ASYNCHRONOUS ELECTRIC DRIVE SUPPLIED BY A RECTANGULAR VOLTAGE

I. F. Kalinin, V. I. Panteleev, B. P. Soustin

(Presented by the scientific-technical seminar of the Department of Electrical Equipment, TPI)

In recent years, owing to the appearance of comparatively simple and reliable frequency converters (FC), the asynchronous electric drive with frequency control has been developing rapidly. The waveform of the output voltage of an FC differs considerably from a sinusoidal one; therefore, it is of interest to investigate the dynamics of an induction motor (IM) supplied by such voltages.

In many FCs the waveform of the output voltage is close to rectangular; therefore, for the investigations precisely this voltage waveform was chosen, corresponding to operation of the FC with a \(180^\circ\) conduction angle, Fig. 1a.

The behavior of an IM supplied by rectangular voltage was investigated on an analog computer of the MN-14 type using the parameters of the DAT250-8 motor.

As the initial system for modeling the system of differential equations of the IM, the “\(\psi_s\)—\(\psi_r\)” system [1] was adopted:

\[ \left. \begin{aligned} \frac{d\Psi_{\alpha s}}{dt} &= U_{\alpha s} - \alpha'_s \Psi_{\alpha s} + k_r \alpha'_s \Psi_{\alpha r},\\[4pt] \frac{d\Psi_{\beta s}}{dt} &= U_{\beta s} - \alpha'_s \Psi_{\beta s} + k_r \alpha'_s \Psi_{\beta r},\\[4pt] \frac{d\Psi_{\alpha r}}{dt} &= -\alpha'_r \Psi_{\alpha r} + k_s \alpha'_r \Psi_{\alpha s} - \omega \Psi_{\beta r},\\[4pt] \frac{d\Psi_{\beta r}}{dt} &= -\alpha'_r \Psi_{\beta r} + k_s \alpha'_r \Psi_{\beta s} + \omega \Psi_{\alpha r},\\[4pt] M &= \frac{k_r}{\sigma L_s}\left(\Psi_{\beta s}\cdot \Psi_{\alpha r} - \Psi_{\alpha s}\cdot \Psi_{\beta r}\right),\\[4pt] \frac{d\omega}{dt} &= \frac{1}{I}\left(M - M_C \operatorname{sign}\omega\right). \end{aligned} \right\} \tag{1} \]

The modeling was carried out in a coordinate system stationary with respect to the stator.

If it is assumed that the system of zero-sequence voltage and current vectors is absent, then \(U_{\alpha s}\) and \(U_{\beta s}\) are determined by formulas from (1):

\[ \begin{gathered} U_{\alpha s}=U_a,\\ U_{\beta s}=\frac{U_b-U_c}{\sqrt{3}}, \end{gathered} \tag{2} \]

where \(U_a\), \(U_b\), and \(U_c\) are the stator phase voltages.

The form of the components \(U_{\alpha s}\) and \(U_{\beta s}\) is shown in Fig. 1, a.

In addition, it is assumed that steel saturation and losses in the steel are absent.

In the study, the electromagnetic processes of the frequency converter were not taken into account in any way; only the operation of an induction motor supplied by rectangular voltage was investigated.

Fig. 1

The analog model of the induction motor, compiled according to the system of equations (1), is generally known and therefore is not presented. The rectangular-voltage source was assembled from the computing blocks of an analog computer, Fig. 1, b, and consists of a source of sinusoidal voltages, operational relays, and amplifiers.

After adjustment of the model, frequency starting of the induction motor was investigated with variation of the voltage frequency according to an exponential law and

\[ \frac{U}{f}=\text{const}. \]

The variation of the supply voltage with time is shown in Fig. 2, a.

A picture was obtained of the dynamic variation of the flux linkage \(\psi_s\) in the complex plane, Fig. 2, b, from which it is seen that the vector \(\psi_s\) changes discretely, in steps of \(60^\circ\). A similar picture for the rotor flux linkage \(\psi_r\) has the form of a circle, Fig. 2, c, and not of a hexagon, and

Fig. 2

indicates that the magnetic circuit of the IM is a kind of filter for higher harmonics.

Superimposing these plots on one another, Fig. 2, d, makes it possible to judge the degree of approximation introduced by the last of the assumptions stated above. As can be seen from this comparison, the maximum error (which may be judged from the difference between \(\bar{\psi}_s\) and \(\bar{\psi}_r\)) due to neglecting the saturation effect occurs in the initial portion of starting; the minimum occurs in the steady-state regime. Obviously, this error is primarily an error in determining the amplitude of the first torque pulsations. For accurately determining the response time it is not of such substantial importance.

Comparing \(\bar{\psi}_s\) and \(\bar{\psi}_r\), it is easy to note that the most significant differences between them in the steady-state operating regime occur when the vector \(\bar{\psi}_s\) changes, i.e., 6 times per period. This explains the presence of high-frequency pulsations on the torque curve, Fig. 2, e. The frequency of these pulsations is \(f_n = 6 f_1\). High-frequency torque pulsations have almost no effect on the response time, since the value of the torque, averaged over a certain interval of time, remains unchanged.

Obviously, a more accurate study of the dynamics of an IM supplied from a FC requires modeling of the electromagnetic processes in the FC, as well as methods of reactive-power compensation specific to each type of FC. The present work did not set itself such an objective. On the basis of the investigations carried out, the following conclusions may be drawn:

1. On an analog computer it is possible to investigate an asynchronous electric drive supplied from a FC. In doing so, it is possible to use only the solving blocks of the analog computer.

2. Accurate investigation of the values of the transient torque requires the introduction of variable parameters in order to take into account the effect of steel saturation, which is especially noticeable in the initial portion of starting, i.e., in the region of large currents.

3. For investigating the FC–IM system, it is necessary to develop FC model circuits that are simple to adjust and operate and that accurately model the physical processes in the FC.

The following notation was adopted in the work:

\[ r_s r_r \text{ — active resistances of the stator and rotor;} \]

\[ L_s L_r L_m \text{ — total inductances of the stator and rotor and the equivalent three-phase mutual inductance;} \]

\[ k_s = \frac{L_m}{L_s}, \quad k_r = \frac{L_m}{L_r} \text{ — coupling coefficients of the stator and rotor;} \]

\[ \sigma = 1 - \frac{L_m^2}{L_s L_r} \text{ — Blondel leakage coefficient;} \]

\[ \alpha'_s = \frac{r_s}{L_s}\sigma;\quad \alpha_r = \frac{r_r}{L_r \sigma} \text{ — damping coefficient of the stator (rotor) with} \]

\[ \text{the rotor (stator) winding short-circuited;} \]

\[ I \text{ — moment of inertia of the drive, referred to the shaft of the IM and expressed in relative units.} \]

REFERENCES

1. K. P. Kovach, I. Rácz. Transient Processes in Alternating-Current Machines. Gosenergoizdat, 1963.

2. Sattler Ph. K., Ulrich B., “Untersuchung der stromrichtergespeisten Asynchronmaschine am Analogrechner,” ETZ-A, 1968, H. 2, S. 25–31.