We present an analytical method for calculation of no-load magnetic field distribution in the slotted air gap of a surface permanent-magnet (PM) motor with radial or parallel magnetization. The method introduces the notion of complex relative air-gap permeance, calculated from the conformal transformation of the slot geometry, to take into account the effect of slotting. As a result, an accurate solution of both radial and tangential components of the flux density can be obtained which shows excellent agreement with the results of finite-element simulations. As an example of the effectiveness of the model, we present calculations of the back electromotive force and the cogging torque waveforms in a surface PM motor.

Analytical calculation of magnetic field distribution in the slotted air gap of a surface permanent-magnet motor using complex relative air-gap permeance

The paper presents an accurate analytical subdomain model for computation of the open-circuit magnetic field in surface-mounted permanent-magnet machines with any pole and slot combinations, including fractional slot machines, accounting for stator slotting effect. It is derived by solving the field governing equations in each simple and regular subdomain, i.e., magnet, air-gap and stator slots, and applying the boundary conditions to the interfaces between these subdomains. The model accurately accounts for the influence of interaction between slots, radial/parallel magnetization, internal/external rotor topologies, relative recoil permeability of magnets, and odd/even periodic boundary conditions. The back-electromotive force, electromagnetic torque, cogging torque, and unbalanced magnetic force are obtained based on the field model. The relationship between this accurate subdomain model and the conventional subdomain model, which is based on the simplified one slot per pole machine model, is also discussed. The investigation shows that the proposed accurate subdomain model has better accuracy than the subdomain model based on one slot/pole machine model. The finite element and experimental results validate the analytical prediction.

An Accurate Subdomain Model for Magnetic Field Computation in Slotted Surface-Mounted Permanent-Magnet Machines

Cogging torque in permanent-magnet machines causes torque and speed ripples, as well as acoustic noise and vibration, especially in low speed and direct drive applications. In this paper, a general analytical expression for cogging torque is derived by the energy method and the Fourier series analysis, based on the air gap permeance and the flux density distribution in an equivalent slotless machine. The optimal design parameters, such as slot number and pole number combination, skewing, pole-arc to pole-pitch ratio, and slot opening, are derived analytically to minimize the cogging torque. Finally, the finite-element analysis is adopted to verify the correctness of analytical methods.

Analytical Methods for Minimizing Cogging Torque in Permanent-Magnet Machines

This paper proposes a coaxial magnetic gear that offers higher torque density, lower cogging torque, and lower iron losses than its counterparts. The key is to newly employ a Halbach permanent-magnet (PM) array to constitute the PM poles in the inner rotor and a partial Halbach array (two segments per pole) for the outer rotor. The corresponding magnetic field distributions, torque transmission, and torque ripples are analytically discussed. Then, the corresponding performances are quantitatively assessed by using the finite-element method.

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A Coaxial Magnetic Gear With Halbach Permanent-Magnet Arrays

We present an analytical method for the calculation of cogging torque in surface permanent-magnet (PM) motors. The cogging torque is calculated by integrating the Maxwell stress tensor inside the air gap. The principle of complex relative air-gap permeance derived from conformal transformation of the slot geometry is used to take into account the effect of slotting and to calculate the radial and tangential components of the air-gap flux density required for integration of the tangential component of the Maxwell stress tensor. We implemented the proposed analytical solution on a 7-kW four-pole surface PM motor and compared the results with finite-element solutions. We present an example of finding the optimal magnet angular span to yield minimum cogging torque as an example of the effectiveness of the method.

Analytical Solution for Cogging Torque in Surface Permanent-Magnet Motors Using Conformal Mapping

This paper presents an analytical method of modeling permanent magnet (PM) motors. The model is dependent only on geometrical and materials data which makes it suitable for insertion into design programs, avoiding long finite element analysis (FEA) calculations. The modeling procedure is based on the calculation of the air gap field density waveform at every time instant. The waveform is the solution of the Laplacian/quasi-Poissonian field equations in polar coordinates in the air gap and takes into account slotting. The model allows the rated performance calculation but also such effects as cogging torque, ripple torque, back-EMF form prediction, some of which are neglected in commonly used analytical models.

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Analytical model for permanent magnet motors with surface mounted magnets

This paper presents an analytical method of modeling permanent magnet motors. The model is dependent only on geometrical and materials data making it suitable for insertion into design programs, avoiding long finite element analysis. Modeling is based on the calculation of the air gap field density waveform at every time instant. The model allows the rated performance calculation but also such effects as cogging torque, ripple torque, back-EMF form prediction, some of which are neglected in commonly used analytical models.

A. El-Antably

Papers

Analytical model for permanent magnet motors with surface mounted magnets

Analytical model for permanent magnet motors with surface mounted magnets