  Energy beamforming (EB) is a key technique to enhance the efficiency of
wireless power transfer (WPT). In this paper, we study the optimal EB under
per-antenna power constraint (PAC) which is more practical than the
conventional sum-power constraint (SPC). We consider a multi antenna energy
transmitter (ET) with PAC that broadcasts wireless energy to multiple randomly
placed energy receivers (ER)s within its cell area. We consider sum energy
maximization problem with PAC and provide the optimal solution structure for
the general case. This optimal structure implies that sending one energy beam
is optimal under PAC which means that the rank of transmit covariance matrix is
one similar to SPC. We also derive closed-form solutions for two special cases
and propose two sub-optimal solutions for general case, which performs very
close to optimal beamforming.
