292-hill-jp-1983-2145 - J . Phys. C: Solid State Phys., 16...

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J. Phys. C: Solid State Phys., 16 (1983) 2145-2156. Printed in Great Britain Theoretical basis for the statistics of dielectric breakdown R M Hill and L A Dissado The Dielectrics Group, Chelsea College, University of London, Pulton Place. London SW6 5PR, UK Received 30 September 1982 Abstract. The statistics of dielectric breakdown processes are presented and compared with those derived from consideration of the fluctuation components of the dielectric response. It is shown that the Weibull statistic parameters have a direct physical meaning and that breakdown processes can be considered as cooperative events. 1. Introduction Weibull statistics have generally been applied to the failure of materials under the action of external forces such as mechanical stress (Trustrum and Jayatilaka 1979) and electrical stress (Metra et a1 1975, Stone and Lawless 1979). It is the intention here to examine the basis of the Weibull approach, to show its applicability to dielectric breakdown and to derive the Weibull form in terms of many-body cooperative fluctuations within the dielectric medium. The Weibull statistic (Gumbel 1958) is based on a stability postulate. If Nsamples, each of size n, are taken from the same population then in each sample there is a smallest value, and the smallest value of the total Nn observations must also be the smallest value obtained from the samples of size n. Asymptotic expansions for the probability function lead to three similar solutions of the form F(Z) = exp[ - ( + zk)] (1) for the cumulative probability distribution of the smallest value of 2. Of interest here is the solution in which k is positive. In this case the distribution F(z) is limited and possesses all moments. The parameter z is a normalised variable and can be transformed. The distribution function of z is given in the normal manner by the differential of as g(z) dz = kzk-' exp [(zk)] dz. (2) For k less than unity decays monoronically and for k greater than unity the distri- bution is peaked at the value z = [(k - l)/k]"k. (3) @ 1983 The Institute of Physics 2145
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2146 R M Hill and LA Dissado A general cumulative distribution function of the Weibull form has been postulated for the process of dielectric breakdown in the form that the probability of breakdown occurring under the action of an electric field E, at a time tis given by 1 - P(E,, t) = 1 - exp ( - La b‘t“-l(€a/€o)b dr) (4) where is the probability of a breakdown not occurring, L is the thickness of the sample and a, b and Eo are experimentally determinable parameters. Equation can be evaluated for particular experimental conditions: (i) Constant electric field. If is constant with respect to time the integration is trivial and leads to = exp[ - L~’(EJE~)~] (5) which is of the Weibull form in both time to breakdown and in the breakdown field at constant time.
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292-hill-jp-1983-2145 - J . Phys. C: Solid State Phys., 16...

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