
Heating parameters: 
aluminum copper 14 kHz 105 mm 40 mm 125 mm 62 mm 60 s 
Fig. 1 – Geometry 
In this case the induction coil is inside the workpiece. Because the geometry is almost symmetrical, the 3D case can be simplified to 2D. First an electromagnetic harmonic analysis is performed. It calculates where the Joule heat is generated (Fig. 2). The Joule heat density is proportional to the square of the current density. The current in the workpiece flows in the opposite direction than in the coil (Lenz's law). The skin effect can be observed in the figure  the current flows most at the surface of the conductor. Also the proximity effect can be observed: The currents of opposite direction tend to flow close to each other, the currents of the same direction tend to flow far apart. Especially in the lower part of the coil where the diameter is smaller the ring effect is evident. The current tends to flow in the shortest way which is the inner diameter of the coil. Electrical efficiency can be calculated as the ratio of the Joule heat in the housing and the total Joule heat. The electrical efficiency is 40% in this case. More heat is generated in the coil than in the housing. The low electrical resistivity of aluminum impairs the electrical efficiency. It is also true that internal inductors are less efficient than commonly used external inductors. The housing heats up less at the smaller diameter because the gap between the coil and the housing is too large relative to the diameter of the housing.
Electromagnetic analysis is usually sufficient to optimize the shape of the coil. If it is desirable to calculate the temperature, thermal analysis must be added and thermal properties of materials, heating time and heat loss to the environment must be defined. Since this is a 2D case the effect of the cooling fins cannot be simulated but it can be approximately replaced by a severalfold increase in the heat transfer coefficient on the housing surface. Fig. 3 shows the temperature distribution after 60 seconds of heating. Heat generated locally in small areas is quickly distributed throughout the housing due to the high thermal conductivity of aluminum.
Finally a structural analysis is added to the simulation, which calculates the deformation of the housing caused by thermal expansion (Fig. 4). The color scale shows the radial component of the displacement. The larger diameter increased by 0.8 mm and the smaller diameter increased by 0.3 mm.
Fig. 2 – Joule heat 
Fig. 3  Temperature 
Fig. 4 – Deformation 