## Machines with One Winding on the Stator and One winding on the Rotor

The majority of electrical machinery that constitute the backbone of the industry (synchronous generators, induction motors and DC machines) belong to this group. In this kind of doubly-excited machines:
1) Stator and rotor each accommodates one winding.

2) Only heteropolar windings are used.
3) The windings might have any number of phases with equal number of poles.
4) The cores could have smooth or toothed surfaces, but the main attempt is to make the cores surfaces as smooth as possible (by using semi-closed slots), as the change of the self and mutual inductance of the windings due to the teeth of the cores are of the secondary importance; they would only be sources of noise and high frequency losses.

The electromechanical energy conversion is based on the change in the relative position of the windings and the change in their mutual-inductances as the rotor rotates. Figure 1. An elementary 4-pole synchronous machine

The locations and the directions of currents in the coils of the winding, determine the number of poles. Generally, when they set up a p-pole field, the mutual-inductance L12 undergoes a complete cycle of change when the rotor rotates through two pole-pitches (π radians in Figure 1).
It’s important to notice that as we relate the pole-pitch to the winding span, we always define one pole pitch equal to 180° and not to 360° (Unlike singly-excited machines, where one tooth pitch was equivalent to 360°).:
yp = 2π/p          (1)

For example, a four-pole machine has 4 pole-pitch (each of 180°), not two pole-pitch (each of 360°).

Again unlike singly-excited machines, the period of mutual-inductance is determined by the windings pole pitch only and the tooth pitch of the stator or rotor has no effect on the period of the mutual-inductance.
If the rotor rotates at angular speed ωm, the mutual-inductance will alternate with a period equal to:
Te = 2.ypm = 4π / (p.ωm) = 2 / (p . fm)          (2)

Where the frequency (and angular frequency) of the mutual-inductance are given by the following formulas:
fe = 1/Te = p.fm / 2
ωe = p/2 . ωm          (3)

yp is the pole-pitch of the winding, Te the time period of L12, p the number of poles, fe the frequency of L12 and fm the rotational frequency of the rotor.

The shape of the plot for L12 for a round core with q = 1 is almost triangular and the EMF has an almost square shape. As the number of slots per phase belt increases, the pattern of change in L12 takes on a stepwise shape close to a sinusoidal waveform.

A practically sinusoidal pattern of change in L12 can be obtained with q = 1 as well, if the air gap at the edges of the rotor teeth are bigger than the air-gap at the teeth axes. In this case the air gap at each tooth axis is chosen two-third to one-half of the air gap at the tooth edge. This type of rotor called Salient-Pole and is usually used in low speed synchronous machines.

As was shown in singly-excited machines, the frequency of the mutual-inductance wave was ZR.fm. Therefore the frequency of self-inductance alternation in a singly-excited machine is 2.ZR/p of frequency of the mutual-inductance of a doubly-excited machine with the same rotor speed.

In synchronous machines, both windings carry currents that their angular frequencies are fixed in advance. The rotor might have a single-phase heteropolar winding (like in a wound-rotor salient-pole machine).

Induction machines are actually singly-excited, but they have two windings each on one element where the stator winding is excited by an AC current and the rotor winding is either short circuited or connected across an impedance, and the current in this winding is built up by electromagnetic induction. The rotor current frequency is different from the stator current frequency.

Induction machines could have wound rotors or squirrel cage type which constitute a large percentage of induction motors. The squirrel cage rotor consists of metal bars (Aluminum or Copper) pushed through the rotor slots and shorted at both ends by rings.

Not similar to synchronous and induction machines that have rotating magnetic fields in the air gap, both windings in DC machines set up fixed and stationary fields that stay perpendicular to each other at all speeds. Perpendicularity of these two fields is maintained by the action of a mechanical device called Commutator that accomplishes the switching of individual armature coils in a well established manner. The continuous perpendicularity of these two fields is the main reason why power density of DC machines is usually higher that those of synchronous and induction machines.

## Machines with Two Windings on the Stator and A Toothed Rotor (Inductor Machines)

These machines are usually used in special purpose applications and have the following unique features:
1) Both windings are on only one the stationary element, so no need to slip rings or brushes.
2) Both windings could be heteropolar with the same or different number of poles.
3) Both windings could be homopolar.
4) A heteropolar and a homopolar winding could be combined.
5) Depending on the design, the stator might be toothed or smooth, but the rotor must always be toothed.

As compared to a conventional synchronous machine, an inductor machine has usually a substantially larger size and weight. The flux density in the toothed surface of the stator (or rotor) in an inductor machine varies only in magnitude, whereas in a conventional synchronous machine it is rotating and varies in both magnitude and direction. Given the same geometrical dimensions and maximum tooth flux density, the peak value of the fundamental flux in an inductor machine would be smaller than its value in a conventional synchronous machine. Therefore, the application of inductor machines is justified only in those cases where the desired frequency is difficult to obtain by a conventional multipole or a claw-pole synchronous machine.

Among their unique features, they are the only kind of mechanoelectrical devices that without any field on their rotors (and without any auxiliary devices), are able to work as a generator. This makes them as a unique solution in very special applications.

The electromechanical energy conversion is due to the variations in the mutual-inductance L12 of  the two windings  as the toothed rotor rotates relative to them. variations of self-inductances are of secondary importance, although this change might be even higher than the mutual-inductance. Like singly-excited machines, one cycle of the change in the mutual-inductance is completed as the rotor rotates through one tooth pitch.

If the rotor is rotating at an angular speed ωm, the time period and angular frequency of the mutual inductance can be found by the same relations that were obtained here for singly-excited machines:
Te = α/ ωm = 2π / (Z. ωm) = 1 / (Z. fm)
fe = 1 / Te = ZR . fm
ωe = ZR . ωm          (4)

## Practical Designs

There are three different kinds that depend on the type of windings on the stator:
1. Both windings are heteropolar.
2. One winding is heteropolar and the other homopolar.
3. Both windings are homopolar.
In all the above three cases the rotor is toothed.

1. The mutual-inductance between the two heteropolar windings on the stator is made to vary owing to the rotation of a toothed rotor core. The windings usually different number of poles and the higher-pole winding plays the role of the carrier signal of the lower-pole winding. Remembering the synchronous reluctance machine that discussed in singly-excited machines,  where we made the rotor with an equal number of teeth as the stator winding poles, here we should have the rotor teeth equal to P1 + P2 or P1 – P2. In this arrangement, as both winding are on stator, its saliency is of minor significance. As the permeance wave is constant (due to smoothness of the stator core), it’s quite difficult to find all acceptable combinations. Here is a brief guideline (assuming P> P2):

i) If P1/P2 = 2K (an even number)
L12 has a zero average and all odd number of teeth work, but a rotor with number of teeth equal to ZR = P1/2±P2/2 has the largest variations in the mutual-inductance (and hence biggest EMF). Figure 2. Two heteropolar 4 & 2 pole windings, ZR = (4+2)/2 = 3

ii) If P1/P2 = 2k+1 (an odd number)
L12 has a non-zero average and all even number of rotor teeth work, but a rotor with number of teeth equal to ZR = P1±P2 has the largest variations in the mutual-inductance (and hence biggest EMF). Figure 3. Two heteropolar 6 & 2 pole windings, ZR = 6 – 2 = 4

iii) If P1/P2 is a fractional number
Then L12 has a zero average and only two combinations ZR = (P±P2)/2 and ZR = (P1±2P2)/2 have acceptable inductance waveforms. Figure 4. Two heteropolar 6 & 4 pole windings, ZR = (6+2×4)/2 = 7

Number of rotor teeth ZR is uniquely fixed by the specified frequency and the rotor speed. Usually it is very large and to satisfy the above relations, winding 1 must have a low number of poles and winding 2 a large number of poles.

1. The mutual-inductance between a heteropolar and a homopolar winding is made to vary by the rotation of a toothed rotor core relative to a smooth or toothed stator core. Figure 5. One homopolar and one 6-pole heteropolar windings, ZR = 9+6/2 = 12

## Singly-Excited Machines (Nature of Excitation Currents)

In the former post of singly-excited machines, some elementary machines were studied where their self inductances were functions of the rotor position and would change periodically with the rotor rotation.

Now we study the conditions under which the currents ij and ik must change in time so that not only the instantaneous but also the average value of Tem becomes large enough.

We start with the electromagnetic torque equation presented here and consider it for a single-winding machine. In that equation, the self-inductance of the single winding L11 was a periodic function of the angular position of the rotor. To have a constant unidirectional energy conversion, the currents time variations must lead to a non-zero average of the electromagnetic torque. As the currents are periodic, we just need to calculate the torque averaged over a period:

From the electromagnetic torque equation, for a singly-excited machine, we have j = k = 1, and the electromagnetic torque developed by that machine would be:

As you remember the time period of self-inductance change in a singly-excited machine was:
T = 2π / (Z. ωm) = 2π / ω
ω = ZR . ωm          (3)

Remembering that the self-inductance in singly-excited machines had a non-zero average and considering only the main harmonic of L11, the self-inductance equation must have a general form like:
L11 = L110 + L11m . cos(ω.t)          (4)

L110 is the non-zero average of the self-inductance and ω is its angular velocity. The derivative of the self-inductance with respect to the angular position of the rotor would be:

dL11/dθ = (dL11/dt) . (dt/dθ) = – L11m. ω . sin(ω.t). (1/ωm)

From equation (3):

dL11/dθ = – L11m. ZR . sin(ω.t)          (5)

As we assumed, in a real-world machine the current in the winding must be alternative. Using Equation (1), we see that if the winding carries a constant current, the mean torque Tavg would be zero. Focusing on the fundamental component of the current which develops the largest mean electromagnetic torque, the general equation would be of the shape:
i1 = I1m . cos(ω1.t + φ)          (6)

We have to find those values of ω1 and φ that make Tavg maximum. Using (2), (5) and (6), the mean torque Tavg is equal to:

After some trigonometric manipulations in the integrand, it is reduced to: (8)

The first term is a sinusoidal function and has a zero average. The second term can be rewritten as:

To have a unidirectional energy conversion, this equation must average to a non-zero number and it will happen only when ω – 2ω1 = 0 which gives the value of ω1 as:
ω1 = ω / 2          (10)

This means that, the frequency of the current is half the frequency of the self-inductance. From equation (2), the torque is proportional to the square of current which has the same frequency as the self-inductance.

Tavg = (L11m . Z. I21m/8) . sin(2φ)          (11)

For φ = ±π/4, the average developed torque is maximum.

Therefore, for a unidirectional energy conversion, a singly-excited machine must be fed by a current at an angular frequency ω1 equal to half the angular frequency of the changes in the self-inductance:
ω1 = ω / 2 = ZR . ωm / 2          (12)

The direction of energy conversion depends on the phase angle of φ. When φ = π/4, the machine will be operating as a motor. When φ = -π/4, it will be operating as a generator. The angular velocity of the machine is proportional to the angular frequency of the current in the winding connected to the electrical system:
ωm = 2.ω1 / ZR          (13)

When the angular velocity of an electric machine is proportional to the angular frequency of the flowing current in its winding, it is called a synchronous machine otherwise it is called an asynchronous one.
From equation (13) it follows then that all singly-excited machines are synchronous machines.

## Singly-Excited Machines (Nature of Inductance Changes)

Some fundamental windings and core designs capable of producing a spatially periodic magnetic field were discussed Here.
We are now interested in finding those configurations that while their self (or mutual) inductances depend on the rotor position, they change periodically as the rotor changes position.

Considering a drum machine, we will study that in how many different ways and how an electric machine must be configured so that the self or mutual inductances of its windings to be functions of the angular position of the rotor and change periodically as the rotor rotates. Meanwhile we will find out some rules for maximizing the developed inductances.

## Smooth and Toothed Cores

The currents flowing in the windings, produce magnetic field (poles N and S) in the core carrying the windings; where the core’s teeth only affect the shape of the field in the air gap. Therefore, as long as a core carries at least one winding, with either a smooth or toothed surface, it would experience changes in the self-inductance of its winding (or mutual inductance between its windings). When a core accommodates winding (or windings), a smooth surface (with semi-closed slots) always performs better (due to saturation, closed slots behave like semi-closed slots in practice).
If the core (a rotor for example) doesn’t accommodate a winding, it can not have its own magnetic field and its magnetic field (N and S poles) must be created by the stator. To the stator magnetic field (disregarding how many poles it has), all positions of a smooth rotor would be the same. Therefore the rotor must be toothed where the number of teeth, play an important role in the characteristic and performance of the machine even if the stator has physically closed slots, since as was mentioned, saturation effects open the closed slots.
Rotation of a toothed rotor, will change the reluctance of the paths of flux lines linking the stator windings. In some cases when the insertion of the windings requires an open slot design for the stator core, we need to pay attention that with toothed cores, in most cases, the permeance of the air gap is changed in space or/and time, which leads to undesirable effects if not addressed properly.

## Basic Machines Types

In the following elementary models, only the motoring mode of operation will be considered, disregarding if they are directly reversible or not. In addition, as electric machines have usually two magnetic elements (stator and rotor), we consider maximum two windings.
With two elements of stator and rotor and maximum two windings, the following three combinations could exist:

1. A singly-excited machine with one winding on the stator and no winding on the rotor. The stator could have smooth or toothed surface but the rotor must have a toothed surface. The periodic self-inductance has a non-zero average value.
2. Two doubly-excited machines where we could have any of the following combinations:
i) A machine with one winding on the stator and one winding on the rotor. Both stator and rotor could have smooth or toothed surfaces. The periodic mutual inductance has a zero average value.
ii) A machine with two windings on the stator. The stator could have smooth or toothed surface but the rotor must have a toothed surface. The periodic mutual inductance could have a zero or non-zero average value depending on the design.

The magnetic field in the air gap may be either heteropolar or homopolar according to the winding used. Homopolar windings could only be single-phase and operate on AC or DC, but heteropolar windings could have any number of phases and operate on AC or DC depending on the configuration and interfaces.

## Wavelength Equation In Rotational Motion

In linear motion, the relation between a wavelength λ, its speed v and time period T is:
λ = v . T          (1)
The relation between an angle θ and the corresponding arc is: S = R.θ (where R, θ and S are radius, angle of rotation and the subtended arc respectively). If arc S is equal to wavelength λ (λ = R.θ), then in rotational motion, the wave equation (1) transforms to:
R.θ = v . T ⇒ θ = v/R . T

Since v = R.ω, we will have the familiar equation:
θ = ω . T          (2)

Equation (2) is the basic equation used in all of the following topics.

## Heteropolar Integer-Slot Windings (Reluctance Machines)

In a machine with one winding on the stator and a toothed rotor core, electromechanical energy conversion is the result of variations in the self-inductance of the winding as the toothed rotor rotates and changes the reluctance of the magnetic circuit (thereby self-inductance L11 changes) where the name reluctance machine came from.
The winding in Figure 1 is a concentrated single-phase winding, but it could have any number of phases. Figure 1. A single-phase 4-pole reluctance machine (ZS=8 and ZR=8)

In a singly-excited machine, both stator and rotor could be toothed. But the self-inductance of the stator winding is maximum when the stator has a smooth surface (semi-closed slots). In stators with semi-closed slots, not only the developed EMF is higher but also the EMF shape is closer to a sinusoidal one.
L11 undergoes a complete cycle of change when the rotor rotates through one tooth pitch αR.
It is important to note that the pole pitch and tooth pitch of the stator have no effect on the period of the self-inductance (and the speed of motor).

If the rotor rotates at angular speed ωm then the time period and frequency of the self-inductance is calculated by the following equations:

Ti = α/ ωm = 2π/ (Z. ωm)
fi = 1/Ti = Z. ωm / 2π = ZR . fm
ωi = ZR . ωm          (3)

Ti is the time period of L11, ZR the number of rotor teeth, fi the frequency of the self-inductance and fm the rotation frequency of the rotor.
As we will see later in doubly-excited machines (like synchronous machines) the relation between the frequency of the mutual-inductance and the rotor is: fi = (p/2) . fm where we conclude that in reluctance machines, the self-inductance frequency is 2ZR/p of mutual-inductance frequency in an equivalent synchronous machine of the same number of poles and rotor speed.

To have a regular steady-state torque (without cogging and radial torques), the permeance wave must be periodic so have a constant shape and amplitude in all poles. To meet this requirement, the rotor and stator teeth must occupy the same relative positions at every pole in every position of the rotor. It can easily be shown that to fulfill this condition, the number of stator and rotor teeth must follow the following rule:

ZS/p – ZR/p = ±k
ZS – ZR = ±k.p         (4)

Where p stands for the number of stator winding poles and k is an integer number k=1, 2, 3, …. The ± sign is for taking into consideration that the stator teeth could be more or less than rotor teeth. In using Equation (4), we must exclude, all those combinations where in them ZS/ZR or ZR/ZS is an integer number, where they lead to a large variation in air-gap permeance; for example, combination ZS=12 and ZR=4.

Therefore, to have a satisfactory operation, not only the the rotor and stator pole-pitches must be integer numbers, but also they must differ by a non-zero integer number. Then at every pole, there would be an equal area of opposite tooth-tooth surfaces. For example, if a stator made of ZS = 16 teeth has a 4-pole winding , then the rotor could have 4, 8 and 12, 20, 24 etc teeth. If we use a distributed winding the change in in self-inductance and the induced voltage will be close to a sinusoidal shape: Figure 2. A single-phase 4-pole reluctance machine (ZS=16 and ZR=12)

Equation (4) not only assures a smooth permeance wave in the air gap, but also maximum variations in the self-inductance and consequently maximum developed EMF. But the ingenuity of the design engineer plays the most important role in choosing the best configuration to guarantee the highest performance.
Assuming the same number of poles and frequency, the highest rotor speed can be attained when the rotor has the minimum number of teeth or when ZR = number of stator winding poles. Such a rotor is called Salient Pole and the machine is called Synchronous Reluctance Machine (SRM). Figure 3. A 4-pole reluctance machine with salient pole rotor (ZS=24, ZR=4)

The time period, frequency of the self-inductance and rotor speed are still given by Equations (3).

## Heteropolar Fractional-Slot Windings (Reluctance Machines)

Despite small phase group slots (q), fractional-slot windings develop the best shapes for L11 and the induced voltages (almost sinusoidal), but as (ZS/p) is a fractional number, in no condition they can produce a constant permeance wave in the air gap. Figure 4 shows a single-phase winding of an 18/4 slots/pole synchronous reluctance machine with sinusoidal L11 and EMF. With open slots, the permeance wave has about 20% fluctuation along the air gap and the motor will experience a light mode 2 radial vibration.

Figure 4. A 4-pole SRM with single-phase, single-layer fractional-slot winding  (ZS=18, ZR=4)

With semi closed slots, the fluctuation of permeance wave could be reduced to under 10%.

## Homopolar Windings

To create a field with the same number of poles as the number of teeth we can either use a heteropolar winding with the same (or half) the number of coils or just use a homopolar winding (or coil) that builds up a field with a number of poles twice as many as the number of slots (or teeth) and it’s done by a simple coil that embraces the rotor (the internal core).
As explained here, for each pole on the outer surface of the rotor core, there is a pole with opposite polarity on the inner surface of the stator core. It means that in singly-excited machines, we cannot choose different number of slots (or teeth) for the stator and the rotor when using homopolar winding, so Z= ZS.
When the number of teeth on the cores is the same, the permeance wave has the highest fluctuations in time. The time period, frequency and rotor speed of the self-inductance are still found by equations (3).

## AC Heteropolar Windings

Before a discussion of drum windings that constitute the majority of the winding types used in the industry nowadays, we need to get familiar with varieties of windings from different points of views.
In this discussion, only distributed three-phase windings (q > 1) will be considered. Windings with q=1 are called concentrated windings and since all space harmonics have a winding factor of 1, are rarely used in practice.
In addition, in the diagrams, the connections between groups are considered series where could be parallel without affecting our discussions.

## Types of Classifications

As discussed here, a heteropolar winding creates alternative magnetic poles on the side of the core facing the air gap. The distance between the first and last slots that accommodate a coil’s sides is called coil throw. For example, if one side of a coil is in slot number 5 and the other side in slot number 15, then the coil throw is 15-5 = 10 slots.
The pole-pitch is defined as: And its unit is slot. For example in a 24/4 slots/pole winding, the pole-pitch is 24/4 = 6 slots.

In a winding, disregarding the throw of the coils and whether the coils are similar or have different throws, the average coil span is a unique number and is considered as the winding span. When it is equal to the pole pitch, the winding is called a full-pitched winding.

Hetero-polar winding can be categorized in different ways:
1) Whether they are closed or open
2) Their topology (ring or drum)
3) If the coils are pre-formed or inserted strand by strand (form or mush)
4) Number of layers in each slot
5) The type of end connections (spiral, lap or wave)

### 1. Open and Closed

All heteropolar windings belong to one or the other of the two types, one with an open geometry or a closed one.
An open winding is the one in which, starting with any conductor and tracing progressively through the winding, a ” dead-end ” is finally reached.
Whereas in a closed winding, the starting point will finally be reached after having passed through all, or some sub-multiple of the coils.

Nowadays, DC machines are always constructed with armature winding of the closed type.
In AC machines with odd number of phases, depending on the phase connection, the winding could be closed or open. In three-phase windings connected in Y, the winding is of the open type, while in ∆ connected ones, the winding is of the closed type.
A ∆ (or Y) winding interface with the three-phase supply is at three points separated by 120º, which is definitely different from the way a DC machine commutator serves as an interface between the armature winding and the single-phase DC power supply.
AC machines with even number of phases, always use the open type winding.

### 2. Ring and Drum

The ring winding (Gramme ring) is one in which, the coils are wound in and out around the core in helical fashion.
In this winding, the coils are usually connected successively to each other so as to form a continuous circuit. The main disadvantage of ring winding is that there are conducting wires outside of the core (if we use it on stator side) which do not cut the flux lines and which do not, therefore, contribute to the EMF buildup and electromagnetic energy conversion, Figure 1. Figure 1. Ring and drum coils

The drum winding (Figure 1) was developed (by Hefner-Alteneck and Siemens in 1873) to reduce the amount of dead wire in a ring winding. Each coil is wound on the inner surface of the stator (and the outer surface of the rotor), with both coils sides inserted in the slots on the same side of the core along the shaft.

The drum armature may be thought of as evolved from the ring type by moving the outer connections of the coils to the inner surface, at the same time stretching the coil circumferentially until the spread of the coil is approximately a pole pitch.

Ring windings are not used anymore, but in a short stack with a very large stator bore and low number of poles (especially with a relatively narrow stator yoke), using ring winding could be justifiable.
Ring windings are especially useful for educational purposes in DC machines and in future posts, I will use them often. But in AC windings discussion we don’t talk about them any more.

### 3. Formed and Mush (Random)

In designing an AC machine, we cannot constantly increase the power rating just by increasing the geometric dimensions. To keep the amount of flux density (while increasing the dimensions), analytic calculation shows that the coils turns have to be reduced, where with more increase in dimensions, they become even smaller than one (without a physical meaning). To keep the number of coils turns in an appropriate level (for more easily winding too), we have two options:
1) Reducing the working frequency
2) Increasing the working voltage level
The frequency is dependent on the network and cannot be generally changed (except in inverter driven machines). So the only option would be increasing the voltage level where the coils need to have stronger (and thicker) electrical insulation and a definite clearance (depending on voltage level) from each other. That’s when we use form (or preformed) coils. The coils are made of enameled rectangular copper wires (wrapped by layers of insulation), with a diamond shape and are usually built by special equipment. The stators must have open slot, so the coils can be inserted. Figure 2. Form winding

Mush winding is made of enameled round copper wires and the coils are inserted strand by strand (manually or automatically) in semi-closed stator slots. As the magnet wires are randomly inserted in the slots, so often called random winding. Figure 3. Mush (Random) winding

### 4. Number of Layers

All full-pitched windings can be of a single-layer type. As the harmonics cannot be fully controlled in them, single-layer windings are not used for large electric machines: Figure 4. Single-Layer Winding

To reduce space harmonics, in addition to using a distributed winding, we can use a double-layer winding where we have the option of choosing a winding span different from the pole pitch (usually smaller). In double-layer windings, with some slots and poles combinations, we can even remove a specific annoying space harmonic.
In integer-slot winding, with a winding span different from the pole pitch, we cannot use a single-layer type at all, but in fractional-slot winding (where the winding span is always different from the pole pitch), in some cases we can use the single-layer type.
A single-layer winding can always be replaced by the double-layer type, but the opposite is not true. Figure 5. Double-Layer Winding

### 5. Types of End Connections

The end connections of a distributed winding may be arranged in several ways, all electrically identical, as shown in Figures 6-8. All these diagrams are of half-coiled (will be explained soon) type. The coils throw, the order they are connected together and grouped, give rise to three different configuration: Figure 6. Spiral Figure 7. Lap Figure 8. Wave

As was mentioned in Basic Rules, as long as you keep the positions and current flowing directions, you can make any other pattern, but probably won’t be of any practical use.

## Practical Winding Patterns

Differed mainly by the type of the end-connection, the following configurations constitute the majority of the windings used nowadays.

### 1. Concentric

Generally only mush windings are of concentric type (form windings could be of concentric type too, but are not recommended at all). It could be half-coiled (unbifurcated) or whole-coiled (bifurcated), where both can be done with single or double-layer configurations.
All concentric windings use the spiral end connection type. Due to the shape of end winding that resembles concentric circles, it is usually called concentric.

#### i) Single-layer

As was already mentioned, in a single-layer winding the average coil-span is equal to the pole-pitch.  It could be made of half-coiled groups or whole-coiled ones which could have major manufacturing differences.

In half-coiled winding (or group per pole-pair), each group of coils, uses the whole phase-group slots (so has as many coils as the number of phase groups slots) and creates two poles.
This configuration is not used in two-pole windings. Figure 9. Half-coiled concentric 24/4 slots/pole

In whole-coiled winding (or group per pole), the phase group splits to two usually equal sub-groups where each sub-group accommodates half of the coils and creates one pole. Figure 10. Whole-coiled concentric 24/4 slots/pole

The end-winding dimension in half-coiled type is axially longer but radially narrower. The order of groups insertion could be planned somehow so the end winding comprises two concentric circles.
The end-winding dimension in whole-coil type is axially shorter but radially thicker and it comprises three concentric circles.

#### ii) Double-layer

Double-layer winding is made of two interleaved single-layer windings; before interleaving, depending on the winding span, one would be shifted one or few slots.
double-layer concentric winding can be used when we want to simulate a lap (or wave) winding with short-pitched coils and instead of manual insertion of coils, use automatic equipment to achieve faster manufacturing time.
Figures 11 and 12 show concentric windings equivalent of a lap winding with coil spans = 5/6: Figure 11. Two single-layer whole-coiled concentric (shifted 1 slot) Figure 12. Two single-layer half-coiled concentric (shifted 1 slot)

All double-layer windings are of whole-coiled type, so the connections between the groups are done similar to Figure 10: Figure 13. Double-layer concentric 24/4 slots/pole (coil span = 5/6)

### 2. Lap

Figure 14 shows a single-layer lap ( the successive coils lap back over each
other) winding which is not used in AC machines, but with some change it is used in universal motors armature, where the geometric symmetry is of importance: Figure 14. Single-layer lap 24/4 slots/pole

The same way that we interleaved two single-layer concentric type to a double-layer one, we can use two single-layer lap winding and build a double-layer lap winding: Figure 15. Two single-layer half-coiled lap (shifted 1 slot)

Figure 16 shows the resulting double-layer winding where all coils are similar and not only geometrically but also magnetically symmetrical (if one side of a coil lays at a slot bottom, the other side will lay at another slot’s top). In other words, the slot and end winding leakage reactances of all phases are the same. Both mush and form coils can use lap winding. To control the space harmonics, the coils span is usually shorter than the pole-pitch which both reduces the harmonics and helps with lower coil resistance and less copper loss. Figure 16. Double-layer lap 24/4 slots/pole (coil span = 5/6)

### 3. Wave

In wave winding, the coils progress continuously in a wave fashion around the periphery of the armature. The procedure from a single-layer to double-layer is like a lap winding, so only the resulting double-layer is shown: Figure 17. Double-layer wave 24/4 slots/pole (coil span = 5/6)

In DC machines, lap and wave winding have different features and depending on the application, one might be preferred to the other.
In AC machines, they are almost the same. They only differ in the total length of the wire required to make the coils and coils connections. With a large number of coils turns and phase belt slots, the effect of coils ends is insignificant and the total wire would practically be the same. With a small number of phase group slots (2 or 3), a large number of poles and a small number of coil turns (especially when it is one), wave winding is more attractive. Then the saving in copper might be 5-10%.

Posted in Heteropolar, Windings | | 2 Comments

## Necessary & Sufficient Conditions For Continuous & Unidirectional Electromagnetic Energy Conversion

Consider that there are totally N parallel circuits in the stator and rotor slots of an elementary drum type electric machine. The windings are embedded inside the slots on the inner surface of the stator and outer surface of the rotor. Each circuit may consist of many coils connected in series.
From classic topics on the electromagnetic energy conversion, assuming that the permeability of the magnetic circuit is very high (a linear material), we have the following general formula for the electromagnetic torque development: J refers to the currents on the stator and k to the currents on the rotor. Ljk refers to the self-inductance of the winding j (when k = j) and mutual inductance of the windings j and k (when k ≠ j).
In motoring mode, Tem is positive and acts in the direction of rotation where electrical energy is converted to mechanical. In generating mode, it is negative and acts against the rotation where mechanical energy is converted to electrical.
From the above formula, a necessary condition for a non-zero torque is a change in the self or mutual inductances of the coils as the rotor rotates. In other words, an electric machine will perform its function if the derivative of at least one quantity (self or mutual inductance) with respect to the angular position of the rotor is non-zero: In real world with a practical design, none of the magnetic fields, flux linkages, self or mutual inductances are monotonically rising or dropping functions of currents or rotor position; so the only possible case is when these quantities change periodically as functions of θ, where the derivative dLjk/dθ changes periodically too.
For Ljk to be a periodic function of θ, it is essential that the current flowing in the stator coil k sets up a spatially periodic magnetic field along the periphery of the air gap. To meet this condition, the conductors must be spatially distributed in such a way that the resulting magnetic field be periodic along the periphery of the air gap.

This is a necessary condition, but not a sufficient one for a continuous, unidirectional electromechanical (or mechanoelectrical) conversion of energy. In addition to the periodicity of magnetic field along the periphery of the air gap, the currents on the stator and rotor, must change in such a way that not only the instantaneous value but the average value of the Tem be sufficiently large.

While a single-phase AC current flowing in a winding, produces a pulsating magneto-motive force (MMF) in the air gap (resulting a zero average and hence a zero torque), two or more spatially shifted windings supplied with the same number of timely-shifted currents result in creating a rotating magneto-motive force with a constant and non-zero average. The time shift of the currents must be corresponding to the space shift of the windings. The time and space shifts have specific quantities for a winding with a specific number of phases.

Therefore the necessary and sufficient conditions for conversion of electromagnetic energy are:

1. The self or mutual inductances of the machine must be a function of the rotor position and change periodically as the rotor rotates (or moves).
2. The currents ij and ichange somehow in time that not only the instantaneous but also the average value of Tem becomes large enough.

Now some of the windings and core designs capable of producing a spatially periodic magnetic field are shortly discussed.

## Heteropolar Winding

It creates alternative magnetic poles on the sides facing the air gap. The current flowing in the conductors laid in the slots on the side facing the air gap, alternates in direction periodically (due to the interconnection between groups of the coils). This gives rise to a periodic variation of magnetic field in space and the core side facing the air gap is magnetized with alternative pole circumferentially: N, S, N, S etc.
In other words, the flux density in different poles of a heteropolar field is the same and only the poles signs (direction of the flux lines) change which implies the fact that the saliency or smoothness of cores is not important.

The width of each pole, along the periphery of the air gap is called pole pitch. If we designate the pole pitch as yp, then the number of pole pairs will be given by:

pp = πD / (2yp)         (1)

where pp stands for pole-pair and D is the air gap diameter.

## Homopolar (Acyclic) Winding

A ring coil that encloses the shaft of an electric machine produces a homopolar field in the air gap. If the outer surface of the rotor core has polarity N, the inner surface of the stator core will have polarity S.
Not similar to a heteropolar winding that creates alternate poles (that have the same flux density but alternate flux directions) in either a smooth or toothed core, a homopolar winding creates a unidirectional field with alternate flux densities. To create different flux densities, the core must be toothed. The number of the regions with different flux densities is determined by the number of the teeth on the rotor surface.

The magnetic flux density in the air gap changes with a space period equal to the tooth pitch, or spacing between adjacent teeth, tZ. The number of pole pairs will then be:

pp = πD / tZ = Number of core teeth          (2)

In comparing (2) to (1), we notice that in homopolar winding:

1. The resultant magnetic field undergoes twice alternations per revolution compared to a heteropolar design.
2. The number of poles is determined by the number of teeth (or slots), where each tooth (and slot) represent a pole pair. Because the conductors need not be laid in slots (the ring winding is external to the core), there is no limit to the slots size and they may be however small. While in heteropolar winding, the pole width is determined at the design stage and disregarding the number of slots, is usually spans few teeth (or slots).

Therefore with homopolar winding we are able to build machines with much higher number of poles compared to a heteropolar design. As a generator, they can produce high frequencies while rotating at moderate speed and as motor they can rotate at low speeds while supplied by moderate frequency.

## Homopolar Winding And Claw-Shape Core

As was seen, a heteropolar winding created a heteropolar magnetic field and a homopolar winding a homopolar field.
But a homopolar winding can be used to create a heteropolar field too.

To create a heteropolar magnetic field by a homopolar coil, we consider this fact that any source of a magnetic field creates both N and S at the same time, so we just need to direct the flux lines in each direction to alternately pass the air gap. This has been done in a configuration that is called claw-shape core whereby a heteropolar field is produced by a simple coil interleaved between two disks with claw-shape edges. By somehow different design of the rotor core, the homopolar coil can be moved to the stator side (so stationary) which eliminates a need for slip rings.

## Facts

Students of electrical engineering at the first days of their first electric machines course learn that rotating electric machines are classified into DC, induction and synchronous. They might get familiarized with switched reluctance (SR), stepping, PM brushless and axial-flux machines too.
In an utmost classic effort so far, the electric machines have been categorized into DC and AC; then DC divided into brushed and brushless, AC into synchronous and asynchronous and each into single and three phase ones. Except DC and induction machines, all the other above mentioned types are classified as synchronous machines.
Actually, from geometrical point of view, all except axial-flux machines belong to the category of cylindrical/drum or radial-flux types.
From electromagnetic point of view, all except switched-reluctance and stepping motors (singly-excited types), belong to the same category: the first type of doubly-excited machines.

The classic criterions of classification do not establish any logical basis to help us shape our accumulated knowledge of electric machines; it doesn’t present the big picture to know what is already found and what is missing, so direct us with a logical approach for inventing new ones.
We need a novel classification to understand their fundamental differences and similarities in the entire perspective. So we can build a methodological approach to help invent new species of electric machines.
First, some agreements: consider two solid elements, one stationary and one capable of moving or rotating around an axis, (called stator and rotor respectively). Without missing generality of the discussion, we assume that in rotating case, the stator is the outside element and the rotor is the inside one; and for any kind of motion, the moving element is called rotor.
Second, the electromagnetic energy conversion is the result of interaction of the following three fundamental concepts (variables) that are perpendicular to each other:
♦  Current
♦  Flux
♦  Torque

## Classification Types

The classification can be done from two general points of views:
♦  Geometric Viewpoint
Depending on the geometric preferences and the axis that we prefer the torque to be developed along, the number and perpendicularity of these three variables, make them each be appropriately assigned to one of the axes in one of the coordinate system types. This leads to diverse topologies, one in Cartesian, six in cylindrical and three in spherical coordinate systems.

♦  Electromagnetic Viewpoint
Where and how to setup the sources of the above three variables to properly create interacting magnetic fields for developing a continuous and unidirectional torque (or force) leads to three major types. All familiar electric machines belong to any of these three types.

### Geometric Classification

Depending on the geometrical features of an electric machine, it could fit in one of the three coordinate systems:

1. Cartesian coordinate system: When flux is along Z axis and current along Y, the moving part of the device will have a motion along X. As all three axes have the same properties, the axes could be exchanged without any change in the topology of the machine. Therefore only one combination is possible. Linear machines (like linear induction motors and MHD DC machines) fit within this system.
2. Cylindrical coordinate system: The three axes (R, θ and Z) have different properties and the exchange of the axes of the coordinate systems leads to six different kinds classified under radial, axial and toroidal flux machines. Drum type machines (like DC, synchronous, induction and so called linear generators) and axial-flux motors fit within this system.
3. Spherical coordinate system: Among three axes (R, θ and φ), θ and φ have the same properties; so the exchange of the axes of the coordinate systems leads to three different kinds. I’m not aware of any real machine in this category, but it’s quite possible to design and build a spherical electric machine.

For now, only those machines are considered that fit within cylindrical coordinate system, as the majority of electric machines have been built in this category.
As mentioned, different assignments of the three fundamental variables to the three axes, lead to six different configurations (Table 1). The configurations in green color are the ones that have been developed so far. To be able to visualize the transformation of different types to each other, we have to keep in mind the following facts:
1. The active component of flux crosses the air gap between the stator and the rotor.
2. The currents close along their distribution direction.
3. The developed torque acts along the currents distribution.

It’s important to note that in this classification only the effective path of each variable has to be considered.
For example, in the first column, the flux flows through laminations teeth (R component) using the stator and rotor yokes (ɵ component), but only the R component cuts the wires. Currents flow through the wires along Z axis (Z component) and close using the end windings (ɵ component), but only the Z component is cut by the flux lines.

In Table 1, the type of machine is related to the direction of the flux; therefore drum machines could be called radial-flux machines too. The diligent reader could find out if any more configurations other than the first three can be built and how.

In next posts, before classifying the electric machines from electromagnetic point of view, the necessary and sufficient conditions for electromagnetic energy conversion will briefly be explained.
Then we choose one configuration (Drum type) and see how differently we can set up the three main variables sources within that configuration for a successful unidirectional and continuous electromagnetic energy conversion.

## Background

Fractional-slot windings were invented for some purpose that was totally different and limited from what they were later used for.

Assume that the armature of a three-phase salient-pole synchronous generator is designed with nine slots per pole (like in a 108/12 slots/pole stator with q = 3 slots) and that the rotor pole arc embraces 60% of the pole pitch. As the rotor poles rotate relative to the armature, the number of slots under each pole face will be alternately five and six, which leads to a pulsation in the permeance of the main magnetic circuit and thereby set up a high frequency harmonic in the induced voltage. Problem may arise when this harmonic has such amplitude as to cause interference with telephone circuits in the vicinity of the lines supplied by the generator. To solve the problem, the generator must be redesigned with 99/12 slots/pole instead, where the pole face sees the same number of stator slots in all times. The new stator design would require a fractional-slot winding with q = 11/4 = 2.75.

This kind of winding is the most complicated and interesting type and has some unique uses like in PAM, BLAC/DC motors and when the same lamination wanted to be used for variety of poles. Especially in high-pole windings where the phase groups cannot have more than two slots, fractional-slot winding is a universal choice.

Nowadays, electronically commutated motors (ECM or so called BLAC/DC) that utilize fractional-slot windings, constitute the most interesting and demanded ones where q is not only a fractional number, but also it is smaller than one, which leads to a winding with coils span = 1.

Due to difference in poles widths in fractional-slot windings, they produce even harmonics in addition to odd harmonics. Except few, a large class of fractional-slot windings produce a number of harmonics that are not integer multiples of the main harmonic. These harmonics called sub-harmonics, are usually stronger than the slot and space harmonics and could lead to an uncontrollable level of noise and vibration.

## Analysis

As an example of a fractional-slot winding, consider a stator of 18/4 slots/pole where the number of slots per each phase group is equal to:

Or 3 coils per a pole pair.

In integer-slot windings, q (the number of slots per phase group) is an integer number (the denominator is one) and there are an integer number of slots per each phase group. Therefore all phase groups have the same number of slots.
In the above example, because 3 slots cannot be shared between 2 poles equally, the only approach would be assigning 2 slots for one pole and 1 slot for the next. We define a ‘unit’ as the number of slots that accommodates 2 poles (the denominator). Later we see that in fractional-slot windings ‘units’ play the role of poles as in integer-slot windings.
The slot pitch is 4×180⁰/18 = 40⁰. Here is the arrangement of slots angles for this winding: Figure 1. Slot angles in a winding with 18/4 slots/pole

By subtracting 180⁰ from all those slot angles that are bigger than 180⁰ (as many times as required until the angle is less than 180⁰) the following layout results. As is seen, slots 10 to 18 have the same angles as slots 1 to 9 which means our units are made of 9 slots. From q=3/2 we conclude that each unit consists of 2 poles (and 3×3=9 slots). Figure 2. Slot angles after subtraction of multiple 180⁰

The problem now changes to finding a winding for a 9/2 slots/pole configuration. The favorite 18/4 slots/pole is made of 2 similar units of 9/2 slots/pole: Figure 3. Slot angles in one unit

In  integer-slot windings, the angles of slots were: 0, αs, 2αs, 3αs, 4αs, etc (αs is the slot pitch). But the above diagram shows something different; the slot angle increases sequentially up to slot 5 and then drops to 20º (half the slot pitch); and then continues with the same order.
The above diagram indicates that the winding poles don’t span 180° anymore but only 90°, which is the result of reduction of the slot pitch αfrom 40° to 20° (half).

Rearranging the slots according to their angles knowing that each phase consists of 3 coils, we have: Figure 4. Dividing the unit to three equal groups

To distinguish three phases, each set of three consecutive slots have different colors. With Red for A, Green for B and Blue for C, notice that the order of AZB has been followed. In other words, the first 3 slots (1, 6 and 2) belong to phase A, the next 3 slots (7, 3, 8) to phase Z (end of phase C) and slots 4, 9 and 5 belong to phase B.

From Figure 4, we can find the rule for the layout of a fractional-slot winding.
Supposing that the consecutive slots of the same phase are separated by equal distances; then we have the following relation (x is the distance (per number of slots) from the former slot in the same phase. n is denominator of the q and K is an integer number):

Remember what we did to get the layout in Figure 4; we subtracted as many 180º from the slot angles until the angles were between 0 and 180º. After this subtraction, the distance between any two consecutive slots were αs/2

After solving for x we have: It is clear that 180n/αs is equal to the number of slots in a unit. From the other side, we know that number of slots in a unit is equal to 3m and we finally have:

Where it means: to find the distance of phase group slots in a fractional-slot winding we have to find the smallest integer number x by trying K=0, 1, 2, 3, etc in the above formula. From the above equation it is seen that if n=3, then we cannot find any integer number for K to make x integer. It means the necessary condition for a three-phase fractional-slot winding to be balanced is that n≠3k (k = 1, 2, 3, …).

In this example the smallest integer x is equal to:

As it’s shown below, the sequential slots (of the same phase) from their angle point of view, are spaced by 5 slots: Figure 5.  Rearranging the coils according to equation (5)

The arrangement is the same as the original one in Figure 3 except for the colors that separates phases from each other. Therefore second slot of phase R is 1+5 = 6 and the third slot of phase R is 6+5 = 11 which in the diagram is equal to 11 – 9 = 2.

So far, we were just concerned about obtaining a diagram that specifies the location of phases’ slots (and the directions of the corresponding flowing currents). The above procedure can be used for building every kind of three-phase double-layer fractional-slot windings directly. A small family of double-layer fractional-slot windings can be reduced to single-layer ones which require some manipulation. I leave it to the diligent readers to find the procedure.

Posted in Fractional-Slot Windings | Tagged , , | 1 Comment