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Direct drives with linear permanent magnet synchronous motors (LPMSMs) are recently attracting the attention of both industry and academia. On the one hand such electric drives permit to reduce size and increase reliability thanks to the lack of mechanical reduction and transmission devices. On the other hand precision positioning requires linear position sensing with a measuring range (and size) equal to the motor allowed travel. It is clear the advantage of sensorless control in such applications in terms of reduced hardware complexity, cost and maintenance requirements. This paper presents a position sensorless control scheme based on high frequency signal injection. A pulsating voltage is superimposed to the control voltage along the estimated d-axis direction. Then a novel demodulation procedure implemented in stationary coordinates is proposed to extract position information. The procedure has a reduced computational cost if compared to the alternatives already proposed in the related literature and requires no tuning effort. A demonstration of the algorithm convergence valid in transient conditions, and a novel method to measure the high frequency motor impedance are also presented. The proposed approach is well suited for motors with reduced magnetic saliency such as tubular LPMSM. The above considerations are validated by extensive experiments.