When approaching the study of electric machines in transient condition, it is customary to discard the real physical vector quantities like magnetic field and magnetic induction and to define fake vectors like flux linkages, currents and voltages (among which there also induced electromotive forces) which are directed along the magnetic axis of a certain coil (for instance in a three phase winding system) and have the magnitude equal to the actual scalar value.

I have read that they are not intended exactly as physical vectors, rather as complex numbers, lying on a complex plane which is the shadow of the actual plane defined by a cross section of the machine: for instance the real axis is aligned with the phase 'A' magnetic axis and the other two phases 'B' and 'C' windings correspond to a couple of complex conjugate points; then, such quantities are used, without any particular derivation, as actual vector lying in the cartesian space (for instance the flux has component along two orthogonal axes, usually called \$\alpha, \beta\$ or \$d,q\$).

I perfecly understand, that, in order to simplify the analysis and improve the understanding of a complex system like an electromagnetic device, a certain abstraction, which requires to define also artificial entities, is needed, but I cannot see the reason why such 'abuse of notation' is physically legitimized, in fact I am not sure if it is still an abuse of notation or directly a misuse of notation.

The most confusing situation is the anisotropic (salient pole) rotor used inside synchronous machines, where, for instance, the flux produced by a sinusoidally distributed winding in the stator, linked with another distributed winding (which can be the same phase or another) is simply computed using direct flux and quadrature flux as fluxes produced when the rotor is aligned or in quadrature with that winding.

I think that, when dealing with physical phenomena, before using powerful mathematical tools like these, one should prove that they produce the same result of a strict physical approach (just like when Maxwell equations are replaced by other theoretical tools, like circuit theory or transmission lines, after showing that this is actually possible, not only sought). I hope someone can help me understand this formalism.

EDIT: In order to better explain the problem, if a winding of a certain phase, no matter what, is distributed along the stator inner periphery, it is accustomed to consider the field it produces as sinusoidal (considering just the first angular harmonic) and the coil distribution can be approximated as sinusoidal as well, centred about the winding magnetic axis. Now to compute the mutual inductance between two stator phases, one should compute the flux linkage of a differential coil (in the angular span between \$\theta\$ and \$\theta +\mathrm{d} \theta\$) produced by the sinusoidal field produced by another winding, then integrate this infinitesimal contribution along all the flat angle, whereas often the problem is roughly solved drawing a flux vector (which is not a vector) and decomposing it along the axes of the other windings, like there were just single loop coil for each phase (non distributed windings).

  • \$\begingroup\$ Have you studied the development of this type of analysis in more than one textbook? If not, it might help to look at how this is explained by several authors. I have some understanding, but I don't think it is sufficient for me to be able to contribute to the understanding of anyone else. At minimum, several illustrations would be required. It would be better to have a live presentation. You might be able to find a recorded lecture. \$\endgroup\$
    – user80875
    Jun 19, 2016 at 20:39
  • \$\begingroup\$ You should read Carver Mead's book on Collective Electrodynamics: mitpress.mit.edu/books/collective-electrodynamics Chapter 1~3 will get you fixed up. \$\endgroup\$
    – b degnan
    Jun 20, 2016 at 0:51
  • \$\begingroup\$ What makes Maxwell's Equations a good foundation? According to your approach, those first need to be derived from string theory. (And perhaps we might as well give up altogether until a grand unified theory is worked out) Engineering models are useful because they align with reality, independently of whether there is an elegant formal description. \$\endgroup\$
    – Ben Voigt
    Jun 20, 2016 at 4:44
  • \$\begingroup\$ Actually, from your question I can't tell which direction your approach is trying to go... Your first paragraph seems to aim for more measurable less theoretical models, while your last paragraph requires the measurable model to be derived from the foundational theoretical model, which uses the abstract quantities you start out complaining about. \$\endgroup\$
    – Ben Voigt
    Jun 20, 2016 at 4:49
  • 1
    \$\begingroup\$ I'm voting to close this question as off-topic because it is not completely clear what is being asked and seems to have generated a discussion rather than moving towards a definite answer. \$\endgroup\$
    – user80875
    Jun 22, 2016 at 12:42

1 Answer 1


Perhaps your difficulty is that the frame of reference or the "space" containing the magnetic flux and force vectors is rotating rather than fixed to points on the structure of the machine. In induction motors, the frame of reference rotates with respect to both the stator and the rotor, moving at the synchronous speed with respect to the stator and at the slip speed with respect to the rotor. In synchronous motors, the frame of reference is fixed with respect to the rotor, but rotates at the synchronous speed with respect to the stator. Thus the vector representations of the fluxes and forces have fixed positions with respect to the phase-vector representations of the AC voltages and currents.

I believe that this representation is developed by first examining the vectors with respect to the windings in the machine without considering the motion of the rotor and then examining the machine operating at a steady-state speed.

Equivalent Circuits

The analysis and control of electric machines makes use of equivalent electric circuits that are mathematically analogous to electromagnetic machines. They have a basis in physics, but that basis may not be as firm and direct as you might like. Engineers generally use the tools that will provide answers with a level of accuracy that is suitable for the ultimate purpose. Scientists strive to explain the laws of nature as thoroughly and accurately as humanly possible. Engineers strive to design products that are useful to human endeavors and economically affordable.


Not all engineering terminology is carefully thought out. Even terminology that is carefully created is often simplified in everyday use. As a result, some terminology can have more than one meaning depending on context. I suspect that engineers are more tolerant of imprecise terminology than scientists may be.

  • \$\begingroup\$ I have already studied both the machines dynamic model using the most appropriate reference frame, my hesitation rises only when physical vectors such as magnetic field and magnetic flux density, forces and velocities are confused with electromagnetic scalar quantities, namely voltages, currents, induced electromotive forces (motional and transformer ones) and of course, magnetic fluxes and flux linkages. \$\endgroup\$
    – Vexx23
    Jun 22, 2016 at 14:09
  • \$\begingroup\$ I have read also books that do not use this thoughtless approach, preferring to derive the same concepts using applied electromagnetism laws, the same used for studying non rotating electric machines like transformers for instance. Anyway, I am very sorry that my question has not been proved enough clear and that you are going to vote to close it, I hoped it could instill a debate among the engineers or the scientists in general. \$\endgroup\$
    – Vexx23
    Jun 22, 2016 at 14:11
  • \$\begingroup\$ This site is not intended for debate. Only questions that are asking for design assistance or help with a question about theory are invited here. There is a chat area chat.stackexchange.com/?host=electronics.stackexchange.com You need to have 20 reputation points to post there. \$\endgroup\$
    – user80875
    Jun 23, 2016 at 1:08
  • \$\begingroup\$ Nevertheless I formulated a question, and answering to questions is not off-topic; its level of clearness is another matter, I would be more than glad to improve it or to add the requested details, but I do not think the question is a void curiosity. Of course, answers aimed just to contest the reasoning without addressing the topic generate only discussions, but my idea of debate was essentially bound to the technical aspects of the problem regarding electric machinery. \$\endgroup\$
    – Vexx23
    Jun 29, 2016 at 10:40

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