Electric Vector Potential Research Articles

Electromagnetic Modeling of Superconducting Bulks in Applied Time-Varying Magnetic Field

An integrodifferential model formulated in terms of the electric vector potential is developed for the 3D numerical modeling of the electromagnetic field in superconducting bulks, for AC losses evaluation. The Newton Raphson method is applied to accelerate the convergence. The model is validated on a benchmark. The comparison results show the accuracy of the model and its performances in terms of computation time compared to classical approaches.

Spin-dependent conductance and magnetoresistance ratio in two magnetic-barrier structures

An improved transfer matrix method is developed with modular matrices for rectangular electric and magnetic vector potentials, as well as the δ-function like potentials. As an example, the spin-dependent conductance in two-dimensional electron gas under modulation of two magnetic barriers are recalculated and compared with those illustrated in (Wang et al. Chin. Phys. B 19, 037301(2010)). The previous results are found inaccurate, their transfer matrix was derived for slowly varying magnetic vector potential, which is inappropriate for δ-function like Zeeman interaction potentials. Furthermore, the last transfer matrix of eq. (4) is also found incorrect, and the Zeeman interaction is wrongly expressed and mistakenly enhanced about (1/0.067 ≈ 15) times. Our improved method here is general and is demonstrated very fast forevaluating various spin-dependent transport properties in complex hybrid magnetic–electric structures.

Nondestructive Testing of Multilayer HTS Tapes by Using Eddy Current Thermography

This article presents numerical and experimental investigations of the structural control of multilayer high temperature superconducting (HTS) tapes, using eddy current thermography at room temperature. A rotating magnet wheel is used as inductor to avoid the thermal disturbance of current supplied inductors. A 3-D magnetothermal modeling approach based on an integro-differential formulation in terms of the electric vector potential is developed for eddy current computation in the HTS tapes; associated to a thermal modeling using the finite difference method. The source magnetic field is calculated using the magnetic surface charge model. The simulation results are confirmed by the measurements, both highlighting the potential of such control to detect structural defects, such as delamination.

Active Electric Dipole Energy Sources: Transduction via Electric Scalar and Vector Potentials.

The creation of electromagnetic energy may be realised by engineering a device with a method of transduction, which allows an external energy source, such as mechanical, chemical, nuclear, etc., to be impressed into the electromagnetic system through a mechanism that enables the separation of opposite polarity charges. For example, a voltage generator, such as a triboelectric nanogenerator, enables the separation of charges through the transduction of mechanical energy, creating an active physical dipole in the static case, or an active Hertzian dipole in the time-dependent case. The net result is the creation of a static or time-dependent permanent polarisation, respectively, without an applied electric field and with a non-zero vector curl. This system is the dual of a magnetic solenoid or permanent magnet excited by a circulating electrical current or fictitious bound current, respectively, which supplies a magnetomotive force described by a magnetic vector potential and a magnetic geometric phase proportional to the enclosed magnetic flux. Thus, the active electric dipole voltage generator has been described macroscopically by a circulating fictitious magnetic current boundary source and exhibits an electric vector potential with an electric geometric phase proportional to the enclosed electric flux density. This macroscopic description of an active dipole is a semi-classical average description of some underlying microscopic physics, which exhibits emergent nonconservative behaviour not found in classical closed-system laws of electrodynamics. We show that the electromotive force produced by an active dipole in general has both electric scalar and vector potential components to account for the magnitude of the electromotive force it produces. Independent of the electromagnetic gauge, we show that Faraday’s and Ampere’s law may be derived from the time rate of change of the magnetic and dual electric geometric phases. Finally, we analyse an active cylindrical dipole in terms of scalar and vector potential and confirm that the electromotive force produced, and hence potential difference across the terminals is a combination of vector and scalar potential difference depending on the aspect ratio (AR) of the dipole. For long thin active dipoles (AR approaches 0), the electric field is suppressed inside, and the voltage is determined mainly by the electric vector potential. For large flat active dipoles (AR approaches infinity), the electric flux density is suppressed inside, and the voltage is mainly determined by the scalar potential.

On gauge invariance and results for spin and orbital angular momenta and helicity densities in optical fields

The spin (sn) and orbital (ln) angular momenta and helicity (hn) densities of the electromagnetic fields are defined by introducing two types of electric (C) and magnetic (A) vector potentials defined as D = ∇ × C and B = ∇ × A for the electric (D) and magnetic (B) induction fields. The gauge invariances of sn, ln and hn for the optical field are achieved without gauge fixing by canceling out the contributions from four nablas ∇ acting on two gauge functions (GFs) (u and υ) and two scalar potentials (SPs) (Φ and ψ) which are generated by introducing A and C. As a result, it is found that sn, ln and hn can be quantized without gauge fixing, where θ and α are phase differences between A and C and between ∇u and ∇Φ or ∇υ and ∇ψ, respectively. In the fundamental aspect of gauge invariance, it was elucidated that the gauge invariance of ln works to make the phase differences θ and α to ± π/2 and 2nπ/ℓ, respectively. As its applied aspect, it is presumed to be useful in studying the behavior of the entanglement of electromagnetic waves. A new interrelationship between four nablas acting on two GFs and two SPs is established near the beam waist in paraxial approximation.

Calculation of the Eddy Current Losses in a Laminated Open-Type Transformer Core Based on the A→,T→−A→ Formulation

Losses due to eddy currents in an open-type transformer core are significantly reduced by the lamination of the transformer core. In order to further reduce the eddy current losses, the open-type core often has a multi-part structure, i.e., it is composed of several more slender cores. The complete homogenization of such a core is not possible when an A→,V−A→ formulation is used, where A and V represent the magnetic vector potential and electric scalar potential, respectively. On the other hand, an A→,T→−A→ formulation, where T represents the electric vector potential, enables the complete homogenization of the general open-type core, but the simulation converges poorly due to the large number of degrees of freedom. By eliminating the redundant degrees of freedom, the convergence rate is significantly improved, and is at least twice as good as the convergence rate of the simulation based on the A→,V−A→ formulation. In this paper, a method for the calculation of the eddy current losses in an open-type core based on the A→,T→−A→ formulation with the elimination of redundant degrees of freedom is presented. The method is validated by comparison with a brute force simulation based on the A→,V−A→ formulation, and the efficiency of the method is determined by comparison with the standard homogenization method based on the A→,V−A→ formulation.

Physical optics approach to wave diffraction by a perfect electromagnetic conductor half-plane

The scattering of electromagnetic plane waves by a perfect electromagnetic conductor is investigated. The method of physical optics is used for the analysis of the problem. The reflected fields by a whole-plane are taken into account. The surface electric and magnetic current densities are constructed with the aid of the incident wave and the reflected field from the whole-surface. The scattering integrals are obtained for the electric and magnetic vector potentials. The scattered electric and magnetic fields are expressed in terms of the vector potentials. The scattering integrals are evaluated asymptotically for large values of the wave-number. Some numerical results are given.

Efficient Computation of Partial Elements in the Full-Wave Surface-PEEC Method

The partial element equivalent circuit (PEEC) method provides an electromagnetic model of interconnections and packaging structures in terms of standard circuit elements. The surface-based PEEC (S-PEEC) formulation can reduce the number of unknowns compared to the standard volume-based PEEC (V-PEEC) method. This reduction is of particular use in the case of high-speed circuits and high-switching power electronics, where the bandwidth extends from low frequencies to the GHz range. In this article, the S-PEEC formulation is revised and cast in a matrix form. The main novelty is that the interaction integrals involving the curl of the magnetic and electric vector potentials are computed through the Taylor series expansion of the full-wave Green's function, leading to analytical forms that are rigorously derived. Therefore, the numerical integration is avoided, with a consequent reduction of the computation time. The proposed formulas are studied in terms of the frequency, size of the mesh, and distance between the basis function domains. Three examples are presented, confirming the accuracy of the proposed method compared to the V-PEEC method and surface-based numerical methods from literature.

A relativistic electron interacting with electromagnetic waves

A relativistic electron, in an electromagnetic wave is studied. The relativistic energy-momentum relation is taken into consideration for the derivation of a wave equation for spin-1/2 particle. The electromagnetic wave is represented by the electric scalar and magnetic vector potentials. Thus the spin information of the electron is integrated into the relativistic differential equation. A relativistic quantum wave equation is derived by using the derivatives with respect to time and spatial coordinates with respect to the kinetic energy and momentum of the electron. It is shown that the obtained relativistic equation directly reduces to the Pauli equation in the non-relativistic limit.

Electric vector potential formulation in electrostatics: analytical treatment of the gaped surface electrode

Electric vector potential $\Theta(\boldsymbol{r})$ is a legitimate but rarely used tool to calculate the steady electric field in free-charge regions. Commonly, it is preferred to employ the scalar electric potential $\Phi(\boldsymbol{r})$ rather than $\Theta(\boldsymbol{r})$ in most of the electrostatic problems. However, the electric vector potential formulation can be a viable representation to study certain systems. One of them is the surface electrode SE, a planar finite region $\mathcal{A}_{-}$ kept at a fixed electric potential with the rest grounded including a gap of thickness $\nu$ between electrodes. In this document we use the \textit{Helmholtz Decomposition Theorem} and the electric vector potential formulation to provide integral expressions for the surface charge density and the electric field of the SE of arbitrary contour $\partial\mathcal{A}$. We also present an alternative derivation of the result found in [M. Oliveira and J. A. Miranda 2001 Eur. J. Phys. 22 31] for the gapless ($\nu=0$) surface electrode GSE without invoking any analogy between the GSE and magnetostatics. It is shown that electric vector potential and the electric field of the gapped circular SE at any point can be obtained from an average of the gapless solution on the gap. Keywords: Electric vector potential, surface-electrode, Helmholtz Decomposition, Green's theorem.

Model order reduction techniques applied to magnetodynamic T-Ω-formulation

PurposeThe purpose of this paper is to use different model order reduction techniques to cope with the computational effort of solving large systems of equations. By appropriate decomposition of the electromagnetic field problem, the number of degrees of freedom (DOF) can be efficiently reduced. In this contribution, the Proper Generalized Decomposition (PGD) and the Proper Orthogonal Decomposition (POD) are used in the frame of the T-Ω-formulation, and the feasibility is elaborated.Design/methodology/approachThe POD and the PGD are two methods to reduce the model order. Particularly in the context of eddy current problems, conventional time-stepping algorithms can lead to many numerical simulations of the studied problem. To simulate the transient field, the T-Ω-formulation is used which couples the magnetic scalar potential and the electric vector potential. In this paper, both methods are studied on an academic example of an induction furnace in terms of accuracy and computational effort.FindingsUsing the proposed reduction techniques significantly reduces the DOF and subsequently the computational effort. Further, the feasibility of the combination of both methods with the T-Ω-formulation is given, and a fundamental step toward fast simulation of eddy current problems is shown.Originality/valueIn this paper, the PGD is combined for the first time with the T-Ω-formulation. The application of the PGD and POD and the following comparison illustrate the great potential of these techniques in combination with the T-Ω-formulation in context of eddy current problems.

Marching On-In-Time Unstructured PEEC Method for Electrically Large Structures with Conductive, Dielectric, and Magnetic Media

The Marching On-In-Time (MOT) unstructured Partial Element Equivalent Circuit (PEEC) method for time domain electromagnetic problems is presented. The method allows the transient analysis of electrically large electromagnetic devices consisting of conductive, dielectric, and magnetic media coupled with external lumped circuits. By re-formulating PEEC following the Coulombian interpretation of magnetization phenomena and by using electric and magnetic vector potentials, the proposed approach allows for a completely equivalent treatment of electric and magnetic media and inhomogeneous and anisotropic materials are accounted for as well. With respect to the recently proposed Marching On-In-Time PEEC approach, based on the standard (structured) discretization of PEEC, the method presented in this paper uses a different space and time MOT discretization, which allows for a reduction in the number of the unknowns. Analytical and industrial test cases consisting in electrically large devices are considered (e.g., the model of a Neutral Beam Injector adopted in thermonuclear fusion applications). Results obtained from the simulations show that the proposed method is accurate and yields good performances. Moreover, when rich harmonic content transient phenomena are considered, the unstructured MOT–PEEC method allows for a significant reduction of the memory and computation time when compared to techniques based on Inverse Discrete Fourier Transform applied to the frequency domain unstructured PEEC approach.

Cyclic Symmetry in Volume Integral Formulations for Eddy Currents: Cohomology Computation and Gauging

This contribution addresses the solution of eddy-current problems by means of a volume integral formulation based on the electric vector potential on a computational domain that exhibits a cyclic symmetry. Even if grids discretizing the domain are typically composed of tetrahedral or hexahedral elements, the proposed approach also works for general polyhedral meshes, such as those ones obtained by subgridding . In this article, an algorithm to compute a set of suitable cohomology generators needed when the conductors are not simply connected is introduced first. Besides being purely combinatorial, with linear-time worst case complexity and suitable with polyhedral meshes, it reuses a code that computes generators for triangular surface meshes, with obvious advantages concerning the implementation effort. Second, the formulation and the algorithm for cohomology computation are tweaked to be able to solve eddy-current problems with cyclic symmetry reserving specific attention to the construction of suitable tree–cotree decomposition for the problem gauging.

A Volume PEEC Formulation Based on the Cell Method for Electromagnetic Problems From Low to High Frequency

A new and general partial element equivalent circuit (PEEC) formulation based on electric and magnetic vector potentials and the cell method is presented. The purpose of this paper is manifold. First, a comprehensive review of well-established PEEC method is given and the state-of-the-art of the method with focus on the inclusion of magnetic media in the formulation is discussed. Then, a new and general formulation of the PEEC method based on integral equations is proposed. The formulation is valid for conductive, dielectric, and magnetic media and its advantages with respect to the state-of-the-art PEEC formulations are underlined. Unlike the existing PEEC methods, this novel formulation, thanks to the reinterpretation of PEEC in the context of integral equation methods, allows for maintaining a circuit interpretation for both electric and magnetic media. Thus, a standard Spice-like solver can in principle be adopted for the solution of electromagnetic (EM) problems also when magnetic media are involved. The imposition of the divergence condition for the current density vectors is guaranteed and also the case of inhomogeneous and anisotropic media is considered. A thorough discussion on the suitability of the method for the study of both low- and high-frequency EM problems is also presented. Finally, the case study of a near-field communication antenna is considered as an example of the applicability of the formulation to problems of industrial interest. Comparisons with commercial softwares, measurements, and the state-of-the-art PEEC method are given in order to demonstrate the accuracy and the advantages of the proposed PEEC formulation.

Finite Difference Generated Transient Potentials of Open-Layered Media by Parallel Computing Using OpenMP, MPI, OpenACC, and CUDA

The implementation of time-domain Green’s functions (TDGFs) in the graphics processing unit (GPU) and the central processing unit (CPU) using a finite-difference scheme is shown. The TDGFs represent the transient electric scalar and magnetic vector potentials due to a horizontal electric dipole (HED) in open-layered media. The layered media is bounded with a perfectly matched layer (PML), symmetry axis, and perfect electric conductor (PEC). We adopted four different parallel approaches as follows: 1) open multiprocessing (OpenMP) CPU implementation; 2) message passing interface (MPI) CPU implementation; 3) open accelerators (OpenACC) GPU implementation; and 4) compute unified device architecture (CUDA) GPU implementation. The accuracy and efficiency of the utilized programming models are validated by comparing and verifying the obtained results using a sequential CPU implementation. Compared to single threaded CPU implementation, speed-ups obtained by the OpenMP, MPI, OpenACC, and CUDA programming models are $4.8\times $ , $6.12\times $ , $45.97\times $ , and $96.53\times $ higher, respectively. The final result shows that GPU implementation leads to a considerable speed-up while the solution’s accuracy is fixed.

Full-Wave Analysis and Computation of Radiation Characteristics for Reconfigurable Plasma Antennas

A full-wave analytical theory has been developed and numerically solved for finding the radiation characteristics of the plasma column antenna for both guided- and leaky-mode configurations using electric and magnetic field vector potentials and the polarization current method. In the guided mode, the wave propagates through the plasma column axially and is radiated in the endfire direction, while in the leaky mode, the wave leaks along the broadside of the column in the normal direction and is attenuated axially. The radiation regions for these modes are classified by the dispersion ( $\omega $ – $\beta $ ) diagram that determines an axial phase propagation constant as a function of frequency. The analytical radiation characteristic findings are validated computationally using CST-MW Studio software. The effective axial length of the plasma generated inside the tube can be gradually increased or decreased as a function of input power, which, in turn, helps to tune/retune the resonant plasma antenna to the desired frequency. This type of antenna is useful for RADAR and communication systems where a continuous change in frequency could be required.

Application of Whitney elements for the reconstruction of electric arc current density in low-voltage circuit breakers

Purpose This paper aims to present the mathematical formulations of a magnetic inverse problem for the electric arc current density reconstruction in a simplified arc chamber of a low-voltage circuit breaker. Design/methodology/approach Considering that electric arc current density is a zero divergence vector field, the inverse problem can be solved in Whitney space W2 in terms of electric current density J with the zero divergence condition as a constraint or can be solved in Whitney space W1 in terms of electric vector potential T where the zero divergence condition naturally holds. Moreover, the tree gauging condition is applied to ensure a unique solution when solving for the vector potential in space W1. Tikhonov regularization is used to treat the ill-posedness of the inverse problem complemented with L-curve method for the selection of regularization parameters. A common mode approach is proposed, which solves for the reduced electric vector potential representing the internal current loops instead of solving for the total electric vector potential. The proposed inversion approaches are numerically tested starting from simulated magnetic field values. Findings With the common mode approach, the reconstruction of current density is significantly improved for both formulations using face elements in space W2 and using edge elements in space W1. When solving the inverse problem in space W1, the choice of the regularization operator has a key role to obtain a good reconstruction, where the discrete curl operator is a good option. The standard Tikhonov regularization obtains a good reconstruction with J-formulation, but fails in the case of T-formulation. The use of edge elements requires a tree-cotree gauging to ensure the uniqueness of T. Moreover, additional efforts have to be taken to find an optimal regularization operator and an optimal tree when using edge elements. In conclusion, the J-formulation is to be preferred. Originality/value The proposed approaches are able to reconstruct the three-dimensional electric arc current density from its magnetic field in a non-intrusive manner. The formulations enable us to incorporate a priori knowledge of the unknown current density into the solution of the inverse problem, including the zero divergence condition and the boundary conditions. A common mode approach is proposed, which can significantly improve the current density reconstruction.

The microstrop step discontinuity analysis by transverse resonance technique: method of boundary value problem algebraization

The two-dimensional eigenfunctions of the magnetic and electric vector potentials are constructed in the form of their expansion in series of Chebyshev orthogonal polynomials of 1st and 2nd kind to describe the current density in a strip transmission line with a step discontinuity of finite length. The solving boundary value problems algorithms for the resonant frequencies of a symmetric microstrip resonator with a capacitive section in it are developed and investigated on convergence. The account of the field behavior on thin edge in a nonuniform strip line provides the fast convergence of algorithm and small orders of solvable linear algebraic equations system. As examples of application, the transverse resonance method was used to calculate the scattering characteristics on 2-plane symmetric discontinuities consisting of a capacitive section of a microstrip line and slot resonators of complex shape in its ground plane.

Electrodynamics with magnetic monopoles: Photon wave mechanical theory

The microscopic Maxwell-Lorentz equations show an intrinsic symmetry if magnetic charge and current densities are included. Starting from the symmetrized set of field equations a propagator approach is used to describe the transverse (photon) electrodynamics in space-time. Two dyadic propagators are needed: (i) a transverse electric propagator with a near-field part and (ii) a magnetic propagator with a midfield part. The first quantized photon wave mechanical theory, based on the analytical parts of the two transverse Riemann-Silberstein vectors, is extended to include magnetic monopole dynamics. The dynamical equations for the photon helicity eigenvectors and the local energy conservation in the photon field are discussed. The Dirac string concept and the fiber bundle approach are avoided using the double-potential formalism of Cabibbo and Ferrari. The transverse ($T$) parts of the electric (${\mathbf{A}}_{T}^{e}$) and magnetic (${\mathbf{A}}_{T}^{m}$) vector potentials relate in a simple manner to our propagator theory if this is formulated in terms of the Huygens scalar propagator and the transverse electric and magnetic current densities. A transformation of the transverse vector potentials, which is nonlocal in space and time, allows one to express the transverse parts of the electric and magnetic fields as curls of combinations of the original (${\mathbf{A}}_{T}^{e},\phantom{\rule{0.16em}{0ex}}{\mathbf{A}}_{T}^{m}$) and the transformed (${\mathsc{A}}_{T}^{e},\phantom{\rule{0.16em}{0ex}}{\mathsc{A}}_{T}^{m}$) transverse vector potentials. The momentum of the particle-photon system is discussed, and it is shown that the electromagnetic parts of the canonical electric (charge $e$) and magnetic (charge $g$) particle momenta are given by $e({\mathbit{A}}_{T}^{e}+{\mathsc{A}}_{T}^{m})$ and $g({\mathbit{A}}_{T}^{m}\ensuremath{-}{\mathsc{A}}_{T}^{e})$, respectively. The angular momentum of the particle-photon system is analyzed and contact is made to the well-known nonretarded Saha-Wilson orbital angular momentum for an ($e,g$) pair. The near field of a magnetic monopole is studied based on the overlooked fact that the magnetic near field contains both longitudinal ($L$) (with $\mathbf{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{B}}_{L}=0$) and transverse vector field components. The sum of the two parts always is Einstein retarded, with proper account of the limitation caused by the lack of complete spatial photon localization. Finally, the relativistic electron-photon Hamiltonian in an external (prescribed) magnetic monopole field is discussed, paying particular attention to a determination of the transformed transverse magnetic vector potential, ${\mathsc{A}}_{T}^{m}$, and its positive-frequency part.

An Implicit Gauge Condition for Auxiliary Potentials in Inhomogeneous Media

Auxiliary potentials are indispensable mathematical tools in classical electromagnetic theory and other branches of physics. The curl of the electric and magnetic vector potentials is defined through relationships to field variables; this is ambiguous because the curl is unchanged by adding the gradient of an arbitrary scalar potential. These redundant degrees of freedom are eliminated through gauge fixing, wherein a relationship between the vector potential’s divergence and the scalar potential is specified. Explicitly, defining this relationship leads to difficulties in inhomogeneous media. This communication presents an implicit gauge that avoids these difficulties and generalizes the Lorenz gauge to inhomogeneous propagation media.