*3.4. Foster RC Ladder Compact Thermal Models*

For the generation of the circuit compact thermal models (CTMs), the time constant spectra for the heating LED were divided, as previously discussed, into four individual segments corresponding to the diode package, the interface to the MCPCB, the conduction through the board and the heat exchange with the ambient. This procedure yielded CTMs in the form of four stage Foster RC ladder models. In the case of other LEDs, the models had only one RC stage, which will then be used for the computation of mutual thermal couplings.

Consistent with the method described in [29], initially, the thermal resistor values in all the CTM stages were determined by separately accumulating the respective thermal resistances in each section. Then, the time constant values were found by minimizing the simulation errors with respect to the measured temperature values within each time constant range. Finally, the capacitor values were computed by dividing the respective time constant and resistor values. The heating curves obtained with these models for all the considered diodes in the case of the test module with the standard size

thermal pads are compared with the measured values in Figure 5. The lighter lines (simulated (SIM)) are used for the simulated values whereas the measured ones (MES) are represented by the black ones.

**Figure 5.** Comparison of measured (MES) and simulated (SIM) heating curves for the LED module with the standard size thermal pads: (**a**) in the different diode locations when only diode D2 is heating; and (**b**) in diode D2 for the different number of heating diodes.

When power was dissipated only in the diode D2, see Figure 5a, the simulation results were very accurate for the time instants over 1 s, where the errors did not exceed 0.7 K. These differences in the time range from 0.1 ms to 1 ms even reached 2.5 K, but the simulation accuracy could be further improved by dividing the first section of the CTM into more segments corresponding to the two distant peaks in the time constant spectra in this region, or by changing the method used to remove the electrical transients from the registered thermal responses. The generated CTMs were also used for simulations when the module was heated by more than one diode, i.e., diodes D2 and D1 (denoted in Figure 5b as D21), and then with the diode D6 (marked as D216) as the third heat source. As can be seen, the thermal coupling between the devices becomes visible only after a few seconds, but the two remote LEDs contribute 50 K to the overall temperature rise of diode D2, which is almost the same value as due to the self-heating.

Generally, the diode heating curves presented in Figure 5 can be generated employing the formula given in Equation (1), where the first component represents the four stage RC model of the *k*-th diode and the second component describes the mutual heating by neighboring devices with the one-stage CTMs. *P* denotes here the respective heating powers of each device:

$$T\_k(t) = P\_k \sum\_{i=1}^4 R\_{ki} \left( 1 - \exp\left(-t/\tau\_{ki}\right) \right) + \sum\_{j=1}^5 P\_j R\_j \left( 1 - \exp\left(-t/\tau\_j\right) \right) \tag{1}$$
