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).
Best Answer
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.
Terminology
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.