Fig. 1 End-view of a 2-layer 3-phase 8-pole AC winding in 36 slots

Mystery is not confined to novels. We have it in abundance in engineering. I remember as an apprentice in the 1960s working alongside a service engineer on a new DC drive system, and he had some commissioning documents with control system diagrams and circuit diagrams. On one of these I noticed some hand-written calculations containing expressions like *R* + j*X*. This was my first encounter with complex numbers on the factory floor, and it was truly a mystery. I asked the service engineer the meaning of *R* + j*X*, and he ‘explained’ that j was the square root of minus one, and left it there. The mystery deepened!

Later, as a student looking through some old examination papers in power-systems analysis, I noticed a diagram in which voltages were expressed in ‘p.u.’ Naturally as this was an examination paper, no explanation was given as to the meaning of ‘p.u.’ Another mystery!

Yet both of these — I suppose we could call them elements of our engineering language — had something more than mystery about them. I was captivated, although also filled with trepidation (fear of exams!) There was a kind of magic in the idea that here were powerful elegant concepts with wide or even universal application, cryptically expressed, simple, unadorned, and known only to the *cognoscenti* of the world of engineering.

And that is precisely what they turned out to be. In the one case *R* + j*X* was a fundamental introduction to the wide world of AC circuit analysis, on which (at least in the mind of an EE student) the whole of civilization depends. And in the other case the simple expression ‘p.u.’ or ‘per-unit’ was a fundamental introduction to the wide world of *normalization*, a fascinating topic that reveals much about our understanding of processes and devices, and is much more than a convenient way to get the numbers into a manageable range of values.^{1}

In the field of AC machine design and manufacture, there has to be a continuous thread of knowledge and understanding between different aspects of the operation, and between the technical specialists concerned with these aspects. For example the winding shop must have procedures for inserting coils into slotted stator cores, and connecting them to the right phase terminals with the right polarities. Fig. 1 shows an example of a 3-phase stator with 36 slots, wound for 8 poles. The diagram is incomplete, because it does not show the interconnectors between the coils. To the finite-element analysis specialist, the interconnectors are of little importance: all that is needed is to be able to specify the ampere-conductors in every slot at any instant of time. But to the winding-shop engineer, the interconnectors are essential, because the winding cannot be assembled without them. Both these specialists are interested in the number of ampere-conductors in the slots, but the analyst will often be concerned with the instantaneous values while the winding engineer will be concerned only with the effective (RMS) values.

Anyone who has worked in a factory will know that there is plenty of room for mystery and misunderstanding in these operations. What one engineer does is often a mystery to another!

In the case of the ampere-conductors in the slots, there needs to be a *mapping* between the instantaneous values and the effective (RMS) values, and this is just a fancy abstract way of expressing part of the calculation and its relation to a possible measurement. Mappings are common in engineering, not only in analysis but also in the definition of processes. The windings of AC machines provide an example.

In the winding-shop we can imagine that the coils will be inserted in an orderly fashion, often by automated tools. ‘Orderly fashion’ implies that each coil will have a number, as well as a number of turns, and the slot-numbers of its two coil-sides. Beyond that it is necessary to have information about the *connection* of each coil, a subject that is treated very sparsely in textbooks, yet which is amenable to treatment by *connection matrices*.^{2} These are themselves mappings — non-square matrices — from the phase terminals to the terminals of each and every individual coil.

Now if we have that mapping we can determine the current in every coil, and the number of ampere-conductors in every coil-side; but we need a further *mapping* to find the total number of ampere-conductors in every *slot*, bearing in mind that one slot may contain several coil-sides from different coils and even from different phases (two coil-sides per slot being common, as in Fig. 1). For the convenience of the analyst, it sometimes helps if the slot-ampere-conductors are given in order of the slot number, so there may be a need for a *sorting* function. It is also sometimes desirable to work in the opposite direction, from a given pattern of slot ampere-conductors to an arrangement of interconnected coils — a reverse mapping, if you will. On a point of detail, the slot-numbering scheme and the coil-numbering scheme must both start at a definite point: for example, in Fig. 1 slot 1 is on the *x*-axis while coil 1 has its *axis* on the *x*-axis, so its coil-sides are in slots 35 and 3. Then we have to distinguish between the ‘go’ coil-side and the ‘return’ coil-side, and which will be in the bottom of the slot and which in the top. The complete definition becomes quite complicated. And the winding-shop may very well have their own slot-numbering and coil-numbering schemes that at first sight appear to be remote from the analyst’s schemes, or even incompatible with them. Mappings needed!

The mappings don’t stop there. They can be developed to assist in the analysis of the spatial harmonics of the winding distribution, and the analysis of faults and parasitic losses; but that is too far ahead of this *Diary*. The concluding observation or final comment here is that when we put all this together, it’s *magic*. The mappings unravel the mysteries; and by association, they link the respective members of the team.

References :

[1] Harris M.R., Lawrenson P.J. and Stephenson J.M. ,*Per-unit Systems*, Cambridge University Press, 1970

[2] Blue book,

*Design Studies in Electric Machines*, sales@Motordesignbooks.com 2022.

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