Windings with Various Numbers of Turns per Phasor
Boris Dotz1, Dieter Gerling2
1
2
FEAAM GmbH, Neubiberg, Germany, [email protected]
Universitaet der Bundeswehr Muenchen, Germany, [email protected]
Abstract— Although windings with various numbers of turns
per coil have been presented in several papers in the past, a
closed theory on applying different turn ratios in electrical
machine windings has not been published yet. Conditions under
which different turn ratios can be used to improve the MMF
spectrum of electrical machines are therefore discussed in this
paper. Analytical expressions for winding factors as well as
expressions for turn ratios are derived and several examples are
presented. It is shown that winding topologies presented in
earlier publications possess a similar structure and can therefore
be easily described by a zone factor or a group factor with two
distinct turn numbers.
A+
a)
A+
b)
A–
A–
Fig. 1. Exemplary stars of phasors a) unequal zones b) equal zones.
Keywords— Electrical machines; fractional slot concentrated
windings; winding factors; MMF harmonics
I.
∑
INTRODUCTION
The idea of using various numbers of turns per coil to
improve the MMF spectrum of electrical machines goes back
to at least 1935. In [1] Heller and Kauders discussed a
distributed winding named after Michael Surjaninoff. The
suggested winding topology utilizes two turn numbers,
whereby all odd/even slots have two coil sides with /
turns respectively and the turn ratio is given by ⁄ =
2⁄√3. As the benefit for distributed windings was rather low
compared to increased manufacturing costs, the winding did
not appear in later publications. During the last years fractional
slot concentrated windings (FSCW) received increasing
attention due to some advantages over distributed windings [2].
As stator teeth are often wound individually in FSCW, using
various numbers of turns per coils becomes a more practical
possibility. Cistelecan et al. presented a four-layer winding for
12 slots and 10 poles per base winding (12/10) showing that
different numbers of turns can improve the MMF spectrum [3].
In [4] a two-layer 12/10 winding with different numbers of
turns per coil side was discussed. For distributed concentric
windings a study of different numbers of turns per coil was
presented in [5]. It has been shown that the THD can be
minimized by applying different numbers of turns and that
therefore the MMF function can be improved. Despite above
publications, a detailed discussion on different numbers of
turns including winding factor analysis has not been published
yet. Therefore the aim of this research is threefold: First,
classical winding factor theory is used to discuss conditions
under which various numbers of turns can lead to an improved
MMF. Based on those findings, closed analytical expressions
for winding factors are derived. Second, it is shown that
already known windings can be divided into two groups: Those
utilizing the zone factor and others using the group factor.
b)
a)
∑
Fig. 2. Exemplary voltage phasor configuration with
and
turns per
coil for a) main spatial harmonic and b) parasitic spatial harmonic. The
sum denotes the geometric sum of all considered phasors.
Third, applying the developed theory new windings are
presented briefly. This paper is structured as follows: After
preliminary notes in section II, the main theory is presented in
section III. The fourth section discusses the zone factor and its
application to known and new windings. Section V derives the
group factor and gives examples on its application. Although
main examples focus on FSCW, the developed terms can be
easily applied to distributed windings as well.
II.
PRELIMINARY NOTES AND MAIN IDEA
The following research uses well-known theory of induced
voltages per coil side, per coil, or per coil group. The relative
phase shift of induced voltages with respect to each other can
be depicted using the star of phasors, which is sometimes
referred to as the star of slots. To apply the theory at its
broadest spectrum, latter term is omitted. The more general
term star of phasors is used instead, where each phasor can
represent the induced voltage either per coil side (single-layer
winding), per coil (double-layer winding), per coil pair (fourlayer winding), per coil group or per winding zone. A phase
winding can be divided into 2 zones, one + and one – zone
978-1-5090-4281-4/17/$31.00 ©2017 IEEE
for each phase. In general the zones do not need to be of equal
length, meaning that the + zone can have more or less phasors
than the – zone. Fig. 1 a) depicts a configuration where the
+ zone consists of two phasors while the – zone consists of 4
phasors. To calculate the distribution factor + and – zones of
one phase need to be considered. Doing so, the distribution
. In contrast, Fig. 1 b)
factor is referred to as the zone factor
depicts a configuration, where + and – zones are of equal
length. To calculate the distribution factor it is therefore
sufficient to consider only one group of three phasors.
Following this approach the distribution factor is referred to as
the group factor
. Please note that the configuration in Fig.
1 a) can also be described by groups of two phasors, where
+ and – zones consist of 1 and 2 groups respectively. Therefore
the most common description of the distribution factor is given
by [6]
,
=
,
⋅
,
,
(1)
where denotes the ordinal number of the spatial harmonic. To
comprehend the main idea of compensating MMF harmonics
Fig. 2 depicts an exemplary configuration of voltage phasors
for a given winding. As can be seen, resulting
and
phasors point in the same direction for the main working wave,
while they may point in different directions for a parasitic
spatial harmonic. Choosing a proper turn ratio
= ⁄
will then allow the geometrical sum of
and
phasors to be
of equal length leading to a zero induced voltage per winding
phase, i.e. leading to a zero winding factor for the considered
spatial harmonic.
III.
THEORY ON VARIOUS NUMBERS OF TURNS
All considerations in this research are performed for base
windings. A base winding is defined as the smallest number of
poles 2 so that the phasor diagram does not repeat. For
windings utilizing the first spatial harmonic = 1 as its
working harmonic the base winding includes two poles only.
The angle between two adjacent phasors is given by
= (2 )⁄ , where denotes the total number of slots. The
fictive commutator pitch allows for calculating the difference
of adjacent phasor positions and is given by
=
⋅
,
(2)
where is the smallest integer for which becomes an integer.
For distributed windings = 1 per base winding so = 1 .
The winding factor can be expressed by
,
=
∑
|
|
∑
⋅
,
(3)
denotes the number of conductors, the current
where
magnitude and direction for phasor . The symmetry axis is
given by the angle .
= arg(∑
⋅
)
With the number of slots per pole and phase being
the following theorem can be stated.
(4)
=
⁄
Theorem 1: Considering a symmetrical
phase winding
excited by symmetrical currents with a phase shift of 2 ⁄
each and neglecting those harmonics which cannot induce the
winding (line-line) in case of a star connection, the following
holds true: The winding under investigation has at most
different absolute winding factors.
Proof 1: Because the phasor distribution of the star of phasors
is the same after steps, there are at most different winding
factors ∈ 1, . For ∈ ℕ
, is given by = 2
(per base winding) and the ordinal number is given by
= +2
, where ∈ ℤ . So there are ⁄(2 ) =
winding factors left. For
∈ℕ , =
, = +
and again ⁄ = .
For a full proof utilizing symmetrical components please refer
to [7].
Theorem 2: To compensate a winding factor with
and
and
phasors of each
turns per phasor the distribution of
phase must have the same symmetry axis.
Proof 2: , given by (3) must equal zero for a given ordinal
number. Therefore ∑ cos(
) + ∑ sin(
) =
0. Consequently each term in square brackets must equal zero.
Assuming (without limitation) positive phasors and
distinguishing
between
and
leads
∑ cos(
) = 0 for the first
) + ∑ cos(
to
∑ sin(
) + ∑ sin(
) = 0 for the
and
second bracket. If sine or cosine sums are zero the symmetry
axis is either the real or the imaginary axis respectively. In
)⁄∑ cos(
)
and
general
= − ∑ cos(
)⁄∑ sin(
= − ∑ sin(
) , which leads to
)⁄∑ cos(
) = ∑ sin(
)
∑ sin(
)⁄∑ cos(
Finally tan( ) = tan( ) and
=
+ , ∈ ℤ.
This is the first main result of this work. It limits the number of
possible distributions of
and
phasors to the symmetrical
ones (e.g. Fig. 2). Please note that distributions providing a low
THD in [5] are symmetrical ones. It can be further shown that
for symmetrical distributions of
and
phasors
given
by (4) can be calculated using
=
,
(5)
counts the phasors under consideration. From
where
theorem 2, the following can be stated
Theorem 3: A winding factor can be set to zero if theorem 2
and N > 0 are satisfied.
Proof 3: Follows directly from theorem 2 and its proof. N >
0 simply states that the resulting phasors must point in opposite
directions (Fig. 2).
When more than two turn numbers shall be utilized to
compensate more than one winding factor, theorem 2 can be
generalized to
Theorem 4: Let denote the integer of different turn numbers
,
, …,
. To compensate ( − 1) different winding
factors, all
, ∈ 1;
phasors must have the same
symmetry axis.
Proof 4: Without limitation the proof is shown for = 3. Let
( ) denote the vector of the geometrical sum of
phasors
for ordinal number . To compensate a winding factor ,
A+ (North Pole)
a)
A– (South Pole)
A+
b)
A–
Fig. 3. Star of phasors for a 12/10 winding. a) Without b) with transposed
coils from north to south pole and various turn numbers.
( )+ ( )
+
the following must hold true
( )
=0. The same applies for
, it is ( ) +
,
( )
+ ( )
=0. Both equations can only be
satisfied when all ( ) are linearly dependent, i.e. point along
the same axis.
Again all resulting turn ratios must be positive similar to
theorem 3. In general it can be seen that various configurations
can be used to set distinct winding factors to zero. In the
following some examples are described in detail.
ZONE FACTOR WITH VARIOUS NUMBERS OF TURNS PER
GROUP
IV.
A. First Grade Windings
First grade windings are defined by
∈ℕ
[8].
Consequently the number of phasors per zone is equal for north
and south pole regions. Fig. 3 shows a 12/10 winding with two
phasors each pole (“pole” refers to + or – phasors respectively
which is in accordance with the Tingley Chart). To achieve a
symmetrical configuration according to theorem 3 one north
pole phasor needs to be transposed to the south pole region
∈ℕ
. For
∈ℕ
two phasors shall be
when
transposed from the north pole to the south pole. Afterwards
and
turns are assigned alternately around the star of
phasors. The zone factor is given by
,
=
1
1
cos
+
−
(
) cos(
)−
(6)
(
+ Γ) ,
+ Γ) cos(
where
/
counts the phasors with
/
turns
respectively. The first sum takes into account all north pole
phasors, while the second represents all south pole phasors.
The distance from the north to the south pole region is given by
Γ=
( −
+
)⁄2 [9]. Treating (6) as a geometric
series the zone factor can be simplified. Considering only odd
harmonics (6) results in
,
=
4
⋅ (1 +
sin
)
4
sin(να )
cos
+cos(
−
+2
4
−
4
(7)
−2
).
A four-layer winding is modelled as two double-layer windings
which are shifted with respect to each other. The winding
Fig. 4. Normalized MMF function and its reconstruction for
= = −0.5 p.u. and
= √3⁄2.
=
∙
factor is then given by
the pitch factor. The shifting factor
,
∙
is
⋅
=sin(
= 1 p.u.
, where
denotes
),
(8)
when the current direction is kept unchanged. ℎ counts the
number of phasors the second winding is shifted by. When the
current direction is reversed (7) becomes
,
⋅
=sin(
).
(9)
To get to the final four-layer winding, the second winding
needs to be shifted by ℎ = /2 steps and the current direction
needs to be reversed. Consequently, the total winding factor
results in
,
=
cos
⋅
sin
4
⋅ (1 +
4
sin(να )
)
−
+2
4
⋅
−
+ cos(
4
−2
) sin
=
To compensate the winding factor for
results in
( )=−
cos
cos
−
−
4
4
(10)
2
the turn ratio
−2
+2
.
(11)
Eq. (10–11) provide the second contribution of this research.
= 1,
= 3 and = 1 the turn
Setting = 2⁄5 with
= √3⁄2 . Thus, the winding presented in
ratio results in
[3] is fully described.
Fig. 4 shows a reconstruction of the MMF applying (10–
11). As the phasors represent coils, the resulting winding is a
four-layer winding with various numbers of turns per coil.
Assuming two single layer windings, each phasor will
represent a coil side. Applying the same procedure, the
resulting winding will be a two-layer winding with various
numbers of turns per coil side. While the pitch factor becomes
= sin( ∙ ⁄2) for the latter case (10–11) are still valid and
the winding presented in [4] is fully covered as well. For all
12/10 windings
= √3⁄2 provides minimal THD and the
cancellation of the first subharmonic (Fig. 5). THD and
winding factors are summarized in Table 1. The harmonic
distortion is calculated according to
Fig. 5. Normalized MMF spectrum for a 12 slots 10 or 14 poles fourlayer winding.
∑
∈ℕ
,
=
Fig. 7. Normalized MMF spectrum for an 18 slots 14 poles four-layer
winding.
,
,
.
,
(12)
A new winding can be designed by choosing = 3 ⁄ 7 with
= 1 and
= 5 (Fig. 6). The turn ratio will now allow to
compensate the winding factor either for
=1
= 1.3473) or for = 5 (
= 2.5321). Fig. 7 shows the
(
= 1. Besides cancelling
MMF for the first case and for
=
winding factors, the THD can be minimized setting
( =
1.9390 (Table 2). It can be shown from (7) that (
)) ⁄ (
) ≠ 0 and (
( = )) ⁄ (
) < 0 . So
the main winding factor increases for
> 1. As
and
are assigned alternately slot current loading is the same for all
slots. Additionally when ⁄ 2 ∈ ℕ
each stator tooth will
and one with
turns. So only one type
carry one coil with
of pre-wound teeth is needed (Fig. 8). Studies on four-layer
windings with equal turn numbers can be found in [10, 11].
…
C- A+
A- B+
B- B-
B+ C-
C+ A-
A+ A+
A- B+
B- C+
C- C-
A+ A+
A- A-
A+ C+
C- C-
C+ C+
C- B-
B+ B+
B- B-
B+ A+
1
2
3
4
5
6
7
8
9
/2
Fig. 8. Winding topology of an 18 slots 14 poles four-layer winding with
various numbers of turns per coil. Only the first half of the base winding is
depicted.
TABLE 2: : WINDING FACTORS FOR AN 18 SLOTS 14 POLES FOUR-LAYER
WINDING WITH VARIOUS TURN RATIOS
18/14 FL
,
,
,
THD
TABLE 1: WINDING FACTORS FOR A 10 SLOTS 12 OR 14 POLES FOUR-LAYER
WINDING WITH VARIOUS TURN RATIOS
12/10 FL
,
, ,
THD
=1
0.0173
0.9012
0.8959
= √3/2*
0
0.8966
0.8906
…
=1
0.0066
0.1041
0.8457
0.8220
= 1.94
0.0076
0.0276
0.8648
0.8010
= 1.35
0
0.0686
0.8556
0.8070
From Fig. 6 it can be seen, that five south pole and one north
pole phasors allow to form three symmetrical pairs. Let the
outer two south pole phasors represent
turns, the inner two
and the middle north and south pole phasor
turns. The
zone factor is then given by
* Cistelecan et al. [3]
To emphasize theorem 4 one more example on four-layer
FSCW shall be discussed briefly.
A+
A+ (North Pole)
a)
A– (South Pole)
b)
A–
Fig. 6. Star of phasors for an 18/14 winding. a) Without b) with transposed
coils from north to south pole and various turn numbers.
,
=
cos(0
cos(8
) + cos(10
) − cos(9
) − cos(7
) −
) + cos(11
) .
(13)
As = = 7 is the main working wave, it is possible to
cancel winding factors for = 1 and = 5 setting
=
= 1.2660. The main winding factor including
1.8794 and
the pich factor results in , = 0.8610. The MMF spectrum
is depicted in Fig. 9. It can be seen that all but slot harmonics
are cancelled. From above it can be stated that satisfying
theorem 3 and 4, first grade windings allow to compensate all
except the main winding factors by utilizing different turns
per coil in four-layer and per coil side in double-layer
configuration. Furthermore for full pitched distributed
windings different numbers of turns per coil can be applied
directly, as the pitch factor cancels even harmonics. It is
interesting to note that consequently double-layer full pitched
Fig. 9. Normalized MMF spectrum for an 18 slots 14 poles four-layer
winding with three different turn numbers.
Fig. 11. Normalized MMF spectrum for a 9 slots 8 poles double-layer
winding.
windings can theoretically perform similar to cage windings
with phases.
(14)
windings, all remaining harmonics are then increased. Similar
to section IV-A the THD can be minimized. The needed turn
= 1.0642 providing only marginal
ratio results in
improvement. Fig. 11 shows the MMF spectra of the doublelayer configuration with equal turns per coil and with
=
1.5321. Again the winding topology can be improved by using
a four-layer configuration. Utilizing (8–9) and (14) two
possibilities are then given to shape the MMF of the resulting
winding: the turn ratio
and the shifting distance ℎ. Setting
= 1 and h = 4 in (9) the winding presented in [3] can be
reproduced. With
= 1.5321 the first spatial subharmonic
is cancelled additionally. Discussed configurations are
summarized in Table 3. Because
and
phasors are not
assigned strictly alternately for second grade windings, slot
current loading is not equal for all slots, making it further
necessary to adjust slot width.
Eq. (13) shows that specific 12 result in a vanishing winding
factor. In contrast to first grade winding, this is even possible
for coil turns in two-layer configurations.
Please note that all windings discussed so far possess a
similar structure, which is given by various numbers of turns
per group and unequal zones. This is exactly the winding
structure named after Surjaninoff in [1].
B. Second Grade Windings
Second grade windings are defined by ∈ ℕ
[8]. Fig.
10 shows the star of phasors for a base FSCW with 9 Slots and
8 poles, = 3/8. It can be seen that the distribution of phase
A phasors is symmetrical to the vertical axis. Assigning north
pole phasors to
and south pole phasors to
allows to
cancel harmonics according to theorem 2 and theorem 3. Klima
derived a closed analytical term for the zone factor valid for
distributed and fractional slot windings with equal number of
turns [7]. With
for (A+) and
for (A–) the zone factor can
be rewritten to
,
=
(
(
( )=
)
(
)
(
)
(
(
/ )
(
/ )
/ )
)
.
(15)
As
= 3, there are three different winding factors present,
which besides the main winding factor are given by = 1,2.
Because the winding factor for the fourth and the fifth spatial
harmonic is the same, unbalanced radial pull cannot be avoided
completely. Nevertheless the influence of the first and the
second winding factor can be avoided by compensating either
the first or second winding factor. Opposing first grade
A+ (North Pole)
A– (South Pole)
Fig. 10. Star of phasors for a 9/8 winding. Phase A phasors (blue) are
symmetrical to the vertical axis.
TABLE 3: WINDING FACTORS FOR A 9 SLOTS 8 POLES WINDING TOPOLOGY
WITH VARIOUS TURN RATIOS
9/8
,
,
, ,
THD
Double-Layer
1
0.0607
0.1398
0.9452
1.0871
1.53
0
0.2156
0.9512
1.1190
1.06
0.0522
0.1504
0.9460
1.0862
Four-Layer ℎ = 4
0.71
1*
1.53
0
0.0357
0.0207
0.1386
0.0548
0.0899
0.9367
0.9266
0.9367
1.0306
1.0550
1.0252
* Cistelecan et al. [3]
V.
GROUP FACTOR WITH WITH VARIOUS NUMBERS OF
TURNS PER PHASOR
Fig. 12 a) depicts a base winding with = 18 slots and
= 3. Instead of zones, the illustrated winding can also be
composed of 6 groups with 3 phasors each. According to Fig. 1
and theorem 2, it follows that utilizing different turn ratios each
group can be designed to compensate winding factors. The
distribution of
and
phasors must now be symmetrical
within each group (Fig. 12 b). The group factor can be
A+ (North Pole)
a)
A– (South Pole)
A+
b)
A–
Fig. 12. Star of phasors for a base winding with = 18 and
= 3.
a) with equal turn numbers b) with two different turn numbers assigned
alternately within each group.
Fig. 14. Normalized MMF spectrum for a 18 slots 2 poles single-layer
distributed concentric winding.
calculated by
,
∑
=
(
) cos(
),
(16)
counts the number of phasors per group. When
where
and
are assigned alternately within each group and
∈ℕ
the group factor (16) reduces to
,
=
1
+1
+
2
−1
2
⋅
⋅
+1
+ sin
2
)
sin(
sin
−1
2
(17)
.
Again distinct spatial harmonics can be compensated by
( )=
(
(
)
)
.
(18)
Two examples shall now be discussed in detail. Setting =
18 , = ⁄ = 3⁄7 a double-layer FSCW is designed.
Assigning
and
according to Fig. 12 b) either the first or
the fifth spatial harmonic can be compensated. In contrast to
the four-layer case remaining parasitic harmonics are then
increased. Fig. 13 shows the normalized MMF spectrum for
= 1 and
= 2.8794. The THD is minimal for
=
are summarized in
0.945. Winding factors for different
Table 4. The second example on various turn numbers applying
the group factor considers a distributed single-layer concentric
winding with = 18, = ⁄ = 3⁄1 . This example has
Fig. 13. Normalized MMF spectrum for an 18 slots 14 poles double-layer
winding.
been discussed previously in [5]. The total winding factor is
given by
,
=
,
.
(19)
Again several spatial harmonics can be compensated by
choosing a turn ratio according to (18). Fig. 14 illustrates the
MMF spectrum where the winding factor for the seventh
harmonic is set to zero. Fig. 15 shows the THD over the turn
. It can be seen that although stated differently in [5] it
ratio
is indeed possible to improve total harmonic distortion. The
minimum THD is given at
= 1.0620. Harmonic distortion
can be improved by using a turn ratio of e.g. 20/19, a ratio not
considered in [5]. Please note that setting
= 1 all presented
terms reduce to the classical ones derived by Klima [6].
TABLE 4: WINDING FACTORS FOR AN 18 SLOTS 14 POLES WINDING TOPOLOGY
WITH VARIOUS TURN RATIOS
=1
0.0378
0.1359
0.9019
0.8823
18/14 DL
,
,
,
THD
VI.
= 2.879
0
0.3867
0.8914
1.0505
= 0.6527
0.0582
0
0.9076
0.9159
APPLICATION RELATED REMARKS
Regarding broad applications the authors do not think that a
winding generally not well suitable for a specific problem
becomes an attractive solution by employing various turn
numbers. Main challenges remain the more complex
manufacturing balanced off by small improvements on the
MMF spectrum. On the other side, once a winding is selected,
this study allows to easily check if further improvement is
for an 18 slots 2 poles distributed
Fig. 15. THD over turn ratio
single-layer winding.
numbers of turns per group and unequal zones or by
equal zones and various numbers of turns per phasor.
Fig. 16. Zone factors for an 18 slots 14 poles four-layer winding over turn
ratio
.
possible. Although very distinct turn ratios have been discussed
above, derived analytical equations are not limited to those
cases. In fact, several turn ratios can lead to an improved
performance. Fig. 16 shows the winding factors over turn ratio
for an 18 slots 14 poles winding (refer to Fig. 8). It can be seen
∈ 1; 1.841 leads to decreased parasitic
that any turn ratio
spatial harmonics
, , and to an increased main winding
factor
.
Likewise,
in case of machine problems arising
,
from specific spatial harmonics e.g. magnetic noise, this
research enables to quickly check if the problem can be dealt
with directly through adjusting turn numbers.
VII. CONCLUSIONS
This research presented a systematic approach on windings
with various turn numbers. The main results can be
summarized as follows
•
Applying
different turn numbers, − 1 winding
factors can be set to zero. Therefore all but the main
winding factor can be set to zero in the most general
case.
•
To allow for a cancellation of winding factors,
phasors with different turn numbers must have the
same symmetry axis.
•
Windings with various turn numbers known so far
possess a similar structure, which is given by various
•
This allows to easily find similar yet unpublished
windings. This includes fractional slot as well as
distributed windings for single-, double- or multilayer topologies.
•
Zone- and group factors for alternately assigned turn
numbers are derived. It is shown that turn numbers
allow for several optimization targets, i.e. cancellation
of specific winding factors or the minimization of the
total harmonic distortion.
REFERENCES
[1]
F. Heller and W. Kauders, “The Görges Polygon (Das Görgessche
Durchflutungspolygon),” Archiv für Elektrotechnik, vol. 29, no. 9, pp.
599–616, 1935.
[2] A. M. EL-Refaie, “Fractional-slot concentrated-windings synchronous
permanent magnet machines: Opportunities and challenges,” IEEE Trans.
Ind. Electron, vol. 57, no. 1, pp. 107–121, 2010.
[3] M. Cistelecan, F. J. T. E. Ferreira, and M. Popescu, “Three phase toothconcentrated multiple-layer fractional windings with low space harmonic
content,” IEEE Energy Conversion Congress and Exposition, pp. 1399–
1405, 2010.
[4] G. Dajaku, Wei Xie, and D. Gerling, “Reduction of low space harmonics
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