On the Variation of Output Parameters of Asynchronous Electric Motors

O. P. Muravlev, E. K. Strelbitskii

(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering.)

At the present time the importance of the problem of manufacturing accuracy for the further development of the national economy is widely recognized [1]. However, the introduction of accuracy calculations into the practice of design and technological bureaus is still extremely insignificant. In particular, in the design of electrical machines no accuracy calculation of output parameters is performed.

The establishment of tolerances is carried out either by directive or experimentally. The existing standards for permissible deviations of the output parameters of asynchronous motors are assigned almost identically for the entire range of torque ratios, speeds, etc. As a result, some motors have unjustifiably large accuracy margins, while in other types of motors large variations of the parameters, even within the tolerance limits, significantly worsen operating characteristics, in particular reliability.

The harmful effect of a large scatter of parameters is not limited to an increased probability of rejects. A large scatter in the value of the starting torque leads to a large scatter in the service life of the stator-winding insulation, especially under conditions of frequent starts under load. Suppose that the starting torque has decreased by 20% relative to the rated starting torque. For this case, the reduction in the service life of the insulation as a function of the load on the shaft of an A61-2 motor is shown in Fig. 1.

Fig. 1. Reduction in insulation service life as a function of shaft load for an A61-2 motor \((M_{\mathrm{p}} = 0.8 M_{\mathrm{pn}} = 0.96 M_{\mathrm{n}}\), the maximum torque of the unit is equal to the maximum torque of the motor). \(\Delta p\) is the reduction in service life relative to the case \(M_{\mathrm{p}} = M_{\mathrm{pn}} = 1.2 M_{\mathrm{n}}\). \(M_{\mathrm{s}}\) is the static torque on the shaft; \(M_{\mathrm{pn}}\) is the catalog starting torque.

As shown by calculations carried out by the method of I. A. Syromyatnikov [2], for asynchronous electric motors of series A, under overloa-

in the stator-winding phases on the order of \(30\)—\(50^\circ\mathrm{C}\) in intermittent-duty operation, insulation wear occurs mainly during starts.

The increase in insulation wear when the starting torque is reduced depends strongly on the speed of rotation, the heating time constant of the winding, the mechanical time constant, the magnitude of the starting current, and the starting and maximum torques.

Figure 2 shows the dependence of the reduction in insulation service life \(\Delta \rho\) on the mechanical time constant of the motor—mechanism unit \(T_j\). The data for the calculation are: winding heating time constant \(T = 140\) sec, stator-winding temperature rise in rated operation \(\tau_{\mathrm{n}} = 40^\circ\mathrm{C}\), ambient temperature \(\Theta_{\mathrm{amb}} = 35^\circ\mathrm{C}\), starting-current ratio

\[ \frac{I_{\mathrm{p}}}{I_{\mathrm{n}}} = 5, \]

starting-torque ratio

\[ \frac{M_{\mathrm{p}}}{M_{\mathrm{n}}} = 1.2, \]

maximum-torque ratio

\[ \frac{M_m}{M_{\mathrm{n}}} = 2.5. \]

The reduction in insulation service life was calculated for a 20% reduction in starting torque relative to the catalog value.

Fig. 2. Reduction in insulation service life as a function of the mechanical time constant for different loads

\[ 1 — \frac{M_c}{M_{\mathrm{n}}} = 0 \]

\[ 2 — \frac{M_c}{M_{\mathrm{n}}} = 0.5 \]

\[ 3 — \frac{M_c}{M_{\mathrm{n}}} = 0.9 \]

It can be shown that scatter in losses also leads to a sharp reduction in the service life of the insulation of the windings in some machines and, consequently, to a decrease in reliability.

Thus, manufacturing accuracy is a decisive factor in improving the quality and reliability of electric machines.

In the present work a method is proposed for calculating the influence on the accuracy and reliability of asynchronous motors of various design solutions. For definiteness, we shall agree to call the output

data characterizing the operation of the machine (starting torque, short-circuit current, efficiency, etc.), the parameters of the equivalent circuit—the electrical parameters of the machine—and, as input parameters, the principal and local dimensions, as well as the characteristics of the materials used (air-gap size, wire diameter, resistivity of aluminum, etc.).

The output and input parameters of an electrical machine, in the most general form, may be related by the following expression:

\[ y_i=f_i(x_1,\ldots,x_j,\ldots,x_m), \tag{1} \]

where \(y_i\) is the output parameter;
\(x_j\) is the input parameter.

The equation for relative errors has the form

\[ \frac{\Delta y_i}{y_{i0}} = \sum_{j=1}^{m} \frac{\partial f_i(x_1,\ldots,x_j,\ldots)}{\partial x_j} \cdot \frac{x_{j0}}{f_i(x_1,\ldots,x_j,\ldots)} \frac{\Delta x_j}{x_{j0}}, \tag{2} \]

where \(y_{i0}\) and \(x_{j0}\) are the nominal values of the parameters;
\(\Delta y_i\) and \(\Delta x_j\) are the deviations of the parameters from the nominal value.

The coefficients in equation (2) that precede the errors of the input parameters and determine the degree of their influence on the errors of the output parameters are called influence coefficients.

In the real case, the accuracy of the machine output parameters will be described by a system of \(l\) error equations, where \(l\) is the number of output parameters under consideration. These equations are the starting point for calculating tolerances.

Using the error equations, one can calculate the deviations of an output parameter from specified tolerances on the input parameters, and calculate the permissible deviations of the input parameters from a specified tolerance on the output parameter.

It follows from expression (2) that a reduction in the scatter of an output parameter can be achieved in two ways:

1) by varying the influence coefficients,
2) by reducing the scatter of the input parameters.

The best solution is obtained when both methods are used. We shall consider only the first, i.e., we shall investigate the influence of the machine design data on the sensitivity of the output parameters of various machines to identical technological errors.

The influence coefficients can be determined by various methods. If simple equations (1) are available, the influence coefficient of the \(j\)-th input parameter on the \(i\)-th output parameter can be determined by the analytical method:

\[ a_{ij} = \frac{\partial f_i(x_1,\ldots,x_j,\ldots)}{\partial x_j} \cdot \frac{x_j}{f_i(x_1,\ldots,x_j,\ldots)}. \tag{3} \]

Substituting the calculated nominal values of the parameters into expression (3), we obtain the numerical value of this coefficient.

For complex functions (1), this method is laborious. Moreover, an analytical function (1) is not always available, for example, for winding overheating.

For any output parameter it is not difficult to write an error equation with unknown influence coefficients, which can be calculated by a numerical method.

In a number of cases, the influence coefficients can be found experimentally if it is possible to keep all input parameters unchanged except one.

For brevity of exposition, in what follows we shall use the concepts of the theory of linear vector spaces.

\[ \overline{y}_0=(y_{10},\ldots,y_{i0},\ldots,y_{l0}), \tag{4} \]

where \(\overline{y}_0\) is the vector of nominal output parameters,
\(y_{i0}\) is the \(i\)-th coordinate of the vector \(y_0\),
\(l\) is the number of output parameters under consideration.

We shall consider the following output parameters:
\(y_{10}\)—starting torque,
\(y_{20}\)—short-circuit current,
\(y_{30}\)—sum of active losses,
\(y_{40}\)—sum of reactive powers.

\[ \overline{\Delta y}=(\Delta y_1,\ldots,\Delta y_i,\ldots,\Delta y_l), \tag{5} \]

where \(\overline{\Delta y}\) is the vector of deviations of the output parameters,
\(\Delta y_i\) is the \(i\)-th coordinate of this vector.

\[ \overline{X}_0=(x_{10},\ldots,x_{j0},\ldots,x_{m0}), \tag{6} \]

where \(\overline{X}_0\) is the \(m\)-dimensional vector of parameters of the equivalent circuit,
\(x_{10}\)—active resistance of the stator winding,
\(x_{20}\)—reduced active resistance of the rotor winding,
\(x_{30}\)—leakage reactance of the stator winding,
\(x_{40}\)—reduced reactance of the rotor winding,
\(x_{50}\)—mutual-induction reactance.

We represent the influence coefficients \(a_{ij}\) by the matrix

\[ A= \begin{vmatrix} a_{11}\ldots a_{1j}\ldots a_{1m}\\ \ldots\ldots\ldots\ldots\ldots\ldots\\ a_{i1}\ldots a_{ij}\ldots a_{im}\\ \ldots\ldots\ldots\ldots\ldots\ldots\\ a_{l1}\ldots a_{lj}\ldots a_{lm} \end{vmatrix} \tag{7} \]

Let us introduce the vectors

\[ \overline{z}_0=(z_{10},\ldots,z_{k0},\ldots,z_{n0}), \]

\[ \overline{\Delta z}=(\Delta z_1,\ldots,\Delta z_k,\ldots,\Delta z_n), \tag{8} \]

where \(z_1\)—number of turns of a phase of the stator winding,
\(z_2\)—diameter of the stator-winding conductor,
\(z_3\)—length of the half-turn of the stator winding,
\(z_4\)—specific resistance of the material of the rotor squirrel cage,
\(z_5\)—magnitude of the air gap,
\(z_6\)—length of the stator stack,
\(z_7\)—length of the rotor stack,
\(z_8\)—number of bars of the rotor squirrel cage.

In the new notation the system of error equations takes the following form:

\[ \overline{\Delta y}=A\,\overline{\Delta x}. \tag{9} \]

Further, one may write

\[ \overline{\Delta x}=B\,\overline{\Delta z}, \tag{10} \]

where \(B\) is the matrix of the coefficients of influence of the vector \(\overline{\Delta z}\) on the vector \(\overline{\Delta x}\)

\[ B= \begin{vmatrix} \beta_{11}\ldots\beta_{1k}\ldots\beta_{1n}\\ \ldots\ldots\ldots\ldots\ldots\ldots\\ \beta_{j1}\ldots\beta_{jk}\ldots\beta_{jn}\\ \ldots\ldots\ldots\ldots\ldots\ldots\\ \beta_{m1}\ldots\beta_{mk}\ldots\beta_{mn} \end{vmatrix} \tag{11} \]

The influence coefficients \(\beta_{jk}\) are determined by analogy with \(\alpha_{ij}\). Obviously,

\[ \overline{\Delta y}=C\,\overline{\Delta z}, \tag{12} \]

where

\[ C=A\cdot B. \tag{13} \]

The use of equation (12) makes it possible to establish the proportional influence of design and technology on the accuracy and reliability of the machine, which facilitates the development of measures to improve reliability [3].

The matrix \(C\) can be calculated directly, having analytical dependences between \(\overline{y}\) and \(\overline{z}\), but it is more convenient to use expression (13), since in this case cumbersome transformations can be avoided.

Expression (12) will be of primary interest to us, since the errors \(\overline{\Delta y}\) are ultimately determined by the errors \(\overline{\Delta z}\).

Ways of Solving Accuracy Problems

From equations (3) and (12) it may be concluded that the practical solution of accuracy problems, depending on the importance of the task, the stage of design and manufacture, and also on the possibility of changing the adopted design, technological, and techno-economic solution, can be carried out along the following lines:

1. Given \(C\) and \(\overline{\Delta z}\), it is required to determine \(\overline{\Delta y}\). In other words, for a given design solution and specified technological accuracy indices, a verification calculation must be performed for the accuracy of the output parameters.

2. Given \(\overline{\Delta y}\) and \(C\). It is required to determine \(\overline{\Delta z}\). The solution will be unique if \(l=n\), and mathematically indeterminate if \(l<n\). In the latter case, from the set of solutions it is necessary to choose such a vector \(\overline{\Delta z}\) that would ensure the minimum cost of providing the tolerances \(\overline{\Delta z}\). This problem can be solved by methods of mathematical programming, if the dependences between the magnitude of the tolerance and the cost of ensuring it are known. Such an approach is expedient under conditions of established production of a well-developed design.

3. Given \(\overline{\Delta z}\). It is required to determine \(C\) under the condition of minimizing the vector \(\overline{\Delta y}\), or its individual components, without exceeding certain constraints on the components of the matrix \(C\) imposed by the design solution. In other words, it is required to change the design solution, within the limits of economic commensurability, in such a way as to obtain a design having minimum sensitivity of the output parameters to actual fluctuations of the input parameters. This approach is acceptable at the initial stage of design, when we are still only slightly bound by design decisions.

4. Determine \(\Delta y\), \(C\), and \(\Delta z\) from the condition of minimum actual cost of the machine. This approach presupposes the simultaneous consideration of technical conditions, requirements for minimizing material consumption, production cost, operating cost, and the desired accuracy conditions. The solution in this case is more complex, but also the most effective, since, although large expenses are incurred at the beginning of development, the design obtained is the most perfect at a minimally low cost.

Analysis of Matrices of Influence Coefficients for Asynchronous Squirrel-Cage Electric Motors of Series A, Frame Sizes 3...6

By analytical and numerical methods, coefficients were calculated for a number of machines. Coefficients of the same name vary from machine to machine, and for convenience of analysis the statistical characteristics of these coefficients are presented below: the mean \(C_m\), the standard deviation \(C_\sigma\), and the coefficient of variation

\[ C_v=\frac{C_\sigma}{C_m}, \]

calculated by ordinary methods [4].

Matrix of mean values of the matrix coefficients

\[ C_m = \left| \begin{matrix} -1{,}732 & 0{,}536 & -0{,}759 & 0{,}761 & 0{,}490 & -0{,}242 & -0{,}103 & -0{,}150 \\ -1{,}870 & 0{,}268 & -0{,}380 & -0{,}121 & 0{,}247 & -0{,}171 & -0{,}400 & 0{,}581 \\ 0{,}921 & -0{,}969 & 0{,}486 & 0{,}220 & 0{,}000 & 0{,}000 & 0{,}154 & 0{,}286 \\ 1{,}188 & -0{,}0436 & -0{,}0323 & 0{,}0191 & -0{,}420 & 0{,}712 & -0{,}0769 & 0{,}1020 \end{matrix} \right| \]

Matrix of standard deviations

\[ C_\sigma = \left| \begin{matrix} 0{,}1595 & 0{,}292 & 0{,}297 & 0{,}121 & 0{,}1305 & 0{,}1365 & 0{,}0312 & 0{,}112 \\ 0{,}0802 & 0{,}146 & 0{,}167 & 0{,}0606 & 0{,}0653 & 0{,}0703 & 0{,}0725 & 0{,}0484 \\ 0{,}067 & 0{,}101 & 0{,}052 & 0{,}034 & 0{,}000 & 0{,}000 & 0{,}0185 & 0{,}0466 \\ 0{,}345 & 0{,}0168 & 0{,}088 & 0{,}0553 & 0{,}0847 & 0{,}0885 & 0{,}026 & 0{,}0455 \end{matrix} \right| \]

Matrix of coefficients of variation

\[ C_v = \left| \begin{matrix} 0{,}092 & 0{,}544 & 0{,}392 & 0{,}159 & 0{,}266 & 0{,}3[[unclear: digit]]9 & 0{,}303 & 0{,}746 \\ 0{,}0428 & 0{,}545 & 0{,}488 & 0{,}501 & 0{,}267 & 0{,}411 & 0{,}181 & 0{,}0831 \\ 0{,}0725 & 0{,}104 & 0{,}107 & 0{,}154 & 0{,}000 & 0{,}000 & 0{,}317 & 0{,}161 \\ 0{,}291 & 0{,}385 & 2{,}730 & 2{,}900 & 0{,}202 & 0{,}124 & 0{,}338 & 0{,}446 \end{matrix} \right| \]

In the general case we are interested not in the influence coefficients themselves, but in their products by the corresponding deviations of the input parameters, since it is precisely these products that determine the structure of the errors of the output parameters. Without knowing \(\Delta z\) and \(\Delta x\), we cannot begin to develop economically justified recommendations for increasing accuracy. Nevertheless, taking into account that for geometrically similar machines under conditions of similar technological processes at one plant, \(\overline{\Delta z}\) and \(\overline{\Delta x}\) for different machines are close, we can estimate the sensitivity of individual machines and individual output parameters to deviations of the components of \(\overline{\Delta z}\) and \(\overline{\Delta x}\).

Analyzing the matrices \(C_m\) and \(C_v\), the following conclusions may be drawn:

1. Individual components of \(\overline{\Delta z}\) affect \(\Delta y\) in different ways. Consequently, in order to analyze the actual scatter and to develop recommendations for increasing accuracy, even for a segment of the series as a whole, it is necessary, in addition to the influence coefficients, to know the vector \(\overline{\Delta z}\).

2. When all components of \(\Delta z\) change by 1%, the fluctuations of the starting torque amount to 4.38%, of the short-circuit current to 4.04%, of active losses to 3.04%, and of reactive power to 2.59%. Other conditions being equal, the starting torque is the most sensitive to deviations, which agrees qualitatively with the available statistical data [5].

3. From the structure of the matrix \(C_v\) it follows that, in the general case, the development of recommendations for increasing accu-

not as a whole for a segment of the series, but for a specific machine because of the presence of variations in the influence coefficients.

1. In the variation of like-named influence coefficients, no rigid functional dependence on power, speed, and the ratio of the principal dimensions has been found. This variation is random in character and is caused by deviations from ideal proportionality in design, due to the desire to satisfy the specified technical conditions.

Since the influence coefficients determine the sensitivity of the output parameters to fluctuations of the input parameters, it seems expedient to study the possibilities for optimizing the matrix of these coefficients in order to obtain a machine with minimum sensitivity to the quality of the technological process.

The investigation has shown that, for specified technical requirements (number of poles, power factor, efficiency, ratios of starting and maximum torques, and starting current), such quantities as \(r_k\), \(x_k\), and \(x_m\) are determined almost uniquely. This circumstance imposes a number of relations on the influence coefficients \(\alpha_{ij}\). For example, for the first row of the matrix \(A\) (for the starting torque) we have the following relations:

\[ \begin{gathered} \alpha_{11}+\alpha_{12}+\alpha_{13}+\alpha_{14}=-1,\\ |\alpha_{12}|-|\alpha_{11}|=\frac{x_k^2-r_k^2}{r_k^2+x_k^2},\\ |\alpha_{13}|+|\alpha_{14}|=\frac{2x_k^2}{r_k^2+x_k^2}. \end{gathered} \tag{14} \]

Thus, it is possible only to redistribute the active resistances of the rotor and stator within the limits of the specified \(r_k\), and the minimum of the sum \(|\alpha_{11}|\) and \(|\alpha_{12}|\) occurs when \(r_1=0\) and \(r'_2=r_k\).

The presence of relations of type (14) sharply limits the possibilities for minimizing the vector \(\Delta \bar{y}\) of variations of the influence coefficients.

In the case where \(x_k\) and \(r_k\) are not specified but are to be selected in design with allowance for the requirements of ensuring a prescribed accuracy, there are considerably greater possibilities for minimizing the coefficients of the matrix \(A\). It has been established that, when \(r_k\) and \(x_k\) are varied, the coefficients \(\alpha_{ij}\) change to different degrees; namely, \(\alpha_{11}\) changes more strongly than \(\alpha_{12}\), \(\alpha_{13}\), and \(\alpha_{14}\). In addition, the magnitude of the change in \(\alpha_{ij}\) strongly depends on the ratio \(\frac{r_k}{x_k}\), i.e., on the power of the machine. In particular, for the starting torque,

\[ \sum_{1}^{4}|\alpha_{1j}|=\frac{r_k^2+3x_k^2}{r_k^2+x_k^2}. \tag{15} \]

For the machines considered, \(\frac{r_k}{x_k}=1.045\ldots0.289\), while \(\sum_{1}^{4}|\alpha_{1j}|=2.18\ldots2.86\), i.e., with increasing power, on average, the sensitivity of the starting torque to fluctuations of the parameters of the equivalent circuit increases.

For the second row of the matrix \(A\) (for the starting current), we have the following relations with the first row:

\[ \begin{gathered} \alpha_{21}=0.5\,\alpha_{11},\\ \alpha_{23}=0.5\,\alpha_{13},\\ \alpha_{24}=0.5\,\alpha_{14}. \end{gathered} \tag{17} \]

Moreover,

\[ \sum_{1} a_{2j} = -1. \tag{17} \]

Since the number of relations is equal to the number of coefficients in the second row, for fixed \(\Delta z\) the second row is rigidly connected with the first.

For the third row the mean values of the coefficients are

\[ a_{31}=0.486;\quad a_{32}=0.220. \]

In a proportioned machine these coefficients should be equal. The difference is explained by the deviation of the current densities in the windings from the ideal ones. For a given frame size, as the length of the machine increases, the sum \(a_{31}\) and \(a_{32}\), as a rule, decreases.

Of the forty elements of the matrix \(B\), 22 elements are equal to zero, 9 are constant, and only 9 depend on the nominal values of the input parameters. The magnitude of the last elements is determined by the local geometry and electromagnetic loadings.

Thus, the use of matrices of coefficients of influence of input and output parameters makes it possible to outline ways of solving questions of increasing accuracy.

Conclusions

1. Ensuring the accuracy of the electrical and mechanical quantities of electric machines is a decisive factor in improving quality and reliability.

2. Ensuring accuracy is a complex problem, which must be solved stage by stage, beginning with preliminary design and ending with product inspection. The most effective approach appears to be accounting for accuracy requirements already at the initial stage of design.

3. The application of the mathematical apparatus of error theory to the analysis of the accuracy of electric machines makes it possible to establish interrelations between input and output parameters, and to choose, on a scientific and economically justified basis, the path for increasing accuracy and reliability in each specific case.

4. In choosing the path for increasing accuracy, a differentiated approach is necessary, depending on the stage of design and production and on the possibility of changing design and technological decisions.

5. The possibilities for increasing accuracy by varying the coefficients of influence are strongly limited by the specified output parameters and electromagnetic loadings, since in this case the coefficients of influence depend little on the local geometry.

6. Different machines and parameters, even within a section of a series, have different sensitivity to fluctuations of the input parameters; in particular, with an increase in \(\dfrac{r_k}{x_k}\), the sensitivity of the output parameters decreases for a given technological process.

7. The use of the matrix \(C\), representing the relation between input and output parameters, substantially facilitates the analysis of the accuracy of asynchronous motors and makes it possible to assign scientifically and economically justified tolerances for electrical and technological characteristics.

8. The applicability of the methodology used is not limited to asynchronous motors. It can also be successfully applied to other types of machines, especially to micromachines of auto-

matics and individual units, for example, in analyzing the reliability of the commutator-brush assembly.

REFERENCES

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2. I. A. Syromyatnikov. Operating Modes of Asynchronous and Synchronous Electric Motors. GEI, 1963.

3. O. P. Muravlev, D. I. Sannikov, E. K. Strelbitskii. On a theoretical-probabilistic method for studying the reliability of electrical machines. Vestnik elektropromyshlennosti, No. 3, 1963.

4. N. V. Smirnov, I. V. Dunin-Barkovskii. A Short Course in Mathematical Statistics for Technical Applications. Fizmatgiz, 1959.

5. G. G. Soroker, O. D. Goldberg. Statistical quality control of asynchronous electric motors in batch production. Vestnik elektropromyshlennosti, No. 5, 1956.