STUDY OF ELECTROMAGNETIC PROCESSES IN THE FORMATION OF TORQUE IN AN ASYNCHRONOUS MOTOR

V. P. Lanzburg, V. I. Panteleev, B. P. Soustin

(Presented by the scientific and technical seminar of the Scientific Research Institute of AEM)

In relay systems for controlling the torque of an asynchronous motor (a.m.) under dynamic load, the problem of forming torque over a wide dynamic range often arises. In this case the duty cycle of the pulses varies from 0 to 1, that is, the duration of the pulses may be measured in units of periods of the supply voltage. In addition, the requirement of economical regulation must be met. The principles of economical regulation developed for ordinary drives with continuous control in the minimum-current mode [1] are not applicable in this case.

In studying such pulsed systems it is usually assumed that, when the stator is connected to a power source with constant voltage, a torque of constant amplitude arises instantaneously in the motor for the entire duration of the pulse; that is, the form of the torque repeats the form of the control pulses. Evidently this assumption is valid if the time of the control pulse is many times greater than the duration of the transient process. When the duration of the control pulse is reduced, it is necessary to take into account the influence of transient electromagnetic processes, which has not at present been covered in the literature.

Theoretical and experimental studies have shown that when the duty cycle of the control pulses is less than unity, the electromagnetic processes during the pause decay, and each new switching-on occurs under zero initial conditions. This condition may fail to hold when the duty cycle is very close to unity; however, in that case the duration of the transient process, as a rule, amounts to fractions of a percent of the pulse duration, and one should make use of the assumption that electromagnetic transient processes may be neglected.

In the analysis it is assumed that the power source may form either a sinusoidal voltage or a sinusoidal current; the slip of the a.m. remains unchanged during the action of the pulse, since the frequency of the supply voltage and the rotational speed of the rotor do not change because of the large reduced moment of inertia.

To study the behavior of the a.m. when supplied from a source with sinusoidal voltage, the motor equations are written through the flux linkages of the stator and rotor for two axes $\alpha$ and $\beta$ [2]:

\[ \frac{d\Psi_{\alpha s}}{dt} = U_{\alpha s} - z_s' \Psi_{\alpha s} + k_r z_s' \Psi_{\alpha r} + \omega_k \Psi_{\beta s}; \]

\[ \frac{d\Psi_{\beta s}}{dt} = - z_s' \Psi_{\beta s} + k_r z_s' \Psi_{\beta r} - \omega_k \Psi_{\alpha s}; \]

\[ \frac{d\Psi_{\beta s}}{dt} = -\alpha_s' \Psi_{\alpha r} + k_s \alpha_r' \Psi_{\beta s} +(\omega_k-\omega)\Psi_{\beta r}; \]

\[ \frac{d\Psi_{\beta r}}{dt} = -\alpha_r' \Psi_{\beta r} + k_s \alpha_r' \Psi_{\beta s} +(\omega_k-\omega)\Psi_{\alpha r}; \]

\[ i_{\alpha s}=\frac{\Psi_{\alpha s}-k_r\Psi_{\alpha r}}{\sigma L_s}; \qquad i_{\beta s}=\frac{\Psi_{\beta s}-k_r\Psi_{\beta r}}{\sigma L_s}; \]

\[ M=\frac{k_r}{\sigma L_s} (\Psi_{\beta s}\Psi_{\alpha r}-\Psi_{\alpha s}\Psi_{\beta r}); \]

\[ P_{\mathrm{es}}=\frac{m}{2}(i_{\alpha s}^{2}+i_{\beta s}^{2})r_s; \qquad P_{\mathrm{er}}=M\cdot \Delta\omega . \]

Here

\[ k_s,\ k_r \]
are the rotor and stator flux-linkage coupling coefficients;

\[ U_{\alpha s},\ i_{\alpha s},\ i_{\beta s},\ \Psi_{\alpha s},\ \Psi_{\beta s},\ \Psi_{\alpha r},\ \Psi_{\beta r} \]
are the voltage, stator currents, and stator and rotor flux linkages along the \(\alpha\) and \(\beta\) axes;

\[ \alpha_s',\ \alpha_r' \]
are the stator and rotor damping coefficients;

\[ L_s,\ L_r,\ L_m,\ r_s,\ r_r \]
are the relative motor parameters according to the T-shaped equivalent circuit;

\[ \sigma \]
is Blondel’s leakage coefficient.

Solving the system of equations presented above by the usual method involves considerable difficulties; therefore the differential equations were solved on an analog computer. For this purpose, two MN-7 machines connected in series were required.

In the analysis, the parameters of a three-phase motor having the following data were used:
\[ U_n=27\ \mathrm{V},\quad I_f=2.88\ \mathrm{A},\quad r_1=0.807\ \Omega, \]

\[ r_2'=0.862\ \Omega;\qquad L_s=\frac{x_1+x_0}{Z_b}=3.12;\qquad L_r=\frac{x_2+x_0}{Z_b}=3.14; \]

\[ L_m=\frac{x_0}{Z_b}=2.99;\qquad Z_b=\frac{U_n}{I_n}=9.38\ \Omega . \]

Figure 1 presents a model for computing all intermediate and principal quantities when a rotating motor with a decaying field is connected to a constant sinusoidal stator voltage. The following quantities were oscillographically recorded: the motor torque, the electrical losses in the stator \(P_{\mathrm{es}}\), in the rotor \(P_{\mathrm{er}}\), and their sum \(P_{\mathrm{e}}\), and the useful power \(P_{\mathrm{useful}}\).

In the analysis it is assumed that the iron losses are constant for all slip values and are insignificant; therefore they were not taken into account. To compute the energy of the losses, it is necessary to include the blocks joined by a dashed line.

To evaluate the efficiency of source utilization, the following criterion is introduced:

\[ K_{\mathrm{iu}}= \frac{\displaystyle\int_{0}^{t_n} M(t)\,dt} {\displaystyle\int_{0}^{t_n} P_{\mathrm{e}}(t)\,dt} \]

the ratio of the mean value of the motor torque to the mean value of the electrical losses during the action time of the control pulse.

Obviously, in the steady-state regime the source-utilization coefficient is equal to

\[ K_{\mathrm{iu}}=\frac{M}{P_{\mathrm{e}}}. \]

\[ K_{\text{in}}=\frac{M}{P_{\text{el}}}. \]

Fig. 1.

Figure 2a shows the time dependence of the torque when the motor is connected to a sinusoidal voltage for various slips; Fig. 2b shows the electrical losses in the motor at the same slips; Fig. 3 shows the source utilization factor.

2a            2b

From Figs. 2 and 3 it can be seen that, at the moment the motor is connected to a source with sinusoidal voltage, a negative torque arises with considerable losses in the motor; and, although the transient process ends quickly, the initial segment has a large influence on the average values of the torque, the total electrical losses in the motor, and the source utilization factor. It is also evident from Fig. 3 that

Fig. 3.

with a known width of the control pulse, in order to obtain the maximum coefficient \(K_{\text{и}}\), it is necessary to select an appropriate slip. The assumption that the average value of the torque remains constant during the action of the pulse will be valid for a pulse duration of more than 150 periods of the supply voltage, and with a smaller error at higher slip.

The optimum slip from the standpoint of minimum losses in the steady-state mode is \(0.06\)—\(0.09\) (Fig. 4). At the moment the motor is connected to the voltage source, it is impossible to obtain a ratio of torque to total losses greater than in the steady-state mode at any slip.

Fig. 4.

Fig. 5.

Fig. 6.

When the motor is connected to a sinusoidal-current source, a model analogous to Fig. 1 is constructed [3, 4]. The dependence of the torque on time for this case, at the rated stator current, is shown in Fig. 5; Fig. 6 shows the dependence of the ratio of the torque to the total electrical losses under current control. From Figs. 5 and 6 it is seen that there is no negative overshoot in the torque curve, and that the dependences \(M=f(t)\) and \(K_{\mathrm{ip}}=f(t)\) have a maximum depending on the magnitude of the slip. The optimal slip for the motor considered in dynamics proves to be close to \(s=0.075\). The coefficient of utilization of the source in dynamics proves to be even greater than in the steady-state regime under voltage control.

Increasing the voltage in the first case reduces the utilization of the source in dynamics because of the considerable increase in negative torque; in statics, however, increasing the voltage increases \(K_{\mathrm{ip}}\). Thus, for \(s=0.1\), at \(U=1.5U_N\), \(K_{\mathrm{ip}}=4.8\) instead of 3.2 at \(U=U_N\). When the current is increased, the dynamic indices improve. The experimental data prove to be lower than those presented because losses in the steel and the nonlinearity of the magnetization curve were not taken into account.

Connecting an induction motor to sources with a nonsinusoidal output introduces additional errors and requires further investigation.

Conclusions

Under pulse control of an induction motor, electromagnetic transient processes must necessarily be taken into account. Control pulses with a duration of less than 2–3 periods of the supply voltage are inadmissible when the induction motor is connected to a sinusoidal-voltage source, since in this case a negative effect may occur. In all cases of pulse control, current control is preferable. To increase the utilization of the power-supply source with narrow control pulses, the magnitude of the slip must be selected as a function of the duration of this pulse. The assumption of a rectangular torque waveform under pulse control is valid only for a considerable pulse duration (at \(s=s_N\), no less than 150–200 periods of the supply voltage).

LITERATURE

1. A. A. Bulgakov. Frequency Control of Asynchronous Electric Motors. Moscow, “Nauka,” 1966.

2. K. P. Kovach and I. Rats. Transient Processes in Alternating-Current Machines. Moscow–Leningrad, GEI, 1963.

3. V. I. Ivanchura, B. P. Soustin, Yu. A. Shurygin. Study of bridge inverters in the formation of sinusoidal current. In: Increasing the Efficiency of Power-Converter Devices, Part 3. Kiev, “Naukova Dumka,” 1972.

4. I. F. Kalinin, V. I. Panteleev, B. P. Soustin. Study of the dynamics of starting an asynchronous motor under frequency-current control. Proceedings of the Jubilee Scientific and Technical Conference of the Faculty of Automatic Systems. Tomsk, TTU Publishing House, 1970.