MATHEMATICAL MODELING

OF THE EQUATION OF MOTION

OF A SYNCHRONOUS RELUCTANCE MOTOR

E. V. Kononenko, A. F. Fink

(Presented by the scientific seminar of the Department of Electrical Machines
and General Electrical Engineering)

Electromechanical transient processes of a synchronous reluctance motor (SRM) are described by the nonlinear system of Park–Gorev differential equations [1].

In a number of cases the influence of electromagnetic transient processes may be neglected, since they proceed so rapidly that during this time the rotational speed of the machine rotor does not have time to change appreciably. Therefore, the processes of pulling into synchronism, as well as sudden load variation, can be studied by investigating the equation of motion of the SRM rotor, which may be represented in the form:

\[ H \frac{d^{2}\Theta}{dt^{2}} + K \frac{d\Theta}{dt} + M_s = M_c, \tag{1} \]

where \(H\) is the mechanical constant;

\(\Theta\) is the angle formed by the transverse axis of the rotor with the voltage vector \(U\);

\(K\) is the proportionality coefficient, which in the general case depends nonlinearly on the slip magnitude

\[ S = \frac{d\Theta}{dt}; \]

\(M_c\) is the resisting torque;

\(M_s\) is the synchronous torque, which, taking into account the active resistance of the stator winding \(r\), may be represented in the form:

\[ M_s = \frac{U^2 (x_d - x_q)} {2 (x_d \cdot x_q + r^2)^2} \left\{ (x_d \cdot x_q - r^2)\sin 2\Theta \right. \]

\[ \left. {} - r \left[ (x_d - x_q) + (x_d + x_q)\cos 2\Theta \right] \right\} = A \sin 2\Theta - B \cos 2\Theta - C, \tag{2} \]

where \(x_d, x_q\) are the synchronous inductive reactances along the \(d\)- and \(q\)-axes.

In studying the processes of entering synchronism and sudden load variation, i.e., processes that occur at small slips, it may be assumed that \(K\) is a constant quantity and determines the steepness of the starting characteristic.

Equation (1) can be solved by the phase-plane method, as presented in [2, 3]. However, taking into account the laboriousness of this method and its comparatively low accuracy, the equation under consideration

it is more expedient to solve on a continuous-action mathematical machine.

The computing elements that implement the operations for solving equation (1) are shown in Fig. 1. In studying the asynchronous mode, the circuit uses a periodizer of the angle \(\Theta\), which is assembled on a highly sensitive polarized relay.

When equation (1) is studied on the EMU-10 analog computer, by varying the motor parameters \(H\), \(K\), \(M_c\), the limiting conditions for reliable pull-in of the rotor into synchronism are determined. In an analogous way, the limiting values of the increase in resisting torque are determined at which the motor continues to operate in synchronous mode. The solution of equation (1) was represented both in phase coordinates and as a function of time.

Fig. 1. Block diagram for modeling the equation of motion of a synchronous reluctance motor.

Fig. 2 presents phase portraits illustrating:

a) stable asynchronous operating mode;
b) reliable pull-in into synchronism;
c) limiting case of pull-in into synchronism;
d) process of oscillation of the SRM rotor under a sudden change in load.

a          b

v                g

Fig. 2.

REFERENCES

1. A. A. Gorev. Transient Processes of a Synchronous Machine. GEI, 1950.

2. E. V. Kononenko, A. F. Fink. Investigation of the equation of motion of the rotor of a synchronous reluctance motor by the phase-plane method. Proceedings of TPI, vol. 139, 1965.

3. E. V. Kononenko, A. F. Fink. Pull-in into synchronism of a synchronous reluctance motor. Proceedings of TPI, vol. 145, 1966.