Enclosure voltage rise of the gas insulated high voltage switchgears is the result of the propagation and transition of internal very fast overvoltages on to the outside enclosure and grounded parts of the switchgears. The phenomenon is mostly evident at switching small capacitive loads by disconnector, when the multiple back flashovers on the disconnector contacts appear. The high frequency weakly damped transient phenomenon occurs at each back flashover, which, due to its wave character, creates the external overvoltages on the discontinuity points of the enclosure and as the result the temporary enclosure voltage rise. Due to the high frequency characteristics, the voltage rise significantly depends on the way the enclosure is connected to the switchgear grounding system. On the stressed discontinuity and inconvenient connection configurations, the voltage rise can be up to several times ten kV and the undesirable flashovers on the external part of the switchgear are possible. The greatest part of the dissertation is dedicated to the experimental method of determining the transient voltage rise on one of the stressed discontinuity points (at the connection of the enclosure and the power cable) and all the researches are done on a typical enclosed SF6 switchgear, voltage level 123 kV. The statistical approach has been implemented due to the random occurrence of the initial transient phenomenon. By multiple repetition of switching operation on the section disconnector and by measuring the transient voltage rise on the enclosure and cable sheath (on the observed discontinuity point), the greater number of the measuring samples for the further statistical elaboration has been obtained. Four different configurations of the connection of the enclosure and the cable sheath to the grounding system have been researched. The measuring results and statistical elaboration show that, beside connecting the enclosure and the cable sheath to the grounding system, it is useful to bridge over the cable insulating flange using the jumpers or non linear resistors. In such manner, too high voltage differences are significantly and efficiently decreased and very probable external flashover is prevented. The measurements require comprehensive preparation and modern measuring system, with high upper frequency limit, great sampling speed and very deep working memory.