Biot-Savart Research Articles

Toward a Maneuverable Miniature Robotic Fish Equipped With a Novel Magnetic Actuator System

Most existing robotic fish have a large body size driven by servo motor system, while conventional small-sized actuators hardly generate a high swimming performance. This paper reports a miniature untethered robotic fish, whose body length is 69 mm. In particular, a newly designed magnetic actuator system (MAS) is equipped, which guarantees both small-sized dimension and flexibility of the robot. More specifically, the magnetic field generated by a permanent magnet is first investigated based on Biot–Savart law. Then, a novel tail-beating rhythm called magnetically actuated pulse width modulation (MAPWM) is modeled for the new actuator system. Further, an MAPWM-based control method is presented, in which the duty ratio of MAPWAM is innovatively utilized to realize the turning maneuvers for the first time. In addition, Lagrangian method is employed to establish the dynamic model to assess the MAPWM-based control method and the turning performance of the robotic fish. To further improve the maneuverability, the effect of a shape-variable caudal fin is analyzed based on computational fluid dynamics and the built dynamic model. Finally, combined with the MAS, the MAPWM-based control method, and the optimally selected caudal fin, extensive aquatic experiments are conducted on the robotic prototype. The results indicate that the developed miniature robotic fish achieves a considerably higher level of maneuverability in terms of turning radius when compared to swimming robots with equivalent dimensions.

Analytical solution for self-induced motion of a helical vortex with a Gaussian core

The paper presents an analytical solution for helical vortices with a Gaussian vorticity distribution in the core, which is confirmed by experimental and numerical simulations. This result is obtained by extending the Dyson method to the Biot–Savart law. Previously, analytical solutions were found and studied only for vortices with constant vorticity distribution in the core (a Rankine-type vortex core). One of the important issues raised during the discussion is the difference between self-induced movements of helical structures with both types of vortex core. The proposed solutions are important for the fundamental understanding and description of the behavior of helical eddy flows in various fields of industry and in nature. Examples include tip vortices behind the rotors of wind or hydro turbines, tornadoes, or axial vortices in aerodynamic devices such as vortex apparatuses and generators; cyclone separators, combustion chambers, etc.

Dynamic multicoil technique (DYNAMITE) MRI on human brain.

Spatial encoding for MRI is generally based on linear x, y, and z magnetic field gradients generated by a set of dedicated gradient coils. We recently introduced the dynamic multicoil technique (DYNAMITE) for B0 field control and demonstrated DYNAMITE MRI in a preclinical MR environment. In this study, we report the first realization of DYNAMITE MRI of the in vivo human head. Gradient fields for DYNAMITE MRI were generated with a 28-channel multicoil hardware arranged in 4 rows of 7 coils on a cylindrical surface (length 359mm, diameter 344 mm, maximum 5A per coil). DYNAMITE MRIs of a resolution phantom and in vivo human heads were acquired with multislice gradient-echo, multislice spin-echo, and 3D gradient-echo sequences. The resultant image fidelity was compared to that obtained with conventional gradient coil technology. DYNAMITE field control enabled the realization of all imaging sequences with average gradient errors ≤ 1%. DYNAMITE MRI provided image quality and sensitivity comparable to conventional gradient technology without any obvious artifacts. Some minor geometric deformations were noticed primarily in the image periphery as the result of regional field imperfections. The imperfections can be readily approximated theoretically through numerical integration of the Biot-Savart law and removed through image distortion correction. The first realization of DYNAMITE MRI of the in vivo human head has been presented. The obtained image fidelity is comparable to MRI with conventional gradient coils, paving the way for full-fledged DYNAMITE MRI and B0 shim systems for human applications.

Application of the Biot–Savart law to parabolic vortex segments using elliptic integrals

The Biot-Savart law is used in aerodynamic theory to calculate the velocity induced by curved vortex lines. Explicit formulas are developed, using multivariate Appell hypergeometric functions, for the velocity induced by a general parabolic vortex segment. The formulas are derived by constructing a particular pencil of elliptic curves whose period integrals provide the solution to the induced velocity. We use numerical integration and a perturbation expansion to evaluate the validity of our formulas.

Regularized vortex approximation for 2D Euler equations with transport noise

We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot–Savart kernel and the same common noise. The approximation happens by sending the number of particles [Formula: see text] to infinity and the regularization [Formula: see text] in the Biot–Savart kernel to [Formula: see text], as a suitable function of [Formula: see text].

A target field approach to optimize cylindrical shim coils for superconducting MRI system

Target field method is used to design a specific form of the magnetic field which has been verified in many works of literature. And to restrict the current distribution on a rectangular or cylindrical surface with finite size, the current density is usually expanded by Fourier series with some unknown coefficients, which can be determined by setting the target-field points over the imaging region of interest, and then comparing the calculated field values from the Biot–Savart law and the ideal ones. The stream-function method is adopted to discrete the current density to generate winding patterns. But some design results can’t match the predetermined ones. Based on it, this paper proposed a new strategy for cylindrical shim coil design. Finally it can make the final design values meet the ideal fields with limited errors. This method can also be used to design gradient coils for high field superconducting MRI systems.

Vibrational and stability analysis of membrane-like current-carrying nanowires under action of longitudinal magnetic fields

The dynamical behavior of a set of nanowires carrying electric current under action of a longitudinally applied magnetic field using the cable model is of our concern. To obtain the nanosystem’s governing equations, we use the Gurtin–Murdoch surface elasticity theory, and by applying the Lorentz formula and the Biot–Savart law, the total Lorentzian force applied on each nanowire from the existing nanosystem and longitudinal magnetic field is appropriately evaluated. Then, using the Galerkin method based on the assumed modes method, we solve the obtained equations of motion for the transverse frequencies. We also calculate the exact vibration modes using the analytical method. The concepts of synchronous and asynchronous vibrational modes are presented, and the validity of the Galerkin method based on the assumed modes is displayed by comparison with the exact approach. Finally, the influences of important parameters such as electric current intensity, magnetic field strength, interwire distance, surface effect, and population of the nanosystem on the free vibration response and the stability of the nanosystem are discussed.

Closed-form expressions for the magnetic fields of rectangular and circular finite-length solenoids and current loops

A summary of closed-form expressions for the magnetic fields produced by rectangular- and circular-shaped finite-length solenoids and current loops is provided altogether for easy reference. Each expression provides the magnetic field in all space, except locations where a current of infinitesimal thickness is considered to exist. The closed-form expression for the magnetic field of a rectangular-shaped finite-length solenoid is derived using the Biot–Savart law. Closed-form expressions for the magnetic fields of solenoids and current loops can be used to avoid approximations in analytical models and may reduce computation time in computer simulations.

An Improved Self-Consistent Model and Its Application to Estimate the Critical Current of REBCO Magnet

There is a growing interest in using HTS magnet to improve the performance of the electric devices such as motor/generator, NMR/MRI and EDS train. For the HTS magnet, the critical current is of importance since it determines the current-carrying capacity, i.e., the ability to generate magnetic field. So, it is imperative to estimate the critical current of HTS magnet. However, the numerical modelling of the magnet is not straightforward due to the field-dependent critical current density, a large number of turns and the ultra-high aspect ratio, e.g., of REBCO (REBa2Cu3Ox, where RE = Rare Earth) coated conductor. To make the estimation of the critical current accessible, this paper aims at developing an improved self-consistent model. First, the analytical expressions of the self-field of a generalized racetrack magnet are derived from the Biot-Savart's law. Secondly, the self-consistent model is introduced and validated by comparing the estimated critical current of a racetrack REBCO magnet with the measured one. Thirdly, based on the homogenized technique, the efficiency of self-consistent model is significantly improved with accuracy preserved. Lastly, we estimate the critical current of a generalized racetrack REBCO magnet for EDS train use with the improved self-consistent model.

3D Analytical Model of Permanent Magnet and Electromagnetic Hybrid Halbach Array Electrodynamic Suspension System

In this paper, a 3D analytical model of permanent magnet (PM) and electromagnetic hybrid Halbach array electrodynamic suspension (EDS) system for a conductive plate is presented. When the hybrid Halbach array moves to cut the conductive plate, the 3D eddy current forces are derived using the second-order vector potential (SOVP) based on y-component of the source magnetic flux density, which is derived using the Biot–Savart law and surface current method. The accuracy of the derived equations is verified by a 3D finite-element analysis (FEA) model. Furthermore, the effects of the main characteristic parameters on the levitation force are analysed.

Measurement of internal magnetic flux density distribution in air-core toroidal transformer under high-frequency excitation.

An investigation of air-core toroidal transformers has been undertaken to enable power converters to achieve megahertz operation. The characteristics of the air-core toroidal transformer govern the behaviors of the internal magnetic flux of the transformer. Understanding the behaviors at an above-megahertz level is a key issue in the design of the air-core toroidal transformer. This paper discusses the internal magnetic flux density distribution in the air-core toroidal transformer under high-frequency excitation. We propose a measurement method for the distribution and show the obtained results under a sinusoidal excitation with an amplitude of 1.5 A and a frequency of 1 MHz. The results reveal a non-uniform distribution in the tangential and normal directions. The cause of this non-uniformity was found to be the structure of the air-core toroidal transformer. In addition, the validity of the proposed measurement method was confirmed by comparing the experimentally obtained results with a numerical estimation using the Biot-Savart law. These results suggest that the proposed measurement method is capable of investigating the distributed characteristics of air-core toroidal transformers under megahertz excitation.

Tratamento experimental do protótipo de um motor de corrente contínua

Esse trabalho tem como objetivo apresentar os resultados do estudo analítico e experimental do protótipo de um motor de corrente contínua desenvolvido experimentalmente. Esse motor tem como princípio da ação motora, as forças repulsoras entre ímãs ou eletroímãs. Teve como base fundamental para análise das forças, campo e torque atuante no rotor as leis da magnetostática, tais como a Lei de Ampére e a Lei de Biot Savart, além de conceitos relacionados ao princípio de funcionamento de um motor de corrente contínua. Para modelagem do experimento utilizou-se o software SketchUp, uma ferramenta que permite a construção de modelos geométricos, modelos físicos e por fim, um protótipo, permitindo dar um posicionamento estratégico as bobinas, rotor e estator, tendo como objetivo principal da modelagem, apresentar um experimento com características de um motor elétrico e que sua estrutura apresente semelhança com as exemplificações feitas em livros didáticos. Na construção do protótipo foi utilizado elementos de baixo custo, aproveitando componentes de equipamentos descartados. Os resultados obtidos através dos estudos analíticos e testes experimentais permite visualizar um adequado funcionamento do protótipo, possibilitando sem muitas complicações visualizar diversos tópicos abordados no curso de física e engenharia, tais como, campos magnéticos gerado por bobinas, força magnética, torque em uma espira, força magnética em um fio percorrido por corrente, velocidade de um rotor composto por bobinas imerso em um campo magnético e momento dipolar, além de sua estrutura permitir chegar a resultados matemáticos acerca de seu funcionamento, usando princípios básicos do eletromagnetismo..

Magnetic Design of an Alternating Gradient Canted-Cosine-Theta Superconducting Magnet for Proton Therapy

Canted-Cosine-Theta (CCT) magnets exhibit superior field quality and structural compactness in superconducting magnet schemes. When combined with an alternating gradient (AG) field, CCT magnets can be applied to proton therapy gantries to achieve a significantly increased momentum acceptance. This paper introduces design considerations for a 67.5 degree AG-CCT magnet for a proton therapy gantry, including the design and optimization of a magnetic field combined with an AG function. The calculation of the magnetic field used to simulate the CCT coil with multiple lines based on the Biot-Savart law is discussed in detail. Technical issues covering the choice of superconducting coils and layering scheme are also discussed.

Semi-Analytical Modeling and Analysis of Halbach Array

Analysis of Halbach array placed in open space by using finite element method involves substantial consumption of memory, time, and cost. To address this problem, development of a mathematical modeling and analytic analysis method for Halbach array can be a solution, but research on this topic is currently insufficient. Therefore, a novel mathematical modeling and analytic analysis method for Halbach array in open space is proposed in this study, which is termed as the Ampere model and the Biot–Savart law (AB method). The proposed AB method can analyze the Halbach array rapidly and accurately with minimal consumption of memory. The usefulness of the AB method in terms of accuracy and memory and time consumption is verified by comparing the AB method with finite element method in this paper.

Untold secrets of the slowly charging capacitor

The slowly charging capacitor is the standard example used to illustrate that the displacement current density is needed in Ampere's law if we want to correctly determine the magnetic field between capacitor plates. However, in any quasi-static situation the magnetic field can also be determined using the Biot-Savart law including only the real current densities. In this work, we will numerically calculate the magnetic field due to the surface currents on the capacitor plates and add it to the magnetic field due to the charging wire and show how they combine to create the correct magnetic field thoughout all space. For regions to the left or right of the capacitor, we find the surprising result that the surface currents replicate the magnetic field that would have been created by the missing section of the charging wire between the plates. For points between the capacitor plates, the magnetic field due to the surface currents mostly cancels the magnetic field from the near-infinite length charging wire, resulting in the well-known reduced field in that interior region. We will also illustrate the impact of finite capacitor plates on these results and briefly comment on how textbook and/or classroom discussions could be improved by carefully discussing these details.

Simultaneous correction of sensitivity and spatial resolution in projection-based magnetic particle imaging.

The purpose of this study was to develop a method to simultaneously correct the spatial resolution and inhomogeneous sensitivity of a receiving coil in projection-based magnetic particle imaging - and to investigate its efficacy through simulation and experimental studies. Magnetic particle imaging (MPI)images were reconstructed using the simultaneous algebraic reconstruction technique (SART), and simultaneous corrections to sensitivity and spatial resolution were performed by incorporating the sensitivity map of the receiving coil and the system function into the SART algorithm. After each SART update, the regularization method - with total variation (TV) minimization - was used to suppress noise amplification and artifact generation. For comparison, MPI images were also reconstructed using the filtered backprojection (FBP) method and the FBP-truncated singular value decomposition (TSVD) method, in which the system function was deconvolved from the projection data using TSVD. In simulation studies, the sensitivity map of a second-order, gradiometer-type receiving coil was generated using the Biot-Savart law, while the system function was obtained by calculating the MPI signals induced by magnetic nanoparticles at various distances from a field-free line (FFL), using a lock-in-amplifier model. The effects of a regularization parameter for TV minimization (α), number of iterations (N), and signal-to-noise ratio (SNR) of the MPI signals on the reconstructed MPI images of a numerical phantom were evaluated, using the image profiles and percent root mean square error (PRMSE). Experimental studies involved the calculation of the system function using a tube phantom. Projection data for an A-shaped phantom were acquired using our MPI scanner, and their MPI images were reconstructed from the projection data, as described above. When both the sensitivity and spatial resolution were corrected (SART-SR), the quality of the reconstructed images was seen to have improved, compared to when the spatial resolution was not corrected - or when the FBP and FBP-TSVD methods were used. When SNR was low (20), a larger α value yielded a better image. The minimum PRMSE occurred at N≈200-400 and increased with increasing N thereafter. When SNR was high (100), the image quality was generally not dependent on the α value within its studied range. The PRMSE decreased slowly with increasing N, and tended to converge to a constant value. The full width at half maximum (FWHM) of the profile was obtained from the A-shaped phantom, reconstructed using the SART-SR algorithm with α=0.05 and N=1000. The FWHM value of the tube (2mm diameter) in the A-shaped phantom image was found to be 2.2mm on average, whereas those calculated from the images obtained by the FBP and FBP-TSVD methods were 4.4 and 3.0mm on average, respectively. Spatial resolution improved when using the FBP-TSVD method as compared to the FBP method but image distortion and artifacts were observed. Although further studies are necessary to optimize the parameters used in the SART algorithm and in TV minimization, the present results suggest that the proposed method is useful for improving the image quality of projection-based MPI.

An Algorithm for Toroidal Field Harmonics Computation in Arbitrary Magnetic Configurations

Toroidal magnetic configurations are widely exploited in industry and scientific research, involving a vast spectrum of applications, such as thermonuclear fusion, particle detectors, SMES systems and medical devices. To properly design and analyse these systems, it is crucial to determine the magnetic field generated by different configurations. The multipole expansion theory can be applied to the analysis of toroidal conurations, by solving the Laplace equation for the magnetic scalar potential in toroidal coordinates. Contrarily to the case of accelerator magnets with straight axis, in this case the correlation between the current distribution and the field harmonics cannot easily be identified. This paper proposes a methodology for the computation of field harmonics in toroidal coordinates, which is validated by comparison with the results obtained through the Biot-Savart law. This work was carried out in the frame of the GaToroid project ongoing at CERN.

Magnetic field detection using a highly sensitive FBG probe

A partially unclad fiber Bragg grating (FBG) coated with Fe3O4 nanoparticles as a magnetic field sensor is experimentally demonstrated. A series of six FBGs reflecting different wavelengths at an efficient length of 30 mm are fixed on the top and outside of a cylindrical glass chamber. The chamber is equipped with a solenoid that acts as a magnetic field generator and is filled up with air, saline solution, and crude oil separately to detect the changes in the magnetic field. The magnetic–hysteresis loops confirm the super-paramagnetic properties of the synthesized Fe3O4 nanoparticles with size smaller than 20 nm. The sensor response time of ∼21 s confirms the high reliability and repeatability of the sensing scheme. The change in the magnetic field strength depended from FBG–solenoid distance, propagating at different media and function generator frequency leads to shift in the reflected wavelength of each single FBG’s accordingly. The magnetic field strength outside the solenoid obeys the inverse-cubic law, and the decrease in the wavelength shift with an increase in the FBG–solenoid distance is in excellent agreement with the Biot–Savart law. The shift is caused by the interference of different propagating modes that are reflected from the core-cladding and cladding-magnetite layer interfaces; these modes have different phases because of the changes in the refractive index of the magnetite layer resulting from the change in the magnetic field. Our precise fabrication of the FBG probe with a maximum sensitivity of ∼11 000 pm mT−1 and the proposed design of experiment may be appropriate for detecting small changes in magnetic fields in the oil industry.

Three‐dimensional rotatable omnidirectional MCR WPT systems

Recently, Wi-Fi wireless power transfer (WPT) has attracted increasing attention. In this study, a rotatable three-dimensional (3D) transmitter (Tx) is first designed which only needs to be powered by a single source without current amplitude or phase control. Mutual inductance formulas of the Tx and a receiver (Rx) during the Tx's 3D rotation are derived based on Biot-Savart law and rotating matrix, and the maximum coupling areas are obtained. A method of achieving equal coupling of Rxs with the Tx which simultaneously charges two Rxs is proposed. A real rotatable 3D Tx and an omnidirectional magnetically coupled resonant (MCR) WPT prototype are fabricated, and the theoretical analysis is verified by simulation and experiment. Experiment displays that the maximum or desired coupling of single or two Rxs in 3D space can be obtained by calculated rotation combinations. The maximum power transfer efficiency (PTE) of the system can reach 90.2% with the transmission distance of 1.5 times the radius of Tx. Finally, the PTE can be maintained at 73% with the transmission distance varying from 18 to 27 cm by a novel method. The proposed 3D technique could be applied to many fields including Wi-Fi WPT home devices, remote sensing and some mechanical systems.

A Mixed Multiscale FEM for the Eddy-Current Problem With T, Φ–Φ in Laminated Conducting Media

A novel mixed multiscale finite-element method for the eddy-current problem is presented to avoid the necessity of modeling each laminate of the core of electrical devices. The method is based on a current vector potential $T$ and a reduced magnetic scalar potential (RMSP) $\Phi $ and copes with the 3-D problems. The edge effect is considered. Material properties are assumed to be linear. Hence, the method is developed for the frequency domain. External currents are represented by the Biot–Savart field serving as excitation. The planes of symmetry are exploited. Numerical simulations are presented, showing excellent accuracy at minimal computational costs.