Showing papers on "Magnetic potential published in 2015"
TL;DR: In this paper, a nonlocal geometrically nonlinear beam model is developed for magneto-electro-thermo-elastic (METE) nanobeams subjected to external electric voltage, external magnetic potential and uniform temperature rise.
136 citations
TL;DR: In this article, the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto-electro-thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Karman geometric nonlinearity was investigated.
Abstract: This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Karman geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.
79 citations
TL;DR: In this paper, a simple proof of concept of an innovative and active technology which allows the individual manipulation of single droplets is reported. But it still requires complex chips such as microelectrode arrays, or equipment, such as laser-based sorting.
Abstract: Droplet microfluidics is currently undergoing an explosive development due to its ability to compartmentalize samples in picolitre to nanolitre volumes, transport them without dispersion and perform high-throughput analysis. The precise manipulation of single droplets, however, still requires complex chips, such as microelectrode arrays, or equipment, such as laser-based sorting. We report here a very simple proof of concept of an innovative and active technology which allows the individual manipulation of single droplets. This technology combines ferromagnetic rails and magnetic nanolitre droplets. Ferromagnetic rails are used to locally create magnetic potential wells. When the field is turned OFF, the hydrodynamic drag force transports the magnetic droplets according to the flow velocity profile. By switching ON the magnetic field, droplets experience a magnetic force that affects their trajectory when passing over the magnetized rail. The combination of the drag force exerted by the oil flow and the magnetic force resulting from the magnetized rail leads to a deflection force that guides the droplet along the rail, thus imposing a deterministic trajectory. The magnetic rails networks offer a spatially and temporally addressable guidance and sorting of individual magnetic droplets by synchronizing field activation and droplets positions. Numerical simulations were performed to evaluate spatial distribution of both drag and magnetic forces within the microdevice. The influence of different parameters such as magnetic flux density magnitude, flow rate and orientation of the rail has been investigated. Finally, selective droplet sorting, parking and merging were demonstrated and the monitoring of parallelized enzymatic reactions was performed.
47 citations
TL;DR: In this paper, an assimilative procedure for mapping high-latitude ionospheric electrodynamics is developed for use with plasma drift observations from the Super Dural Auroral Radar Network (SuperDARN) and magnetic perturbation from the Active Magnetosphere and Planetary Electrodynamic Response Experiment (AMPERE) to infer complete distributions of electrostatic potential and vector magnetic potential.
Abstract: An assimilative procedure for mapping high-latitude ionospheric electrodynamics is developed for use with plasma drift observations from the Super Dural Auroral Radar Network (SuperDARN) and magnetic perturbation observations from the Active Magnetosphere and Planetary Electrodynamics Response Experiment (AMPERE) This procedure incorporates the observations and their errors, as well as two background models and their error covariances (estimated through empirical orthogonal function analysis) to infer complete distributions of electrostatic potential and vector magnetic potential in the high-latitude ionosphere The assimilative technique also enables objective error analysis of the results Various methods of specifying height-integrated ionospheric conductivity, which is required by the procedure, are implemented and evaluated quantitatively The benefits of using both SuperDARN and AMPERE data to solve for both electrostatic and vector magnetic potentials, rather than using the data sets independently or solving for just electrostatic potential, are demonstrated Specifically, solving for vector magnetic potential improves the specification of field-aligned currents (FACs), and using both data sets together improves the specification of features in regions lacking one type of data (SuperDARN or AMPERE) Additionally, using the data sets together results in a better correspondence between large-scale features in the electrostatic potential distribution and those in the FAC distribution, as compared to using SuperDARN data alone to infer electrostatic potential and AMPERE data alone to infer FACs Finally, the estimated uncertainty in the results decreases by typically ∼20% when both data sets rather than just one are included
45 citations
TL;DR: In this paper, the covariant Duffin-Kemmer-Petiau (DKP) equation in the curved space-time of a cosmic string was studied in the presence of a magnetic vector potential in the Coulomb gauge and the analytical solutions for the modified spectrum of the system were reported.
Abstract: We study the covariant Duffin-Kemmer-Petiau (DKP) equation in the curved space-time of a cosmic string. The calculations are done in the presence of a magnetic vector potential in the Coulomb gauge and the analytical solutions for the modified spectrum of the system are reported.
44 citations
TL;DR: The proposed method provides more realistic predictions for the vortex dynamics of the TDGL equations in nonsmooth domains, while the vortex motion influenced by a defect of the domain is of high interest in the study of superconductors.
44 citations
TL;DR: In this article, an axially symmetric E-cored coil with a circular air gap inside the core column located above a two-layered conductive half-space is presented.
Abstract: The problem of an axially symmetric E-cored coil with a circular air gap inside the core column located above a two-layered conductive half-space is presented. The truncated region eigenfunction expansion method is used to derive expressions describing the magnetic vector potential of the filamentary coil. The final expressions for the impedance of the rectangular cross-sectional coil are obtained, and calculations for various frequency values are carried out. The results are compared with those from the COMSOL package, which shows a very good agreement.
37 citations
TL;DR: In this article, a new approach for finite element simulations of the time-dependent Ginzburg-Landau equations (TDGL) in a general curved polygon, possibly with reentrant corners, was introduced.
35 citations
TL;DR: In this article, the two-dimensional sliding frictional contact between a rigid perfectly conducting flat punch and a functionally graded magneto-electro-elastic material (FGMEEM) layered half-plane was investigated.
Abstract: This paper presents the two-dimensional sliding frictional contact between a rigid perfectly conducting flat punch and a functionally graded magneto-electro-elastic material (FGMEEM) layered half-plane. The electric potential and magnetic potential of the punch are assumed to be constant within the contact region. The magneto-electro-elastic (MEE) material properties of the FGMEEM layer vary as an exponential function along the thickness direction, and the Coulomb type friction is adopted within the contact region. By using the Fourier integral transform technique, the problem is reduced to coupled Cauchy singular integral equations of the first and second kinds for the unknown surface contact pressure, electric charge, and magnetic induction. An iterative method is developed to solve the coupled equations numerically and obtain the surface MEE fields. Then, the interior MEE fields are also obtained according to the surface MEE fields. Numerical results indicate that the gradient index and friction coefficient affect both the surface and interior MEE fields significantly.
34 citations
TL;DR: In this article, the eigenvalues of a magnetic Schrodinger operator of the Aharonov-Bohm type depend on the singularities of its magnetic potential, and the connection of this eigenvalue problem to the problem of finding spectral minimal partitions of the domain is made.
Abstract: We study how the eigenvalues of a magnetic Schrodinger operator of Aharonov-Bohm type depend on the singularities of its magnetic potential We consider a magnetic potential defined everywhere in ℝ2 except at a finite number of singularities, so that the associated magnetic field is zero On a fixed planar domain, we define the corresponding magnetic Hamiltonian with Dirichlet boundary conditions and study its eigenvalues as functions of the singularities We prove that these functions are continuous, and in some cases even analytic We sketch the connection of this eigenvalue problem to the problem of finding spectral minimal partitions of the domain
33 citations
TL;DR: In this paper, a transceiver transceiver with three identical windings located 120° apart on the same physical axis is proposed to estimate the position of a defect by analyzing the amplitude and phase of the sum of the three terminal voltage signals.
Abstract: Metallic tube inspection techniques using eddy current probes have evolved over the years from those employing a single bobbin coil to rotating coils and arrays, in an attempt to improve the speed and reliability of inspection. This paper presents a novel eddy current transceiver probe design that uses a rotating electromagnetic field. The transceiver coils consist of three identical windings located 120° apart on the same physical axis. A three-phase sinusoidal current source is used for exciting the coils. The phase voltages are identical in amplitude, but 120° apart in phase. The rotating magnetic field generated by the three-phase current is sinusoidal in space and time and so are the induced eddy currents in the tube wall. The sensor achieves mechanical rotating probe functionality by electronic means and eliminates the need for mechanical rotation. The terminal voltages of the three-phase windings can be measured during the scan. The defect’s axial and angular position can be estimated by analyzing the amplitude and phase of the sum of the three terminal voltage signals. The probe is sensitive to defects of all orientations and is as effective as conventional rotating pancake coil probes while offering the advantages of high inspection speed and greater reliability, since the probe does not rotate mechanically. A 3-D finite-element model based on reduced magnetic vector potential Ar, $ {V}$ -Ar formulation was developed to simulate and predict the response of the probe to a variety of defects. A prototype unit consisting of a probe connected to a three-phase constant current source and data acquisition system was developed and tested. Experimental results validating the simulation model and demonstrating the feasibility concept are presented.
TL;DR: In this paper, the authors studied the magnetic field at a short wavelength and simulate its relaxation toward equilibrium on two and three dimensional periodic domains, in both helical and non-helical settings.
Abstract: Freely decaying relativistic force-free turbulence is studied for the first time. We initiate the magnetic field at a short wavelength and simulate its relaxation toward equilibrium on two and three dimensional periodic domains, in both helical and non-helical settings. Force-free turbulent relaxation is found to exhibit an inverse cascade in all settings, and in 3D to have a magnetic energy spectrum consistent with the Kolmogorov $5/3$ power law. 3D relaxations also obey the Taylor hypothesis; they settle promptly into the lowest energy configuration allowed by conservation of the total magnetic helicity. But in 2D, the relaxed state is a force-free equilibrium whose energy greatly exceeds the Taylor minimum, and which contains persistent force-free current layers and isolated flux tubes. We explain this behavior in terms of additional topological invariants that exist only in two dimensions, namely the helicity enclosed within each level surface of the magnetic potential function. The speed and completeness of turbulent magnetic free energy discharge could help account for rapidly variable gamma-ray emission from the Crab Nebula, gamma-ray bursts, blazars, and radio galaxies.
TL;DR: In this article, the infimum of the Ginzburg-Landau functional in a 2D simply connected domain with an external magnetic field allowed to vanish along a smooth curve was studied.
Abstract: We study the infimum of the Ginzburg–Landau functional in a two dimensional simply connected domain and with an external magnetic field allowed to vanish along a smooth curve. We obtain energy asymptotics which are valid when the Ginzburg–Landau parameter is large and the strength of the external field is below the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field. Our results complete former results obtained by K. Attar and X.B. Pan–K.H. Kwek.
11 May 2015
TL;DR: In this article, the authors developed an analytical model to predict the power factor of the FM-PMSM and investigate the essence of the low power factor, which has a good agreement with the commercial finite element method (FEM) to quickly obtain the electromagnetic solution.
Abstract: In recent years, flux modulation is increasingly receiving attention since it is capable of adjusting the pole-pair number and rotational speed of the magnetic field. This principle has already been applied to magnetic gear (MG) to realize the low-speed high-torque operation [1]. Flux-modulated permanent magnet synchronous machine (FM-PMSM) is an integration of a coaxial MG with a conventional outer rotor PM machine. Compared with the coaxial MG, a stator with armature windings is employed in the FM-PMSM to replace the inner rotor of the coaxial MG [2-3], as shown in Fig. 1 (a). A modulation ring with a number of ferromagnetic pole-pieces is sandwiched between the outer rotor and stator, and two air-gaps are thereby formed to separate the above three components from each other. The modulation ring could act as a mediator to match the pole-pairs differential between the two magnetic fields generated by the PMs and armature windings, respectively. The advantage of the FM-PMSM is that it could provide high torque at low speed directly without employing mechanical gearbox, mechanical issues, such as friction loss and mechanical fatigue, are thus avoided and high reliability could be achieved. However, it is found that the power factor of the FM-PMSM is relatively low [4], and few details in the published literatures are provided to explain the intrinsic causes of this problem. Therefore, this paper will develop an analytical model to predict the power factor of the FM-PMSM and investigate the essence of the low power factor. Because of the existence of the flux modulation, many high frequency harmonics are introduced into the magnetic field of the FM-PMSM. It is important to have a good knowledge of the magnetic field distribution in the FM-PMSM in order to accurately predict its power factor. This paper gives an alternative to the commercial finite element method (FEM) to quickly obtain the electromagnetic solution. This approach could provide many physical insights into the electromagnetic solution. In this paper, partial differential equations are used to describe the magnet field behavior in terms of magnetic vector potentials. The whole calculation domain is partitioned into six regions, viz. the stator slot and slot opening (Region I_i and Region II_i), the inner and outer air-gaps (Region III and Region V), the slots between modulation pole-pieces (Region IV_j), and the PMs (Region VI), as shown in Fig. 1 (b). In Region II_i, III, IV_j and V, the magnetic vector potentials are governed by Laplace's equations. And in Region I and Region VI, magnetic vector potentials are of Poisson's equations due to the existence of current and PMs, respectively. By applying the constraints on the interfaces between these regions, the solution of the magnetic field could be derived. Fig. 2 (a) compares the analytically and numerically calculated distribution of the radial flux density in the inner air-gap due to the PMs on the outer rotor. As can be seen, the analytical results have a good agreement with the FEM calculation, which indicates the developed analytical model could be employed as a powerful tool for predicting and analyzing the power factor of the FM-PMSM. With the knowledge of the solution of the magnetic field, the flux linkage of each phase winding could be obtained through the surface integral of the magnetic vector potential in Region I_i. Then, the self and mutual inductances of each winding could be deducted, which are 4.06 mH and 2.02 mH, respectively. Similarly, the analytically and numerically calculated inductances have a good agreement. Substituting the inductances, currents and resistances into the voltage equation, the voltage of each phase winding could be obtained as well. Fig. 2 (b) presents the waveforms of the voltage and current of phase-A winding versus the time at the rated power. Finally, the active and reactive power could be calculated by expressing the flux linkage in terms of impedances, and the predicted power factor of the studied FM-PMSM is 0.6, which is quite low compared with conventional PM machines. And the low power factor will inevitably lead to the increase of the capacity, volume and loss of the inverter.
TL;DR: In this article, the existence of axi-symmetric stationary solutions for the Euler-Poisson system of compressible fluids coupled to a magnetic field is proved by a variational method.
Abstract: The existence of magnetic star solutions which are axi-symmetric stationary solutions for the Euler–Poisson system of compressible fluids coupled to a magnetic field is proved in this paper by a variational method. Our method of proof consists in deriving an elliptic equation for the magnetic potential in cylindrical coordinates in \({\mathbb{R}^3}\), and obtaining the estimates of the Green’s function for this elliptic equation by transforming it to 5-Laplacian.
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TL;DR: In this article, the non-inertial effects on the covariant DKP equation in the curved space-time of a cosmic string were studied in the presence of a magnetic vector potential in the Coulomb gauge.
Abstract: We study the non-inertial effects on the covariant DKP equation in the curved space-time of a cosmic string. The calculations are done in the presence of a magnetic vector potential in the Coulomb gauge and the analytical solutions for the modified spectrum of the system are reported.
TL;DR: An in-house code is developed to simulate the inductively coupled plasma (ICP), and it is demonstrated that there is an overall increase in the plasma potential when current is extracted from ICP with a biased wall electrode.
TL;DR: In this article, a numerical solution of nonlinear ferromagnetic problems is formulated using the three-dimensional time-domain finite element method (TDFEM) combined with the inverse JilesAtherton (J-A) vector hysteresis model.
Abstract: In this work, a numerical solution of nonlinear ferromagnetic problems is formulated using the three-dimensional time-domain finite element method (TDFEM) combined with the inverse JilesAtherton (J-A) vector hysteresis model. After a brief introduction of the J-A constitutive model, the second-order nonlinear partial differential equation (PDE) is constructed through the magnetic vector potential in the time domain, which is then discretized by employing the Newmark-β scheme, and solved by applying the Newton-Raphson method. Different Newton-Raphson schemes are constructed and compared. The capability of the proposed methods is demonstrated by several numerical examples including the simulation of the physical demagnetization process, the prediction of the magnetic remanence in the ferromagnetic material, and the generation of higher-order harmonics.
11 May 2015
TL;DR: In this article, the analysis of eddy-current loss of a magnetic coupling that uses a Halbach array permanent magnet (PM) is presented, where the authors obtain the magnetic field solutions of the Halbach magnetized PM based on the magnetic vector potential and 2-D polar coordinate system and derive analytical solutions for eddy current loss using the equivalent electrical resistance that is calculated from the magnet volume and conductivity.
Abstract: This paper reports on the analysis of eddy-current loss of a magnetic coupling (MC) that uses a Halbach array permanent magnet (PM). A MC is a kind of noncontact magnetic device (NCMD). When the conductive materials of NCMDs are exposed to time-varying magnetic fields, eddy currents are induced in the conductive materials, leading to losses. An analysis of the eddy-current loss is therefore required, and for MCs, the analysis of eddy-current loss is particularly more important. We obtain the magnetic field solutions of the Halbach magnetized PM based on the magnetic vector potential and 2-D polar-coordinate system. In addition, using the derived magnetic field solutions, we derive analytical solutions for eddy-current loss using the equivalent electrical resistance that is calculated from the magnet volume and conductivity. A 2-D finite-element analysis is employed to confirm the validity of the analytical results of the eddy-current density and eddy-current loss. Finally, we investigate the influence of the number of Halbach segments on the eddy-current loss induced on the MC.
TL;DR: In this article, the Laplace equation of a scalar magnetic potential is used to calculate the general vector field inside a real (finite) solenoid, using only the magnitude of the field along the symmetry axis.
Abstract: In a region free of currents, magnetostatics can be described by the Laplace equation of a scalar magnetic potential, and one can apply the same methods commonly used in electrostatics. Here, we show how to calculate the general vector field inside a real (finite) solenoid, using only the magnitude of the field along the symmetry axis. Our method does not require integration or knowledge of the current distribution and is presented through practical examples, including a nonuniform finite solenoid used to produce cold atomic beams via laser cooling. These examples allow educators to discuss the nontrivial calculation of fields off-axis using concepts familiar to most students, while offering the opportunity to introduce themes of current modern research.
TL;DR: An original and powerful volume integral formulation for solving electromagnetic problems is proposed in this article, which is based on a facet interpolation for representing the magnetic flux in magnetic regions and the current density in conducting regions.
Abstract: An original and powerful volume integral formulation for solving electromagnetic problems is proposed. It is based on a facet interpolation for representing the magnetic flux in magnetic regions and the current density in conducting regions. This formulation is particularly well adapted for solving electromagnetic problems where air is preponderant, whatever the frequency. First results obtained are very encouraging, in terms of efficiency and accuracy.
TL;DR: In this article, a new method for numerically solving the equations of electromagnetic induction in conducting materials using native, primary variables and not a magnetic vector potential is presented, which allows the meshed domain to cover only the processed material rather than extend further out in space.
TL;DR: In this paper, a mixed analytical modeling method for linear oscillating motor with silicon steel stator is proposed, which combines Magnetic Potential Vector (MPV) and 3D equivalent magnetic circuit (3D-EMC) methods.
Abstract: Linear oscillating motor receives much attention recently due to direct and efficient linear motion output. In order to achieve high thrust and eliminate eddy current effect, circumferential silicon steel stack is usually utilized in stator as a universal topology design. However, circumferential arrangement of silicon steel introduces 3D magnetic circuit, which complicates the design process and cannot be solved using conventional modeling method. In addition, it leads to severe magnetic saturation effect and must be taken into accounts in the analytical model for accurate results. In this paper, a novel mixed analytical modeling method for linear oscillating motor with silicon steel stator is proposed. Magnetic potential vector (MPV) and 3D equivalent magnetic circuit (3D-EMC) methods are combined together in this mixed model. Magnetic potential vector method sets up the coarse flux density distribution, while 3D equivalent magnetic circuit method is used for refinement considering saturation. Moreover, the performance between radial and parallel magnetized structure is compared based on this model. For validation, FEM and experimental results are compared with the proposed model and proves its effectiveness and accuracy, which can be used in further motor structure optimization. It may also act as a universal guidance to other PM motor design.
Patent•
28 Oct 2015
TL;DR: In this article, an analytical method based permanent magnet motor field analysis and torque calculation method was proposed, which comprises the following steps of: step 1: establishing a magnetic vector potential mathematical model of a gap zone and a permanent magnet region; step 2: establishing the potential of a slot area; step 3: calculating a cogging torque; and step 4: analyzing and comparing an analytical calculating result.
Abstract: Provided is an analytical method based permanent magnet motor field analysis and torque calculation method. The method comprises the following steps of: step 1: establishing a magnetic vector potential mathematical model of a gap zone and a permanent magnet region; step 2: establishing a magnetic vector potential mathematical model of a slot area; step 3: calculating a cogging torque; and step 4: analyzing and comparing an analytical calculating result. The invention provides an analytical method based permanent magnet motor field analysis and torque calculation method. According to the structural features of a surface mounted radially magnetized permanent magnet motor model, the analytical model is divided into three sub-regions, namely a permanent magnet region, a gap zone and a slot area; a magnetic vector Laplace equation or a Poisson equation is established in each region; sub-region equations are connected through junction conditions, so as to obtain the analytical expression of a load air gap field and a slot field by using a direct analytical method.
TL;DR: An improved extended finite element method for modeling electromagnetic devices with multiple nearby geometrical interfaces by incorporating multiple derivative discontinuous fields based on the partition of unity method results in the high accuracy in the approximation field and derivative field.
Abstract: This paper proposes an improved extended finite element method for modeling electromagnetic devices with multiple nearby geometrical interfaces. In regions near these interfaces, the magnetic vector potential approximation is enriched by incorporating multiple derivative discontinuous fields based on the partition of unity method such that the interfaces can be represented independent of the mesh. The support of a node or an element can be cut by several interfaces. This method results in the high accuracy in the approximation field and derivative field. Numerical examples applied to the iron core in 1-D eddy current field involving level set-based parts, error analysis, and electromagnetic field computations are provided to demonstrate the utility of the proposed approach.
TL;DR: The proposed method, being based on the search of a scalar magnetic potential in the non-conducting region, has the advantage of making use of a reduced number of unknowns.
TL;DR: A setup that simultaneously measures the three components of the magnetic field with high spatial resolution and takes a novel approach to determine the magnetic vector potential via volume integration of the measured field reproduction with very good accuracy is implemented.
TL;DR: In this article, a solution to the double curl equation with generalized Coulomb gauge is proposed based on the vectorial representation of the magnetic vector potential, which can be expanded by the edge elements, where its vectorial nature is retained and only the tangential continuity is required.
Abstract: In this paper, a solution to the double curl equation with generalized Coulomb gauge is proposed based on the vectorial representation of the magnetic vector potential. Traditional Coulomb gauge is applied to remove the null space of the curl operator and hence the uniqueness of the solution is guaranteed. However, as the divergence operator cannot act on edge elements (curl-conforming) directly, the magnetic vector potential is represented by nodal elements, which is too restrictive, since both the tangential continuity and the normal continuity are required. Inspired by the mapping of Whitney forms by mathematical operators and Hodge (star) operators, the divergence of the magnetic vector potential, as a whole, can be approximated by Whitney elements. Hence, the magnetic vector potential can be expanded by the edge elements, where its vectorial nature is retained and only the tangential continuity is required. Finally, the original equation can be rewritten in a generalized form and solved in a more natural and accurate way using finite-element method.
TL;DR: In this paper, the authors developed analytical expressions for the magnetic vector potential and the magnetic field of modulated double helical coils (MDHCs) under the assumption that the coils are tightly wound and can be approximated by two infinite thin current sheets.
Abstract: We develop analytical expressions for the magnetic vector potential and the magnetic field of modulated double helical coils (MDHC). It is assumed that the coils are tightly wound and can be approximated by two infinite thin current sheets. From each coil parametric equation, the surface current density is derived by considering the turn-to-turn spacing. The field inside and outside each coil is then calculated by decomposing the surface current density into three contributions. By summation, the field of any MDHC can be obtained. As a validation, we evaluate the field of a superconducting double helical quadrupole. A good agreement is found between analytical and numerical results. From the field, the main coil parameters can be quickly estimated with a fair accuracy. This can be useful for the design an optimization of advanced magnets, both conventional and superconducting.