The tensor equations of electrical machines

In the analysis of modern electrical machine systems there has been a trend towards the use of various components of the variables and parameters of the system, examples being the use of symmetrical components and direct- and quadrature-axis quantities. Such quantities may be considered to be different ¿reference frames¿ to which the various currents, fluxes, etc., are referred. Kron has shown that such transformations may be expressed in a general way when the equations of a network or machine are written in tensor form. Transformations may then be carried out to any one of a number of reference axes, which may be stationary or rotating with respect to the windings of the machine or network. In particular it has been found that in certain cases there are advantages in analysing machines with respect to axes rotating with the flux. The purpose of the present paper is to investigate the tensor form of the equations of electrical machines, to demonstrate the differences between tensor and non-tensor terms and to show how these terms are interpreted in application to simple cases. The equations of primitive machine are examined in both stationary and rotating axes and the equations of a 3-phase series impedance and a 3-phase induction-motor are derived in both systems of reference.