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. L_jk 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, T_em 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 dL_jk/dθ changes periodically too.
For L_jk 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 T_em 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 i_j and i_k change somehow in time that not only the instantaneous but also the average value of T_em 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 y_p, then the number of pole pairs will be given by:

pp = πD / (2y_p)         (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, t_Z. The number of pole pairs will then be:

pp = πD / t_Z = 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.