Motor windings: Cycle sequence and winding composition of fractional slot windings

1 Cycle sequence of fractional slot winding

1.1 What is the cycle sequence of fractional slot winding?

The slot potential star diagram can be used to determine the slot number and connection rules of each phase of the fractional slot winding, check the symmetry of the winding, etc. For fractional slot motors with a relatively small number of slots , this method is simple, easy and intuitive. Especially in micro motors, the so-called "fractional slot concentrated winding" with a large number of slots q < 1 is often used. It is most suitable to use the slot potential star diagram to divide the phase and determine the winding connection relationship. However, when the number of slots is large, it is very troublesome to draw the slot potential star diagram. For example, some large hydroelectric generators, direct-drive wind turbines, etc., have hundreds or thousands of slots, and drawing the slot potential star diagram is time-consuming and laborious. In fact, there are many ways to divide the phases of each slot, and the slot potential star diagram method is only one of them. Next, we will introduce a phase division method commonly used in practice, the "cyclic number sequence".

The number of slots per pole and per phase of the fractional slot winding is a fraction, which means that the number of slots allocated to each phase under each pole is not equal. This is mainly manifested in that the number of slots occupied by different phases under the same pole is not equal, with some phases having one more slot and some phases having one less slot; the number of slots allocated to the same phase under different poles is also different, with some poles having one more slot and some poles having one less slot. These two manifestations ultimately come down to the different sizes of the phase belts. Some phase belts have one more slot, which we call large phase belts; some phase belts have one less slot, which we call small phase belts. The number of slots allocated to each phase belt along the entire circumference of the stator forms a digital sequence, which is periodic along the circumference of the stator and repeats every d (the denominator of q) digits. We call this digital sequence composed of the number of slots allocated to each phase in a cycle along the circumference of the stator the cyclic number sequence of the fractional slot winding. This may be too abstract and difficult to understand. Next, we will still use the fractional slot winding with Z1=30 slots and 2p=8 poles mentioned in the previous article as an example to specifically talk about the cyclic number sequence of the fractional slot winding and how to use the cyclic number sequence to phase the fractional slot winding.

First, let’s show the slot potential star diagram of the fractional slot winding with Z1=30 slots and 2p=8 poles mentioned in the previous article again as shown in Figure 1 below.

According to the phase division result of Figure 1a, the number of slots divided for each phase under each pole is tabulated as follows:

Comparing Figure 1a and Table 1, it can be seen that the two are completely corresponding. Phase A under the N1 pole is divided into two slots (1# slot and 2# slot), phase Z (-C) is divided into one slot (3# slot), phase B is divided into one slot (4# slot), and phase X (-A) under the S1 pole is divided into one slot (5# slot). This series of numbers (a total of d=4 numbers): 2, 1, 1, 1, is called the cyclic number sequence of the fractional slot winding. Repeating the cyclic number sequence composed of these four numbers three times will give the phase belt allocation result of a unit motor. In other words, the cyclic number sequence is a series of numbers that represent the distribution law of the number of slots of each phase under each pole in the motor. In this example, the cyclic number sequence is 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1... represents the number of slots allocated to each phase belt A-Z-B-X-C-Y... starting from the first slot along the stator circumference. In this way, as long as the cyclic number sequence is determined, the phase band division can be performed without drawing the slot potential star diagram. Therefore, using the cyclic number sequence for phase band division is a simple phase division method, which is particularly suitable for fractional slot windings with a large number of slots. It should be noted that since the motor stator is a closed circle and the position of the first starting slot can be defined arbitrarily, for this example, the cyclic number sequence can be intercepted from any section of 2, 1, 1, 1, 2, 1, 1, 1... (a total of d numbers) as the cyclic number sequence, that is: 1, 1, 1, 2; 1, 1, 2, 1; 1, 2, 1, 1, etc. can all be used as the cyclic number sequence of the fractional slot winding. In other words, the cyclic number sequence of the same fractional slot winding is not unique, and there can be multiple combinations. The starting point when intercepting the cyclic number sequence can be different, but the order of the numbers cannot be interchanged.

1.2 Relationship between cyclic number sequence and q

Now that we know what the cyclic number sequence of fractional slot windings is, let's introduce the relationship between the cyclic number sequence and the number of slots per pole per phase q, that is, how to determine the cyclic number sequence after knowing q. It is not difficult to see from Table 1 that when the cyclic number sequence is used to divide the phase belt along the circumference of the stator, the number of slots allocated to each phase belt is different. One part of the phase belt includes 2 slots, and the other part includes 1 slot, that is, there are two phase belts with different widths. We call the phase belt with one more slot a large phase belt, and the phase belt with one less slot a small phase belt. Without loss of generality, for a fractional slot winding with q=b+c/d, in order to achieve an average number of slots per phase belt of q, it is necessary to make a part of the phase belt (large phase belt) have b+1 slots, and the other part of the phase belt (small phase belt) have b slots. Since q=b+c/d=(bd+c)/d=N/d, each phase belt is first divided into b slots, then d phase belts are divided into bd slots, and c slots are left unallocated. In order to achieve an average number of slots per pole per phase of q, one more slot must be added to each of the c phase belts, that is, c large phase belts and dc small phase belts are formed. It can be concluded that when the number of slots per pole per phase is known, q, the number of cyclic number sequence digits is d (the denominator of q), and the total number of slots included is N (the numerator of q), that is, N slots are divided into d parts. Since d and N are irreducible, and each part can only be an integer number of slots, it cannot be divided equally, and some parts can only have one more slot than others. Some distribution combinations formed in this way can be used as the cyclic number sequence of the fractional slot winding.

For example, this is equivalent to dividing N yuan into d red envelopes. Each red envelope must contain an integer amount of yuan, and a large red envelope can only have one yuan more than a small red envelope. In this way, the numerical sequence composed of the amount of money in each red envelope is a cyclic number sequence.

Still taking the 8-pole, 30-slot, q=1+1/4=5/4 fractional slot winding as an example, the way to divide the 5 slots into 4 parts can be q=(2+1+1+1)/4, so the cyclic number sequence is 2, 1, 1, 1. Of course, it can also be distributed in combinations of 1, 2, 1, 1; 1, 1, 2, 1; 1, 1, 1, 2, etc. These combinations can all be used as the cyclic number sequence of the fractional slot. It should be noted that not all fractional slot windings can be divided into d parts as described above, and all distribution combinations can be used as their cyclic number sequence. Only when c=1 or c=d-1 can all distribution combinations be used as cyclic number sequences. When c≠1 or c≠d-1, according to the above method, only a part of the distribution combinations can be used as cyclic number sequences. Therefore, this method of determining the cyclic number sequence is still limited, and it is necessary to seek other methods to determine the cyclic number sequence.

1.3 Method for determining cyclic number sequence

As mentioned above, there are many kinds of cyclic number sequences for the same fractional slot winding. There are also many methods for determining the cyclic number sequence in practice. Here are some commonly used ones:

①For fractional slot windings with q=b+c/d, where c=1 or c=d-1

The cyclic number sequence consists of d numbers, with a total of bd+c=N slots, including c large phase bands, each with (b+1) slots; and dc small phase bands, each with b slots.

When c=1, the order of numbers in the cyclic sequence is generally that the number of slots of (dc) small phase bands is in the front, and the number of slots of c=1 large phase band is in the back, for example:

When q=2+1/4, the cyclic number sequence is (2, 2, 2, 3)…

When q=1+1/5, the cyclic number sequence is (1, 1, 1, 1, 2)…

When q=2+1/5, the cyclic number sequence is (2, 2, 2, 2, 3)…

When c=d-1, the slots of the c large phase belt are usually arranged in front, and the slots of the dc=1 small phase belt are arranged in the back, for example:

When q=1+3/4, the cyclic number sequence is (2, 2, 2, 1)…

When q=2+3/4, the cyclic number sequence is (3, 3, 3, 2)…

When q=2+16/17, the cyclic number sequence is (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2)…

Of course, the cyclic number sequence mentioned above can also start from any number and sequentially select d numbers as a cyclic number sequence.

②For fractional slot windings with q=b+c/d, where c≠1 and c≠d-1

The cyclic sequence of this fractional slot winding can be obtained by the tabulation method. There are two tabulation methods: