Generalization of Maxwell Stress Tensor Method for Magnetically Anisotropic Regions

Spałek, D.

doi:10.1109/tmag.2019.2937283

Cited by 5 publications

(12 citation statements)

References 8 publications

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“…both Lorentz and material forces. Such the situation appears when conducting and magnetic ball levitates in an alternating magnetic field [2–4, 16, 21 ]. In such a case, the total force could be either positive (the ball could levitate) or negative or zero depending on the conductivity and magnetic permeabilities of the ball.…”

Section: Stability Free and Damped Oscillationsmentioning

confidence: 99%

“…The levitation force is evaluated using the four following methods: a) Maxwell stress tensor [2, 3, 5, 14, 16 ]

{bold-italicσ}_{z} = - E_{z} D + {bold-italici}_{z} \frac{1}{2} bold-italicE bold-italicD,

method leads to the levitation force

F_{z} = F_{Mz} + normalΔ F,

where integral over the sphere surrounding the ball equals to

F_{M z} = 2 π R^{2} \int_{0}^{π} \{\frac{D_{z} D_{r}}{ε_{0}} - ()(, \frac{D_{r}^{2}}{2 ε_{0}} + ε_{0} \frac{D_{θ}^{2}}{2 ε_{θ θ}^{2}}) \cos θ\} \sin θ normald θ,

and Δ F is the volume integral for Maxwell stress tensor generalised method (Appendix 1). b) Co‐energy method [5, 14, 16 ] requires to calculate derivative of electric filed strength as follows:

F_{C z} = \frac{normal∂ W_{C}}{normal∂ z} = - \int_{V} D \frac{normal∂ E}{normal∂ z} normald V

hence

\begin{matrix} right leftthickmathspace.5em \\ F_{C z} & = π R^{2} \int_{0}^{π<...>} \end{matrix}

…”

Section: Electrostatic Levitationmentioning

confidence: 99%

“…It should be also pointed out that residual component Δ F (Maxwell stress tensor generalised method volume integral [2, 3 ]) vanishes for the isotropic ball (central abscissa point of each curve).…”

Section: Electrostatic Levitationmentioning

confidence: 99%

“…Let us consider a magnetically anisotropic and conducting ball in an axially‐symmetric stationary field (Fig. 5 ) in which the axial component (far from the ball along the z ‐axis) is given by polynomial of N items (extension of [2, 3 ]) as follows:

B_{z} = B_{0} + \sum_{k = 1}^{N - 1} G_{k} z^{k},

where B 0 and G k are given constants.…”

Section: Magnetostatic Levitationmentioning

confidence: 99%

“…Levitation driven by magnetic alternating field is often applied in many technological solutions and enables, e.g. spotless melting [1–4 ]. There is also a certain interest in electrostatic levitation forced by electric charge [5 ].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Electrostatic and magnetostatic levitations of ball – forces, stability and oscillations frequencies

Spałek

2020

IET Science, Measurement &Amp; Technology

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The study presents approaches to both electrostatic and magnetostatic levitations. For anisotropic balls, both dielectric and magnetic field distributions are derived in analytical forms by means of the variable separation method. Distributions are given by power functions and polynomials either Legendre's (for the electrostatic problem) or the associated polynomials (for the magnetostatic problem). The levitation forces are evaluated by means of four methods, i.e. Maxwell stress tensor generalised method, co‐energy method, material force density, and equivalent dipole analysis for both levitations. Levitation forces versus either electric permittivity or magnetic permeability are presented. The stability of the equilibrium points is investigated. The frequencies of free and damped oscillations are evaluated.

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Section: Stability Free and Damped Oscillationsmentioning

confidence: 99%

“…The levitation force is evaluated using the four following methods: a) Maxwell stress tensor [2, 3, 5, 14, 16 ]

{bold-italicσ}_{z} = - E_{z} D + {bold-italici}_{z} \frac{1}{2} bold-italicE bold-italicD,

method leads to the levitation force

F_{z} = F_{Mz} + normalΔ F,

where integral over the sphere surrounding the ball equals to

F_{M z} = 2 π R^{2} \int_{0}^{π} \{\frac{D_{z} D_{r}}{ε_{0}} - ()(, \frac{D_{r}^{2}}{2 ε_{0}} + ε_{0} \frac{D_{θ}^{2}}{2 ε_{θ θ}^{2}}) \cos θ\} \sin θ normald θ,

and Δ F is the volume integral for Maxwell stress tensor generalised method (Appendix 1). b) Co‐energy method [5, 14, 16 ] requires to calculate derivative of electric filed strength as follows:

F_{C z} = \frac{normal∂ W_{C}}{normal∂ z} = - \int_{V} D \frac{normal∂ E}{normal∂ z} normald V

hence

\begin{matrix} right leftthickmathspace.5em \\ F_{C z} & = π R^{2} \int_{0}^{π<...>} \end{matrix}

…”

Section: Electrostatic Levitationmentioning

confidence: 99%

Section: Electrostatic Levitationmentioning

confidence: 99%

B_{z} = B_{0} + \sum_{k = 1}^{N - 1} G_{k} z^{k},

where B 0 and G k are given constants.…”

Section: Magnetostatic Levitationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Electrostatic and magnetostatic levitations of ball – forces, stability and oscillations frequencies

Spałek

2020

IET Science, Measurement &Amp; Technology

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Electrostatic levitations of anisotropic dielectric oblate and prolate spheroids

Spałek

2023

Electr Eng

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The paper presents analytical approach to electrostatic field for dielectric spheroids problem. For both oblate and prolate spheroids fields distributions are derived. The variable separation method is applied. Electrostatic force is evaluated with the help of the Maxwell stress tensor generalized method, material force density, coenergy and equivalent dipole. Levitation forces for both oblate and prolate dielectric spheroids versus axis permittivities and spheroids height are presented.

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Analysis and calculation of electromagnetic force of stator winding and damping bar during dynamic loading of pumped storage generator/motor

Tao,

Dai,

et al. 2024

IET Electric Power Appl

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Frequently and rapidly dynamic conversion is one of the main differences between pumped‐storage motor and general hydro‐generator. In the process of changing operating conditions, in order to achieve fast dynamic conversion, the loading rate requirement of pumped storage motors is often much higher than that of conventional generators, which brings great impact on the key structural components of generators, it affects the safe operating life of motors directly. To solve the impact of dynamic loading rate on key structural components of pumped storage motors, for a pumped storage motor as an example, investigating different loading rates during the dynamic conversion of two main operating conditions on pumped storage motors (power generation and electric power) of pumped storage motors are investigated. Calculating accurately the distribution laws of electromagnetic forces of key structural components which contains stator tooth walls and damping strips during dynamic loading. The results show that the different loading rates will affect the force distribution of the key position directly, under the condition of generator state and motor state, the distribution of force on stator winding and damping bar present different laws, it will provide reference for the determination of multi‐state conversion loading rate of pumped storage motor and the improvement of pumped storage motor structure scheme.

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Generalization of Maxwell Stress Tensor Method for Magnetically Anisotropic Regions

Cited by 5 publications

References 8 publications

Electrostatic and magnetostatic levitations of ball – forces, stability and oscillations frequencies

Electrostatic and magnetostatic levitations of ball – forces, stability and oscillations frequencies

Electrostatic levitations of anisotropic dielectric oblate and prolate spheroids

Analysis and calculation of electromagnetic force of stator winding and damping bar during dynamic loading of pumped storage generator/motor

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