Abstract—This paper represents a winding design technique for multi-phase, especially three-phase machines. The winding distribution, the winding factor and some winding qualification characteristics like THD and differential leakage are calculated on the basis of general rules of symmetrical AC windings.

Index Terms—Windings, Design Rules, Permanent Magnet Machines, Electric Machines

I. NOMENCLATURE p number of poles m number of phases N number of slots t number of source windings q number of slots per pole and per phase z numerator of q n denominator of q W coil span ξ winding factor

II. INTRODUCTION The design of symmetrical multi-phase – in general

three-phase – integral-slot windings means no great challenge to electrical engineers, because only two features can be varied, the number of slots per pole and per phase and the coil span. The design of fractional-slot windings, however, and their analysis is already more complicated. More ambitious are also tooth-wound coils recently used especially in permanent magnet synchronous machines. They are understood as – in general symmetrical – fractional-slot windings which are subject to the same methods of design and analysis. The methods described below have been realized in the program WET (Wicklungs-Entwurfs-Tool = Winding Design Tool), recently included in the finite element program FEMAG for an automated design. The paper describes the rules of winding design and their computational realisation for several examples.

III. DESIGN RULES OF SYMMETRICAL MULTI-PHASE WINDINGS OF ELECTRICAL MACHINES

The design of all types of windings for electrical machines is just subject to a few, but severe rules. Simple and obvious is the design of multi-phase windings, having equally sized winding zones, recurrent in each pole pair. The sometimes complex windings of a fractional-slot winding are nevertheless subject to the

same winding rules. [1], [2] The major task of an automatic winding design is to

assign the coil sides to the individual slots for a given configuration of number of phases, number of poles, number of coil sides per slot and of the coil span, so that a symmetrical winding with a maximum winding factor of the fundamental field and with a minimum of differential leakage is achieved.

A. Rules of the Winding Zones Adjacent coil sides of a phase are designated as

geometrical zone of a winding. The zone width is characterised by the number of poles per slot and per phase q.

2

qN zpm n

= = (1)

If the number of poles per slot and per phase is an integer, we call it an integral-slot winding; in the case of a fraction number, we talk about a fractional-slot winding.

In order to assign the coil sides to the single phases, symmetrical multi-phase systems must be defined in their phase position first. As a polygon winding (not possible for some numbers of phases) and a neutral conductor with preferably no load or better without any load are requested, the multi-phase systems shown in fig. 1 are used. In the case of uneven numbers of phases, except for the single-phase system, all requirements are fulfilled

Fig. 1 Multi-phase systems for symmetrical supply of multi-phase

windings (except fig. on top left side: single-phase system for supply of single-phase windings)

Design and Analysis of Windings of Electrical Machines

Jörn Steinbrink University of Hannover, Institute for Drive Systems und Power Electronics, Welfengarten 1, D 30167 Hannover

(Germany), Tel.: (+49) 511 762 2864, Fax: (+49) 511 762 3040, e-mail: steinbrink@ial.uni-hannover.de

SPEEDAM 2008International Symposium on Power Electronics,Electrical Drives, Automation and Motion

717978-1-4244-1664-6/08/$25.00 ©2008 IEEE

without restrictions by a shift of 2π/m (radially symmetrical muti-phase systems). In the case of even numbers of phases, it has to be distinguished between numbers of phases being a power of 2 and those with an uneven divisor mu. In the case of uneven divisors, there are always m/mu groups, the corresponding phases being shifted by π/m against each other, which are arranged like multi-phase systems with mu phases. The request for a neutral conductor is thus fulfilled. These systems also meet the demand for polygon windings by m/mu separated polygons, being composed of phases with the same shift (e.g. for m=6: 1-3-5 and 2-4-6). For numbers of phases of the power of 2, these demands cannot be met; neutral loading can only be minimised.

B. Demand for Symmetrical Windings A winding is called symmetrical, when a symmetrical

voltage system is induced under the effect of an air-gap field especially the fundamental field. These requirements are met under the following symmetry conditions:

pq ∈ (for single-layer winding) (2a)

2 pq ∈ (for two-layer winding) (2b)

Nmt

∈ (for single-layer winding) (3a)

2Nmt

∈ (for two-layer winding) (3b)

For fractional-slot windings, equation (2)

pn

∈ (for single-layer winding) (4a)

2 pn

∈ (for two-layer winding) (4b)

and equation (3) lead to the rule, that m and n must not have a common divisor. [1]

C. Principle Approach Each coil consists of two coil sides. A rotational field

with the number of pole pairs p is theoretically impressed in the air gap. The rotational field induces a voltage in each coil side, being called slot voltage. Provided that the slots are distributed symmetrically, the phasors of the slot voltages represent the so-called slot star, which may be repeated several times, in case of integral-slot winding 2p times around the circumference (see fig. 2).

When the phasors of the slot star result in a polygon, the winding factor can easily be derived from the geometrical/arithmetical sum of the ratio of the phasors of one phase. The Goerges polygon (GD) allows to evaluate the harmonic content generated by windings. The GD represents the phasor diagram of the ampere turns per slot, which determine the flux density in the air gap between two adjacent slots and consequently the air-gap field itself provided that the air gap is constant. The GD is a simple means to determine the generated field spectrum and the differential leakage as a measure for the

Fig. 2 Slot star diagram

Fig. 3 Example for a simple winding: distribution diagram, slot star, GD

and voltage phasors per phase of a distributed two-layer winding for N=12; p=1; W=5; ξp=0,9330; σd=2,354%

Fig. 4 Example for the parameter list for winding generation in FEMAG

spatial harmonic content. It visualises the deviation of the resulting air-gap field which contains all spatial harmonic fields from the circle of the air-gap field which contains the fundamental field only. The winding design is exemplarily described for a simple distributed winding (see fig. 3). The winding distribution in a FE model of FEMAG [3] is shown in the examples given in the following section. The winding generation in FEMAG is based on the parameter list (figure 4).

First of all, the distribution diagram for one layer or coil side is derived from the slot star. Then, the second half of the winding is completed, staggered around the circumference by the coil span. In the GD, the centre of mass of which is equal to the circle centre, the distance

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Fig. 5 Distribution diagram, slot star, GD and voltage phasors per phase

of a tooth-wound single-layer winding for N=12; m=3; p=4; W=1; ξp=0,8660; σd=192,43%; THD=67,49%

Fig. 6 Distribution diagram, slot star, GD and voltage phasors per phase

of a tooth-wound two-layer winding for N=9; m=3; p=4; W=1; ξp=0,9452; σd=118,21%; THD=20,81%

between this point and the displayed points corresponds to the flux density of the air gap field between two adjacent slots. The circle represents the flux density corresponding to the fundamental field. The differential leakage is determined to the deviation of the points from the circle. The voltage phasors illustrate for one single field, here the fundamental field is shown, the winding factor, the symmetry, and the sense of rotation of the field. For all examples, the winding factor of the fundamental field, the differential leakage and the spatial harmonics generated by the windings are indicated. In this context, a representation of the winding factors and spatial harmonic fields calculated by the program remains out of consideration. The winding factor is analytically calculated as the product of the distribution factor and the

Fig. 7 Distribution diagram, slot star, GD and voltage phasors per phase

of a tooth-wound two-layer winding for N=15; m=3; p=4; W=1; ξp=0,7109; σd=38,854%; THD=15,02%

Fig. 8 Distribution diagram, slot star, GD and voltage phasors per phase

of a PAM winding for N=36; m=3; p=3; ξp=0,8797; σd=5,799%

Fig. 9 Distribution diagram, slot star, GD and voltage phasors per phase

of a PAM winding for N=36; m=3; p=2; ξp=0,7198; σd=2,876%

chording factor as well as geometrically from the phasor of a phase. The value given for the THD from the 100 first harmonics of the oscillation of the line-to-line terminal fundamental voltage is a limit which would occur for all spatial harmonics of identical field amplitudes.

IV. EXAMPLES The winding design is exemplarily described by some

different windings. Figure 5 to 9 show windings of three- phase machines as distributed winding and tooth-wound coils. Tooth-wound coils are a special variety of distributed windings, the coil span of which is equal to

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Fig. 10 Distribution diagram, slot star, GD and voltage phasors per

phase of a distributed two-layer winding for N=36; m=6; p=1; W=17; ξp=0,9861; σd=0,258%; THD=4,46%

Fig. 11 Distribution diagram, slot star, GD and voltage phasors per

phase of a tooth-wound single-layer winding for N=14; m=7; p=6; W=1; ξp=0,9749; σd=90,72%; THD=25,91%

one slot pitch. The figures show the schematic layout of the distribution diagram instead to the real connecting diagram of the winding. Furthermore the figures show the corresponding FE-model of FEMAG of these machines. In addition, any unsymmetrical winding can be defined and analysed with the program WET; the field spectrum and the winding factors are determined separately for each phase, analysing their symmetrical components. The analysis of a complicated winding is carried out for a pole changing winding based on the principle of pole amplitude modulation at figure 8 and 9 (PAM).

Fig. 10 and 11 show further examples with the

winding design for other numbers of phases. In general, windings can be generated with any number of phases. In FEMAG, this can be used among others to generate a nearly ideal air-gap field with extremely high numbers of phases for one air-gap winding, which is practically free from spatial harmonics. By this means, the effects of fluctuations of magnetic permeances can be investigated numerically by slotting.

V. CONCLUSIONS With the programs WET and FEMAG, a simple and

generally suitable method to determine the winding distribution has been implemented. WET offers designers of electrical machines the possibility to compare different windings in detail and evaluate their suitability for a definite task. The conversion of algorithms in FEMAG permits a quick entry of the winding parameters into a numerical model, enabling an automated winding design, too. Thus the time-consuming and error-prone manual entry of the winding can be avoided, and users can concentrate on their proper task, the determination of machine parameters.

REFERENCES

[1] Vogt K.: Elektrische Maschinen, Berechnung rotierender elektrischer Maschinen, 4rd ed., Verlag Technik Berlin, 1988.

[2] Sequenz, H.: Die Wicklungen elektrischer Maschinen, 1. Band, Wechselstrom-Ankerwicklungen. Springer Verlag, 1950

[3] Reichert, K.: User’s Manual FEMAG. www.ial.uni-hannover.de/femag_download.html

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