# Multi-Domain Modelling of LEDs for Supporting Virtual Prototyping of Luminaires

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Context of this Work

#### 1.2. The Wider Background

#### 1.3. Overall Considerations Regarding Multi-Domain Chip Level Modelling of LEDs Aimed at Use in Industrial Application Design Flows

#### 1.4. Expectations for a Chip Level Multi-Domain LED Model

**A**(anode) and

**C**(cathode) electrical terminals and the fluxes, leaving the LED as a function of a forced forward current, ${I}_{F}$, and junction temperature, ${T}_{J}$ (Figure 3). The fluxes we are interested in are:

- The heat-flux (also known as the real heating power, ${P}_{H}$ leaving the device through the thermal junction node denoted by
**J**. - The total emitted radiant flux, ${\Phi}_{\mathrm{e}}$ [36] (also known as the optical power, ${P}_{opt}$), leaving the device through the optical port of the model denoted by
**R**. - The total emitted luminous flux, ${\Phi}_{\mathrm{V}}$ [12], leaving the device through the optical port of the model denoted by
**L**.

## 2. The Starting Point: Basic Spice Model

## 3. An Analytical, Quasi Black-Box Multi-Domain Model of an LED Chip

#### 3.1. The Basic Concept: Splitting the Total Forward Current into Two Components

#### 3.2. Model Equations at a Fixed Reference Temperature, ${T}_{ref}$, in the Current Driven Mode

#### 3.3. Implementation of the Temperature Dependence of the LED Chip Model

#### 3.4. Modelling the Emitted Luminous Flux

_{F}and ${\Phi}_{\mathrm{e}}$ points. However, with this mathematic apparatus, better accuracy can be achieved with the concept presented in Section 4.

## 4. An LED Chip Multi-Domain Model Closer to Physics

#### 4.1. The Physical Roots

- Number of generated electrons/holes and their recombination through different mechanisms within the section. These effects can be thoroughly treated by the collision theory, but a simplified quantity called the generation and recombination rate can be used in the treatment of the statistical mechanics, too.
- Injection from the adjacent sections, determined by the band structure near the interface.

**A. Monomolecular recombination**, where the electrons are recombined at irregularities of the crystalline structure, which are present in a steady number. Such centers of recombination can be dislocations, section boundaries, etc.

**B. Bimolecular recombination**, where the electrons and holes participate in a band to band recombination, in the same section. The recombination rate can be written as $R=B\xb7n\xb7p$.

**C. Auger recombination**, an electron and a hole recombine in a band-to-band transition and give off the resulting energy to another electron (or hole).

- Diffusion of minority carriers at low concentrations, that is, in low numbers compared to the density of the majority carriers. This yields ${I}_{{F}_{d}}\left({V}_{F}\right)={I}_{{0}_{d}}\xb7\mathrm{exp}\left({V}_{F}/{V}_{T}\right)$, corresponding to Equation (1) with $m=1$.
- Diffusion of both types of carriers at high concentrations, when the quasi-neutrality principle makes the density of both carriers equal. This yields a current constituent in an ${I}_{{F}_{q}}\left({V}_{F}\right)={I}_{{0}_{q}}\xb7\mathrm{exp}\left({V}_{F}/2{V}_{T}\right)$ form.

#### 4.2. Modeling the Electrical Characteristics

**+**markers) and the ${R}_{S}$ value which yields the best logarithmic fit on the ${V}_{F}$–${V}_{R}$ relationship in the 10 mA to 1000 mA range is searched for. Herein, this optimization method is referred to as OPT1.

**x**markers in Figure 11a show the fitted ${R}_{S}$ series resistance values generated by the OPT1 method at the given temperatures. The temperature dependence in this range is rather “flat”, so a good fit can be achieved by a second order polynomial approximation. For the model parameter, $m$, the optimization yields several values around $m=2.5$ (the blue

**x**markers in Figure 11b). Figure 12a presents the ${I}_{0}$ values for the best fit.

**+**markers in Figure 11a,b, and Figure 12b).

^{−22}and $a=$ 0.138 in this particular example. The fit is correct in the given range with an ${R}^{2}$ determination coefficient of 0.9972.

#### 4.3. Modeling the Light Output Characteristics

#### 4.4. The Predictive Power of the Physics Based Model

_{F}is compared to the model based on the median device. Still, the error remains in the 0% to 2% range.

#### 4.5. The Actual Implementation of the Physics Based Model

## 5. Multi-Domain Modelling of LEDs for Real Design Tasks

#### 5.1. Modeling an “LED type” and Parameter Extraction

#### 5.2. Applying the Quasi Black-Box Model for a Real Luminaire Design Task

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Delphi4LED workflow and design persona for the generation and application of digital twins (simulation models) of LED packages, modules, and luminaire with which luminaire level design optimization is performed in order to find the best solution to meet lighting design requirements (such as required light output properties) under application conditions.

**Figure 2.**A typical LED efficacy diagram with forward current values suggested for measurement of isothermal IVL (current-voltage-light output) characteristics [35] to cover the LED operating domain relevant for LED luminaire design.

**Figure 4.**Application of a chip level multi-domain LED model in a relaxation type implementation of an electro-thermal-optical solver called an LED luminaire design calculator [3].

**Figure 6.**(

**a**) Temperature dependence of the forward voltage of a diode at constant forward current; (

**b**) Modelling the effect of self-heating for both the pn-junction and the series resistance by a single temperature dependent source.

**Figure 7.**The generic Spice implementation of the chip level multi-domain LED model in LTspice shown as a schematic (resulting voltages for an actual parameter set and an actual setting of the forward current are also indicated in the circuit diagram). With an appropriate dynamic compact model of the environment (referred to as the Z

_{th}RC model in the circuit diagram), the model also describes LED operation under alternating current driving conditions.

**Figure 9.**Current ranges of a pn-junction, I—recombination in the depletion layer, II—minority carrier diffusion, III—ambipolar diffusion, IV—serial resistance: (

**a**) approximation of an LED I-V characteristic in the current ranges I–IV by means of theoretical values of the ideality factor m and an assumed series resistance R

_{S}(

**b**) real LED I-V characteristic covering the current ranges I–IV.

**Figure 10.**Measured forward characteristics of a power LED (

**+**markers) and the logarithmic equation of best fit.

**Figure 11.**Results of the OPT1 and OPT2 optimization methods: (

**a**) ${R}_{S}$ series resistance; (

**b**) for the $m$ ideality factor—fixed at $m=2.5$ in the case of the OPT2 method.

**Figure 12.**Results obtained for the ${I}_{0}$ current coefficient (saturation current): (

**a**) obtained by the OPT1 method; (

**b**) obtained by the OPT2 method.

**Figure 14.**${V}_{F}$ and ${V}_{rad}$ values of a royal blue LED sample measured at ${T}_{J}=$ 30, 50, 70, and 85 °C: (

**a**) Shockley approximation of ${V}_{F}$, log-quadratic fitting curves of ${V}_{rad}$ and their determination coefficient are also shown; (

**b**) comparison of a royal blue (RB) and white (W) LED sample measured at ${T}_{J}=$ 50 and 85 °C; (

**c**) comparison of a red (R) and an amber (A) LED measured at ${T}_{J}=$ 50 and 85 °C.

**Figure 15.**Temperature dependence of parameters $A$, $B$, and $C$ used in Equation (29), for all LED colors: (

**a**) parameter A; (

**b**) parameter B; (

**c**) parameter C.

**Figure 16.**(

**a**) Measured ${I}_{F}$–${V}_{F}$ characteristics; (

**b**) measured radiant efficiency, ${\eta}_{e}$, of the median sample of a red LED population, at various junction temperatures, ${T}_{J}$.

**Figure 17.**Relative error of the ${I}_{F}$–${V}_{F}$ model of red samples, (

**a**) measured ${V}_{F}$ of the median sample; (

**b**) measured ${V}_{F}$ of the sample with minimum ${V}_{F}$, both compared to the calculated ${V}_{F}$ of the model, fitted on the parameters identified for the median sample.

**Figure 18.**Relative error of the ${I}_{F}$–${\Phi}_{\mathrm{e}}$ model of red samples, (

**a**) measured ${\Phi}_{\mathrm{e}}$ of the median sample; (

**b**) measured ${\Phi}_{\mathrm{e}}$ of the sample with minimum ${V}_{F}$, both compared to the calculated ${\Phi}_{\mathrm{e}}$ values of the model, fitted to the parameters identified for the median sample.

**Figure 19.**Predictive power of the ${I}_{F}$–${\Phi}_{\mathrm{e}}$ model for white LEDs: relative error, measured vs. modeled ${\Phi}_{\mathrm{e}}$, of the median sample.

**Figure 20.**Predictive power of the ${I}_{F}$–${\Phi}_{\mathrm{e}}$ model for white LEDs: measured ${\Phi}_{\mathrm{e}}$ of the minimum ${V}_{F}$ sample vs. the ${\Phi}_{\mathrm{e}}$ model generated from the median sample.

**Figure 21.**LTspice implementation of the electrical characteristics of LEDs based on our proposed new model, with controlled sources.

**Figure 22.**Spice implementation of the LED as a heat source and a simplified equivalent Foster chain representing the device and its environment in the thermal domain.

**Figure 23.**Transient simulation, LED switched on to a ${I}_{heat}=$ 1 A heating current. Values of ${I}_{F}$, ${T}_{J}$, ${V}_{F}$, and ${\Phi}_{\mathrm{e}}$ shown, simulated with the LTspice equivalent circuit implementations of the model equations and combined with the simple dynamic compact thermal model shown in in Figure 22.

**Figure 24.**A snapshot of the user interface of the parameter identification tool developed for the quasi black-box multi-domain LED chip model, showing the LED families (two white, a red, an ember, and three blue LED types) and all individual LED samples for which isothermal IVL characteristics were measured. With this tool, parameter sets, as shown in Table 1, can be extracted for selected individual LED samples, or for average characteristics of the given LED population.

**Figure 25.**Luminous flux as the function of ${I}_{F}$ and ${T}_{J}$, calculated by the Visual Basic macro implementation of the quasi black-box model: (

**a**) a phosphor converted white LED; (

**b**) a red LED.

**Figure 26.**The complete parameter sets and fitting errors of the forward voltage and the emitted fluxes of the quasi black-box multi-domain LED model for a set of five samples of Cree XPG3 phosphor converted white LEDs. The physical prototypes of the luminaire aimed for project demonstration were built by using this LED type [3,4].

**Figure 27.**The user interface of the luminaire design calculator Excel spreadsheet application with the design input settings and calculated results of the final, optimized version of the first Delphi4LED demonstrator design.

Input quantities:${I}_{F},{T}_{J}$ | |||

Output quantities:V_{F}, P_{H}, Φ_{e}, Φ_{V} | |||

Model parameters | Symbol in Spice diode model | ||

Thermal | ${T}_{ref}$, ${V}_{T}$ | $\mathrm{TNOM}$, $\mathrm{UT}$ | |

Electrical | entire LED | radiative branch only | |

diode internal pn-junctions at${T}_{ref}$ | ${I}_{0}$, $m$ | ${I}_{{0}_{rad}}$, ${m}_{rad}$ | $\mathrm{IS}$, $N$ |

resistors at${T}_{ref}$ | ${R}_{S}$ | ${R}_{R}$ | $\mathrm{RS}=0$ to switch of Spice’s model of the series resistance |

Coefficients of the model of the “$\Delta {V}_{F}$generators” | ${a}_{el}$, ${b}_{el}$, ${c}_{el}$, ${d}_{el}$, ${e}_{el}$, ${f}_{el}$ | ${a}_{rad}$, ${b}_{rad}$, ${c}_{rad}$, ${d}_{rad}$, ${e}_{rad}$, ${f}_{rad}$ | - |

Coefficients of the efficacy of radiation model | ${a}_{Kap}$, ${b}_{Kap}$, ${c}_{Kap}$, ${d}_{Kap}$, ${e}_{Kap}$, ${f}_{Kap}$, ${g}_{Kap}$, ${h}_{Kap}$, ${i}_{Kap}$ | - |

Color | Code | ${\mathit{\lambda}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}\text{}\left(\mathbf{nm}\right)$ | ${\mathit{V}}_{\mathit{g}}\text{}\left(\mathbf{V}\right)$ |
---|---|---|---|

Royal Blue, White | W, RB | 450 | 2.76 |

Blue | B | 470 | 2.64 |

Green | G | 515 | 2.41 |

Amber | A | 600 | 2.07 |

Red | R | 640 | 1.94 |

Code | ${\mathit{R}}_{\mathit{S}}\text{}\mathbf{max}\text{}\left(\mathsf{\Omega}\right)$ | ${\mathit{R}}_{\mathit{S}}\text{}\mathbf{min}\text{}\left(\mathsf{\Omega}\right)$ | $\mathit{m}$ | $\mathsf{\vartheta}$ |
---|---|---|---|---|

RB | 0.662 | 0.413 | 2.5 | 6.6531 |

W | 0.824 | 0.453 | 1.8 | 5.8421 |

B | 0.610 | 0.277 | 3.0 | 7.7791 |

G | 0.553 | 0.298 | 4.0 | 11.1289 |

A | 0.508 | 0.525 | 1.5 | 8.8466 |

R | 0.773 | 0.688 | 1.5 | 6.5855 |

**Table 4.**Simulated and measured major parameters of the variant of the first Delphi4LED project demonstrator performed in the design style of an SME (small and medium sized enterprise).

Property | Simulated | Measured |
---|---|---|

Total input electric power | 11.71 W | 10.7 W |

Total emitted luminous flux | 1302 lm | 1339 lm |

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## Share and Cite

**MDPI and ACS Style**

Poppe, A.; Farkas, G.; Gaál, L.; Hantos, G.; Hegedüs, J.; Rencz, M.
Multi-Domain Modelling of LEDs for Supporting Virtual Prototyping of Luminaires. *Energies* **2019**, *12*, 1909.
https://doi.org/10.3390/en12101909

**AMA Style**

Poppe A, Farkas G, Gaál L, Hantos G, Hegedüs J, Rencz M.
Multi-Domain Modelling of LEDs for Supporting Virtual Prototyping of Luminaires. *Energies*. 2019; 12(10):1909.
https://doi.org/10.3390/en12101909

**Chicago/Turabian Style**

Poppe, András, Gábor Farkas, Lajos Gaál, Gusztáv Hantos, János Hegedüs, and Márta Rencz.
2019. "Multi-Domain Modelling of LEDs for Supporting Virtual Prototyping of Luminaires" *Energies* 12, no. 10: 1909.
https://doi.org/10.3390/en12101909