This paper presents a two-dimensional (2-D) analytical model for plannar linear permanent-magnet (PM) synchronous motors. The general and particular solutions are presented for a slotless single-sided linear PM synchronous motor as a case study. The governing partial differential equations (PDEs) of the motor are formulated for seven regions: mover-side exterior, mover back iron, PMs, air-gap, winding, stator back-iron and stator-side exterior. Four different magnetization patterns, i.e. parallel, ideal Halbach, 2-segment Halbach and bar magnets in shifting directions magnetization patterns are considered to calculate the tangential and normal components of the open-circuit magnetic flux density, and armature reaction of the motor under the study. The winding induced voltage, flux linkage, self and mutual inductances, normal and tangential components of forces are computed. The validity of the proposed technique is confirmed by comparing the analytical results with the results obtained from 2-D finite-element analysis.

Consequent-pole permanent-magnet machines are of particular interest owing to their special features. They have alternate hard and soft magnet poles. In this paper, a two-dimensional (2-D) analytical model is proposed for outer-rotor slotted-stator consequent-pole brushless PM (CPBLPM) machines. The model is developed from the quasi-magneto-static Maxwell's equations, which are analytically solved. The magnetic flux density expressions are initially derived and then other quantities such as the no-load induced voltage, electromagnetic torque, reluctance torque, cogging torque, self and mutual inductances as well as unbalanced magnetic forces (UMF) are computed. The suggested model is generic and applicable to CPBLPM machines with any number of poles and phases, type of magnetization patterns and current waveforms. A case study is presented to evaluate the performance of the proposed model. The accuracy of the 2-D analytical solutions is verified against those obtained from the finite element method (FEM). The proposed method can replace the FEM in designing and optimization of outer-rotor consequent-pole permanent-magnet motors.

This paper presents a two-dimensional analytical model of outer rotor permanent magnet machines equipped with surface inset permanent magnets. To obtain the analytical model, the whole model is divided into the sub-domains, according to the magnetic properties and geometries. Maxwell equations in each sub-domain are expressed and analytically solved. By using the boundary/interface conditions between adjacent sub-regions, integral coefficients in the general solutions are obtained. At the end, the analytically calculated results of the air-gap magnetic flux density, electromagnetic torque, unbalanced magnetic force (UMF), back-electromotive force (EMF) and inductances are verified by comparing them with those obtained from finite element method (FEM). One of the merits of this method in comparison with the numerical model is the capability of rapid calculation with the highest precision, which made it suitable for optimization problems.

Slotless double-sided outer armature permanent-magnet (PM) linear motors (SDOPMLs) have high efficiency and low detent force. Despite their simple control strategy and easy manufacturing process, finding an accurate model of these motors to calculate the machine quantities is challenging. It is particularly critical for obtaining the optimum design of these machines which may include too many iterations in a short time. To overcome this challenge, a 2-D analytical model based on the subdomain method is presented to determine the magnetic flux density components for the motor under the study. According to this analytical procedure, the motor cross-section is divided to 11 sub-regions, then the superposition theorem is utilized to analyze the flux density distribution in all sub-regions due to various magnetization patterns, (i.e., parallel, two-segment Halbach, ideal Halbach, and bar magnet in shifting directions) as well as armature reaction current, respectively. According to the calculated magnetic flux density components, machine quantities like flux linkage, induced voltage, inductances, and electromagnetic force components are explained. Also, the obtained analytical results are compared with those of the finite-element method (FEM) to confirm the accuracy of the proposed model. The proposed model can be used in the design and optimization stage of the linear slotless motor against the numerical model to save time. Finally, a comparative study between the performance of the single-sided and double-sided slotless PM linear motors in the same volume is implemented. This comparison shows the advantage of the double-sided motor in terms of the unbalanced magnetic force (UMF).

This paper estimates the magnetic flux density components in the slotless single-sided axial flux permanent-magnet synchronous machines (SAFPMSMs). For this purpose, a 2-D analytical model based on the sub-domain method is utilized in which the cross-section of the presented machine is divided into the seven sub-regions such as stator side exterior, stator, winding, air-gap, permanent-magnets (PMs), mover and mover side exterior. Based on the Maxwell equations, the related partial differential equations (PDEs) of magnetic flux density components are formed in each sub-region which are identified as the essential step for obtaining the machines quantities. According to the superposition theorem, two separate steps are implemented for calculating the magnetic flux density components. In the first step, open circuit analysis includes various type of magnetization patterns, i.e. parallel, ideal Halbach, 2-segment Halbach and bar magnet in shifting direction is investigated and armature currents are zero and in the second step PMs are inactive and the magnetic flux density components are originated due to only armature reaction. Eventually, 2-D finite element method (FEM) is determined to confirm the accuracy of the presented analytical approach and an acceptable agreement between the analytical and FEM models can be observed.

Analytical approaches, if possible, are suggested for saving the simulation time in the design stage of the electrical machines. This benefit is highlighted when the optimization issues including too many iterations are desired. Hence, this paper presents a 2-D analytical model for magnetic field distribution based on the sub-domain method in a slotless double-sided axial flux permanent-magnet (PM) brushless machines (AFPMBMs) with internal-rotor-external-stators. According to this method, the machine cross-section is divided into the appropriate number of sub-regions and the related partial differential equations (PDEs) extracted from Maxwell equations are formed for magnetic vector potential in each sub-region. Applying curl on the obtained results leads to calculating the magnetic flux density components in each sub-region. Based on the superposition theorem, the analytical procedure is utilized in the two separate steps where in the first step the magnetic flux is originated by only PMs with various magnetization patterns (i.e., parallel, ideal Halbach, 2-segment Halbach and bar magnet in shifting magnetization patterns) and the armature currents are zero. In the second step, all PMs are inactivated and only armature currents affect the magnetic flux distribution. Finally, the obtained analytical results are compared with those of the Finite element method (FEM) to confirm the accuracy of the proposed analytical model. The extracted results reveal the benefit of the analytical model for replacing instead of the FEM to predict the magnetic flux density component in the presented AFPMBMs in a shorter time.