Skin Effect Problems Research Articles

BIE formulation for skin and proximity effect problems of parallel conductors

The boundary integral equation (BIE) is used to determine the current distribution in the cross sections of a group of parallel conductors. Magnetic vector potentials at conductor surfaces are calculated by using the boundary element technique. The known current in each conductor is used as a constraint in the formulation. A source component (AA/sup s/) is introduced for the magnetic vector potential in order to maintain the balance between unknowns and equations and to transform Helmholtz's equation into a homogeneous equation. The choice of A/sup s/ is preferred over any other choice because it has physical meaning and its simplifies the calculation of various quantities such as voltage drop along the conductor, power loss, and line impedance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

FEM Analysis of Inverter-Induction Motor Rotor Conduction Losses

A direct method to simulate the steady-state stator and rotor currents in an inverter-induction machine of known geometry and winding design is developed, and the FEM (finite element method) analysis of the skin effect problem is used to determine the variation of rotor resistance with rotor current frequency. The rotor conduction loss can be calculated from these two sets of information. The total losses incurred for a particular mode of inverter operation are determined both experimentally and using the analytical procedure developed here. The total calculated loss is only 3.5% greater than the measured amount, which confirms the validity of the theory developed and its computer implementation. Furthermore, experimental and simulation results of the standstill condition with sinusoidal input were also found to be in close agreement. Finally, the proposed procedure yields information on the instantaneous current distribution throughout the machine, which permits the accurate determination of the flux pattern and optimization of the lamination design. >

An evaluation of the direct boundary element method and the method of fundamental solutions

The advantages and limitations of the direct boundary element method (DBEM) and the method of fundamental solutions (MFS) are compared, will illustrating examples. The salient features of the two methods are discussed. The formulations are described with the aim of illustrating the relative merits of each. The evaluation covers robustness, numerical accuracy, model development and implementation, as well as the ease of applying the impedance boundary condition (IBC). As a test problem, the case of two circular conductors carrying a total impressed current per conductor is considered. A wide range of frequencies and separating distances is examined to show the behavior of each method for skin and proximity effect problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Boundary integral equation formulation of skin effect problems in multiconductor transmission lines

The inductive and resistive skin effect in multiconductor transmission lines is discussed with respect to the analysis and design of the interconnections in high-speed digital circuitry. A boundary integral equation formulation is proposed to reduce the two-dimensional electromagnetic field problem to one-dimensional unknowns along the conductor boundaries. The method of moments is then applied to solve the equation and find the desired resistance and inductance matrices. This approach is suitable for multiple conductors of arbitrary shapes at low as well as high frequencies. Some examples are included to illustrate the method. Results are validated by comparison to known solutions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Direct BIE formulation for skin and proximity effect calculation with and without the use of the surface impedance approximation

The authors illustrate a new, direct BIE (boundary integral equation) formulation for skin and proximity effect calculations. The main advantage of the present formulation is that it is a logical extension of the well-known TM (transverse magnetic) induction formulation. The authors introduce the impedance boundary condition (IBC) approximation of the skin and proximity effect problems in terms of the conventional Fredholm boundary integral equations. Unlike previous formulations, the IBC approximation can be used with the external electric field equation, so the skin and proximity effect problem is formulated as an exterior problem. The method has been applied to test problems; reliable results were obtained both with and without the use of the IBC approximation. >

FEM analysis of inverter-induction motor rotor conduction losses

A direct method to simulate the steady-state stator and rotor currents in an inverter-induction machine of known geometry and winding design is developed, and the FEM (finite element method) analysis of the skin effect problem is used to determine the variation of rotor resistance with rotor current frequency. The rotor conduction loss can be calculated from these two sets of information. The total losses incurred for a particular mode of inverter operation are determined both experimentally and using the analytical procedure developed here. The total calculated loss is only 3.5% greater than the measured amount, which confirms the validity of the theory developed and its computer implementation. Furthermore, experimental and simulation results of the standstill condition with sinusoidal input were also found to be in close agreement. Finally, the proposed procedure yields information on the instantaneous current distribution throughout the machine, which permits the accurate determination of the flux pattern and optimization of the lamination design.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical calculation of steady-state skin effect problems in axisymmetry

A numerical method based on the boundary-element method is described for solving steady-state, axisymmetric skin effect problems. The method can be used to treat problems with very small skin depth. The computer program has been developed with special attention to accuracy and validation. Besides the current density distribution, which is graphically displayed, all global magnitudes can be immediately obtained as final results from the program.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Eddy currents modelling in synchronous machines during starting accounting for the nature of damper end connections

A model is developed for calculating eddy current in electric machines, taking into account the end connections of the conducting materials. The method is based on the combination of classical finite elements, representing the saturated parts of the machine, with a special type of element (conductor macroelements) simulating the subdomains where eddy currents are developed. The end connections of the conductors are considered by incorporating the corresponding circuit equations into the field equations. This technique offers a good accuracy involving reduced meshes and is particularly suitable for thin skin effect problems. The method is used to model the effect of damper configuration on the startup of a 4.5 MVA synchronous machine.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High frequency losses in multi-turn coils using a boundary element model

Skin and proximity effect problems in single and multiple conductors are formulated using an integro-differential expression and are solved using Boundary Element techniques. The formulation is in terms of the vector potential and its normal derivative together with an average potential A <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> to account for integral constraint conditions. The solution accuracy offered by simple constant and linear elements is examined, as is the effect of discretization. The losses in single and multiple turn, solid and hollow conductor coils are examined as a function of turn spacing and separation for values of a/δ as high as 40; a is a characteristic dimension of the coil conductor.

Complementary variational finite-element solution of Eddy current problems using the field variables

A variational finite element method, based upon a functional related to the classical Lagrange density and originally embodied into a program intended for high frequency problems with loss, has been tested on a simple skin-effect eddy current problem. Either of the dual variables E or H may be used subject to complementary boundary constraints. The results so far have been worked in terms of the H variable and show good agreement with the theoretical skin-effect result.

One step finite element formulation of skin effect problems in multiconductor systems with rotational symmetry

This paper presents a new formulation for the classical steady state eddy current problem in multiconductor systems with rotational symmetry. The formulation given in this paper consists of obtaining the magnetic vector potential and the current distribution from the projective solution of the diffusion equation and Ampere's law given the total measurable current flowing in the conductors. This procedure provides an efficient, one step solution technique for the multipath eddy current problem in axisymmetric geometries.

The Skin-Effect at High Frequencies

During the last years, monolithic integrated circuits have been used more and more in microwave techniques. As a result, the metallization thickness of the planar circuits became of the order of the skin depth even at very high frequencies, so that the approximate methods for loss calculations used until recently must be revised. In this paper, a variational formulation of the skin-effect problem for calculating the losses as well as the inner inductances of components in such circuits will be described, and the first numerical results of the method will be discussed.

Dual simple-layer source formulation for two-dimensional eddy current and skin effect problems

The integrodifferential expression resulting from a reformulation of the time-harmonic diffusion equation to accommodate skin effect problems is further transformed, via an approximate average potential A0, to allow solution using a boundary integral technique. Conducting and nonconducting regions are initially separated into subregions. On each subregion boundary a simple-layer distribution of sources is prescribed resulting in a first kind Fredholm integral equation. Dual simple-layer sources are therefore distributed on subregion interfaces. The 2n equations derived from enforcing continuity of normal B̄ and tangential H̄, using point collocation, are added to the current conservation equation for direct solution of the 2n+1 unknowns. This method allows subregion solutions to be decoupled once the sources are determined since the fields in any subregion are adequately prescribed by the corresponding source layer. Curved interfaces are modeled accurately by fourth-order boundary elements. Numerical results, including vector potential distributions and loss ratio and inductance variation with frequency, for a single conductor of circular cross section modeled by four quartic elements are compared with the exact solution. The extension of this technique to noncontacting multiconductor geometries is discussed. It is shown that asymmetric block-sparse matrices are generated for problems made up of more than two subregions.

A new approach to the calculation of three-dimensional skin effect problems

This paper presents a new formulation for skin effect problems in current carrying conductors. The specific difficulties arising in such problems stem from the fact that total measurable currents in conductors rather than local current density distributions are required to be given. These difficulties are particularly significant in the case of three-dimensional (3-D) problems. A new approach presented in this paper attempts to circumvent these difficulties by means of special modification of field equations outside the conductors. As a result of such a modification, the total currents in the conductors appear explicitly in the formulation through the boundary conditions.

FINITE‐ELEMENT SOLUTION OF STEADY‐STATE SKIN‐EFFECT PROBLEMS IN STRAIGHT FLAT CONDUCTORS

Using an integrodifferential approach to steady‐state skin effect problems, the current density distribution in straight flat conductors is solved by the finite‐element method. The approach takes into account a combination of one‐dimensional finite elements corresponding to the flat conductors and triangular finite elements for the remaining domain outside conductors. The results obtained for a flat conductor placed inside a ferromagnetic medium are compared with analytical solutions provided by finite Fourier transforms. As a final output, besides current density distribution, one can calculate parameters useful to designers such a a.c. resistance and reactance of the straight flat conductors.

Open boundary finite elements for axisymmetric magnetic and skin effect problems

Finite element techniques are described for calculating magnetic fields and eddy current distributions in problems with open boundaries. A new matrix recursion algorithm is derived for axisymmetric problems and is applied to the case of a circular coil in free space. Also discussed are open boundary skin effect problems, including an example of three double-channel bus bars. Solutions using the open boundary finite elements are shown to agree closely with classical solutions or with measurements.

A One-Step Finite Element Method for Multiconductor Skin Effect Problems

A one-step procedure is developed to solve the skin effect problem in multiconductor busbars. The procedure differs from previous solutions in that it treats the source (or quasi-static) current density in each conductor as an unknown. It couples the solution of the diffusion equation to Ampere's law, and allows for the finite element solution of skin effect problems to be obtained directly in one step from the currents imposed by the power system in each conductor.

Use of the Bubnov-Galerkin method to skin effect problems for the case of unknown boundary conditions

A method based on coupling the Bubnov-Galerkin method and the separation of variables method is described for the solution of alternating field problems in which the region of prime interest is embedded in an infinitely extending region where Laplace's equation holds. The proposed method can also be used when the stored energy associated with the exterior region and the boundary conditions at infinity are infinite. As an example illustrating the use of the method, the inner impedance of an elliptic conductor is calculated. By means of iterative numerical processes, the method enables the exact values of impedance to be obtained.

New finite element techniques for skin effect problems

This paper describes fwo new techniques for analyzing linear, steady-state, skin effect phenomena in multi-conductor systems by the finite element method. The first of these is based on a linear superposition principle for the field solution, while the second one is based on an integrodifferential equation field formulation. The techniques are applied to single phase sheet-wound transformer problems in which the current windings are modeled by thin axisymmetric sheets.

Integrodifferential finite element formulation of two-dimensional steady-state skin effect problems

A novel integrodifferential approach to steady-state skin effect problems is applied to two-dimensional geometries using traditional triangular finite elements. The forcing function in the new formulation is the total measurable current in the conductors. An explanation of why the new approach works is included. The application of Galerkin's criterion to the integrodifferential equation is outlined and it is shown how to proceed with the finite element discretization. The methods used in the computation of performance measuring quantities useful to designers are described in detail. Simple examples illustrate the method. Results are validated by comparison to known solutions.