## 3 Results and Discussion
### Temperature Rise with Excitation Frequencies in the 2.5 to 7 Hz Range  
At very low frequencies, a ferromagnetic material will be magnetised in the well-known form of a hysteresis loop. The area in the loop is equal to the power released per cycle of the excitation, the hysteresis loss. However, the shape of the hysteresis curve is frequency dependent, as Barkhausen jumps have a characteristic time constant and eddy currents are generated locally around moving domain walls [25]. With increasing induction frequency, hysteresis curves under sinusoidal excitation deform increasingly more to an elliptical shape. Initially, the eddy currents around the Bloch walls lead to additional (excess) heat release; afterward, the macroscopic eddy current losses on a scale beyond the length of a domain wall (classical losses) dominate, allowing for approximation of the magnetic behaviour by a relative permeability \(\mu_{R}\), as introduced in Section 2.1. At frequencies lower than 50 Hz, the losses in ferritic steel are often dominated by hysteresis losses [25]. It has been shown that the release of heat within one hysteresis cycle occurs with two peaks in time [26]. This means that a doubling of the electromagnetic excitation frequency should occur, not only in the domain of macroscopic eddy current losses, but also in the hysteresis domain. A recorded temperature rise in the first second of excitation averaged over the evaluation area is shown in Figure 5a. The recorded temperature signal over 20 s was processed by a pulse phase thermography algorithm (a Fourier transform for each pixel) in order to obtain the frequency spectra shown in Figure 5b. In Figure 5a, one can see that the thermal response at the induction frequency \(f=2.5\) Hz exhibits a periodicity at 5 Hz. It is superimposed by a background signal close to 23 Hz. This also appears in the amplitude frequency spectrum shown in Figure 5b. The 23 Hz component is always present in the spectra, independent of the induction frequency and even without any inductive excitation and is probably due to vibrations in the IR camera. The frequency of this changes through time, probably due to a temperature dependence, up to 25 Hz within one hour. There is no peak at the induction frequency of 2.5 Hz, but a  
Figure 5: (**a**) Measured temperature signal in digital levels DL as a function of time on the sample surface of the ferromagnetic sample over 1 s of excitation at an induction frequency of 2.5 Hz. (**b**) The corresponding frequency amplitude spectrum obtained for a recording time of 20 s.  
strong peak at 5 Hz. A higher harmonic signal at 10 Hz is weakly visible. The origin of the tiny peak at 8.2 Hz is unknown. The rise in the spectrum towards zero frequency is due to the general linear rise of temperature, as expected for \(\mu_{T}<\delta\). The dashed line in Figure 5a is a fit to this temperature rise. Owing to the small \(2f\) signals, care has to be taken to exclude other effects as origin. Some of these might be:  
(a) Mechanical vibration of the sample and/or magnet or magnetostriction due to the strong magnetic fields. Some vibrations of the system magnet-sample are present. They are visible at \(2f\) at the sample edges, which cause an edge-contrast enhancement. Vibration and magnetostriction might also cause such effects in the evaluation area. However, the signal from the evaluation area remains unchanged when the camera lens is defocused, which causes significant spatial averaging. Measurement of the actual mechanical vibration using an acoustic transducer was not successful, as the stray field close to the magnet generated electromagnetic interference in the transducer and its connections. (b) Direct electromagnetic interaction of the magnetic stray fields with the IR camera. Such artefacts are occasionally observed in induction thermography experiments. However, when the sample is covered by a plate of rubber with a thickness of 2 mm, the camera signal is largely damped.