Complex Relative Air-gap Permeance Research Articles

Semi-Analytical Calculation of No-Load Radial and Tangential Electromagnetic Force Waves of a Non-Salient Pole Synchronous Generator

For a marine electrical machine, its high power density and large electromagnetic (EM) load tend to bring high saturation to the magnetic circuit. When the non-linearity of ferromagnetic material is not taken into consideration, large errors will appear in the calculation of air-gap flux density and EM force wave. Considering the radial and tangential EM force waves that cause the stator core to vibrate and deform are the primary sources of EM vibration of a non-salient pole synchronous generator (NSPSG), this paper introduces a semi-analytical method of calculating the radial and tangential EM force waves, with the non-linearity of ferromagnetic material taken into account. The method includes the distributed magnetic circuit method (DMCM), the separation variable method (SVM) and the complex variable conformal transformation method (CVCTM). First of all, the radial air-gap flux density is obtained by using the DMCM to calculate the no-load magnetic circuit of the NSPSG and dealing with the non-linearity of ferromagnetic material by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B-H</i> curve. Then, the tangential air-gap flux density is calculated by the SVM and the complex relative air-gap permeance is calculated by the CVCTM to determine the effect of slotting. Finally, the radial and tangential EM force waves are obtained by Maxwell stress tensor method. The proposed semi-analytical method is verified through measuring vibration acceleration of the NSPSG.

Analytical Model of Open-Circuit Air-Gap Field Distribution in Interior Permanent Magnet Machines Based on Magnetic Equivalent Circuit Method and Boundary Conditions of Macroscopic Equations

To accurately and quickly predict the open-circuit air-gap magnetic field in interior permanent magnet synchronous machines (IPMSMs), an analytical model based on magnetic equivalent circuit (MEC) method is proposed. The analytical model can provide radial and tangential components of open-circuit magnetic field in the slotless air gap. The radial component is obtained from the MEC model, and the tangential component is obtained from boundary conditions of macroscopic equations. Then, a complex relative air-gap permeance is applied to take the effect of slots into account. As a result, an accurate solution of both radial and tangential components of the flux density in the slotted air gap is obtained. Additionally, the no-load back electromotive force (EMF) and cogging torque are calculated based on the open-circuit air-gap magnetic field. All the analytical results are verified by the finite element (FE) analysis and they are well matched. In the end, a direct measurement experiment of open-circuit air-gap magnetic field is proposed to verify the validity of both radial and tangential open-circuit air-gap magnetic fields.

Analytical Method for Calculating the Magnetic Field of Spoke-Type Permanent Magnet Machines Accounting for Eccentric Magnetic Pole

This article proposes an analytical method to calculate the magnetic field, back electromotive force (BEMF), inductance, and torque of spoke-type permanent magnet machine (PMM) accounting for eccentric magnetic pole. The magnetic field in the slotted spoke-type PMM is predicted according to conformal mapping, flux continuity theorem, double reaction theory, superposing principle, and complex relative air-gap permeance. Then, BEMF, inductance, and torque are obtained according to the results of magnetic field. A spoke-type PMM with eccentric magnetic pole is used to verify the analytical model. The harmonics of BEMF and torque ripple of spoke-type PMM with eccentric magnetic pole are very slight. Results of magnetic field, BEMF, inductance, and torque by the analytical model are verified by finite element. The validation to calculate BEMF and electromagnetic torque with analytical model is confirmed by experimental result. The analytical method proposed in this article can be applied to the spoke-type PMM with all kinds of magnetic-pole shape.

A novel analytical calculation of magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping

This paper presented a novel analytical method for calculating magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping. Firstly, flux density without slots and complex relative air-gap permeance of slotted air gap are derived from conformal transformation separately. Secondly, they are combined in order to obtain normalized flux density taking account into the slots effect. The finite element (FE) results confirmed the validity of the analytical method for predicting magnetic field and back electromotive force (BEMF) in the slotted air gap of spoke-type permanent-magnet machines. In comparison with FE result, the analytical solution yields higher peak value of cogging torque.

Analytical Model of Permanent Magnet Linear Synchronous Machines Considering End Effect and Slotting Effect

An effective analytical model of permanent magnet linear synchronous machine (PMLSM) is proposed in this paper, which considers the end effect by adding virtual slots and the slotting effect by adopting relative air-gap permeance. To employ this model, the open-circuit air-gap flux density of a slotless PMLSM with virtual slots is firstly predicted by Subdomain (SD) method, and then the complex relative air-gap permeance is introduced based on conformal mapping. By multiplying these two quantities, the open-circuit air-gap flux density of actual slotted PMLSM can be obtained. The phase flux linkage, back-EMF and detent force can be predicted accordingly. Moreover, the detent force is more easily depressed by optimizing the end teeth and slots via the proposed method. The predicted results are validated by 2-D nonlinear finite-element (FE) and prototype measurements.

Magnetic Field Analysis of Notched SPMSM Using Conformal Mapping

We present an analytical method for applying conformal mapping to the notched geometry of a surface permanent magnet synchronous motor to calculate the cogging torque. First, the complex relative air-gap permeance (CRAP) of the notched air-gap is calculated by multiplying the CRAP of a slotted air-gap with the notch sensitivity coefficient, which is defined by the relation of slot and notch shapes. Secondly, the no-load air-gap flux density (AFD) is derived from the CRAPs of the notched and slotted air-gaps. Finally, the cogging torque of the notched motor is calculated by the no-load AFD and is verified using the 2D finite element analysis (FEA) result. A 4-pole and 12-slot surface permanent magnet synchronous motor with the notch is used herein.

Analytical Method for Overhang Effect of Surface-Mounted Permanent-Magnet Motor Using Conformal Mapping

In this paper, an analytical method for predicting the rotor overhang (OH) effect is presented using two types of conformal mappings (CMs), which are the ${r}$ – ${\theta }$ and zr plane CMs. First, the no-load air-gap flux density (AFD) of a slotted air gap without OH is determined using the ${r} {\theta }$ plane CM. Second, the complex relative air-gap permeance (CRAP) in the zr plane is obtained by the zr plane CM. This helps in taking the OH effect into account. Finally, the no-load AFD of the slotted and OH air gaps in cylindrical coordinates is derived from the no-load AFD in the ${r} {\theta }$ plane with CRAP along the ${z}$ -axis. The accuracy of the presented method is verified by comparing the results of no-load analysis with those of 3-D finite-element analysis.

Improved method for field analysis of surface permanent magnet machines using Schwarz–Christoffel transformation

Air gap field solution for an outer rotor permanent magnet (PM) machine is derived based on numerical Schwarz–Christoffel (SC) transformation of a single slot geometry and analytical solution of a point wire field in the rectangular canonical domain. This approach takes into account geometrical distortions of PM edges and field evaluation points located along the air gap centre, which occur due to conformal mapping of the slotted air gap into a slotless domain and impair the accuracy of the existing analytical field solutions using SC transformation and complex relative air gap permeance. The increased accuracy of the improved solution is confirmed by comparing the air gap flux density waveforms produced by the PMs and armature winding, cogging torque, and total torque with the results of time-stepping transient finite element simulation.

Analytical Calculation of Magnetic Field Distribution and Stator Iron Losses for Surface-Mounted Permanent Magnet Synchronous Machines

Permanent-magnet synchronous machines (PMSMs) are widely used in electric vehicles owing to many advantages, such as high power density, high efficiency, etc. Iron losses can account for a significant component of the total loss in permanent-magnet (PM) machines. Consequently, these losses should be carefully considered during the PMSM design. In this paper, an analytical calculation method has been proposed to predict the magnetic field distribution and stator iron losses in the surface-mounted permanent magnet (SPM) synchronous machines. The method introduces the notion of complex relative air-gap permeance to take into account the effect of slotting. The imaginary part of the relative air-gap permeance is neglected to simplify the calculation of the magnetic field distribution in the slotted air gap for the surface-mounted permanent-magnet (SPM) machine. Based on the armature reaction magnetic field analysis, the stator iron losses can be estimated by the modified Steinmetz equation. The stator iron losses under load conditions are calculated according to the varying d-q-axis currents of different control methods. In order to verify the analysis method, finite element simulation results are compared with analytical calculations. The comparisons show good performance of the proposed analytical method.

Magnetic Field Analysis in Eccentric Surface-Mounted Permanent-Magnet Motors Using an Improved Conformal Mapping Method

This paper presents an improved conformal mapping (ICM) method for magnetic field analysis in one typical surface-mounted permanent-magnet (SMPM) motor considering the static, dynamic, and mixed rotor eccentricities. The ICM method accurately accounts for the slotting effect, the winding distribution, the armature reaction, the magnetic saturation effect, and the working point variation of PMs throughout their volume. The ICM method is also very rigorous to separate the onload air gap field components. First, the slotless model of an eccentric SMPM motor is mapped to one concentric model using a new form of bilinear conformal mapping, and the field solution is then obtained through applying Hague's solution in the main canonical domain. Second, the slotted air gap magnetic field is obtained through the modulation of air gap magnetic field in the slotless physical domain using the complex relative air gap permeance. The ICM method is also used to investigate the influence of rotor eccentricity on the flux linkage of the stator coils. This developed model is validated through comparing with the corresponding results obtained through finite element method and the frozen permeability method. This method can also be used as a tool for design and optimization of electrical machines.

Analytical Calculation of No-load Magnetic Field Distribution in the Slotted Airgap of a Permanent Magnet Synchronous Motor

An analytical method for calculation of no-load magnetic field distribution in the slotted airgap of a surface permanent magnet synchronous motor has been presented. The method involves the solution of the governing field equations in polar coordinates in the annular airgap region of a slotless motor in which the magnets are assumed to have uniform radial magnetization and constant relative recoil permeability. Then it introduces the notion of complex relative airgap permeance calculated from the conformal transformation to account the effect of slotting. As a result, an accurate solution of both radial and tangential components of the flux density can be obtained.

Magnetic Field Analysis of External Rotor Permanent-Magnet Synchronous Motors Using Conformal Mapping

This paper presents analytical and numerical conformal mapping (CM) to analyze magnetic fields originating from permanent magnets and armature winding currents in a slotted air-gap of a surface mounted radial permanent-magnet synchronous motor (SPM), taking into account the effect of arbitrarily curved motor surfaces. We also studied the slotless configuration of the external rotor permanent-magnet motor for the purpose of calculating the complex relative air-gap permeance. The model is able to predict the air gap field and torque with high accuracy, which cannot be achieved using the previously available analytical methods in both cases: internal and external rotor permanent-magnet motor. We obtained one of the conformal mappings used in this study, the numerical solution of the Schwarz-Christoffel (SC) integral, by using the Matlab SC Toolbox. We used this field solution to calculate cogging torque and electromagnetic torque by integrating the Maxwell stress tensor inside the air gap. We verified the accuracy of the developed method by comparing its results with those obtained from the developed numerical finite-element (FE) model.

Analytical Model of Slotted Air-Gap Surface Mounted Permanent-Magnet Synchronous Motor With Magnet Bars Magnetized in the Shifting Direction

An analytical model is presented, which uses two-dimensional field theory in polar coordinates to determine the flux density distribution, cogging torque, back EMF and electromagnetic torque in the slotted air gap of permanent-magnet motors with surface mounted magnet bars which are magnetized in shifting direction. The magnet arc to pole pitch ratio in the motor is not necessarily equal to unity like in the case of Halbach array magnetization. The effect of stator slots is introduced by modulating the magnetic field distribution in the slotless stator by the complex relative air-gap permeance. With this complex permeance, the radial and tangential components of flux density are calculated. In the analytical and numerical study a finite number of magnet bars, which is considered sufficient to get a sinusoidal magnetization, is used. The influence of the number of magnet bars on magnetization is also investigated. The accuracy of the developed model is verified by comparing its results with those obtained from experimental measurement and previously validated linear and nonlinear numerical finite element code.

Analytical and numerical calculation of magnetic field distribution in the slotted air-gap of tangential surface permanent-magnet motors

This paper deals with the analytical and numerical analysis of the flux density distribution in the slotted air gap of permanent magnet motors with surface mounted tangentially magnetized permanent magnets. Two methods for magnetostatic field calculations are developed. The first one is an analytical method in which the effect of stator slots is taken into account by modulating the magnetic field distribution by the complex relative air gap permeance. The second one is a numerical method using 2-D finite element analysis with consideration of Dirichlet and anti-periodicity (periodicity) boundary conditions and Lagrange Multipliers for simulation of movement. The results obtained by the analytical method are compared to the results of finite-element analysis.

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

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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. </para>

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

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.