Analytical Modeling of Magnetic Field Distribution at No Load for Surface Mounted Permanent Magnet Machines
Abstract
:1. Introduction
2. Magnetic Field between Smooth Ferromagnetic Surfaces
2.1. Linear Conductor with Infinitesimal Section
- i
- Stator and moving surfaces are supposed to be smooth and separated by an air gap of uniform width.
- ii
- Magnetic field is just 2D and lies in the paper plane.
- iii
- The machine is supposed to be so long into the paper plane direction, so end effects are negligible.
- iv
- Relative recoil permeability is considered to be equal to the vacuum permeability ().
- v
- Iron permeability is assumed to be infinite so the method of superimposition can be used.
- vi
- The iron parts are perfectly laminated thus no eddy current can be induced in the iron by the varying magnetic field.
- vii
- The conductors placed inside the air gap extend indefinitely perpendicular to the paper plane.
- viii
- Skin effect due to alternating currents flowing in the conductors is neglected.
2.2. Permanent Magnet Current Sheet Model
2.3. Magnetic Field due to a Sequence of Permanent Magnets
3. Magnetic Field in Case of Slotted Ferromagnetic Surfaces
3.1. Slot Opening Model
3.2. Magnetic Field due to a Sequence of Permanent Magnets for Slotted Surfaces
4. Complex Integral Quantities
4.1. Complex Vectors and Mechanical Actions
4.2. Maxwell Stress Complex Tensor and Forces
- evaluation of the forces applied to each tooth, typically considered in NVH calculations;
- evaluation of the forces applied to the global system.
- FEM and analytical waveforms are in fair agreement;
- the normal component amplitude is predominant with respect to the tangential one;
- the force waveform acting on a generic tooth tip has a frequency that is double the fundamental frequency ().
- on this peripheral extension, corresponding to the space period, FEM calculation can be also performed by using the Virtual Work Principle;
- the analytically calculated normal component shows a better agreement with respect to the FEM result;
- a wider difference appears for the tangential component: the reason for that could be a massive difference in the order of magnitude of these two components, making the numerical noise error predominant in the analytical calculation of the minor quantity; moreover, the tangent component calculation is affected by local inaccuracies due to fringing effect as shown in Figure 11b.
4.3. Computational Efficiency
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
PM peripheral size, p.u. ratio . | |
tooth tip peripheral size. | |
slot, slot opening peripheral size. | |
conductor, vertical current sheet, SPM double current sheet flux density complex vector. | |
smooth (slotless), slotted flux density complex vector. | |
PM residual flux density, linearly extrapolated coercivity, relative recoil permeability. | |
ideal flux density flat profile reference value. | |
force, force per unit length, Maxwell Tensor complex vector. | |
magnetic field strength complex vector. | |
PM, mechanical air-gap, equivalent air-gap height. | |
current sheet length. | |
tooth tip, slot, slot opening height. | |
conductor current, linear current sheet, vector current density. | |
Carter’s factor. | |
ℓ | lamination stack length. |
total slot m.m.f. | |
N° of time samples, spatial samples, slotting function harmonics. | |
N° of stator teeth, permanent magnets. | |
N° of stator teeth counted from left to right. | |
U | ideal magnetic voltage drop. |
parameter time, electrical period. | |
generic vector tangential and normal component. | |
exploration coordinate inside the air-gap, middle air-gap y coordinate . | |
generic point p of complex coordinates where the conductor is positioned and its conjugate. | |
generic point of coordinate inside the air-gap complex plane and its conjugate. | |
slotting, permeance complex function. | |
generic exploration segment at , middle mechanical air-gap segment at inside the air-gap. | |
p.u. ratios and . | |
PM pitch, tooth pitch. |
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Parameters | |
---|---|
residual flux density, | 1.1 T |
N° of teeth, ; N° of PM, | 9; 8 |
PM pitch, ; p.u. ratio, | 113.39 mm; 0.9 |
PM height, ; air-gap height, | 10 mm; 1.5 mm |
slot height, ; slot width, | 20.5 mm; 44.3 mm |
tooth tip height, ; slot opening width, | 2 mm; 6 mm |
pole shoe height, ; pole shoe width, | 12 mm; 94.79 mm |
stack length, ℓ | 61.25 mm |
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Di Gerlando, A.; Ricca, C. Analytical Modeling of Magnetic Field Distribution at No Load for Surface Mounted Permanent Magnet Machines. Energies 2023, 16, 3197. https://doi.org/10.3390/en16073197
Di Gerlando A, Ricca C. Analytical Modeling of Magnetic Field Distribution at No Load for Surface Mounted Permanent Magnet Machines. Energies. 2023; 16(7):3197. https://doi.org/10.3390/en16073197
Chicago/Turabian StyleDi Gerlando, Antonino, and Claudio Ricca. 2023. "Analytical Modeling of Magnetic Field Distribution at No Load for Surface Mounted Permanent Magnet Machines" Energies 16, no. 7: 3197. https://doi.org/10.3390/en16073197