## 2 LED Modeling: The Governing Equations
### The Delphi4 LED Chip-Level Multi-Domain LED Model [31]
In the current driven modeling approach, the ideal diode behavior and the effect of the electrical series resistance are jointly described by  
\[V_{F}=m_{e}\cdot U_{T}\cdot ln\left(\frac{I_{F}}{I_{0e}}+1\right)+I_{F}\cdot R _{S} \tag{2}\]  
where \(V_{F}\) is the forward voltage, \(m_{e}\) is the diode's ideality factor, \(U_{T}=k_{B}T/q\) is the thermal voltage, \(I_{F}\) is the forward current, \(I_{0}\), is the saturation current, and \(R_{S}\) is the series resistance, the temperature dependence of which is merged into the overall model of the diode's temperature sensitivity. It is a known fact that, under forced constant forward-current operating conditions, the forward voltage of the diode diminishes with the increasing junction temperature. This temperature-dependent forward voltage change is represented by the \(\Delta V_{e}\) voltage-controlled voltage generator connected in series with the temperature independent diode and \(R_{S}\) models. The controlling voltage of this voltage generator is the voltage of the junction node, \(T_{J}\), as indicated by the red route in Figure 1. The temperature dependence of the voltage of the \(\Delta V_{e}\) generator is approximated as follows:  
\[ \begin{split}\Delta V_{e}=\left(a_{e}\cdot I_{F}^{2}+b_{e}\cdot I _{F}+c_{e}\right)\cdot\left(T_{J}^{2}-T_{ref}^{2}\right)\\ +\left(d_{e}\cdot I_{F}^{2}+e_{e}\cdot I_{F}+f_{e}\right)\cdot \left(T_{J}-T_{ref}\right)\end{split}  \tag{3}\]  
where \(T_{ref}\) is the reference temperature for which the parameters of the ideal diode model describing the \(I_{F}-V_{F}\) relationship and the constant value of the \(R_{S}\) series resistance were identified. Note that this temperature dependence of the change in the diode's overall forward voltage corresponds to the quadratic relationship of the so-called K-factor function proposed by the most recent thermal testing standard, JESD51-51A, published by JEDEC [33]. Symbols \(a_{e}-f_{e}\) used in Equation (3) are model parameters to be identified. The model of the optical branch (Figure 1c) is based on the idea that the electrons comprising the total forward current (\(I_{F}\)) can be divided into two parts, according to the recombination processes to which they are subject to in the active area of the LED's pn-junction. In this way, we separate the total forward current into a component associated with the radiative recombination processes (direct bandgap transitions of the charge carriers) giving rise to photon emission, and into another component that is associated with non-radiative recombination processes, eventually resulting in heat generation. The current-voltage characteristics of these forward current components can both be described by [PERSON]'s diode equation [31]. To improve model the optical losses, the ideal light-emitting diode through which the charge carriers pass to perform radiative recombination processes is complemented by a series resistance, denoted as \(R_{r}\) in Figure 1. Thus, similarly to the model of the overall electrical behavior of the LED, the optical branch also comprises a temperature independent electrical series resistance (\(R_{r}\)) and a temperature independent ideal diode model, supplemented by a voltage-controlled voltage generator (\(\Delta V_{r}\)). The controlling voltage of this generator, again, is the voltage representing the junction temperature of the LED. The model equation is similar to Equation (2), but the characteristic here is presented in a voltage-controlled manner, describing the dependence of this radiative diode on the \(V_{r}\) forward voltage of the internal pn-junction of the LED (excluding the effect of the \(R_{S}\) electrical series resistance):  
\[I_{r}=I_{0r}\cdot\left[exp\left(\frac{V_{r}-\Delta V_{r}-I_{r}\cdot R_{r}}{m_{ r}\cdot U_{T}}\right)-1\right] \tag{4}\]  
where \(I_{r}\) is the radiative current component of the overall forward current of the entire LED, \(I_{0r}\) is the saturation current, and \(m_{r}\) is the ideality factor of the radiative diode.