EVALUATION OF THE SENSITIVITY OF THE TECHNICAL CHARACTERISTICS OF ASYNCHRONOUS ROLLER-TABLE ELECTRIC MOTORS TO TECHNOLOGICAL ERRORS

V. V. Dneprovskii, O. P. Muravlev

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

The need for an individual drive for the rollers of transport and working roller tables, as well as for other drives of the metallurgical industry of the USSR, is mainly met by motors of the AR series, manufactured according to GOST 10283-69. The selection of motors for working roller tables is carried out according to the dynamic constant [1] and according to the torque of rotation required for the desired starting and braking time. The acceleration of the flywheel masses must occur in the shortest possible time; therefore, a high average acceleration torque during starting is desirable.

Large scatters in the values of the dynamic constant and the torque lead to deviations from the prescribed technological process of rolling, which affects the quality of the products manufactured. The error obtained in determining the starting and braking times, due to the selection of motors according to catalog data that differ from the actual data, affects the calculation of the average losses per cycle and the expected overheating of the motor. For the drive of transport roller tables, motors are selected in accordance with the load of the conveyor for continuous operation or for the relative duration of switching-on corresponding to the operating conditions.

The motors of working and transport roller tables in most cases operate in groups. It is known that the actual values of the design-and-technological parameters of motors of the same type produced by a plant differ somewhat from the nominal ones, owing to which the motor characteristics themselves also differ from the nominal ones. This circumstance causes inequality of the loads of motors operating in parallel in one drive system, which is usually undesirable.

In this connection, it is of practical interest to evaluate the influence of the scatter of technological errors on the stability of technical characteristics and to create engineering methods for calculating the tolerances of technical characteristics at the design stage.

The main technical characteristics (output parameters) of roller-table motors will be considered the following: starting and maximum torque, starting current, efficiency, cosφ, dynamic constant, and slip.

By independent variables (input parameters) we shall understand the design-and-technological parameters, chosen as the number of turns, wire diameter, length of the half-turn, specific resistance of the rotor cage material, magnitude of the air gap,

the lengths of the stator and rotor stacks and the number of rods in the rotor winding. (The designations of the input and output parameters, as well as the parameters of the equivalent circuit, are given in the Appendix).

One of the principal requirements imposed by the State Standard (GOST) on asynchronous rolling-mill motors is the accuracy of their technical characteristics. The output parameters \(y_i\) may be regarded as functions of the input parameters \(x_j\)

\[ y_i=\varphi_i(x,\ldots,x_j,\ldots,x_m),\quad (i=1.2,\ldots,l;\ j=1,2,\ldots,m); \tag{1} \]

In manufacturing products, the input parameters and, consequently, the output parameters, owing to the influence of technological factors and the properties of the materials used, have deviations from the nominal values. In this case, based on the fundamental provisions of accuracy theory [2, 3] and taking into account that the quantities \(x_j\) and \(y_i\) are random [4], the deviation of the output parameter for relative quantities can be represented in the form

\[ \frac{\Delta y_i}{y_i} = \sum_{j=1}^{m} \frac{\partial \varphi_i(x_1,\ldots,x_j,\ldots,x_m)}{\partial x_j} \cdot \frac{x_j}{\varphi_i(x_1,\ldots,x_j,\ldots,x_m)} \cdot \frac{\Delta x_j}{x_j}; \tag{2} \]

Expression (2) functionally relates the total deviation of the output parameter to the partial deviations of the input parameters.

The coefficients in expression (2) that stand before the errors of the input parameters and determine the degree of influence of these errors on the error of the output parameter are called influence coefficients

\[ C_{ij} = \frac{\partial \varphi_i(x_1,\ldots,x_j,\ldots,x_m)}{\partial x_j} \cdot \frac{x_j}{\varphi_i(x_1,\ldots,x_j,\ldots,x_m)}. \tag{3} \]

To obtain the numerical value of an influence coefficient, the calculated nominal values of the input parameters must be substituted into expression (3). To obtain tolerances for the output parameters by the probabilistic method [5], it is necessary to know the numerical values of the coefficients of the equations for the principal errors and the dispersions of the input parameters.

The aim of the present work is the determination and analysis of the coefficients of influence of the input parameters on the output parameters for asynchronous rolling-mill motors.

When there are simple analytical expressions of type (1) for an output parameter as a function of the input parameters, the calculation of the influence coefficients is carried out directly from expression (3).

For asynchronous rolling-mill motors, the analytical expressions relating the output parameters \(y_i\) to the input \(x_j\) will be complicated functions. In addition, in the technical literature use is made mainly of analytical equations relating the output parameters to the parameters of the equivalent circuit, and of equations relating the parameters of the equivalent circuit to the input parameters.

\[ z_k=\varphi_k(x_1,\ldots,x_j,\ldots,x_m),\quad (k=1,2,\ldots,n;\ j=1,2,\ldots,m), \tag{4} \]

where

\(z_k\) — parameters of the equivalent circuit.

In this case the determination of the coefficients \(C_{ij}\) is carried out as follows:

1. The matrix \(A\) of the coefficients of influence of the equivalent-circuit parameters on the output parameters is computed.

2. The matrix \(B\) of the coefficients of influence of the input parameters on the equivalent-circuit parameters is computed.

1. The sought coefficients of matrix \(C\) are determined by multiplying the coefficients of matrices \(A\) and \(B\).

\[ C=A\cdot B . \tag{5} \]

To determine the coefficients of influence of the parameters of the equivalent circuit on the dynamic constant, the following analytical dependence was used [6]

\[ D= \frac{ Q_{\mathrm{dop}}-3600\left[\dfrac{3U_1^2 r_1}{(x_1+x_m)^2}+P'_0\right]\mathrm{PV} }{ \dfrac{n_c^2}{730} \left[ 1+\dfrac{r_1}{r'_2} \left( 1+\dfrac{2x_k}{x_1+x_m} \right) \right] }, \tag{6} \]

where \(Q_{\mathrm{dop}}=3600\cdot \theta_{\mathrm{cu}}\cdot S_{\mathrm{cool}}\cdot \alpha\) is the permissible amount of heat dissipated by the motor per hour;

\[ S_{\mathrm{cool}}=\pi D_a\left(\frac{D_a}{2}+l_1\right) \]

is the cooling surface over the active dimensions of the iron;

\(D_a\) — outer diameter of the stator stack;
\(l_1\) — length of the stator stack;
\(\alpha\) — heat-transfer coefficient;
\(U_1\) — phase voltage;
\(P'_0=p_{\mathrm{st}}+p_{\mathrm{mech}}\) — sum of steel and mechanical losses;
\(\mathrm{PV}\) — duty-cycle duration;
\(n_c\) — synchronous rotational speed.

Using formulas obtained on the basis of (3, 6), and known analytical expressions for the interrelation of the technical characteristics with the parameters of the equivalent circuit and equations relating the parameters of the equivalent circuit to the design and technological parameters, the coefficients of influence of matrices \(A\), \(B\), and \(C\) were calculated.

Below are given the matrix of mean values of the coefficients \(C_{ij}\) over the entire series and the matrices of coefficients of variation.

Matrix of mean values of the coefficients \(C_{ij}\)

Matrix of coefficients of variation \(C_v\)

\[ C_v = \begin{array}{rrrrrrrr} 0{,}0756 & 0{,}219 & 0{,}196 & 0{,}620 & 0{,}287 & 0{,}193 & 0{,}268 & 0{,}193\\ 0{,}0341 & 0{,}212 & 0{,}191 & 0 & 0{,}240 & 0{,}170 & 0{,}172 & 0{,}132\\ 0{,}0309 & 0{,}219 & 0{,}195 & 0{,}114 & 0{,}286 & 0{,}192 & 0{,}0804 & 0{,}0652\\ 1{,}6640 & 0{,}523 & 0{,}515 & 0{,}462 & 0{,}850 & 0{,}864 & 0{,}522 & 0{,}527\\ 0{,}2890 & 0{,}809 & 0{,}909 & 0{,}429 & 0{,}283 & 0{,}257 & 0{,}675 & 0{,}718\\ 0{,}2780 & 0{,}191 & 0{,}190 & 0{,}149 & 0{,}572 & 0{,}582 & 0{,}132 & 0{,}134\\ 0{,}0953 & 0{,}571 & 0{,}576 & 0{,}089 & 0{,}663 & 0{,}606 & 0{,}113 & 0{,}113 \end{array} \]

The matrix of mean values of the coefficients \(C_{ij}\) represents the mean values of the coefficients of influence of the input parameters on the out—

similar over the entire series of rollgang motors; therefore, the deviations of the output parameters calculated on the basis of these matrices will reflect the general picture for the whole series. From the matrix of mean values of the coefficients one can see that the various output parameters of the series respond differently to the same deviations of the input parameters. In addition, most of the influence coefficients \(C_{ij}\), as is evident from the matrix \(C_v\), have a large variability depending on the motor sizes; therefore, the adoption of design decisions aimed at minimizing the sensitivity of the output parameters to changes in the input parameters must be carried out for specific types of motors.

Taking into account a number of design features of asynchronous rollgang motors (a wide variety of motor types in terms of the number of poles, casting of rotors with a special alloy of reduced electrical conductivity, etc.), as well as the introduction, in comparison with motors of standard design, of one of the principal technical characteristics—the dynamic time constant—the following influence coefficients were additionally determined.

The influence coefficients \(C_{49}\), \(C_{4,10}\) of losses in steel and mechanical losses on the efficiency, no-load losses on the power factor, and the dynamic time constant \(C_{5,14}\), \(C_{6,14}\), the duration of starting, total losses in steel and mechanical losses, and the heat-transfer coefficient on the dynamic time constant \(C_{6,11}\), \(C_{6,12}\), \(C_{6,13}\). The sensitivity of the remaining output parameters does not depend on changes in the above input parameters, and the corresponding influence coefficients are identically equal to zero.

The derivation of formulas for determining the influence coefficients \(C_{4,9}\), \(C_{4,10}\), \(C_{5,14}\), \(C_{6,14}\), \(C_{6,11}\), \(C_{6,12}\), \(C_{6,13}\) was carried out from expressions (3, 6), and also on the basis of well-known analytical relationships from the course of electrical machines. The mean values of these influence coefficients for the whole series and the coefficients of variation are given in Table 1.

Table 1

Conclusions

1. Analytical relationships have been obtained and the influence coefficients of input parameters on output parameters for rollgang motors have been calculated; these can be used to calculate tolerances for input parameters and to reassess the existing tolerance values for technical characteristics.

2. The calculated influence coefficients make it possible, for specified tolerances on the output characteristics, to adopt technically justified tolerances for design-and-technological parameters. This will make it possible to stabilize the values of the technical characteristics and more justifiably

again approach the assignment of the nominal values themselves with the aim of raising the technical level of the motors.

Appendix

For the parameters of rolled asynchronous motors, the following designations have been introduced.

1. Output parameters

\(y_1—M_{\text{p}}\) — starting torque;
\(y_2—M_{\text{m}}\) — maximum torque;
\(y_3—I_{\text{k}}\) — short-circuit current;
\(y_4—\eta\) — efficiency;
\(y_5—\cos\varphi\) — power factor;
\(y_6—T\) — dynamic constant;
\(y_7—S\) — slip.

2. Parameters of the equivalent circuit

\(z_1—r_1\) — active resistance of the stator winding;
\(z_2—r_2'\) — reduced active resistance of the rotor winding;
\(z_3—x_{\text{k}}\) — short-circuit reactance;
\(z_4—x_{\text{m}}\) — magnetizing reactance.

3. Input parameters

\(x_1—\omega_1\) — number of turns of a phase of the stator winding;
\(x_2—d_{\text{pr}}\) — diameter of the bare wire of the stator winding;
\(x_3—l_{\text{m}}\) — length of the half-turn of the stator winding;
\(x_4—\rho\) — specific resistance of the material of the rotor squirrel cage;
\(x_5—\delta\) — size of the air gap;
\(x_6—l_1\) — length of the stator stack;
\(x_7—l_2\) — length of the rotor stack;
\(x_8—Z_2\) — number of complete bars of the rotor winding;
\(x_9—p_{\text{st}}\) — iron losses;
\(x_{10}—p_{\text{mech}}\) — mechanical losses;
\(x_{11}—\text{PV}\) — duty cycle;
\(x_{12}—p_{\text{st}}+p_{\text{mech}}\) — sum of iron and mechanical losses;
\(x_{13}—\alpha\) — heat-transfer coefficient;
\(x_{14}—p_0\) — no-load losses.

References

1. GOST 10283-69. Three-phase squirrel-cage asynchronous rolled motors of the AR series.

2. N. A. Borodachev. Analysis of Quality and Accuracy of Production. Mashgiz, Moscow, 1946.

3. V. P. Gusev, A. V. Fomin, et al. Calculation of Electrical Tolerances for Radio-Electronic Equipment. “Sovetskoe radio,” Moscow, 1963.

4. I. P. Isaev. Tolerances for the Characteristics of Electric Locomotives. Transzheldorizdat, Moscow, 1958.

5. O. P. Muravlev, E. K. Strelbitskii. Ensuring the required accuracy in the production of asynchronous motors. — Elektrotekhnika, 1966, No. 7.

6. Technical Information OTE 143.001. Asynchronous rolled electric motors of the AR2 series with starting torques from 1 to 55 kg·m at rotational speeds from 150 to 1500 rpm. Tomsk, 1967.