A typical assumption in most mathematical models is that the system differential equations or transfer functions have real coefficients. In a few applications, this assumption is not valid. In models of rotating machines, the evolution of the vectors governing the system with time depends on their space orientation relative to fixed inertial axes. For an induction motor, assuming symmetry, two stator fixed axes are used as the reference frame: the direct axis (d) in the horizontal direction and the quadrature axis (q) in the vertical direction. The terms in the quadrature direction are identified with a (j) coefficient that is absent from the direct axis terms. The two axes are shown in Figure P7.15.

We write the equations for the electrical subsystem of the motor in terms of the stator and rotor currents and voltages. Each current is decomposed into a direct axis component and quadrature component, with the latter identified with the term (j). The s-domain equations of the motor are obtained from its equivalent circuit using Kirchhoff's laws. The equations relative to the stator axes and including complex terms are

(a) Write the state equations for the induction motor.

(b) Without obtaining the eigenvalues of the state matrix, show that the two eigenvalues are not complex conjugate.