###### Abstract  
This paper presents an analytical model for calculating the detailed air-gap magnetic field distribution in interior permanent magnet synchronous motors (IPMSMs) with mixed eccentricity. In order to improve the efficiency of model solving, a modeling strategy combining the equivalent magnetic circuit network method and the subdomain method was adopted. Specifically, the magnetic field distribution of the rotor was modeled by using the magnetic circuit analysis method, and the magnetic field distributions of the air-gap, slot opening and stator slot along the radial direction were modeled in different regions according to their structure ruler. Then, the influence of bridge saturation was considered. Moreover, based on the analysis of the air-gap geometric structure with mixed eccentricity, a detailed spatiotemporal analytical model of the air-gap magnetic field was established, which provides a more accurate description of the mixed eccentricity composed of static and dynamic rotor eccentricities of different severity. The analytical models were compared with the corresponding models established by the finite element method, which proved the accuracy and validity of the models established in this paper. Finally, some key features related to radial and tangential air-gap flux density were extracted, which can significantly reflect the characteristics of eccentricities. The main findings reported in this paper will be of benefit for developing methods for early identification and diagnosis of eccentricity faults in IPMSMs. IPMSMS; mixed eccentricity; bridge saturation; air-gap magnetic field; analytic calculation +
Footnote †: journal: Applied Sciences  
and the consumption of computing resources is large. Therefore, it has been mostly used in the design optimization and structural verification of the motor [8; 9]. In contrast, the analytical methods can reveal the coupling law between mechanical/electrical parameters and magnetic field through mathematical modeling, the motor and has a fast running speed [3; 10]. Therefore, analytical method has been widely used in the research of motor fault diagnosis mechanisms. However, there are many factors affecting the spatial and temporal distribution of the air-gap magnetic field in IPMSMs. The main factors include the load characteristics, the cogging effect formed between the permanent magnet and stator teeth, and the magnetic saturation in the bridge region. In addition, in the study of the air-gap magnetic field under rotor eccentricity, single static eccentricity, or dynamic eccentricity, has been considered in previous research [11; 12; 13], which is different from the mixed eccentricity in actual operation. Therefore, the establishment of the analytical model of air-gap magnetic field accounts for the above factors is a key issue to accurately calculate the spatial and temporal distribution of the air-gap magnetic field under the general operation of IPMSMs. At present, the analytical methods for modeling the air-gap magnetic field of permanent magnet synchronous motors (PMSMs) include the equivalent magnetic circuit network method [14; 15; 16; 17], the subdomain model method [18; 19; 20; 21], and the magneto-motive force-magnetic permeability method [22; 23; 24; 25; 26]. In [14; 15], the equivalent magnetic circuit network method was used to model and analyze the rotor magnetic field of PMSM to verify the effect of the optimal design. The same method was used to model the armature magnetic field of the permanent magnet linear synchronous motor analytically in [16]. By combining the equivalent magnetic circuit network method and the Carter coefficient method, the influence of the cogging torque on the distribution of the air-gap magnetic field in the IPMSM was analyzed in [17]. However, the equivalent magnetic circuit network method can only calculate the radial component of the air-gap magnetic field and cannot calculate the tangential component of the air-gap magnetic field.