## 2 Materials and Methods
### Temperature Compensation and Luminous Flux Regulation
Figure 7 shows the scaled data and dynamic model behavior of the thermal resistance MP-5050-240E, where the expression (40) provides a satisfactory performance. It is observed that the thermal stabilization of the LED occurs around 70,000 s, i.e., 1 h and 57 min. In addition, the change in the temperature of the LED is calculated by multiplying the dynamic thermal resistance and the power of the LED. \[RR_{th}=0.4980\cdot\log(t)^{2}+13.9903\cdot\log(t)+39.1209 \wedge \max(RR_{th})=100 \tag{39}\] \[DR_{th}=\frac{R_{th}\cdot RR_{th}}{100} \tag{40}\]  
Figure 7 shows that the dynamic thermal resistance of the LED increases over the operation time, which reduces the effective luminous flux \(Efx_{LED}\) as observed in Figure 6. The simulation of the dynamic behavior of the LED is performed as follows:  
1. Set the LED current \(i_{LED}\) to the desired value. 2. Calculate the LED voltage \(v_{LED}\) using (35) and the LED power as \(P_{LED}=i_{LED}\cdot v_{LED}\). 3. The dynamic thermal resistance \(DR_{th}\) is calculated from (39) and (40) for the particular time instant of the simulation. Figure 4:0 LED relative luminous flux and effective luminous flux vs. temperature. The increment in the LED temperature is calculated as \(\Delta T_{LED}=DR_{th}\cdot P_{LED}\). Then, the LED temperature is calculated by adding the ambient temperature \(T_{a}\) as \(T_{LED}=\Delta T_{LED}+T_{a}\). 5. The theoretical luminous flux \(Fx_{LED}\) of the LED is calculated from (36). 6. The relative luminous flux \(Rfx_{LED}\) caused by the LED temperature is calculated from (37). 7. Finally, the effective luminous flux \(Efx_{LED}\) produced by the LED, due to the temperature effect, is calculated from (38). Figure 6 put into evidence that keeping constant the effective luminous flux produced by the LED requires to increase the LED current over the operation time. Therefore, each LED must be start operating below its maximum rating; for the LED MP-5050-240E it is 800 mA. To achieve the desired 2000 lm in each LED group requires providing 666.67 lm in each parallel-connected LED (3 LEDs form a group), and the current in each LED group is calculated multiplying by 3 expression (36). Moreover, since the LEDs in each group are close, the temperature increase is equal to multiplying by 3 the \(\Delta T_{LED}\) produced by a single LED. In order to produce the desired 2000 lm it required a theoretical group LED current equal to 1.65 A, which is calculated from (36). Setting such a current, and executing the previous simulation steps, produce the dynamic simulation of a LED group reported in Figure 4:1 This figure shows the dynamic thermal resistance, without a logarithmic axis, that reaches the expected steady-state value, that is, the manufacturer \(R_{th}\). The LED group temperature \(T_{LED}\) starts at the ambient temperature, which in this simulation is considered \(T_{a}=30\)\({}^{\circ}\)C, reaching a steady-state value equal to 50 \({}^{\circ}\)C. This temperature increase reduces the relative luminous flux \(Rfx_{LED}\) of the LED group, also reducing the effective luminous flux \(Efx_{LED}\) produced by the LED group. The previous simulation confirms that the current of the LED group must be adjusted to compensate for the temperature effect on the effective luminous flux \(Efx_{LED}\) produced by the LED group. From simulation data between 0 and 1 s, the dynamic relation between \(Efx_{LED}\) and \(i_{LED}\) is calculated as the following transfer function:  
Figure 4:2  
Figure 4:3 Dynamic behavior of the thermal resistance.