I. Introduction
An absolute encoder is a device for measuring the rotation angle of a rotational motor in industrial applications. The two typical types of absolute encoders are optical and magnetic. Among these, the absolute magnetic encoder (AME) is widely used in applications because of its benefits, which include low cost and superior performance in industrial environments. It can achieve the same resolution and accuracy as an optical encoder. In recent years, extensive studies have been conducted on vernier-type absolute encoders [1]–[3]. In this article, we investigate a vernier absolute magnetic encoder (VAME), which is a multipolar dual track magnetic encoder. It has master and nonius tracks. The master track was located outside and split to pole pairs, while the nonius was located inside and split to pole pairs. Each pole pair of each track will generate a pair of sine and cosine signals and each period of the sine/cosine signal corresponds to a phase of a saw tooth wave in the range of rad. Two phase values are extracted from the master and nonius tracks. The absolute information of any point on the two tracks can be obtained by the vernier principle [2], [3]. In this structure, the accuracy of the phase obtained from the master and nonius in each pole pair is important for improving the accuracy of the VAME. In practice, the sine/cosine signals from the master and nonius tracks are contaminated by nonideal factors, such as different amplitudes, dc-offsets, phase shifts, and random noise [4], [5]. Moreover, the harmonics existing in the encoder signals significantly affect the accuracy of the multipolar magnetic encoder [6], [7]. Therefore, a method to overcome the error factors is needed to obtain the absolute information.