Fig. 1

It is probably fair to say that most of us do not have much to do with the mathematics of transformation theory, yet much of what we do relies on the theory of one or more transformations. One of the commonest is the Fourier transform, which arises in many branches of engineering. The theory of electrical machines has many famous transformations of its own: for example, Park’s transform, the symmetrical components transformation, the forward-and-backward-fields transformation, and the turns-ratio transformation. In the solution of electromagnetic problems, we can say that the finite-element method is a dramatic series of transformations from calculus into algebra and thence into binary arithmetic; and from there into vivid visualizations on our screens.

The theory of transformations is quite mathematical and it may be daunting to many. It may help to realise that in many cases the essential idea is not purely mathematical, but has a physical origin. In such cases, the physical vision probably originated in the mind’s eye of the engineer before the mathematical form evolved. Over time, mathematicians added such things as *necessary and sufficient conditions*, considerations of *uniqueness*, *symmetry*, and *power invariance*, and of course their *lemmas* and *tensors*. As engineers we may be tempted to regard these accretions as academic baggage, and in the teaching of the practical use of transformations they may present an insurmountable barrier to the student and instructor alike. But make no mistake: mathematical rigour is indispensable to the engineer, even if it is contributed by someone we can’t fully understand, and even if it appears to be scorned by giants such as Oliver Heaviside (1850−1925).

Heaviside’s famous example is the transform which he formulated for the solution of circuit equations, differential equations which could also be tackled by the Laplace transform. Laplace’s mathematics was of impeccable pedigree, while Heaviside might once have been characterized as a “mere” telegraph engineer; and although the genius of Heaviside was fully recognized over time, it took *decades* for his transform to acquire the seal of formal mathematical approval. In electric machine theory, Gabriel Kron was another famous name whose vast work on practical and useful transformations was not acknowledged to be rigorous at the time of its first publication. Kron was motivated to facilitate the calculation of complex circuits including electric machine windings and power systems, and although he worked long before the computer age, his methods proved themselves in computer analysis thirty or forty years later.

I would like to mark this article with a quotation from Oliver Heaviside:

*Vigour before Rigour.*

He doesn’t say “vigour *instead of* rigour”, or “vigour *without* rigour”, but merely “vigour *before* rigour”. It represents the spark of genius — the vision in the mind’s eye, which often attains a practical embodiment before it is put on a firm theoretical basis (often by others, specialists in mathematics and theory). The English word *before* can also imply *more important*; so we have to be rather careful. While *Heaviside* might put vigour before or above rigour in importance, I’m quite sure *I* would not be so brave: for rigour surely provides intellectual comfort even when we might not fully understand it.

*Heaviside ‘proved’ nothing ; he simply made sure he was right. To make sure of being right he appealed to experiment and measurement…*” (—Josephs [1946]; emphasis added)

I wonder if this quotation might apply to the engineers and mathematicians (including software specialists) who prepare numerical analysis tools like the finite-element method. If so, they are in very good company. But we’re digressing. . .

Because we are continually learning and teaching and studying and reflecting on our methods and theories, the need to express the meaning of transformations and other processes never goes away. So in the remaining paragraphs I would like to try to explain some of the common transformations in electric machine theory in plain language *without any mathematics*. In doing this, I hope to describe the vigorous original physical ideas; and although I have used almost no mathematics, I do believe the need for mathematical rigour is self-evident.

**The turns-ratio transformation**. This arises in transformer calculations, but in electric machines the commonest instance is in the equivalent circuit of the induction motor. In the physical induction motor, the voltages and currents and the frequency are all at completely different levels in the stator and rotor circuits. What the turns-ratio transformation does (with some help from Faraday’s law and Ampère’s law) is to represent the rotor quantities in that part of the circuit that belongs to the stator. It performs this function not only for voltages, currents, and frequency, but also for power, reactive power, and impedance. The resulting simplification in calculation is *enormous*. It also enhances physical visualization.

**The commutator transformation**. The name of this transformation obviously derives from the DC commutator machine, but it is also used with synchronous machines where it is often seen as *Park’s transform*. The transformation is used when one “member” (the stator or the rotor) carries DC (or permanent magnets) while the other carries AC, and it represents the AC quantities on the DC side. Again, the resulting simplification in calculation is *enormous*. It also enhances physical visualization.

It is common to express transformations in terms of a matrix equation, **u**_{dq0} = [C]**u**_{abc}, where, for example, **u** represents a vector of voltages or currents in the *abc* frame of reference on the AC side, or the *dq0* frame of reference on the DC side; and [C] is the transformation matrix. The properties of the transformation matrix are open to mathematical scrutiny, and these properties have vital physical significance concerning “power invariance”, the appearance of apparently arbitrary coefficients, and even the appearance of non-reciprocal mutual inductances. When electrical engineers apply a particular transform, they should ideally understand the mathematical properties of the matrix : for example, as to whether the inverse is equal to its transpose or the transpose of its complex conjugate. Without the mathematical analysis, unexpected results can arise, and this can make it harder to detect errors. Sometimes arbitrary coefficients such as √3 are introduced to force power-invariance, but this makes it difficult to reconcile the results with test data as ordinarily measured. The whole process, however executed, requires extreme care!

**The symmetrical components transformation**. In its original form [Fortescue, 1918] this was developed to simplify the calculation of unbalanced polyphase circuits, but when it is applied to AC induction motors it turns out that the positive- and negative-sequence components are closely associated with fields and ampere-conductor distributions that rotate in opposite directions. It is therefore close to the **forward-and-backward components transformation** which is central to the theory of single-phase and unbalanced induction motors. Again, the resulting simplification in calculation is *enormous*. The forward-and-backward components provide enhanced physical visualization, although the original (mathematical) symmetrical components transformation could be said to be less easy to visualize.

**Space vectors**. Although we don’t usually think of space vectors in terms of transformations, they are precisely the result of a transformation from instantaneous three-phase quantities (voltages, currents, flux-linkages) to equivalent complex numbers known as space-vectors. When these are associated with the fundamental space-harmonic of the winding distribution in an AC machine, they acquire phenomenal power in the concise expression of the workings *and the control* of such machines. Not only that, but the voltages and currents at the terminals of an inverter can also be expressed as space-vectors, taking advantage of the fact that space-vectors are *instantaneous* quantities that can respond instantly to the changes in the switching state of the inverter. In this essential regard, space-vectors differ fundamentally from phasors which represent the *steady-state* magnitude and phase of sinusoidally-varying waveforms of voltage, current, and flux-linkage at a fixed frequency. Again, the simplification in calculation is enormous, while the association of a space-vector with the *orientation* of the flux or the ampere-conductor distribution in an AC machine (as well as its magnitude) is immensely valuable. The space-vector is not as old as the other transformations [see Kovács, 1984], but in many ways it is the most extraordinary one: not only is it mathematically elegant and simple, but it also unites the theory of the AC machine with the control theory of the inverter, including the rapidly changing states of its power-electronic switches (transistors). It originated more than 20 years before field-oriented control [Blaschke, 1971] and other AC machine/inverter control schemes; and it was ready when the time came.

**Connection matrix**. The transformation matrix (such as [C]) is usually square and non-singular, so it can readily be inverted to be used in the inverse transformation, **u**_{abc} = [C]^{−1}**u**_{dq0}. In many cases [C] is symmetric so that its inverse is equal to (or at least proportional to) its transpose, a feature that is convenient not only for inversion but also in processing the power equation. But not all transformation matrices are square. An important example is the *connection matrix*, which transforms a set of “primitive” voltages and currents into a smaller number of voltages and currents representing an interconnected network. This type of transformation is at the heart of many algorithms used in circuit analysis and in the calculation of machine windings (which are made up from a multiplicity of interconnected coils). Since the transformation matrix is not square, it cannot be inverted. But this is entirely consistent with the fact that, for example, if we know the voltage across the phase terminals of a machine, we do not immediately know the voltages across the individual coils. It appears that there might be a loss of information, but if we didn’t have the connection matrix we wouldn’t be able to solve even for the terminal voltage and current, and we wouldn’t have *any* information!

When you press the “FFT” button on a digital oscilloscope, you’ll be aware that you are executing a transformation of a signal record between the time domain and the frequency domain. Why do we do that? We want to “look at the frequency content, the spectrum”. This is the common language of the laboratory, even the factory floor. It rests not only on the theory of Fourier’s transform but also on the efforts of the engineers and mathematicians who developed the discrete and then the fast Fourier transform, and on the many specialist experts who programmed all this to provide the functionality of that FFT button. But do you ask yourself if the transformation can be inverted or “undone”, or whether there are any unexpected coefficients to be taken into account in the final result, or whether the result is valid for the signal you’ve entered? Such questions arise with all transformations, and the answers can ultimately be found only by a rigorous mathematical appraisal of the properties of the transform. Pressing the button represents the “vigour”, but let’s remember the “rigour”.

#### Some references

- Blaschke, F.,
*Das Prinzip der Feldorientierung, die Grundlage für die TRANSVEKTOR-Regelung von Asynchronmaschinen*, Siemens Zeitschr.**45**, p. 757, 1971. - Cooley J.W., Lewis P.A.W. and Welch P.D.,
*Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals*, I.E.E.E. Trans.,**AU-15**, pp. 79-84, 1967. - Fortescue C.L.,
*Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks*, Transactions A.I.E.E., Vol. 37, No. 2, pp. 1027−1140, 1918. - Harris M.R., Stephenson J.M. and Lawrenson P.J.,
*Per-Unit Systems*, Cambridge University Press, 1970. - Jones C.V.,
*The Unified Theory of Electrical Machines*, Butterworths, London, 1967. - Josephs H.J.,
*Heaviside’s Electric Circuit Theory*, Methuen, London, 1946. - Kline M.,
*Mathematical Thought from Ancient to Modern Times*, Oxford University Press, 1972. - Kovács P.,
*Transient Phenomena in Electrical Machines*, Elsevier, Amsterdam, 1984. - Kron G.,
*Tensor Analysis of Networks*, John Wiley & Sons, New York, 1939. - Ku Y.H.,
*Electric Energy Conversion*, Ronald Press, New York, 1959. - Retter G.J.,
*Matrix and Space-Phasor Theory of Electrical Machines*, Akadémiai Kiadóm Budapest, 1987.