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Direct-drive wave energy converters using linear generators (LGs) are attractive due to their high efficiency and reliability. Air-cored LGs, in particular, are receiving increasing attention due to the elimination of attraction forces between the stator and translator and the resulting reduction in structural mass. In this paper, details of the construction of a novel air-cored LG are presented. A custom test rig is constructed for testing the LG, particularly with zero overlap between its stator and translator at the stroke ends. Predictive control for maximum power transfer from the generator is first proposed and tested as a sensorless alternative to methods employing linear position feedback and electromotive force (EMF) estimation with sense coils. The control strategy is verified through simulation and measurements. It is also shown that allowing for zero stator-translator overlap at the stroke ends can improve the power-to-weight ratio compared to complete stator-translator overlap during the entire stroke.