## 2 Induction Heating and Numerical Simulations
### Analytical Solution  
For geometrically simple symmetrical inductors and workpieces, calculations of the distributions of magnetic field, eddy currents and heat sources can be performed analytically. However, prior to such an analytical solution, numerous simplifications have to be introduced. The equations applied to characterize the distributions of the investigated parameters are in this case calculated using the Bessel functions, the solution of which is rather complicated and can lead to results of insufficient reliability for complicated tasks [44]. For this reason, the characterization and numerical expression of the parameters affecting the workpiece and its vicinity during the induction heating process are based on the general equations for the characterization of the magnetic field--the Maxwell equations (see, e.g., [45]).  
The Maxwell equations are then supplemented with equations characterizing the heated material (Equations (2)-(4)),  
\[\overset{\rightarrow}{J} =\sigma\cdot\overset{\rightarrow}{E} \tag{2}\] \[\overset{\rightarrow}{B} =\mu\cdot\overset{\rightarrow}{H}\] (3) \[\overset{\rightarrow}{D} =\varepsilon\cdot\overset{\rightarrow}{E} \tag{4}\]  
where \(B\) is magnetic induction (T), \(E\) is the electric field intensity (V-m\({}^{-1}\)), \(D\) is electric induction (C\(\cdot\)m\({}^{-2}\)), \(\varepsilon\) is electric permittivity (F\(\cdot\)m\({}^{-1}\)), \(H\) is magnetic field intensity (A\(\cdot\)m\({}^{-1}\)), and \(J\) is current density (A\(\cdot\)m\({}^{-2}\)).  
The fact that especially the \(\sigma\) and \(\mu\) parameters influence the behavior of the heated metal in the outer alternating magnetic field needs to be pointed out. For the induction heating of ferromagnetic materials featuring high \(\mu_{r}\) values (hundreds to thousands), the magnetization process is non-linear. Such materials intensify the magnetic field, and thus the induction effect generating the eddy currents is already substantial at low frequencies. The design and analyses of induction heating are typically based on amplitude magnetizing characteristics. Nevertheless, both these characteristics are strongly temperature-dependent up to the Curie temperature, at which ferromagnetism diminishes. Beyond this critical temperature, the material, including a ferromagnetic one, further behaves as paramagnetic. Crossing this critical temperature also results in the modification of the characteristic hysteresis loop. In other words, the non-linear dependence changes into linear and magnetic permeability are further expressed by a constant value approaching the limit of 1. However, this does not apply for paramagnetic materials [46]. Tungsten heavy alloys (THA), being paramagnetic, feature the permeability value  
Figure 1: General principle of induction heating: schematic depiction of coil and heated workpiece (**a**); detailed depiction of eddy currents (**b**); graphical depiction of skin effect (**c**).  
of 1, i.e., their \(B\!/\!H\) characteristics are linear; see Figure 2a (to be able to thoroughly characterize the magnetic properties of THAs, the magnetic susceptibility, \(\sim\!\!7.8\times 10^{-5}\) for THAs, would also have to be considered). Similarly, \(\mu_{r}\) is also constant, expressed by the value of 1, for this material (see Figure 2b).