TORQUE GENERATION IN AN INDUCTION MOTOR UNDER VOLTAGE CONTROL

A. V. Al’kin, V. I. Panteleev, B. P. Soustin

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

Drives whose main task is the control of kinetic torque have large flywheel masses on the shaft and differ from general-purpose drives in the need to regulate torque at a rotational speed close to nominal. For energy-related reasons such drives are built according to the scheme inverter—induction motor with pulse control. Continuous control methods are not used because of a significant increase in losses in the inverter–motor system. With short control pulses, the time of the transient electromagnetic process may occupy a considerable part of the pulse, and neglecting transients leads to significant errors. The assumption that the torque is rectangular is valid if the pulse duration \(t_i \geq 150 \div 200\) periods of the supply voltage, and with a smaller error at greater slip.

The connection of an induction motor rotating at constant speed to a sinusoidal voltage source is accompanied by a transient process having a complex time dependence determined by the initial rotational speed and by the motor parameters. For illustration, Fig. 1 gives the solution on an AVM of the equations of an induction motor. It is seen that after the motor is connected to the source, a negative torque of considerable magnitude arises, the physical nature of which is shown in [1], and, although the transient pro-

Fig. 1

cess of an induction motor. It is seen that after the motor is connected to the source, a significant negative torque arises, the physical essence of which is shown in [1] and, although the transient pro-

process ends quickly, the initial segment has a significant influence on the average value of the torque and on the magnitude of the total losses.

Below, the influence of the free components of the currents on the magnitude of the average torque during the action of the control pulse is shown, and dependences are given that determine the amplitude of the negative undershoot in the torque curve.

To estimate the influence of the free components of the currents on the magnitude of the average torque, a coefficient of reduction of the average torque due to the influence of the free components, \(K_M\), is introduced—the ratio of the average torque during the action of the control pulse to the steady-state value

\[ K_M = \frac{ \frac{1}{t_{\mathrm{u}}}\int_{0}^{t_{\mathrm{u}}} M\,dt }{ M_{\mathrm{ust}} }. \tag{1} \]

In Fig. 2 the dependences of \(K_M\) for different slips are presented; from them it is seen that in the initial segment \(K_M\) is negative, while for small pulses \(t_{\mathrm{u}}\) its value is only slightly greater than zero. Since the negative torque undershoot has the greatest influence on \(K_M\), it is necessary to carry out a detailed analysis of this phenomenon.

Fig. 2

To determine the amplitude of the negative undershoot, let us write the dependence of the electromagnetic torque of an induction motor on time at constant rotational speed [2]:

\[ \frac{M(t)}{M_{\mathrm{ust}}}\bigg|_{\substack{s\ne 0\\ s\ne 1}} = 1 + m_{a_1}\exp(-2\alpha_1\omega_0 t) + m_{a_2}\exp(-2\alpha_2\omega_0 t) + \]

\[ + (m_{a_2}-m_{a_1}-1)\cos\omega_1\omega_0 t\cdot \exp(-\alpha_1\omega_0 t) + (m_{a_1}-m_{a_2}-1)\cos\omega_2\omega_0 t \times \]

\[ \times \exp(-\alpha_2\omega_0 t) + (1-m_{a_1}-m_{a_2})\cos(\omega_1-\omega_2)\omega_0 t \times \]

\[ \times \exp[-(\alpha_1+\alpha_2)\omega_0 t] + K\{-\exp(-\alpha_1\omega_0 t)\sin\omega_1\omega_0 t + \]

\[ + \exp[-(\alpha_1+\alpha_2)\omega_0 t]\sin(\omega_1-\omega_2)\omega_0 t + \exp(-\alpha_2\omega_0 t)\sin\omega_2\omega_0 t\}, \tag{2} \]

where \(m_{a_1}\), \(m_{a_2}\), \(K\) are coefficients determined through the parameters of the motor;

\[ \alpha_1,\ \alpha_2,\ \omega_1,\ \omega_2 \]

are the coefficients and frequencies of damping of the free components of the currents,

\[ M_{\mathrm{ust}} \]

is the steady-state value of the torque.

To determine the amplitude of the negative undershoot it is necessary to determine the extremum of (2). In view of the difficulty of carrying out the mathematical operations, it is reasonable first to consider each component of (2) from the standpoint of the coefficients and frequencies of damping of the free components.

Fig. 3

Figure 3 shows the dependences \(\alpha_1\), \(\alpha_2\), \(\omega_1\), \(\omega_2\), from which it is evident that at small slips \(s \leq 0.1\) the value of \(\omega_2\) is close to zero and its influence may be neglected. In addition, the time at which the transient negative torque reaches its maximum is approximately \(25\%\) of the transient-process time. If it is assumed that the damping of all components in (2) occurs with the same intensity, then by the time the maximum is reached its value will be approximately \(62\%\). Taking this into account, the expression will take the form

\[ M(t)=-0.76\,m_{a_1} M_{\text{st}}(1-\cos\omega_1 t), \tag{3} \]

that is, the torque has the maximum value of the negative overshoot when \(\cos \omega_1 t=0\), or

\[ t=\frac{n\pi}{2\omega}. \]

Hence, knowing the value of \(\omega_1\), one determines

\[ t=\frac{n\pi}{2\omega_1}. \tag{4} \]

The time required to reach the maximum negative torque varies inversely with \(\omega_1\), i.e., for \(\omega_1\) corresponding to a smaller slip, it has a larger value (Fig. 2). The amplitude of the negative torque overshoot (Fig. 1, dotted line), calculated according to (2), for slips 0.05 and 0.1 is 20 and 15% greater than the amplitudes obtained by simulation on an AVM. The obtained expression can be used for slips \(0<s<s_{\text{cr}}\). If, in order to determine the magnitude of the negative overshoot when operating with slips smaller than the critical one, it is assumed that \(\omega_1\) and \(\omega_2\) are commensurable and damping is neglected, then expression (2) takes the form

\[ \frac{M(t)}{M_{\text{st}}} = 1+m_{a_1}+m_{a_2}+(m_{a_2}-m_{a_1}-1)\cos\omega_1 t+ \]

\[ +(m_{a_1}-m_{a_2}-1)\cos\omega_2 t+(-m_{a_1}-m_{a_2}+1)\cos(\omega_1-\omega_2)t . \tag{5} \]

The last term in expression (2) may be neglected, since, according to a calculation obtained on a digital computer, its influence on the formation of the amplitude of the negative torque is insignificant.

Let us investigate expression (5) for extrema; for this purpose, taking the first derivative, we obtain

\[ \frac{d[M(t)]}{dt} = - M_{\text{st}} \left[ A\omega_1\sin\omega_1 t + B\omega_2\sin\omega_2 t + C(\omega_1-\omega_2)\sin(\omega_1-\omega_2)t \right], \tag{6} \]

where

\[ \begin{aligned} A &= (m_{a_2} - m_{a_1} - 1),\\ B &= (m_{a_1} - m_{a_2} - 1),\\ C &= (1 - m_{a_1} - m_{a_2}), \end{aligned} \tag{7} \]

whence

\[ A\omega_1 \sin \omega_1 t + B\omega_2 \sin \omega_2 t + C(\omega_1 - \omega_2)\sin(\omega_1 - \omega_2)t = 0. \]

Equation (6) is transcendental, and its solution is usually found by numerical methods. However, considering the curves of variation of the damping frequencies up to the critical slip, one may note that \(\omega_1 \approx \omega_2\); then

\[ (A\omega_1 + B\omega_2)\sin \omega t = 0, \tag{8} \]

\[ (A\omega_1 + B\omega_2) \ne 0,\quad \text{i.e. } \sin \omega t = 0, \]

\[ \omega t = n\pi. \tag{9} \]

The expressions obtained—the simultaneous solution of (3) and (4) when the motor operates with slips smaller than the critical slip, and (7), (8), (9) when it operates with slips greater than the critical slip—make it possible, without the use of computers or laborious calculations on a logarithmic slide rule, to estimate, with an admissible error of up to 15%, the amplitude of the negative overshoot.

References

1. A. V. Al’kin, V. P. Lanzburg, V. I. Panteleev, B. P. Soustin. Formation of torque of an induction motor in a transient regime. Izvestiya TPI, vol. 283, Tomsk, Tomsk Polytechnic Institute Press, 1973.

2. M. M. Sokolov, L. P. Petrov, L. B. Masandilov, V. A. Ladenzon. Electromagnetic transient processes in an asynchronous electric drive. Moscow, “Energiya,” 1967.