# Super-Frequency Sampling for Thermal Transient Analysis
## 4 Results and Discussion  
### Thermal Transient Reconstruction  
The above-described setup is used to measure the thermographic image of an SMD diode, as shown in Figure 4 (left). The diode with the dimensions of \(2.3\,\mathrm{mm}\) by \(1.4\,\mathrm{mm}\) is indicated by a dashed line. The algorithm is evaluated based on the temperature difference between the mean of the \(11\times 11\) test area on top of the diode (black square) and the mean temperature of the two reference areas, which also have \(11\times 11\) pixels in size (white squares). Figure 4 (right) shows the temporal behavior of the temperature difference for the test area, with the diode excited by a square wave signal. The true sampling period, \(\tau_{\mathrm{true}}\), is then extracted from the measurement data in the temporal range from \(0\,\mathrm{s}\) to \(220\,\mathrm{s}\), as described in Section 2.2.  
Figure 3: Sketch of the experimental setup. The heating current from the power supply is switched by a MOSFET, which is controlled by a frequency generator. A voltmeter is used for voltage measurements.  
Figure 5 shows the cost function for the initial (left) and for the final search range (right). In the first iteration, the cost function oscillates between two branches but converges to a clear minimum. The refinement of the search interval, \(\tau_{\text{min}}\,\ldots\,\tau_{\text{max}}\), results in the formation of plateaus, as shown in Figure 5 (right). Transitions between these plateaus occur at critical periods at which the sequence of the reconstructed measurement points changes. Within the plateaus, only the distance between the points varies, which has no impact on the cost function. The algorithm converges in this example in seven iterations, resulting in a true period value of \(\tau_{\text{true}}=20\,\text{s}\,+\,1.7895\cdot 10^{-4}\,\text{s}\) with an uncertainty of \(\Delta\tau_{\text{true}}\,=\,0.0005\cdot 10^{-4}\,\text{s}\). The uncertainty is given by the width of the plateau (indicated as the gray shaded area).  
Figure 4: **(left)** Thermographic image of the diode in the on-state. The marked areas indicate the \(11\times 11\) test area as well as the two \(11\times 11\) reference areas. **(right)** Temporal behavior between the mean temperature difference of the test area at the center of the diode and the two reference areas.  
Figure 5: Cost function for the reconstructed temperature signal. **(left)** Initial search range. **(right)** Final search range with a global minimum in the gray range and the obtained true sampling value, \(\tau_{\text{true}}\), marked with a solid black line. On the \(x\)-axis, the difference between the used signal period for the reconstruction, \(\tau\), and the set signal period, \(\tau_{\text{set}}\), is shown.  
Having extracted the true period, \(\tau_{\text{true}}\), the recorded thermal transient (Figure 4 Right) is collapsed. The starting (end) point of the heating curves is defined as the first significant increase (decrease) in temperature. The impedance, \(Z_{\text{th}}(t)\), is then calculated via  
\[Z_{\text{th}}(t)=\frac{T(t)-T_{0}}{P}. \tag{4}\]  
The initial temperature, \(T_{0}\), of the heating (cooling) is equal to the end temperature of the cooling (heating) curve. The power amounts to \(P=0.9\,\text{A}\cdot 1\,\text{V}=0.9\,\text{W}\).  
The calculated thermal impedance of the heating phase is shown in Figure 6. The points, derived from an analysis of the first period only, are depicted as red dots. With super-frequency sampling (eleven periods), however, additional points (green) within each decade are obtained. The temporal resolution is increased and allows for the analysis of the thermal behavior in the range between \(10^{-4}\) and \(10^{-3}\).