Magnetic resonant coupling (MRC) is an efficient method for realizing the near-field wireless power transfer (WPT). The use of multiple transmitters (TXs) each with one coil can be applied to enhance the WPT performance by coherently combining the magnetic fields induced by all TX coils in a beam toward the receiver (RX) coil, a technique termed "magnetic beamforming". In this paper, we study the optimal magnetic beamforming design for an MRC-WPT system with multiple TXs and a single RX. We formulate a problem to jointly optimize the currents flowing through different TXs so as to minimize the total power drawn from their voltage sources, subject to the minimum power required by the RX load as well as the practical constraints on the peak voltage and current at all TXs. For the special case of identical TX resistances and without the peak voltage and current constraints, we show that the optimal current at each TX should be proportional to the mutual inductance between its TX coil and the RX coil. In general, the problem is a non-convex quadratically constrained quadratic programming (QCQP), which is reformulated as a semidefinite programming (SDP) with rank-one constraint. We show that the semidefinite relaxation (SDR) of the reformulated problem is tight and hence the problem is solved optimally. Numerical results show that the optimal magnetic beamforming design significantly enhances the deliverable power as well as the power efficiency over the uncoordinated WPT benchmark with equal current allocation over TXs.
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