STUDY OF HEATING OF ELECTRIC MOTORS IN RANDOM OPERATING MODES

Yu. M. Bashagurov, D. I. Sannikov

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

An important point in the study of heating and insulation wear of electric-motor windings in random modes is obtaining the law of temperature distribution. In some cases these laws can be obtained by applying the theory of random functions and considering the motor as a dynamic system. Such an approach is the most natural and, to some extent, universal in thermal studies of electric motors in random modes. However, analytical determination of the distribution law at the output of a dynamic system is, as a rule, associated with complicated calculations, since application of the general theory of random processes in this case requires knowledge of the multidimensional distribution laws of the input process.

The properties of the distribution law at the output of a dynamic system can be judged in advance on the basis of qualitative considerations concerning the normalization of a random process passing through a linear dynamic inertial system [1]. Their meaning, as applied to asynchronous electric motors, is essentially as follows: the distribution law of the ordinates of the output process $\Theta(t)$ will be the closer to normal, the greater the heating time constant of the winding is and the faster the correlation dependence of the input process $I(t)$ decreases.

The arguments presented in [1] on the normalization of the input process are often qualitative in character and only clarify the essence of the matter. In order to be able to regard the process $\Theta(t)$ of the stator winding of an asynchronous motor as normal with confidence, it is necessary to carry out an experiment confirming the correctness of the adopted assumptions.

The task of the experiment is to obtain the distribution law of the temperature rise of the windings of asynchronous motors for known distribution laws and correlation functions of the load.

For the study, modes of the most thoroughly studied group of loads—asynchronous motors of metal-cutting equipment—were selected and simulated [2]. The load processes under consideration have an exponential distribution law

$$ p(I)=\lambda \exp(-\lambda I), \tag{1} $$

where

\[ \lambda = \frac{1}{m(I)} \]
—is the distribution parameter,

\(m(I)\)—the mean value of the load,

and normalized correlation functions of the form

\[ R_I(\tau)=e^{-\alpha |\tau|}, \tag{2} \]

where

\(\alpha\) is the parameter of the correlation function (in the indicated modes it lies within the range \(0.01 \div 0.1\ \text{s}^{-1}\)).

The specific features of the exponential distribution of the load determine the selection of an asynchronous motor not from the heating condition, but only from the condition of ensuring the maximum electromagnetic torque. For more effective use of asynchronous motors under an exponential load distribution, the problem arises of making maximum use of their overload capacity and, consequently, of investigating heating and insulation wear of the windings under limiting modes (with respect to \(M_{\max}\)).

Taking the above into account, when testing asynchronous motors the parameters of the exponential law were chosen so that the maximum peak current loads reached values \(I_{\max}\) corresponding to a value of the maximum torque not lower than the catalog value.

Load realizations were obtained on a BESM-4 electronic computer by the statistical modeling method [3] for different magnitudes of the autocorrelation relationship.

Two-, four-, and six-pole asynchronous motors of the same 4A series with a shaft-axis height of 112 mm were tested. The average overheating of the stator winding was measured by the method of thermocouples embedded in the winding. The temperature-versus-time graph was recorded using a self-recording millivoltmeter of type H39, with a chart-tape feed rate of 600 mm/h.

The specified mode \(I(t)\) was supplied to the input of the motor under test in the following way.

Using a self-recording ammeter of type H376, connected into the circuit of the electric motor, the specified mode \(I(t)\), modeled with respect to current and time and plotted on chart tape, was “copied.” The load was varied using the independent-excitation regulator of the load generator.

Naturally, it is impossible to reproduce exactly the load mode specified and recorded on the chart tape. It should be added that the very method of modeling random processes described in [3] does not make it possible to obtain random processes whose ordinates will be exactly distributed according to the specified law, since it is known that a random variable obtained as the sum of random independent (or weakly dependent) numbers distributed according to any law will no longer strictly obey the specified law, but will approach the normal law as closely as desired [4]. Therefore, in order to monitor and refine the parameters of the specified mode, the newly recorded mode \(I(t)\) was processed.

A preliminary analysis of the temperature diagrams showed that the values of the minimum periods of load oscillations \(T_{\min}\), corresponding to the high-frequency components of the process \(\omega_{\max}\), vary within the range \(100 \div 150\) s. Processing of the process on the electronic computer was carried out, according to [4], with a sampling interval \(\Delta t = 15\) s. Taking into account the recommendations of [5], the number of level-quantization intervals was taken to be 12.

The required length of the realizations was determined by the method indicated in [6]. Setting the value of half the confidence interval \(L=0.1\) and the confidence probability \(\beta_c=0.99\), from the tables \(\beta_c=f(L,n)\) we obtain the sample size \(n=450\). The length of the load realization \(T\) at \(\Delta t=15\) sec is, in this case, 2 hours.

To eliminate the error due to the influence of the nonstationary initial segment of the process \(\Theta(t)\), the tests of the motors in the specified mode were begun at the mean temperature, which may be approximately determined by the formula

\[ \tilde{\theta}=\theta_{\mathrm{n}}\left(\frac{\bar I_1}{I_{\mathrm{n}}}\right)^2, \tag{3} \]

where

\(\Theta_{\mathrm{n}}\) is the temperature rise of the winding under the nominal regime, °C;

\(\bar I_1\) is the specified mean load, a.

Table

As a result of statistical processing of the realizations of the process \(\Theta(t)\) on an electronic digital computer, histograms, mean values, and the second, third, and fourth central moments were obtained.

Proceeding from the assumption that the experimental distributions were normal, their deviation from the normal law was determined

\[ p(\theta)=\frac{1}{\sigma(\theta)\sqrt{2\pi}}\exp\left[-\frac{(\theta-\bar{\theta})^2}{2\sigma^2(\theta)}\right] \tag{4} \]

by means of an approximate goodness-of-fit criterion based on the calculation of the variances of the skewness \(D(A)\) and kurtosis \(D(E)\). According to this criterion, the observed distribution is considered normal if the inequalities

\[ |A|<3\sqrt{D(A)} \quad \text{and} \quad |E|<5\sqrt{D(E)}. \tag{5} \]

are satisfied.

The results of the tests and of the goodness-of-fit check are given in Table 1. The values of the absolute magnitudes of the skewness that do not satisfy conditions (5) are marked with an asterisk.

Conclusions

When considering the electric motor as a dynamic system, it has been experimentally established that the temperature distribution of the winding of asynchronous motors in the operating modes of metal-cutting equipment may be regarded as obeying a normal law.

References

1. A. A. Sveshnikov. Applied Methods of the Theory of Random Functions. Moscow, “Nauka,” 1969.

2. Yu. M. Bashaturov, E. K. Strelbitskii. Accounting for load regimes in the design of asynchronous motors. Izvestiya TPI, vol. 212, 1971.

3. N. P. Buslenko. Mathematical Modeling of Production Processes. Moscow, “Nauka,” 1964.

4. E. S. Wentzel. Probability Theory. Moscow, Fizmatgiz, 1962.

5. G. Korn. Modeling of Random Processes on Analog and Analog-Digital Machines. Moscow, “Mir,” 1968.

6. R. Manly. Analysis and Processing of Vibration Records. Moscow, “Mashinostroenie,” 1972.