Calculation of Permissible Parameter Value Regions for Control Tests of Three-Phase Asynchronous Electric Motors
Abstract Generated abstract
This paper proposes a method for calculating permissible regions of control-test parameters for three-phase asynchronous electric motors as an alternative to labor-intensive procedures requiring type-test results. The approach uses linear relationships between deviations of input, output, and control-test parameters, expressed through influence-coefficient matrices, so that allowable zones can be derived at the design stage together with tolerance calculations. The method is illustrated for AO 32-4A motors by transforming a permissible input-parameter polygon into allowable regions for no-load current, no-load losses, short-circuit current, and short-circuit losses. The example shows how the resulting control-test zones can be linked to constraints on starting torque, maximum torque, efficiency, and power factor.
Full Text
CALCULATION OF ALLOWABLE PARAMETER VALUE ZONES FOR CONTROL TESTS OF THREE-PHASE ASYNCHRONOUS ELECTRIC MOTORS
O. P. MURAVLEV, E. K. STRELBITSKII
(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
The conformity of the output parameters of electrical machines to the tolerances of GOST 183-66 may be established from the results of very labor-intensive type tests or from simpler control tests. In the latter case it is necessary to know an algorithm that maps the boundaries of the values of the output parameters into the boundaries of the values of the parameters of the control tests.
In the practice of electrical-machine-building plants, the zones developed by O. D. Goldberg [1] are used as these boundaries. A substantial drawback of these zones is the need, in order to calculate them, for the results of type tests. Their calculation is very labor-intensive.
In the present work a method is proposed for calculating zones that is free of the indicated drawbacks.
This method is based on the procedure for calculating tolerances developed at TPI and has a number of computational operations in common with it. This considerably simplifies the calculation of zones and makes it possible to carry it out at the design stage simultaneously with the calculation of tolerances.
In works [2, 3, 4] it was shown that there is a linear relationship between deviations of input and output parameters, determined by the matrix of influence coefficients \(C\). It can be shown that the relationship between the parameters of control tests and the input parameters is also linear:
\[ \overline{\Delta U}=F\overline{\Delta V}, \tag{1} \]
where
\[ \overline{\Delta U}=(\Delta I_0,\ \Delta P_0,\ \Delta I_k,\ \Delta P_k) \]
is the vector of deviations of the parameters of the control tests, whose components are the deviations of the no-load and short-circuit current and losses in control tests;
\[ \overline{\Delta V}=(\Delta \delta,\ \Delta \rho_{\mathrm{Al}},\ \Delta(P_{\mathrm{st}}+P_{\mathrm{mech}})) \]
is the vector of deviations of the significantly influencing input parameters.
Here and below the notation of work [2] is used.
Expression (1) in expanded form
\[ \left. \begin{aligned} \Delta I_0 &= f_{11}\cdot \Delta\delta,\\ \Delta P_0 &= f_{23}\cdot \Delta(P_{\mathrm{st}}+P_{\mathrm{mech}}),\\ \Delta I_k &= f_{31}\cdot \Delta\delta + f_{32}\cdot \Delta\rho_{A1},\\ \Delta P_k &= f_{41}\cdot \Delta\delta + f_{42}\cdot \Delta\rho_{A1} \end{aligned} \right\} \tag{2} \]
Elements of the matrix \(\mathbf F\)
\[ \begin{aligned} &f_{11}=d_{13}\cdot l_{31}; \qquad f_{23}=d_{24}\cdot l_{44}; \qquad f_{31}=d_{32}\cdot l_{21};\\ &f_{32}=d_{31}\cdot l_{12}; \qquad f_{41}=d_{42}\cdot l_{21}; \qquad f_{42}=d_{41}\cdot l_{12}; \end{aligned} \]
\[ d_{13}=-\frac{x_m}{x_1+x_m}; \qquad d_{24}=\frac{P_{\mathrm{st}}+P_{\mathrm{mech}}}{P_0}; \qquad d_{31}=-\frac{r_k^2}{r_k^2+x_k^2}; \]
\[ d_{32}=-1-d_{31}; \qquad d_{42}=2d_{32}; \qquad d_{41}=-1-d_{42}; \]
\[ l_{12}=\frac{c_1^2 r_2'}{r_k}; \qquad l_{31}=-1; \qquad l_{44}=1; \]
\[ l_{21}=-\frac{c_1 x_1}{x_k}\cdot\frac{\lambda_{k1}}{\sum \lambda_1} -\frac{c_1^2 x_2'}{x_k}\cdot\frac{\lambda_{k2}}{\sum \lambda_2}. \]
We shall present the proposed method using the example of calculating the zones of allowable values of the parameters of control tests of AO 32-4A motors. The values of the elements of the matrix \(c\) and the required tolerances for the input parameters are taken from works [2, 3, 4].
All calculations are carried out in relative units.
To calculate \(\overline{\Delta U}\), it is necessary to know only the tolerances for three quantities: \(\delta\), \(\rho_{A1}\), and \(P_{\mathrm{st}}+P_{\mathrm{mech}}\). Neglecting the remaining input parameters gives, for the \(i\)-th output parameter, a root-mean-square error at a confidence level of 0.997 equal to
\[ \delta_i= \sqrt{ c_{i1}^2\delta_{w_1}^2 +c_{i2}^2\delta_{d_{\mathrm{pr}}}^2 +c_{i3}^2\delta_{l_1}^2 +c_{i6}^2\delta_{l_r}^2 +c_{i7}^2\delta_{l_2}^2 +c_{i8}^2\delta_{z_2}^2 } \tag{3} \]
Initial data for calculating the zones
Tolerances for the output parameters:
\[ \delta_{M_{\mathrm{p}}}=-20\%;\quad \delta_{M_{\mathrm{m}}}=-10\%;\quad \delta_{I_k}=+15\%;\quad \delta_{\eta}=-4.88\%;\quad \delta_{\cos\varphi}=-4.98\%. \]
\[ \delta_{w_1}=1\%;\quad \delta_{d_{\mathrm{pr}}}=3.0\%;\quad \delta_{l_m}=1.0\%;\quad \delta_{l_1}=4.15\%;\quad \delta_{l_2}=3.60\%;\quad \delta_{z_2}=0. \]
Matrix of coefficients of the influence of input parameters on output parameters \(\|c_{ij}\|\)
| −1.543 | 0.904 | −0.634 | 0.748 | 0.703 | −0.462 | 0.030 | −0.203 | 0.000 |
| −1.570 | 0.860 | −0.508 | 0.000 | 0.310 | −0.204 | −0.265 | 0.290 | 0.000 |
| −1.772 | 0.456 | −0.317 | −0.126 | 0.352 | −0.231 | −0.407 | 0.473 | 0.000 |
| −0.033 | 0.261 | −0.130 | −0.0347 | −0.0556 | 0.0829 | −0.0288 | 0.0394 | −0.0568 |
Mean values of the parameters of the control tests (they may be calculated or taken from experience):
\[ \overline{I}_0=1.525\ \mathrm{a};\quad \overline{P}_0=128.4\ \mathrm{w};\quad \overline{I}_k=2.27\ \mathrm{a};\quad \overline{P}_k=186\ \mathrm{w}. \]
Equations of the boundaries of the zones for the input parameters
\[ c_{i4}\delta_{\rho A1}+c_{i5}\delta_{\delta}+c_{i9}\delta P_{\mathrm{st}}+P_{\mathrm{mech}} =\delta_{yi}\pm\delta_i, \tag{4} \]
\[ i=1,2,\ldots,5. \]
The sign of \(\delta_i\) is taken opposite to the sign of the tolerance \(\delta_{yi}\).
Rewrite system (4) in numerical form.
\[ \begin{gathered} 0.748\delta_{\rho A1}+0.703\delta_{\delta}=-20+13.9=-16.27\\ 0.310\delta_{\delta}=-10+10.9=-6.70\\ -0.126\delta_{\rho A1}+0.352\delta_{\delta}=15-8.18=12.14\\ -0.0347\delta_{\rho A1}+0.0556\delta_{\delta}=0.0568\delta_{\rho \mathrm{st}}+P_{\mathrm{mech}}=-4.80+0.76=\\ =-4.01-0.236\delta_{\delta}=-4.98+2.95=-3.26. \end{gathered} \tag{5} \]
On the basis of system (5), Fig. 1 shows the boundaries of the region of permissible values of \(\delta\) and \(\rho_{A1}\). With respect to \(\rho_{A1}\) the region is practically unbounded because of the small influence of this parameter on the efficiency. On the basis of the actual maximum value \((\Delta \rho_{A1})_{\max}+40\%\) [3], we conditionally bound the region from above by the line \(CD\).
Fig. 1.
The equations of the boundaries of the permissible values of the parameters of control tests (2) in numerical form are
\[ \left. \begin{aligned} \Delta I_0 &= 0.937\delta_{\delta}\\ \Delta P_0 &= 0.624\delta P_{\mathrm{st}}+P_{\mathrm{mech}}\\ \Delta I_k &= 0.354\delta_{\delta}-0.126\delta_{\rho A1}\\ \Delta P_k &= 0.704\delta_{\delta}+0.105\delta_{\rho A1} \end{aligned} \right\} \tag{6} \]
Expression (6) is an algorithm for transforming the vertices of polygon \(ABCD\) (Fig. 1) onto the planes \(I_0-I_k\), \(I_k-P_k\), \(P_k-P_0\). Connecting the points obtained in the same order, we obtain new polygons that bound the zones of permissible values \(I_0\), \(I_k\), \(P_0\), \(P_k\). The permissible zones for the parameters of control tests are presented in Fig. 2. Below, the calculation of the coordinates of the polygon vertices in the axes \(I_0-I_k\), \(I_k-P_k\), and \(P_k-P_0\) is given.
Zone \(I_0-I_k\)
\[ I_{0A}=I_{0D}=\overline{I}_0+\overline{I}_0\cdot 10^{-2}\cdot 0.937(-21.6)=\overline{I}_0(1-0.202)=1.22a, \]
\[ I_{0B}=I_{0C}=\overline{I}_0+\overline{I}_0\cdot 10^{-2}\cdot 0.937\cdot 13.8=\overline{I}_0(1+0.130)=1.72a, \]
\[ I_{kA}=\overline{I}_k+\overline{I}_k\cdot 10^{-2}\cdot 0.35\cdot(-21.6)-0.126\cdot(-1.47)=\overline{I}_k(1-0.0741)=2.10a, \]
\[ I_{kB}=\overline{I}_k+\overline{I}_k\cdot 10^{-2}\cdot 0.352\cdot 13.8-0.126\cdot(-34.7)=\overline{I}_k(1+0.0922)=2.48a, \]
Fig. 2. Allowable zones for AO 32-4A electric motors.
Visible labels in the figure:
- Zone \(I_0 - I_k\)
- Zone \(I_k - p_k\)
- Zone \(p_k - p_0\)
Axis labels and point labels shown include \(I_0/a\), \(I_k/a\), \(p_k/\mathrm{W}\), \(p_0/\mathrm{W}\), and points \(A\), \(B\), \(C\), \(D\).
\[ I_{kC}=\overline{I}_{k}+\overline{I}_{k}\cdot 10^{-2}\cdot 0.352\cdot 13.8-0.126\cdot 40 =\overline{I}_{k}(1-0.0019)=2.27\text{ A}, \]
\[ I_{kD}=\overline{I}_{k}+\overline{I}_{k}\cdot 10^{-2}\cdot 0.352\cdot(-21.6)-0.126\cdot 40 =\overline{I}_{k}(1-0.1264)=1.98\text{ A}. \]
Zone \(I_k\)—\(P_k\)
The coordinates of the points along the \(I_k\) axis are the same as for the \(I_0\)—\(I_k\) zone.
\[ P_{kA}=\overline{P}_{k}+\overline{P}_{k}\cdot 10^{-2}\cdot[0.704\cdot(-21.6)+0.105(-1.47)] = \]
\[ =\overline{P}_{k}(1-0.1536)=157.46\text{ W}. \]
\[ P_{kB}=\overline{P}_{k}+\overline{P}_{k}\cdot 10^{-2}\cdot[0.704\cdot 13.8+0.105(-34.7)] =\overline{P}_{k}(1+0.0606)=197.3\text{ W}. \]
\[ P_{kC}=\overline{P}_{k}+\overline{P}_{k}\cdot 10^{-2}\cdot[0.704\cdot 13.8+0.105\cdot 40] =\overline{P}_{k}(1+0.0139)=211.8\text{ W}. \]
\[ P_{kD}=\overline{P}_{k}+\overline{P}_{k}\cdot 10^{-2}\cdot[0.704\cdot(-21.6)+0.105\cdot 40] = \]
\[ =\overline{P}_{k}(1-0.00110)=165.6\text{ W}. \]
Zone \(P_k\)—\(P_0\)
The \(P_k\)—\(P_0\) zone is not limited with respect to the minimum value of \(P_0\). The limitation of the region with respect to the maximum permissible value of \(P_0\) can be made if the limiting values are taken for \(P_k\) (points A and C).
From the equation for the efficiency (5),
\[ \delta_{P_{\mathrm{st}}+P_{\mathrm{mech}}} = -\frac{4.01-0.0347\delta_{P_{A1}}-0.0556\delta_{\delta}}{0.0568}. \]
Then
\[ P_{0A}=0.624\overline{P}_{0}\left(1+\delta_{P_{\mathrm{st}}+P_{\mathrm{mech}}}^{(A)}\right) =0.624\overline{P}_{0}(1+0.578)=202.5\text{ W}. \]
\[ P_{0C}=0.624P_{0}\left(1+\delta_{P_{\mathrm{st}}+P_{\mathrm{mech}}}^{(B)}\right) =0.624\overline{P}_{0}(1+0.204)=154.6\text{ W}. \]
For the \(I_0\)—\(I_k\) and \(I_k\)—\(P_k\) zones, the lines \(AB\) represent the limitation by starting torque, \(BC\) by power factor, \(DA\) by maximum torque, and the line \(DC\) is drawn conventionally.
REFERENCES
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O. D. Goldberg. A method for calculating technical standards for permissible parameter values in control tests of three-phase asynchronous electric motors. Information and Technical Collection, issue 19(103), 1956, TsBTI, MEP.
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O. P. Muravlev, E. K. Strelbitsky. The influence of manufacturing technology on the output parameters of asynchronous electric motors (in the same collection).
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O. P. Muravlev. Investigation of the influence of the accuracy characteristics of the technological process on the quality and reliability of asynchronous electric motors. Dissertation, Tomsk, 1966.
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O. P. Muravlev, E. K. Strelbitsky. Ensuring the required accuracy in the production of asynchronous motors. Elektrotekhnika, No. 7, 1966.