The electromagnetic phenomena of induction heating are described, mathematically, by Maxwell equations.

Maxwell's equations, in differential form, can be written as:

(from Ampere's law) [1]

(from Faraday's law) [2]

(from Gauss' law) [3]

(from Gauss' law) [4]

where

• is the magnetic field intensity,
• is the conduction current density,
• is the electric flux density,
• is the electric field,
• is the magnetic flux density,
• is the electric charge density, and
• is time.

and can be related to and through the electromagnetic material properties of permittivity, , and magnetic permeability, , according to the following equations:

[5]

[6]

The Maxwell equations can be further reduced using Ohm's law:

[7]

Substituting equations [5] and [7] into equation [1], and noting that for current frequencies less than 10 mHz, the induced current, , is greater than the displaced current density, , hence this term can be ignored, then equation [1] can be written:

[8]

Since the magnetic flux density, , satisfies zero divergence from equation [3], it can be expressed as a magnetic vector potential, , such that:

[9]

Substituting equation [9] into equation [2] gives:

[10]

Therefore,

[11]

where is the electric scalar potential. Equation [7] now becomes:

[12]

where is the amplitude of the source current density in the coil and is given by:

[13]

Substituting equations [6], [9] and [12] into equation [8] gives:

[14]

Now, using the triple product vector identity equation [15]

[15]

on equation [14] gives:

[16]

Now noting that for one component vector potential fields

[17]

then equation [16] reduces to

[18]

For the sinusoidal steady state with angular frequency = 2f, and units (rad/s), equation [18] becomes:

[19]

Once the time harmonic magnetic vector potential, , is solved the magnetic field flux density can be found from equation [9]. The time harmonic induced eddy currents, , in the conductors is given by equation [20]:

[20]

From which the Joule heat, , in the conductors can be found :

[21]

The Joule heat is the volumetric heat source, with units (W/m3), that is induced by the Eddy currents in the conductor.

The heat transfer phenomenon taking place in induction heating is the heat conduction within the conductor and is described by the transient heat conduction equation that is used in all the simulations.

[22]

where

• is the temperature,
• is the density,
• is the specific heat capacity,
• is the thermal conductivity of the material, and
• is the Joule heat from equation [21].