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This paper presents experimental and modelling results of an ultra-compact self-rectifier flux pump energizing a superconducting coil. The device fits inside a volume of 65x65x50~mm and generates up to 320~A dc through the coil and a peak output voltage up to 60~mV. We also develop and present a full electromagnetic effective circuit model of the flux pump and compare its predictions to the experimental results. We show that our model can reproduce accurately the charging of the load coil and that it reproduces the systematic dependence of the maximum load coil current on the input current waveform. The experiments and modelling together show also the importance of dc-flux offsets in the transformer core on the final achievable current through the coil. The miniaturization possible for this class of flux pump and their minimal heat-leak into the cryogenic environment from thermal conduction make them attractive for applications with demanding size, weight and power limitations. Our effective circuit model is a useful tool in the understanding, design and optimization of such flux pumps which will accelerate their progression from research devices to their application.