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 MaxwellAmpere'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 c_{o}, 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 finiteelements method, which has been implemented in a large number of computer software packages,
such as ANSIS, FEMM, COMSOL MULTIPHYSICS and others.




Modeling of the static characteristics of an AC electromagnet by the 3D FEM 
Reference Book: Magnetic Materials


TransformersTheory

Time dependent model for analysis of induction motors by the FEM 

Method and computer program for calculation of the magnetic circuit of electrical machines with variable air 



Supplyvoltage optimization by the frequency control of highpower induction motors 



Created by Ilonka T. Lilianova (part of Ph.D. Thesis) and Hristofor Tahrilov

INTRODUCTION
Classical cases of induction heating  cylinder and rectangular prism systems
inductordetail 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 N_{det}=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 selfinduction and mutual induction for each section. R means the active inductor's resistance; from R_{1}, to R_{13}  active resistance of detail's sections. The active resistance are calculated in accordance to R = .I_{i}/S_{i}, where:  resistivity; I_{i} and S_{i}  length and square. The inductive resistance of inductor's sections are: X_{L} = .L, where: = 2..f, f  frequency (f=50Hz). The inductance L is calculated by the known rule: L=(_{0}/8.).W^{2}.d., where: W  windings of coil, d=(d_{i}+d_{i+1})/2  medial diameter of coil,  function from k=(d_{i+1}  d_{i})/(d_{i+1} + d_{i}). So are calculated all inductive resistance X_{LA} to X_{LF} for the inductor and X_{L1} to X_{L13} for the detail. The inductive resistance of mutual inductivity between inductor's sections are X_{MAB}, X_{MAC}, .... X_{MEF}; between detail's section are X_{M12}, X_{M13} .... X_{M131}; between both inductor's and detail's section are X_{MA1}, X_{MA2} .... X_{ME13}. They are calculated with the mutual inductivity M_{ij} by Taylor's row method:
M_{ij}=((W_{1},W_{2})/6).(M_{Q1}+M_{Q2}+M_{Q3}+M_{Q4}+M_{P5},+M_{P6}+M_{P7}+M_{P8}2M_{PQ})
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 I_{1} .... I_{13}  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
Regimelodness  Regimeloading
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
Regimelodness  Regimeloading
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 