We consider the problem of grid-forming control of power converters in low-inertia power systems. Starting from an average-switch three-phase power converter model, we draw parallels to a synchronous machine (SM) model and propose a novel converter control strategy which dwells upon the main characteristic of a SM: the presence of an internal rotating magnetic field. In particular, we augment the converter system with a virtual oscillator whose frequency is driven by the DC-side voltage measurement and which sets the converter pulse-width-modulation signal, thereby achieving exact matching between the converter in closed-loop and the SM dynamics. We then provide a sufficient condition asserting existence, uniqueness, and global asymptotic stability of a shifted equilibrium, all in a rotating coordinate frame attached to the virtual oscillator angle. By actuating the DC-side input of the converter we are able to enforce this condition and provide additional inertia and damping. In this framework, we illustrate strict incremental passivity, droop, and power-sharing properties which are compatible with conventional power system operation requirements. We subsequently adopt disturbance-decoupling and droop techniques to design additional control loops that regulate the DC-side voltage, as well as AC-side frequency and amplitude, while in the end evaluating them with numerical experiments. (C) 2018 Elsevier Ltd. All rights reserved.