## 2 LED Modeling: The Governing Equations
### Measurement  
The parameters of the Delphi4 LED model were fitted to a sufficiently detailed set of so-called isothermal IVL characteristics. These characteristics were obtained using compliant procedures outlined in JEDEC standards JESD51-51A and JESD51-52A [33; 34], which involved combined thermal and radiometric/photometric measurements. During these tests, the LEDs under test (DUT) had their junction temperatures set and kept at a pre-specified value while the LEDs were driven by a pre-specified value of the forward current. Thermal impedance, light output characteristics (total emitted radiant flux and spectral power distribution), and forward voltage were measured. Forward current values covered almost three decades, from the mA range up to the 1000 mA range, and junction temperatures varying from room temperature up to 100-120 \({}^{\circ}\)C.  
The measurement setup utilized SIEMENS SIMCENTER T3 Ster, Budapest, Hungary and SIMCENTER, TeraLED equipment (compliant with JEDEC JESD51-51A and JESD51-52A standards [33; 34]), completed with a spectroradiometer (Instrumentsystems CAS-140 CT, Munich, Germany).  
According to the standards, the measurements were conducted in the hot steady-state of the DUT LED, when the junction temperature stabilized after powering. Waiting for the thermal steady state was time-consuming (minutes for each measurement point), but it was necessary to characterize the diode under stable conditions as required by the CIE's recommendations for the optical testing of LEDs [35].  
The measurement control software of the joint T3 Ster-TeraLED setup enabled achieving the LEDs' hot steady state at a pre-specified value of their \(T_{J}\) junction temperature and capturing the IVL characteristics at such pre-defined, constant-junction temperature values. The problem with such a temperature control was that, once the forward current of the DUT LED changed, its heating power also changed; thus, the temperature of the cold plate holding the DUT LED had to be adjusted in order to maintain the pre-defined junction temperature. The settling time of this temperature control mechanism took several minutes, representing a significant portion of the overall measurement time to characterize a single LED package. One of the goals of this research is to find a way to establish the LED chip-level model without the requirement of the pre-defined junction temperatures during the measurement. A sample of a measurement result is shown in Figure 2. In subsequent sections, the forward voltage (\(V_{\rm F}\)), the radiative flux (\(\Phi_{e}\)), and the luminous flux (\(\Phi_{V}\)) will be analyzed at each forward current (\(I_{\rm F}\)) and junction temperature (\(T_{junction}\)). This measurement was aligned with the aforementioned standards. Therefore, the junction temperature goal was set according to the reference (first column). Not all reference temperatures can be achieved by the measurement apparatus, though. For instance, a junction temperature of 110 \({}^{\circ}\)C cannot be achieved with a 100 mA current due to the LED's limited heating power. In the original extraction method, such data points were excluded, as discussed in Section 3.1. One other goal of this research is to be able to use these points as well for model extraction purposes.  
As detailed in Section 3.2, the new parameter extraction method allows using isothermal characteristics in terms of the ambient temperature control, (\(T_{a}\)), instead of the characteristics measured at a constant junction temperature, (\(T_{J}\)). The benefit is that the temperature control loop of the measurement system is much simpler, resulting in shorter overall measurement times, while the level of detail and accuracy of the measurements are maintained, and all the measured data can be used in the extraction phase. Since the chip-level model lacks a thermal component, the junction temperature measurement is still essential.