Travelling magnetic waves in electrical machines described by rotating vectors

Abstract

In the special case of machines with rotor dimensions small compared with the wavelength, the travellingwave description of the magnetic field gives little insight into the forces and torques that might be expected. It is shown that a more local description of the field in terms of contrarotating vectors can lead to a simple explanation of the behaviour of these machines. v 1 IntroductionThe travelling-or rotating-wave approach to conventional machines is well established, and has led to the development of equivalent circuits which include the effects of space-harmonic fields or of unbalanced operation. 1 The approach can also be applied to certain unconventional machines, e.g. linear induction motors, and in cases when detailed analysis is extremely difficult, it can often be used intuitively to predict directions of resultant forces or torques. Occasionally, however, and particularly when 'rotor' dimensions are extremely small compared with the wavelength of the field, this intuitive application of the travelling-wave approach seems to break down. 2 It is in these 'small-rotor' cases that the rotating-vector approach developed in this paper is particularly effective. 2Basis of rotating-vector concept 2-dimensional cylindrical configurationConsider a smooth, uniform air gap, the inner cylinder being of high-permeability low-conductivity material. Assume that the system has no variation in the z direction and is excited by a sinusoidally distributed, rotating-current sheet on the stator surface. The field in the air gap must satisfy Laplace's equation, V 2 H = 0, and if the current distribution produces p pairs of poles, the solutions of this equation are of the form 3 -4 and // e = {/4r<"-'> -cos (w o t -pd) sin (co 0 / -pd) (1)where H r and H % are the radial and circumferential components of field strength. The constants A and B can be determined from the boundary conditions, which are that, at the inner surface, // ft is zero, and at the outer surface, // e is equal to the surface current.Referring to Fig. la, it will be seen that at any point r, 6 in the air gap, the resultant field-strength vector H can be resolved into its radial and tangential components, or into two contrarotating vectors, one of magnitude Ar^~] ) and the other of magnitude Br~{ p+l ). The angular velocity of these vectors is o> 0 (i.e. 2n x supply frequency) and not a> o lp, the wave velocity.Alternatively, since Hi 2 b 2 where a = Ar^~]) + Br and b = Ar p~] -Br-(