Biot-Savart Research Articles

An Analytical Determination of the Magnetic Field in a System of Finite-Length Ribbon Busbars

Using the analytic method based on the Biot–Savart law for the electromagnetic field, the distribution of the magnetic field of a ribbon busbar of finite length was determined. The analytical formulas describing the magnetic field in all areas of the considered ribbon busbars were obtained. The Mathematica program was used to visualize the solutions obtained. The Mathematica programme is a good and convenient tool for analytical measurements using the integration function and conversion of the analytical solutions, for the determination of field quantities, and for the graphical visualisation of the obtained final solutions. This allowed for quick field analysis to be conducted after changes were made in the geometrical or electrical parameters of the systems under examination.

Resource-saving microprocessor-based reed switch current protection

We suggest designing a cutoff and an overcurrent protection for 6–35 kV electrical installations with switchgear cubicles on the basis of reed switches and a microprocessor without current transformers. A reed switch is a small glass tube with plates inside, which contact under the action of the corresponding magnetic field. Two reed switches are mounted near each busbar inside a switchgear cubicle at a safe distance using specially designed structures. The magnetic induction which acts on them is determined by the Biot–Savart law with experimentally found coefficients. The effect of currents in neighboring electrical installations and in switchgear cubicle casings is taken into account. We have developed and describe here an algorithm of protection operation and a technique for selecting parameters and estimating protection sensitivity. Such protections are noise-resistant due to detection of short circuits by the third actuation of a read switch; they detect a failed component with the help of built-in diagnostics. The test diagnostics is performed by means of supplying current simultaneously to the control windings of two reed switches, and the functional diagnostics, by means of supplying current to them in turn. The efficiency of such protections is confirmed by the simulation results.

Research on Experimental Bench Testing of Wear Metal Particles in Lubricating Oil Based on Self-Excitation and Self-Induction of Coil

The on-line monitoring of wear metal particles in lubricating oil plays an important role in equipment fault diagnosis technology. According to the principle of electromagnetic induction and the Biot-Savart law, model structure design, coil production, etc., different from traditional multi-coil excitation detection methods and compared to traditional methods that are only sensitive to large particles, this paper designed a self-excitation and self-induction metal particle monitoring sensor in lubricating oil based on LC resonance. At the same time, a sensor experimental bench is built by combining the production of ferromagnetic and non-ferromagnetic metal particle standards. In addition, through experimental verification, the sensor can achieve monitoring sensitivity for ferromagnetic particles greater than 41.28 μm and non-ferromagnetic (copper) particles greater than 61.53 μm under the 4.2 mm aperture flow channel, realizing the sensitivity on monitoring of small wear ferromagnetic particles and wear metal non-ferromagnetic particles, the experiment showed that it is particularly sensitive to small non-ferromagnetic particles. Based on a wide pipeline radius, the accuracy and sensitivity of the designed single-coil sensing system have been demonstrated.

Modeling the propagation of coronal mass ejections with COCONUT: Implementation of the regularized Biot-Savart law flux rope model

Context. Coronal mass ejections (CMEs) are rapid eruptions of magnetized plasma that occur on the Sun. They are known to be the main drivers of adverse space weather. The accurate tracking of their evolution in the heliosphere in numerical models is of the utmost importance for space weather forecasting. Aims. The main objective of this paper is to implement the regularized Biot-Savart law (RBSL) method in a new global corona model, called COCONUT. This approach has the capability to construct the magnetic flux rope with an axis of arbitrary shape. Methods. We present the implementation process of the RBSL flux rope model in COCONUT, which is superposed onto a realistic solar wind reconstructed from the observed magnetogram around the minimum of solar activity. Based on this, we simulate the propagation of an S-shaped flux rope from the solar surface to a distance of 25 R⊙. Results. Our simulation successfully reproduces the birth process of a CME originating from a sigmoid in a self-consistent way. The model effectively captures various physical processes and retrieves the prominent features of the CMEs in observations. In addition, the simulation results indicate that the magnetic topology of the CME flux rope at around 20 R⊙ deviates from a coherent structure and manifests as a mix of open and closed field lines with diverse footpoints. Conclusions. This work demonstrates the potential of the RBSL flux rope model in reproducing CME events that are more consistent with observations. Moreover, our findings strongly suggest that magnetic reconnection during the CME propagation plays a critical role in destroying the coherent characteristics of a CME flux rope.

Calculation of the velocities induced by the trailing vorticity in the rotor plane of a horizontal-axis turbine or propeller

Lifting line (LL) analysis of propellers and horizontal-axis turbines requires the axial and circumferential velocities induced by the vortex system representing the blades and the trailing vorticity. If the blades are straight and radial, the induced velocities along the LLs are due only to the trailing vorticity. Accurate two-term approximations for these velocities have been developed from the exact Kawada–Hardin (KH) equations for the velocity field of a doubly infinite helical vortex of constant pitch and radius, Wood et al. (Ocean Engineering, 2021, 235). This paper describes a straightforward extension of the approximations to give the induced velocities anywhere in the equivalent of the rotor plane for a doubly infinite helix. The third term in the approximation of the KH equations is derived and compared to an alternative third term due to Okulov (Journal of Fluid Mechanics, 2004, 521, 319–342). Both three-term approximations produce a small improvement in accuracy over the two-term approximations for a range of operating conditions for turbines and propellers. Okulov’s third term is superior. To determine the induced velocities for a singly infinite trailing vortex behind a rotor, an additional equation is derived from the Biot–Savart law. Numerical examples show that the resulting equations provide accurate estimates for the induced velocities over the rotor plane. The main application of the analysis is to account for blade sweep and coning by including the angle between the vortex origin and the control point at which the velocities are required, often the center of each blade element.

Robust Reconstruction of the Void Fraction from Noisy Magnetic Flux Density Using Invertible Neural Networks.

Electrolysis stands as a pivotal method for environmentally sustainable hydrogen production. However, the formation of gas bubbles during the electrolysis process poses significant challenges by impeding the electrochemical reactions, diminishing cell efficiency, and dramatically increasing energy consumption. Furthermore, the inherent difficulty in detecting these bubbles arises from the non-transparency of the wall of electrolysis cells. Additionally, these gas bubbles induce alterations in the conductivity of the electrolyte, leading to corresponding fluctuations in the magnetic flux density outside of the electrolysis cell, which can be measured by externally placed magnetic sensors. By solving the inverse problem of the Biot-Savart Law, we can estimate the conductivity distribution as well as the void fraction within the cell. In this work, we study different approaches to solve the inverse problem including Invertible Neural Networks (INNs) and Tikhonov regularization. Our experiments demonstrate that INNs are much more robust to solving the inverse problem than Tikhonov regularization when the level of noise in the magnetic flux density measurements is not known or changes over space and time.

Rotation and Confined Eruption of a Double Flux-rope System

We perform a data-constrained simulation with the zero-β assumption to study the mechanisms of strong rotation and failed eruption of a filament in active region 11474 on 2012 May 5 observed by Solar Dynamics Observatory and Solar Terrestrial Relations Observatory. The initial magnetic field is provided by nonlinear force-free field extrapolation, which is reconstructed by the regularized Biot–Savart laws and magnetofrictional method. Our simulation reproduces most observational features very well, e.g., the filament large-angle rotation of about 130°, the confined eruption, and the flare ribbons, allowing us to analyze the underlying physical processes behind observations. We discover two flux ropes in the sigmoid system, an upper flux rope (MFR1) and a lower flux rope (MFR2), which correspond to the filament and hot channel in observations, respectively. Both flux ropes undergo confined eruptions. MFR2 grows by tether-cutting reconnection during the eruption. The rotation of MFR1 is related to the shear-field component along the axis. The toroidal field tension force and the nonaxisymmetry forces confine the eruption of MFR1. We also suggest that the mutual interaction between MFR1 and MFR2 contributes to the large-angle rotation and the eruption failure. In addition, we calculate the temporal evolution of the twist and writhe of MFR1, which is a hint of probable reversal rotation.

Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems

We adapt the work of Jabin and Wang (2018) to show the first result of uniform in time propagation of chaos for a class of singular interaction kernels. In particular, our models contain the Biot–Savart kernel on the torus and thus the 2D vortex model.

An Arduino-based magnetic polarity detector and field intensity display

We have designed a simple Arduino-based device, which runs as magnetic polarity detector and field intensity display. It uses an analog Hall sensor together with the Arduino electronic board. The sign of the magnetic flux density allows us to identify the pole of a magnet or electromagnet, while the absolute value allows us to build an intensity display. This device allows us to visualize, among other topics, the basic functional dependences included into the Biot–Savart law, and particularly the changes in magnetic flux density with distance and current for a current-carrying coil or electromagnet.

Fully Microscopic Treatment of Magnetic Field Using Bogoliubov–De Gennes Approach

This work introduces an algorithm designed to solve the Bogoliubov–de Gennes equations of superconductivity theory. What sets this algorithm apart is its remarkable ability to precisely and consistently consider the impact of an external magnetic field, all within the microscopic approach. The computation scheme’s convergence is guaranteed by addressing the Biot–Savart equation for the field where the vector potential appears on both of its sides. To showcase the capabilities of this approach, we provide several key examples: the Abrikosov lattice, vortex core states, and the vortex structure in the intermediate mixed state of a superconductor. This method promises to offer valuable insights into the microscopic physics of intertype superconductivity.

Fast Analytic–Numerical Algorithms for Calculating Mutual and Self-Inductances of Air Coils

This paper deals with a method of calculating the mutual and self-inductances of various air coils located arbitrarily in space. Known elementary solutions (the Biot–Savart formulas) were used to determine the magnetic field of infinitely thin current loops and infinitely thin wires of finite length magnetically linking other coils. Unlike commonly used algorithms, these elementary solutions were not extensively transformed analytically but were used to perform calculations via direct numerical integration. This enabled the very quick and accurate obtaining of the self-inductance values, as well as determining the dependence of mutual inductances on the positions of both coils. This method allows for the analysis of different coil configurations (misaligned coils, inclined to each other, etc.) that other methods do not cover. It also enables the determination of the forces acting on the coils, as well as the calculation of the magnetic field distribution from any coil configuration. The obtained results were compared with those presented by other authors (both computational and measurement results).

Biot–Savart law with a smartphone: Phyphox app

Biot–Savart law with a smartphone: Phyphox app

基于Simdroid软件对毕奥–萨伐尔定律验证实验的辅助教学

基于Simdroid软件对毕奥–萨伐尔定律验证实验的辅助教学

What is the Effect of a Magnetic Field on Dye Adsorption onto Graphite Carbon?

This research explores the impact of magnetic fields on dye adsorption onto graphite carbon, utilizing electric currents to generate varying magnetic field strengths, as determined by the Biot-Savart law. The study demonstrates that even with small current magnitudes typically used in physics laboratories, the generated magnetic fields significantly influence dye adsorption. Through experiments with currents ranging from 1.5 A to 7.5 A, resulting in magnetic fields from 1.54 µT to 4.63 µT, we observed enhanced adsorption for congo red, methylene blue, and methyl orange. In contrast, phenol red exhibited a unique desorption pattern due to electrostatic repulsion. Temperature variations were noted but were considered to have a negligible effect on the adsorption behavior. The findings highlight the crucial role of magnetic energy density and the charge of dye molecules in the adsorption process, leading to the conclusion that magnetic fields, indeed, play a significant role in influencing dye adsorption onto graphite carbon, with potential applications in environmental conservation and industrial waste management.

A Rigorous Explicit Expression for the Mutual Inductance of Two Co-Axial Thin-Wire Coil Antennas Placed above a Layered Ground

This paper presents a quasi-analytical method that allows the derivation of a rigorous series-form representation for the mutual inductance of two co-axial coil antennas located above an arbitrarily layered earth structure. Starting from Biot–Savart law, which gives the integral representation for the primary vector potential generated by the source coil, the potential reflected by the layered ground is derived, and the resulting total vector potential is then integrated along the external circumference of the receiving coil to give the mutual inductance of the two antennas. The obtained representation for the flux is then evaluated analytically through the usage of the Gegenbauer addition theorem once an accurate, rational approximation is used in place of the factor of the integrand that exhibits branch cuts. It is shown how the resulting explicit solution exhibits the same degree of accuracy as purely numerical approaches like the finite-difference time-domain (FDTD) method and conventional numerical quadrature schemes, while it is less time-demanding than the latter methods.

Analytical modeling and experimental verification of a multi-DOF spherical pendulum electromagnetic energy harvester

Low-frequency vibration energy harvesting is a promising technology for providing floating buoys and small marine robots with sustainable energy from ocean waves. A spherical vibration energy harvester with multiple degrees of freedom has the advantages of good multi-directional adaptability and compact structure. However, its magnetic field distribution in three-dimension space is much more complex due to the combined effect of a series of cylindrical permanent magnets mounted on the surface of a gimbaled pendulum. The traditional finite element method is too time-consuming to perform efficient parameter design and optimization. To solve this problem, this work develops an analytical electromagnetic model for the spherical vibration energy harvester based on Biot-Savart Law. This model can be used to predict spatial magnetic flux distribution and electromotive force for given magnet and coil parameters. The accuracy and computational cost of the analytical model is verified by the comparison with the finite element method. Furthermore, the optimization of electromagnetic parameters is implemented to achieve maximum energy conversion efficiency. Finally, an optimized prototype is fabricated and tested in both laboratory and real sea. The resonant frequency and energy conversion efficiency of the prototype reaches 1.0 Hz and 25.45 %, respectively. It is shown that the experimental results are in good accordance with the analytical results.

Research on the coupling relationship between metal particles in lubricating oil and inductive coils based on self-excitation single coil

Characterizing the operational wear status of mechanical equipment based on online monitoring of metal particles in lubricating oil plays a significant role in the effect of equipment fault diagnosis technology. According to the theoretical analysis of the electromagnetic induction principle and Biot–Savart’s law, combined with the finite element simulation analysis, systematic study and analysis were conducted on the distribution of magnetic field inside a single inductance coil, the coupling relationship between metal particles in lubricating oil, and the inductance coil. This paper analyzed the effects of four important design parameters that were the excitation frequency, the inside diameter of the coil, the turns number of the coil, and the length of the coil. On magnetic field uniformity and monitoring sensitivity, experiments demonstrated that: the correlation between the uniformity of the magnetic field on the testing surface of the sensor inductance coil and the coil radius and length is the highest, followed by the correlation with the number of coil turns. The correlation with the excitation frequency can be ignored in a negligible manner. In addition, experimental results indicated that if the ratio between the inside diameter of the inductance coil and the length of the coil (R/L) was less than 0.41, the magnetic field uniformity would reach over 91%. Meanwhile, results showed that the sensitivity of the sensor to ferromagnetic and non-ferromagnetic particles increased with the decrease in the coil inner diameter and inductance and increased with the increase in the excitation frequency. There is a specific length that can make the sensor coil monitoring, which is the most sensitive.

Data-constrained Magnetohydrodynamic Simulation of an Intermediate Solar Filament Eruption

Solar eruptive activities could occur in weak magnetic field environments and over large spatial scales, which are especially relevant to eruptions involving intermediate or quiescent solar filaments. To handle the large scales, we implement and apply a flux rope embedding method using regularized Biot–Savart laws in the spherical coordinate system. Combined with a potential field source surface model and a magneto-frictional method, a nonlinear force-free field comprising a flux rope embedded in a potential field is constructed. Using the combined nonlinear force-free field as the initial condition, we then perform a zero-β data-constrained magnetohydrodynamic (MHD) simulation for an M8.7 flare at 03:38 UT on 2012 January 23. The MHD model reproduces the eruption process, flare ribbon evolution (represented by the quasi-separatrix layer evolution), and kinematics of the flux rope. This approach could potentially model global-scale eruptions from weak field regions.

Two-dimensional Axisymmetric Modeling of the Magnetic Field for High-voltage Gas Circuit Breakers

High fidelity numerical analyses of high-voltage gas circuit breakers have been conducted at Hitachi Energy Research with an in-house CFD-based arc simulation tool. The tool extends the capability of a commercial flow solver (ANSYS Fluent) to represent physical phenomena at play in a high voltage circuit breaker during a breaking operation, such as magnetostatics, polymeric and metal evaporation, and arc-network interaction. This work describes the implementation of the Biot-Savart law for computing the magnetic field generated by an electric arc under the magnetostatic approximation and in two--dimensional axisymmetric conditions. The implementation is compared to the reference one based on the magnetic vector potential formulation of the Amp`ere’s law in the Coulomb gauge. The limitations of the two formulations are discussed and their numerical accuracy compared.

Energy harvesting analysis of the magneto-electric and fluid-structure interaction parametric excited system

This research proposes an innovative design of a fluid-solid coupling vibration energy harvesting system (VEH system) that includes a downstream waterwheel driven by the flow field, which, in turn, drives gears and connecting rods to rotate a wheel equipped with magnets to generate electricity by changing the magnetic field. A piezoelectric patch (PZT) is installed upstream of the pipeline with a magnet attached to it. The repulsive force between the magnet on the wheel and the magnet on the PZT generates additional force while also creating vibration through fluid-solid coupling of the pipeline. The study derives a theoretical model of the nonlinear vibrating beam and couples it with the piezoelectric and magneto-electric equations to simulate the vibration of the fixed-fixed elastic pipe. The method of multiple scales (MOMS), fixed points plots, phase plots, and Poincaré maps are employed to verify the theoretically predicted parametric excitation properties of the system. The study uses the Biot-Savart Law to calculate the theoretical magnetic force and combines it with the fluid-conveying nonlinear beam and the PZT to create a magneto-electric coupling fluid pipeline vibration energy harvesting model. The study conducts a simple experiment to verify the feasibility of the theoretical model and demonstrates that the repulsive force of the magnet significantly enhances the electric generation benefit of the system. Regardless of whether the PZT is located in the curved or flat area (straight part) of the nonlinear beam, the addition of magnets to the system significantly increases voltage generation efficiency by more than 190 % when compared to systems without magnets.