Insulation Wear of Induction Motors under Steady-State Operating Conditions
Abstract Generated abstract
The article analyzes thermal insulation wear in induction motors used on metal-cutting equipment under statistically stationary operating loads. It formulates expected wear per unit time from winding temperature rise and ambient temperature, using an exponential distribution for utilized motor power, a quadratic approximation relating current to load, and a normal model for shop temperature variation. The calculation suggests that, because such motors often operate at low average load, actual insulation service life can greatly exceed nominal assumptions, making rated-load heating relatively unimportant in this application. The authors argue that motor design for machine-tool drives should account for operating-load statistics and reliability, including possible reductions in winding wire diameter, lower slot-fill factors, or strengthened insulation.
Full Text
Insulation Wear of Induction Motors under Steady-State Operating Conditions
E. K. Strelbitskii, A. S. Gitman
One of the principal tasks of technical progress is to increase the reliability of equipment and, in particular, of the most widespread type of drive—induction motors. It is obvious that the reliability of induction motors can be increased by improving the quality of the materials used in manufacture (primarily insulating materials) and by improving manufacturing quality. At the same time, another path is possible: the search for reserves for constructing a sufficiently reliable system from unreliable elements. If one takes into account that the quality of materials and manufacture cannot be improved abruptly, then the second path may already at present prove to be a radical means of increasing reliability.
In this connection, studies carried out at the Tomsk Polytechnic Institute are of interest; they revealed a significant correlation between the value of the slot-fill factor of random-wound windings and the value of the breakdown voltages of the insulation. From this it follows that one of the reserves for increasing reliability is a reduction in the slot-fill factor and, accordingly, a reduction in the cross section of the winding wire. For motors operating with variable load, if correlation functions of the load are available, it is possible to find such a wire cross section that the insulation wear under the actual variable load will be equal to the insulation wear under the nominal design condition, i.e., as was proposed for cable products in [1].
The purpose of the present article is to investigate only the magnitude of the actual insulation wear of induction motors installed on metal-cutting equipment, according to statistical data [2]. The process of load variation is assumed to be stationary in the sense accepted in the theory of random functions.
The insulation life of a motor \(r\) during operation is determined by the expression:
\[ r(t)=r\cdot \exp\left(-\frac{t}{T}\right)-t\exp\left[b(\Delta \Theta_{\mathrm{dv}}+\Theta_{\mathrm{okr}}-\Theta_{\mathrm{dop}})\right], \tag{1} \]
where \(r\) is the factory initial life,
\(t\) is the operating time,
\(\Delta \Theta_{\mathrm{dv}}\) is the excess of the motor winding temperature over the ambient temperature,
\(\Theta_{\mathrm{okr}}\) is the ambient temperature,
\(\Theta_{\mathrm{dop}}\) is the maximum permissible temperature for the given class of insulation,
\(T\) — the time constant of natural aging of the insulation,
\(b\) — a coefficient depending on the insulation class.
Excluding natural aging of the insulation from consideration, we shall find the mathematical expectation of insulation wear during the operating period per unit time. Taking into account that the quantities \(\Delta \Theta_{\mathrm{dv}}\) and \(\Theta_{\mathrm{okr}}\) are stochastically independent, we obtain:
\[ \begin{aligned} M[r(t)] &= M\{\exp[b(\Delta \Theta_{\mathrm{dv}}+\Theta_{\mathrm{okr}}-\Theta_{\mathrm{dop}})]\}=\\ &= \exp[-b\Theta_{\mathrm{dop}}]\cdot M\{[\exp(b\Delta \Theta_{\mathrm{dv}}+\Theta_{\mathrm{okr}})]\}=\\ &= \exp[-b\Theta_{\mathrm{dop}}]\cdot M[\exp(b\Delta \Theta_{\mathrm{dv}})]\,M[\exp(b\Theta_{\mathrm{okr}})]. \end{aligned} \tag{2} \]
To compute \(M[\exp(b\Delta \Theta_{\mathrm{dv}})]\), it is necessary to find the probability density of \(\Delta \Theta_{\mathrm{dv}}\), or the probability density of another variable related to \(\Delta \Theta_{\mathrm{dv}}\) by a functional dependence.
According to the data of [2], the probability density of the utilized power of induction motors installed on metal-cutting equipment may be represented by the exponential distribution law
\[ \varphi(P)=\lambda \exp(-\lambda P), \tag{3} \]
where \(P\) is the power in fractions of the rated value,
\(\lambda\) is the distribution parameter.
Since the recalculation of winding overheating is performed according to the square of the current, let us find the dependence of current on power. First introduce a system of relative units, based on the values of current and power at rated load:
\[ P_{\mathrm{nom}}=i_{\mathrm{nom}}=1 \tag{4} \]
For motors installed on metal-cutting equipment, with overload protection taken into account, the change in power occurs within the limits from 0 to 1.1. In this case the current varies from the no-load current \(i_0\) to 1.1. To find the functional dependence \(i=f(P)\), a quadratic approximation is sufficient.
Starting from the boundary condition
\[ P=0,\quad i=i_0; \]
we obtain the general form of the equation
\[ i=i_0+a_1P+a_2P^2 \tag{5} \]
Using the second boundary condition
\[ P=1.1,\quad i=1.1 \]
and the condition \(P_{\mathrm{nom}}=1,\ i_{\mathrm{nom}}=1\), we obtain a system of equations for finding the coefficients \(a_1\) and \(a_2\):
\[ \left. \begin{aligned} i_0+a_1+a_2&=1\\ i_0+1.1a_1+1.21a_2&=1.1 \end{aligned} \right\} \tag{6} \]
From (6) we obtain
\[ a_1\simeq 1-2i_0;\quad a_2\simeq i_0. \]
Equation (5) is transformed into the form:
\[ i=i_0+(1-2i_0)P+i_0P^2 \tag{7} \]
or
\[ i=P+i_0(P-1)^2 \tag{8} \]
Knowing the overheating of the motor winding at rated current \(\Delta \Theta_{\mathrm{nom}}\), one can obtain the overheating for another load current:
\[ \Delta \Theta_{\mathrm{dv}}=\left(\frac{i}{i_{\mathrm{nom}}}\right)^2\Delta \Theta_{\mathrm{dv.nom}} =i^2\Delta \Theta_{\mathrm{dv.nom}}. \tag{9} \]
Substituting into (9) the value of \(i\) from (8), we obtain:
\[ \Delta \Theta_{\mathrm{dv}}=[P+i_0(P-1)^2]^2\Delta \Theta_{\mathrm{dv.nom}}. \tag{10} \]
Having obtained \(\Delta \Theta_{\mathrm{dv}}\) in the form (10) and knowing the probability density of the power used (3), one can find
\[ \mathrm{M}\left[\exp\left(b\Delta \Theta_{\mathrm{dv}}\right)\right] = J = \]
\[ = \lambda \int_{0}^{1.1} \exp\left\{b\,\Delta \Theta_{\mathrm{dv.nom}}\left[P+i_0(P-1)^2\right]^2\right\} e^{-\lambda P}\,dP \tag{11} \]
Although the distribution (3) is truncated, the normalizing factor may be neglected, since, in accordance with [2], the distribution parameter \(\lambda \geq 3\). Replacing the product of the constants \(b\) and \(\Delta \Theta_{\mathrm{dv.nom}}\) by \(K\), we transform (11).
Let us calculate \(\mathrm{M}[\exp(b\Theta_{\mathrm{okr}})]\). Assuming that the temperature variation in shops where metal-cutting equipment is located obeys a normal law with parameters \(\overline{\Theta}_{\mathrm{okr}}\) and \(\delta\), we obtain
\[ \mathrm{M}\left[\exp\left(b\theta_{\mathrm{okr}}\right)\right] = \exp\left[ b\left( \overline{\Theta}_{\mathrm{okr}}+\frac{b\delta^2}{2} \right) \right]. \tag{12} \]
Thus, the mathematical expectation of insulation wear (2) per unit time is
\[ M=\exp[-b\Theta_{\mathrm{dop}}]\cdot \mathrm{M}\left[\exp\left(b\Delta\Theta_{\mathrm{dv}}\right)\right]\, \mathrm{M}\left[\exp\left(b\theta_{\mathrm{okr}}\right)\right]= \]
\[ =J\exp\left[ b\left( \overline{\Theta}_{\mathrm{okr}}-\Theta_{\mathrm{dop}}+\frac{b\delta^2}{2} \right) \right], \tag{13} \]
where \(J\) is determined from formula (11).
Taking (13) into account, formula (1) assumes the form:
\[ r(t)=r\exp\left[-\frac{t}{T}\right]-Mt. \tag{14} \]
Equating \(r(t)\) in (14) to zero and solving the equation with respect to \(t\), we find the insulation service life
\[ r\exp\left(-\frac{t}{T}\right)-Mt=0 \tag{15} \]
Expanding \(\exp\left(-\frac{t}{T}\right)\) in a series and restricting ourselves to two terms of the expansion, we obtain
\[ t=\frac{1}{\dfrac{1}{T}+\dfrac{M}{r}}. \tag{16} \]
It is not difficult to show that, for our case, insulation wear depends almost not at all on the temperature of the stator winding. Without taking natural wear into account, the insulation service life is determined by the expression
\[ t=\frac{r}{M} \tag{17} \]
Taking \(r=10000\) h, \(\Theta_{\mathrm{dop}}=105^\circ\mathrm{C}\), \(\Theta_{\mathrm{okr}}=20^\circ\mathrm{C}\), \(\sigma=5^\circ\mathrm{C}\), \(k=6\), \(\lambda=5\), \(i_0=0.5\), we obtain
\[ M=9.73\exp\left[ 0.0866\left( 20-105+\frac{0.0866\cdot 5^2}{2} \right) \right]= \]
\[ =\frac{10000}{6.8\cdot 10^{-3}} =1.47\cdot 10^6\ \text{hours}\simeq 170\ \text{years}. \]
In [2], on the basis of a study of the operation of a large fleet of metal-cutting equipment, it was found that the average load of asynchronous moto-
…is about 0.2 of the rated value (\(\lambda=5\)), and the conclusion is drawn that, in the near future, increasing the rated power of the electric motors of the main drives of general-purpose lathes and milling machines is inadvisable. From the standpoint of electrical-machine design, taking into account the low reliability of the motors currently being produced, other conclusions may be drawn:
-
In motors supplied for the machine-tool industry, the diameter of the winding wire can be substantially reduced. For these motors, increased heating at rated load is not dangerous, even if this heating exceeds that permissible for the given insulation class.
-
When the diameter of the winding wire is reduced, reliability can be increased either by reducing the slot fill factor, or by strengthening the insulation while maintaining the fill factor.
-
The method for calculating the optimal parameters of induction motors must be corrected. It must take into account reliability and the low average load during operation.
REFERENCES
-
Kh. I. Brakmann. Determination of the design maximum according to the conditions of thermal wear of conductor insulation. Elektrichestvo, No. 3, 1965.
-
R. M. Pratusevich. Operating load regimes of general-purpose machine tools, Stanki i instrument, No. 6, 1960.