THE EQUATION OF THE “LIFE CURVE” OF ELECTRICAL INSULATION

V. S. Dmitrevskii, V. F. Grafov

(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)

The design of electrical insulation and the determination of its service life require knowledge of the dependence of the time to breakdown on the field strength (the equation of the “life curve”).

Experimental investigation of the “life curve” of insulation has been carried out by many authors [1—8]. In the present work one of the possible ways of theoretically obtaining the equation of the “life curve” of insulation is shown.

The fluctuation theory of the destruction of materials under the action of a mechanical load gives [9]:

\[ \tau=\tau_0 \exp \left\{ \frac{ W-\dfrac{e\alpha\sigma a}{N\left(1-\dfrac{2m}{m_{\mathrm{c}}}\right)} \ln \dfrac{2WN\left(1-\dfrac{2m}{m_{\mathrm{c}}}\right)}{\alpha\cdot a\sigma e} }{ 2kT } \right\}. \tag{1} \]

where \(\tau\) is the time to rupture of the specimen;
\(\tau_0\) is the relaxation time of vibrations of an atom in a molecule;
\(W\) is the energy of rupture (dissociation) of a bond;
\(m\) is the molecular weight of the end of a chain that does not take up load;
\(m_{\mathrm{c}}\) is the molecular weight of the chain;
\(\alpha\) is the coefficient of ordering of polymer chains, varying according to [10] from 1 to 3;
\(2kT\) is the average energy of two atoms forming one chemical bond.

\[ N=\frac{l\cdot N_A\cdot \rho}{M}. \tag{2} \]

where \(N_A\) is Avogadro’s number,
\(\rho\) is the density,
\(M\) is the molecular weight of the monomer unit,
\(l\) is the bond length,
\(N\) is the number of bonds per unit cross section of the specimen,

. \(a\) characterizes the curvature of the potential curve near its minimum. For the \(C—C\) bond

\[ a=\frac{l}{3.22}. \tag{3} \]

The destruction of the material is determined by the magnitude of the supplied energy and does not depend on its nature [11].

Under operating conditions, electrical insulation is subjected to the action of an electric field and mechanical load. The increase in free energy in this case is

\[ \Delta W=\Delta W_{\mathrm{e}}+\Delta W_{\mathrm{m}} =\frac{\varepsilon E_{\mathrm{m}}^{2}}{2} +\frac{\sigma^{2}}{2G_{\mathrm{m}}} =\frac{\sigma_{\mathrm{e}}^{2}}{2G_{\mathrm{m}}}, \tag{4} \]

where \(\varepsilon\) is the dielectric permittivity;
\(E_{\mathrm{m}}\) is the electric-field strength;
\(\sigma\) is the mechanical stress;
\(G_{\mathrm{m}}\) is the elastic modulus (Young’s modulus);
\(\sigma_{\mathrm{e}}\) is the equivalent mechanical stress, which produces in the material the same increase in free energy as the simultaneous action of electrical and mechanical loads.

Substituting into 1 the value of \(\sigma_{\mathrm{e}}\), determined from 4 with allowance for 2, 3, and denoting

\[ \gamma=\frac{eM\alpha}{3.22N_A\rho\left(1-\dfrac{2m}{m_{\mathrm{c}}}\right)}; \]

\[ Ae^{-bT}=\frac{\alpha eM\sqrt{\varepsilon G_{\mathrm{m}}}}{3.22N_A\rho\left(1-\dfrac{2m}{m_{\mathrm{c}}}\right)}, \]

we obtain

\[ \tau=\tau_0\exp \left\{ \frac{ W-\sqrt{\gamma^2\sigma^2+\left(Ae^{-bT}E_{\mathrm{m}}\right)^2} \ln \dfrac{2We}{\sqrt{\gamma^2\sigma^2+\left(Ae^{-bT}E_{\mathrm{m}}\right)^2}} } {2kT} \right\}. \tag{5} \]

Real dielectrics contain various inhomogeneities that lead to the occurrence of local field strengths. The maximum value of the local field strength for an ellipsoidal inhomogeneity will be determined [12] as

\[ E_{\mathrm{m}}=\frac{4\nu}{4+\chi+\chi\nu}\cdot E; \tag{6} \]

\[ \nu=\frac{\dfrac{1}{\rho_1}+j\omega\varepsilon_1} {\dfrac{1}{\rho_2}+j\omega\varepsilon_2}, \tag{7} \]

where \(\rho_1,\ \varepsilon_1\) and \(\rho_2,\ \varepsilon_2\) are the specific volume resistivity and dielectric permittivity of the inhomogeneity and the material, respectively;

\(E\) is the average electric-field strength.

If the ratio of the size of the inhomogeneity in the direction of the field to its diameter \(y\) is greater than 1, the coefficient \(x\) is found as

\[ x=\frac{2}{y^{2}-1}\left(2-\frac{y}{\sqrt{y^{2}-1}}\ln \frac{y+\sqrt{y^{2}-1}}{y-\sqrt{y^{2}-1}}\right). \tag{8} \]

For simplicity, let us write

\[ E_{\mathrm{m}}=\beta E, \tag{9} \]

where \(\beta\) is the coefficient of concentration of local field stresses due to inhomogeneities.

One type of inhomogeneity in dielectrics is gas inclusions. Ionization in gas inclusions leads to a reduction in the service life of insulation as a result of the following processes:

1) reduction of the insulation thickness due to its destruction by erosion from the surface;

2) change in the structure of the material, causing a decrease in its electrical strength;

3) creation of high local field stresses in the dielectric.

Arzbarger’s work has shown that breakdown of insulation at a voltage amounting to only 30% of the initial breakdown voltage caused a reduction in its thickness due to erosion by only 5–6% [13]. Under prolonged action of ionization in a gas inclusion, owing to chemical processes, the electrical conductivity of polyethylene increased by 4 orders of magnitude, while the short-time electrical strength decreased by only 10–15% [14].

Failures of insulation under prolonged application of voltage cannot be explained only by erosion of the material or by a change in its structure under the action of ionization in gas inclusions.

The conductivity of the ionized gas is high, and \(v\) may be taken as equal to infinity. Then the magnitude of the local field stress is determined only by \(y\) and by the average field strength. The value of \(y\) is found approximately as

\[ y=\frac{h}{2r}, \tag{10} \]

where \(h\) is the depth of the gas inclusion in the direction of the field,

\(r\) is the radius of the gas inclusion, or the radius of the anode spot when the dimensions of the gas inclusion along the equipotential surface are large.

In the case of a flat pore, the radius of the anode spot is found as [15]

\[ r=\sqrt{2Dt}, \tag{11} \]

where \(D\) is the electron diffusion coefficient;

\(t\) is the time taken by an electron to traverse the distance between the two walls of an inclusion of depth \(h\).

The time \(t\) is determined as

\[ t=\frac{h}{U_{e}E_{\mathrm{pr}}}, \tag{12} \]

where \(U\) is the electron mobility;
\(E^{*}\) is the breakdown field strength in the gas.

From [16] it is known that

\[ \frac{D}{U_e}=\frac{kT_{\mathrm{e}}}{q_0}. \tag{13} \]

Solving 10, 11, 12, 13 jointly, we obtain

\[ y=\sqrt{\frac{hE_{\mathrm{pr}}q_0}{8kT_{\mathrm{e}}}}. \tag{14} \]

In these expressions:
\(k\) is Boltzmann’s constant;
\(T_{\mathrm{e}}\) is the electron temperature in the gas discharge;
\(q_0\) is the electron charge.

Taking \(T\) equal to \(2000^\circ\) K, let us determine the value of \(y\) for different \(h\). The results of the calculations are given in Table 1.

Table 1

Expression (14) gives \(y\) on the assumption that the radius of the anodic spot is equal to the radius of the electron avalanche. This phenomenon is observed for small \(h\). For large \(h\), a significant tangential component of the electric-field strength arises along the pore wall, leading to the appearance of surface discharges. In this case the radius of the anodic spot will be determined by the radius of the brush discharge over the surface, which we determine [17] as

\[ r=\eta C_{\mathrm{p}}^{2}U^{5}\sqrt[4]{\frac{dU}{dt}}, \tag{15} \]

where \(\eta\) is a coefficient depending on the form of the applied voltage \(U\),
\(C\) is the specific surface capacitance,
\(\dfrac{dU}{dt}\) is the rate of change of voltage.

The table also gives the value \(y_1\), calculated from the condition for the development of the discharge over the pore surface.

It is clear from the table that, as the thickness of the gaseous inclusion increases, \(y\) increases, while \(y_1\) decreases. Apparently, for inclusions with a thickness of less than \(100\,\mu\mathrm{m}\), discharges over the pore surface are absent, and the magnitude of the local field strength is determined by the gas discharge across the thickness of the pore. When the pore thickness is \(200\,\mu\mathrm{m}\) or more, the local field strength is determined by the surface discharge in the inclusion. From this assumption, \(\beta\) was calculated by formulas (6–9).

As can be seen from Table 1, in a dielectric in the presence of a gaseous inclusion in which the ionization process occurs, a high local field strength \(E\) is created.

Substituting \(E_{\mathrm{m}}\) from (9) into (5), we obtain

\[ \tau=\tau_0 \exp \left\{ \frac{ W-\sqrt{\gamma^2\sigma^2+\left(Ae^{-bT}\beta E\right)^2}\, \ln \dfrac{2We}{\sqrt{\gamma^2\sigma^2+\left(Ae^{-bT}\beta E\right)^2}} }{2kT} \right\}. \tag{16} \]

If \(\sigma\) is equal to 0, equation (16) will have the form

\[ \tau=\tau_0 \exp \left\{ \frac{ W-Ae^{-bT}\beta E \ln \dfrac{2We}{Ae^{-bT}\beta E} }{2kT} \right\}. \tag{16a} \]

In processing the results of experiments, the dependence of \(\ln \tau\) on \(E\) is often used. From relation (16a), for \(T=\mathrm{const}\) we find

\[ \ln \tau = B - CE \ln \frac{D}{E}; \tag{16b} \]

\[ B=\ln \tau_0+\frac{W}{2kT}; \]

\[ C=\frac{A\beta e^{-bT}}{2kT}; \qquad D=\frac{2We^{bT+1}}{A\beta}. \]

In Fig. 1, according to equation (16b), the dependence of \(\ln \tau\) on \(E\) is plotted. In the general case a concave curve is obtained. Figures 2 and 3 show experimental “life curves” according to the data of [1–8]. The theoretical and experimental dependences agree qualitatively well with each other. Quantitative comparison is difficult, since \(\beta\) is a random quantity and depends on the type of inhomogeneities and their sizes and on the conditions under which the experiments are carried out.

Fig. 1. Theoretical dependence \(\ln \tau=f(E)\).

As the coefficients \(\beta\) increase, the lifetime of the insulation, as is evident from (16a), decreases. The values of \(\beta\) given in the table vary with the thickness of the gas inclusion according to a curve with a maximum. It should be expected that the lifetime of the insulation as a function of the thickness of the gas inclusion should vary according to a curve with a minimum. The experiments presented in [18] confirm the stated assumption.

Fig. 2. Dependence \(\lg \tau=f(E)\) in an alternating field for films of PE (1), PS (2), PETP (3), PTFE (4) at normal temperature according to data from [1].

Fig. 3. Dependence \(\lg \tau=f(E)\) in an alternating field: 1, 4—PE according to data from [5, 4], 6, 7, 8, 9, 10—PTFE according to [8, 7, 2, 4, 3], 2—PS according to [8], 3, 11—PETP according to [7, 3], 5—impregnated capacitor paper KON-1 according to [6], 12—triacetate film according to [3].

Thus, the lifetime of electrical insulation is determined by the local field strength created by the inhomogeneity of its structure. Ionization in gas inclusions is a particular case of inhomogeneities with a nonlinear characteristic.

The derived equation of the “life curve” can be used for processing experimental results on the lifetime of materials and for predicting the service life of electrical insulation.

References

1. N. N. Barabanov, S. N. Koikov, V. A. Fomin, A. N. Tsikin. Ionization aging of polymer films over a wide range of temperature, voltage, and frequency. “Elektromekhanika,” 1963, No. 12, 15—19.

2. Sh. T. Abzianidze. Investigation of the properties of fluoroplastic film with the aim of using it for the insulation of electrical machines. Scientific-technical bulletin of LPI, “Elektromekhanika,” 1957, No. 11.

3. N. V. Aleksandrov, E. A. Kalinina, S. G. Trubachev. Determination of the corona resistance of electrical insulating materials by various methods. “Elektrichestvo,” 1961, No. 4.

4. A. R. Hippel. Dielectrics and Their Applications. Gosenergoizdat, 1959.

5. R. G. Rhudy, H. E. Mazanek. Accelerated voltage endurance tests. “Power Apparatus and Systems,” 1960, No. 49, 576—580.

6. D. S. Varshavsky. On the question of the electric strength of a multilayer dielectric made of impregnated capacitor paper under prolonged application of a voltage of frequency 50 Hz. Izv. vuzov, “Energetika,” 1964, No. 1, 19—26.

7. N. G. Sakhno, A. S. Romanov. Influence of frequency on the resistance of certain polymer films to the action of electrical voltage. Collection “Exchange of Experience in the Electronics Industry,” 1967, issue 1, 61—73.

8. Sh. T. Abzianidze. On the long-term electric strength of certain insulating materials and the influence of discharge on them. Proceedings of the Scientific-Research Electrical Engineering Institute of the Council of National Economy of the Georgian SSR, 1961, 1, 75—83.

9. A. I. Gubanov, A. D. Chevychelov. On the theory of tensile strength of solid polymers. FTT, 1962, 4, No. 4, 928—933.

10. F. Bueche. Tensile strength of plastics below the glass-transition temperature. Physics of Polymers (collection of articles). Izd-vo IL, 1960, 49—57.

11. A. A. Vorob’ev. Analogies and differences in the temperature, mechanical, thermal, and electrical destruction of solid dielectrics. Collection “Breakdown of Dielectrics and Semiconductors.” Izd-vo “Energiya,” 1964.

12. G. I. Skanavi. Physics of Dielectrics (the region of strong fields). Publishing House of Physico-Mathematical Literature, 1958.

13. I. Artbauer, I. Griac. Der Durchschlag von Kunststoffen unter Einwirkung von Glimmentladungen. “Electric,” 1963, 17, No. 4, 120—124.

14. V. G. Sotnikov. Influence of partial discharges in an air layer on certain characteristics of polyethylene. Collection “Electrical Apparatus and Electrical Insulation.” Moscow, “Energiya,” 1970, 467—475.

15. J. Meek, J. Craggs. Electrical Breakdown in Gases. Izd-vo IL, 1960.

16. Yu. N. Vershinin. Electrical Breakdown of Solid Dielectrics. Izd-vo “Nauka,” Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk, 1968.

17. M. E. Ierusalimov, N. N. Orlov. High-Voltage Engineering. Publishing House of Kiev University, Kiev, 1967.

18. N. S. Il’chenko, I. N. Onishchuk. Service life of insulation under rectified voltage. Collection “Ionization Destruction, Corona Resistance, and Electric Strength of Organic High-Voltage Insulation of Radio-Electronic Electrical Equipment (abstracts of reports).” Publishing House of the Kiev House of Scientific and Technical Propaganda, Kiev, 1969.