The problem of electromagnetic analysis on a macroscopic level is that of solving Maxwell's equations subject to certain boundary conditions. Maxwell's equations are a set of equations, written in differential or integral form, stating the relationships between the fundamental electromagnetic quantities. These quantities are:     E - the electric field intensity;     D - The electric displacement or electric flux density;     H - The magnetic field intensity;     B - The magnetic flux density;     J - The current density;     - The electric charge density;     You can formulate the equations in differential or integral form. This discussion presents them in differential form because it leads to differential equations that the finite element method can be handle.     For general time-varying fields, Maxwell's equations are:     The first two equations are also referred to as Maxwell-Ampere's law and Faraday's law, respectively. The last two are forms of Gauss' law in the electric and magnetic form, respectively.     Another fundamental relationship is the equation of continuity:     Out of these five equations only three are independent. The first two combined with either the electric form of Gauss' law or the equation of continuity form an independent system. Constitutive Relationships     To obtain a closed system, you need the constitutive relationships describing the macroscopic properties of the medium. They are: where o is the permittivity of vacuum, o is the permeability of vacuum, and is the electrical conductivity. In the SI system the permeability of a vacuum is 4.10-7 H/m. The velocity of an electromagnetic wave in a vacuum is given as co, and you can derive the permittivity of a vacuum from the relationship:     The electric polarization vector P describes how a material is polarized when an electric field E is present. It can be interpreted as the volume density of electric dipole moments. P is generally a function of E. Some materials can have a nonzero P in the absence of an electric field.     The magnetization vector M similarly describes how a material is magnetized when a magnetic field H is present. It can be interpreted as the volume density of magnetic dipole moments. M is generally a function of H. One use of the magnetization vector is to describe permanent magnets, which have a nonzero M when no magnetic field is present.     For linear materials the polarization is directly proportional to the electric field, P = o. e.E, where e is the electric susceptibility. Similarly, in linear materials the magnetization is directly proportional to the magnetic field, M = m.H, where m is the magnetic susceptibility. For such materials the constitutive relations are: where r is the material's relative permittivity, and r is its relative permeability. Usually these are scalar properties but can, in the general case, be 3X3 tensors when the material is anisotropic. The properties and (without subscripts) are the material's permittivity and permeability.     At the present time it is possible to avoid the labor manual work using modern computer technologies. A number of methods have been offered to solve directly Maxwell's equations for various electromagnetic configurations. An easy for use is the finite-elements method, which has been implemented in a large number of computer software packages, such as ANSIS, FEMM, COMSOL MULTIPHYSICS and others. Generalized Constitutive Relationships     For nonlinear materials, a generalized form of the constitutive relationships is useful. The relationship used for electric fields is:     The field Dr is the remanent displacement, which is the displacement when no electric field is present.     Similarly, a generalized form of the constitutive relationship for the magnetic field is: where Br is the remanent magnetic flux density, which is the magnetic flux density when no magnetic field is present. You can generalize J = .E by introducing an externally generated current Je. This relationship is then: Potentials     Under certain circumstances it can be helpful to formulate a problem in terms of the electric scalar potential V and magnetic vector potential A. They are given by the equalities which are direct consequences of the magnetic case of Gauss' law and Faraday's law, respectively.

 The sections contain useful information: references, articles and so on with regard to the relevant section. The Physical Process Modeling team does not claim authorship. The sources are given here. Modeling of the static characteristics of an AC     electromagnet by the 3D FEM Reference Book: Magnetic Materials Abstract In the present paper the static characteristics of a voltage-fed AC electromagnet are modeled by the three dimensional Finite Element Method. Utilized is a time-dependent model, taking into account the time variation of the supply voltage. Analysed are the exciting current and the eddy currents in the short circuited coils. The attractive electromagnetic force is computed by the Maxwell Stress Tensor Method. The obtained numerical results are compared with experimental data.

 Transformers-Theory Time dependent model for analysis of induction motors by the FEM Abstract The paper presents a time dependent model based on the Finite Element Method for analysis of the magnetic field and parameters of the induction motors at steady-stale. The excitation sources are modeled by the line voltages. Obtained are the distribution of the magnetic field and the time variations of the line and phase currents. The electromagnetic torque and the consumed power are computed by the results from the numerical solution.

 Method and computer program for calculation of the magnetic     circuit of electrical machines with variable air Abstract The problems of the electromagnetic field calculation of the "wave" motors are considered in this work. At a steady state, the rotor designed as a flexible steel cylinder is deformed into an ellipse. The magnetic flux arising from the symmetric stator windings is closed through a variable air gap. Axes are defined giving an electromagnetic field definition of the motor. The suggested algorithm for calculation of the asymmetric magnetic circuit involves a solution on the basis of Pung's method for a half-pole pitch and a generalized solution for a whole-pole pilch.

 Supply-voltage optimization by the frequency     control of high-power induction motors Abstract Analyzed is the PMW supply voltage of high-power induction motors. An optimization problem is defined for the determination of the switching-angles set with minimization of the voltage distortion. By means of own software packages, the sets of switching angels of a number 4 to 14 within a half-period are determined. The results are analyzed on the basis of the flux-linkage vector and compared to results of other authors.

INTRODUCTION

Classical cases of induction heating - cylinder and rectangular prism systems inductor-detail are researched in details and are shown with help of mathematical models, in accordance with mathematical description and methods for numerical analysis. Specifications are made by their practical realization of induction devices with different measures and parameters.
In some partial cases induction devices for flat detail's heating are researched on the bases of experimental dependencies because of the complication of theoretic mathematical description. The main problem consists in distribution of the electromagnetic field and the processes connected with it.
In such type of systems other method for analyze can be applied. Definition of system's electromagnetic parameters and with them connected distribution of electromagnetic field can be accomplished with integral dependence, applied for partial elements.
The aim of this research is the theoretical current's distribution, flowing through circle formed detail to be obtained and compendious characters of system flat inductor-detail to be compared with the experimental results of laboratory model.

MATHEMATICAL MODEL

Based on existing methods for electromagnetic fields' analysis questions from the numerical analysis of the system flat inductor-detail are elaborated and these are connected with current distribution in inductor and detail.
We suppose that the problem is considered for sinusoid quantities, and if they are not like mentioned above we accept they are sinusoid. The load (detail) is made of nonferromagnetic material.
A flat circle formed inductor consists of many sections (N). Indexes [A,B,C,D,E,F] describe six sections follow from center to periphery. The selection of these sections is like the experimental model and results are compared with this model. From the numerical results for the mathematics model is established that the minimal number sections are Ndet=2.N+1.
The nonferromagnetic detail in accordance to the inductor is separated in 13 sections from center to periphery - fig. 1. The currents' calculations are done with the method of contour currents.

Fig. 1

The inductor is considered with consistent equivalent scheme with elements -active and inductive resistance - from self-induction and mutual induction for each section.
R means the active inductor's resistance; from R1, to R13 - active resistance of detail's sections.
The active resistance are calculated in accordance to R = .Ii/Si, where: - resistivity; Ii and Si - length and square.
The inductive resistance of inductor's sections are: XL = .L, where: = 2..f, f - frequency (f=50Hz). The inductance L is calculated by the known rule: L=(0/8.).W2.d., where: W - windings of coil, d=(di+di+1)/2 - medial diameter of coil, - function from k=(di+1 - di)/(di+1 + di).
So are calculated all inductive resistance XLA to XLF for the inductor and XL1 to XL13 for the detail.
The inductive resistance of mutual inductivity between inductor's sections are XMAB, XMAC, .... XMEF; between detail's section are XM1-2, XM1-3 .... XM13-1; between both inductor's and detail's section are XMA-1, XMA-2 .... XME-13. They are calculated with the mutual inductivity Mij by Taylor's row method:

Mij=((W1,W2)/6).(MQ1+MQ2+MQ3+MQ4+MP5,+MP6+MP7+MP8-2MPQ)

and the mutual inductivity for 10 coaxial circle formed contours are defined, where in according to:

and F is defined from

a - stretch between the coaxial circle formed contours along their total axis; r1, r2 and W1, W2 - radius and windings for the circle formed contours and they dependent from each other.     For the inductor's loading regime a system of 14 equations is worked out based on the method with contours' currents for complex values of the quantities about I - current through the inductor and I1 .... I13 - currents through the detail's sections.
For the inductor's contour:

For first detail's contour:

Rest 12 equations are obtained like the mentioned above with cyclical change of indexes.
The results of the theoretical researches for two regimes - loudness and loading are shown in table 1, and the graphic performance of the current distribution - on the fig. 2.

Fig2

Table 1
 I[A] U[V] R[] X[] I[A] U[V] R[] X[] 14,5 3,3 0,18 0,14 14,6 3,7 0,18 0,14

Magnetic induction for each section of the detail is quality in accordance with the obtained current distribution.

EXPERIMENTAL RESEARCHES

The experimental researches are carried out on a model (fig. 3) and the mathematical model with consistent connected sections is made for it. Each section consists of 8 windings, which are uniform distributed, with radius r, to r2. On each section border there are drills, which measure the corresponding magnetic flows.

Table 2
 I[A] U[V] R[] X[] I[A] U[V] R[] X[] 15,0 3,3 0,18 0,12 15,0 3,7 0,18 0,14

Table 2 shows the obtained results of the experimental model for 2 regimes -loudness and loading with constant voltage U. Loading is made with a sheet of aluminum thickness 1,4 mm and diameter equal to the external diameter of the inductor. The results of the electrical moved voltage E of the drills are in graphic on fig. 4.
The magnetic flows and their corresponding induction are mean quantities for each section by corresponding data for E.

Fig. 3.

Fig. 4.

SUMMARY

The theoretical and experimental researches of electromagnetic parameters of this device make possible to draw some important conclusions:
- as a result of used analytic dependencies for the definition of system flat inductor-detail's parameters are obtained results close to the experimental data;
- the theoretical distribution and current's values in the detail's contours are produced and this makes possible the thermal processes in the system to be research;
- the experimental dependencies for electric moved voltage in the drills, respectively magnetic flows are shift in comparison with the theoretic maximum.

The obtained exactness for theoretical and experimental currents' values in inductor for regimes of loudness and loading show:
- the used method for theoretical analysis is possible can be applied for researches of induction devices with different switching way of sections;
- with applied of method can be made different in configuration electromagnetic fields which corresponding to determinate limited conditions.

Electromagnetic Module. Modeling, Analysis and Design.

Experiments
with
inductors