Sensorless control of permanent-magnet synchronous motors at low velocity remains a challenging task. A now well-established method consists of injecting a high-frequency signal and using the rotor saliency, both geometric and magnetic-saturation induced. This paper proposes a clear and original analysis based on second-order averaging of how to recover the position information from signal injection; this analysis blends well with a general model of magnetic saturation. It also proposes a simple parametric model of the saturated motor, based on an energy function which simply encompasses saturation and cross-saturation effects. Experimental results on a surface-mounted motor and an interior magnet motor illustrate the relevance of the approach.