Mihai Maricaru

Modelling eddy currents in thin shields

Purpose – The purpose of this paper is to present a simplified rigorous mathematical formulation of the problem of electric currents induced in thin shields with holes yielding more efficient numerical computations with respect to available methods.Design/methodology/approach – A surface integral equation satisfied by the current density was constructed, which is, subsequently, represented at any point by linear combinations of novel vector basis functions only associated with the interior nodes of the discretization mesh, such that the current continuity is everywhere insured. The existence of the holes in the shield is taken into account by associating only one surface vector function with each hole. A method of moments is then applied to compute the scalar coefficients of the vector functions employed.Findings – It was found that the induced current distribution for shields with holes having the complexity of real world structures can be determined with a satisfactory accuracy utilizing a moderate size...

Novel Solution to Eddy-Current Heating of Ferromagnetic Bodies With Nonlinear

An efficient solution is presented for coupled nonlinear eddy currents-thermal diffusion problems. Applying the fixed-point polarization method to the nonlinear eddy-current field problem, with the magnetization dependent on magnetic induction and on temperature, allows the field computation to be performed for each harmonic separately. Since the fictitious permeability can be chosen to be everywhere that of free space, the matrices of the linear systems to be solved at each iteration remain unchanged even when the nonlinear characteristic changes with the temperature. A simple integral equation is used to compute the eddy currents, the inversion of the matrices corresponding to the harmonics being performed only once, before starting the iterative process. The heat conduction-diffusion equation is solved at each time step by the finite-element method. Three illustrative examples are also presented.

{\mbi B}\hbox{–}{\mbi H}

A new procedure for the study of the evolution of the solid phase in a moving solidifying ferromagnetic metal is proposed. The temperature distribution is controlled using eddy currents induced by a coil that covers partially the crucible surface and by cooling the rest of it, with an imposed crucible velocity. Analysis of the thermal field requires the solution of the time-periodic eddy-current problem coupled with the thermal diffusion problem. The nonlinearity of the B-H relation within the ferromagnetic material of the yoke and inside the solidified material cooled below the Curie point, as well as its dependence on temperature, are taken into consideration. Application of the polarization fixed point method allows the construction of an integral equation for eddy currents and always ensures the convergence of the iterative solution. At each time step, the heat diffusion equation is solved through a standard finite element technique, with the thermal conductivity and the specific heat capacity dependent on temperature.

Characteristic Dependent on Temperature

The piezoelectric effect occurs in materials for which an externally applied elastic strain causes a change in electric polarization which produces a charge and a voltage across the material. The converse piezoelectric effect is produced by an externally applied electric field, which changes the electric polarization, which in turn produces an elastic strain. The most known piezoelectric material is quartz crystal. Many other natural crystalline solids, as Rochelle salt, ammonium dihydrogen phosphate, lithium sulfate, and tourmaline as well as some man-made crystal as gallium orthophosphate, aluminium nitride (AlN), and langasite exhibit piezoelectric properties. A lot of artificial ceramics as barium titanate, lead titanate, lead zirconate titanate (PZT), potassium niobate, lithium niobate, and lithium tantalite have similar properties. The most known technical application is the piezoelectric transducer. In the last years electromechanical AlN resonators emerged as a very efficient solution for mobile communications filters due to the possibility to be integrated at a relatively low cost together with CMOS circuits in systems on a chip and systems in a package. In most applications the piezoelectric devices have a linear behaviour. In Section 2 the linear and nonlinear equations of the piezoelectric effect are described, a new iterative procedure for solving the nonlinear equations is given, and some aspects of the Finite Element solution are discussed. An electromechanical field analysis of a displacement transducer is presented in Section 3. Sections 4 and 5 show some recent applications in the mobile communication technology. The field analysis of a bulk acoustic wave (BAW) resonator using 3D linear models is presented in Section 4. Some nonlinear effects in power BAW resonators together with their circuit models are discussed in Section 5.

Efficient Analysis of the Solidification of Moving Ferromagnetic Bodies With Eddy-Current Control

Solution of the equation of motion for conductors in external magnetic fields requires the knowledge of the magnetic forces due to the induced eddy currents that, in turn, can be determined if the position and the velocity of the bodies are known. An iterative technique is adopted, where, at each time step, an initial value of the magnetic force is used to determine the position and the velocity of the body at the end of the time step and, then, the value of the force is corrected. To reduce the computational effort in the case of thin metallic sheets, it is proposed to use the surface integral equation of the induced eddy currents, with a supplementary term added to account for the motion. A sinusoidal with time variation of the excitation field is considered and a phasor representation of various physical quantities is employed. An average magnetic force over each period is used to solve the equation of motion.

Modeling and Simulation of Piezoelectric Devices

Purpose – The purpose of this paper is to present three novel techniques aimed at increasing the efficiency of the polarization fixed point method for the solution of nonlinear periodic field problems.Design/methodology/approach – Firstly, the characteristic B‐M resulting from the constitutive relation B‐H is replaced by a relation between the components of the harmonics of the vectors B and M. Secondly, a dynamic overrelaxation method is implemented for the convergence acceleration of the iterative process involved. Thirdly, a modified dynamic overrelaxation method is proposed, where only the relation B‐M between the magnitudes of the field vectors is used.Findings – By approximating the actual characteristic B‐M by the relation between the components of the harmonics of the vectors B and M, the amount of computation required for the field analysis is substantially reduced. The rate of convergence of the iterative process is increased by implementing the proposed dynamic overrelaxation technique, with th...

Analysis of the Motion of Conducting Sheets in Magnetic Fields

Application of the current sheet integral equation for an efficient analysis of the electromagnetic field in the presence of thin shields is limited to the case of nonferromagnetic and homogeneous media. An extension of this method to thin shields in the vicinity of ferromagnetic bodies is proposed in this paper for time-harmonic fields. The fixed-point polarization technique is used, where the nonlinear material is replaced by a free space and a distribution of fictitious polarization whose magnitude is corrected iteratively in terms of the magnetic induction. A Fourier decomposition of the polarization is employed and the integral equation is solved for each harmonic. The procedure has a guaranteed convergence and is more and more rapid as the number of harmonics retained decreases. In the beginning only the fundamental harmonic is considered and, afterward, higher order harmonics are added to increase the computation accuracy.

Convergence acceleration in the polarization method for nonlinear periodic fields

The B-H characteristic of an iron body material influences the magnetic field measured in the air. On principle, one can pose the problem of B-H relation determination, by making measurements of the magnetic induction in the neighbourhood of the body. Unfortunately, we have an ill-posed inverse magnetic field problem, for which there is possible that, big variations of the BH characteristic to produce only very small modifications of the magnetic field in the air. It is essential to use a sufficiently sensitive computation procedure in order to produce credible results. This paper proposes a device for the B-H characteristic evaluation, admitting that inside the ferromagnetic bodies the magnetic field distribution is not uniform.

Field Analysis for Thin Shields in the Presence of Ferromagnetic Bodies

To analyze the effects of small variations of the electric machine airgaps due to rotor eccentricity it is necessary to compute the magnetic field highly accurately. An efficient iterative integral technique is proposed, where the material nonlinearity is treated by the polarization method, with the magnetic field determined at each iteration by superposing the contributions of the given electric currents and the polarizations. This computation technique has great advantages over the finite element based procedures, namely, the change in the rotor position does not require the construction of a new discretization mesh, very small airgaps can be taken into consideration without increasing the amount of computation, and the calculated magnetic field in the air is divergenceless and curlless, thus eliminating the introduction of spurious forces. As well, the phase voltages and the magnetic forces are easily calculated from the magnetic field quantities.

In Situ Evaluation of Ferromagnetic Bodies Magnetic Characteristics

A new method for solving the periodic steady state computation of circuits with nonlinear resistors and linear dynamic elements is presented. Any nonlinear resistor is replaced by equivalent sources controlled by its terminal voltages or currents. The convergence of this procedure is assured, unlike the cases of the harmonic balance and shooting methods. Its high convergence speed allows the fast computation of the steady state solutions. An example is given for illustration.