On a Mathematical Model of Insulation Wear in the Windings of Electrical Machines and Experimental Determination of Its Parameters
Abstract Generated abstract
This paper proposes a mathematical model for predicting wear and service life of insulation in electrical machine windings under multiple operating factors, including temperature, mechanical stress, humidity, electrical effects, and aggressive environments. Drawing on physicochemical laws of thermal, mechanical, and chemical destruction, the authors express insulation aging through an exponential lifetime relation and transform it into an incomplete quadratic polynomial suitable for regression analysis. They show that the model coefficients can be determined using linear orthogonal multifactor experimental designs, substantially reducing the number of required machine tests compared with conventional nonlinear experimental planning. The approach is presented as applicable to studies of armature winding insulation life in series-II direct-current machines.
Full Text
ON A MATHEMATICAL MODEL OF INSULATION WEAR IN ELECTRICAL MACHINE WINDINGS AND EXPERIMENTAL DETERMINATION OF ITS PARAMETERS
A. Ya. TSIRULIN, V. N. POPOV, I. A. VOLKOMIRSKII
(Presented by the scientific seminar of the Department of Electrical Machines)
The longevity of electrical-machine windings depends on the rate of the processes of insulation aging under the influence of the operating factors acting upon them. The rate of aging, in turn, depends on the levels of the acting factors. Prediction of the durability parameters of windings is impossible if the mathematical model of insulation aging is not known as a function of the output parameters of the insulation, characterizing its state, in terms of the input operating factors. The theoretical determination of this model is impossible within a reasonable time; therefore an experimental search method must be used.
The input parameters of the insulation-wear model are design and operating factors: electrodynamic forces acting on the winding during operation of the machine, vibrations, centrifugal forces, winding temperature, ambient humidity, electrical overvoltages during starts and reversals, the presence in the surrounding medium of caustic vapors and dust, etc. When there are several, and especially many, input parameters, finding a mathematical model experimentally can be accomplished only by setting up a statistically planned multifactor experiment. Since the dependence of insulation service life on many factors is nonlinear, a nonlinear experimental design is required. The form of the insulation-wear model is not known. In this case, the theory of experimental design provides for specifying the model by a polynomial of the second, third, or higher degree. Even when planning for the second degree, with four input variables a central composite design includes 31 runs, which requires testing at least 31 machines if one machine corresponds to each run. In order to obtain the insulation-aging function with the time coordinate, the design must be implemented for at least three values of the test duration. In all, 93 runs must be performed, which requires 93 machines. This is expensive and labor-consuming. With a larger number of input parameters, or when the model is specified by a third-degree polynomial, the experiment becomes excessively cumbersome and costly. The scope of the experimental investigation can be significantly reduced if one uses data from the physicochemical theory of material destruction and represents the insulation-wear model in a form corresponding to this theory.
The general physicochemical regularities of the processes of thermal, mechanical, and chemical destruction of metals, dielectrics, polymers, semiconductors, and other materials are described, as a rule,
was an exponential function [1]. Thus, the “lifetime” of a material specimen from the beginning of application of a mechanical load until failure is a function of the form
\[ \tau=\tau_0\cdot \exp\left(\frac{U_0-\gamma\cdot\sigma}{R\cdot T}\right), \tag{1} \]
where
\[ \begin{aligned} &\tau_0,\ U_0,\ \gamma &&\text{are parameters characterizing the strength properties of the material;}\\ &T &&\text{is temperature, }{}^\circ\mathrm{K};\\ &R &&\text{is the universal gas constant.} \end{aligned} \]
The rate \(v\) of the crack-formation process can be determined from the expression
\[ v=v_0\cdot \exp\left(-\frac{U_0-\gamma\cdot\sigma}{kT}\right). \tag{2} \]
The kinetics of the decomposition of chemical bonds in the process of mechanical failure of polymers is such that the rate of accumulation of fragments of macromolecules (free radicals) is equal to [2]
\[ \dot n=2n_v\cdot\omega_0\cdot \exp\left(-\frac{U_0-\gamma\cdot k\cdot\sigma(n_s)}{RT}\right), \tag{3} \]
where
\[ \begin{aligned} &n_v,\ \omega_0 &&\text{are constants,}\\ &k &&\text{is the stress-concentration coefficient.} \end{aligned} \]
According to the Arrhenius equation, the dependence of the rate constant \(K\) of a chemical reaction on the magnitude of the activation energy \(E\) is determined by the expression
\[ K=B\cdot \exp\left(-\frac{E}{RT}\right). \tag{4} \]
The service life of polymeric materials in which aging processes occur, associated with thermal, oxidative, and mechanical destruction, is established according to the maximum permissible change in the basic properties. If one introduces a generalized characteristic \(S\) of the basic properties of the insulation of an electrical-machine winding—mechanical, physical, dielectric—then the dependence of the value \(S\) on time may be written in the following form [1]:
\[ \ln S=-K\cdot t+\ln S_0. \tag{5} \]
From this expression one can determine the limiting lifetime of the winding insulation, during which the characteristic \(S\) becomes equal to the specified maximum permissible value \(S_{\mathrm{pr}}\), limiting the possibility of operation of the product:
\[ \ln \tau_{\mathrm{pr}}=\ln[\ln S_{\mathrm{pr}}-\ln S_0]-\ln K. \tag{6} \]
The rate constant of the processes of decomposition and destruction of insulation under the action of temperature and physico-mechanical factors, according to the general laws of the kinetics of destruction processes, may in the general case be represented in the form
\[ \ln K=\ln K_0-\frac{1}{RT}\cdot\left(a_0+\sum_{i=1}^{n} a_i\cdot x_i\right), \tag{7} \]
where
\[ \begin{aligned} &a_0,\ a_i &&\text{are constants;}\\ &x_i &&\text{are operating factors: temperature, mechanical actions, humidity, aggressiveness of the medium, and others.} \end{aligned} \]
If \(-\dfrac{1}{T}\) is denoted by \(x_{n+1}\) and the parentheses are expanded, we obtain
\[ \ln K=\ln K_0+\left(a_0\cdot x_{n+1}+\sum_{i=1}^{n} a_i\cdot x_i\cdot x_{n+1}\right)\frac{1}{R}. \tag{8} \]
This is an incomplete quadratic polynomial, containing terms only of the first degree and with pairwise products of the variables. In the general case, the dependence of the destruction-rate constant may be sought in the form
\[ \ln K=b_0+\sum_{i=1}^{n} b_i x_i+\sum_{i<j}^{n} b_{ij}x_i x_j . \tag{9} \]
Taking the logarithm of expression (5) and substituting into it the value of \(K\) from (9), we obtain
\[ \ln t=\ln\ln\frac{S}{S_0}-b_0-\sum_{i=1}^{n} b_i x_i-\sum_{i<j}^{n} b_{ij}x_i x_j . \tag{10} \]
The limiting operating time \(\tau_{\mathrm{pr}}\) is obtained from this expression if the specified \(S_{\mathrm{pr}}\) is substituted for \(S\). Expression (10) is a mathematical model of winding-insulation life. The coefficients of the model \(b_0\), \(b_i\), \(b_{ij}\) must be determined experimentally. For this purpose, over a time \(t=\tau_1\), the machines under investigation must be tested under the conditions of action of the factors \(x_i\). The levels of the factors \(x_i\) in each experiment correspond to the experimental design. In new, unused machines it is necessary to determine the initial value \(S_0\) of the characteristic of the insulation state, and in those tested for the time \(\tau_1\), the insulation parameter \(S_1\) for each experiment. By regression analysis of the experimental results, the parameters \(b_0\), \(b_i\), \(b_{ij}\) can be found as the coefficients of the polynomial
\[ y=\ln\ln\frac{S_1}{S_0}=\ln\tau_1+b_0+\sum_{i=1}^{n} b_i x_i+\sum_{i<j}^{n} b_{ij}x_i x_j = b'_0+\sum_{i=1}^{n} b_i x_i+ \]
\[ +\sum_{i<j}^{n} b_{ij}x_i x_j . \tag{11} \]
Thus, it has been shown that the exponential mathematical model of insulation life, whose exponential form follows from the regularities of the processes of thermal, mechanical, and chemical destruction, can be transformed into the form of an incomplete quadratic polynomial, and its coefficients can be found from a linear orthogonal design of a multifactor experiment. This leads to a great reduction in the scope of the experiment and in the number of machines required for testing. Thus, with three and four input factors, when fractional orthogonal designs are used, eight machines are sufficient, and with five, six, and seven factors, sixteen machines are sufficient. This method is used by us in conducting a study of the life of the insulation of armature windings of series-II direct-current machines.
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I. M. Melamedov. Physical Foundations of Reliability. Leningrad, “Energiya,” 1970.
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E. E. Tomashevsky. Study of the Kinetics of Mechanical Destruction of Polymers. Author’s abstract of dissertation. PTI, Academy of Sciences of the USSR, 1966.