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This paper proposes a circuit topology for a matrix converter with a boost-up AC chopper in the input stage. In order to suppress the input filter resonance, the authors have developed a stabilization control for the input current using a V-connection AC chopper. However, the input current is required as the signal for the damping control. In this paper, the damping control using the existing filter capacitor voltage is proposed. In order to verify effectiveness of the proposed damping control, the proposed circuit is demonstrated by a 1.8-kW prototype. As a result, it was confirmed that the input current THD can be suppressed approximately 72.5% improved by applying the input current-type or the filter capacitor voltage-type damping controls.