In magnetic resonant coupling (MRC) enabled multiple-input multiple-output(MIMO) wireless power transfer (WPT) systems, multiple transmitters (TXs) eachwith one single coil are used to enhance the efficiency of simultaneous powertransfer to multiple single-coil receivers (RXs) by constructively combiningtheir induced magnetic fields at the RXs, a technique termed "magneticbeamforming". In this paper, we study the optimal magnetic beamforming designin a multi-user MIMO MRC-WPT system. We introduce the multi-user power regionthat constitutes all the achievable power tuples for all RXs, subject to thegiven total power constraint over all TXs as well as their individual peakvoltage and current constraints. We characterize each boundary point of thepower region by maximizing the sum-power deliverable to all RXs subject totheir minimum harvested power constraints. For the special case without the TXpeak voltage and current constraints, we derive the optimal TX currentallocation for the single-RX setup in closed-form as well as that for themulti-RX setup. In general, the problem is a non-convex quadraticallyconstrained quadratic programming (QCQP), which is difficult to solve. For thecase of one single RX, we show that the semidefinite relaxation (SDR) of theproblem is tight. For the general case with multiple RXs, based on SDR weobtain two approximate solutions by applying time-sharing and randomization,respectively. Moreover, for practical implementation of magnetic beamforming,we propose a novel signal processing method to estimate the magnetic MIMOchannel due to the mutual inductances between TXs and RXs. Numerical resultsshow that our proposed magnetic channel estimation and adaptive beamformingschemes are practically effective, and can significantly improve the powertransfer efficiency and multi-user performance trade-off in MIMO MRC-WPTsystems.
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