Comparative PEEC Modeling of Magnetic Materials Under Different Equivalent Sources

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.

The partial element equivalent circuit (PEEC) formulation is an integral equation based approach for the solution of combined circuit and electromagnetic (EM) problems. In This work, a surface-based PEEC formulation is presented as an alternative to the existing volume-based method. With the rise in the operating frequencies and the increasing complexity of test structures on boards, packages and chips, a surface based formulation is more efficient for many problems in terms of the number of unknowns generated. The composite conductor dielectric modeling is based on the PMCHWT formulation which is transformed into a PEEC representation using equivalent magnetic and electric circuits connected by mutual coupling. We are interested in both time and frequency domain analyses similar to a SPICE type circuit solver.

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.

Spatial distribution of magnetic field and persistent currents in Condon domain phase

Coatings containing Fe-Si particles were electrodeposited on 3.0wt% Si steel sheets under magnetic fields. The effects of magnetic flux density (MFD), electrode arrangement and current density on the surface morphology, the silicon content in the coatings and the cathode current efficiency were investigated. When a magnetic field was applied parallel to the current and when the MFD was less than 0.5 T, numerous needle-like structures appeared on the coating surface. With increasing MFD, the needle-like structures weakened and were transformed into dome-shaped structures. Meanwhile, compared to results obtained in the absence of a magnetic field, the silicon content in the coatings significantly increased as the MFD was increased for all of the samples obtained using a vertical electrode system. However, in the case of an aclinic electrode system, the silicon content decreased. Furthermore, the cathode current efficiency was considerably diminished when a magnetic field was applied. A possible mechanism for these phenomena was discussed.

Current density distribution generated inside a volume conductor by externally applied currents can be calculated by using spatial distribution of its magnetic flux density, . The imaging modality used to reconstruct images of the current density distribution is known as magnetic resonance current density imaging (MRCDI). In MRCDI, spatial distribution of the current‐induced magnetic flux density is measured on a magnetic resonance imaging (MRI) platform. Calculation of current density distribution from magnetic flux density measurements requires all three components of . As only the component of which is in parallel with the main field of the MRI system can be measured, rotation of the subject is necessary to be able to acquire all three components of for a given current injection pattern. Rotating the subject inside the magnet is not trivial even for small‐sized objects and remains as a strong limitation to clinical applicability of the technique. In this study, a novel MRCDI reconstruction algorithm using only one component of is proposed to eliminate the need for subject rotation. Reconstruction performance of the proposed algorithm is evaluated on numerical and experimental models. Images obtained by the proposed and the conventional MRCDI algorithms, which utilizes three components of , are compared.

Partial element equivalent circuit (PEEC) is a convenient method for modeling transient behavior of high temperature superconductors (HTS) magnets made of metal-as insulation windings. In HTS coils, especially those made of tapes, screening currents are known to lower the magnetic field, to generate losses, and to contribute to the mechanical stress. Including screening current calculation in PEEC method can be achieved by subdividing the superconducting tape along its width, leading to larger models and much longer computation time. Herein, we propose a finite element (FE) model for screening currents calculation, which can be combined with PEEC model for transport current addition. The FE model is defined as thin shell using formulation with normal component of electric vector potential T as nodal variable, calculated by an iterative procedure. Source term in the FE model is obtained from variation of radial flux density. Critical current densities calculated in the FE model are fed back to the PEEC model at each step of transport current calculation. This work presents in detail the FE model and its coupling with PEEC model. The model is validated by comparison with other 3D FE models and models with axial symmetry. The effect of reducing the size of FE model, by averaging screening currents in adjacent turns, is also presented in this study. Finally, we present simulation results for current ramps and a quench of a double-pancake magnet, obtained by PEEC with and without adding FE screening current model.

Current density imaging (CDI) was developed with the aim of determining the three-dimensional distribution of externally applied electric current pathways inside a conductive medium, using measurements of magnetic flux density [Formula: see text] data. While the B field may be measurable using instruments such as a magnetometer, in magnetic resonance current density imaging (MR-CDI), an MRI scanner is used to measure the magnetic flux density data induced by current flow. In MR-CDI, the object must be rotated inside the MRI machine to find all three components of the B-field, as only the component of B parallel to the magnet main magnetic field can be measured. In principle, once the all three components of the B field have been obtained from an MR imaging experiment, the current density distribution [Formula: see text] can be reconstructed from Ampere's law [Formula: see text]. However, the need to rotate the object within the MRI scanner limits the usability of this technique. To overcome this problem, researchers have investigated the current density reconstruction problem using only one component of the magnetic flux density Bq, where q = x, y, z. In this chapter, we discuss numerical algorithms developed to reconstruct the distribution of J information from the measured B-field.

Models with magnetic materials can be a part of different interesting partial element equivalent circuit (PEEC) solutions. The chapter deals with magnetically closed loops such that the magnetic part may be treated separately from the rest of the problem. For this class of problems, the solution may result in a simpler model. The chapter considers how such a model is applied in a PEEC environment. In fact, integrating such a part may be very simple since it can be included as circuit models. The second class of problems consists predominantly of magnetic materials, where most of the solution is based on magnetic materials. The key assumption is that the majority of the magnetic flux is in the magnetic material rather than through the air. Since the computation of the reluctance resistance is a key part of the magnetic circuit approach, the chapter gives a specific example for the computation of a piece of magnetic material.

This paper details the impact of partial element accuracy on quasi-static partial element equivalent circuit (PEEC) model stability in the time domain. The potential sources of inaccurate partial element values are found to be poor geometrical meshing and the use of unsuitable partial element calculation routines. The impact on PEEC model stability of erroneous partial element values, and the coefficients of potential and partial inductances, are shown as theoretical constraints and practical results. Projection meshing, which is a discretization strategy suitable for the PEEC method, is shown to improve calculated partial element values for the same number of unknowns, thus improving model stability

PV systems are considered one of the most important and popular forms of renewable energy sources worldwide. Due to the installation of PV modules on rooftops and outdoor areas, they are frequently subjected to lightning strikes, leading to a deterioration in PV systems. To provide an efficient lightning protection system, the accurate modeling of PV systems during lightning strikes is required. The main objective of this paper is to assess the transient behavior of PV systems when struck by lightning, where the PV system is modeled using the partial element equivalent circuit (PEEC) and finite difference time-domain (FDTD) methods. The PEEC method is one of the accurate methods used to represent the PV system by its equivalent RLC circuit to study lightning transients. Moreover, the FDTD method provides high accuracy when studying lightning transients by analyzing the electric and magnetic fields inside the PV system. The PEEC and FDTD methods are used to calculate the transient overvoltages at different points of the PV system. Then, a small-scale PV system is tested in the laboratory using an impulse generator representing the lightning strike to verify the correctness of the results obtained by simulation. Most of the existing research works introduce low accuracy in PV system modeling during lightning transients by neglecting the effect of the mounting system, metal frame, and mutual coupling, which are considered in this work. Moreover, the high-frequency nature of lightning current is considered when modeling the elements of the grounding system. Verifying the results by laboratory experiments can guarantee the application of the simulation model on large-scale PV systems.

The Partial Element Equivalent Circuit (PEEC) method is a well recognized integral-equation (IE) technique to solve Maxwell’s equations. Similarly to the method of moments (MoM), the electromagnetic (EM) interactions between currents and between charges are described in terms of integrals. In contrast to the standard MoM, the PEEC method keeps the electric and magnetic coupling phenomena separate, which leads to different interaction integrals to be computed. These integrals admit simplified solutions for the case of the static free-space Green’s function and orthogonal geometries but their applicability is limited to electrically small problems only. When the full-wave free-space Green’s function is considered, the integrals are typically computed in the frequency domain (FD) by resorting to Gaussian quadrature schemes. The accuracy and efficiency of such schemes is a delicate issue. Therefore, recent works have investigated the possibility of applying the Cagniard–DeHoop (CdH) technique to calculate the interaction integrals for zero-thickness elementary domains. In this paper, we close the loop and shall apply the CdH technique to calculate the partial-inductance between two elementary bricks as prescribed by the PEEC technique exactly in the time domain (TD). The analytical approach is demonstrated on the interaction between two bricks as it occurs in the modeling of the magnetic field coupling between volumetric currents. The accuracy of the proposed approach is (successfully) tested for two representative cases.

Chapter 4.1 - Maxwell’s equations and numerical electromagnetic modeling in the context of the theory of differential forms

This paper presents the unbalanced torque generated in No-Insulation (NI) REBa2Cu3Ox (REBCO, RE = Rare Earth) pancake coils after a normal-state transition. The NI REBCO pancake coil has a high thermal stability, and it is desired to apply to ultra-high field magnets. When an NI REBCO pancake coil locally transitions into a resistive state, the operating current bypasses a normal-state-transitioned turn to adjacent turns in the coil-radial direction, and the amount of heat generation can be reduced. That is an inherent feature of NI REBCO pancake coils. However, when a radial current flows in a magnetic field, a Lorentz force is generated toward the coil-circumferential direction, i.e. torque. In case that the coil radius and the external magnetic field are very large, the extremely large torque is generated, and it could damage the magnet itself and/or the joints to terminals with a mechanical factor. Therefore, it is necessary to investigate the behavior of the torque generated in the NI REBCO pancake coil after the quench.In this paper, we present the numerical simulation results of the behaviors of the torque generated in the NI REBCO double pancake coil after quench. Their current, thermal, and mechanical behaviors are simulated by a partial element equivalent circuit (PEEC) method and 2-D thermal finite element method (FEM). The behaviors are different depending on the coil size. In addition, the behavior is presented when the operating current is suddenly shut down after quench.

This study presents a numerical and computational investigation into the energy fluctuations of three diatomic molecules—carbon monoxide (CO), hydrogen (H₂), and scandium hydride (ScH)— under a varying temperature parameter β based on two – dimension Schrödinger equation governed by Nikiforov – Uvarov Functional Analysis (NUFA) approach under the influence of magnetic field and magnetic flux. The numerical analysis indicates that when no external magnetic field is applied B = 0T and AB = 0, the energy spectrum experiences a slight increase. However, a higher energy spectrum is observed when an external field is introduced at ∅AB = 9 and B = 9 T. Examining the individual effects reveals that ∅AB = 9 and B = 0 T have a somewhat stronger impact on the energy spectrum compared to B = 0T, and ∅AB = 0 has a limited effect on the overall energy spectrum. Additionally, it was numerically demonstrated that the energy levels remain consistent across three diatomic molecules, regardless of changes in the magnetic quantum number. Furthermore, shifts were noted in orbitals (s, p, d, f), which are uniformly affected by external variables. The values of these orbitals decrease as the orbital quantum number (n) increases, following the expected sequence: s > p > d > f. Computational findings, derived from a thermodynamic approach using partition analysis, show a reduction in the partition function when plotted against temperature, influenced by the magnetic field and flux. It was also observed that both molecules exhibit heightened energy fluctuations as the magnetic field strength and flux increase. The study underscores the critical role of external magnetic factors in shaping molecular energy behaviour through entropy, internal energy, free energy, specific heat capacity and energy fluctuation across various temperature ranges.