Analytical Prediction of Cogging Torque in Surface-Mounted Permanent-Magnet Motors

We present a new approach to computing cogging torque based on only one analytical field solution in surface-mounted permanent-magnet (PM) motors. We use conformal transformation to compute the magnetic field created by the permanent magnets in the air gap with a single slot. Then, we derive the cogging torque due to the single-slot effect at different rotor positions directly from the 2-D analytical field solution. The total cogging torque is synthesized from the contribution of each slot. We have applied our approach to predicting the cogging torque of two surface-mounted PM motors, one with 4 poles, 24 slots and the other with 42 poles, 36 slots. The predicted cogging torques for both applications agree well with those obtained from 2-D finite-element analysis.      

Analytical Methods for Minimizing Cogging Torque in 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 Solution of Air-Gap Field in Permanent-Magnet Motors Taking Into Account the Effect of Pole Transition Over Slots

We present an analytical method to study magnetic fields in permanent-magnet brushless motors, taking into consideration the effect of stator slotting. Our attention concentrates particularly on the instantaneous field distribution in the slot regions where the magnet pole transition passes over the slot opening. The accuracy in the flux density vector distribution in such regions plays a critical role in the prediction of the magnetic forces, i.e., the cogging torque and unbalanced magnetic pull. However, the currently available analytical solutions for calculating air-gap fields in permanent magnet motors can estimate only the distribution of the flux density component in the radial direction. Magnetic field and forces computed by the new analytical method agree well with those obtained by the finite-element method. The analytical method provides a useful tool for design and optimization of permanent-magnet motors.     

Iron Loss Model for Permanent-Magnet Synchronous Motors

Iron losses in permanent-magnet synchronous machines form a larger portion of the total losses than in induction machines and, hence, more importance should be given to the iron losses. Previously, models have been presented for the calculations of these losses, but these models still rely on finite-element simulations to obtain correction factors, which are substantial, to apply to the theoretically derived formulas in order to obtain good agreement with the experimental data. This paper points out the source of this correction factor: the neglect of the excess eddy-current loss component. In many cases, this loss component dominates the total iron losses and needs to be incorporated in the theoretical considerations. The paper also provides a more complete model of iron loss, which greatly reduces the need for calculating the correction factors using the finite-element method (FEM). This more complete model reduces design time, especially when a number of candidate designs need to be analyzed. Otherwise, the calculation of the correction factors using FEM would be cumbersome, as the correction factors tend to be nonlinear.     

In-Depth Study of the Torque Constant for Permanent-Magnet Machines

The torque constant, together with the back-EMF constant, was originally used in permanent-magnet dc commutator (PMDC) motors to couple electric circuit equations with mechanical equations. It is still an open question how these two constants should be defined in brushless dc (BLDC) motors and permanent-magnet (PM) ac motors. We present an in-depth study of these two constants, and conclude that the definition based on the $dq$ axes is applicable for the dynamic motion control of not only PM ac motors, but also BLDC motors.      

Analytical Solution for Rotor Eddy-Current Losses in a Slotless Permanent-Magnet Motor: The Case of Current Sheet Excitation

We present an analytical solution for the magnetic field, induced eddy currents, and the corresponding losses generated in the rotor of a slotless permanent-magnet (PM) motor. The field excitation is a current sheet placed at the stator interior surface. The solution is based on an analytical solution of the diffusion equation using double Fourier series and in the complex domain. The first and the second Fourier series correspond to the time and space harmonics, respectively. We have analyzed the effect of each time harmonic in detail, and verified the results with FEM software.     

Minimizing Thrust Fluctuation in Moving-Magnet Permanent-Magnet Brushless Linear DC Motors

We present the results of our research on the factors that cause thrust fluctuation in moving-magnet-type permanent-magnet brushless DC linear motors (PMBLDCLM). We combined Fourier transforms and finite-element models to obtain the power spectra of three components of the detent force. We developed a method of optimizing magnet width to minimize the detent force on the basis of harmonic analysis. To verify this method, we designed several motor models with different magnet widths and analyzed them by finite-element methods. The calculations and experimental results prove that thrust fluctuation of the motor can be effectively reduced with our method     

An Improved Magnetic Equivalent Circuit Model for Iron-Core Linear Permanent-Magnet Synchronous Motors

The aim of this work is to establish an accurate yet simple method for predicting flux density distribution and iron losses in linear permanent-magnet synchronous motors (LPMSMs) for iterative design procedures. For this purpose, an improved magnetic equivalent circuit for calculation of the teeth and yoke flux densities in the LPMSMs is presented. The magnetic saturation of iron core is considered by nonlinear elements and an iterative procedure is used to update these elements. The armature reaction is also taken into account in the modeling by flux sources located on the teeth of motors. These sources are time dependent and can model every winding configuration. The relative motion between the motor primary and secondary is considered by wisely designing air gap elements simplifying the permeance network construction and preventing permeance matrix distortion during primary motion. Flux densities in different load conditions are calculated by means of the proposed model. The effects of saturation and armature reaction on the flux density distribution are shown in detail. Using these flux densities, iron losses in the motor are examined and its variations versus motor parameters are then studied. All results obtained by proposed model are verified by finite-element method based on an extensive analysis.     

Magnetic Field Analysis of Inset and Surface-Mounted Permanent-Magnet Synchronous Motors Using Schwarz–Christoffel Transformation

In this paper, we apply an original numerical Schwarz–Christoffel (SC) transformation to analyze magnetic field originating from permanent magnets and the armature winding currents in a slotted air gap of an inset permanent-magnet synchronous motor. We obtained the solution of the SC integral numerically using Matlab SC Toolbox. We used this field solution to calculate both cogging torque and electromagnetic torque by integrating the Maxwell stress tensor inside the air gap. The case without inter-polar piece, which is equivalent to a surface-mounted permanent-magnet motor, is also treated. The accuracy of the developed method is verified by comparing its results with those obtained from the developed numerical finite-element models.      

Detection of Demagnetization Faults in Permanent-Magnet Synchronous Motors Under Nonstationary Conditions

This paper presents the results of our study of the permanent-magnet synchronous motor (PMSM) running under demagnetization. We examined the effect of demagnetization on the current spectrum of PMSMs with the aim of developing an effective condition-monitoring scheme. Harmonics of the stator currents induced by the fault conditions are examined. Simulation by means of a two-dimensional finite-element analysis (FEA) software package and experimental results are presented to substantiate the successful application of the proposed method over a wide range of motor operation conditions. Methods based on continuous wavelet transform (CWT) and discrete wavelet transform (DWT) have been successfully applied to detect and to discriminate demagnetization faults in PMSM motors under nonstationary conditions. Additionally, a reduced set of easy-to-compute discriminating features for both CWT and DWT methods has been defined. We have shown the effectiveness of the proposed method by means of experimental results.      

Design Consideration to Reduce Cogging Torque in Axial Flux Permanent-Magnet Machines

In designing new topologies for permanent-magnet machines based on rare earth magnets, it is necessary to diminish the undesired cogging torque. This paper presents a 3-D finite-element analysis to evaluate the effect of magnet shape and stator displacement on cogging torque reduction, for axial flux machines. It analyzes the final electromagnetic torque for the proposed configurations. Finally, it presents the resultant cogging torque waveform for a 5.0 kW prototype, based on our optimization techniques.     

Analytical Modeling of a Permanent-Magnet Synchronous Machine in a Flywheel

We develop an analytical model for a radial-flux external-rotor permanent-magnet synchronous machine (PMSM) without slots in the stator iron and with a shielding cylinder. The machine is part of an energy storage flywheel, to be used as the peak-power unit in a hybrid electric passenger bus. To reduce the induced no-load losses due to the high rotational speed of the flywheel, the slots in the stator are made not of iron but of a nonmagnetic plastic material. This results in an air gap winding with a stator yoke consisting of stacked circular laminations. The analytical model includes the effect of the winding distribution on the field, the fact that it is in the air gap, and the effect of the eddy-current reaction field of the shielding cylinder. The two-dimensional magnetic field is solved in six defined machine layers and useful machine quantities are derived directly from it, leading to the machine voltage equation. We built a prototype flywheel machine. The locked-rotor machine resistance and inductance predicted by the analytical model was compared with the experimentally determined values. The values showed good agreement, thereby validating the analytical model of the machine.      

Programmable Design of Magnet Shape for Permanent-Magnet Synchronous Motors With Sinusoidal Back EMF Waveforms

The characteristics of a permanent-magnet synchronous motor (PMSM) are influenced greatly by the back-electromotive force waveforms in the motor, which are directly related to the magnet shape. We assume that the outer surface of a magnet in a PMSM can be modeled as one part of a cylinder. Using an ANSOFT model, we optimized the radius of the magnet with respect to number of poles, rotor size, and magnet thickness when the total harmonic distortion is below 1%. We then set up a lookup table using the optimization results so that the desired outer radius of a magnet of any new PMSM motor could be derived, subject to the parameter limits. We applied this method to some PMSM prototypes, and confirmed that the calculation results are substantially accurate.      

A New Analytical Model for Switched Reluctance Motors

We present a new analytical model for switched reluctance motors (SRMs). The model uses four coefficients, which are rotor position dependent and calculated from the flux-current-angle data using numerical curve fitting with least squares method. The four coefficients are further represented with sixth degree polynomials, which can be calculated effectively with Qin Jiushao's method. Simulated results show that the magnetic characteristics calculated with our model agree well with the original flux-current-angle data. The instantaneous electromagnetic torque derived from the developed model matches well with the numerically calculated torque. Finally, we compare dynamic simulations based on the model with experimental results and find good agreement.      

Analytical Calculation of No-Load Voltage Waveforms in Machines Based on Permanent-Magnet Volume Integration

We develop an analytical expression for predicting electromotive force (EMF) waveforms resulting from permanent magnets (PMs) in electrical machines. The expressions for the flux linkage are based on a volume integral over the magnet volume, rather than the usual surface integral over the coil. The proposed method consists of applying a virtual current in the coil of the machine and calculating the magnetic field generated inside the PM volume. The EMF waveform is obtained by taking the derivative of the flux linkage with respect to time. We present analytical expressions of the EMF for various PM shapes and Halbach magnetization patterns. We tested a total of four configurations of PMs, and the experimental waveforms confirmed the validity of the expressions obtained theoretically.     

Using Modular Poles for Shape Optimization of Flux Density Distribution in Permanent-Magnet Machines

A modular configuration for a type of permanent-magnet pole is proposed for use in permanent-magnet (PM) machines. The pole consists of three or more permanent-magnet pieces. The main objective of this configuration is to shape the air-gap flux density distribution produced by the pole. An optimization procedure based on an analytical model is then used to determine the optimal pole specifications. A finite-element method is finally carried out to evaluate the proposed configuration. Extensive investigations on a linear permanent-magnet motor demonstrate that the proposed configuration reduces flux density as well as back electromotive force harmonics. This pole configuration also results in more efficient use of PM materials.      

Analysis of Relationship Between Abnormal Current and Position Detection Error in Sensorless Controller for Interior Permanent-Magnet Brushless DC Motors

We have analyzed the characteristics of terminal voltages used to detect rotor position in interior permanent-magnet (IPM) brushless DC (BLDC) motors in sensorless control by employing a two-phase conduction method. We found that a detected zero crossing point (ZCP) of the open phase voltage advances the real rotor position due to additional voltage components caused by variation of inductance with rotor position. We conclude that the amount of position error is related to rotor saliency, load, and motor speed. We also present the relationship between abnormal currents and amount of position errors in the sensorless controller by using simulations that take into account additional voltages in our model of terminal voltages and motor neutral voltage as well as experiments with a position sensor. We verified the validity of our analysis on IPM motors having various motor parameters under sensorless control.      

Analytical Force Determination in an Electromagnetic Actuator

We present an original analytical determination of the force acting on the moving coil in an electromagnetic actuator. The actuator magnetic field is solved analytically by Schwarz-Christoffel (SC) mapping. The SC integral is solved analytically using the elliptic functions. We verified the results by finite-element methods.      

Analytical Modeling of Open-Circuit Air-Gap Field Distributions in Multisegment and Multilayer Interior Permanent-Magnet Machines

We present a simple lumped magnetic circuit model for interior permanent-magnet (IPM) machines with multisegment and multilayer permanent magnets. We derived analytically the open-circuit air-gap field distribution, average air-gap flux density, and leakage fluxes. To verify the developed models and analytical method, we adopted finite-element analysis (FEA). We show that for prototype machines, the errors between the FEA and analytically predicted results are $≪$1% for multisegment IPM machines and $≪$ 2% for multilayer IPM machines. By utilizing the developed lumped magnetic circuit models, the IPM machines can be optimized for maximum fundamental and minimum total harmonic distortion of the air-gap flux density distribution.