ON THE CHOICE OF THE LAW OF FREQUENCY AND VOLTAGE VARIATION IN FREQUENCY CONTROL OF AN INDUCTION MOTOR

V. I. PANTELEEV, B. P. SOUSTIN

(Presented by the Scientific and Technical Seminar of the Scientific Research Institute of AEM)

The dynamic characteristics of a frequency-controlled induction drive are determined not only by the ratio of voltage to frequency, but also by the law of variation of these quantities in time and by the duration of the transient mode. In this direction, recommendations for controlling impact torque, i.e., obtaining smoothness [1], have been considered rather fully, but such an important question as obtaining high speed of response with a prescribed smoothness in open-loop systems has not yet been posed. In closed-loop control, the formation of any mechanical dynamic characteristics is possible, but not every drive permits the construction of closed-loop systems.

Ensuring smoothness and high speed of response in transient modes are different problems, but they are achieved in the same way—by eliminating periodic components from the curve of transient electromagnetic torque. In solving the problem of high speed of response, one usually relies on experimental determination of laws for a specific drive [2], without obtaining general recommendations.

Six components of the expression for transient electromagnetic torque have a periodic decaying character [3]; moreover, they can be grouped, according to damping coefficients and oscillation frequencies, into three groups.

\[ M_1=\bigl[(m_{a2}-m_{a1}-1)\cos\omega_1 t-k\sin\omega_1 t\bigr]e^{-\alpha_1 t}, \tag{1} \]

\[ M_2=\bigl[(m_{a1}-m_{a2}-1)\cos\omega_2 t+k\sin\omega_2 t\bigr]e^{-\alpha_2 t}, \tag{2} \]

\[ M_3=\bigl[(-m_{a1}-m_{a2}-1)\cos(\omega_1-\omega_2)t+k\sin(\omega_1-\omega_2)t\bigr]e^{-(\alpha_1+\alpha_2)t}, \tag{3} \]

where

\[ m_{a1}= \frac{(s-\omega_1)(\alpha_2^{2}+\omega_2^{2})} {s\bigl[(\alpha_1-\alpha_2)^2+(\omega_1-\omega_2)^2\bigr]}; \qquad m_{a2}= \frac{(s-\omega_2)(\alpha_1^{2}-\omega_1^{2})} {s\bigl[(\alpha_1-\alpha_2)^2+(\omega_1-\omega_2)^2\bigr]}; \]

\[ k= \frac{\alpha_1}{s-\omega_1}m_{a1} - \frac{\alpha_2}{s-\omega_2}m_{a2} + \frac{2(\alpha_1\omega_1-\alpha_1\cdot\omega_2)} {(\alpha_1-\alpha_2)^2+(\omega_1-\omega_2)^2}; \]

\(s\) is the relative slip, and \(\alpha_{1,2}\) and \(\omega_{1,2}\) are the real and imaginary parts of the roots of the characteristic equation of the induction motor, which determine the damping rate and oscillation frequency of the periodic components of the torque.

\[ \begin{aligned} \alpha_1&=0.5[(\alpha_s' + \alpha_r')-a],\\ \alpha_2&=0.5[(\alpha_s' + \alpha_r')+a], \end{aligned} \qquad \begin{aligned} \omega_1&=0.5[(1+s)\mp b],\\ \omega_2&=0.5[(1+s)\pm b], \end{aligned} \]

where \(\alpha_s'\) and \(\alpha_r'\) are the damping coefficients of the stator (rotor) with the rotor (stator) winding short-circuited; the quantities \(a\) and \(b\) are functions of the relative slip and are determined when solving the characteristic equation.

Fig. 1

The dependence of \(\alpha_1\), \(\alpha_2\) and \(\omega_1\), \(\omega_2\) on slip in the starting region for a DAT 250-8 motor is shown in Fig. 1, from which it is seen that the damping time of the free oscillations of the electromagnetic torque is determined by the coefficient \(\alpha_1\), that is, in frequency starting of an IM the time during which the frequency and voltage change must be approximately equal to the damping time \(\alpha_1\). This is confirmed on an analog model of frequency starting with exponential and linear laws of frequency and voltage variation for DAT 250-8, A51-6, and MTK11-6 motors. The parameters of these

a)

b)

Fig. 2

motors have the maximum difference in the values of the damping coefficients.

In Fig. 2,a, b are shown the dependences of the relative starting time $\tau_p$ and the relative smoothness $\Pi$ on the damping coefficients for the exponential law (curves 1—4) and on the slope coefficients for the linear law (curves $1'—4'$) for the DAT 250-8 motor; the scale along the abscissa axis is chosen so that the time of variation of the frequency and voltage from zero to the steady-state values is the same for both laws. The relative starting time $\tau_p$ was determined as the ratio of the frequency-starting time to the starting time with direct connection to the supply network.

Since smoothness is a quantity inverse to the derivative of the electromagnetic torque at the instant of time equal to zero [4], the relative smoothness was determined as the ratio of the derivative of the torque during starting by direct connection to the supply network to the derivative of the torque during frequency starting.

Conclusions

In frequency starting, in order to obtain maximum speed of response with sufficient smoothness, it is necessary that the time of variation of the frequency and voltage be determined by the smaller of the damping coefficients of the free components of the electromagnetic torque. When the starting regime is made heavier, the desired value of the coefficient shifts into the region of larger values of slip $s$. In this case the lightest regimes are ideal no-load and fan load, while the heaviest are $M=\mathrm{const}$ and acceleration of considerable flywheel masses. The best speed of response is provided by the exponential law, and the best smoothness by the linear law.

LITERATURE

1. A. D. Gildebrand, V. M. Kirpichnikov. Analysis of the initial segment of frequency starting of an asynchronous machine. Izv. vuzov, “Elektromekhanika,” No. 4, 1970.

2. I. F. Kalinin, V. I. Panteleev, B. P. Soustin. Investigation of rational methods of frequency control of asynchronous motors. Izv. TPI, vol. 225, Tomsk State University Press, Tomsk, 1970.

3. M. M. Sokolov et al. Electromagnetic Transient Processes in an Asynchronous Electric Drive. “Energiya,” 1967.

4. I. I. Petrov, A. M. Meister. Special Operating Modes of an Asynchronous Electric Drive. “Energiya,” 1968.