In this paper, a novel extended permanent magnet synchronous motor model is presented that incorporates a quadratic flux-current function to represent the polarity-dependent saliency. The proposed model enables the design of sensorless polarity detection algorithms required by the initial position detection of permanent magnet synchronous motors. The novelty of the model is that it integrates the polarity-dependent saliency into the traditional machine model and introduces a new machine parameter, the polarity-dependent saliency coefficient, to specify the Hessian matrix of the flux-current function. A measurement method is presented for determination of the elements of the Hessian and the polarity-dependent saliency coefficient. The solution of the model is given for high-frequency sinusoidal pulsating voltage injection. Experimental results show that the proposed extended model accurately predicts the amplitudes and phases of the second harmonics of the motor currents, which are the carriers of the polarity-dependent information. This information enables a current measurement-based polarity detection algorithm using the phase difference between the fundamental and second harmonic of the apparent d-axis current. Both the presented measurement data and the proposed model show that injection in the d-direction is optimal for polarity detection.