Analysis of the Manufacturing Process Accuracy of Explosion-Proof Induction Motors Rated 7.5–100 kW
Abstract Generated abstract
This paper analyzes how manufacturing tolerances affect the performance scatter of explosion-proof induction motors rated 7.5 to 100 kW in the VAO series. Using an analytical tolerance propagation method based on influence coefficients for input dimensions and material characteristics, the authors calculate the expected variation in starting torque, maximum torque, starting current, efficiency, and power factor for multiple frame sizes and pole numbers. The results indicate that rotor aluminum resistivity, air-gap magnitude, and steel plus mechanical losses are the dominant contributors to output variation. Comparison with GOST 183-66 tolerances suggests that the existing manufacturing process generally satisfies requirements for maximum torque, starting current, efficiency, and partly power factor, while starting torque scatter often exceeds the specified limit.
Full Text
ANALYSIS OF TECHNOLOGICAL ACCURACY IN THE MANUFACTURE OF EXPLOSION-PROOF INDUCTION MOTORS RATED 7.5–100 kW
Yu. M. Grinberg, O. P. Muravlev
(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
The accuracy of the technological process for manufacturing induction motors is characterized by the scatter of input parameters, by which we mean the main and local dimensions, as well as the characteristics of the materials used (the magnitude of the air gap, the lengths of the stator and rotor cores, the resistivity of the aluminum of the rotor winding, etc.). Scatter of the input parameters causes scatter of the output parameters (starting and maximum torques, starting current, efficiency, and power factor).
The correspondence between the tolerances for output parameters according to GOST 183–66 and the scatter of these parameters can be established by processing the results of a large number of type tests or analytically on the basis of accuracy indicators of the technological processes. The first method is very labor-intensive and is impossible at the design stage, while the second does not have these disadvantages.
The dependence between the scatter of input and output parameters has the form [1]
\[ \delta_{y_i}^{2}=\sum_{j=1}^{n} c_{ij}^{2}\delta_{x_j}^{2}, \tag{1} \]
where
\[ \begin{aligned} \delta_{y_i}\ \text{and}\ \delta_{x_j} &\quad \text{are half the scatter range, in percent, for the output and input parameters;}\\ c_{ij} &\quad \text{is the coefficient of influence of the } j\text{-th input parameter on the } i\text{-th output parameter. It is determined according to [1].} \end{aligned} \]
The accuracy indicators of the technology for the plant “Kuzbasselektromotor,” where the motors under consideration are produced, have the following values:
\[ \delta_{lm}=0.8\text{—}1.1\%; \qquad \delta_{w1}=0; \]
\[ \delta_{l1}=4.79-0.01\,l_1; \]
\[ \delta_{l2}=3.57-0.0044\,l_2, \qquad \text{where } l_1 \text{ and } l_2 \text{ are in mm;} \]
\[ \delta_{\rho A1}=19.2+\frac{268}{P_{\mathrm{H}}}, \qquad \text{where } P_{\mathrm{H}} \text{ is in kW;} \]
\[ \delta_o = 15{,}7 + 1{,}02 \cdot 2p; \]
\[ \delta_{P_{\mathrm{st}}+P_{\mathrm{mech}}} = 40{,}9 + \frac{10{,}8}{P_{\mathrm{st}}+P_{\mathrm{mech}}}, \quad \text{where } P_{\mathrm{st}}+P_{\mathrm{mech}} \text{ is in kW.} \]
\[ \delta_{z_2}=0; \quad \delta_{d_{\mathrm{pr}}} \text{ according to } [2]. \]
The method for determining \(\delta_{x_j}\) is given in [3], and the symbols used are in [1, 3].
The coefficients of the influence of input parameters on the output parameters for induction motors are calculated analytically on the basis of the verification-calculation method [1]. To simplify the calculation, the dependence
\[ \left\|c_{ij}\right\|=\left\|a_{ik}\right\|\cdot\left\|b_{kj}\right\|, \tag{2} \]
where
\[ \left\|a_{ik}\right\| \text{ is the matrix of coefficients of influence of the parameters of the equivalent circuit on the output parameters;} \]
\[ \left\|b_{kj}\right\| \text{ is the matrix of coefficients of influence of the inputs on the parameters of the equivalent circuit.} \]
For 31 frame sizes of VAO electric motors of sizes 6–9, the elements of all matrices were calculated.
Matrix of mean values \(a_{ik}\).
| \(r_1\) | \(r_2^{1}\) | \(x_k\) | \(x_m\) | |
|---|---|---|---|---|
| \(M_{\mathrm{p}}\) | \(-0{,}167\) | \(0{,}911\) | \(-1{,}745\) | \(0{,}000\) |
| \(M_m\) | \(-0{,}229\) | \(0{,}000\) | \(-0{,}771\) | \(0{,}000\) |
| \(I_{\mathrm{p}}\) | \(-0{,}83\) | \(0{,}043\) | \(-0{,}872\) | \(0{,}000\) |
| \(\eta\) | \(-0{,}047\) | \(0{,}018\) | \(0{,}002\) | \(0{,}017\) |
| \(\cos\varphi\) | \(-0{,}006\) | \(0{,}000\) | \(-0{,}065\) | \(0{,}210\) |
Matrix of mean values \(b_{kj}\)
| \(w_1\) | \(d_{\mathrm{pr}}\) | \(l_m\) | \(\rho_{\mathrm{Al}}\) | \(\delta\) | \(l_1\) | \(l_2\) | \(z_2\) | |
|---|---|---|---|---|---|---|---|---|
| \(r_1\) | \(1\) | \(-2\) | \(1\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) |
| \(r_2^{1}\) | \(2\) | \(0\) | \(0\) | \(1\) | \(0\) | \(0\) | \(0{,}736\) | \(-1\) |
| \(x_k\) | \(2\) | \(0\) | \(0{,}466\) | \(0\) | \(-0{,}202\) | \(0{,}402\) | \(0{,}496\) | \(-0{,}646\) |
| \(x_m\) | \(2\) | \(0\) | \(0\) | \(0\) | \(-0{,}665\) | \(1\) | \(0\) | \(0\) |
Table 1
Mean values of the coefficients \(c_{ij}\)
| \(j\) | Input parameter | \(M_{\mathrm{p}}\) \(i=1\) |
\(M_m\) \(i=2\) |
\(I_{\mathrm{p}}\) \(i=3\) |
\(\eta\) \(i=4\) |
\(\cos\varphi\) \(i=5\) |
|---|---|---|---|---|---|---|
| 1 | \(w_1\) | \(-1{,}838\) | \(-1{,}771\) | \(-1{,}900\) | \(-0{,}049\) | \(0{,}299\) |
| 2 | \(d_{\mathrm{pr}}\) | \(0{,}333\) | \(0{,}459\) | \(0{,}167\) | \(0{,}094\) | \(-0{,}014\) |
| 3 | \(l_m\) | \(-0{,}983\) | \(-0{,}592\) | \(-0{,}492\) | \(-0{,}047\) | \(-0{,}023\) |
| 4 | \(\rho_{\mathrm{Al}}\) | \(0{,}911\) | \(0{,}000\) | \(-0{,}043\) | \(-0{,}181\) | \(0{,}000\) |
| 5 | \(\delta\) | \(0{,}351\) | \(0{,}155\) | \(0{,}176\) | \(-0{,}011\) | \(-0{,}126\) |
| 6 | \(l_1\) | \(-0{,}695\) | \(-0{,}305\) | \(-0{,}346\) | \(0{,}017\) | \(0{,}185\) |
| 7 | \(l_2\) | \(-0{,}199\) | \(-0{,}386\) | \(-0{,}469\) | \(-0{,}013\) | \(-0{,}033\) |
| 8 | \(z_2\) | \(0{,}225\) | \(0{,}504\) | \(0{,}611\) | \(0{,}018\) | \(0{,}043\) |
| 9 | \(P_{\mathrm{st}}+P_{\mathrm{mech}}\) | \(0{,}000\) | \(0{,}000\) | \(0{,}000\) | \(-0{,}0357\) | \(0{,}000\) |
Analysis of the coefficients of influence shows that the elements of the matrices have a large scatter. For approximate calculations the following relationships may be used:
\[ a_{ik}=b_{0ik}+b_{1ik}P_{\mathrm{n}}; \tag{3} \]
\[ b_{ki}=b_{0kj}+b_{1kj}\cdot 2p. \tag{4} \]
Table 2 presents the data required for calculation by formulas (3) and (4).
Table 2
| Coefficient of influence | \(b_0\) | \(b_1\) |
|---|---|---|
| \(a_{11}\) | −0,297 | 0,00352 |
| \(a_{12}\) | 0,857 | 0,00144 |
| \(a_{13}\) | −1,616 | −0,00275 |
| \(a_{23}\) | −0,662 | −0,00295 |
| \(a_{41}\) | −0,0388 | −0,00064 |
| \(a_{42}\) | −0,0234 | 0,00014 |
| \(a_{54}\) | 0,231 | −0,00080 |
| \(b_{27}\) | 0,53 | 0,0405 |
| \(b_{35}\) | −0,127 | −0,0146 |
| \(b_{36}\) | 0,335 | 0,0132 |
| \(b_{37}\) | 0,383 | 0,0222 |
All the remaining coefficients, for whose calculation there are no data in Table 2, are determined from the matrices of mean values, with the exception of \(a_{21}\), \(a_{31}\), \(a_{32}\), \(a_{33}\), and \(b_{45}\) \((a_{21}=-1-a_{23},\ a_{31}=0{,}5a_{11},\ a_{33}=\)
\[ =0{,}5a_{13},\quad a_{32}=-1-a_{31}-a_{33},\quad b_{45}=-\frac{1}{k_{\mathrm{n}}}, \]
where \(k_{\mathrm{n}}\) is the saturation coefficient.
The coefficients of the matrix \(\lVert c_{ij}\rVert\) are calculated by the formula
\[ c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\ldots+a_{in}b_{nj}; \tag{5} \]
\[ i=1,2,3,4,5;\quad j=1,2,\ldots,8;\quad c_{19}=c_{29}=c_{39}=c_{59}=0; \]
\[ c_{49}=-\frac{P_{\mathrm{st}}+P_{\mathrm{mech}}}{P_2+\sum P}. \tag{6} \]
To estimate the share effect of the scatter of the input parameters according to equation (1), calculations were performed for three motors of each polarity and of different frame sizes. Table 3 gives the results of the calculations.
Table 3
| Output parameters | Input parameters | \(2p=2\) | \(2p=4\) | \(2p=6\) | Calculation by means: \(2p=8\) | Calculation by means: \(c_{ij}\) and \(\delta_{x_j}\) |
|---|---|---|---|---|---|---|
| \(M_{\mathrm{p}}\) | \(\rho_{Al}\) | 0,939 | 0,833 | 0,960 | 0,962 | 0,931 |
| \(M_{\mathrm{p}}\) | \(\delta\) | 0,054 | 0,161 | 0,035 | 0,034 | 0,063 |
| \(M_{\mathrm{p}}\) | \(\Sigma\) | 0,993 | 0,994 | 0,995 | 0,996 | 0,994 |
| \(M_{\mathrm{m}}\) | \(\delta\) | 0,718 | 0,923 | 0,656 | 0,715 | 0,816 |
| \(I_{\mathrm{p}}\) | \(\rho_{Al}\) | 0,141 | 0,013 | 0,288 | 0,283 | 0,098 |
| \(I_{\mathrm{p}}\) | \(\delta\) | 0,667 | 0,896 | 0,500 | 0,559 | 0,761 |
| \(I_{\mathrm{p}}\) | \(\Sigma\) | 0,808 | 0,909 | 0,788 | 0,842 | 0,859 |
| \(\eta\) | \(\rho_{Al}\) | 0,012 | 0,099 | 0,152 | 0,145 | 0,084 |
| \(\eta\) | \(\delta\) | 0,000 | 0,003 | 0,020 | 0,177 | 0,015 |
| \(\eta\) | \(P_{\mathrm{st}}+P_{\mathrm{mech}}\) | 0,985 | 0,892 | 0,819 | 0,725 | 0,893 |
| \(\eta\) | \(\Sigma\) | 0,997 | 0,994 | 0,991 | 0,987 | 0,992 |
| \(\cos\varphi\) | \(\delta\) | 0,950 | 0,911 | 0,966 | 0,973 | 0,986 |
As can be seen from the table, the scatter of the starting torque is due to \(\rho_{\mathrm{Al}}\) and \(\delta\), with the influence of the air-gap magnitude being insignificant. The scatter in the air-gap magnitude determines the scatter of the maximum torque, the power factor, and, to a considerable extent, the starting current.
The scatter of the efficiency depends entirely on \(\rho_{\mathrm{Al}}\), \(\delta\), \(P_{\mathrm{st}} + P_{\mathrm{mech}}\), but the relationship between the influence of these parameters is not the same: as the number of poles increases, the influence of \(\rho_{\mathrm{Al}}\) and \(\delta\) increases.
Thus, the equivalent specific resistance of the short-circuited rotor winding, the magnitude of the air gap, and the losses in steel and mechanical losses determine the scatter of the output parameters.
For the existing manufacturing technology of induction motors of the VAO series, sizes 6–9, the scatter of the output parameters was calculated. The results are given in Table 4.
Table 4
| Output parameter | Limits of \(\delta_{y_i}\), % |
|---|---|
| \(M_{\mathrm{p}}\) | 20.4—45.6 |
| \(M_{\mathrm{m}}\) | 2.5—6.3 |
| \(I_{\mathrm{p}}\) | 2.5—7.1 |
| \(\eta\) | 1.4—2.8 |
| \(\cos\varphi\) | 0.9—5.8 |
Comparison of the scatter of the output parameters with the tolerances according to GOST 183—66 shows that the adopted manufacturing technology ensures compliance with the standard for such parameters as \(M_{\mathrm{m}}\), \(I_{\mathrm{p}}\), \(\eta\), and partially \(\cos\varphi\).
The scatter of the starting torque for most motors considerably exceeds the 20% established by the standard.
Thus, the studies carried out make it possible to evaluate the accuracy of the technological process for manufacturing explosion-proof induction motors, to calculate the scatter of the output parameters both in production and at the motor design stage, and to assess the relationship between the capabilities of the technological process and the GOST tolerances for the output parameters.
REFERENCES
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O. P. Muravlev, E. K. Strelbitskii. Calculation of tolerances for the parameters of induction motors. Elektrotekhnika, 1968, No. 11.
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GOST 2112—62. “Round copper electrical wire.”
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O. P. Muravlev. Study of the influence of the accuracy characteristics of the technological process on the quality and reliability of induction electric motors. Dissertation, Tomsk, 1966.