About the mathematical description of electromagnetic parameters of system flat inductor-detail

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.

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FEM MODELLING OF AN INDUCTION HEATING SYSTEM

The paper presents investigation of axisymmetrical induction heating system, consisting of two-layer disc-type inductor and heated detail. Two - dimensional coupled electromagnetic and temperature fields were analysed using FEM. The problem was solved as nonlinear and transient. The obtained numerical results have been compared to the experimental data for the magnetic flux and temperature distribution.
The present paper deals with numerical and experimental investigation of an induction heating system. It consists of two-layer disc-type inductor and heated detail. The numerical models of the coupled electromagnetic and temperature fields are based on the finite element method (FEM) and electromagnetic and temperature distribution have been obtained using ANSYS 9.0 software package. The results were compared to the experimental data for the temperature distribution in the laboratory experimental setup.
The aim of the work is investigation of possibilities for creating adequate field models at some control points of the induction system. They can be used in the next work, concerning special requirements for temperature distribution in the heating detail. So it is a first step for future solving of the optimization problem.
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Simulation of the process

 Movie 1     Movie 2     Movie 3     Movie 4 Movie 5     Movie 6     Movie 7     Movie 8 Movie 9     Movie 10     Movie 11 Movie 12     Movie 13     The electromagnetic problem is quasistationary and the field model with respect to the magnetic vector potential A is based on the equation: where is angular frequency, is electric conductivity, is magnetic permeability and J is current density.     The transient thermal field is modeled by: where is thermal conductivity, T is temperature, is density, c is specific heat and q is power density. Equation is solved under convection and radiation boundary condition.     The coupled problem is solved using indirect coupling of the quasistationary electromagnetic and transient thermal problem. The electromagnetic problem is solved in a domain consisting of the whole system and a wide buffer zone around it. The thermal problem is solved only in the heated detail..... Theory background

Induction heating - modeling, analysis, simulation and experiment

Induction heating treatments are used on steel components to produce surface hardening by self-quenching of the heated surface. Coaxial coils are used to treat components with cylindrical symmetry. High frequency alternating current is made to flow through the inductor coils. Through the action of the associated fluctuating magnetic .eld, oscillating eddy currents are induced in the treated component without the need of electrical contact. The induced currents together with the electrical resistance of the material result in local- ized heating by Joule e.ect. The resulting temperature field is then directly related to the electro-magnetic parameters of the system. The objective of modeling is to produce a mathematical representation of the induction heating process by .rst determining the induced current distribution in the component. Ultimately, one would like to produce a predictive capability capable of assisting in process optimization and new process design. The formulation of the problem requires statement of the electromagnetic field (Maxwell's) equations in the time-harmonic form neglecting displacement fields:

where E, H, B and J are, respectively, the electric field, the magnetic field, the magnetic flux density and the current density vectors, is the frequency. Further, since the magnetic field density can be represented in terms of a magnetic potential A by B = X A one has:

where is the permeability. From the above the vector potential inside a volume u can be expressed as:

Integrating the first Maxwell equation over the area a, using Stokes theorem and the above yields:

where Uapp is the externally applied scalar electromagnetic potential, = J/E is the electrical conductivity and L is the length of a current carrying path. This equation is valid for each closed current carrying path Lk.
Introducing the mutual inductance Mj,k defined as:

where rik is the straight line distance between points i and k, the above equation becomes:

where the subscripts i and k refer to current carrying loops i and k, respectively.
The above is an integral equation that can be solved numerically by first subdiving the domain of interest into a finite number of current carrying loops and then adding all individual contributions. Specifically, consider an axisymmetric system consisting of a part to be induction treated and the coaxial inducting coils. Next, subdivide the workpiece and the coil into a set of n current carrying loops in the r-direction and a set of m loops in the z-direction. For simplicity, let all current carrying loops have the same square cross sectional area h2.
Let the current density passing through the loop labelled i, k be Ji,k = J(ri, zk). From the above, this current plus the result of the collective influence of the currents passing through all other loops in the system must equal the scalar potential at the loop, i.e.

Since, there is no applied potential in the workpiece, the equations there are:

The mutual inductance in this case is given in closed analytical form by

and Kp and E(p) are the elliptic integrals of .rst and second order, respectively. The self-inductance is given by:

The above is a system of algebraic equations that can be solved by standard numerical linear algebra methods.