  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
focusing the magnetic fields from all TX coils in a beam toward the receiver
(RX) coil, termed as "magnetic beamforming". In this paper, we study the
optimal magnetic beamforming for an MRC-WPT system with multiple TXs and a
single RX. We formulate an optimization problem to jointly design 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 TXs' constraints on the peak voltage and current. For the special
case of identical TX resistances and neglecting all TXs' constraints on the
peak voltage and current, we show that the optimal current magnitude of each TX
is 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) problem, which is reformulated as a semidefinite programming
(SDP) problem. We show that its semidefinite relaxation (SDR) is tight.
Numerical results show that magnetic beamforming significantly enhances the
deliverable power as well as the WPT efficiency over the uncoordinated
benchmark scheme of equal current allocation.
