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In this research we present a short distance capacitively coupled wireless power transfer (WPT) system with a self-oscillating half-bridge converter, which uses a positive feedback signal from the WPT system. By this novel implementation we find additional features of the system compared to linear amplifier based systems. With the DC bias voltage supplying the power converter, we achieve a controllable output power, tunable self-oscillation frequency and extension of the self-oscillation range. We present the analytical expressions for the converter waveforms. The theoretical analysis is confirmed by circuit simulation results and measurements on a prototype featuring a half-bridge converter with Gallium-Nitide (GaN) HEMTs. We achieve a system efficiency of 93% at a load power of 83W and 150kHz switching frequency.