Universal Curves for Determining Maximum Transient Torques and Currents of Roller Table Motors
Abstract Generated abstract
The paper addresses the estimation of maximum transient currents and torques in AR series squirrel cage induction motors used for roller table drives, where frequent starting, reversal, and braking can produce large short term stresses. After noting limitations of existing approximate formulas for general industrial motors, the authors formulate the electromechanical transient equations and argue that impact values depend mainly on motor parameters rather than load torque or inertia under the considered conditions. They divide the motor series into four parameter groups and use analog computer calculations to generate universal curves relating impact current and torque to stator and rotor active resistances for starting and reversal. Verification against representative motors indicates approximate errors not exceeding about 18 percent for torque and 10 percent for current, making the curves a practical method for rapid engineering estimates.
Full Text
UNIVERSAL CURVES FOR DETERMINING THE MAXIMUM TRANSIENT TORQUES AND CURRENTS OF ROLLER-TABLE MOTORS
E. V. Kononenko, B. A. Danchinov
Roller-table asynchronous motors with a squirrel-cage rotor of the AR series are intended chiefly for the individual drive of the rollers of working and transport roller tables at metallurgical enterprises. In accordance with the requirements of the technological process of rolling and transporting metal, they operate under conditions of frequent starts, reversals, and braking, accompanied by transient processes. In this, the torques and currents may for short periods attain considerable values.
Knowledge of at least approximate values of the impact currents and torques developed by roller-table motors during starts and reversals is of practical importance for assessing the reliability of roller-table drives. In addition, a specific feature of roller-table drives is the limitation on the magnitude of the maximum accelerations, due to the inadmissibility of slipping of the rollers relative to the metal being rolled or transported. Knowledge of the maximum transient torque makes it possible to judge the magnitude of the maximum acceleration.
A number of works [1–5] give approximate formulas for determining the maximum transient torques of asynchronous motors of general industrial design. The simplest formulas for calculations are given in [1, 2, 5]. However, they have certain limited ranges of application. In particular, the formula presented in [1] is valid for \(\alpha'_s+\alpha'_r \leq 0.7\), where \(\alpha'_s(\alpha'_r)\) is the damping coefficient of the stator (rotor) winding with the rotor (stator) winding short-circuited. For roller-table motors, however, \(\alpha'_s+\alpha'_r > 1\).
According to the formula of K. P. Kovács and I. Rácz [2], the ratio of the maximum transient torque to the established starting torque is always greater than two. In reality, however, there may be cases in which this ratio proves to be less than two.
According to the formula given in [5], the maximum torque is determined only by the inductive resistances of the motor and does not depend on the active resistances, although it is known that the influence of the active resistances of the motor on the maximum torque is very substantial. For example, when additional active resistances are connected into the stator circuit, the torque is significantly reduced.
The formula for the maximum torque in [3] is derived for the case of a preassigned time at which the transient torque reaches its maximum. This time is taken equal to \(\frac{3}{2}\pi\) electrical radians, which corresponds to motors of general industrial design.
Determining the maximum torque according to [4] requires a rather laborious calculation of the damping coefficients of the free components of the transient torque. In addition, it is necessary to know the time \(t_m\) at which the transient torque reaches its maximum value. An error in determining \(t_m\) leads to an error in calculating \(M_{\max}\). The method proposed by the author for graphical determination of the maximum torque from the curve \(\dot M_{\max} = f(t_m)\) increases the laboriousness of the calculations.
In connection with the above, the problem arises of finding a relatively simple method for determining the maximum transient currents and torques specifically for roller-table motors of the AP series.
When determining the maximum currents and torques, electromagnetic transients must be taken into account.
The electromechanical transients in an asynchronous squirrel-cage motor, in relative units in synchronously rotating axes, are described by the following system of equations:
\[ p\psi_{s\alpha}=U_{s\alpha}-r_s i_{s\alpha}+\psi_{s\beta}; \]
\[ p\psi_{s\beta}=U_{s\beta}-r_s i_{s\beta}-\psi_{s\alpha}; \]
\[ p\psi_{r\alpha}=-r_r y_s\psi_{r\alpha}+r_r y_m\psi_{s\alpha}+\psi_{r\beta}-\omega\psi_{r\beta}; \]
\[ p\psi_{r\beta}=-r_r y_s\psi_{r\beta}+r_r y_m\psi_{s\beta}-\psi_{r\alpha}+\omega\psi_{r\alpha}; \]
\[ i_{s\alpha}=y_r\psi_{s\alpha}-y_m\psi_{r\alpha}; \]
\[ i_{s\beta}=y_r\psi_{s\beta}-y_m\psi_{r\beta}; \]
\[ i_s=\sqrt{i_{s\alpha}^{2}+i_{s\beta}^{2}}; \]
\[ p\omega=\frac{1}{J_\Sigma}(M-M_c), \]
where
\[ M=\frac{3}{2}y_m(\psi_{r\alpha}\psi_{s\beta}-\psi_{s\alpha}\psi_{r\beta}); \]
\[ y_s=\frac{x_s}{x_s x_r-x_m^{2}}; \]
\[ y_r=\frac{x_r}{x_s x_r-x_m^{2}}; \]
\[ y_m=\frac{x_m}{x_s x_r-x_m^{2}}. \]
In these expressions the following notation has been adopted:
\(u_{s\alpha}, u_{s\beta}\) — components of the stator-voltage vector along the \(\alpha\) and \(\beta\) axes;
\(\psi_{s\alpha}, \psi_{s\beta}\) \((\psi_{r\alpha}, \psi_{r\beta})\) — components of the stator (rotor) flux-linkage vector along the \(\alpha\) and \(\beta\) axes;
\(i_{s\alpha}, i_{s\beta}\) — components of the stator-current vector along the \(\alpha\) and \(\beta\) axes;
\(i_s\) — resultant vector of the stator current;
\(r_s\) \((r_r)\) — active resistance of the stator (rotor);
\(x_s\) \((x_r)\) — total inductive reactance of the stator (rotor);
\(x_m\) — mutual-inductance reactance;
\(\omega\) — angular velocity of the rotor;
\(M\) — rotating torque of the motor;
\(M_c\) — load resisting torque;
\(J_\Sigma\) — moment of inertia referred to the motor shaft;
\(t\) — time;
\[ p=\frac{d}{dt} \]
— symbol of differentiation with respect to time.
As can be seen from this system of equations, the quantitative and qualitative character of the course of electromechanical transients is determined by the parameters of the motor \((r_s, r_r, y_s, y_r, y_m)\) and of the drive \((J_\Sigma, M_c)\).
The impact values of the current and torque, as the investigations carried out have shown, practically do not depend on the moment of inertia (for \(J_{\Sigma} > 3J_{\mathrm{rot}}\)) or on the load resistance torque. Consequently, the magnitudes of the impact current and torque will be determined only by the parameters of the motor.
As a result of an analysis of the parameters of roller-table motors, it was established that the entire series can be divided into four groups. In this case the parameters \(y_s\) and \(y_r\) should be adopted as the criterion for division into groups. The ranges of variation of these parameters within each group are given in Table 1.
Table 1
| Group | I | II | III | IV |
|---|---|---|---|---|
| \(y_s,\ y_r\) | up to 4 | 4–5 | 5–6 | Over 6 |
To determine \(i_{\mathrm{imp}}\) and \(M_{\mathrm{imp}}\), it is necessary to solve the above system of nonlinear equations. An analytical solution can be obtained by linearizing these equations, i.e., assuming the parameters and the rotor speed to be constant. Such a solution, when determining only impact currents and torques, is quite acceptable; however, even under these assumptions the calculation is very labor-intensive. Therefore, the most expedient approach is to regard the solution of the nonlinear system of equations on an analog computer as preferable.
In our case the calculations of \(M_{\mathrm{imp}}\) and \(i_{\mathrm{imp}}\) were carried out on an MN-14 analog computer. As the calculation parameters \(y_s\), \(y_r\), and \(y_m\), their values averaged for each group were adopted. The active resistances of the stator and rotor windings were varied within the range of values possible for the given group. The total moment of inertia reduced to the motor shaft was taken equal to \(3J_{\mathrm{rot}}\) for the motor having the largest rotor moment of inertia in the given group. The load resistance torque was taken equal to zero (\(M_c = 0\)).
Table 2 gives the calculated motor parameters for each group.
Table 2
| Groups | \(y_s\) | \(y_r\) | \(y_m\) | \(r_s\) | \(r_r\) | \(J_{\Sigma}\) |
|---|---|---|---|---|---|---|
| I | 3.73 | 3.58 | 3.13 | 0.05–0.3 | 0.2–0.7 | 116 |
| II | 4.33 | 4.28 | 3.93 | 0.04–0.15 | 0.2–0.4 | 120 |
| III | 5.55 | 5.45 | 5.12 | 0.03–0.1 | 0.1–0.3 | 339 |
| IV | 6.21 | 6.16 | 5.85 | 0.03–0.1 | 0.1–0.3 | 351 |
Table 3
| Groups | Motor type | \(y_s\) | \(y_r\) | \(y_m\) |
|---|---|---|---|---|
| I | AR 64-16 | 3.96 | 3.79 | 3.32 |
| I | AR 42-12 | 2.53 | 2.47 | 2.12 |
| II | AR 52-8 | 4.83 | 4.72 | 4.46 |
| II | AR 74-16 | 4.18 | 4.0 | 3.58 |
| III | AR 63-12 | 5.72 | 5.56 | 5.19 |
| III | AR 73-12 | 5.36 | 5.22 | 4.91 |
| IV | AR 43-4 | 6.42 | 6.48 | 6.23 |
| IV | AR 74-10 | 6.07 | 5.98 | 5.75 |
As a result of the calculations carried out, for each group of motors a family of universal curves was obtained, expressing the dependence of impact currents and torques on the motor parameters.
Fig. 1. Dependences \(M_{\mathrm{ud}} = f(r_s, r_r)\) during starting.
The universality of these curves consists in the fact that, with their aid, one can determine \(M_{\mathrm{ud}}\) and \(i_{\mathrm{ud}}\) for any motors having approximately identical values of the parameters \(y_s\), \(y_r\), and \(y_m\), and differing in their active resistances.
Fig. 2. Dependences \(M_{\mathrm{ud}} = f(r_s, r_r)\) during reversing.
As an example, Figs. 1–4 show such curves for group IV.
To estimate the possible error, verification calculations were performed for motors having the largest and smallest values within each-
...of the group, the values of the parameters \(y_s\), \(y_r\), and \(y_m\) (which are given in Table 3) for different values of the active resistances \(r_s\) and \(r_r\).
The maximum errors established in this way are given in Table 4, from which it is seen that, when the universal curves are used, they do not exceed:
a) in determining impact torques — 18%;
b) in determining impact currents — 10%.
For purely approximate purposes, such error values may be considered quite acceptable.
Fig. 3. Dependences \(i_{\mathrm{ud}} = f(r_s,\ r_r)\) during starting.
Fig. 4. Dependences \(i_{\mathrm{ud}} = f(r_s,\ r_r)\) during reversal.
When determining the impact currents and torques of any motor, it is first necessary to determine the values of the parameters \(y_s\), \(y_r\), \(y_m\) and to establish to which group it belongs. Then, from the values of the active resistances \(r_s\) and \(r_r\), using the curves \(i_{\mathrm{ud}} = f(r_s,\ r_r)\) and \(M_{\mathrm{ud}} = f(r_s,\ r_r)\) corresponding to the group of the given motor, \(i_{\mathrm{ud}}\) and \(M_{\mathrm{ud}}\) are determined.
Table 4
| Groups | Groups | Groups | Groups | ||
|---|---|---|---|---|---|
| I | II | III | IV | ||
| Starting | \(\Delta M_{\mathrm{ud}}\ \%\) | \(+8,3\) \(-11,5\) |
\(+14,3\) \(-9,7\) |
\(+3,56\) \(-1,12\) |
\(+1,98\) \(-7,14\) |
| Starting | \(\Delta i_{\mathrm{ud}}\ \%\) | \(+8,6\) \(-2,3\) |
\(5,9\) \(-9,15\) |
\(+2,35\) \(-5,4\) |
\(+3,53\) \(-1,06\) |
| Reversal | \(\Delta M_{\mathrm{ud}}\ \%\) | \(+10,3\) \(-11,2\) |
\(+17,8\) \(-15,2\) |
\(+6,3\) \(-0,96\) |
\(+3,25\) \(-8,7\) |
| Reversal | \(\Delta i_{\mathrm{ud}}\ \%\) | \(+8,0\) \(-2,06\) |
\(+4,9\) \(-3,64\) |
\(+2,84\) \(-1,27\) |
\(+1,76\) \(-3,1\) |
It is appropriate to note that, when determining the parameters \(y_s\), \(y_r\), and \(y_m\), the saturated values of the inductive reactances must be used.
It should also be noted that the maximum currents and torques under reversal are determined under the condition that, during the time between disconnection and reconnection with the changed phase sequence, the rotor magnetic field has time to decay completely. For roller-table motors such an assumption is quite acceptable [6].
Thus, using the universal curves obtained, one can determine, relatively simply and quickly, approximate values of the maximum currents and torques of any roller-table motor.
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E. V. Kononenko, B. A. Danchinov. Starting and reversal of roller-table motors at various initial speeds. Article in the present collection.