## 2 LED Modeling: The Governing Equations
### The Delphi4 LED Chip-Level Multi-Domain LED Model [31]  
The Delphi4 LED multi-domain model is aimed at modeling the electrical, optical, and thermal properties of LEDs at the chip level. The model is an improved version of the original model proposed earlier [21]. It relies on [PERSON]'s model of the IV characteristics of pn-junctions. It is a SPICE-like model, meaning that it can be implemented using standard netlist components of an electrical-only, generic version of the SPICE nonlinear circuit simulation program. As mentioned earlier, the electrical operation depends on the temperature, while the generated heat depends on the electrical operating point and the amount of light emitted. The optical behavior depends primarily on the electrical conditions, but it is also influenced by the LED's operating temperature. The optical losses, eventually, also result in heat generation within an LED package. As the temperature sensitivity of the electrical and optical behavior is high, the model must reflect the link between these domains and the thermal domain. In the Delphi4 LED model, the electrical, optical, and thermal domains are represented as branches (Figure 1). These branches are represented by electrical circuits that can be simulated by any version of the SPICE program. However, according to the modular modeling approach suggested and used in the Delphi4 LED project, the model of the thermal impedance of the junction-to-ambient heat-flow path of the LED package (denoted as \(Z_{Th}\) in Figure 1) is not part of the chip-level LED model. The circuit model representation of the branches basically represents their governing equations that we briefly describe below. Each of the elements can be found in Figure 1.  
The thermal branch (Figure 1a) is simple, consisting of a heat source terminated by an external model of the \(Z_{Th}\) thermal impedance of the package. From the point of view of the steady-state value of the LED chip's junction temperature, the steady-state value of this thermal impedance should be considered: \(R_{ThJA}=Z_{Th}(t=\infty)\). The junction temperature can be determined by solving the following energy conservation equation:  
\[P_{H}\big{(}T_{J},I_{F},V_{F}\big{)}-\frac{T_{J}-T_{a}}{R_{ThJA}}=0 \tag{1}\]where \(P_{H}\) is the total heat generated in the LED package (also known as the heating power), which depends on the electrical operating point determined by the applied forward current \((I_{F})\), the total forward voltage drop (\(V_{F}\)), and the junction temperature (\(T_{J}\)). \(T_{a}\) is the ambient temperature. For modeling the thermal impedance, different approaches have emerged [9], also within the Delphi4 LED project [31; 32]. The heating power can be calculated from the applied electrical power (\(I_{F}\cdot V_{F}\), yellow route in Figure 1) subtracting the optical energy leaving the system (\(\Phi_{e}\), orange route in Figure 1). The electrical branch (Figure 1b) consists of a current source, a series resistance, an ideal diode, and a voltage generator. This represents the LED as a semiconductor diode, seen barely as an electrical component. Since, in practice, LEDs are typically driven by a current source, our multi-domain LED model follows this and is based on a current-driven approach; thus, as indicated in Figure 1, it is driven by the \(I_{F}\) current source, forcing the forward current into the electrical model. The next netlist item in the model is a built-in SPICE diode model, characterized by [PERSON]'s ideal diode equation. In the SPICE diode model used, the calculation of the secondary effects is switched off. The most important effect is represented by the inherent electrical series resistance, \(R_{S}\).