Analysis of the Accuracy of Calculating the Parameters of Asynchronous Roller Table Motors
Abstract Generated abstract
This paper evaluates the accuracy of calculated parameters for AR series asynchronous roller table motors, using standard test data for frame sizes 4 to 7 at 50 Hz. The authors distinguish systematic errors from random technological scatter and apply statistical confidence intervals, based on small samples of 8 to 17 motors per size, to compare calculated and measured values for starting, rated, no load, loss, temperature, speed, and resistance parameters. The analysis identifies parameters whose calculated values fall within confidence limits and others with systematic bias, including understated efficiency and stator winding temperature rise and overestimated rotor cage resistance affecting rated speed. The paper also derives a method for estimating the minimum number of motors required to verify a specified calculation accuracy while accounting for technological scatter, which may reach about 30 percent for some parameters.
Full Text
ANALYSIS OF THE ACCURACY OF CALCULATING THE PARAMETERS OF ASYNCHRONOUS ROLLER-TABLE MOTORS
V. V. DNEPROVSKII, O. P. MURAVLEV, S. A. SHELEKHOV
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
Existing methods for calculating electrical machines make it possible to approach the optimal variant fairly closely; however, the establishment of specific guaranteed values for most parameters is carried out on the basis of experience because of the error of the calculation, the magnitude of which is unknown. The accuracy of the calculation is characterized by the degree of agreement between the actual values of the parameters of the manufactured electrical machine and the calculated parameters.
In the present article an analysis is made of the accuracy of calculating a number of parameters of asynchronous roller-table motors of the AR series, frames 4–7, for a frequency of 50 Hz. To determine the error in calculating electric motors, statistical material based on data from standard tests was used. The number of motors in the sample was 8–17 units for each type size considered. Such a volume of statistical material makes it possible, with sufficient reliability, to determine the error in calculating individual motor parameters.
The discrepancy between calculated technical characteristics and experimental ones is due to two kinds of causes: systematic and random. The systematic error is caused by the inaccuracy of the calculation method, which is the result of the use of empirical formulas, various coefficients, and a simplified treatment of the physical processes occurring in the machine.
The random error of the parameters is due to instability in the properties of the materials used, errors in manufacturing technology, design tolerances, measurement errors, etc.
The series of asynchronous roller-table motors with a squirrel-cage rotor is intended for operation in continuous, intermittent-periodic, and mixed duty. This series differs from ordinary series of general-purpose asynchronous machines by its high starting qualities, a wide range of rotational speeds, an increased air gap, a rotor cage made of an aluminum–magnesium alloy whose electrical conductivity is 2.1 times lower than that of pure aluminum, and so on, which introduces specific features into the determination of individual parameters. Determining the greatest discrepancies between the calculated and actual values of the parameters, caused by the particular features of these motors, will make it possible to use this in refining the calculation method.
It is known that the parameters of asynchronous motors are random quantities with a normal distribution law [1, 2].
The scatter of parameters can be characterized by half the width of the scatter field:
\[ \delta_x=\frac{300\overline{s}}{\overline{x}},\% \tag{1} \]
where \(\overline{s}\) is the unbiased estimate of the standard deviation of the population;
\(\overline{x}\) is the mean actual value of the parameter.
The quantity \(\delta_x\) is an accurate indicator of the manufacturing technology of roller-table motors and is given in Table 1 for the principal parameters.
Table 1
| Parameter | \(M_k\) | \(I_k\) | \(P_k\) | \(D_{120}\) | \(\eta\) | \(\cos\varphi\) | \(\Sigma P\) |
|---|---|---|---|---|---|---|---|
| \(\delta_x,\%\) | 25.43 | 18.90 | 22.24 | 21.17 | 7.68 | 8.11 | 25.37 |
| Parameter | \(I_n\) | \(\Delta t_{\mathrm{cu}}\) | \(n_n\) | \(I_0\) | \(P_0\) | \(R_{15}\) | |
| \(\delta_x,\%\) | 10.13 | 29.40 | 5.65 | 13.93 | 26.0 | 6.18 |
As can be seen from Table 1, some of the parameters have a large technological scatter.
The confidence limits for the mean value of the population \(X_0\) are:
\[ \overline{x}-t\frac{\overline{s}}{\sqrt{n}}<X_0<\overline{x}+t\frac{\overline{s}}{\sqrt{n}}, \tag{2} \]
where \(n\) is the number of electric motors in the sample;
\(t\) is Student’s coefficient, characterizing the deviation of the mean value of a small sample from the mean value of the normal population. The coefficient \(t\) for the given sample size \(n\) and the selected confidence probability is found from [3]. In our calculations the confidence probability was taken as \(\alpha=0.95\).
In the statistical processing of the results of type tests, for each parameter considered the limiting interval was determined from
\[ \overline{x}_{\min}=\overline{x}-t\frac{\overline{s}}{\sqrt{n}} \]
to
\[ \overline{x}_{\max}=\overline{x}+t\frac{\overline{s}}{\sqrt{n}}, \]
and the calculation error was computed from the limiting values:
\[ \Delta_{\max}=x_p-\overline{x}_{\min}=x_p-\overline{x}+t\frac{\overline{s}}{\sqrt{n}} \quad \text{and} \]
\[ \Delta_{\min}=x_p-\overline{x}_{\max}=x_p-\overline{x}-t\frac{\overline{s}}{\sqrt{n}}, \tag{3} \]
where \(\Delta_{\max}\) and \(\Delta_{\min}\) are the limiting calculation errors;
\(x_p\) is the calculated value of the parameter.
In the absence of technological scatter, or with an infinitely large sample size, the mean value of the parameter determined from the sample would also be the mean value of the general population, i.e., \(\overline{x}=X_0\), and the difference between the mean value and the calculated value would indicate the systematic calculation error. Usually we have a limited sample size, while the parameters, owing to the entire
for a number of random reasons, exhibit scatter. Therefore, for the mean value of the general population only a confidence interval can be found, which will contain \(X_0\) with the adopted confidence probability. This additionally introduces a certain uncertainty into the determination of the calculation error.
Fig. 1
Let us consider three possible cases that may occur when comparing the calculated values of the parameters with the limiting values \(x_{\min}\) and \(x_{\max}\) (Fig. 1).
In case “a,” the calculated value of the parameter \(x_p\) falls within the confidence interval, so that the condition is satisfied
\[ \bar{x}_{\min} \leq x_p \leq \bar{x}_{\max}. \tag{4} \]
This indicates that the error of the calculation method does not exceed the value of the confidence interval for the mean value of the total population at the selected confidence probability.
In case “b,” the calculated value of the parameter is shifted toward smaller values and falls outside the permissible interval, so that
\[ x_p < \bar{x}_{\min}, \tag{5} \]
and in case “c”—toward larger values, so that
\[ x_p > \bar{x}_{\max}. \tag{6} \]
In the last two cases there is a systematic error in the calculated value relative to the actual mean value at the selected confidence probability. Knowledge of the magnitude of the bias makes it possible to introduce the corresponding corrections into the calculated values of the motor parameters.
Table 2 gives, for several frame sizes of motors of the AR series, the maximum \(\delta_{\max}\) and minimum \(\delta_{\min}\) calculation errors, determined for various parameters by the formulas:
\[ \delta_{\max} = \frac{100\Delta_{\max}}{\bar{x}}, \% \quad \text{and} \quad \delta_{\min} = \frac{100\Delta_{\min}}{\bar{x}}, \%. \tag{7} \]
An analysis of the maximum and minimum calculation errors of the principal parameters of several frame sizes of AR-series motors, presented in Table 2, shows that all cases of the position of the calculated value relative to the confidence interval that were considered in Fig. 1 are encountered.
Errors \(\delta_{\max}\) and \(\delta_{\min}\) having different signs for one and the same parameter and motor frame size correspond to case “a.”
Thus, for motor parameters satisfying case “a,” the calculation error does not exceed the value of the confidence interval at confidence probability \(\alpha = 0.95\).
Let us consider Table 2 in more detail for each parameter.
- Parameters characterizing the short-circuit mode, intermittent duty, and mixed operating modes of the motors (initial starting torque \(M_k\), initial starting current \(I_k\), short-circuit power \(P_k\), and dynamic constant \(D_{120^\circ}\) at PV25% for a stator-winding temperature rise of \(120^\circ\mathrm{C}\)).
For most motor frame sizes, the limiting calculation errors for these parameters lie within the confidence interval. For some motor frame sizes, there is a bias in one direction (only “plus” or only “minus”) for the parameters under consideration.
- Parameters characterizing continuous operation.
a. Efficiency \(\eta\). For this parameter there is a clearly expressed bias in one direction—the calculation gives a reduced value compared with the actual mean value. This is explained by the inaccuracy of determining individual components of the total losses \(\Sigma P\).
b. For AR 4.5 gab. motors, the calculated value of the power factor \(\cos\varphi\) is overestimated, while for AR 6.7 gab. it is underestimated compared with the actual mean value. That is, a systematic—
Table 9
| AP motor type | $M_p$ $\delta_{\max}$, % | $M_p$ $\delta_{\min}$, % | $I_k$ $\delta_{\max}$, % | $I_k$ $\delta_{\min}$, % | $P_k$ $\delta_{\max}$, % | $P_k$ $\delta_{\min}$, % | $D\,120$ $\delta_{\max}$, % | $D\,120$ $\delta_{\min}$, % | $r_1$ $\delta_{\max}$, % | $r_1$ $\delta_{\min}$, % | $\cos\varphi$ $\delta_{\max}$, % | $\cos\varphi$ $\delta_{\min}$, % |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 42-10 | +5,02 | −2,28 | +2,32 | −3,47 | −0,43 | −10,6 | +6,07 | −5,22 | −24,3 | −28,2 | +5,73 | +3,52 |
| 43-6 | +5,15 | −0,645 | +0,681 | −3,8 | +2,35 | −1,92 | +4,13 | −1,22 | −2,78 | −7,05 | +13,8 | +9,57 |
| 43-8 | +6,00 | −1,365 | +7,47 | +0,277 | +3,3 | −3,86 | +20,5 | −10,41 | −2,93 | −9,4 | +1,5 | +0,62 |
| 43-10 | +5,66 | −2,34 | +2,47 | −2,47 | +0,595 | −13,9 | +2,4 | −13,9 | −4,4 | −9,12 | +8,34 | +4,1 |
| 53-10 | +11,0 | +1,83 | +10,3 | +3,88 | +6,88 | −0,66 | +2,44 | −0,66 | −2,49 | −6,64 | +3,98 | +1,21 |
| 53-12 | +10,02 | +1,58 | +7,3 | +0,232 | +5,38 | +0,53 | +4,8 | +0,53 | −3,4 | −5,26 | +7,94 | −5,05 |
| 64-10 | +4,33 | −10,7 | +17,4 | +0,133 | +7,05 | −11,5 | −3,66 | −11,5 | −1,42 | −2,97 | 0 | −5,8 |
| 64-16 | +14,60 | −17,1 | +19,45 | +7,98 | +26,80 | −5,1 | +3,32 | −5,1 | −1,23 | −5,1 | −1,37 | −10,12 |
| 73-10 | −7,03 | −22,7 | +4,25 | −4,23 | −8,55 | −9,24 | +0,67 | −9,24 | −0,562 | −3,18 | −2,74 | −5,08 |
| 73-16 | +8,55 | −13,2 | +8,93 | +1,18 | +1,75 | −8,25 | −2,54 | −8,25 | −3,09 | −6,62 | −4,25 | −10,18 |
| 74-10 | −10,45 | −19,02 | +1,92 | −2,56 | −9,33 | +4,57 | +12,56 | +4,57 | −0,377 | −3,14 | −3,36 | −6,46 |
| 74-15 | −10,40 | −19,1 | −0,397 | −6,98 | −10,95 | +2,15 | +12,1 | +2,15 | −6,69 | −9,11 | −3,3 | −6,84 |
Continuation of Table 2
| $\Sigma P$ $\delta_{\max}$, % | $\Sigma P$ $\delta_{\min}$, % | $I_k$ $\delta_{\max}$, % | $I_k$ $\delta_{\min}$, % | $\Delta t_{\mathrm{st}}$ $\delta_{\max}$, % | $\Delta t_{\mathrm{st}}$ $\delta_{\min}$, % | $m_{\mathrm{H}}$ $\delta_{\max}$, % | $m_{\mathrm{H}}$ $\delta_{\min}$, % | $I_0$ $\delta_{\max}$, % | $I_0$ $\delta_{\min}$, % | $P_0$ $\delta_{\max}$, % | $P_0$ $\delta_{\min}$, % | $R_{15}$ $\delta_{\max}$, % | $R_{15}$ $\delta_{\min}$, % |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| +22,8 | +13,5 | +5,1 | +1,18 | +28,7 | +18,6 | −4,9 | −6,46 | −4,4 | −9,75 | +4,75 | −5,6 | +3,86 | +1,57 |
| +36,5 | +20,7 | +4,87 | −0,23 | +28,2 | +14,6 | −4,38 | −7,35 | −10,5 | −18,5 | −0,46 | −9,8 | +3,36 | +0,20 |
| +34,8 | +15,1 | +4,9 | +1,64 | +34,4 | +28,4 | −4,27 | −6,0 | −8,65 | −10,0 | +5,4 | −1,3 | +3,48 | +0,68 |
| +21,6 | +11,1 | +2,63 | −1,46 | +21,7 | +14,05 | −3,51 | −5,84 | −4,25 | −10,1 | +1,7 | −0,66 | +3,20 | +0,98 |
| +24,8 | +9,8 | +7,15 | −1,67 | +29,4 | +11,3 | −1,35 | −4,05 | +0,74 | −7,8 | −2,4 | +0,2 | +3,90 | +0,97 |
| +15,2 | +10,0 | −0,63 | −3,9 | +34,8 | −23,5 | −2,69 | −3,58 | −5,53 | −10,35 | +5,06 | −5,42 | +4,40 | +0,78 |
| +13,3 | +6,15 | +7,15 | +2,38 | +16,4 | +9,6 | +0,37 | −1,1 | +6,85 | −1,43 | −7,72 | −9,22 | +7,38 | +4,84 |
| +17,2 | +3,63 | +5,5 | +4,95 | — | — | +4,33 | −1,8 | +0,99 | −0,55 | +11,2 | −11,2 | +2,28 | 0 |
| +16,0 | +3,08 | +7,4 | +7,32 | +20,3 | +7,32 | 0 | −2,27 | +8,02 | −2,04 | +11,7 | −0,39 | +4,42 | +1,20 |
| +21,7 | +10,9 | +6,4 | +16,2 | +26,8 | +16,2 | 0 | −2,43 | +10,6 | +3,08 | +20,4 | +13,0 | +3,82 | +2,73 |
| +15,6 | +2,46 | +8,75 | +3,27 | +10,25 | +3,27 | −0,19 | −2,44 | +10,3 | +4,34 | +15,2 | −5,0 | +2,99 | +1,90 |
| +33,7 | +24,6 | +15,6 | +11,8 | +24,8 | +14,6 | −2,38 | −4,17 | +6,7 | +2,82 | +27,8 | +19,0 | +3,70 | +1,96 |
displacement of the calculated value either toward higher values or toward lower values.
c. All three cases (see Fig. 1) of deviations of the calculated value of the rated current \(I_{\mathrm{n}}\) from the average actual value occur, since this is associated with deviations of \(\eta\) and \(\cos\varphi\).
d. The actual temperature rise \(\Delta t_{cu}\) of the stator winding for all the motors considered is significantly higher than the calculated value (by 25% on average).
e. The calculated value of the rated speed \(n_{\mathrm{n}}\) is lower in comparison with the average actual value because of the inaccurate (overestimated) determination of the resistance of the rotor cage.
- No-load parameters.
a. For AR 4.5-gage motors, the calculated value of the no-load current \(I_0\) is underestimated, while for AR 6.7-gage motors it is overestimated in comparison with the average actual values.
b. For most of the motors considered, the error in calculating the no-load losses \(P_0\) lies within the confidence interval, with the exception of AR 73-16 and AR 74-16, for which the calculated value is displaced toward higher values.
- The error in calculating the active resistance of the stator winding \(R_{15}\) at a temperature of \(15^\circ\mathrm{C}\) falls within the permissible value of \(\pm 5\%\), but the calculation gives a somewhat overestimated resistance in comparison with the average actual value.
For calculated values of parameters displaced toward lower and higher values (cases “b” and “c”), i.e., lying outside the confidence interval, the corresponding corrections must be introduced in order to bring the calculated parameter values closer to the average actual values and thereby place the calculated values within the confidence interval. Knowing the technological scatter of a parameter (Table 1) and the calculation error (Table 2) for this same parameter, if the calculated value lies within the boundaries of the confidence interval (case “a” in Fig. 1), it is possible to determine the required number of motors that must be tested in order to confirm, with the required accuracy, the agreement between calculated and experimental data.
For the normal distribution law, the minimum number of such motors (case “a” in Fig. 1) for the required probability is determined according to [4]:
\[ n=\frac{4 z_{\alpha}^{2}\delta_x^{2}}{9\delta^2}, \tag{8} \]
where \(\delta\) is the maximum permissible error of the calculation method for the parameter of interest, %;
\(z_{\alpha}\) is the argument of the Laplace function \(\Phi_0(z_{\alpha})\) at the selected confidence probability \(\alpha\).
In cases “b” and “c,” the calculated parameter values should be corrected so that they satisfy case “a,” and the number of motors \(n\) should be calculated by formula (8).
For the confidence probability adopted by us in the calculation, \(\alpha=0.95\), \(z_{\alpha}=1.96\), formula (8) takes the form:
\[ n=1.71\left(\frac{\delta_x}{\delta}\right)^2 . \tag{9} \]
It is evident from this expression that the minimum number of motors that must be tested to confirm the accuracy of the calculation,
depends on the ratio \(\left(\dfrac{\delta_x}{\delta}\right)^2\). As the scatter of the parameters increases, for one and the same calculation accuracy the number of motors tested that is required to confirm this accuracy increases. It should be noted that, when determining the required number of motors, there is a certain error caused by the limited sample.
For most parameters of AP-series roller-table motors (Table 1), the technological scatter lies within the range from 5 to 30%, and the calculation accuracy is up to 15%.
Table 3 gives the required number of motors as a function of the calculation accuracy and the technological scatter of the parameter for the above-indicated limits, at confidence probability \(\alpha = 0.95\).
Knowing, from preliminary data, the technological scatter \(\delta_x\) and the approximate calculation accuracy for the parameter of interest, one can select from Table 3 the required number of motors for testing.
Table 3
| \(\delta_x,\ \%\) | \(\delta,\ \% = 1\) | \(\delta,\ \% = 2\) | \(\delta,\ \% = 5\) | \(\delta,\ \% = 10\) | \(\delta,\ \% = 15\) |
|---|---|---|---|---|---|
| 5 | 43 | 11 | 2 | 1 | 1 |
| 8 | 110 | 28 | 5 | 2 | 1 |
| 10 | 171 | 43 | 7 | 2 | 1 |
| 15 | 385 | 97 | 16 | 4 | 2 |
| 20 | 683 | 171 | 28 | 7 | 4 |
| 25 | 1070 | 265 | 43 | 11 | 5 |
| 30 | 1540 | 385 | 62 | 16 | 7 |
From the test results, the mean value of the parameter \(\bar{x}\) is determined; comparing it with the calculated value \(x_p\), we obtain \(\delta_{\min}\) and \(\delta_{\max}\).
If the condition
\[ |\delta_{\min}| + |\delta_{\max}| \leq \delta, \tag{10} \]
is satisfied, then it may be considered that the calculation accuracy \(\delta\) has been confirmed, for which the required number of motors was selected and, consequently, the calculated value falls within the interval
\[ \bar{x} - \frac{\delta \bar{x}}{200} \leq x_p \leq \bar{x} + \frac{\delta \bar{x}}{200}. \tag{11} \]
In the case where the sum \(|\delta_{\min}| + |\delta_{\max}| \ll \delta\), the value \(n\) for a higher accuracy \(\delta\) is taken from Table 3; that is, an additional number of motors is tested. An analogous calculation is carried out, but the newly adopted accuracy is checked according to (10), (11).
Conclusions
-
The calculation error of the main parameters of certain frame sizes of AP-series motors has been determined.
-
The calculated values of the parameters of roller-table motors have different errors. Motor parameters for which the calculation errors \(\delta_{\min}\) and \(\delta_{\max}\) have the same sign have a systematic bias.
In this case, the calculation procedure must be corrected in such a way that the calculated value approaches the mean actual value.
- When determining the minimum number of motors to be tested in order to confirm the corresponding calculation accuracy at the selected confidence probability, it is necessary to take into account the technological scatter, which for individual parameters of roller-table motors reaches 30%.
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