Influence of the Stator Slot Opening on the Characteristics
of Windings with Concentrated Coils
D. Gerling
Institute of Electrical Drives, University of Federal Defense Munich
Werner-Heisenberg-Weg 39
85579 Neubiberg, Germany
Abstract-In this paper the influence of the stator slot opening
on the characteristics of windings with concentrated coils is
investigated. The analysis is based on the computation of the
fourier coefficients of the electric loading distribution. The
components of the working wave and the harmonics are evaluated
in dependency of the stator slot opening.
I.
will be ϕ0 =
m
( m being the number of phases), the
geometrical shift between two phases will be α 0 =
2π
pS ⋅ m
(in
mechanical degrees, pS being the number of stator pole pairs)
INTRODUCTION
Permanent Magnet Machines with concentrated windings
gain more and more attention because of some outstanding
advantages:
• The end windings are much shorter than for machines with
distributed windings, resulting in lower stator resistance,
less copper weight, and larger active length of the machine
for a given overall machine length.
• The winding manufacturing and machine assembly is
much easier, resulting in less costs.
• The maximum obtainable copper filling factor is higher
than for a distributed winding, resulting in better machine
performance.
• The danger of phase-to-phase short circuits is much lower.
Nevertheless, there is a main drawback of such a
concentrated winding: The harmonics of the electric loading
and magneto-motive force (MMF) distribution cannot be
controlled by typical means used with distributed windings:
• The concentrated winding is already a special kind of a
short-pitch winding, so that this degree of freedom cannot
be used to influence the harmonics.
• The distribution in more than one slot per pole per phase
(like used for distributed windings) is not possible,
because this contradicts the requirement to have a coil
winding around a single tooth. Of course, having coil
windings belonging to one phase on at least two
neighboring teeth is possible for the concentrated winding
(this will be called “slot per pole per phase”, even if it has
a different influence on the harmonics like the “slot per
pole per phase” for a distributed winding).
or β 0 =
2π
(in electrical degrees). It could be shown in [1],
m
that even for a different geometrical phase distribution a
constant torque can be achieved, but the most efficient version
is that one described before. Therefore, in this paper the phase
distribution is limited to the conventionally used multi-phase
winding.
The following figure 1 gives an impression, why the
symmetrical phase shift between the different phase currents
like described above is advantageous. In addition, it gives a
hint, that the phase shift for even number of phases is different
from that for odd number of phases: If the phase shift is
realized like shown in the following figure, the best situation
concerning symmetry is achieved.
2π
m
2π
2m
1
1
4’
2 3’
2’
3 2
3
3’
4 a)
2’
1’
1’
b)
Figure 1. Current phasors for multi-phase windings:
a) odd number of phases (example: m=3);
b) even number of phases (example: m=4).
II. NUMBER OF PHASES
III. STATOR WINDING
As three-phase machines are most relevant for practical
cases, the following analysis will be limited to odd phase
numbers greater than 1. The phase shift between two currents
978-1-4244-4252-2/09/$25.00 ©2009 IEEE
2π
A. MMF of Every Slot
There are Electrical Machines with concentrated windings
that do not use the fundamental MMF wave for torque
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production, because the amplitude of the fundamental MMF
wave is lower than the amplitude of a higher harmonic wave
(see e.g. [1]). Therefore, for these machines the “main” or
“working” MMF wave is different from the fundamental wave.
In the following such a machine with a two-layer
concentrated winding and odd number of phases (larger than 1)
is assumed. The nomenclature “two-layer winding” in the
context of a concentrated winding means, that every stator
tooth contains a coil (in contrary, “single-layer winding”
means that every other tooth contains a coil). In both cases
there are wires in every slot. It is quite easy to extend the
following investigation to single-layer windings and even
number of phases (just for the sake of clarity of reading, these
cases are not regarded in the following analysis). The current
in the k -th phase ( k = 1… m ) can be calculated as:
2π ⎞
⎛
(1)
I k = ˆI ⋅ sin ⎜ ωt − ( k − 1)
⎟
m⎠
⎝
As we are regarding a two-layer winding, the magnetomotive force in every slot consists of two parts: Two different
coils from neighbouring teeth are energized in every single
slot. In the following, the slots shall be numbered with n , so
that the slots of the first pole are numbered by
Z
(2)
n = 1… S
2pS
with ZS being the number of stator slots (equal to the number
of stator teeth). Slot number 1 shall be defined in the following
as that slot, where the first coil of the first phase is started. In
this slot we always have in addition the wires of the last coil of
the last phase, see the following figure 1 for the example of a
three-phase winding. Furthermore, the following figure 1
shows a winding with a number of slots per pole per phase of
q = 2 ; the orientation of such two directly neighboring coils of
a single phase is always opposite.
Figure 2. Exemplary winding topology; for the sake of simplicity
the original round stator lamination has been wound off into a straight version.
The magneto-motive force of every slot (number of ampereturns) of the first pole can be described as follows (assuming
the same number of turns N for each coil; the function int ( x )
means that the largest integer less than or equal to x is taken):
N ⋅ ( − I m + I1 )
if
n =1
⎧
⎪
⎛n⎞
⎛ n +1 ⎞ ⎞
∆Θ n = ⎨ ⎛
ZS
int ⎜ ⎟ −1
int ⎜
n −1
⎟ −1
if 1 < n ≤
⎪ N ⋅ ⎜⎜ Iint ⎛⎜ n ⎞⎟ ⋅ ( −1) ⎝ q ⎠ + Iint ⎛⎜ n +1 ⎞⎟ ⋅ ( −1) ⎝ q ⎠ ⎟⎟ ⋅ ( −1)
2 ⋅ pS
⎩ ⎝ ⎝q⎠
⎠
⎝ q ⎠
(3)
2
Because of the regarded symmetry, the magneto-motive
force of the slots of the remaining poles becomes:
∆Θ n = ∆Θ
⎛ n −1 ⎞
n − int ⎜
⎟⋅m⋅q
⎝ m⋅q ⎠
⋅ ( −1)
⎛ n −1 ⎞
int ⎜
⎟
⎝ m ⋅q ⎠
,
if
ZS
2 ⋅ pS
1
+ 1 ≤ n ≤ ZS (4)
For the exemplary winding topology that is regarded in this
paper ( m = 3 , q = 2 , pS = 2 , resulting in ZS = 2pS qm = 24 )
the following magneto-motive force of every slot can be
deduced (assumptions: peak current Î = 1A , number of turns
per coil N = 1 ):
∆Θn
0
1
2
0
5
10
15
20
n
Figure 3. Magneto-motive force of every slot versus slot number
for the exemplary winding topology for the time t = 0 .
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25
B. Electric Loading of Every Slot
The idea of the electric loading separates the field in the
slots of the electrical machine from the field in the air-gap [2]:
Representing the energized coils in the stator slots by an
electric loading on a smooth stator surface gives (nearly) the
same air-gap field like in reality, but simplifies the calculation
by far.1
To investigate the effect of the slot opening, this slot
opening will be regarded as a parameter in the following,
handling the slot opening as a percentage of the stator slot
pitch τS .
With
1 ≤ z1 ≤ 80
and
z1
s z1 =
⋅ τS ,
100
As the locations and amplitudes of every step of the electric
loading characteristic are known, a fourier analysis according
to [3] can be performed. The advantage of this method against
e.g. FFT-methods is the following:
• There is no limitation in the number of coefficients that
can be calculated.
• There is no requirement to the symmetry of the data
points.
10
5
(5)
τS =
A (α)
2π
0
(6)
ZS
the slot opening s z1 varies between 1% and 80% of the stator
5
slot pitch τS .
Having the electric loading as the number of ampere-turns in
every slot divided by the slot opening, the change of amplitude
of the electric loading on the circumference of the machine can
be calculated as follows (as the amplitude varies at the begin
and end of the slot opening, the control variable will be
1 ≤ n1 ≤ 2 ⋅ ZS ):
∆A ampln1
1
⎧ ∆Θ
⎪ int ⎛⎜ n1 ⎞⎟ +1 ⋅ s
z1
⎪
⎝ 2⎠
=⎨
⎪ −∆Θ ⋅ 1
n1
⎪⎩
2 s z1
⎛ n1 ⎞ ≠ n1
⎟
⎝2⎠ 2
if
int ⎜
if
⎛ n1 ⎞ n1
int ⎜ ⎟ =
⎝2⎠ 2
(7)
10
0.6
0.8
1
α
2π
Figure 4. Electric loading versus circumferential angle for 10% slot opening
(red, additional amplitude factor 0.1) and 80% slot opening (blue).
∆A anglen1
ai =
bi =
−1
∑ ∆A ampl
i ⋅ π n1=1
1
∑ ∆A ampl
i ⋅ π n1=1
⎛ n1 ⎞ = n1
⎟
⎝2⎠ 2
ϕ Ai = arctan ⎜
if
int ⎜
n1
2⋅ZS
Ai =
int ⎜
0.4
2⋅ZS
⎛ n1 ⎞ ≠ n1
⎟
⎝2⎠ 2
if
2
n1
(
)
(
)
⋅ sin i ⋅ ∆A anglen1
⋅ cos i ⋅ ∆A anglen1
(9)
2
a i + bi
⎛ ai ⎞
⎟
⎝ bi ⎠
(8)
Now the electric loading versus the circumferential angle
can be plotted. In the following figure 4 the electric loading
distribution for 10% and 80% slot opening (measured against
the stator slot pitch) is shown in red and blue, respectively. It
has to be emphasized that the amplitude of the characteristic
for 10% slot opening is divided by 10 in this figure, just to
have both characteristics shown in one graph (because of the
much smaller slot opening, the electric loading amplitude is
much higher at constant current).
1
0.2
If the ordinal number of the fourier coefficients is called “i”,
for example the first 1000 coefficients can be calculated as
follows:
1 ≤ i ≤ 1000 :
These variations take place at the following angles:
⎧⎛ int ⎛ n1 ⎞ + 1 ⎞ ⋅ τ − s z1
⎪⎪⎝⎜ ⎝⎜ 2 ⎠⎟ 2 ⎠⎟ S 2
=⎨
⎪⎛⎜ int ⎛⎜ n1 ⎞⎟ − 1 ⎞⎟ ⋅ τ + s z1
⎪⎩⎝ ⎝ 2 ⎠ 2 ⎠ S 2
0
There are two effects which are not covered by this method: The slot
leakage and the field deformation because of the stator slotting are
neglected.
The following figure 5 shows the amplitudes A i of the
electric loading versus ordinal number for closed slots and a
50% slot opening (measured against the stator slot pitch) in red
and blue, respectively.
The following can be clearly deduced:
• The fundamental wave does not have the highest
amplitude, i.e. the fundamental wave should not be chosen
as working wave.
• For closed slots (which is equivalent to the simple model
that the electric loading is concentrated in the centre of the
slot opening) there are always pairs of harmonics with
maximum amplitude (these are the so-called “slotharmonics”).
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•
For open slots (e.g. 50% slot opening, please refer to
figure 5) the harmonics are damped more and more with
increasing ordinal number.
These findings give the strong recommendation that the
wave with the lowest ordinal number of all “slot harmonics”
should be chosen as working wave.2
opening (measured against the stator slot pitch) the
characteristic is shown in red and blue, respectively.
10
5
8
A (α) 6
0
Amplitude 4
5
Ai 2
0
10
0
10
20
30
40
0
0.2
0.4
α
2π
0.6
0.8
1
Figure 6. Electric loading versus circumferential angle for 10% slot opening
(red, additional amplitude factor 0.1) and 80% slot opening (blue); calculated
from the first 1000 fourier coefficients.
50
ordinal number i Figure 5. Fourier coefficients of the electric loading versus ordinal number
(red: closed slots; blue: 50% slot opening).
1
C. Electric Loading and MMF Distribution
To check the calculation, the electric loading distribution can
be calculated from the fourier coefficients (as just a limited
number of coefficients is calculated, this will give an
approximation to the exact function). Considering the first
1000 coefficients the graph in figure 6 can be drawn from equ.
(10), which is in good agreement with figure 3:3
A (α) =
0.5
Θ (α) 0
1000
∑ ( a i ⋅ cos ( i ⋅ α ) + bi ⋅ sin ( i ⋅ α ) )
(10)
0.5
i =1
As the MMF distribution can be calculated from the
integration of the electric loading distribution (see e.g. [2]), this
MMF distribution can be calculated as follows:4
1000
1
⎛1
⎞
(11)
Θ ( α ) = ∑ ⎜ a i ⋅ sin ( i ⋅ α ) − b i ⋅ cos ( i ⋅ α ) ⎟
i
⎠
i =1 ⎝ i
The result for the MMF distribution is shown in figure 7
(please have in mind that figure 3 shows the magneto-motive
force of every slot ∆Θ and not the magneto-motive force
versus circumferential angle Θ ( α ) ). For 10% and 80% slot
2
It has to be mentioned that in reality the saturation of the tooth tips
will change these characteristics slightly, but the general effects
remain unchanged.
3
Please have in mind that figure 6 shows the electric loading of every
slot and figure 3 shows the magneto-motive force of every slot; these
characteristics differ in amplitude.
4
Again, per unit values are calculated, as the bore radius is assumed
being 1 in equ. (11).
1
0
0.2
0.4
α
2π
0.6
0.8
1
Figure 7. Magneto-motive force versus circumferential angle for 10% slot
opening (red) and 80% slot opening (blue); calculated from the first 1000
fourier coefficients.
D. The Effect of Stator Slot Opening
To judge the possibilities of harmonic reduction by means of
special design of the stator slot opening, the two main
harmonics (ordinal number 10 and 14, see figure 4) will be
regarded in the following.
The following figure 8 shows the 10th and 14th harmonic as a
function of stator slot opening in per unit values, i.e.
normalized to the value at closed slots (as the 10th and 14th
harmonic is a “slot harmonic” pair, the normalization value is
the same for both harmonics). It can be clearly seen that the
harmonic with the higher ordinal number is damped more
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intensely, like it is expected. To give an example: If the 14th
harmonic is damped by about 10%, the 10th harmonic (which
shall be used as working wave) is damped by about 5%.5
Another relevant finding is demonstrated in figure 9: It can
be seen that the relative damping of the 14th harmonic
(measured against the 10th harmonic) reveals practically the
same value like the per unit value of the 10th harmonic (which
is the working wave). In other words: The more relative
damping of the main troublesome harmonic (14th) is achieved,
the more relative damping of the working wave has to be
accepted.
1
0.95
per unit value 0.9
0.85
0.8
0
10
20
30
40
50
60
70
80
relative slot opening in % Figure 8. Per unit value of the 10th (red) and 14th (blue) harmonic versus
relative slot opening.
1
IV. CONCLUSION
It is well-known that for influencing the harmonics of the
MMF distribution, the distribution factor and the short-pitch
factor can be exploited. Having a concentrated winding, the
short-pitch factor is already disposed to realize this special
kind of winding. Making use of the distribution factor would
mean, distributing one coil side into more than one slot. This
contradicts the requirement of a concentrated winding. In
summary, two of the most relevant means of reducing MMF
harmonics cannot be used with concentrated windings.
Nevertheless, there are two other means that can be helpful:
skewing and slot opening. In this paper, the focus is on the slot
opening.
All means to influence the MMF harmonics mentioned
above are in some way proportional to the sinus of the
harmonic number [2]. For machines with distributed windings,
where the working wave is the fundamental one, this is quite
advantageous, because the harmonic waves can be damped.
For machines with concentrated windings, where the working
wave is not the fundamental one, the situation is much
different. The electric loading harmonics have the
characteristic, that always a pair (with just a small difference in
the harmonic number) has the same amplitude (these pairs are
called the “slot harmonics”). Only one harmonic of such a pair
can be selected to become the working wave, the other should
be damped as much as possible. The amount of damping can
be only relatively small, because the difference in the harmonic
number is small. It could be shown in this paper that it is
advantageous to select the smaller harmonic number from the
relevant “slot harmonic” pair to become the working wave,
because the higher one can be damped better. A calculation
method and an exemplary analysis are given in this paper.
0.95
REFERENCES
per unit value 0.9
[1]
0.85
[2]
[3]
0.8
0
10
20
30
40
50
60
70
80
relative slot opening in % Figure 9. Per unit value of the 10th harmonic (blue circles) and the ratio of 14th
to 10th harmonic (red line) versus relative slot opening.
5
This situation is much worse than known from distributed windings:
As here the working wave is the fundamental one, the working wave
is much less damped when influencing the harmonics.
2077
D. Gerling, “Analysis of the Magnetomotive Force of a Three-Phase
Winding with Concentrated Coils and Different Symmetry Features,”
18th International Conference on Electrical Machines and Systems
(ICEMS2008), Wuhan, China, October 17-20, 2008
D. Gerling, “Electrical Machines and Drives“, Lecture, Institute for
Electrical Drives and Actuators, University of Federal Defence Munich,
Germany (in German)
G. Koehler, A. Walther, “Fouriersche Analyse von Funktionen mit
Sprüngen, Ecken und ähnlichen Besonderheiten” Archiv für
Elektrotechnik, XXV. Band, 1931 (in German)