On the Rate of Temperature Rise under Emergency Conditions and Cooling Laws of AO2 Series Motors
Abstract Generated abstract
This paper examines stator winding heating and cooling in AO2 series induction motors under emergency conditions, especially locked-rotor three-phase short circuits and operation stopped on two phases. Using resistance measurements recorded continuously with an oscillograph, the authors experimentally determine temperature-rise and short-circuit current laws for several commonly used motor types, processing the data by least squares. The results show that the rate of temperature rise is not constant during a short circuit, that commonly used calculation formulas can produce substantial errors, and that heating is better represented by quadratic time functions with a time-varying rate. The paper also proposes an empirical cooling expression with a variable time constant, providing coefficients that improve estimates needed for protection selection and insulation-wear calculations.
Full Text
ON THE RATE OF TEMPERATURE RISE UNDER EMERGENCY CONDITIONS AND THE COOLING LAWS OF AO2 SERIES MOTORS
B. A. ITKIN, E. K. STREL'BITSKII
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The rate of temperature rise in the stator winding of an induction motor with a locked rotor—$T_{v3}$—is one of the principal characteristics of the machine. In order to increase reliability, in electric motors of the new AO2 series the rate of temperature rise has been reduced to $7^\circ\mathrm{C}/\mathrm{s}$ [1]. For electric-drive designers, an equally important quantity when selecting protection is the rate of temperature rise in the stator winding when the motor is stopped on two phases—$T_{v2}$. When evaluating the influence of motor wear from emergency conditions on its operational reliability, in addition to the values $T_{v3}$, $T_{v2}$, it is necessary to know the actual cooling law of the winding, $\Theta_{\text{cool}} = f(t)$, after the short circuit has been interrupted. If approximate values of $T_{v3}$ for some motors of the new series are known, then data on $T_{v2}$ and $\Theta_{\text{cool}} = f(t)$ are entirely lacking.
This article gives the actual values of the rate of temperature rise, heating, and cooling for the most widely used motors of the new series AO2-52-4; AS2-42-4; AO2-32-4; AOL2-11-4.
Usually the calculated value of the rate of temperature rise with a locked rotor is determined by the well-known VNIIEM formula
\[ T_{v3} = \frac{\Delta^2}{200}, \tag{1} \]
where $\Delta$ is the current density during a short circuit of the motor on 3 phases.
The experimental value of $T_{v3}$ is found from the expression
\[ T_{v3} = \frac{\Theta}{t}, \tag{2} \]
where $\Theta$ is the measured value of the overheating in the stator winding over the time $t$ of action of the 3-phase short circuit.
It should be noted that in a number of cases the values $T_{v3}$ determined from (1) and (2) do not give the required accuracy in calculating heating and do not fully reflect the real physical process occurring in the stator winding under the condition considered. Thus, formula (1) gives an unambiguous value of $T_{v3}$ and was obtained on the assumption of the absence of
heat transfer and the constancy of the stator current and resistance during the short-circuit interval. From (2), the mean value of the rate of temperature rise over the time interval under consideration is determined. Our experimental studies of the heating of AO2 asynchronous motors under emergency conditions have shown that, in the general case, it is impossible to assume that the rate of temperature rise remains constant throughout the entire short-circuit process.
For \(t > 5 \div 10\) seconds, \(T_v\) decreases during the short-circuit process because of the action of the heat-transfer factor and the change in the current. Therefore the value \(T_{v3}\), determined from (1), may be regarded as the initial rate of temperature rise \(T_{v0}\). The experimental values of \(T_{v0}\) and the values calculated from (1), when the actual current density is substituted, differ by no more than \(2 \div 3\%\).
The absence of exact data on \(T_{v3}\) and \(T_{v2}\) is explained by the difficulty of measuring the winding temperature during an emergency condition. There is a circuit [2] that makes it possible to measure the average temperatures of the windings of electrical machines, \(R_{\mathrm{ob}}\), by the resistance method without disconnecting the electric motor from the 380-volt supply. Since the rate of temperature rise in short-circuit conditions is large, measurement of \(R_{\mathrm{ob}}\) by pointer instruments can introduce a significant error.
We supplemented the circuit [2] by installing galvanometers of an N700 oscillograph, which made it possible to make a continuous record of the ohmic resistance and helped to increase the accuracy of measurement. Processing the oscillogram data by the method of least squares showed that the change in temperature rise during a short circuit can be represented by a second-degree polynomial in time,
\[ \Theta = bt - at^2 + c, \tag{3} \]
where \(a\), \(b\), and \(c\) are the coefficients of equation (3). On the basis of the physical meaning of the process, the coefficient \(b\) is the initial rate of temperature rise \(T_{v0}\).
The coefficient \(c\) is the initial temperature rise of the winding. The law of variation of the rate of temperature rise is found by differentiating (3):
\[ T_v(t) = \frac{d\Theta}{dt} = T_{v0} - 2at. \tag{4} \]
Table 1 gives the actual laws of variation of the temperature rise in the stator winding for three- and two-phase short circuits, both from the cold state and from the hot state of the motors investigated, obtained by processing the experimental data by the method of least squares.
Oscillograph recording of the short-circuit current and subsequent mathematical processing of the data showed that the variation of the current during an emergency condition is described by the equation
\[ I(t) = I_{\mathrm{init}} - \kappa_1 t + \kappa_2 t^2, \tag{5} \]
where \(I_{\mathrm{init}}\) is the initial short-circuit current,
\(\kappa_1, \kappa_2\) are the coefficients of the equation.
Table 2 gives the laws of variation of the current under emergency conditions, obtained by processing the experimental data by the method of least squares.
Table 1
| Motor type | Type of short circuit | Temperature-rise equation, from cold | Temperature-rise equation, from hot |
|---|---|---|---|
| AO2-52-4 | 3-phase | $6.2t - 0.0192t^2 + 0.83$ | $6.189t - 0.0125t^2 + 41.6$ |
| AO2-52-4 | 2-phase | $3.667t - 0.0048t^2 + 4.6$ | $3.554t - 0.00207t^2 + 42.7$ |
| AO2-42-4 | 3-phase | $5.863t - 0.0117t^2$ | $5.343t - 0.00762t^2 + 43.9$ |
| AO2-42-4 | 2-phase | $3.249t - 0.073t^2 - 0.163$ | $3.056t - 0.00737t^2 + 51.6$ |
| AO2-32-4 | 3-phase | $4.512t - 0.0159t^2 + 1.36$ | $4.407t - 0.0193t^2 + 18.4$ |
| AO2-32-4 | 2-phase | $3.31t - 0.0151t^2$ | $3.278t - 0.0148t^2 + 39.8$ |
| AOL2-11-4 | 3-phase | $3.975t - 0.0215t^2 + 2.1$ | $3.658t - 0.0197t^2 + 49$ |
| AOL2-11-4 | 2-phase | $3.02t - 0.0137t^2 + 0.42$ | $2.545t - 0.0128t^2 + 44.7$ |
Table 2
| Motor type | Type of short circuit | Short-circuit current equation, from cold | Short-circuit current equation, from hot |
|---|---|---|---|
| AO2-52-4 | 3-phase | $146.17 - 0.92t + 0.0079t^2$ | $140 - 0.634t + 0.0015t^2$ |
| AO2-52-4 | 2-phase | $124.48 - 0.5t + 0.00199t^2$ | $118.87 - 0.457t + 0.0156t^2$ |
| AO2-42-4 | 3-phase | $85.8 - 0.438t + 0.00221t^2$ | $82.1 - 0.817t + 0.0151t^2$ |
| AO2-42-4 | 2-phase | $63.7 - 0.109t + 0.00055t^2$ | $59.46 - 0.138t + 0.00072t^2$ |
| AO2-32-4 | 3-phase | $41.48 - 0.232t + 0.00139t^2$ | $37.08 - 0.166t + 0.0003t^2$ |
| AO2-32-4 | 2-phase | $34.56 - 0.113t + 0.00055t^2$ | $31.92 - 0.105t + 0.00042t^2$ |
| AOL2-11-4 | 3-phase | $7.84 - 0.0523t + 0.00032t^2$ | $7.2 - 0.051t + 0.00034t^2$ |
| AOL2-11-4 | 2-phase | $6.8 - 0.036t + 0.00022t^2$ | $6.06 - 0.026t + 0.00012t^2$ |
It should be noted that the error in calculating the temperature rise during a motor short circuit by the commonly used formula [3]
\[ \Theta = (235 + \vartheta_{\mathrm{n}})(l^{K_1 \Delta^2 t} - 1), \tag{5} \]
where $\vartheta_{\mathrm{n}}$ is the initial temperature of the winding; $K_1 = 1.98 \cdot 10^{-5}$ (for copper windings), and the experimental data may reach 20–30%. When calculating $\Theta$ by (3), the error is no more than several percent, due to the possible scatter of the motor parameters in Tables 1 and 2 and of the motor of the same type under investigation.
To calculate the insulation wear of electrical machines under emergency conditions, it is necessary to know the winding temperature both during heating and during cooling. The question of the cooling law of the stator winding of an induction motor after the cessation of the short-
of a short circuit is little studied. In [3] it is assumed that cooling, in the first approximation, is expressed as
\[ \Theta_{\text{cool}}=\Theta_{\max} l^{-\frac{t}{T}}, \tag{6} \]
where \(\Theta_{\max}\) is the maximum temperature rise of the winding during a short circuit;
\(T\) is the time constant of the winding.
The investigations we carried out showed that calculation of the cooling curve by (6) leads to large errors. It is rather difficult to find analytically the dependence \(\Theta_{\text{cool}}=f(t)\). The value \(T\) changes during the cooling process and depends on the final temperature of the winding and on the temperature difference between the stator iron, the insulation, and the winding copper.
Mathematical processing of the experimental data showed that the law of variation of the winding temperature rise after emergency operation is expressed with sufficient accuracy by the equation
\[ \Theta_{\text{cool}}=\Theta_{\max}\cdot l^{-\frac{t}{T_0+mt-nt^2}}, \tag{7} \]
where \(T_0\) is the initial value of the cooling time constant of the stator winding,
\(m\) and \(n\) are coefficients of the equation.
For the motors under consideration the values are
\[ T_0=0.8\div 2.2\ \text{min.};\qquad m=1.4\div 2.5;\qquad n=0.5\div 0.9. \]
Analytical calculations of the rate of temperature rise, of the temperature rise during a short circuit, and of cooling after emergency operation of the stator winding of an induction motor by existing methods do not give the desired accuracy.
The equations obtained by us, (3), (4), (5), (7), with experimental values of the coefficients, make it possible to increase considerably the accuracy of these calculations for the motors under consideration, which is extremely important in selecting protection and in calculating motor wear.
REFERENCES
-
O. D. Gol’dberg. Increasing the operational reliability of electric motors of series A and AO with power up to 100 kW. New series of alternating-current electrical machines. TsINTI, 1963.
-
A. M. Sharipov. A device for measuring the ohmic resistances of windings of electrical machines in the presence of alternating e.m.f. Collection TsITEIN, No. 3-61-72/21, 1961.
-
G. Gotter. Heating and cooling of electrical machines. GEI, 1961.