## 2 Materials and Methods
### Temperature Rise in Time Domain Due to the Static Heating Term  
Using (7), the static heating term in (15) can be written as  
\[q_{DC}(z)=\frac{H_{0}{}^{2}}{2}\mu_{0}\mu_{R}\omega\exp(-\frac{2z}{\delta}) \tag{16}\]  
When induction heating starts at the time \(t=0\), the resulting surface temperature change \(\Delta T\) at the surface can be calculated based on a result in [17] (p. 80):  
\[\Delta T=\frac{H_{0}{}^{2}}{2}\mu_{0}\mu_{R}\frac{\omega}{\lambda}\delta^{2} \bigg{(}\frac{1}{2\sqrt{\pi}}\frac{\mu_{T}}{\delta}-\frac{1}{4}\Big{(}1-\exp( \Big{(}\frac{\mu_{T}}{\delta})^{2}\Big{)}erfc\Big{(}\frac{\mu_{T}}{\delta} \Big{)}\Big{)}\bigg{)} \tag{17}\]  
where  
\[\mu_{T}(t)=2\sqrt{\pi t} \tag{18}\]  
is the thermal penetration depth, the depth of the thermal propagation for a given observation time \(t\) after heating begins. \(\lambda\) and \(\alpha\) are the thermal conductivity and the thermal diffusivity of the conducting material, respectively. The time dependence of \(\Delta T\) in Equation (17) is implicitly given by \(\mu_{T}\) (Equation (18)). The temperature change is governed by the ratio of thermal penetration depth and electromagnetic skin depth. Equation (17) can be used to compare the thermal response of a material after a certain heating time caused by a given induction field at its surface (in the following called 'induction heating efficiency'). Table 1 shows examples for material parameters and penetration depths and for selected materials in induction thermography. Table 1 lists the surface temperature rise \(\Delta T\) resulting from a magnetic induction \(B_{0}=0.02\) T in air or a magnetic field of \(H_{0}=1.59\times 10^{4}\) A/m at an induction frequency of 100 kHz after 0.1 s of inductive heating. Figure 1: Power density profiles \(q(z,t)\) in the conducting material for selected times \(t\). The dashed line represents the time average. Four groups of materials can be identified in Table 1. Owing to their high magnetic permeability, the ferromagnetic metals have a very small skin depth compared to the thermal penetration even after a short time. Only a thin surface layer generates the heat. The nonmagnetic metals in the second group are often good electrical conductors, but their relative permeability is close to one. Their skin depth is still one order below the typical thermal penetration. In the third group, there are metallic alloys with relatively poor electrical and thermal conduction. Here, the electromagnetic and thermal penetration depths are of comparable size for the given parameters. Finally, the materials of the last group have an electromagnetic skin depth significantly larger than the thermal penetration. Table 1 shows that the ferromagnetic metals have the highest induction heating efficiency. On the other hand, the induction heating efficiency in the second group is poor. In particular, the aluminium alloys have an efficiency about a factor of 30 below that of magnetic steel, rendering testing of these materials difficult. This is further complicated by their often low infrared emissivity. In spite of this, successful applications on aluminium were reported [18]. If the thermal penetration depth is larger than the skin depth, \(\mu_{T}>>\delta\), Equation (17) can be simplified to  
\[\Delta T(t)=\frac{H_{0}}{2}\mu_{0}\mu_{R}\frac{2}{\sqrt{\pi}}\omega\delta\frac {\sqrt{t}}{\sqrt{(\lambda\rho c)}}. \tag{19}\]  
Here, \(\rho\) and \(c\) are the density and the specific heat capacity of the material, respectively. This describes a classical surface heating process, which is characterised by a temperature rise proportional to the square root of time. As the skin depth \(\delta\) is proportional to \(\omega^{-0.5}\), the temperature rise \(\Delta T\) is proportional to the square root of the induction frequency. Owing to their larger skin depth, the alloys with \(\mu_{T}/\delta\cong 1\) represent a transition from surface heating to volume heating. The induction heating efficiency is much better than that of the second group.