APPLICATION OF THE EMU-10 COMPUTER TO THE STUDY OF TRANSIENT ELECTROMECHANICAL PROCESSES IN SYNCHRONOUS RELUCTANCE MOTORS

E. V. Kononenko, A. F. Fink

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

In recent years, in connection with the rapidly developing process of mechanization and automation of labor, the tendency toward increasing the operational reliability of power installations and, in particular, toward creating highly reliable electrical machines that are simple to operate has become increasingly apparent.

Synchronous reluctance motors, which are ordinary synchronous salient-pole machines without an excitation winding on the rotor, are finding wide application in a number of branches of industry. Synchronous reluctance motors are usually designed with asynchronous starting. Taking into account that starting is a transient electromechanical process, the behavior of the motor in this mode is described by the complete system of Park–Gorev differential equations [1]. These equations, under the condition that the electromagnetic torque and slip are taken as positive in the motor operating mode, and that the \(q\)-axis leads the \(d\)-axis by an angle of \(90^\circ\), using a system of relative units in operator form, may be represented as

\[ \begin{aligned} p\psi_d &= -U\sin\theta - ri_d + (1-S)\psi_q;\\ p\psi_q &= U\cos\theta - ri_q - (1-S)\psi_d;\\ p\psi_{rd} &= -r_{rd}\cdot i_{rd};\\ p\psi_{rq} &= -r_{rq}\cdot i_{rq};\\ Hp^2\theta &= M_c - M_{\text{em}}, \end{aligned} \tag{1} \]

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

\(S\) is the slip;

\(H\) is the mechanical constant;

\(M_c\) is the resisting torque;

\(M_{\text{em}}=\psi_q i_d-\psi_d i_q\) is the electromagnetic torque.

The system of equations (1), with variable rotational speed, is nonlinear and has no regular methods of solution. However, these equations can be solved by applying analog computing machines (ACMs), which are gaining increasing recognition as a tool for solving a wide range of technical problems.

The advantages of these machines include their compactness, the ease of introducing the necessary changes into the problem conditions, and the clarity of the results obtained. The application of ACMs makes it possible

to optimize transient electromechanical processes with respect to one or another motor parameter, which is very important in prospective design.

The use of an analog computer considerably expands the range of questions that can be determined by calculation without carrying out experimental investigations on an actual motor or on a physical model.

The subject of the present investigations is the development of a method for mathematical modeling of the asynchronous mode, the process of pulling into synchronism, and the sudden application of load to a synchronous reluctance motor.

When solving on an analog computer, it is expedient to determine, from the first four equations of system (1), the flux linkages \(\psi_d\), \(\psi_q\), \(\psi_{rd}\), \(\psi_{rq}\) [2], on the assumption that the angle \(\Theta\) and, consequently, its trigonometric functions \(\sin\theta\) and \(\cos\theta\), as well as the currents \(i_d\), \(i_q\), \(i_{rd}\), \(i_{rq}\), have been obtained from the solution of the other equations. The equations for the currents of a synchronous reluctance motor may be represented in the form:

\[ \begin{aligned} i_d &= \frac{1}{x_d}\psi_d - \frac{x_{ad}}{x_d}i_{rd};\\ i_q &= \frac{1}{x_q}\psi_q - \frac{x_{aq}}{x_q}i_{rq};\\ i_{rd} &= \frac{1}{x_{rd}}\psi_{rd} - \frac{x_{ad}}{x_{rd}}i_d;\\ i_{rq} &= \frac{1}{x_{rq}}\psi_{rq} - \frac{x_{aq}}{x_{rq}}i_q. \end{aligned} \tag{2} \]

The elements implementing the operations necessary for solving the systems of equations (1) and (2) are shown in Fig. 1. In order to reduce the number of operational amplifiers, the currents of the rotor circuits \(i_{rd}\) and \(i_{rq}\) are excluded from systems of equations (1) and (2).

Fig. 1. Structural diagram for modeling a synchronous reluctance motor.

In studying asynchronous operating conditions, the angle \(\Theta\) varies over very wide limits, which creates great difficulties in setting up the function generators. Therefore the model must be equipped with a device for periodizing the angle \(\Theta\). Figure 2 shows a circuit for periodizing the angle \(\Theta\), assembled on a highly sensitive polarized relay. The circuit uses three diode function generators adjusted to sine and cosine dependences within the range of variation of the argument \(\pm \pi\), which ensures high setting accuracy.

Fig. 2. Circuit for periodizing the angle \(\Theta\).

In studying the operating conditions of a synchronous reluctance motor, all parameters entering equations (1) and (2) must be known (calculated or determined experimentally). The studies were carried out for a motor whose parameters are presented in Table 1.

Table 1

For the known motor parameters it is not difficult to calculate the initial steady-state condition. When studying the resynchronization of a synchronous reluctance motor, in order to find the initial conditions of the solving elements of the circuit, it is convenient to take the ideal no-load operation of the motor as the initial condition. For the specified parameters this operating condition of the motor is characterized by the quantities presented in Table 2.

Table 2

The solution was carried out in the following order. By an instantaneous application of a torque \(M_c\) considerably exceeding the maximum synchronous torque, the motor was brought out of synchronism. When it reached a slip approximately equal to 0.1, the torque \(M_c\) was reduced to

value corresponding to the specified load value. If the torque developed by the motor is, in this case, greater than \(M_c\), the rotational speed increases.

For the specified parameter values, the motor is pulled into synchronism or continues to operate in the established asynchronous mode. By varying the motor parameters, the limiting conditions for pulling into synchronism are determined. Figure 3 shows oscillograms corresponding to the asynchronous mode and to pulling into synchronism, presented in phase coordinates \(S = f(\theta)\).

a            b

Fig. 3. Results of calculations of electromechanical transients:
a) asynchronous mode,
b) pulling into synchronism.

Similarly, a sudden application of load was investigated. During operation in the established mode, the torque \(M_c\) was instantaneously increased by the specified amount. Under certain initial conditions and load-increase magnitude, the motor remained in synchronism or passed into the asynchronous mode. By varying the parameters, the limiting conditions under which the motor does not fall out of synchronism are determined.

Conclusions

The application of analog computers to the study of transient electromechanical processes makes it possible, with relative simplicity, to establish the influence of design parameters on the process of entry into synchronism and on the dynamic stability of synchronous reluctance motors.

References

1. A. A. Gorev. Transients of a Synchronous Machine. GЭI, 1950.
2. N. I. Sokolov et al. Application of Analog Computers in Power Systems. Energiya Publishing House, 1964.