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In this paper, the evolution of a nonsteady ion sheath in a plasma-filled diode with cylindrical protrusions present on the cathode has been investigated by numerical methods. The values of bursts of electric field strength (E C) and ion current density (J C) at the cathode surface that accompany the formation of the ion sheath have been predicted. It has been shown that the bursts of E C and J C at protrusion tips are greater in amplitude than those at the plane part of the cathode. Approximation formulas have been derived for the plane part of the cathode (or for a protrusion-free plane cathode) which allow one to predict the amplitudes of E C and J C bursts as functions of voltage-rise rate, peak voltage, ion mass and charge, and plasma-electron density and temperature. It has been shown that the considerably increased J C at protrusion tips is due to the ion-flow focusing related to the fact that the plasma boundary around a protrusion is curved during a short time interval at the stage of growth of the ion sheath. The calculation results correlate with the measured delay times to the initiation of explosive electron emission at the multiwire cathode of a plasma-filled diode.