We consider the problem of grid-forming control of power converters in low-inertia power systems. Starting from an average-switch three-phase inverter model, we draw parallels to a synchronous machine (SM) model and propose a novel grid-forming 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 assuring existence, uniqueness, and global asymptotic stability of equilibria in a coordinate frame attached to the virtual oscillator angle. By actuating the DC-side input of the converter we are able to enforce this sufficient condition. In the same setting, we highlight strict incremental passivity, droop, and power-sharing properties of the proposed framework, which are compatible with conventional requirements of power system operation. We subsequently adopt disturbance decoupling techniques to design additional control loops that regulate the DC-side voltage, as well as AC-side frequency and amplitude, while in the end validating them with numerical experiments.