STUDY OF ELECTROMECHANICAL TRANSIENT PROCESSES OF LOW-SPEED SYNCHRONOUS REACTIVE MOTORS

A. A. BUIMOV, E. V. KONONENKO, A. F. FINK

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

Analysis of electromechanical transient processes, even in the simplest case of operation of low-speed synchronous motors when supplied from a network of infinitely large power, is extremely difficult. This is explained by the fact that, with variable rotor speed, the system of differential equations describing the transient electromechanical processes is nonlinear and has no regular methods of solution. Using the operational form of notation, the equations in the system of relative units may be written as:

\[ \begin{aligned} -U\sin\Theta &= p\Psi_d-\Psi_q(1-s)+r\cdot i_d,\\ U\cos\Theta &= p\Psi_q+\Psi_d(1-s)+r\cdot i_q,\\ M_{\mathrm{st}} &= H\cdot ps+(\Psi_d\cdot i_q-\Psi_q\cdot i_d), \end{aligned} \tag{1} \]

where

\[ H=\frac{4l\omega_1^3}{P_\delta Z_2^2},\qquad \Psi_d=x_d\cdot i_d,\qquad \Psi_q=x_q\cdot i_q. \]

This system of equations, for given parameters and voltage, contains five variables: \(\Theta, \Psi_d, \Psi_q, i_d, i_q\). Time \(t\) is the independent variable, and the slip \(s\) is determined by the equation

\[ s=\frac{1}{\omega_1}\frac{d\Theta}{dt}. \]

To solve this system of equations, an EMU-10 analog computer was used. The structural diagram of the connections of the solving elements, corresponding to the system of equations (1), is presented in Fig. 1. To expand the range of variation of the angle \(\Theta\), it includes a periodization device. By varying the coefficients of the machine equations, it is possible to carry out an extensive program of investigations of transient electromechanical processes of low-speed synchronous reactive motors.

Starting and Pulling into Synchronism

The conditions for starting and pulling into synchronism depend on the load on the shaft and the motor parameters, as well as on the inertial constant of the rotating masses.

In the study of starting and pulling into synchronism, we assume that at the initial instant of time the rotor is stationary and nominal voltage is applied to the stator winding. The results of the solution can be obtained

Fig. 1. Structural diagram of the model

in the form of oscillograms representing the dependences of the quantities of interest to us (electromagnetic torque, angle $\Theta$, currents, flux linkages, etc.) on slip or time for different parameter values.

Figure 2 shows the mechanical characteristics calculated during the starting of a motor having different inertial constants:
a) $H=3.5$, b) $H=4.0$, c) $H=4.5$. The motor had the following parameters: $x_d=1.3$, $x_q=0.9$, $r=0.1$. As follows from these oscillograms, an increase in the inertial constant leads to a deterioration of the starting properties. At $H=3.5$ the motor is reliably pulled into synchronism, and the oscillations about the equilibrium position quickly decay. At $H=4.0$ the start becomes unstable, and upon being pulled into synchronism prolonged oscillations about the equilibrium position are observed. At $H=4.5$ starting of the motor becomes impossible. If transient electromagnetic processes are neglected, then the conditions under which pull-in to synchronism will occur can be determined by solving the rotor equation of motion:

\[ Hps+M_m\operatorname{Sin}2\Theta=M_{\mathrm{cm}}, \tag{2} \]

where $M_m$ is the maximum electromagnetic torque.

Analyzing equation (2) and using [2], one can obtain a simplified criterion for pull-in to synchronism in the form

\[ f=k\sqrt{\frac{M_m z_2}{2\pi GD^2}}, \tag{3} \]

where

\[ \begin{aligned} f&\text{ is the maximum frequency at which pull-in of the motor rotor into synchronism is possible,}\\ z_2&\text{ is the number of rotor teeth,}\\ k&=1.8\text{ is an empirical coefficient obtained on the basis of experimental data.} \end{aligned} \]

Figure 3 gives the experimental and calculated dependences of the supply-network frequency on the moment of inertia of the rotating masses. As follows from the curves presented, they agree fairly well.

Stability of operation of slow-speed synchronous reluctance motors

During operation of slow-speed synchronous reluctance motors, disturbances of the normal mode are possible owing to changes in the paramet-

Fig. 2. Mechanical characteristics of the motor:
a) \(H=3.5\), b) \(H=4.0\),
c) \(H=4.5\)

Fig. 4. Mechanical characteristics of the motor:
a) \(r=0.1\), b) \(r=0.5\)

Fig. 3

of the motor’s electrical circuits, the resisting torque, or the voltages acting at the terminals. The magnitude of the active resistance of the stator-winding circuit has a substantial influence on the stability of operation. Figure 4 shows oscillograms calculated for a motor having the following parameters:

a) \(x_d=1.3,\ x_q=0.9,\ r=0.5,\ H=3.5.\)
b) \(x_d=1.3,\ x_q=0.9,\ r=0.1,\ H=3.5.\)

It follows from these oscillograms that, unlike ordinary synchronous machines, increasing the active resistance of a low-speed synchronous reluctance motor with no windings on the rotor shortens the starting time and makes its operation more stable. Thus, in Fig. 4a, after being pulled into synchronism, the motor has a steady-state oscillation amplitude of the angle \(\Theta=22^\circ\). Increasing the active resistance of the stator winding to \(r=0.5\) makes it possible to obtain stable operation of the motor after pull-in to synchronism.

In the presence of sharply variable motor loads, as well as frequent starts and reversals, it is necessary to investigate dynamic stability. In these modes the motor must not fall out of synchronism and, in some cases, must have a limited, rapidly decaying oscillation amplitude. In considering dynamic stability, one may restrict oneself to solving the rotor equation of motion (2), which, taking into account the asynchronous torque \((k_p\Theta)\) and with \(M_{\mathrm{cm}}=0\), can be written in the form

\[ Hp^2\Theta + k_p\Theta + M_m\sin 2\Theta = 0 . \tag{3} \]

After integrating this equation and carrying out a number of mathematical transformations, we obtain

\[ \frac{d\Theta}{dt}=\sqrt{a\Theta + b\cdot\cos\cdot 2\Theta + c}, \tag{4} \]

where

\[ a=-\frac{2k}{H},\quad b=\frac{M_m}{H},\quad c=-(a\Theta_0+b\cos 2\Theta). \]

The motor will operate stably if there exists a maximum angular deviation at which the angular velocity of the rotor relative to the rotating magnetic field is equal to zero, i.e.,

\[ \frac{d\Theta}{dt}=0. \]

Consequently, the condition of dynamic stability can be expressed by the analytic equation

\[ a\Theta + b\cos 2\Theta + c = 0 \]

or

\[ \frac{a}{b}\Theta + \frac{c}{b} = -\cos 2\Theta . \tag{5} \]

The possible solutions of equation (5) depend on the quantities \(\frac{a}{b}\) and \(\frac{c}{b}\).

By investigating the limiting cases of possible solutions, one can establish the values of the quantities \(\frac{a}{b}\) and \(\frac{c}{b}\) at which the system still remains stable. These values determine the boundary curve of stability.

If we set \(k = 0\), then equation (4) takes the form

\[ \frac{d\Theta}{dt} = \sqrt{ \frac{M_m}{H} \left(\cos 2\Theta - \cos 2\Theta_0\right) }. \tag{6} \]

Putting \(\frac{d\Theta}{dt} = 0\), we obtain that the motor operates stably at \(\Theta = \Theta_0\), i.e., the slightest effects of a disturbance lead to unstable operation of the motor.

Thus, solving the equation of motion of the rotor in the first approximation makes it possible to estimate the influence of the parameters on the region of stable operation of the motor.

LITERATURE

1. E. V. Kononenko, R. P. Laas, A. F. Fink. Basic equations of synchronous geared motors of the reactive type (SRD-R). Izvestiya TPI, vol. 212, 1971.

2. S. N. Lee. The Vernier motor and its calculation. Electrical Systems and Electrotechnical Equipment, 1963, No. 6.