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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;
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.
 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
 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.
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Modeling of the static characteristics of an AC electromagnet by the 3D FEM |
Reference Book: Magnetic Materials
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Transformers-Theory
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Time dependent model for analysis of induction motors by the FEM |
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Method and computer program for calculation of the magnetic circuit of electrical machines with variable air |
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Supply-voltage optimization by the frequency control of high-power induction motors |
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Created by Ilonka T. Lilianova (part of Ph.D. Thesis) and Hristofor Tahrilov
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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. MATHEMATICAL MODEL
    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
Regime-lodness | Regime-loading
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
Regime-lodness | Regime-loading
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.
The theoretical and experimental researches of electromagnetic parameters of this device make
possible to draw some important conclusions:
The obtained exactness for theoretical and experimental currents' values in inductor for regimes
of loudness and loading show: Electromagnetic Module. Modeling, Analysis and Design. |
Experiments
with inductors          |
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