Verification Calculation of Tolerances for Output Parameters of Asynchronous Electric Motors
Abstract Generated abstract
The paper addresses verification calculation of tolerances for the output parameters of asynchronous motors, including starting and maximum torque, starting current, efficiency, and power factor, as functions of manufacturing scatter in input parameters. It formulates tolerance propagation by both limiting deviation and probabilistic methods, using influence coefficients derived from partial derivatives and quadratic summation of normally distributed errors. Drawing on production measurements and acceptance test data for A, AO, and AO2 series motors, the authors estimate scatter in winding dimensions, resistance, core lengths, rotor aluminum resistivity, air gap, and combined iron and mechanical losses. The work provides empirical relations and a general procedure for using manufacturing statistics to assess whether motor output tolerances are consistent with actual production accuracy.
Full Text
Verification Calculation of Tolerances for Output Parameters of Asynchronous Electric Motors
O. P. Muravlev, E. K. Strelbitskii
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The manufacturing accuracy of electrical machines is an important indicator characterizing quality. Accuracy should be understood as the degree of correspondence of the parameters of the manufactured machine to the specified design parameters.
By the output parameters of an asynchronous motor we mean the starting and maximum torques, starting current, efficiency, and \(\cos\varphi\); by the input parameters—the number of turns of the stator winding \(w_1\), the wire diameter \(d_{\text{pr}}\), the turn length \(l_t\), the specific resistance of the rotor aluminum \(\rho_{\text{Al}}\), the value of the air gap \(\delta\), the lengths of the rotor and stator cores \(l_2\) and \(l_1\), the number of rotor bars \(z_2\), and the sum of the iron and mechanical losses \(P_{\text{st}} + P_{\text{mech}}\).
There are two methods for calculating tolerances: calculation by limiting deviations (the “maximum–minimum” method) and the probabilistic method [1]. The probabilistic method is more accurate. In applying this method, we make the following assumptions.
- The law of distribution of the errors of the input and output parameters is normal, which in fact holds in practice for the parameters we have chosen. Half of the tolerance field \(\delta_i\) is related to the standard deviation \(\sigma_i\) by the relation
\[ \delta_i = 3\sigma_i . \tag{1} \]
- The centers of grouping of the errors coincide with the calculated values of the parameters, shifted by the systematic displacement in the case of an asymmetric tolerance. To carry out the calculations, \(\delta_i\) are taken as percentages of the nominal value of the corresponding parameter.
To determine the tolerances, the error equations may be used.
In general form, the input and output parameters are related by the expression
\[ y_i = f(x_1, \ldots, x_j, \ldots, x_m), \tag{2} \]
where \(y_i\) is an input parameter,
\(x_i\) is an output parameter.
The system of equations for calculating tolerances by the maximum–minimum method in vector form has the form:
\[ \bar{\delta}_y = \bar{C}\bar{\delta}_x, \tag{3} \]
where \(\bar{\delta}_y\) is an \(l\)-dimensional vector of tolerances of the output parameters;
\(\bar{\delta}_x\) is an \(m\)-dimensional vector of tolerances of the input parameters;
\(C\) is the matrix of coefficients of the influence of the input parameters on the output parameters.
The values of the coefficients of the matrix \(C\) can be determined analytically:
\[ c_{ij} = \frac{\partial f_i(x_1,\ldots,x_j,\ldots,x_m)}{\partial x_j} \cdot \frac{x_j}{f_i(x_1,\ldots,x_j,\ldots,x_m)} . \tag{4} \]
When calculating tolerances by the probabilistic method, quadratic addition of quantities characterizing scatter is used. The system of equations in vector form has the form:
\[ (\bar{\delta}_y^2)=F(\bar{\delta}_x^2), \tag{5} \]
where \((\bar{\delta}_y^2)\) is an \(l\)-dimensional vector of the squares of the tolerances of the output parameters;
\((\bar{\delta}_x^2)\) is an \(m\)-dimensional vector of the squares of the tolerances of the input parameters.
The coefficients of the matrix \(F\)
\[ f_{ij}=c_{ij}^2 . \tag{6} \]
The values \(\delta_{x_i}\) can be determined by statistical processing of production errors of the machines being manufactured. Below are data on the scatter of input parameters for a number of motors of the A (AO) and AO2 series manufactured at the Sibelektromotor plant.
The scatter of the mean length of a turn of the stator winding is affected by deviation of the dimensions of the template within the tolerance field from the calculated value, and by different tensioning and placement of the turns when winding the coils. Half the tolerance field for the mean length of a winding turn is
\[ \delta_{l_m}=0.80\ldots 1.20\%, \]
with the larger values applying to motors of lower power.
The deviation of the diameter of the wire used for the stator winding is determined by the permissible deviations established by standards (GOST 2112—62 “Round copper electrical wire” and GOST 6132—62 “Aluminum wire”).
The scatter in the number of turns of the stator winding can be determined by calculation, knowing the deviations of \(r_1\), \(d_{\text{pr}}\), \(l_m\), and the specific resistance of the stator winding \(\rho\). To determine the scatter of \(r_1\), measurements were made of the ohmic resistance of the stator winding under direct current on 100 samples of each frame size. The values of \(\delta_{r_1}\) vary within the range \(3.3\ldots 6.5\%\). The larger values apply to aluminum windings.
The resistance of the stator winding is
\[ r_1=\rho \frac{4w_1l_m}{\pi d_{\text{pr}}^2}. \tag{7} \]
Taking expression (4) into account, one may write
\[ \delta_{r_1}^2=\delta_{w_1}^2+\delta_{l_m}^2+4\delta_{d_{\text{pr}}}^2+\delta_{\rho}^2, \]
whence
\[ \delta_{w_1}^2=\delta_{r_1}^2-\delta_{l_m}^2-4\delta_{d_{\text{pr}}}^2-\delta_{\rho}^2 . \tag{8} \]
Using formula (8), \(\delta_{w_1}\) is calculated for any asynchronous motor. For motors of frame sizes 3–4 of the AO and AO2 series, \(\delta_{w_1}=0\ldots 2.0\%\). The value \(\delta_{\rho}\simeq 2.25\%\) [2].
The lengths of the stator and rotor stacks were measured in two mutually perpendicular directions for motors of frames 3, 4, and 6 of the AO series and frames 3 and 4 of the AO2 series. The measurement results show that the scatter of \(l_1\) and \(l_2\) does not depend substantially on the stack diameter and exceeds the established tolerances in all the motors considered.
From the experimental data, by the least-squares method, the following dependences were obtained:
\[ \delta_{l_1}=5.88-0.0173\,l_1\%, \tag{9} \]
\[ \delta_{l_2}=7.07-0.0347\,l_2\%, \tag{10} \]
where \(l_1\) and \(l_2\) are in mm.
For manufacturing squirrel-cage rotor windings of normal-design asynchronous motors, aluminum of grade A1 is used; its resistivity \(\rho_{\mathrm{Al}}\) is monitored indirectly by determining the amount of impurities (iron and silicon), which guarantees maintenance of the value of \(\rho_{\mathrm{Al}}\) with an accuracy of \(\pm 8.0\%\). The value of \(\rho_{\mathrm{Al}}\) is substantially affected by pulverization of the metal, which promotes the formation of air porosity, and by the absence of refining of the metal, which removes nonmetallic impurities and absorbed gases. Therefore the scatter of \(\rho_{\mathrm{Al}}\) considerably exceeds \(\pm 8\%\).
With a confidence of not less than 0.99, it may be assumed that \(\delta_{\rho_{\mathrm{Al}}}=\delta_{r'_2}\). We determine the value of the actual tolerance field of the rotor resistance \(2\delta_{r'_2}\) from the results of check measurements of the short-circuit power and current \(P_k\) and \(I_k\). The short-circuit resistance is
\[ r_k=c_1r_1+c_1^2r'_2=\frac{P_k}{3I_k^2}, \tag{11} \]
\[ \delta_{r_k}^{2}=\left(\frac{c_1r_1}{r_k}\right)^2\delta_{r_1}^{2} +\left(\frac{c_1^2r'_2}{r_k}\right)^2\delta_{r'_2}^{2}, \tag{12} \]
whence
\[ \delta_{r'_2}^{2}= \frac{ \delta_{r_k}^{2}-\left(\dfrac{c_1r_1}{r_k}\right)^2\delta_{r_1}^{2} }{ \left(\dfrac{c_1^2r'_2}{r_k}\right)^2 }. \tag{13} \]
Here \(c_1, r_1\) are calculated values, and \(\delta_{r_1}=5.38\%\), taking into account the scatter of \(r_1\) and the fluctuation of the winding temperature in the interval \(15\ldots 25^\circ\mathrm{C}\).
On the basis of processing the results of acceptance tests, the following dependence was obtained for motors of power \(1.0\ldots 7.5\) kW:
\[ \delta_{\rho_{\mathrm{Al}}}=45.5-2.05\,P_n, \tag{14} \]
where \(P_n\) is the rated power of the motor, kW.
The magnitude of the air gap \(\delta\) of small- and medium-power asynchronous electric motors is practically not monitored because of the great labor intensity or impossibility of the measurements. However, the magnitude of the scatter of \(\delta\) can be judged from the no-load current \(I_0\), which is measured during acceptance tests. Statistical processing of the results obtained for motors of frames 3–6 of the A (AO) series and frames 3–4 of the AO2 series makes it possible to determine the magnitude of the actual scatter:
\[ \delta_{\delta}=12.0\ldots 15.0\% \quad \text{for } 2p=2, \]
\[ \delta_{\delta}=13.5\ldots 18.5\% \quad \text{for } 2p>2. \]
The larger values apply to motors of lower power. The iron losses \(P_{\mathrm{st}}\) and mechanical losses \(P_{\mathrm{mech}}\) are not separated for the following reasons.
-
The distribution laws of the iron and mechanical losses differ from the normal law, while the sum \(P_{\mathrm{st}}+P_{\mathrm{mech}}\) usually has a normal distribution.
-
The scatter of \(P_{\mathrm{st}}+P_{\mathrm{mech}}\) can be calculated from the results of acceptance tests.
The no-load losses are
\[ P_0=P_{\mathrm{mo}}+P_{\mathrm{st}}+P_{\mathrm{mech}}, \tag{15} \]
where \(P_{\mathrm{mo}}=3I_0^2r_1\) is the loss in the stator winding under no-load conditions.
On the basis of expression (15),
\[ \delta^2_{P_{\mathrm{st}}+P_{\mathrm{mech}}} = \frac{ \delta^2_{P_0} - \left(\dfrac{P_{\mathrm{mo}}}{P_0}\right)^2 \delta^2_{P_{\mathrm{mo}}} }{ \left(\dfrac{P_{\mathrm{st}}+P_{\mathrm{mech}}}{P_0}\right)^2 }, \tag{16} \]
where
\[ \delta^2_{P_{\mathrm{mo}}}=4\delta^2_{I_0}+\delta^2_{r_1}; \]
\(\delta_{P_0}\), \(\delta_{I_0}\), \(P_0\), and \(P_{\mathrm{st}}+P_{\mathrm{mech}}\) are determined from acceptance-test data.
For motors of frame sizes 3 and 4 of the AO and AO2 series, the following dependence was obtained by the method of least squares:
\[ \delta_{P_{\mathrm{st}}+P_{\mathrm{mech}}} = 60-0.59\,(P_{\mathrm{st}}+P_{\mathrm{mech}}). \tag{17} \]
Taking into account that the reject rate due to poor rotor casting is small—on the order of \(0.05\%\)—one may assume \(\delta_{z_2}=0\).
Thus, data have been obtained on the scatter of the input parameters of asynchronous electric motors that can be used for the verification calculation of tolerances. The proposed method for calculating tolerances and determining the scatter of parameters can be used for any electrical machine.
REFERENCES
- N. A. Borodachev. Analysis of the Quality and Accuracy of Production. Mashgiz, 1946.
- E. K. Ganusevich, L. Z. Reut. Ohmic resistance of enamel wires. Elektrotekhnika, 1964, No. 6.