Design of fractional slot windings with coil span of two slots for use in six‐phase synchronous machines

Abstract

The use of a winding step of two slots attempts to strike a balance between traditional distributed windings and fractional slot concentrated windings. It promises less ohmic losses and space for the end windings than distributed windings and less harmonics than fractional slot concentrated windings. This paper gives an introduction into the design of windings with a winding step of two slots for six-phase machines. The design procedure relies on the star of slots. Whereas for a given combination of slots and poles, the winding of a three-phase machine is unique, there exist several winding designs for six-phase machines. They differ by the shape of the phase belts and by the phase shift between the two three-phase subsystems. The definition of two characteristic figures facilitates the comparison of these windings: one describes the content of sub-harmonics in the magnetomotive force, the other is a measure for rotor eddy losses. The comparison reveals that windings with 12 small phase belts are better as long as fault tolerance is not an issue. If this is the case, windings with wider phase belts and a proper choice of pole number and phase shift are superior. 11Introduction When dealing with six-phase synchronous machines, fault tolerance is often in the focus of attention. There are also other reasons for the use of such a design [1]: the lower current rating per phase for a given power is a decisive advantage in all low-voltage high-power applications. Another reason is less torque ripple in large variable speed drives with current source inverter. Finally, the lower space-harmonic content in the mmf is advantageous for smoothness and efficiency. Yet, it is impossible to have all these benefits entirely at the same time. If the requirements on fault tolerance are high, the machine must have a single-layer fractional slot concentrated winding in order to decouple the phases from each other at the best. However, such a winding design entails a huge amount of harmonics in the armature field. If fault tolerance is not the main cause for the choice of a six-phase design a traditional dual-layer chorded winding can keep the harmonics very low. The coil pitch of such distributed windings is normally at least three slots which brings about a high winding resistance and occupies much space for the end windings. The use of windings with a coil span of two slots promises a better balance between ohmic losses and end windings space on the one hand and harmonic content on the other. This idea found an early three-phase implementation in [2] where the authors created such a winding by nesting two fractional slot concentrated windings into each other. A systematic elaboration of three-phase windings with a winding step of two slots takes place in [3, 4]. With respect to six-phase machines, the literature discusses some specific combinations of slots and poles: a design with 24 slots and 10 poles can suppress the first-order harmonic in the mmf [5]. The authors of [6] show for an 18 slots stator and a rotor with 8 or 10 poles how one three-phase winding can split into two three-phase subsystems. Reference [7] suggests a six-phase traction drive with 18 slots and 8 poles which is further analysed regarding demagetisation behaviour in [8]. This paper gives a general insight into the design of six-phase windings with a coil span of two slots. It depicts that for a given combination of slots and poles, the winding is not unique. Several winding designs are possible which differ by the shape of the phase belts and by the phase-shift between the two three-phase subsystems. The paper illustrates the construction of these different designs and analyses their individual mmf harmonic content. The results for each winding are merged into two characteristic figures which facilitate the comparison of these windings: one describes the overall content of sub-harmonics, the other is a measure for rotor losses by eddy currents. The comparison leads to some recommended winding implementations. 22Design of six-phase windings 2.1 Number of slots per pole per phase The term 'fractional slot winding' refers to the fact that the number of slots per pole per phase q is not an integer but a fraction. Doubling the number of phases m compared to a three-phase system halfens the number of slots per pole per phase, which depends also on the number of slots Q s and the pole pair number p: q = Q s 2pm ⇒ q 6 ∼ = q 3 ∼ /2 (1) As six-phase machines are just a combination of two three-phase systems, possible combinations of the number of slots Q s and the number of poles 2p are a subset of the combinations for three-phase windings. Among the numerous combinations which reference [4] lists for three-phase windings, there are only some which have a satisfactory high winding factor and these are within a range of 1/2 < q 3 ~ < 1. Hence, interesting six-phase windings exhibit 1/4 < q 6 ~ < 1/2. The number of stator slots Q s must be a multiple of the number of phases, i.e. Q s = 6, 12, 18, ... for m = 6 phases. Reasonable dual-layer fractional slot windings with even number of slots and a coil span of two slots require at least 18 slots. Table 1 lists suitable combinations of stator slots and rotor poles for six-phase windings and gives the corresponding number of slots per pole per phase q 6 ~. 2.2 Winding design with star of slots The star of slots is a collection of phasors which illustrates all possible phase shifts of the slot emfs. Each of these phasors is valid for t slots, so the star of slots consists of Q b = Q s /t basic phasors. The counter t is the greatest common divisor of the number of slots Q s and the number of pole pairs p. Neighbouring phasors in the star of slots include a phase angle α p = 2π/Q b. The electrical phase shift between neighbouring slot emfs is α e = α p · p/t. In order to label the