## 3 Model and Methods  
### Thermal Resistance of a Semiconductor Device  
PC tests are commonly used for evaluating the fatigue life of LED systems under thermal cyclic loading by simulating the on-/off-switch as in real applications. The measurable physical quantity is the forward voltage \(V_{f}\) at a given forward current \(I_{f}\) across the LED, which can be related to the junction temperature \(T_{j}(t)\) at cycle time \(t\), \(t=1,\ldots,\mathrm{N_{PC}}\). The difference between the ambient temperature \(T_{a}\) and the junction temperature is given as \(\Delta T(t)=T_{j}(t)-T_{a}(t)\). The higher the difference between the ambient temperature and the junction temperature, the higher the resulting mechanical stress over the LED system, leading to failure. The desired imminent physical quantity is the operational thermal resistance \(R_{\mathrm{th}}\) of the LED system. Thermal resistance refers to the ability to dissipate internally generated heat and determines \(T_{j}\)[50]. As shown in Figure 1, the total thermal resistance is composed of different parts: \(R_{\mathrm{th,p}}\) includes all thermal resistances from the LED die to the backside of the PCB, and \(R_{\text{th,TIM}}\) comprises, in addition to the thermal resistance dependent on the material used, the thermal interface resistance from the PCB to the TIM and from the TIM to the heat sink side. \(R_{\text{th,HS}}\) can be negated as long the heat sink has sufficient cooling capacity. The (total) effective thermal resistance is defined as the ratio \(R_{\text{thc}}(t)=\Delta T(t)/P_{\text{el}}\), with electric input power \(P_{\text{el}}\) given as \(P_{\text{el}}=P_{d}+P_{r}\), where \(P_{r}\) defines the electric loss; see [51]. It is assumed that the total electric power fed into the LED system during power cycling is given as \(P_{d}=I_{f}\cdot V_{f}\). Electric input leaves partly as heat and partly as light; thus, \(P_{\text{el}}=P_{\text{opt}}+P_{\text{heat}}\); see Figure 2. \(P_{\text{opt}}\) is the emitted optical power, and \(P_{\text{heat}}\) remains as dissipated heat. Thus, the investigated model assumes that the device has a constant efficiency \(\eta\), and the electric loss \(P_{r}\) can be neglected, i.e., \(P_{\text{el}}=P_{d}\). Therefore, \(P_{\text{opt}}=\eta\cdot P_{d}\), which refers to the optical power the LED emits, and, simultaneously, \(P_{\text{heat}}=P_{d}-\eta\cdot P_{d}=P_{d}\cdot(1-\eta)\). Furthermore, with the residual thermal resistance \(R_{\text{th}}(t)=\Delta T(t)/P_{\text{heat}}\), the effective thermal resistance can be calculated as \(R_{\text{th}_{\text{th}}}(t)=R_{\text{th}}(t)P_{\text{heat}}/P_{d}=(1-\eta)R_ {\text{th}}(t)\). Hence, the temperature step at power cycle time \(t\) can be modelled as  
\[\Delta T(t)=R_{\text{th}}(t)\cdot(1-\eta)\cdot I_{f}\cdot V_{f} \tag{1}\]  
with parameters \(R_{\text{th}}(t)\), \(\eta\), \(I_{f}\) and \(V_{f}\). The temperature swing results in degradation of materials due to thermal fatigue. Cyclic thermal stresses and strains are induced due to temperature changes, spatial temperature gradients and high temperatures under constrained thermal deformation, resulting in a change of the thermal resistance over time. Thermal stresses are produced by cyclic material expansion and contraction when temperature changes under geometric constraints, such as what happens in LED systems under real operating conditions. We focus on the initial temperature swing, \(\Delta T_{t=0}=\Delta T_{0}\), which is vitally important for the increase of the LED junction temperature swing over the number of power cycles (cf., e.g., [52]), i.e.,  
\[\Delta T_{0} = R_{\text{th}}\cdot\eta_{h}\cdot I_{f}\cdot V_{f} \tag{2}\]  
based on the thermal resistance \(R_{\text{th}}\) (at \(t=0\)) and parameters \(\eta_{h}=1-\eta\), \(I_{f}\) and \(V_{f}\). Using a probabilistic framework, we systematically account for different types and sources of uncertainty and variability in the input parameters in (2) to quantify the overall uncertainty in \(\Delta T_{0}\).