## 3 LED Model Parameter Extraction Methods
### Original Extraction Strategy  
The first parameter extraction approach was presented in detail in [22] and briefly described in this section.  
As a first step, the reference junction temperature, \(T_{ref}\), is defined, at which the parameters of the temperature-independent diodes included in the model of Figure 1b,c are valid. At \(T_{J}=T_{ref}\), the dVF_el and dVF_rad temperature-controlled voltage generators, which describe the temperature dependence of the two electrical-only diode models in Figure 1, according to Equations (3) and (5), present 0 as the output voltage.  
The extraction process starts with the electrical branch. At a high forward current, the series resistance dominates in the overall voltage drop across the LED: \(I_{F}\cdot R_{S}\gg m\cdot U_{T}\cdot\ln(I_{F}/I_{0}+1)\); thus, \(I_{F}\cdot R_{S}\approx\ V_{F}\). Therefore, the slope of the high current range of the measured \(I_{F}-V_{F}\) characteristic is a good approximation for the value of \(R_{S}\). Once the serial resistance is estimated, the measured \(I_{F}-V_{F}\) curve can be stripped of its effect, resulting in an almost ideal diode characteristic. The ideality factor and the saturation current can be estimated by curve fitting Equation (2) to the series of \(V_{F}\) values modified this way, yielding the estimated values of \(m_{e}\) and \(I_{0e}\) (values of \(I_{F}\) are pre-defined ones, set during the measurements). This initial estimation of the parameters for Equation (2) is quite rough, so the initially obtained value of \(R_{S}\) has to be fine-tuned in order to achieve a better overall fit of Equation (2) to the measured dataset. This approach requires an exhaustive search in the parameter spaces of \(R_{S}\), \(I_{0e}\), and \(m_{e}\). A good fit at this stage of the parameter extraction procedure is critical since the \(V_{r}\) voltage drop on the entire \"radiative diode\" denoted as branch (c) in Figure 1 is calculated as \(V_{r}=V_{F}-R_{S}\cdot I_{F}\). The \(I_{r}\) current across this \"radiative diode\" can be calculated by solving Equation (4), with \(\Delta V_{r}=0\), corresponding to the \(T_{J}=T_{ref}\) assumption.  
The optical branch is very similar to the electrical one, but for the \(I_{r}-V_{r}\) relationship, Equation (4) provides an implicit formula. The relationship between the \"experimental values\" of \(I_{r}\) and the measured \(\Phi_{e}\) total radiant flux values provided by the optical measurements during the physical LED characterization process is provided by Equation (6).  
The six parameters of the \(\Delta V_{e}\big{(}T_{J}\big{)}\) relationship are determined by quadratic fitting. However, for any \(T_{J}\) different from the \(T_{ref}\) reference temperature, a fitting procedure for the of \(R_{S}\), \(I_{0e}\), and \(m_{e}\) parameters also needs to be performed, requiring substantial computation resources and long execution times, without assuring the best global optimum. The same applies to the parameters of the \(\Delta V_{r}\big{(}T_{J}\big{)}\) relationship and the \(R_{r}\), \(I_{0r}\), and \(m_{r}\) parameters of the \"radiative diode\" denoted as branch (c), Figure 1. The optimal parameters for the entire multi-domain LED model were obtained using a brute-force method [22]. The parameters of the simple model of the luminous flux--Equations (8) and (9)--are determined by quadratic fitting. This process is straightforward, but takes a long time in its 9-dimensional parameter space.