We study two-user multiple-input single-output (MISO) wireless powered communication networks (WPCNs), where the user devices are equipped with non-linear energy harvesting (EH) circuits. We consider time-division duplex (TDD) transmission, where the users harvest power from the signal received in the downlink phase, and then, utilize this harvested power for information transmission in the uplink phase. In contrast to existing works, we adopt a non-linear model of the harvested power based on a precise analysis of the employed EH circuit. We jointly optimize the beamforming vectors in the downlink and the time allocated for downlink and uplink transmission to minimize the average transmit power in the downlink under per-user data rate constraints in the uplink. We provide conditions for the feasibility of the resource allocation problem and the existence of a trivial solution, respectively. For the case where the resource allocation has a non-trivial solution, we show that it is optimal to employ no more than three beamforming vectors for power transfer in the downlink. To determine these beamforming vectors, we develop an iterative algorithm based on semi-definite relaxation (SDR) and successive convex approximation (SCA). Our simulation results reveal that the proposed resource allocation scheme outperforms two baseline schemes based on linear and sigmoidal EH models, respectively.