machine adopted round wires in trapezoidal slots (Fig. 6-8). To estimate this effect, a simplified assumption is made for multiple square wires in a rectangular slot as shown in Fig. 6-8. The equivalent square wire has the same cross section area as the round wires and the dimension is calculated by
2
T Cu
b 4d , (6-15)
where dCu is the round wire diameter.
Fig. 6-8 Designed (left) and equivalent (right) slot shape and wire section (bQm: average slot width)
6.3.2.1 Additional Losses due to Skin and Proximity Effect
The additional losses PCu,add,pr due to skin and proximity effect can be calculated by [52]
2
Cu, add, pr 3 sk s s , prk 1 Cu, DC, k s , prk 1
P I R k P k , (6-16)
with
dCu
bQm bQm
bT
96
Fe s , pr b s , pr
Fe b
k k
l k l
k l l
, (6-17)
T2
s , pr T T
1
k k 3 k
k m
, (6-18)
T
T
T T
T T
sinh 2 sin 2 cosh 2 cos 2
k k
k k
k k
, (6-19)
T
TT T
T T
sinh sin
2 cosh cos
k k
k k
k k
, (6-20)
Tk bT 0 Cu fsk aT b bT Qm
, (6-21)
where aT is the number of horizontally placed wires in one slot and mT is the total num-ber of vertically placed wires in one slot. They can be calculated by (6-22) and (6-23).
bQm is the average slot width. lFe is the length of the laminated iron package, lb is the length of the overhang winding. As stray flux at the winding overhang can be neglected, no current displacement occurs in this part. Thus a corrected coefficient ks , prk is used in (6-17) instead of ks , prk .
Qm fill T
T
b k
a b
(6-22)
c i
T
T
2 N a
m a
(6-23)
In (6-22), kfill is the slot filling factor, which is the ratio of total copper wires section area to the slot area. Nc is number of turns per coil.
The winding of the designed machine has Nc = 11 turns for each coil with ai = 4 paral-lel strands for each turn. Thus in each slot, 44 wires should be arranged for both upper and lower layer. The diameter of the round wire is dCu = 0.75 mm, thus the equivalent dimension of the assumed squre wire, calculated by (6-15), is bT = 0.665 mm. An aver-age slot width bQm = 5.2 mm is obtained from the stator geometry in Fig. 6-5. With the slot filling factor of kfill = 0.438, a feasible arrangement of the wires in one slot is:
aT = 5 and mT = 18 as shown in Fig. 6-9 a).
6.3.2.2 Additional Losses due to Circulating Current in Parallel Wires
The second type additional losses PCu,add,circ occurs when a coil contains several parallel strands. Depending on the positions of these parallel strands in one slot, the induced
6. E-Machine Design
97
a) b) c)
Fig. 6-9 Wire arrangements for the additional losses calculation: a) due to skin and proximity effect, b) due to circulating current: best case, c) due to circulating current:
worst case
voltage in each parallel strands differs and thus leads to a circulating current within these parallel strands. As in real case, the arrangement of the wires are irregular, the losses can be estimated by considering an average of the best case and worst case as shown in Fig. 6-9. As the voltage induced in one single wire varies depending on the vertical position in the slot, the best case is to arrange the parallel strands horizontally as much as possible, thus reducing the voltage potential between two parallel strands and therefore the circulating current. For the worst case, on the contrary, the parallel strands are prioritized to be placed vertically as shown in Fig. 6-9 c).
Concerning the arrangement of parallel strands, the best case is shown in Fig. 6-9 b): 9 turns have horizontally arranged parallel strands and 2 turns have vertically arranged parallel strands. The equivalent hL is (1 9 + 4 2) / 11 bT = 1.55 bT and mL = 18.
The worst case is shown in Fig. 6-9 c): 10 turns have vertically arranged parallel strands and 1 turn has horizontally arranged parallel strands. The equivalent hL is (4 10 + 1 1) / 11 bT = 3.73 bT and mL = 4.
The additional losses PCu,add,circ can be calculated by [52]
2
Cu, add, circ 3 sk s s , circk 1 Cu, DC, k s , circk 1
P I R k P k , (6-24)
with
hL
mL = 18 mL = 4
mT = 18
aT
hL
bT
98
s , circk k 1 k
k , (6-25)
Fe
L 0 Cu s T T Qm
Fe b
k k
h f l a b b
l l
, (6-26)
L 1
2
m
, (6-27)
where hL is the height of vertically placed wires of the parallels strands.
Table 6-6 Calculated copper losses in the winding for the current harmonics calculated in Chapter 6.4.1, winding resistance Rs = 0.06615 at temperature = 150 °C (f: harmonic frequency, ˆIsk: current amplitude, PCu,DC,k: DC resistive losses, PCu,AC,k: AC
resistive losses, PCu,add,pr: additional losses due to skin and proximity effect, PCu,add,circ: additional losses due to circulating current in the winding)
OP1: 12000 min-1 f [Hz] ˆIsk[A] PCu,add,pr/
PCu,DC,k
PCu,add,circ/ PCu,DC,k
PCu,DC,k
[W]
PCu,AC,k
[W]
PCu,AC,k/ PCu,DC,k
best case
worst
case average
400 69.1 1.01 1.02 1.03 1.03 473.15 488.55 1.03 10400 0.8 7.93 14.42 14.58 14.50 0.06 1.19 18.78
11200 1 9.01 16.52 15.80 16.16 0.10 2.10 21.17
12800 0.9 11.40 21.13 18.08 19.60 0.080 2.11 26.25 13600 0.6 12.70 23.63 19.12 21.38 0.036 1.03 28.92 22000 0.3 30.23 57.15 27.02 42.09 0.0089 0.55 62.21 23600 0.2 34.27 64.81 28.07 46.44 0.0040 0.28 69.52 24400 0.2 36.36 68.76 28.56 48.66 0.0040 0.29 73.27 26000 0.2 40.67 76.90 29.47 53.19 0.0040 0.32 80.97
- - - 473.45 496.42 1.05
OP2: 24000 min-1 f [Hz] ˆIsk[A] PCu,add,pr/
PCu,DC,k
PCu,add,circ/ PCu,DC,k
PCu,DC,k
[W]
PCu,AC,k
[W]
PCu,AC,k/ PCu,DC,k
best case
worst
case average
800 65.4 1.04 1.08 1.13 1.11 423.84 478.94 1.13
8800 0.9 5.98 10.67 11.97 11.32 0.08 1.15 14.32
10400 1.1 7.93 14.42 14.58 14.50 0.12 2.25 18.78 13600 0.8 12.70 23.63 19.12 21.38 0.063 1.83 28.92 15200 0.5 15.51 29.04 21.04 25.04 0.025 0.86 34.55 20000 0.3 25.47 48.09 25.56 36.82 0.0089 0.48 53.48 23200 0.2 33.24 62.86 27.82 45.34 0.0040 0.27 67.67 24800 0.2 37.42 70.77 28.79 49.78 0.0040 0.30 75.17 28000 0.2 46.30 87.46 30.51 58.99 0.0040 0.36 90.91
- - - 424.15 486.43 1.15
6. E-Machine Design
99
In case of short pitching, several slots are filled with windings in different phases. The induced voltages of the upper and lower winding in such slots have a phase shift. A cor-rection factor ks,ph is used to consider this effect. Thus (6-16) and (6-24) should be cor-rected by
2
Cu, add, pr Cu, DC, k s, ph s , prk 1
P P k k , (6-28)
2
Cu, add, circ Cu, DC, k s, ph s , circk 1
P P k k . (6-29)
The designed machine adopts a double-layer short pitching winding in 48 slots with w/s = 10/12. The winding arrangement leads to 24 slots filled with different phases.
Therefore, the correction factor ks,ph is 0.933.
The copper losses of the designed machine are calculated in Table 6-6 concerning the current harmonics calculated in Chapter 6.4.1. At the operation speed of 12000 min-1 and rated power, the current displacement effect causes 5 % additional losses. At the operation speed of 24000 min-1 and the rated power, 15 % additional losses will be caused.