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In this paper, we present a novel Z-source sparse matrix converter (ZSMC) and a compensation method based on a fuzzy logic controller to compensate unbalanced input voltages. The ZSMC is developed based on the structure of an SMC to reduce the number of unipolar power semiconductor switches and employs a Z-source network to overcome the inherent limitation of the low voltage transfer ratio of conventional matrix converters. Although the ZSMC is a two-stage converter, it directly connects between a source and a load through a Z-source network, which is designed to have smaller passive components as the only purpose is voltage boosting. Therefore, the output of the ZSMC is directly affected by disturbances of the input-voltage source. The operational principle of the ZSMC is described and its modulation strategy is explained. Simulation and experimental results are shown to verify the feasibility of the ZSMC and its compensation method.