###### Abstract  
The thin wobble motors that are required to hold rating shafs employ an electropermanent magnet. This turns the holding force on and off by applying a momentary electrical pulse. To design the magnet devices without the need for finite element analyses, a theoretical force model is necessary for predicting the attractive force. In this paper, first, a force model is derived by estimating the permeance around the air gap. A magnetic circuit is constructed, employing a relatively simple method to build the model in clouding leakage flux. Thus, the basic structure and driving principle are also presented. Next, an analytical force model is constructed on the basis of distribution parameter analysis between the stator and the rotating shaft. The design of the electromagnet core and the control method are presented. Finally, a prototype model of the motor that is 30 mm in diameter and 7 mm in thick is fabricated. The two models are verified by comparing the results of FEM with the results of the experiments. They can properly predict the attractive force, so the thin wobble motor with holding force can be applied in portable electric equipment. wobble motor; permeance; magnetic circuit; leakage flux; electropermanent magnet; force model +
Footnote †: journal: Applied Physics Letters  
14 January 2021  
Footnote †: journal: Applied Physics Letters  
14 January 2021  
the shaft is halted. To fix the shaft, we introduce a new type of wobble actuator with an electropermanent magnet (EPM). It comprises semi-hard (Alnico 5) and neodymium (NdFeB) magnets, and it can be used to fix the rotating shaft. In this paper, we introduce the two theoretical force models for application with portable electric devices and fabricate a thin electromagnetic wobble motor that is 50 mm in diameter and 7 mm thick including the EPM device. First, it is necessary to predict the magnetic force generated by the magnets of the EPM device. The most accurate method predicting force is used by commercial finite element method software. FEM is one of the ways to solve problems using numerical approximation for problems that are difficult to obtain an accurate theoretical interpretation. In addition, FEM has the advantage of being compatible with Auto CAD or other design programs, and accuracy of the model analysis and user convenience are very high, and high-quality analysis is possible even if there is a lack of professional knowledge about FEM. On the other hand, it will take several hours to obtain the precise analysis of the complex model and variables in the model. However, this comes with the problem of having to perform numerous iterations in the initial design step. On the other hand, a more systematic method is magnetic circuit analysis. An EPM device with two magnets is expressed as a model in which a magnetic flux source and an internal reluctance are connected in parallel. The magnetic flux paths, including the reluctance of the core of the magnetic material and the reluctance in the air gap, are shown as an electric circuit. The method for calculating the magnetic force in a magnetic circuit is similar to calculate the current in an electric circuit and is very simple. Since the fringing effect and the leakage flux around the air gap are not considered, the results of magnetic circuit analysis led to a considerably large degree of error. This is because it is difficult to accurately include them in magnetic circuits. Therefore, we introduce a theoretical force method into the initial design for use in the magnetic circuit. In addition, we design an EPM device including magnets with a flat plate and a constant air gap, and propose a simple 3D model for the leakage flux and fringing effect around the gap. The paths of magnetic flux are modeled using a simplified method consisting of several arcs, straight lines, and a mean flux line. Additionally, we introduce the schematic of the EPM device to validate the proposed theoretical force model, and the basic principle and structure, are presented in a straightforward manner.