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

Created by Physical Process Modeling

    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 U_app 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 L_k.
  Introducing the mutual inductance M_j,k defined as:

  where r_ik 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 J_i,k = J(r_i, z_k). 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 K_p 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.