*3.2. Transient Thermal Response Measuremnts*

This paper will illustrate in detail how the compact thermal model was derived for the diode D2. A similar procedure should be applied for the other LEDs. Initially, during the measurements the constant current of 700 mA was forced through this diode until the thermal steady state was reached, and then, after switching it o ff, transient temperature responses were measured in the heating device and in the diodes D1 and D6. The measurement current was the same as during the calibration, i.e., 10 mA.

The measurement results are represented in Figure 3b as heating curves, which were obtained by subtracting the recorded diode cooling curves from the respective steady state temperature rise values. The black color in this figure denotes the module with the standard thermal pads. The solid, dashed and double lines are used for the diodes D2, D1 and D6, respectively. The forced LED heating current value was equal to 700 mA, and the measured electrical power was 2.23 W with the standard pad, and 2.18 W with the double one.

**Figure 3.** LED measurement results: (**a**) the LED voltage temperature sensitivity; (**b**) the diode heating curves at 700 mA current.

The thermal response in the heating device, as can be seen in the figure, develops almost immediately. The heat diffuses to the other diodes already within a few seconds, reaching 10% of the final temperature rise value in remote devices after more than 20 s. Although the thermal responses in these devices are visibly attenuated, their steady state temperature rise values amount to at least 40% of the maximal temperature rise in the heating diode, hence indicating the existence of the important thermal coupling between the diodes. The measured steady state temperature values in the devices with the larger thermal pads are always at least 10% lower and the influence of the pad in the heating diode becomes visible already after 100 ms. The effect of a larger thermal pad becomes visible already when heat diffuses into the MCPCB in less than a second. This is probably due to the fact that the thermal pad works similarly as the traditional heat spreader, which facilitates the heat removal from the package.

### *3.3. Computation of Thermal Structure Functions and Time Constant Spectra*

According to the JEDEC standards [30,31], all thermal analyses should be always carried out using the real heating power as the input quantity, which can be found by subtracting from the measured electrical power the value of the optical power emitted in the form of light. Thus, it was also necessary to determine the value of the LED optical power, which could be computed using the method described in [32], based on the knowledge of the measured light intensity at a known distance directly over the diode and the spatial light distribution curve provided by the manufacturer in the datasheet. For the LED heating current considered here, the measured optical and real heating power values were equal to 0.4 W and 1.8 W. Unfortunately, without the information concerning the internal diode structure, it was not possible to evaluate how much heat was dissipated in the semiconductor structure itself and how much was dissipated during the wavelength conversion in phosphorus.

Once knowing the LED heating power, the measured curves were processed using the network identification by deconvolution (NID) method offering the entire set of different thermal analysis tools [28]. The thermal cumulative structure functions and the time constant spectra computed using this method are presented in Figure 4. The cumulative thermal structure functions in Figure 4a show the entire heat flow path from the junction (the origin of the co-ordinate system) to the ambient (the steep vertical line at the end). The deflection points in these curves indicate the heat diffusion to another material, and the horizontal plateaus indicate the total accumulated thermal capacitance.

**Figure 4.** Thermal analysis results: (**a**) the cumulative structure functions; (**b**) the time constant spectra.

Normally, the structure functions are computed only for the driving point thermal impedance, i.e., diode D2 in this case, but here we also included the curves computed for the remote diodes, which have only the large plateau corresponding to the PCB capacitance. The curves for the heating diode D2 have three distinct flat sections, which most probably correspond to the LED die, package and board capacitances. The influence of the larger thermal pad only becomes visible when heat diffuses into the substrate, i.e., for the thermal capacitance over 1 mJ/K and for a resistance of around 8 K/W. All the lighter curves at their ends are shifted visibly to the left of their black counterparts, what confirms the earlier observation that the use of a large thermal pad can effectively reduce the total thermal resistance.

The time constant spectra of the thermal responses computed using the NID method are presented in Figure 4b. In order to expose the short time constant components, the spectra presented here were initially integrated over time. As can be seen, the spectra in the figure contain large thermal time constant components related to the heat exchange with ambient, visible large peaks located around 200 s. Short-time constants are present only in the temperature responses of the heating diode. This part of the heat flow path can be analyzed by dividing the spectra into individual sections in the locations of the minima. These sections are additionally indicated in the figure by the text labels. Consequently, the low thermal resistance section from just below a second to around half a minute reflects the conduction of the heat through the MCPCB. Furthermore, the peaks with the maxima in the range of 10÷30 ms correspond to the interface between the package and the board. The remaining part of the spectra below the minima located at a couple of milliseconds describe the heat flow inside the LED package, except for the narrow peaks visible at tens of microseconds, which are just the artefacts after the removal of the electrical transients from thermal responses.
