Poly-Phase Fractional Slot Windings

If the number of slots per pole and phaseqof a winding is a fraction, the winding is called a fractional slot winding. Windings of this type are either concentric or diamond windings with one or two layers. Some advantages of fractional slot windings when compared with integer slot windings are:

rgreat freedom of choice with respect to the number of slots;

ropportunity to reach a suitable magnetic flux density with the given dimensions;

rmultiple alternatives for short pitching;

rif the number of slots is predetermined, the fractional slot winding can be applied to a wider range of numbers of poles than the integral slot winding;

rsegment structures of large machines are better controlled by using fractional slot windings;

ropportunity to improve the voltage waveform of a generator by removing certain harmonics.

The greatest disadvantage of fractional slot windings is subharmonics, when the denomi-nator ofq(slots per pole and phase) isn=2

q= Q 2pm = z

n. (2.64)

Now,qis reduced so that the numerator and the denominator are the smallest possible in-tegers, the numerator beingzand the denominatorn. If the denominatornis an odd number, the winding is said to be a first-grade winding, and whennis an even number, the winding is of the second grade. The most reliable fractional slot winding is constructed by selecting

n=2. An especially interesting winding of this type can be designed for fractional slot per-manent magnet machines by selectingq=1/2.

In integral slot windings, the base winding is of the length of two pole pitches (the distance of the fundamental wavelength), whereas in the case of fractional slot windings, a distance of several fundamental wavelengths has to be travelled before the phasor of a voltage phasor diagram again meets the exact same point of the waveform. The difference between an integral slot and fractional slot winding is illustrated in Figure 2.24.

B B

0 2π α 0 2π ^α p'2π

5... Qs

Qs^,1... p Q_s^,1... 5... Q_s'+1

...

slot positions slot positions

=12

base winding base winding

B

0 α

(a) (b)

Qs = 12

1 2 3 4 5 6 7 8 9 10 11 12 1 slot positions and coil sides of phase U

base winding

1 2 3 4 5 1 p = 5

(c)

π π

Figure 2.24 Basic differences of (a) an integral slot stator winding and (b) a fractional slot winding.

The number of stator slots isQs. In an integral slot winding, the length of the base winding isQs/p slots ((a): 12 slots,qs = 2), but in a fractional slot winding, the division is not equal ((b):qs < 2).

In the observed integer slot winding, the base winding lengthQs = 12 and, after that, the magnetic conditions for the slots repeat themselves equally; observe slots 1 and 13. In the fractional slot winding, the base winding is notably longer and containsQ_sslots. Figure (c) illustrates an example of a fractional slot winding withQs=12 andp=5. Such a winding may be used in concentrated wound permanent magnet fractional slot machines, whereq=0.4. In a two-layer system, each of the stator phases carries four coils. The coil sides are located in slots 12–1, 1–2, 6–7 and 7–8. The air gap flux density is mainly created by the rotor poles

In a fractional slot winding, we have to proceed a distance ofp pole pairs before a coil side of the same phase again meets exactly the peak value of the flux density. Then, we need a number ofQ_sphasors of the voltage phasor diagram, pointing in different directions. Now, we can write

Q_s=pQ_s

p , Q_s<Qs, p> p. (2.65) Here the voltage phasorsQ_s+1, 2Q_s+1, 3Q_s+1 and (t−1)Q_s+1 are in the same posi-tion in the voltage phasor diagram as the voltage phasor of slot 1. In this posiposi-tion, the cycle of the voltage phasor diagram is always started again. Either a new periphery is drawn, or more slot numbers are added to the phasors of the initial diagram. In the numbering of a voltage phasor diagram, each layer of the diagram has to be circledptimes. Thus,tlayers are created in the voltage phasor diagram. In other words, in each electrical machine, there aret electri-cally equal slot sequences, the slot number of which isQs =Qs/tand the number of pole pairsp=p/t.To determinet, we have to find the smallest integers Q_sandp·tis thus the largest common divider ofQsandp. IfQs/(2pm)∈N(Nis the set of integers,Neventhe set of even integers andNoddthe set of odd integers), we have an integral slot winding, andt=p, Q_s=Qs/pandp=p/p=1. Table 2.3 shows some parameters of a voltage phasor diagram.

To generalize the representation, the subscript ‘s’ is left out of what follows.

If the number of radii in the voltage phasor diagram isQ=Q/t, the angle of adjacent radii, that is the phasor angleαz, is written as

αz= 2π

Qt. (2.66)

The slot angleαuis correspondingly a multiple of the phasor angleαz

αu= p

tαz= pαz. (2.67)

Whenp=t, we obtainαu =αz, and the numbering of the voltage phasor diagram pro-ceeds continuously. Ifp>t,αu> αz, a number of (p/t)−1 phasors have to be skipped in the numbering of slots. In that case, a single layer of a voltage phasor diagram has to be cir-cled (p/t) times when numbering the slots. When considering the voltage phasor diagrams of harmonicsν, we see that the slot angle of theνth harmonic isναu. Also the phasor angle is

Table 2.3 Parameters of voltage phasor diagrams

t the largest common divider ofQandp, the number of phasors of a single radius, the number of layers of a voltage phasor diagram

Q=Q/t the number of radii, or the number of phasors of a single turn in a voltage phasor diagram (the number of slots in a base winding)

p=p/t the number of revolutions around a single layer when numbering a voltage phasor diagram

(p/t)−1 the phasors skipped in the numbering of the voltage phasor diagram

ναz. The voltage phasor diagram of theνth harmonic differs from the voltage phasor diagram of the fundamental with respect to the angles, which areν-fold.

Example2.15: Create voltage phasor diagrams for two different fractional slot windings:

(a)Q=27 andp=3, (b)Q=30,p=4.

Solution: (a)Q=27, p=3,Q/p=9∈N,qs =1.5,t=p=3,Q =9,p=1,αu= αz=40^◦.

There are, therefore, nine radii in the voltage phasor diagram, each having three phasors.

Becauseαu=αz, no phasors are skipped in the numbering, Figure 2.25a.

(b)Q=30,p=4,Q/p=7.5∈N,q_s=1.25,t=2=p,Q=15,p=2,Z=Q/t= 30/2=15,αz=360^◦/15=24^◦,αu=2αz=2×24^◦=48^◦, (p/t)−1=1.

In this case, there are 15 radii in the voltage phasor diagram, each having two phasors.

Becauseαu=2αz, the number of phasors skipped will be (p/t)−1=1. Both of the layers of the voltage phasor diagram have to be circled twice in order to number all the phasors, Figure 2.25b.

Figure 2.25 Voltage phasor diagrams for two different fractional slot windings. On the left, the numbering is continuous, whereas on the right, certain phasors are skipped. (a)Q=27,p=3,t=3, Q=9,p=1αu=αz=40^◦; (b)Q=30,p=4,t=2,Q=15,p=2,αu=2αz=48^◦;αuis the angle between voltages in the slots in electrical degrees and the angleαzis the angle between two adjacent phasors in electrical degrees

2.9 Phase Systems and Zones of Windings