A General Design Procedure for Symmetrical Windings of Electrical Machines

A General Design Procedure for Symmetrical Windings of Electrical Machines Massimo Caruso 1,∗ ID , Antonino Oscar Di Tommaso 1 ID , Fabrizio Marignetti 2 ID , Rosario Miceli 1 ID and Giuseppe Ricco Galluzzo 1 ID 1 Department of Energy, Information Engineering and Mathematical Models (DEIM), University of Palermo, viale delle Scienze, Building nr. 9, 90128, Palermo, Italy. 2 Department of Electrical and Information Engineering (DIEI), University of Cassino and South Lazio, via G. Di Biasio, 43, 03043, Cassino, Italy. * Corresponding author: Antonino Oscar Di Tommaso, e-mail: antoninooscar.ditommaso@unipa.it Academic Editor: name Version January 25, 2018 submitted to

G the set of even numbers; U the set of odd numbers.

Introduction
Over the last century, several procedures for the design of the windings of electrical machines have been proposed in the literature.The first studies on the synthesis of windings date back to the first decade of the 20th century and can be attributed to E. Arnold, R. Richter and W. Kauders.The first proposed a systematic method based on the so-called star of slots (the original German term was Nutenstern) to solve design issues on windings of both AC and DC machines [1,2], whereas the second author improved these techniques particularly for AC machines [3].In the same period W. Kauders, who changed his name in W. Klima after 1945, developed a study of two and three-phase windings, also providing general formulas to determine the winding factors [4,5].In the 1950's Richter completed his researches and published a book concerning this topic [6].Thus, it can be stated that a wide part of the literature regarding the winding design was accomplished in Europe mainly by German researchers from late 1890 to early 1960's and many of these works were collected in a three-volumes book, written by the Austrian H. Sequenz [7][8][9].In the same time, French researchers, such as H. de Pistoye [10], followed by German-Czechoslovakians [5,11], Italians [12], Japanese [13] and American scholars, as M.
Liwschitz-Garik, Malti-Herzog, Hellmund and E. M. Tingley [14][15][16][17][18][19], also contributed to the significant development of this area.Other interesting manual and/or automated methods and algorithms were developed later [20][21][22][23][24].Each of these methods can be used for specific design constrains and none can be considered as universally valid.This is mainly due to the fact that, during the first seven decades of the 20th century, researchers were focused only on DC machines and single-phase, 2-phase and 3-phase AC machines.Nowadays, the interest is also directed towards multiphase machines, particularly with 5, 6 and 7-phases [21,22,[25][26][27].
The cited winding design methods provide relevant drawbacks and difficulties if they are applied for these new winding configurations.In this context, we propose a simple, powerful and general procedure suitable for all symmetrical winding configurations, covering also concentrated topologies and even reduced or normal (radial symmetrical) systems [22,28].Indeed, a hint of a similar method was fleetingly introduced by F. Heiles in [29]; however, he never developed it and later researchers, unexpectedly, did not give it an adequate consideration.The proposed method, namely WDT (Winding Distribution Table ), allows the design of all m-phase single or double-layer windings, providing fast and automated winding designs.Moreover, by manipulating the WDT with simple procedures, it is possible to implement optimization techniques, such as: chording (coil pitch shortening), zone widening (coil side shift in a slot), interspersing 1 (or imbrication) [3,6,9,20,30,31] and double chording [16] (consisting on the adoption of both chording and interspersing [9,20]).The procedure for the determination of the WDT and its use are hereafter presented and, in order to validate this procedure, the comparison with the traditional star of slots method is provided with several examples, highlighting the fact that the latter is much more complex to construct because of several geometrical issues.More in detail, this paper is structured as follows: the general conditions allowing the definition of the winding configuration are described in Section 2, the proposed WDT procedure is described in detail in Section 3 and the validation of the procedure through the description of several examples of symmetrical windings is provided in Section 4.

Symmetry Conditions for Winding Design
For an electrical machine with N slots, m phases, i layers and p pole pairs, the winding symmetry can be evaluated by using the factors γ, namely the number of wound coils per phase, and g, which are defined, respectively, as: where η represents the number of empty slots (dead-coil winding) in the case of single layer windings, and the number of unwound coils, in case of double layer windings and: is the greatest common divider between N and p, whereas i = 1, 2 is the number of layers.
The winding symmetry is assessed with the help of the following conditions: 1 i.e. transfer of one or more coils to another zone [28].
γ ∈ N for coil windings γ ∈ {N, N /2} for bar windings, and g ∈ N for symmetrical windings where N is the set of natural numbers.
If conditions (4) and the first of Eq. ( 5) are satisfied, the winding is symmetrical; otherwise, it is asymmetrical.In any case, for the computation of (1), the initial value of the empty slots η is assigned equal to zero.If (4) is not satisfied, then the value of η must be determined through formula where γ is the largest integer less than or equal to γ.In this way, considering empty slots or unwound coils, the first condition of winding symmetry expressed by ( 4) is always satisfied.It must be underlined that ( 6) is valid only if m > 1.
The assignment of η between the different phases can be arbitrary if η = m (even if the distribution of the empty slots should be as uniform as possible among the phase sections).If η = m, the empty slots should be chosen one for each phase (uniformly distributed), so that the corresponding slot EMF phasors are phase-shifted by 2π /m = 360°/ m in case of normal systems and by π /m = 180°/ m in case of reduced systems.
An exception is valid if m = 1 (single phase windings), for which the following equation has to be considered in place of ( 6): where a winding covering 2 /3 of the slots is considered.
The number of wound slots per pole per phase is determined as where a = q and z /i•p is a proper fraction.If q is an integer number, then the winding is called "integral slot winding".Otherwise, if q is a fractional number, the winding is defined as "fractional slot winding".
Moreover, in the cases where η > 0, it is advantageous to define the number of slots per pole per phase given by If η = 0, then Q = q.
In order to recognize basic winding configurations when q is a fractional number and η = 0, the following equation must be taken into account (considering (8)): which gives the numerator of the proper fraction contained in (8).In this equation the case i = 2 can be referred both to double-layer windings and to single-layer bar windings when γ ∈ N /2.In the last case, with η = 0, i = which meets condition (1) for double layer windings.Furthermore, the factor t must be defined for fractional slot windings: If t = 1, the winding is a basic one.Otherwise, the winding is composed by t repetitions of the same basic winding, which is composed by N = N /t slots and p = p /t pole pairs.If η > 0 then it is sometimes convenient to set t = 1 in order to avoid the case η > m (a case in which there would be too much unwound slots or coils).
For integer slot windings (q ∈ N and η = 0) the number of repetitions is equal to t = p, because N and p have the same divisor equal to p itself as can be deduced by (8).

WDT Procedure
The proposed winding distribution procedure is based on the construction of a Winding Distribution Table (WDT), which allows the assignment of a specific stator slot to a winding phase section [6,23].Hereafter the basic rules to determine the WDT are defined.

Determination of WDT
This table, containing N elements, is composed by a number of rows equal to m and a number of columns equal to whose elements are ordered as shown in Table 1.
In the case of a single-layer winding, the WDT elements are referred to phasors associated to each slot or, in other words, to each coil side: a single-layer machine winding is composed by N = 2m • γ coils.In the double-layer winding configuration, the WDT elements are referred to a coil side, whereas the second coil side is defined by the coil pitch y c usually expressed in terms of number of slots.In this case, the machine winding is composed by N = m • γ coils.
The definition of the WDT starts by assigning the slot number 1 to the first cell of Table 1.Then, the slot number 2 is assigned to the cell whose distance is equal to p cells from the first one, the slot number 3 to the cell whose distance is equal to p cells from the second one, following the orientation of the progressive numbering of the cells of Table In order to clarify this procedure, Table 2 reports the WDT referred to a simple integral slot symmetrical winding with N = 24, p = 2, m = 3 and q = 2.The Table is filled in p = 2 filling cycles, each one corresponding to N /p = 12 assignments.In fact, by starting from placing slot nr. 1 in the first cell (row 1, col.1), the slot nr. 2 is placed after p = 2 cells (row 1, col.3), till slot nr. 12 is assigned to the 23-th cell (row 3, col.7), which corresponds to the N /p-th assignment and the first filling cycle is completed.The second cycle starts by counting p elements from cell nr.23, which corresponds to element nr.25.However, due to the periodicity of this winding configuration (N = 24), the element nr. 25 corresponds to the first element.Therefore, the slot nr. 13 will be placed in the adjacent empty second cell (row 1, col. 2).The vertical double bar separates the active conductors with positive EMF from those with negative EMF.In symmetrical windings, the double bar should be located around the middle of the table in order to gain the highest EMF.The star of slots corresponding to this configuration is plotted in Figure 1.The positive EMFs are reported in a thicker width with respect to the negative ones.It can be noticed, by comparing the rows of Table 2 with Figure 1, that the WDT is the tabular and linear representation of the star of slots, therefore there is virtually no need to use the star of slots.Anyway, hereinafter the stars of slots will be represented only for more clarity.The parameter t given in (3) determines the number of superposing EMF stars related to the specific winding configuration.Therefore, in the WDT, the slots associated to the cells from kt + 1 to (k + 1)t (with k = 0, 1, 2, ..., N /t − 1) will have the same phase-angle, by following the WDT path previously defined and shown in Table 2.In particular, each EMF star is composed by N/t phasors, phase-shifted to each other by an electrical angle α equal to By following these considerations it is possible to reconfigure Table 1 and to build Table 3 as used in [3,6], in which the electrical angles ϕ k of each slot-phasor pair are computed by the following formula: Table 3. Electrical angles of the star of slots (j = 1, ..., t).
With reference to the previous example, it can be noticed that each EMF star is composed by N/t = 24/2 = 12 phasors, phase-shifted to each other by an electrical angle α equal to Therefore, t = 2 represents the number of phasors having equal phases.In this example (see Table 2), the pair of phasors 1-13, 2-14, 3-15, 4-16... have the same phase-shift.Moreover, the consecutive pairs are shifted between each other by an angle equal to 30°, as shown in Fig. 1 (i.e., the pair 2-14 is phase-shifted by an angle of 30°with the pair 1-13).Thus, the phase-shifts of all phasors can be easily computed by considering Table 3.The resulting table is shown in Table 4.In truth, by considering that q = 2, the same WDT can be obtained considering the number of repetitions t as defined by formula (11).In fact, this winding can be seen as composed by t = 2 repetitions of the same winding characterized by N = N /t = 24 /2 = 12 slots, m = 3 phases and p = p /t = 2 /2 = 1 pole pair (γ = γ /t = 4) and, therefore, the WDT can be seen as the composition of two basic WDTs as shown in Table 5.In this case the second WDT is created with the same rules above exposed but starting from slot N /2 + 1 till slot N. Now, by considering the two repetitions, this winding could be divided in two parallel connected branches.A comparison of Tables 2 and 5 shows their equivalence.(phase 1: blue, phase 2: green, phase 3: red) It will be demonstrated in the following sections that simple computations allow the determination of the winding distribution even for complex configurations.

WDT Post-processing
After the WDT is created, the winding factors of the ν − th harmonic and the related proper phases for each h − th phase can be computed through the following equations: and where: a h,n is a generic element of the WDT, n = 1, 2, ..., N /m is the n-th column of the WDT and h = 1...m is the h-th phase of the related winding.If the winding is symmetrical, the computation of harmonic winding factors can be limited to only one phase (for example the 1-st phase, i.e. h = 1).When dealing with double layer windings, also the negative coil sides (defined by their coil pitch y c ) have to be inserted in the WDT.

WDT for Symmetrical Non-reduced and Normal Systems
In the case of radially symmetrical or normal systems, in order to visualize, if required, all the positive and the negative phasors of the same phase in the same row, an adequate number of the last WDT columns (these group of columns are indicated in Figure 2 as "right side") must all be shifted in the upper direction by ζ shifts (see Figure 2) and their sign must be changed, where The number of columns to be shifted depends on the type of winding.In the case of a single layer symmetrical winding, N /2m = γ columns must be shifted, due to the fact that the number of positive slots must be equal to the negative ones and the WDT will be divided into two equal left and right sides.In the case of a single layer bar winding, it may happens that γ be a fraction with the denominator equal to 2. Therefore, γ ± 1 = N /2m ± 1 columns should be shifted.For the double-layer configuration, this restriction is not necessary.Therefore, the number of positive and negative coils can be chosen properly, depending on the type of winding design optimization (maximization of the EMF, reduction of the harmonic content, simplicity of realization of end windings, zone widening, etc.).In addition, in any case, a minus sign is added to the shifted columns, so that the coil sides with the negative EMF can be explicitly visualized within the WDT.It clearly appears that the groups of adjacent elements with the same sign located in each row of the WDT represent the winding zones of each phase.It should be noted that, in case of bar or double-layer windings, the number of columns of the "left" and "right" side are, in general, given by N /m and N /m , respectively, or vice versa (see Figure 2).

WDT for Reduced Systems
For reduced systems the WDT is not affected by vertical shifting, but the quadrants named 1 and 3 in Figure 3 are simply swapped.In this case the WDT is valid for all windings with m ∈ G (for example in two-, four-or six-phase reduced system windings).The sign of the slot numbers in Quadrants 1 and 4 are then changed to account for negative coil sides.It should be noted that, in case of bar or double-layer windings, the number of columns of Quadrants 1 and 4 is, in general, given by N /m , whereas the number of columns of Quadrants 2 and 3 by N /m or vice versa.After this operation the rows of the WDT should be reordered following the procedure of Table 6.Reduced system whose number of phases is not a power of 2 are composed by m g groups of m u -phase systems shifted by an angle of where is the greatest prime factor of m, and For example a reduced system winding with m = 10 phases can be decomposed into m g = 2 groups of m u = 5-phase systems; a reduced system winding with m = 12 phases can be decomposed into m g = 4 groups of m u = 3-phase systems.The even groups of a reduced systen must then be  multiplied by −1 in order to make them radially symmetrical.The procedure will be cleared in Section 4.

WDT for Single-phase Systems
For m = 1, the WDT is reduced to a single row composed by four sectors, as shown in Figure 4: the first sector is composed by (N−η) /2 cells, the second by η /2 (or η /2 ) cells (representing empty slots of the first unwound part), the third by (N−η) /2 cells and the last one by η /2 (or η /2 ) cells (representing empty slots of the second unwound part).Another simple possible procedure to solve single phase windings is to consider them as derived from a three-phase winding, but excluding one phase (therefore, the "starting phase" can be located in the unwound slots).An example of this approach will be shown and discussed in the last example of Section 4.

Examples and Procedure Validation
In this Section several examples of winding configurations for electrical machines are reported in order to validate the proposed procedure.These examples can be adequate for the comprehension of the modality of construction of the WDT for each of the proposed electrical machine.Other more complex examples for multiphase windings are also considered and analyzed in the following subsections.The construction of the EMF stars and the representation in plane of the related winding configuration are used as references for the procedure validation.

3-phase Windings
An electrical machine composed by 24 stator slots and 2 pole pairs, equipped with a three-phase winding with q = 2, is used as a first validation test, which is symmetrical because γ ∈ N and g ∈ N.
The corresponding WDT has been already determined in Subsection 3.1.The data needed for the WDT construction and its related properties can be summarized as follows: Therefore, the corresponding WDT is composed by m = 3 rows and N /m = 8 columns, with a cell pitch equal to p = 2, as shown in for each phase and by the superposition of two stars of coil EMFs (t = 2), where each one contains twelve phasors equally phase-shifted between each others by 30°(α = 30°).The EMFs corresponding to the slots 1, 2, 3, ..., 12 will have the same phase angle of the EMFs located in the slots 13, 14, 15, ..., 24, respectively.Therefore, the elements of the Table can be regrouped in 12 groups composed by t = 2 elements and by following the order of the Table, the linear representation of the star of slots can be detected, as it can be seen by comparing Table 2 with Figure 1.
In Table 2, the phasors (or slots) of phase 1 are written in blue color, while the phasors of phase 2 and 3 are reported in green and red colors, respectively.In order to visualize the phasors of the same phase in a single WDT row, the last 4 columns ( N /2m = 4) of the WDT should be circularly shifted in the upper direction by a quantity equal to ζ = 1 as calculated by applying (21) and that can be deduced by analyzing Table 2.The winding scheme is visualized in Figure 5a.
As previously mentioned, the WDT can be referred to both single-layer and double-layer windings.
Indeed, for double layer configuration, the position of the left side of a coil is defined by the number corresponding to the slot, whereas the position of right side is defined by the coil pitch y c , which is equal, in most cases, to 5 /6 • y p (a shortened pitch employed for the limitation of both the 5-th and 7-th harmonics), where y p = N /2p is the pole pitch expressed in terms of number of slots.An example of double-layer winding scheme is plotted in Figure 5b, where y c = 5.Winding with 1 path.Tables 7 and 8 show the WDT referred to a three-phase winding with N = 27, p = 3, for a single-layer and a double-layer configuration, respectively.It can be noticed that, in case of single-layer coil winding, three empty slots are required (η = 3) in order to satisfy the symmetry conditions.The corresponding unwound slots are highlighted by the crossed elements of Table 7.The winding is therefore composed by the same number of positive and negative slots.Here Q = 1 + 1 /2 and q = 2.
The corresponding star of slots achieved from the two WDT Table and the related winding scheme are plotted in Figures 6a and 6b
If it is necessary to fill also the empty slots only a bar winding can be here considered as shown in Figures 7a and 7b.In fact, only in case of single layer bar windings, γ can be a fractional number with denominator equal to 2.

(a)
Winding with 1 path.If necessary, this winding could be divided in three parallel connected paths: then, each path will be identifyed by the three WDTs shown in Table 8. Figure 8b refers to a series connection of the three paths.

(a)
Winding with 1 path.The WDT method is also suitable to design fractional single and double layer concentrated windings.Figures 9a and 9b show, for example, a single layer concentrated winding with N = 24, m = 3, p = 11, γ = N /m = 24 /3 = 8 and q = 4 /11.The related WDT is represented in Table 9.

6-phase Windings
An example of WDT applied to multiphase winding configurations is reported in Table 10, which shows the WDT for a six-phase, non-reduced winding with N = 36, p = 5, γ = 3 and q = 3 /5, whose corresponding star of slot is shown in Figure 10a.
By applying now the swapping procedure to Table 10 (without shifting) and the phases reordering following Table 6 described in Subsection 3.4, and by changing the sign of rows 3 and 4 in order to achieve a radial symmetry (as general rule the WDT rows of the even numbered phase groups are multiplied by −1), it is possible to determine the WDT referred to the reduced winding configuration, shown in Table 11.The star of slots referred to this type of winding is plotted in Figure 10b.It can be noticed that this reduced configuration is composed by two groups (m g = 2) of three-phase winding sections (m u = 3), phase-shifted between each other by an angle corresponding to 30°: the first one is composed by phases 1-5-3, whereas the second group by phases 2-6-4.Tables 10 and 11 can be now used to determine the winding configuration: here a single layer configuration has been chosen for the non reduced system winding, whereas a double layer one with coil pitch y c = 3 has been chosen for the reduced system.The winding diagrams are shown in Figure 11a and in Figure 11b, respectively.In the case of the double layer winding, Table 11 shows only one side of the coils, i.e. the positive one, whereas the position of the other (the negative coil sides) depends on the chosen value of the coil pitch y c .Winding with 1 path.Winding with 1 path.

2-phase Winding
A reduced 2-phase double-layer winding is here considered, with N = 28, p = 1, y c = 12.The WDT is represented in Table 12.
After swapping Quadrant 1 and 3 and changing the signs of Quadrants 1 and 4 the WDT becomes as shown in Table 13. Figure 12a represents the related star of slots and Figure 12b shows the double-layer winding scheme.(ph1: blue, ph2: green) Table 13.WDT for a single-layer, 2-phase winding with N = 28 and p = 1, after swapping.

7-phase Winding
Another example consists in a seven-phase machine with 42 slots and 2 pole pairs.The WDT parameters can be easily determined by taking into account the equations described in Section 2.
Table 14 reports the WDT for this case of study.It is a 7x6 (rows x columns) table with a cell pitch equal to 2, with six coils composing a winding phase section and two stars of 21 EMF phasors, as shown in the diagram of Figure 13.By virtue of ( 11) the WDT can be divided into t = 2 basic tables as shown in Table 15.The related double-layer winding scheme is depicted in Figure 14 with a chosen shortened coil pitch y c = 9.

Procedures for Winding Optimization
The proposed procedure is also suitable for the implementation of winding optimization techniques, such as interspersing for single-layer windings and double chording for double layer windings.For instance, the WDT referred to a three-phase machine with p = 5 and a double-layer winding located into 72 slots is reported in Table 15, whereas the corresponding star of slots and winding scheme are plotted in Figures 15 and 16, respectively.
The double chording with y c = 6 and a three-order single-sided imbrication is obtained by circularly shifting upwards by two cells and by changing the sign of the first 2 • j elements of the left side of the WDT and the first j elements of the right side of the WDT (in this case, the elements of columns nr. 2, 4 and 6 for the left side and the elements of columns nr.Winding with 1 path.both circularly shifted and changed in sign), according to Table 16.Indeed, the corresponding star of slots is plotted in Figure 17, whereas the modified winding scheme is plotted in Figure 18.
Moreover, by shifting the WDT elements in accordance to Table 17, it is possible to implement a double chording technique with y c = 6 and a double-sided imbrication with order equal to 3.This case differs from the previous one (single-sided imbrication) by the fact that the 2 • j columns are circularly shifted for each side of the WDT, as shown by the arrows of Table 17.The corresponding star of slots and winding scheme are depicted in Figures 19 and 20, respectively.
A triple chording can be performed in order to further reduce the winding harmonic content with the following procedures whose results are given in Table 18: 1. the WDT is developed; 2. the zone widening by two slots is performed by shifting the double bar from the middle position to the right by two cells;  Winding with 1 path.The imbrication procedure can now be generalized.
1.For normal or non-reduced system windings the columns of positive slots that are to be shifted are shifted upwards by   Winding with 1 path.cells, whereas the columns of negative slots must be shifted upwards by 2. For reduced system windings both the positive and the negative columns are to be shifted always upwards by 1 cell, i.e. s pos = 1 and s neg = 1 (27) The sign of the cells of the last row must be then changed.
To clarify the procedure to carry out an imbrication in a reduced winding the following example is presented.Here a 6-phase reduced winding with N = 96, p = 5, q = 1 + 2 /10 is considered.
After swapping Quadrants 1 and 3 of Table 23 and reordering it following Table 6 we obtain Table 24.
Now, by shifting upwards both columns 2 and 8 by one cell (s pos = 1 and s neg = 1), and by changing the signs of the cells in the last row of the shifted columns (here row nr. 6) we obtain, finally, Table 25.Here again, in order to realize a radial symmetrical system, the even phase groups must be multiplied by −1 (i.e rows 3 and 4 in Table 25).
The star of slots of the winding configuration without imbrication is shown in Figure 23a, whereas the final configuration is shown in Figure 23a.N = 96, p = 5, m = 6 and q = 1 + 2 /10.

g = iN 2mt for normal and non−reduced systems 2 N
2mt for reduced systems,

Figure 1 .
Figure 1.Plot of the EMF star for a symmetrical winding with N = 24, m = 3 and p = 2.

Figure 3 .
Figure 3. Swapping of the WDT quadrants in case of reduced systems and m ∈ G.

Figure 4 .
Figure 4.A possible WDT for single phase windings.

Figures
Figures 8a and 8b refer to the double layer configuration with a coil pitch y c = 4, which is composed by 3 repetitions of the basic winding with N = N /t = 27 /3 = 9 slots, m = 3 phases and p = p /t = 3 /3 = 1 pole pair.The WDT is composed by the repetition of 3 basic WDTs: the first is

Figure 15 .Figure 16 .
Figure 15.Plot of the EMF star for a three-phase symmetrical winding with N = 72, p = 5 and m = 3.

Figure 23 .
Figure 23.(a) Star of slots of a 6-phase reduced single-layer winding with N = 96, p = 5 and (b) related winding scheme with 1-st order imbrication.

Table 1 .
Order of the WDT elements.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 January 2018 doi:10.20944/preprints201801.0242.v1
1.After N/p (if N/p ∈ N) assignments, the procedure continues by imagining that the last cell of the table is ideally consecutive to the first: therefore, once the N-th table element is counted, the counting continues restarting from the first cell of the table.If the count ends in a filled cell, the adjacent empty cell will be filled with the counted slot.The procedure continues until the table is completely filled by the slot numbers.

Table 2 .
Winding Distribution Table for a winding with N = 24, p = 2 and m = 3.

Table 4 .
Electrical angles of the EMF star phasors for N = 24, p = 2 and m = 3.

Table 5 .
Winding Distribution Table with t = 2 repetitions, for a winding with N = 24, p = 2 and m = 3.

Table 6 .
Reordering of the rows of a WDT for an m-phase reduced system winging with m ∈ G.

Table 2 .
This winding is composed by γ = 8 coils per phase, which means m • γ = 3 • 8 = 24 overall coils.The EMF of each coil is represented by a phasor in the star of slots, as shown in Figure 1, which is, therefore, composed by 8 EMF phasors Preprints (www.

Table 14 .
Winding Distribution Table for a double layer winding with N = 42, p = 2 and m = 7

Table 15 .
Winding Distribution Table for a winding with N = 72, p = 5 and m = 3

Table 18 .
Triple chording with a 3-rd order double-sided imbrication and zone widening by 2 coils for N = 72, p = 5 and m = 3

Table 19 .
Winding factor harmonics with different optimization techniques for N = 72, p = 5 and m = 3 3. the double sided imbrication is made by shifting upwards by two cells the 2-nd, the 4-th and the 6-th negative columns.Finally, Table19summarizes the winding factor harmonics for each of the proposed optimization technique.It can be noticed that the chording allows a significant reduction of both the fifth and the seventh order harmonics, which are almost canceled by applying a double chording technique.Similar results can be appreciated from Table20, which summarizes the winding harmonic factors for different optimization techniques for a single-phase, double-layer winding located into 60 slots and with one pole pair.The corresponding WDT for the proposed example is plotted in Table21.The modified WDT for a double chording with a 2-nd order double-sided imbrication is represented in

Table 22 .
It can be noticed that, in case of imbrication, the WDT follows the same rules of the previous example shown inTable 17and the corresponding star of slots and winding scheme are plotted in Figures21 and 22, respectively.

Table 20 .
Winding factor harmonics with different optimization techniques for N = 60, p = 1 and m = 1 Preprints (www.

Table 21 .
WDT with a double chording and a 3-rd order double-sided imbrication for N = 60, p = 1 and m = 1

Table 22 .
WDT with a double chording and a 3-rd order double-sided imbrication for N = 60, p = 1 and m = 1