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Electrical breakdown in insulators very often initiate near high field regions of the structure, as found near small-radius impurities or at electrode defects. This is attributed to the development of localized space charges. For this reason many efforts have been made to determine such charge. Various techniques are now available, but they are not directly applicable to complex geometries where it is difficult to determine analytically the field configuration and thus the relation between the measured variables and the space charge distribution. To solve this problem, we propose to use a numerical simulation using a finite element method (FEM). In this paper we describe how it can be implemented in the case of the pressure wave propagation (PWP) method. It is shown that measured signals in insulating samples with divergent electric field regions are well fitted by simulations. We show that this allows for the determination of space charge distribution in such samples.

2 AbstractVarious techniques for the determination of the space charge distribution are currently used to analyse the behaviour of insulators submitted to high electric fields such as is found near small radius impurities or electrodes. In such geometries it is difficult to determine analytically the field configuration and thus the relation between the measured variables and the space charge distribution. For this reason, a numerical simulation using a finite element method has been developed. IntroductionIt is now well known that the build up of space charge in diverging field regions of insulating materials can lead to electrical tree growth and thus to breakdown phenomena. The presence of impurities or defects, in a material subjected to a high applied voltage, can be the cause for such high field concentrations. Various methods are currently being used to observe charge injection in diverging field geometries, such as point plane structures. These methods typically involve differential current measurements [ 13 or electroluminescence [2]. They all suggest, or confirm, that charge transfer occurs at the metal-insulator interfaces. In both cases however the charge transfer is deduced from the experiments through very indirect procedures. For this reason, it has been proposed to use presently availablenon destructive methods for direct charge distribution measurements and particularly the Pressure Wave Propagation method [3,4,5]. Several studies have already been carried out in such a geometry [6,7]. However the relationship between the measured signal and the density of space charge is quite complex. For this reason, in order to interpret ' the observed data, we have developed a numerical simulation using a finite element method that is presented in this paper. It gives, for any space charge distribution, the current induced in the external circuit by the propagation of the pressure wave. By measuring the experimental current it is possible to determine, by successive simulations, an unknown space charge distribution. Principle of the PWP methodElectrode J The principle of a PWP measurement [3,4,5] is shown on Fig. 1. Usually a pressure pulse is generated either by the impact of a short laser pulse on a target electrode or by a piezoelectric transducer. During its propagation the charges in the insulator are displaced which slightly modifies the internal electric field and the image charges on the electrodes. In the case of a planar geometry and under shortcircuit conditions the current appearing in the measuring circuit is proportional to the space charge distribution. The relationship between the signal and the space charge has been established and verified [4,5] for plane and coaxial Figure 1. Principle of the PWP methodgeometries. As this method is direct and-non-destructive it has been implemented in order to determine the space charge distribution in insulators submitted to high divergent electric fields [6,7,8]. Geometry of the sampleIn order to be able to use plane front pressure pulses and to provide ...