*Article* **Digital Luminaire Design Using LED Digital Twins—Accuracy and Reduced Computation Time: A Delphi4LED Methodology**

#### **Marc van der Schans \*, Joan Yu and Genevieve Martin**

Signify, High Tech Campus 7, 5656AE Eindhoven, The Netherlands; joan.yu@signify.com (J.Y.); genevieve.martin@signify.com (G.M.) **\*** Correspondence: marc.van.der.schans@signify.com

Received: 16 July 2020; Accepted: 14 September 2020; Published: 22 September 2020

**Abstract:** Light-emitting diode (LED) digital twins enable the implementation of fast digital design flows for LED-based products as the lighting industry moves towards Industry 4.0. The LED digital twin developed in the European project Delphi4LED mimics the thermal-electrical-optical behavior of a physical LED. It consists of two parts: a package-level LED compact thermal model (CTM), coupled to a chip-level multi-domain model. In this paper, the accuracy and computation time reductions achieved by using LED CTMs, compared to LED detailed thermal models, in 3D system-level models with a large number of LEDs are investigated. This is done up to luminaire-level, where all heat transfer mechanisms are accounted for, and up to 60 LEDs. First, we characterize a physical phosphor-converted white high-power LED and apply LED-level modelling to produce an LED detailed model and an LED CTM following the Delphi4LED methodology. It is shown that the steady-state junction temperature errors of the LED CTM, compared to the detailed model, are smaller than 2% on LED-level. To assess the accuracy and the reduction of computation time that can be realized in a 3D system-level model with a large number of LEDs, two use cases are considered: (1) an LED module-level model, and (2) an LED luminaire-level model. In the LED module-level model, the LED CTMs predict junction temperatures within about 6% of the LED detailed models, and reduce the calculation time by up to nearly a factor 13. In the LED luminaire-level model, the LED CTMs predict junctions temperatures within about 1% of LED detailed models and reduce the calculation time by about a factor of 4. This shows that the achievable computation time reduction depends on the complexity of the 3D model environment. Nevertheless, the results demonstrate that using LED CTMs has the potential to significantly decrease computation times in 3D system-level models with large numbers of LEDs, while maintaining junction temperature accuracy.

**Keywords:** compact thermal model; LED; Delphi4LED; digital twin; digital luminaire design; computation time; Industry 4.0

#### **1. Introduction**

Rapid innovation and customization of light-emitting diode (LED)-based lighting products demand shorter design cycles, higher cost efficiency, and more reliable solutions from manufacturers. To meet these demands, digitalization of the design flow, also called an "Industry 4.0" approach, is required. Methods, processes, and tools that facilitate the usage of LED components in a digital design flow were developed and demonstrated in the European project Delphi4LED [1,2]. The proposed approach consists of the generation and implementation of multi-domain LED digital twins to enable fast and reliable computer simulations of LED-based lighting products. Multi-domain LED digital twins are models that accurately mimic the thermal-electrical-optical behavior of a physical LED, and can be integrated in larger system-level models, for example a luminaire-level

model. Additionally, an LED digital twin should not carry proprietary information of the LED manufacturer, such as details related to the LED's construction, materials or production processes. This way, LED manufacturers or vendors can share them with end-users without disclosing their sensitive intellectual property.

An overview of the major steps involved in creating and implementing multi-domain LED digital twins is described in detail by Martin et al. [3]. The steps are summarized in Figure 1 and briefly outlined here. First, a thermal-electrical-optical characterization of the physical LED device is performed (step 1). The testing protocols and methods are discussed in [4–7] and take the established testing standards JEDEC JESD51-14, JESD51-51, JESD51-52, and CIE 225:2017 [8–11] into account. The results of the characterization are so-called iso-thermal current-voltage-flux (IVL) characteristics and thermal transient characteristics of the LED device. This data will be reported in future standard LED electronic datasheets [12] (step 2).

**Figure 1.** The Delphi4LED approach to creating and implementing LED digital twins (multi-domain compact models). The involved steps are indicated from top to bottom. Adapted from [3].

Next, the multi-domain LED digital twin, also referred to as the LED multi-domain compact model (MDCM), is extracted from the characterization data (step 3). The LED MDCM consists of two parts. The first part is a chip-level multi-domain model. It calculates the forward voltage, power dissipation, radiant flux, and luminous flux from the forward current and junction temperature. Poppe et al. [13,14]

discuss several sets of equations that can be used for this purpose. The extraction of the chip-level model is achieved by fitting the parameters of the equations to the IVL characteristics. The second part of the LED MDCM is a package-level compact thermal model (CTM). The LED CTM is a thermal RC-network attached to a simplified geometric representation of the LED. It calculates the relevant operating temperatures of the LED package, such as junction, phosphor and solder temperatures, from the power dissipation. The CTM extraction procedure is described by Bornoff et al. [15,16]. It involves the calibration of a detailed thermal model using the thermal transient characteristics, and the optimization of the RC-network to produce matching thermal dynamic behavior. In a fully realized LED digital twin, the two parts of the LED MDCM are coupled and solved self-consistently.

Finally, the LED digital twin is implemented in the larger system-level model of an LED-based lighting product, for example an LED module or an LED luminaire (step 4). There are different approaches to the system-level model. One option is to generate a compact thermal model of the LED module or LED luminaire. Poppe et al. [17] describe a method to create thermal network compact models for luminaires, which have subsequently been used in Spice-like luminaire simulations [3,18] and in an Excel spreadsheet application [3,19]. Alternatively, model order reduction could be used instead of thermal network compact models [20–23]. Another approach is to perform the LED module or LED luminaire simulations using the LED MDCM directly in a 3D computational fluid dynamics (CFD) model [3,24]. Ultimately, the LED module or LED luminaire model is used for virtual prototyping (step 5).

In this publication, we assess and compare the accuracy and computation time of 3D CFD system-level models equipped with LED CTMs and with LED detailed models. While this study does not include a multi-domain chip-level model, the LED thermal model is most demanding in terms of computation time in this case. First, an LED detailed model and an LED CTM are created according to the Delphi4LED methodology by performing a thermal characterization and LED-level modelling. The obtained LED thermal models are then implemented into 3D CFD software in the system-level model of two use cases: (1) an LED module-level model, and (2) an LED luminaire-level model. In previous research [24], we compared the computation time required to simulate an LED module-level model with up to 22 LEDs using LED detailed models and using LED CTMs. It showed that using the LED CTMs reduces the computation time by approximately a factor 10 for a steady-state, conduction only model. The novelty of this work is that the analysis is extended to the luminaire-level model. Typically, a LED luminaire contains several LED modules, and thus contains larger numbers of LEDs. Moreover, all methods of heat transport must be taken into account and their impact on the computation time is investigated.

#### **2. Materials and Methods**

The Delphi4LED approach as outlined Figure 1 is followed to create an LED detailed model and an LED CTM and to subsequently implement them in an LED module-level model and an LED luminaire-level model. The methods and processes of three of the involved steps are each described in a subsection. The first subsection briefly explains the test procedures used to obtain the required characterization data from physical LED samples (LED device testing). In the second subsection, the creation of the LED detailed model and the LED CTM are described (LED-level modelling). Finally, in the third subsection, the implementation of the LED detailed model and the LED CTM in the LED module-level model and the LED luminaire-level model is specified.

#### *2.1. LED Device Testing*

For the investigations, a phosphor-converted white high-power LED with a color rendering index (CRI) of 70 and correlated color temperature (CCT) of 4000 K is used. Four physical samples of the same type LED are each assembled on an insulated metal substrate (IMS) board for testing. Figure 2a shows the detailed 3D geometry of the LED sample placed on the test board. A cross-sectional view of the LED package geometry is provided in Figure 2b. The LED package and the test board together constitute the device under test (DUT).

**Figure 2.** Detailed geometry of the LED under investigation: (**a**) The LED package assembled on the test board (DUT). (**b**) Cross-sectional view of the LED package geometry.

The device under test is placed on a cold plate. Thermal paste is applied between the test board and the cold plate to provide good thermal contact, and the board is fixed in place with two screws. To ensure reproducibility a torque screwdriver used. Simultaneous radiometric measurements and thermal transient measurements, i.e., the temperature response *T*(*t*) to a power step, are performed using commercially available testing equipment (Simcenter T3ster and TeraLED) [13,25,26]. For these measurements a fixed cold plate temperature of *T*ref = 50 ◦C is used. In these tests, the DUT is first operated at a (total) forward current of *I*<sup>f</sup> = 1400 mA until steady-state is reached. Then a power step is applied by decreasing the current to a measurement current of *I*meas = 10 mA. The measured emitted radiant flux Φ<sup>e</sup> is subtracted from the electrical power *P*el to obtain the total thermal dissipation *P*th = *P*el − Φe. The thermal dissipation is then used to normalize the transient temperature Δ*T*(*t*) = *T*(*t*) − *T*ref to obtain the transient thermal impedance *Z*th(*t*) = Δ*T*(*t*)/*P*th, as well as the corresponding structure function (SF) and differential structure function (DSF). This is the thermal characterization data needed for the LED CTM in the LED-level modelling step.

#### *2.2. LED-Level Modelling*

Generating an LED CTM, in the form of a thermal RC-network, from the thermal transient characterization data involves two parts. In the first part, a detailed thermal model of the LED package is created and calibrated using the characterization data as described in [15]. Then, in the second part, the calibrated LED detailed model is subsequently used to generate training data for the LED CTM. This training data consist of thermal responses under several different boundary conditions. The training data is used to optimize the RC-values of the LED CTM, such that the errors in thermal behavior compared to LED detailed model are minimized [16]. Finally, after the LED CTM is optimized, the achieved accuracy is validated by subjecting both the detailed model and CTM to several additional boundary conditions, which were not used in the training, and determining the errors.

#### 2.2.1. LED Detailed Model

For the LED detailed model, geometrical information is required. The outer dimensions of the LED are provided by the manufacturer. However, in order to have a sufficiently accurate model, additional information is extracted from microscope images, e.g., the chip size and phosphor layer size. For other internal geometrical characteristics, generally not provided by suppliers, an educated guess is made. This is for instance done for the die attach thickness. Minor mismatches in those

thicknesses are later compensated during the calibration process by adjusting the thermal conductivity values. The geometric model of the LED package and test board shown in Figure 2 is used in the calibration process.

In the detailed model, thermal loads are applied to the junction as well as to the phosphor layer. They are considered the main contributors to the total heat dissipation. Indeed, there may be other package losses resulting from trapped light due to total internal reflections. Recently, Alexeev et al. discussed the effects of secondary heat sources on thermal transient analysis in detail [27]. However, since losses related to trapped light are difficult to quantify and localize without elaborate optical modelling, only the junction and phosphor losses are considered in this study. Since only the total dissipation *P*th is known, the power split between the junction and phosphor is included in the calibration as an optimization parameter, together with the thermal conductivity values of the materials in the model.

The LED detailed model is calibrated by minimizing the errors between the modelled and measured *Z*th(*t*) responses and SFs. The model parameters are optimized to best match all four measured samples simultaneously. The calibration is performed using commercially available 3D CFD software (Simcenter Flotherm XT 2019.2).

To produce training data for the LED CTM, the calibrated LED detailed model is virtually taken off the test board and subjected to sets of different boundary conditions. This is done by applying uniform heat transfer coefficients (HTCs) to four selected peripheral faces: the bottom face of the anode solder pad, the bottom face of the cathode solder pad, the bottom face of the thermal solder pad, and the top face of the package/dome. The purpose of using multiple boundary conditions is to ensure that the extracted CTM will be boundary condition independent (BCI). The four sets of heat transfer coefficients that are used to generate the training data for the CTM are listed in Table 1. These HTCs training sets are chosen to represent practical operating environments of LEDs and are in the same range as the HTC sets used for the same purpose in [28]. The generated training data consists of the transient temperature profiles Δ*T*detailed *ih* (*t*) obtained for a power step *P*th. Here, the index *i* indicates the junction layer, the phosphor layer, and each of the four faces to which HTCs are applied. The index *h* indicates the each of the four HTC training sets. This data is exported from the CFD software.


**Table 1.** HTC training sets used to generate training data for the LED CTM optimization.

#### 2.2.2. LED CTM

A CTM, in the form of a thermal RC-network, is optimized to produce matching thermal behavior under the same boundary conditions, i.e., the four imposed HTC training sets. The chosen network topology is illustrated in Figure 3. Each node *i* of the thermal network has a thermal capacitance *C*th,*<sup>i</sup>* to ground. A line between two nodes *m* and *n* indicates that the nodes are connected by a thermal resistor *R*th,*mn*. Compared to the network topology used in earlier studies [16,28], our network topology has an additional node between the junction and phosphor nodes (node 4), and between the phosphor and dome nodes (node 2). It was found by trial and error that those nodes are necessary to better fit the dynamic behavior, particularly of the phosphor and dome nodes.

**Figure 3.** RC-network topology of the LED CTM (**left**), and two isometric views of the simplified 3D geometry of the LED CTM (**right**). The red arrows indicate to which surfaces (in blue) the temperatures of the peripheral nodes of the RC-network model are connected.

The RC-network optimization is performed by code developed in Python and works in a manner similar to that described by Schweitzer [29]. It uses the derivative-free BOBYQA algorithm [30] implementation from the NLopt library [31]. An advantage of performing the optimization separately outside the CFD software is that stand-alone RC-network evaluations are faster, which means that several 10,000 to 100,000 optimization iterations can be made in a few minutes.

First, the training data is imported and the corresponding thermal (transfer) impedances *Z*detailed th,*ih* (*t*) = <sup>Δ</sup>*T*detailed *ih* (*t*)/*P*th are calculated. Subsequently, the RC-network is numerically solved for the same power step *P*th to obtain the CTM temperatures Δ*T*CTM *ih* (*t*) of nodes *i* under HTC training sets *h*. The corresponding thermal (transfer) impedances are again calculated as *Z*CTM th,*ih* (*t*) = <sup>Δ</sup>*T*CTM *ih* (*t*)/*P*th. By varying the *C*th,*<sup>i</sup>* and *R*th,*mn* values, the difference in dynamic thermal behavior between the detailed model and the RC-network CTM is minimized using the following cost function:

$$f\_{\rm cost} = \sum\_{h} \sum\_{i} \sum\_{j} \frac{\left( Z\_{\rm th, ih}^{\rm detailed}(t\_j) - Z\_{\rm th, ih}^{\rm CTM}(t\_j) \right)^2}{Z\_{\rm th, ih}^{\rm detailed}(t\_j)} \tag{1}$$

where index *h* runs over all four HTC training sets, index *i* runs over the nodes included in the optimization, and index *j* runs over the time steps for which the simulation is performed. The included nodes are the junction, phosphor, and peripheral nodes, i.e., *i* = {1, 3, 5, 9, 10}. In this particular case node 11 is not explicitly included due to symmetry.

To assess the accuracy and boundary condition independence of the optimized LED CTM, both the LED CTM and the LED detailed model are tested under twenty additional HTC sets. The twenty HTC testing sets are listed in Table 2. These sets are combinations generated using the design of experiments functionality of the CFD software. For each of the peripheral faces, the lower and upper HTC bounds were set to the minimum and maximum values that occur in the training sets. Since the HTC testing sets were not used to train the model, they provide a better evaluation of the predictive temperature accuracy of the LED CTM compared to the detailed model. In the report on end-user specifications of the Delphi4LED project [32] the required junction temperature accuracy is stated as 2%.


**Table 2.** HTC testing sets used to generate test data for the LED CTM validation.

Finally, to be able to interface with a larger 3D system-level model, the LED CTM is attached to simplified 3D geometric representation of the LED package, as indicated in Figure 3. While the outer contours of the simplified geometry are identical to those of the detailed model, it has no internal structure. The faces of the simplified geometry that are connected to the peripheral nodes of the RC-network have the same surface area as the corresponding faces of the detailed model.

#### *2.3. System-Level Modelling*

Two use cases are investigated to assess the predicted junction temperature accuracy and computation time performance of the LED CTM, compared to the LED detailed model, integrated in a 3D system-level model: (1) an LED module-level model, and (2) an LED luminaire-level model. The thermal environment of the LED CTMs and the LED detailed models is now explicitly simulated in these cases, instead of imposed by uniform HTCs. Please note that both the LED module-level model and the luminaire-level model presented here are solely intended for the purpose of numerically assessing the LED CTM accuracy and performance, compared to the LED detailed model. They are by no means optimized for thermal management or any other actual product requirements.

While the described methods and models can in principle be used in any 3D CFD software, we used Simcenter Flotherm XT 2019.2. All reported computation times are obtained on a workstation laptop with an Intel Core i7-6820HQ (2.7 GHz, 4 cores) processor. Unless stated otherwise, the default computational mesh settings ('Standard Resolution') of the software tool are used.

#### 2.3.1. LED Module-Level Model

The LED module-level model consists of a simplified printed circuit board (PCB) populated with an array of LEDs, as illustrated in Figure 4. The board is 25 mm wide, 90 mm long and has a 1 mm thick dielectric layer (1 W/mK) and a 70 μm thick copper layer (386 W/mK). Several LEDs, *N*LED, are uniformly distributed on top of the copper layer. The number of LEDs along the *x*-axis is *Nx*, and the number of LEDs along the *y*-axis is *Ny*. For *N*LED up to 15, a single row of LEDs is used (*Nx* = 1), for *N*LED between 16 and 30, two rows of LEDs are used (*Nx* = 2), for *N*LED between 31 and 45, three rows of LEDs are used (*Nx* = 3), and for *N*LED greater than 45, four rows of LEDs are used

(*Nx* = 4). The bottom face of the dielectric layer is kept at a fixed uniform temperature *T*ref and only solid conduction is considered in this model.

**Figure 4.** The LED module-level model. On the left an example is shown with *N*LED = 16 LED detailed models (*Nx* = 2 and *Ny* = 8), and on the right an example is shown with *N*LED = 60 LED CTMs (*Nx* = 4 and *Ny* = 15).

#### 2.3.2. LED Luminaire-Level Model

The LED luminaire model consists of a simplified luminaire housing and five instances of the LED module, as illustrated in Figure 5. Each of the LED modules has *N*LED = 12 LEDs (*Nx* = 2 and *Ny* = 6), resulting in total number of 60 LEDs. The luminaire housing is made of an aluminum alloy (140 W/mK) and has fins located above the LED modules for cooling to the surrounding air. In this luminaire-level model, conduction, convection and radiation are all taken into account, increasing the model complexity with flow simulation. All solid-fluid interfaces are assigned a surface emissivity of 0.8, including the anodized surface of the luminaire housing, and the ambient temperature is set at *T*ref = 25 °C.

**Figure 5.** The LED luminaire-level model. On the left, the top part of the luminaire housing with cooling fins is visible, and on the right, the bottom part of the luminaire housing is visible, which supports five LED modules each with 12 LEDs (*Nx* = 2 and *Ny* = 6). The rectangular outlines indicate the computational domain.

#### **3. Results**

In this section, the created LED-level models (detailed model and CTM) and the results of using the LED-level models in a system-level model (module and luminaire) are presented. The first part presents the results of the physical LED device testing. Then, the second part presents the modelling results at LED-level. It includes the calibration of the LED detailed model, and the extraction and validation of the LED CTM. Next, the third part presents the accuracy and performance results of the LED module-level model. Finally, the fourth part presents the accuracy and performance results of the LED-luminaire model.

#### *3.1. LED Device Testing Results*

Subtracting the measured radiant flux from the supplied electrical power *P*el results in the total dissipated thermal power *P*th = *P*el − Φ<sup>e</sup> of the DUT. It is found that *P*th = (2.53 ± 0.01) W. The reported uncertainty of 0.01W indicates the standard deviation between the four measured samples.

Figure 6 shows the transient behavior of the DUT obtained from the measurements. The thermal impedance *Z*th(*t*) is presented in Figure 6a. The corresponding structure functions (SF) and differential structure functions (DSF) are shown in Figure 6b. The four measured samples show good reproducibility. The largest relative deviation in measured *Z*th between the four samples is about 4%, and occurs in the early transient (*t* < 150 μs), where initial correction has to be applied due to electrical transients present in the measurement signal. The measured steady-state *Z*th has a relative deviation of around 0.1%. Using an additional 'dry' thermal transient additional measurement, i.e., without thermal paste applied between board and cold plate, the junction-to-board resistance was determined according to the standard JEDEC JESD51-14 [8]. The junction-to-board thermal resistance of the DUT is found as *R*th,j-b = (6.2 ± 0.1) K/W.

**Figure 6.** Thermal transient characteristics of the measured LED device and the calibrated LED detailed model: (**a**) Thermal transient impedance of the DUT and calibrated detailed model. (**b**) Structure function and differential structure function of the DUT and the calibrated detailed model.

#### *3.2. LED-Level Modelling Results*

First, the model parameters of the detailed model, i.e., the thermal conductivity values of the materials and junction-phosphor power split, were calibrated to match the *Z*th(*t*) and SF of the measured LED devices. The power split in the detailed model that best fits the measurements was found in the calibration as approximately 77% in the junction and 23% in the phosphor. The *Z*th(*t*), SF and DSF of the calibrated LED detailed model are shown together with those obtained from the measurements in Figure 6. Overall, the curves match well. Some deviations between the SFs and between the locations of the peaks and valleys of the DSFs are observed for approximately *R*th > 6 K/W. However, since we are ultimately only interested in the LED package, without the board, no further improvements to matching this part of the SF are required.

Next, training data was generated using the calibrated LED detailed model for four HTC training sets. This training data was subsequently used to optimize the RC-values of the LED CTM. The optimized RC-values are given in Table 3. The *C*th column lists the thermal capacitance to ground for each of the nodes, and the *R*th array lists the thermal resistances between connected nodes of the RC-network. Since *R*th,*mn* = *R*th,*nm* only the lower triangular entries are displayed.


**Table 3.** Optimized thermal capacitance values and thermal resistance values of the LED CTM.

The thermal transient behavior of the calibrated LED detailed model and the optimized RC-network LED CTM are compared for the four HTC training sets in Figure 7. The largest absolute error |*Z*detailed th (*t*) <sup>−</sup> *<sup>Z</sup>*CTM th (*t*)| after optimization is about 0.7 K/W and occurs between approximately 10−<sup>1</sup> s and 10<sup>1</sup> s for the phosphor node in HTC training set 3. For steady-state conditions, the relative errors |*Z*detailed th <sup>−</sup> *<sup>Z</sup>*CTM th <sup>|</sup>/*Z*detailed th are all smaller than 2%, and smaller than 1% for the *Z*th values corresponding to the junction. This results in junction temperatures that match within 0.2 K for the training data.

The relative errors in steady-state temperature rise, |Δ*T*detailed − <sup>Δ</sup>*T*CTM|/Δ*T*detailed, for the twenty HTC testing sets are plotted in Figure 8. The junction temperature errors range from about 0.6% to about 1.7%, remaining within the 2% requirement. All other temperature errors also remain within this limit, with the exception of the dome temperature for five of the twenty HTC testing sets. However, it should be noted that the temperature requirement is only specified for the junction temperature [32].

**Figure 7.** Transient behavior of the LED detailed model (blue) and optimized LED CTM (red) for (**a**) HTC training set 1, (**b**) HTC training set 2, (**c**) HTC training set 3, and (**d**) HTC training set 4. The numbers correspond to the nodes of the RC-network: 1-dome, 3-phosphor, 5-junction, 9-thermal pad, and 10-dome.

**Figure 8.** Relative errors in the steady-state Δ*T* of the LED CTM, compared to the LED detailed model, for each of the twenty HTC testing sets. The dashed line indicates the 2% junction temperature error requirement of the Delphi4LED end-user specifications [32].

#### *3.3. LED Module-Level Model*

To assess the accuracy and performance of the LED CTM, compared to the LED detailed model, implemented in the LED module-level model, it was simulated with multiple numbers of LEDs. Since in each case the LEDs are uniformly distributed over the board, and the bottom face of the board is kept at a uniform temperature *T*ref, the Δ*T*<sup>j</sup> = *T*<sup>j</sup> − *T*ref varies less than 0.2 K between individual LEDs. For this reason only a single value, the average Δ*T*<sup>j</sup> for all LEDs on the board, is reported for each case. The results are presented in Figure 9.

When there is only a small number of LEDs on the board, the distance between the individual LEDs is large enough that no mutual influence, or 'cross-talk', is experienced by the LEDs. This can be observed in the constant Δ*T*<sup>j</sup> for *N*LED up to 4 in Figure 9a. When the number of LEDs is further increased, Δ*T*<sup>j</sup> gradually rises. As mentioned, between *N*LED = 15 and *N*LED = 16, we change from a single row (*Nx* = 1) to two rows (*Nx* = 2) of LEDs. This results in an effective increase in the distance between individual LEDs and causes the drop in Δ*T*j. This occurs again, albeit less pronounced, when the number of rows is increased to three (*Nx* = 3) and to four (*Nx* = 4).

Comparing the average Δ*T*<sup>j</sup> obtained using the LED CTM and the LED detailed model, the same behavior is observed. Nevertheless, the predictions of the LED CTM are systematically lower. The absolute difference in Δ*T*<sup>j</sup> between the LED CTM and the LED detailed model decreases from about 1.6 K for *N*LED = 1 to about 1.0 K for *N*LED = 60. This corresponds to relative errors in Δ*T*<sup>j</sup> between 6.0% for *N*LED = 1 and 2.2% for *N*LED = 60, as shown in Figure 9b.

Inspecting the peripheral faces of the thermal pad, anode pad and cathode pad, reveals discrepancies in the average surface temperatures and in the heat transfer distribution. For example, for *N*LED = 1 the average surface temperature rise, *T* − *T*ref, of the thermal pad and the total heat transfer through the thermal pad in the LED detailed model are 19.6 K and 1.02 W respectively, whereas in the LED CTM they are 17.4 K and 1.18 W respectively. For *N*LED = 60 these differences are smaller, which results in a smaller junction temperature error. In this case the average surface temperature rise of the thermal pad and the total heat transfer through the thermal pad in the LED detailed models are 39.9 K and 1.07 W respectively, whereas in the LED CTMs they are 38.3 K and 1.18 W respectively. To ensure that the observed differences cannot be attributed to mesh convergence issues, the simulations for *N*LED = 1, *N*LED = 16, and *N*LED = 60 were repeated with higher mesh density. The results were reproduced within 0.2 K.

**Figure 9.** Average junction temperature rise of the LEDs in the LED module-level model: (**a**) comparison between the LED detailed model and the LED CTM, and (**b**) the relative error in the LED CTM values for different number of LEDs.

The time required by the central processor unit (CPU) to solve the model is shown as a function of the number of LEDs in Figure 10a. Up to 60 LEDs, the computation time increases approximately linearly for the LED CTMs, while the computation time using the LED detailed models increases super-linearly. The ratio between the CPU times using the LED detailed models and the LED CTMs is plotted in Figure 10b. It can be seen that for a single LED, using the CTM results in about a factor 2 reduction of computation time. For 60 LEDs the required computation time is reduced by nearly a factor 13 when the LED CTM is used instead of the LED detailed model. This is in line with our previous findings [24].

**Figure 10.** Time required to solve the LED module-level model: (**a**) comparison of CPU time between the LED detailed model and the LED CTM, and (**b**) the ratio in CPU time between the LED detailed model and the LED CTM.

#### *3.4. LED Luminaire-Level Model*

In the LED luminaire-level model, the bottom faces of the LED modules are in direct thermal contact with the luminaire housing, instead of being kept at a constant fixed temperature. This results larger gradients in the board temperature, in particular for LED modules 3, 4 and 5. This is illustrated in the temperature plot presented in Figure 11. As a consequence, there are also larger variations in Δ*T*<sup>j</sup> among LEDs on the same board than in the previous case of the LED module-level model. For this reason not only the average Δ*T*<sup>j</sup> for each of the modules is reported, but also the minimum and maximum Δ*T*<sup>j</sup> are listed in Table 4. Since the geometry of the model has a plane of symmetry, the Δ*T*<sup>j</sup> values should be the same for LED module 1 and 2, and for LED module 3 and 5. The values obtained from the model are indeed very similar for these boards, apart from some small variations of 0.1 K between LED module 1 and 2, which can be caused by asymmetries in the computational mesh.

**Figure 11.** Surface temperature of the LED luminaire-level model with LED detailed models (**top**) and LED CTMs (**bottom**). For the LED CTMs, the temperature of the internal junction node is plotted on the LED geometry.


**Table 4.** Minimum, average, and maximum junction temperature rises and relative errors for each of LED modules in the LED luminaire-level model.

As a general note, such high Δ*T*<sup>j</sup> values indicate an improved thermal design would be necessary in practice. Comparing the Δ*T*<sup>j</sup> values between the LED detailed models and the LED CTMs shows that the junction temperatures predicted by the LED CTMs are again systematically lower. The largest absolute error found in this case is 1.3 K, which is in the same range as the errors found previously in the LED module-level model, and occurs for the LEDs with the largest Δ*T*<sup>j</sup> on LED modules 3 and 5. The relative errors in Δ*T*<sup>j</sup> range from about 0.3% to 1.1% in this case. Inspecting the average surface temperature rise of the thermal pad and the total heat transfer through the thermal pad of those LEDs again reveals similar discrepancies as before. For the detailed model they are 114.6 K and 1.03 W respectively, and for the CTM they are 113.2 K and 1.12 W respectively.

The LED luminaire-level model containing the LED detailed models took 9016 s to solve, whereas the model containing the LED CTMs only needed 2153 s, resulting in a reduction of a factor 4.2. This is smaller than the computation time reduction that was found for 60 LEDs in the LED module-level model. Besides that the luminaire-level model comprises a larger geometry than the module-level model, convection and radiation are also simulated in this case. To assess the impact of convection and radiation on the computation time, the simulation of the luminaire-level model with LED CTMs is repeated with convection and radiation each turned off separately. Without radiation the model is solved in 1907 s, and without convection the model is solved in 706 s.

#### **4. Discussion**

An RC-network LED CTM was successfully generated. The steady-state junction temperature error of the LED CTM, compared to the LED detailed model, was evaluated between 0.6% and 1.7% on LED-level under imposed uniform HTCs. This meets the requirement of 2% stated in the Delphi4LED end-users' specifications [32].

A similar RC-network LED CTM, for a different LED package, is reported in [28]. The main differences are that a network topology with only nine instead of eleven nodes was used, the CTM was trained under three instead of four HTC training sets, using and a cost function based on the SF instead of *Z*th. A slightly better relative error range of about 0.6% to 1.2% in the predicted steady-state junction temperature rise is reported there. Nevertheless, in both cases an acceptable accuracy for steady-state thermal behavior is achieved with an RC-network LED CTM. However, both in the present case and in [28] it appears more difficult to also achieve accurate dynamic behavior, in particular for the phosphor node. This indicates that further refinement of the extraction process or used network topology may still be necessary for cases in which the LED CTM is not operated in steady-state conditions. Another development in achieving accurate dynamic LED-level models that should be mentioned here is the BCI reduced order model (BCI-ROM) approach [20,21]. Recently, Bornoff and Gaal [28] compared this approach to the RC-network CTM and discussed its advantages related to extraction (no choices on a network topology have to be made) and accuracy (the required accuracy is prescribed by the user *a priori* for a wide range of HTCs). However, at the moment BCI-ROMs cannot be implemented yet in commercially available 3D CFD software.

When implemented in a 3D system-level, the LED CTM predicted slightly but systematically lower junction temperatures than the LED detailed model. The largest difference was 1.6 K among the two studied use cases. Inspecting the peripheral faces of the thermal pad, anode pad and cathode pad, revealed discrepancies between the LED CTMs and LED detailed models in the average surface temperatures and in the heat transfer distribution. This is likely caused by the fact that the LED CTM is extracted under uniform peripheral conditions, whereas gradients exist in the more realistic 3D thermal environment. This situation could be improved by splitting the anode, cathode and thermal pad surfaces in multiple surfaces and assigning each their own node in the RC-network. However, this is at the cost of making the LED CTM more complex. The error could also be partly related to the choice and number of HTC training sets. In the original DELPHI project for semiconductor CTMs, 38 HTC sets were proposed, and later even larger sets were tested [33]. However, it was shown that smaller subsets of five HTCs can still lead to accurate CTMs [33,34]. The range of thermal operating environments that is relevant for LEDs is much smaller, but to date no studies have been performed involving the number and type of HTC sets to apply for accurate LED CTM extraction.

In the LED module-level model, absolute errors in Δ*T*<sup>j</sup> of up to 1.6 K and relative errors of up to 6.0% were found when comparing the model with LED CTMs and LED detailed models. In the LED luminaire-level model, absolute errors in Δ*T*<sup>j</sup> of up to 1.3 K and relative errors of up to 1.1% were found when comparing the model with LED CTMs and LED detailed models. While the absolute errors are on the same scale in both cases, the relative errors are substantially smaller in the luminaire-level case. This is explained by the larger total thermal resistance to ambient of the luminaire system, resulting in a larger temperature rise. Although the CTM meets the 2% error requirement compared to the detailed model on luminaire-level, in the LED module-level model the relative errors are larger. Hence the aforementioned potential improvements may be necessary, depending on the end-user's needs. It should also be stressed however that we only considered the error of the extracted and implemented LED CTM compared to the LED detailed model. Compared to reality, for example if we were to measure a physical prototype, there may be various additional sources of errors. Some examples include: measurement uncertainties, uncertainties in the thermal dissipation of the components, and uncertainties related to the CFD simulation itself [35]. Additionally, the thermal resistance of the LED package may only represent a small fraction of the entire thermal resistance to ambient, especially at LED luminaire-level. When that is the case, the accuracy of the predicted *T*<sup>j</sup> compared to a physical prototype will also largely depend on the accuracy of the part of the model besides the LEDs.

Regarding the computation time, using the LED CTM in the LED module-level model resulted in a reduction from about a factor 2 with one LED up to almost a factor 13 with 60 LEDs. In the LED luminaire-level model with 60 LEDs, a reduction of about a factor 4 was achieved using the LED CTMs. Of course, the exact computation time will be different in every case and depends on the complexity of the luminaire design. The difference in the achieved reduction between the studied cases can be explained by the fact that the luminaire-level model has a larger 3D environment, more complex shapes, and that convection and radiation are considered. As a result, the LEDs themselves constitute a relatively smaller part of the entire model than in the case of the LED module-level model, and hence their relative impact on the total calculation time decreases. In particular, the flow simulations were found responsible for a large part of the computation time in the studied case. When turned off, the computation time decreased by about a factor 3, from 2153 s to 706 s. Without radiation, the computation time decreased by about 10%. Nevertheless, the factor 4 reduction for the simplified LED luminaire-level model is still significant when a large number of scenarios needs to be simulated in a design parameter optimization. Furthermore, the gains will increase significantly for higher LED counts, as demonstrated by the LED module-level model. As an example, this approach could therefore be highly advantageous when modelling systems containing LED filaments, as each filament may contain several hundreds of LEDs.

#### **5. Conclusions and Outlook**

To summarize, we created an LED detailed model and an LED CTM following the Delphi4LED methodology and assessed the accuracy and computation time of 3D CFD system-level models equipped with these LED models. Compared to previous work [24], the analysis was extended to luminaire-level. This involved including higher number of LEDs, up to 60, and taking all heat transfer mechanisms into account. The cases discussed in this work demonstrate that using Delphi4LED LED CTMs in digital luminaire designs can provide a significant reduction in computation time while maintaining the required accuracy compared to a LED detailed model.

With this approach, any node could be added to the LED model in order to monitor a thermally critical part of the LED. In this case, the temperature of the node of interest does need to be monitored during the LED testing. This way it can be included in the detailed model calibration in order to obtain an accurate model and predictions for this temperature. It would, for example, be interesting to do this for the phosphor temperature. While our LED CTM has a phosphor node, no phosphor temperatures were measured and accounted for in the calibration of the LED detailed model. Therefore, the modeled phosphor temperatures cannot currently be validated. In many cases, it is also not straightforward to monitor the temperature phosphor temperature. In the present case, the silicone dome covering the phosphor precludes measurements using thermocouples or infrared (IR) thermography. However, methods based on the spectral distribution of the converted light [36,37], or specifically prepared phosphors with magnetic nano-particles [38] could in principle be used.

Finally, the present study focused on 3D thermal modeling of the LED-based lighting designs using an RC-network LED CTM. It is expected that a future implementation of LED BCI-ROMs in 3D system-level models will provide an alternative option for the RC-network CTM. Additionally, only the LED CTM was considered. The implemented model will be extended in future work with a chip-level multi-domain model to obtain a fully realized LED digital twin. Another useful addition to the luminaire-level model is to include lifetime prediction [39], which will be the subject of a follow up paper.

**Author Contributions:** Conceptualization, G.M.; methodology, M.v.d.S. and G.M.; software, M.v.d.S.; validation, J.Y. and G.M.; formal analysis, M.v.d.S.; investigation, M.v.d.S.; resources, J.Y.; data curation, M.v.d.S.; writing–original draft preparation, M.v.d.S.; writing–review and editing, G.M. and J.Y.; visualization, M.v.d.S.; supervision, G.M.; project administration, G.M.; funding acquisition, G.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used:


MDCM multi-domain compact model ROM reduced order model SF structure function

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Mixed Detailed and Compact Multi-Domain Modeling to Describe CoB LEDs**

#### **László Pohl \*, Gusztáv Hantos, János Hegedüs, Márton Németh, Zsolt Kohári and András Poppe \***

Department of Electron Devices, Budapest University of Technology and Economics, Magyar tudósok körútja 2, bldg. Q, 1117 Budapest, Hungary; hantos@eet.bme.hu (G.H.); hegedus@eet.bme.hu (J.H.); nemeth@eet.bme.hu (M.N.); kohari@eet.bme.hu (Z.K.)

**\*** Correspondence: pohl@eet.bme.hu (L.P.); poppe@eet.bme.hu (A.P.); Tel.: +36-1-463-2704 (L.P.); +36-1-463-2721 (A.P.)

Received: 6 July 2020; Accepted: 27 July 2020; Published: 5 August 2020

**Abstract:** Large area multi-chip LED devices, such as chip-on-board (CoB) LEDs, require the combined use of chip-level multi-domain compact LED models (Spice-like compact models) and the proper description of distributed nature of the thermal environment (the CoB substrate and phosphor) of the LED chips. In this paper, we describe such a new numerical solver that was specifically developed for this purpose. For chip-level, the multi-domain compact modeling approach of the Delphi4LED project is used. This chip-level model is coupled to a finite difference scheme based numerical solver that is used to simulate the thermal phenomena in the substrate and in the phosphor (heat transfer and heat generation). Besides solving the 3D heat-conduction problem, this new numerical simulator also tracks the propagation and absorption of the blue light emitted by the LED chips, as well as the propagation and absorption of the longer wavelength light that is converted by the phosphor from blue. Heat generation in the phosphor, due to conversion loss (Stokes shift), is also modeled. To validate our proposed multi-domain model of the phosphor, dedicated phosphor and LED package samples with known resin—phosphor powder ratios and known geometry were created. These samples were partly used to identify the nature of the temperature dependence of phosphor-conversion efficiency and were also used as simple test cases to "calibrate" and test the new numerical solver. With the models developed, combined simulation of the LED chip and the CoB substrate + phosphor for a known CoB LED device is shown, and the simulation results are compared to measurement results.

**Keywords:** Light-emitting diodes; power LEDs; CoB LEDs; multi-domain modeling; finite volume method; phosphor modeling

#### **1. Introduction, Related Work**

Commercial LED-based white lighting devices work in the following ways [1]:


The first way ensures the most versatile options for the user to tune the light, but arises a few serious problems: Very complex driving circuitry, bad long term stability, due to the different ageing of the three kinds of LEDs, high production cost and last, but not least, usually provide lower color rendering indexes than phosphor-converted white LEDs. Therefore, nowadays the mainstream lighting

applications are based on the two latter ways, mostly on the last one. The second option enables easy tuning of the resulting light during the production technology. The last option provides the lowest production cost, but also limits the variability of the light the most; such LED devices are called phosphor-converted white LEDs (pc-WLEDs). In the subsequent parts of this paper, we shall also refer to such LEDs simply as white LEDs. This paper deals with multi-chip, large-area packages where the blue LED chips are directly attached to a common ceramics substrate, where this common chip carrier substrate also constitutes the LED package itself, and we focus on pc-WLED devices realized with single yellow phosphor-conversion, and chip-on-board (CoB) assemblies built of them.

Phosphor materials consist of a host compound and optical activator dopant ions. Appropriate phosphor materials used in pc-WLEDs should meet the following six basic criteria [2]:


Although many phosphor materials have been proposed in the literature in recent years, the number of phosphors effectively fulfilling all six requirements is relatively small [3]. Host materials include garnets, sulphides, (oxo-) nitrides, silicates, aluminates, borates, phosphates, and so on. The most frequently used activators are either broad-band emitting transitional metals Eu2<sup>+</sup>, Ce3<sup>+</sup>, Yb2<sup>+</sup> ions, or line-emitting rare-earth ions Ln3<sup>+</sup> and Mn4+, etc. [3,4]. The first commercially available pc-WLEDs invented by Nichia Corporation was fabricated using blue InGaN LED chip and the yellow yttrium aluminium garnet Y3Al5O12:Ce3<sup>+</sup> (YAG:Ce) phosphor.

For efficiency and long-term stability reasons, today, most commercial single-phosphor-converted white LED devices are still based on YAG:Ce [5]. A detailed discussion of the underlying physical effects (4f–5d transition, d-d transition) and of the structural design of phosphor materials can also be found there.

A much higher luminous emittance and conversion efficiency can be achieved by using nano-structured YAG:Ce ceramic phosphor plate and a high power blue laser diode for excitation [6]. The optimal Ce3<sup>+</sup> dopant concentration, resulting in the highest luminous emittance and conversion efficiency was found at 0.5 mol%. Investigation of such solid-state light-sources, however, is beyond the scope of this paper.

The modeling of the phosphor layer of a white LED primarily means optical modeling; following the light-scattering, light absorption and light frequency (wavelength) conversion, which happen inside the phosphor layer. Simulating these processes calculated their thermal effects too, which results in a multi-domain model of the phosphor layer. There are many solutions for modeling these effects, from simple one-dimensional models through using the bidirectional scattering distribution functions to the detailed 3D models. Here we only summarize some examples that use these methods. 1D modeling of light is used in papers [7,8] where the model verification for thin phosphor layers is given. We also used a similar, modified model. This model [7,8] is improved to study the effect of non-homogenous phosphor concentration in Reference [9], although the simulation results, in that case, do not match the measurement results. The expected heat generation in the phosphor layer is calculated in article [10] without comparison to measurement.

The most commonly used method for establishing the optical model of a phosphor layer is to measure the bidirectional scattering distribution function, which gives the relationship between the radiance and emission of the phosphor layer by infinitesimal solid angle for both incoming and outgoing light. Once the bidirectional scattering function is recorded, the optical behavior can be modeled by simple integration. The method is used for phosphor layer modeling with experimental validation [11]. These measurements are made on phosphor plates only. In Reference [12], phosphor-coated LED optical modeling and measurements are reported. This modeling technique is quite accurate for optical modeling, but as the microscopic details are not known, only the macroscopic thermal model can be established.

Detailed models are also used for optical modeling of phosphor layers. There are several commercial software tools available for this purpose. [13] For example in paper [13] Tan et al report about the use of TracePro and ANSYS. For phosphor-converted CoB device modeling, FloTHERM from Mentor Graphics was also used [14]. In this work, an emphasis is put on the thermal aspects. LightTools from Synopsys allows detailed modeling of phosphor layers with user-defined properties [15], with a focus on simulating light properties, but without considering thermal effects. In their paper [16] Alexev et al. describe the combined use of ANSYS and LighTools for the study of single-chip, custom-made white LEDs; optical results of their simulations are checked by luminance measurements in a special test setup while the correctness of the thermal simulations is checked with the help of structure functions extracted from the simulation results and from thermal transient measurements performed by Mentor Graphics' T3Ster equipment [17]. In the special, custom-made mid-power LEDs investigated, they used their own custom-made phosphor composites. To help set up their combined thermal and optical simulation model, they measured the thermal conductivity, as well as the reflection, excitation and emission spectra of these phosphor composites. In their paper [18] Jeon et al. also report their measurements of phosphor properties aimed as input for optical modeling of white LEDs, though, this publication does not provide any information about the temperature dependence of these properties. The paper of Qian et al. [19] provides a detailed review of combined optical-thermal modeling of phosphor-converted white LEDs and presents an example for LED filament bulb. Unfortunately, in none of these publications is the interaction with the electric domain through the blue pump LED chip(s) included.

A few multi-domain models have already been created. For example, an optical-electrical- thermal compact model was published by Ye at al., where the phosphor layer is taken into account with temperature dependence [20]. In this paper, single and multi-chip white LEDs (with contact phosphor layers) and remote phosphor solutions are investigated. The multi-chip structure they studied is very close to the structures of the CoB LED devices. In their model, the electrical behavior of the LED chips is lumped into the energy conversion efficiency.

Compact model for multi-domain purposes with a remote phosphor layer presented in [21], where applying bidirectional resistances showed good agreement with the measurements. A similar solution can be found in Reference [22]. In Reference [21], a large are multi-chip white LED device is studied with a structure close to that of white CoB LEDs, through measured, so-called 'ensemble' characteristics (see later). For modeling heat transfer from the LED chips' junctions to the environment both in Reference [21,22] the so-called bidirectional thermal resistance model is used. The thermal resistance values needed for such a model are identified from thermal transient measurements with the help of the structure functions. In Reference [21], the authors provide a final single equation for the total luminous flux in which both the bidirectional resistance model and the Shockley type model for the IV characteristics are included. The heat dissipation coefficient introduced by the authors, in which the light propagation properties in the phosphor are embedded, is also part of this equation. In summary, the model presented in this paper can be well applied to represent the 'ensemble' characteristics of CoB LEDs, but is not able to provide detailed information on the lateral and vertical temperature distributions in the phosphor layer and cannot provide information on the individual junction temperatures of the blue pump chips of the LED array.

In our current paper, we summarize our multi-domain modeling solution for phosphor- converted LED devices, which is a mixed, compact-detailed model by using one of the "standard" chip-level multi-domain LED models for the description of the operation of the blue pump chips within CoB devices. Only the multi-domain nature of the operation of the blue LED chips can be represented by a compact model; the blue LED chips' thermal environment (substrate, phosphor) of a CoB device has to be considered by a distributed, detailed 3D model. This way detailed studying of thermal phenomena in the phosphor layer (both in the vertical and lateral direction), including the effect of local interaction of the phosphor and the blue pump LED chips within the CoB array is made possible. In Figure 1, we provide a summary of physical processes taking place in the different major structural elements of a phosphor-converted white CoB LED device that we aimed to cover with our simulation approach.

**Figure 1.** Overview of the physical processes in different, major structural elements of a phosphor-converted white chip-on-board (CoB) LED device to be captured by dedicated simulation models.

The organization of our paper is as follows: In Section 2, we describe the context and the goals of this work. Section 3 deals with the major bottleneck: How to identify the material properties of phosphor layers needed for multi-domain simulation. We were inspired by References [16,18] to also prepare stand-alone custom phosphor samples that measure the temperature dependency of the phosphor properties. Moreover, with the same phosphor mixtures, we prepared single-chip white LED samples with well-controlled properties in order to allow us to fine-tune and to validate our simulation methods and models. Details on this part of our work are also provided in Section 3. In Section 4, we introduce our coupled chip + phosphor multi-domain simulation model along with the description of the light and heat propagation models of different complexity. In Section 5, we apply the introduced models to a commercially available CoB LED device that has also been characterized by common thermal and optical measurements as well. The comparison of the simulation and measurement results obtained for this device is provided in Section 6. In Section 7, we provide a summary and conclusions. In the Abbreviations we provide a summary of abbreviations and symbols used in this paper.

This paper, as a significantly extended version of our THERMINIC 2019 conference paper [23] provides a comprehensive summary of our CoB LED multi-domain modeling related work (parts of which have already been published at other conferences as well [24,25]) completed with a few, recent measurement results.

#### **2. Background, Related Own Work**

The work described here has been carried out in the framework of the recently completed European H2020 ECSEL research project Delphi4LED [26]. The major focus of the project was placed on single-chip LED packages and luminaires made thereof, applying a modular approach in the overall luminaire design process [27]. In the Delphi4LED approach, multi-domain behavior is treated on LED chip-level, by means of Spice-like compact (lumped) models [28]. The thermal effect of the LED package physical structure is also described by compact models [29]. To consider farther elements of the thermal environment, there are two options. On the one hand, the luminaire's thermal behavior and the effect of its thermal environment can be represented by yet another compact thermal model [30], and the entire model (including the LED chips models completed with the compact thermal models of their packages) forms a Spice netlist that can be simulated with any Spice compatible circuit simulator. On the other hand, another approach has also been developed within the Delphi4LED project: The 3D thermal environment of the LED chip is considered by a detailed 3D thermal model. In this approach, a 3D thermal simulator is modified in a way that it can iterate between the chip-level multi-domain LED model and the thermal solver, as illustrated in Figure 2. An implementation of such a scheme was used in the Delphi4LED project to demonstrate the use and benefits of the "industry 4.0" like design workflow suggested by the project [31].

**Figure 2.** A chip-level multi-domain LED model embedded in a thermal simulator using a relaxation type iteration in order to realize an electro-thermal-optical solver used for the virtual prototyping of luminaires based on single-chip LED packages [27,29,31]. For the explanation of symbols used in this figure see the Abbreviations.

In the case of phosphor-converted white CoB LEDs, however, the first approach of using compact models only cannot be applied because of the distributed, multi-domain nature of the phosphor layer, involving:


This is illustrated in Figure 1. The thermal effect of the phosphor layer (distributed heat source over the entire area of the CoB device) cannot be separated from the thermal behavior of the rest of the structural elements of a CoB LED, therefore:


**Figure 3.** Application of a chip-level multi-domain LED model in a relaxation type implementation of an electro-thermal-optical solver, with a detailed 3D thermal model of the CoB LED chips' thermal environment completed with a thermo-optical model describing the light conversion and heat generation in the phosphor. For the explanation of symbols used in this figure see the Abbreviations.

Regarding the description of the multi-domain behavior of the blue LED chips, we relied on one of the Spice-like multi-domain LED models we developed earlier [28]. The heat transfer within the bulk of the LED chips, the ceramics substrate and within the phosphor layer is treated by BME's proprietary conduction-mode only thermal field solver [32,33] that has already been successfully adapted to the multi-domain modeling of large-area OLED devices [34,35].

The most important differences and similarities between the former multi-domain OLED simulator and the present approach for CoB LEDs are the following:


The electrical interconnect network of blue LED chips of a CoB device is considered as zero-dimensional electrical nodes (with no voltage drop in the interconnects). The typical electrical configurations of LED chips inside a CoB device are shown in Figure 4.

**Figure 4.** Typical electrical configurations of the arrays of blue LED chips within a CoB LED device package: (**a**) A single string of serially connected LED chips, (**b**) parallel connection of multiple serially connected strings of LED chips. (After [36]).

As seen in Figure 4, none of the electrical configurations of the arrays of blue LED chip inside a CoB device provides individual access to any of the chips within the array. This means that with the common *IF* forward current we power the entire LED array and we can measure the total radiant/luminous flux of the entire array that is the sum of the fluxes emitted by the individual chips and converted by the phosphor. For a single LED string, the situation is the same for the overall forward voltage of the string. These measured characteristics are called 'ensemble' characteristics in the JEDEC JESD 51-51 standard [36].

The relationship between the overall, ensemble characteristics of an LED array and the individual chip characteristics (as illustrated in Figure 5 for the forward voltage) are

$$V\_{\text{F\\_ensemble}} = \sum\_{i=1}^{N} V\_{\text{F\\_j}} \tag{1}$$

$$V\_{F\_{chip}}(I\_{F\_{"\prime}}, T\_{I\_{\prime\prime}}) = V\_{F\_{\prime\prime}}c\_{ensemble}(I\_{F\_{\prime}}, T\_{I\_{\prime\prime}})\_{\text{(2)}} / N \tag{2}$$

$$\Phi\_{\chi\_{\text{\\_ensemble}}} = \sum\_{i=1}^{N} \Phi\_{X\_{-i}} \tag{3}$$

$$\Phi\_{\mathbf{x}\_{\rm chip}}(I\_{\mathbf{F}\_{\prime}} \ T\_{I\_{\cdots \rm combable}}) = \Phi\_{\mathbf{x}\_{\rm acccombable}}(I\_{\mathbf{F}\_{\prime}} \ T\_{I\_{\cdots \rm combable}}) / \ N \tag{4}$$

where Φ*<sup>X</sup>* represents either the radiant flux, Φ*<sup>e</sup>* or the luminous flux, Φ*V*, *N* is the number of the LED chips in the LED string forming the LED array. *VFchip* and Φ*xchip* represent the average forward voltage and flux data that can be related to an individual LED chip within the array.

**Figure 5.** Illustration of the 'ensemble' thermal resistance and forward voltage of a serially connected LED array (as per Figure 4a) in the powering/measurement scheme recommended by the JEDEC JESD 51-51 standard [36] for thermal testing. For the explanation of symbols used in this figure see the main text and the Abbreviations.

This means, that the measured isothermal IVL characteristics of a CoB device need to be post-processed before applying the parameter extraction procedure to obtain the multi-domain chip-level model parameters to be used by the chosen Spice-like multi-domain LED model. Note, that in the case of using a combined thermal and radiometric/photometric test setup as suggested by the JEDEC JESD 51-51 and JESD 51-52 standards [36,37] for CoB LED measurements, there is no way to identify the *TJ*\_*<sup>i</sup>* individual junction temperatures of the blue LED chips within the entire array. As the best approximation, one has to calculate with the *TJ*\_*ensemble* temperature as if it was a uniform temperature for each LED chip within the CoB device. Thus, the set of isothermal IVL characteristics is given by the *VFchip* - *IF*, *TJ*\_*ensemble* and <sup>Φ</sup>*xchip* - *IF*, *TJ*\_*ensemble* data as given by Equation (2) and Equation (4), respectively.

To test the validity of the Equations (1)–(4) we created an array from individual LED packages on a cold plate with a diameter of 12 cm that was attached to a 50 cm integrating sphere. In this setup, the LEDs could be powered and characterized both individually and together as an array of LEDs. It was found that the chip-level characteristics derived from the measurement results of the entire array were very close to the averages of the individually measured voltages and fluxes, suggesting that the above-mentioned approximation for the individual characteristics of the individual LED chips is acceptable.

#### **3. Characterization of the Phosphor**

#### *3.1. Methodology to Set up and to Validate the Multi-Domain Model of the Phosphor*

The major bottleneck in the modeling of phosphor-converted white LEDs like the CoB devices is to get access to the properties of the phosphor layers as manufacturers do not share such information. Regarding the thermal properties, one can find multiple approaches in the literature. Papers [38,39] describe numerical simulation methods for the calculation of the effective thermal conductivity of different phosphor-resin mixtures with different phosphor particle concentrations. In both papers the

simulation results are compared to measurements: In Reference [38], the laser flash method, while in Reference [39] the transient hot-wire method is used to measure the effective thermal conductivity of phosphor layers. A. Alexeev et al. also investigated the effect of phosphor particle concentration on the overall thermal resistance of white LED packages [40]. Wenzl et al. in their paper [41] describe how heat generation in the phosphor layers depends on the extinction coefficient (i.e., light absorption). In this work, besides the thermal conductivity, further material properties of the phosphor layers, such as quantum efficiency are also considered in the simulations. The authors used a simple LED reference structure that inspired our present work (see later). Data available in these papers, unfortunately, did not help us set up our own models, especially regarding temperature dependence of the light conversion.

Though Bachmann in his PhD dissertation [42] provides detailed measurement data of different kinds of phosphors, including a few graphs showing the temperature dependence of luminescence intensity, there is no data on efficiency and thermal properties of phosphor powder-resin composites; the limited data on the temperature dependence of luminescence intensity could not be used for our modeling purposes. As a workaround to the problem of lack of sufficient data on temperature-dependent behavior of phosphors, we had the following approach [25]:


The subsequent (sub)sections of this paper describe the details of the steps listed above.

#### *3.2. Study of the Relevant Properties of Phosphor Layers*

In order to establish a thermo-optical model for the phosphor layer, first, we investigated the features of some phosphor materials. Our initial idea was to derive phosphor properties from measured spectra of blue and phosphor-converted white LEDs of the same LED family (XP-E LEDs of Cree) in the hope that in the white LEDs the same blue chips were used. In the measured spectra the blue peak wavelengths differed; therefore, the assumption, that the only difference between the two kinds of LEDs was the phosphor, was questioned.

As a next step in setting up a proper multi-domain simulation model for CoB LED devices, standard, commercial CoB LED devices in their unpowered state were considered as stand-alone

phosphor samples (Figure 6). We prepared different PDMS/phosphor powder composites to be characterized as stand-alone samples (see Figure 7b) to obtain certain parameters of the phosphors.

**Figure 6.** Commercially available CoB LED devices in their unpowered state were also measured as stand-alone phosphor samples: (**a**) A Lumileds 1202 s CoB LED device; (**b**) multiple such CoB LED devices attached to a temperature-controlled stage to be measured as stand-alone phosphor samples.

**Figure 7.** Test setup for phosphor sample measurements: (**a**) Schematic of the integrating sphere arrangement with excitation blue light source and passive phosphor layers attached to a temperature-controlled stage; (**b**) photograph of a custom-made phosphor sample attached to a temperature-controlled stage.

A 50 cm integrating sphere with dual DUT (device under test) ports was used in an arrangement, as seen in Figure 7, to capture the spectral power distribution (SPD) of the secondary emission of the phosphor samples with a CAS-140CT spectroradiometer. All phosphor samples were attached to a temperature-controlled stage. By sweeping the temperature of that stage spectra, were captured at phosphor temperatures between 15 ◦C and 150 ◦C.

The integrating sphere that we used had two DUT ports facing each other along the equator of the sphere (Figure 7a). This arrangement allowed us to install a blue excitation light source at one port, to focus the excitation blue light on the phosphor sample mounted on a temperature-controlled stage on the other port of the sphere. A cone with a black outer surface with a small aperture was used to decrease the amount of blue light inclining not the sample, but the sphere. This ensured to capture reasonable levels of converted light without saturating the spectroradiometer with the blue excitation.

The samples were exposed to variable intensities of blue light. The base material of the phosphor samples was PDMS. 1 mm thick PDMS medals with different mass fractions of phosphor powders were prepared on a black-painted aluminum plate that was attached to the above-mentioned temperature-controlled stage (as shown in Figure 7b). Also, the whole thermostat was painted black to minimize the backscattering of blue light into the sphere. Figure 8 presents a set of spectra of the blue excitation and the secondary emission of a phosphor sample measured at different phosphor temperatures.

**Figure 8.** Temperature-dependent spectral power distribution (SPD) of a custom-made remote phosphor sample.

As our integrating sphere is not calibrated in the given geometrical arrangement, only an estimated blue reference SPD could be used for temperature-dependent efficiency calculations. For the definition of the different efficiency parameters of the phosphor refer to F. Schubert's widely known book on LEDs [44]. Figure 9 shows the calculated temperature dependence of different properties, such as conversion efficiencies for one of the characterized custom-made PDMS samples.

**Figure 9.** Measured temperature dependence of (**a**) the blue absorption rate, (**b**) the external quantum efficiency, and (**c**) the power conversion efficiency for a phosphor sample prepared from one of the commercially available phosphor powders.

As seen in the diagrams presented in Figure 9, linear or second order relationships can well be used as good approximations for the temperature dependence of the measured properties. The measurement results show that not only the efficiency of the phosphor layer depends on the temperature, but slightly the wavelength of the converted photons (thus, the emission spectra) as well. Because of the experienced linearity in conversion efficiency, this effect can be built in the multi-domain model. Based on these experimental data, we could set up the thermo-optical phosphor model that we built into our thermal

solver; the parameters regarding the temperature dependence can be well fitted to data of other phosphor materials.

Besides the temperature dependence of the light conversion properties, the thermal conductivity of the manufactured PDMS and phosphor-powder mixtures was measured with the DynTIM equipment of Mentor Graphics [45], see Figure 10. The thermal conductivity measurement results for one of the phosphor powder types are shown in Table 1.

**Figure 10.** One of our custom-made phosphor samples on the measurement stage of a Mentor DynTIM (dynamic thermal interface material thermal conductivity measurement) equipment [45].

**Table 1.** Measured thermal conductivity of phosphor layer with different phosphor powder concentration.


#### *3.3. Characterization of Custom-Made Phosphor-Converted White LEDs*

To test the validity of the phosphor model we created custom-made phosphor-converted white LEDs with precisely known structure, both in terms of the bare, packaged blue LED chips and the added phosphor layers. These devices were used as reference structures for fine-tuning the phosphor model attached to our thermal solver. Also, with these single-chip LEDs we avoided all uncertainties associated with the 'ensemble' characteristics of the actual CoB structures, as well as obtained blue LED spectra and white LED spectra where the blue peaks precisely matched.

Figure 11 provides some details of the 'fabrication process' of our own custom-made white LEDs. XPG3 flip-chip power LEDs from Cree were used such, that their original lenses were removed (Figure 11a). At this stage, each blue LED was characterized by isothermal IVL measurements. Phosphor-converted white LEDs were then fabricated by proximate conformal phosphor deposition before forming the clear lens (Figure 11b,c). We used PDMS + phosphor powder mixtures as for the characterization of stand-alone phosphor samples discussed in the previous section. The phosphor powder was mixed with PDMS in 50–50 m/m %, and light conversion layers of four different thicknesses were deposited on the already characterized bare blue LEDs. This way, white LEDs with four different spectral power distributions (thus, four different correlated color temperatures) were achieved. With varying the phosphor thickness, our aim was to convert a different number of photons using the same blue excitation every time. With this technique, we assured that the only differences between the measurement results for the blue and the white LEDs were caused by the phosphor layers themselves.

*Energies* **2020**, *13*, 4051

**Figure 11.** Creating our own white LEDs from Cree XPG3 LEDs: (**a**) Bare blue LED packages with original phosphor and lens removed, (**b**) custom-made phosphor layer with known composition and new lens, (**c**) blue LED packages with the new phosphor + lens structure attached, (**d**) simplified cross-sectional view of the custom-made LED structures used for simulations [24] to validate the simulator. For the explanation of abbreviations used in this figure see the Abbreviations.

The flip-chip assembly of the base LED device was chosen to make sure that while the original dome with phosphor is removed and our own custom-made phosphor layers are added, all electrical connections of the LED chips remain safely untouched.

For each phosphor layer setup, complete isothermal IVL characterization was performed for six forward current values at five junction temperatures.

After measurement of the isothermal IVL characteristics, the custom lenses were dismounted, and cross-sectioned to measure the thickness of the phosphor layers. As we experienced, the temperature of the phosphor layer has a significant impact on the external quantum efficiency, and on blue absorption, which not only affects the efficiency of the LED, but the color of the resulting light too. That is one reason why it is necessary to establish joint compact and detailed multi-domain (thermal, electrical and optical) models in the case of white LEDs, especially for CoB devices.

The thickness dependence of the total dissipated power and the temperature rise of the phosphor layers is shown in Figure 12. This set of data was used to validate the multi-domain phosphor model when applied to the custom-made single-chip reference LED devices.

**Figure 12.** The layer thickness dependence of the major thermal properties of the custom-made phosphor layers identified from the measurements: (**a**) Thickness dependence of the total dissipation in the phosphor layer; (**b**) temperature rise of the phosphor layer.

#### **4. Multi-Domain Modeling: Chip-Phosphor Interaction, Light and Heat Propagation**

From now on, we call the color of the absorbed light of the primary emitter LED chips as blue and the converted and re-emitted light color as yellow. For modeling phosphor layers different effects need to be considered, such as blue absorption, blue scattering, blue-to-yellow conversion (Stokes shift), yellow absorption, yellow scattering and the temperature dependence thereof, if applicable.

We distinguish our phosphor models according to the way the light path is followed in 1D (distance from the source), or in 3D. In certain cases, the absorption and reflection on the LED chip/substrate surface are taken into account. In the following section, we discuss these approaches with their possible limitations.

#### *4.1. A 1D Phosphor Model*

In our 1D model, we consider the blue and yellow absorption and the wavelength conversion. We use simple formulae to approximate the optical (radiant) power of the blue and yellow light, and for the heat generated due to conversion and absorption losses.

When the blue light propagates through the phosphor layer, it may be absorbed or converted by the phosphor particles. Assuming that the particle concentration in the phosphor layer is homogeneous, the blue photon number follows the Lambert-Beer law:

$$N\_B(\mathbf{x}) = N\_\mathbf{c} \cdot \mathbf{c}^{-\mu\_\mathbf{b} \mathbf{x}} \tag{5}$$

where *NB* is the blue photon number at distance *x* (measured from the source), *Ne* is the originally emitted photon number (calculated from the optical power), μ*<sup>b</sup>* is the sum of the attenuation and conversion coefficients μ*<sup>b</sup>* = μ*ba* + μ*bac*, and is proportional to the probability of hitting a phosphor particle. As there is no yellow light emitted from the LED chips, the source of yellow photons is the conversion by a phosphor particle, so the number of yellow photons can be written as:

$$N\_Y(x) = N\_{\mathfrak{e}} \cdot (1 - e^{-\mu\_{\mathfrak{e}}x}),\tag{6}$$

where μ*<sup>c</sup>* is the conversion coefficient.

Considering the yellow attenuation from the distribution of yellow source:

$$N\_Y(\mathbf{x}) = N\_\varepsilon \frac{\mu\_\varepsilon (1 - \varepsilon^{-\mu\_\mathbb{P} x})}{\mu\_\mathbb{y} + \mu\_\mathbb{b}} \tag{7}$$

Note that the converted yellow photons do not follow the direction of blue photons. The direction of propagation of the converted yellow photons can be assumed to be isotropic. Therefore, only half of the converted yellow photons will propagate away from the LED.

The other half starts to move in the direction of the surface of the LED chip. Let the thickness of the phosphor layer be *d* and let the yellow source (*x* ) be above point *x* of the blue source, where the photon number sought can be described as:

$$N\_{\rm ys}(\mathbf{x'}) = N\_{\rm t'} \cdot \mu\_{\rm c} e^{-\mu\_{\rm b} \mathbf{x'}} \tag{8}$$

The attenuation of that source from *x* also given by the Lambert-Beer law:

$$N\_y(\mathbf{x}) = N\_{\mathbf{c}'} \cdot \mu\_{\mathbf{c}} \mathbf{c}^{-\mu\_{\mathbf{b}} \mathbf{x}'} \cdot \mathbf{c}^{\mu\_y (\mathbf{x} - \mathbf{x}')}.\tag{9}$$

With the help of integration, we can get the formula for the photons arriving above point *x*:

$$N\_{y\mathbb{B}}(\mathbf{x}) = \int\_{\mathcal{X}}^{d} N\_{\varepsilon} \cdot \mu\_{\varepsilon} \varepsilon^{-\mu\_{\mathbb{B}} \mathbf{x}'} \cdot e^{\mu\_{\mathbb{B}}(\mathbf{x} - \mathbf{x}')} d\mathbf{x}' = N\_{\varepsilon} \frac{\mu\_{\varepsilon} (\varepsilon^{-\mu\_{\mathbb{B}} \mathbf{x}} - \varepsilon^{-\mu\_{\mathbb{B}} d})}{\mu\_{\mathbb{B}} + \mu\_{\mathbb{B}}}. \tag{10}$$

The number of yellow photons reflected from the LED chip surface is calculated from the power of yellow light incident at the surface. The latter is

$$N\_{\mathcal{Y}}(0) = N\_{\mathfrak{e}} \cdot \frac{\mu\_{\mathfrak{e}} (1 - \mathfrak{e}^{\mu\_{\mathfrak{b}} d})}{\mu\_{\mathfrak{Y}} + \mu\_{\mathfrak{b}}}.\tag{11}$$

Considering also the *r* reflection coefficient (the ratio of reflected and absorbed photons), and the attenuation from the LEDs surface we get:

$$N\_{yR}(\mathbf{x}) = r \cdot N\_{\mathbf{c}} \frac{\mu\_{\mathbf{c}} (1 - e^{\mu\_{\mathbf{b}}d})}{\mu\_{\mathbf{y}} + \mu\_{\mathbf{b}}} \cdot e^{-\mu\_{\mathbf{y}}\mathbf{x}}.\tag{12}$$

Considering that in each direction half of the number of photons is emitted, the final formula for the yellow photon number is:

$$N\_{y}(\mathbf{x}) = \frac{1}{2} \Big( N\_{\varepsilon} \frac{\mu\_{\varepsilon} (1 - \varepsilon^{-\mu\_{b} \ge})}{\mu\_{y} + \mu\_{b}} + N\_{\varepsilon} \frac{\mu\_{\varepsilon} (\varepsilon^{-\mu\_{b} \ge} - \varepsilon^{-\mu\_{b} d})}{\mu\_{y} + \mu\_{b}} + r \cdot N\_{\varepsilon} \frac{\mu\_{\varepsilon} (1 - \varepsilon^{\mu\_{b} d})}{\mu\_{y} + \mu\_{b}} \cdot \varepsilon^{-\mu\_{y} x} \Big). \tag{13}$$

From the known wavelengths of the photons, the energies of the blue and yellow photons and the energy difference, due to wavelength conversion can be calculated as follows:

$$E\_{\rm bu} = h \frac{\varepsilon}{\lambda\_b}, \quad E\_{\rm yu} = h \frac{\varepsilon}{\lambda\_y}, \quad E\_{\rm c} = h \frac{\varepsilon}{\lambda\_b} - h \frac{\varepsilon}{\lambda\_y} \tag{14}$$

With the above energy values, the dissipation density at point *x* can be expressed with the original blue photon number, *Ne*as follows:

$$N\_{\varepsilon} \left( E\_{\text{bar}} \mu\_{\text{ba}} \cdot \varepsilon^{-\mu\_{\text{f}} \text{x}} + E\_{\text{bar}} \mu\_{\text{c}} \cdot \varepsilon^{-\mu\_{\text{f}} \text{x}} + E\_{\text{y} \mu} \mu\_{\text{y}} \cdot \frac{1}{2} \left( \frac{\mu\_{\text{c}} (1 - \varepsilon^{-\mu\_{\text{f}} \text{x}})}{\mu\_{\text{y}} + \mu\_{\text{b}}} + \frac{\mu\_{\text{c}} (\varepsilon^{-\mu\_{\text{b}} \text{x}} - \varepsilon^{-\mu\_{\text{f}} \text{d}})}{\mu\_{\text{y}} + \mu\_{\text{b}}} + r \frac{\mu\_{\text{c}} (1 - \varepsilon^{\mu\_{\text{f}} \text{d}})}{\mu\_{\text{y}} + \mu\_{\text{b}}} \cdot \varepsilon^{-\mu\_{\text{g}} \text{x}} \right) \right) \tag{15}$$

With this analytical formula, we can determine the parameters of the phosphor layer by measurement, although,


• Furthermore, in Equation (14), we assumed a single blue and yellow wavelength and did not deal with the actual spectral power distributions of the original blue and the converted longer wavelength light.

The neglected effects do not result in a significant error in the thermal model, if the phosphor layer is much thinner than the dimension of the light emitting surface of the blue LED chips.

#### *4.2. Simplified 3D Phosphor Model*

The simplified 3D model is an extension of the 1D model to 3D. The scattering is still neglected, but the spatial light distribution is taken into consideration now. To keep this model simple, we do not calculate with the attenuation of the yellow light. The main question to answer with the use of this model is how the overall heat generation distribution will be affected by the losses in the phosphor.

In this approximation the Lambert-Beer law is used again, which gives us the relation between a point on the surface of the LED chip (**r** ) and one in the phosphor layer (**r**):

$$N\_B(\mathbf{r} - \mathbf{r}') = N\_{\rm env} \cdot e^{-\mu\_\hbar(\mathbf{r} - \mathbf{r}')},\tag{16}$$

where *Nen* is the number of nodal (*r* ) emitted photons. Since every location of the lighting surface is considered as a point-like source we have to correct the formula as follows:

$$N\_B(\mathbf{r} - \mathbf{r'}) = \frac{N\_{en} \cdot e^{-\mu\_b(\mathbf{r} - \mathbf{r'})}}{2\pi \left|\mathbf{r} - \mathbf{r'}\right|^2} \tag{17}$$

To get the number of blue photons at location **r**, an integration over the lighting surface is needed:

$$N\_B(\mathbf{r}) = \iint \frac{N\_{\rm eff} \cdot e^{-\mu\_b(\mathbf{r} - \mathbf{r'})}}{2\pi \left|\mathbf{r} - \mathbf{r'}\right|^2} dA' \tag{18}$$

Considering now the Θ(θ) spatial distribution of the blue light we get

$$N\_B(\mathbf{r}) = \iint \Theta(\theta) \frac{N\_{\text{eff}} \cdot e^{-\mu\_b(\mathbf{r} - \mathbf{r}')}}{2\pi \left|\mathbf{r} - \mathbf{r}'\right|^2} dA' \tag{19}$$

The computation for this surface integral for a 100 by 100 mesh lasts a few minutes for each plane that is modeled in the phosphor layer. In order to calculate with the yellow light propagation, we can use the calculated blue photon number, as it is proportional to the source of yellow light:

$$N\_Y(\mathbf{r}) = \int \int \int \frac{\mu\_\mathbf{c} N\_B(\mathbf{r}')}{4\pi \left|\mathbf{r} - \mathbf{r}'\right|^2} \cdot e^{-\mu\_\mathbf{g}(\mathbf{r} - \mathbf{r}')} dV' \tag{20}$$

which increases the computation time to hours for every point. The reflection can be handled by a mirrored blue light source (situated on the backside). This computational cost implies that the analytical formulas should not be used even with a simplified 3D model. The practical workaround to this computational cost issue is to use numerical methods, such as the Finite Volumes Method (FVM).

#### *4.3. Detailed 3D Phosphor Model with a Numerical Approach*

The approach of overall modeling of a CoB device described in the present study was inspired by our previous work that targeted multi-domain simulation of large-area OLEDs [34,35]. For OLED simulations the FVM was applied to get a 3D network of multi-domain elementary cells. This network was solved by the Successive Network Reduction (SUNRED) method [32,33] and the solution provided voltage, temperature, radiance and luminance maps as a response to a given, assumed driving current of the investigated device.

For the simulation of phosphor covered (inorganic) LEDs, we created two special elementary cell models: One for the pn-junction of the LED (describing the multi-domain behavior of the blue LED chips by a Spice-like model) and another multi-domain simulation grid cell type for the phosphor, as illustrated in Figure 13 (for the sake of simplicity for a 2D case). These models will be detailed in the next subsections.

**Figure 13.** Part of the realization of the overall scheme of Figure 3 within our Finite Volumes Method (FVM) numerical solver: The multi-domain FVM cell models depicted in 2D—black circuit elements

represent electrical connections, red circuit elements represent mesh grid cells of the thermal subsystem. The mesh grid level multi-domain phosphor model takes local temperature and material parameters, plus the incident blue and yellow light fluxes as input and provides the transmitted blue/yellow light and converted yellow light fluxes as output. (For a detailed explanation of the symbols used in the drawing refer to the main text). For the explanation of symbols used in this figure see the main text and the Abbreviations.

#### 4.3.1. Junction and Phosphor Cells

A 'junction cell' is basically a special cell that represents the bulk material of the LED chip from thermal perspective, but it also incorporates the new, constant forward current-driven formulation of the Shockley-model based multi-domain LED model that we developed previously within the Delphi4LED project [28]. For every single blue LED chip in the CoB device a local instance of this model is applied, considering the local junction temperature (*TJ*) in the FVM simulation grid cell where this model is attached to, see Figure 13. Such a grid cell is called a 'junction cell'.

This LED chip model is connected to the electrical model of the cell at its interface to other regions of the device, see the black network elements in Figure 13. The two input quantities of the junction model are the constant forward current flowing through it (*IF*) and the temperature of the junction (*TJ*.).

Its four output quantities are the forward voltage (*VF*), the generated heat flux (*PH*), the emitted radiant flux (Φ*e*) and the emitted luminous flux (Φ*V*).

The *PH* power provided by the LED chip multi-domain model instances is the local heat-source of the FVM grid cell, represented by the *Pd* generators in the generic thermal grid cell. The calculated Φ*<sup>e</sup>* radiant flux of a junction cell is split among the blue light rays that are propagated towards the 'phosphor type' simulation grid cells of the FVM solver (see the light blue arrows in Figure 13).

The 'phosphor cells' include the same thermal part as an ordinary structural material or the 'junction cells' (red thermal network elements), and they also include a phosphor multi-domain model, describing the light conversion and propagation and the corresponding thermal losses in the phosphor material.

The calculated thermal loss is the local *Pd* heat-source of the given FVM grid cell. Note, that in Figure 13, symbol *PH* is used both for the multi-domain model of the LED chips and for phosphor regions to denote the local heating power.

#### 4.3.2. Modeling the Light Conversion in the Phosphor

The chip-level multi-domain LED model that we use calculates the total emitted radiant/luminous flux only; it does not provide actual spectral power distribution of the emitted light Therefore, we considered only the blue and converted yellow fluxes, without any details about their actual spectral power distributions. Note, however, that the following discussion is also valid for all colors in the entire spectral range of the emitted converted light, ranging up to red as well and with an additional model for the absorption and emission spectra of phosphors, temperature-induced slight spectral changes like the ones seen in Figure 8 could also be included.

The wavelength conversion results in a loss of energy, which heats up the phosphor. Because the phosphor particles are mixed with transparent but poorly heat-conductive material (e.g., silicone) as a host matrix, their temperature can be significantly, even tens of degrees higher than in other parts of the LED package. The conversion is temperature-dependent, therefore, a multi- domain simulation of the phosphor is required.

Figure 14 shows some light paths in an LED package. Blue light is emitted in different directions from the LED chip, which may reflect from the surfaces, even crossing the phosphor layer several times. Meanwhile, the blue light is partially converted to yellow. The emitted yellow follows different paths than the blue light absorbed by a phosphor particle; it may propagate in any direction. The yellow light can be reflected several times, and it can be absorbed and re-emitted in different directions. The energy loss due to the absorption of yellow light also contributes to the temperature rise of the phosphor layer.

**Figure 14.** Some light paths in an LED: Blue light comes from the chip, yellow light is generated in the phosphor.

We assume again, that the attenuation of the blue and yellow fluxes follows the Lambert-Beer law:

$$
\Phi\_{\mathfrak{c\\_blue\\_out}} = \Phi\_{\mathfrak{c\\_blue\\_in}} e^{-n\_{\mathfrak{blur}}(T)\cdot d} \tag{21}
$$

$$
\Phi\_{\varepsilon\_{\text{-}yellow\\_trans}} = \Phi\_{\varepsilon\_{\text{-}yellow\\_in}\mathcal{E}} \varepsilon^{-n\_{\text{yellow}}(T)\cdot d} \tag{22}
$$

where α*blue* and α*yellow* are the blue and yellow attenuation coefficients, *d* is the effective thickness of the phosphor layer. Equation (22) refers to the case where the yellow light is not produced in the layer but enters from the outside (transmitted yellow). The absorbed blue and yellow fluxes can be calculated as

$$
\Phi\_{\text{c\\_blue\\_absorb}} = \Phi\_{\text{c\\_blue\\_in}} - \Phi\_{\text{c\\_blue\\_out}} \text{ and} \tag{23}
$$

$$
\Phi\_{\mathfrak{e}\_{-}\text{yellow\\_absorb}} = \Phi\_{\mathfrak{e}\_{-}\text{yellow\\_in}} - \Phi\_{\mathfrak{e}\_{-}\text{yellow\\_trans}}.\tag{24}
$$

The yellow flux converted from the absorbed blue flux can be calculated as

Φ*e*\_*yellow*\_*conv* = Φ*e*\_*blue*\_*absorb*·η*conv*(*T*) (25)

where η*conv* is the conversion efficiency. A part of the absorbed yellow may be re-emitted:

$$
\Phi\_{\mathfrak{e}\_{-}\text{yellow\\_re}} = \Phi\_{\mathfrak{e}\_{-}\text{yellow\\_absorb}^{\*}} \eta\_{\text{yellow\\_re}}(T) \tag{26}
$$

where η*yellow*\_*re* is the "yellow-to-yellow conversion efficiency", characterizing the re-emission. The yellow output of a phosphor layer is the sum of the transmitted, converted and re-emitted yellow:

$$
\Phi\_{\mathfrak{e}\_{-}\text{yellow\\_out}} = \Phi\_{\mathfrak{e}\_{-}\text{yellow\\_trans}} + \Phi\_{\mathfrak{e}\_{-}\text{yellow\\_conv}} + \Phi\_{\mathfrak{e}\_{-}\text{yellow\\_rec}} \tag{27}
$$

The heating power due to the conversion loss is the difference between the absorbed blue and the converted yellow radiant fluxes:

$$P\_{\text{loss\\_conv}} = \Phi\_{\text{c\\_blue\\_absorb}} - \Phi\_{\text{c\\_yellow\\_conv}}.\tag{28}$$

The heating power due to the yellow transmission loss is the difference between the input and the transmitted and re-emitted yellow fluxes:

$$P\_{\text{loss\\_yellow\\_turns}} = \Phi\_{\mathfrak{e}\_{-\text{yellow\\_inv}}} - \Phi\_{\mathfrak{e}\_{\text{yellow}\text{rms}\_{\text{frame}}}} - \Phi\_{\mathfrak{e}\_{-\text{yellow\\_rc}}} \tag{29}$$

The full heating power of the phosphor layer is the sum of the absorption and transmission losses:

$$P\_{H\_{-}\text{phosph}v} = P\_{\text{loss\\_conv}} + P\_{\text{loss\\_yellow\\_trans}}\tag{30}$$

#### 4.3.3. Rays split over the FVM Simulation Grid Cells

The FVM simulation calculates the quantities per elementary cell—therefore, the absorption and transmission losses, as well as the blue and yellow output fluxes, are determined for the elementary cells of the phosphor layer. The idea behind the method comes from the ray tracing [46] and path tracing [47] algorithms used in global illumination.

The goal is to create a generic cell model that can handle all-optical and thermal phenomena presented in Section 4.3.2 (as illustrated in Figure 13), and leaves the definition of the light paths to the user. The quantities are determined for each elementary cell individually.

In a cell-split model, light propagates in a beam from the internal volume or from a surface of a cell to the outer surface of another cell. For example, blue light typically propagates from the surface of junction cells to the outer surfaces of a phosphor layer, as illustrated in Figure 15a Handling 3D beams would require integral calculation. To avoid this complicated and time-consuming method, we use 1D rays as in ray tracing methods, see Figure 15b. A ray is defined by the coordinates of its start and end points. In this model, based on Equation (21), the blue output flux for Ray 1 is:

$$
\Phi\_{c\\_blue\\_out} = \Phi\_{c\\_blue\\_in} e^{-a\_{blue\\_1}(T\_1) \cdot d\_1} e^{-a\_{blue\\_4}(T\_4) \cdot d\_4} \tag{31}
$$

where α*blue*\_*n*, *Tn* and *dn* are the attenuation coefficient, temperature and distance traveled by the light in cell *n*, respectively. Φ*e*\_*blue*\_*in* is the flux of the junction cell in the direction of the output surface. As *d* is the distance between the entry and exit points of the ray in the cell, and the temperature and the material are considered homogeneous in a cell, this is a simple calculation. The other equations presented in Section 4.3.2 are similarly simple to re-write for this discretized view.

**Figure 15.** Illustrations for modeling the propagation of the blue light; (**a**) 3D paths of the blue light; (**b**) Single 1D rays of the blue light; (**c**) Multiple 1D rays of the blue light.

If the 3D beam is replaced with a 1D ray, an error is introduced in the calculation that can be reduced by considering multiple rays from the starting surface to the target surface, as illustrated in Figure 15c. (This method is well known in computer graphics for antialiasing purposes).

Different strategies are possible to handle yellow rays. If the 1D light propagation, shown in Section 4.1 is used, the blue ray and the converted yellow will further propagate together. The method is also applicable for 3D blue propagation (Figure 16a), although the result will obviously be inaccurate. A more accurate result can be obtained by determining the flux of the yellow light emitted from a blue ray in a cell and how much of it is radiated towards an outer surface. This will be a yellow ray (Figure 16b). The re-emitted yellow rays can come from the yellow rays in a similar way.

How many rays will be in total if the phosphor region is a rectangular cuboid of *X* × *Y* × *Z* cells with one *XY* surface in contact with the blue chip's junction cell? The number of junction surfaces, in this case, is *NJ* = *X* × *Y*. If 1D light propagation is used, a single ray will leave every elementary junction surface, resulting in a total of *NJ* rays. Using our 3D light propagation model, blue rays start from every junction, one for each outer surface. The number of outer surfaces of the phosphor is *NP* = *X* × *Y* + 2 × *X* × *Z* + 2 × *X* × *Z*, so the number of blue rays *NB* = *NJ* × *NP*. Each blue ray triggers a yellow ray from each intersected cell to each outer surface. If the number of intersected cells is estimated

to be *NC* ≈ 0.7*X* + 0.7*Y* + 0.6*Z*, then for the number of yellow rays we obtain *NY* = *NC* × *NB* × *NP*. If, for example, *X* = *Y* = 10, *Z* = 5, then *NJ* = 100, *NP* = 100, *NB* = 30, 000, *NC* = 17, *NJ* = 153, 000, 000.

The number of yellow rays is by several orders of magnitude greater than the number of blue rays, and the re-emitted yellow rays have not yet been addressed. A realistic model of a real LED device requires a higher spatial resolution of the FVM grid—thus, the number of yellow rays would increase to such a high value that one cannot manage. Therefore, we need a modeling approach where the number of yellow rays to follow is significantly reduced: This is the "indirect yellow ray model" illustrated in Figure 16c. In this model, for each cell, the amount of yellow flux generated by the blue rays passing through and the re-emitted flux generated by the passing yellow rays are cumulated. In a subsequent iteration step, this cumulated flux is considered as the total yellow flux emitted by the cell, and one yellow ray per cell will be started for every outer surface. (Since this uses the flux calculated in the previous iteration, it is not derived from the flux generated by the junction cell in the current iteration, so the calculation is indirect). The number of yellow rays, thus, depends on the number of phosphor cells and the number of the outer surfaces. Using numbers of the previous example the number of the yellow rays is *NYi* = *X* × *Y* × *Z* × *NP* = 150, 000—by three orders of magnitude less, therefore, manageable.

**Figure 16.** Strategies to count the number of yellow light rays: (**a**) Common yellow and blue rays; (**b**) yellow rays triggered by blue rays arriving directly from an elementary junction surface; (**c**) indirect yellow rays.

#### 4.3.4. Phosphor Cell Model

The phosphor multi-domain model is built in the thermal FVM model of the phosphor cells, as shown in Figure 13. The model delivers four output quantities:


Two of these quantities, *PH* and Φ*e*\_*ny* directly influence the operation of the actual simulation grid cell where they were calculated, the other two values (Φ*e*\_*b*, Φ*e*\_*y*) are stored as simulation results.

The model calculates the total output quantities for rays starting from the junction cells, taking into account the temperature and material parameters of each cell crossed by them.

The structure of the phosphor cell model is shown in Figure 17. The output quantities are calculated individually per ray, and then they are summed. There are two types of ray models.

**Figure 17.** Structure of the phosphor cell model. The number of general and yellow ray models can be different.

The general model describes one ray that travels from a junction cell to a particular phosphor cell. It can be a blue, a yellow or a mixed ray (such as in Figure 16a). Blue rays were shown in Figure 15b, yellow rays are shown in Figure 18: the ray starts as blue at the junction, then it is converted in a cell and continues its path to a given destination as a yellow ray.

**Figure 18.** Yellow rays in the general ray model: The eight yellow rays start with the same blue part.

The yellow ray model describes a ray starting from a phosphor cell containing only yellow component, see Figure 16c. The yellow ray model uses less memory than the general ray model. The general model must be used at junction cells, and the yellow model can be used at phosphor cells.

The ray models represent a ray by sections corresponding to individual cells it crosses, see Figure 19. Generally, multiple rays leave the starting cell, the amount of flux in a given ray is controlled by the proper adjustment of the *K*<sup>0</sup> multiplier factor. The internal ray sections calculate the input blue and/or yellow flux of the current cell, the ray model of the cell cascade in the path calculates the output quantities of the current cell.

Each internal ray section corresponds to a cell in the space between the starting cell and the current cell, using the temperature and material parameters of that cell. The structure of the general and yellow ray sections is shown in Figure 20.

If a ray suffers an imperfect reflection at the *i*-th section of its path, we can model it by adjusting the *K*-values of section *i*. In the case of reflection, sections *i* and *i* + 1 usually belong to the same phosphor cell.

The fluxes of the input rays are transferred to the α*blue* and α*yellow* blocks of the model. These blocks represent attenuation according to Equations (21) and (22), as well as absorbed fluxes according to Equations (23) and (24). The α*blue* and α*yellow* attenuation factors are temperature-dependent material parameters of the cell. The η*conv* and η*yellow*\_*re* blocks represent the blue-to-yellow and yellow-to-yellow conversion efficiencies, which are temperature-dependent material parameters, as seen in Equations (25) and (26).

The *K* blocks control the amount of the resulting flux propagated to the next ray section. These blocks correspond to a constant multiplication, most often zero or one.

**Figure 19.** Structure of the ray models. Φ*e*\_*yellow*<sup>0</sup> is the sum new yellow of the starting phosphor cell. *K*<sup>0</sup> is the ratio of the flux treated by the ray to that of the starting cell.

**Figure 20.** Structure of the internal ray sections.

The structures of the general and yellow ray sections are shown in Figure 21. The terminal sections differ from internal ones in that they calculate both new yellow flux and *Ploss*\_*conv* power loss of the conversion as described by Equations (28) and (29).

#### *4.4. The Use of the Simple 1D and of the Complex 3D Models, Extraction of Phosphor Model Parameters*

The 1D model provides fairly good accuracy for thin phosphor layers (much thinner than the lateral dimensions of the layer). Therefore, with an appropriate set of phosphor samples, it can be used for the extraction of phosphor material properties (model parameters), such as absorption rate or conversion efficiency. With the set of model parameters identified, the 3D phosphor model is used for accurate simulations.

As described in Section 3.3, phosphor-converted white LEDs have been created using fully pre-characterized bare blue ones, with five different phosphor layer thicknesses, realizing five different spectral power distributions, thus, realizing light output with different CCTs. (The mass fraction of the phosphor powder has also been varied; see the applied mass fractions in Table 1).

**Figure 21.** Structure of the terminal ray sections.

On the one hand, these LEDs with known properties have been used for extracting material properties for the multi-domain model as mentioned above [25], and on the other hand, for validating the overall simulation approach (with the blue chip multi-domain compact model embedded), simulating them characterized single-chip white LEDs well. The major steps of extracting the phosphor layer parameters were the following:


Note, that the spectral power distribution of the 'yellow' light can be better approximated by assuming multiple wavelength bands of the 'yellow' light (in an extreme case bands correspond to the wavelength resolution to the measured spectra). This would assume, however, as many yellow ray models as many yellow wavelength bands are assumed. (The execution time of the ray model would linearly scale with the number of the assumed yellow wavelength bands).

We measured the SPDs of all custom-made white LEDs. The spectrum measurement was done on five samples with different phosphor layer thicknesses (0 μm (blue LED), 32 μm (cool white LED), 47 μm (neutral white LED), 68 μm (warm white LED), and 99 μm (amber LED)), at five different blue LED junction temperatures (30 ◦C, 50 ◦C, 70 ◦C, 90 ◦C and 110 ◦C) and each with six different forward currents (100 mA, 350 mA, 700 mA, 1000 mA 1500 mA, 2000 mA). From the total of 150 spectrum measurements, we present data for five measurements only, for demonstration purposes (the applied measurement conditions were: Fifty degree Celsius ambient temperature and 1000 mA forward current). The calculated photon numbers are summarized in Table 2.


**Table 2.** Calculated photon numbers for different LED phosphor layer thicknesses.

For the blue photon numbers, we can fit an exponential, with a coefficient of μ*<sup>b</sup>* = 0.039/μm, from the yellow data, we can determine the missing coefficients. As Equation (12) is transcendent there are more solutions, but physically only one is found to be correct: (*r* = 1 ± 0.08; μ*<sup>c</sup>* = 0.038 ± 0.04; μ*<sup>y</sup>* = 0.037 ± 0.006; μ*<sup>b</sup>* = 0.039 ± 0.001), where the length is given in μm, so the dimension of the coefficients is 1/μm.

The measured blue and yellow photon number versus the 1D prediction is shown in Figure 22. As the diameter of the LED chip is 1.6 mm, theoretically, the accuracy range of the 1D model is 0–160 μm in phosphor layer thickness. Comparing the measurement data to simulation results suggests that the method provides sufficient accuracy. The measurements at other LED operating points (junction temperature, forward current) show less difference in the extracted parameters than the inaccuracy of the calculated parameters. For different concentrations of phosphor powder the coefficients can be scaled, as the original relationship between the coefficients and the physical effect was the Lambert-Beet law, where the attenuation coefficient is proportional to the attenuation cross-section, which is proportional to the phosphor powder concentration: μ ∼ σ ∼ *Cphos*. The phosphor temperature rise inside the phosphor layer with respect to the junction temperature can be determined through the integration of the heat distribution.

**Figure 22.** The photon numbers of blue and yellow light emitted at the top surface the phosphor layer, for different phosphor layer thicknesses.

#### **5. Simulation of a Commercially Available CoB Device**

We measured and modeled a Lumileds 1202s CoB LED device which consists of 24 LED chips (two strings of 12 LEDs, connected in parallel) with a lateral dimension of 600 μm × 700 μm each, placed on an aluminium pad with solder layer between them, covered by a phosphor layer, see Figure 23a. The parameters of its phosphor coating were measured as outlined in Section 3.2, see further details in Reference [25]. A 3D model was created for simulation (Figure 23b). In the numerical simulation model the 650 μm thick phosphor was divided into nine layers of equal thickness. The parameters of the multi-domain LED chip model were extracted from the measured 'ensemble' characteristics. The coefficients of the phosphor model were extracted from the measured spectral power distributions.

**Figure 23.** A Lumileds 1202s CoB LED: (**a**) Photograph of a physical device; (**b**) axonometric view of its 3D simulation model (internal structure, without the visualization of the phosphor layer).

#### *5.1. Di*ff*erent Simulation Setups*

We studied seven strategies that describe the propagation of light in the phosphor with different level of approximations, as shown in Figure 24: 1D propagation only (Figure 24a) or propagation in multiple directions with a uniform spatial distribution (Figure 24b–f), as follows:


As it can be seen in Figure 24, even with a small number of cells, the number of possible rays and ray sections is already very high. The full model, shown in Figure 23, contains 720 blue chip junction cells and 20,628 phosphor cells. Due to the symmetry, we can use half of the model with 360 and 10,314 junction and phosphor cells, respectively. If all rays were taken into account in the calculation, we would have an unmanageable number of rays, especially for yellow rays; therefore, we weight the rays by the estimated flux they carry (i.e., we assign 'importance' to them) and sort them by this. The rays with the lowest flux are discarded until the total flux of the remaining rays reaches the desired level. In the simulations presented, we use two *levels of importance*: Ninety percent and ninety-nine percent, that is, rays representing 10% and 1% of the flux are discarded. Of course, the discarded flux is distributed proportionally among the remaining rays so the total flux will be finally propagated by the rays considered. The estimated flux transported by a ray is the product of the input flux of the ray and the solid angle, Ω*output*\_*sur f ace* of the target surface seen from the starting point:

$$
\Phi\_{\text{c\\_estimated\\_transportal}} = \Phi\_{\text{c\\_estimated\\_in}} \cdot \Omega\_{\text{output\\_surface}} \tag{32}
$$

**Figure 24.** Different strategies of modeling light propagation, tested in our CoB simulations: (**a**) 1D; (**b**) general rays to top; (**c**) general rays to top and sides; (**d**) general + yellow rays to top; (**e**) general + yellow rays to top and sides; (**f**) general + yellow rays to all phosphor sides; (**g**) general + yellow rays to all sides with bottom side reflection. (One of the reflected yellow rays was highlighted in red).

In the case of the half model and five rays per junction, for cases (c), (e), (f), and (g) shown in Figure 24, the total number of blue rays was 3,663,900. The number of yellow rays was 3,075,078 at a 99% importance level, while at a 90% level it was 1,648,216 only. Table 3 shows the number of the general (or blue) rays and yellow rays for the different importance levels, the total number of sections of the rays, and the average length of the rays weighted by the estimated flux that they carry.


**Table 3.** General and yellow ray numbers of the half CoB LED model in the case of five blue rays/junction.

<sup>1</sup> In the case of the ray strategy (a), always one blue ray/junction is used, and the importance level is 100%.

For the simulations, we used a workstation with an AMD Threadripper 2920x processor having 12 cores and 32 GB of RAM. With this machine, we achieved the following execution times and memory need.

For the half model the simulation of strategy (a), which contains 360 rays, takes 6.9 s and requires 1.0 GB of memory, while strategy (g), with a 90% importance level and nine million rays, takes 435 s and consumes 5.1 GB of RAM. For the full model, strategy (g) with 90% level and 32 million rays, the execution time is 2215 s and memory need is 20.0 GB. Simulating the full model would have required about 80 GB of RAM at a 99% importance level.

In the simulations, the bottom surface of the models was set to a fixed 25 ◦C, the other sides were modeled with constant convection boundary condition with a heat transfer coefficient of 10 W/m2K, at an ambient temperature of 25 ◦C.

Based on the findings of A. Alexeev and his co-workers [48–50], from the point of view steady-state behavior, the effect of the heat transfer from the top of the phosphor (silicone dome in the case of LED packages with lenses) can be neglected. Therefore, in all simulation scenarios, we neglected radiation—but due to the large, open phosphor surfaces, we assumed cooling by natural convection through the large top surface area of a CoB device. (Alexeev pointed out that in the phosphor, as a secondary heat-path towards the ambient, the heat storage has a significant effect though. Through the thermal capacitance associated with every FVM simulating grid cell, this is inherently accounted for in our thermal simulation model, as it was shown in Figure 13).

The model parameters used in Equations (21)–(30) contain four phosphor material parameters that were extracted from the measurement results, as outlined earlier. We examined two models:


Temperature dependency of the parameters is under 0.02%/ ◦C, therefore, it is considered constant in the simulation.

#### *5.2. Influence of the Chosen Light Propagation Model on the Phosphor Temperature*

With our modeling approach, the main target was to accurately describe the thermal behavior of white CoB LEDs; the accurate calculation of the distribution of emitted light (i.e., radiance/luminance maps of the CoB surface) was a secondary target for us. Since we cannot measure the temperature distribution inside the phosphor, the question is how accurately the simulated temperature distributions obtained with the simpler models match the results obtained with the most accurate model. To reduce the need for computational resources, we took advantage of the symmetry of the CoB LED device, and only half of the detailed model, shown in Figure 23, was used. In these simulations, the driving current of the CoB LED was 200 mA, half of the 400 mA of the full model, which resulted in 7.3 W of electrical input power, and the blue flux radiated by the LEDs was 2.5 W.

Tables 4 and 5 show the simulation results. We obtained roughly 28% higher phosphor temperature rise with the simplest 1D model (ray strategy (a)) than with the most detailed, most accurate model (ray strategy (g)) while the obtained junction temperatures were practically not affected by the chosen light propagation model.

**Table 4.** Simulation results of the half CoB LED model in the case of 400 mA driving current, no yellow absorption, 90% and 99% importance levels, five blue rays/junction.


<sup>1</sup> In the case of the (a) ray strategy, always one blue ray/junction is used and the importance level is 100%.


**Table 5.** Simulation results of the half CoB LED model in the case of 400 mA driving, equal blue and yellow absorption, 90% and 99% importance levels, five blue rays/junction.

<sup>1</sup> In the case of the (a) ray strategy, always one blue ray/junction is used and the importance level is 100%.

#### *5.3. Simulated Temperature, Radiance and Luminance Distributions at 400 mA Driving Current, Using Di*ff*erent Phosphor Models*

With the models described in the previous sections, steady-state simulations have been carried out for the real CoB device. In all simulations, 400 mA forward current was applied, and we used the phosphor models (a–g) and compared them. Figure 25 shows the temperature distribution at the top of the CoB LED. Several degrees of differences develop in the temperature distribution obtained by simulating the full and the half structure, see Figure 25b. At the center of the CoB device, the difference between the results obtained by the full and half models reaches 7.1 ◦C. The difference obtained for the two structures is due to the different light propagation caused by the symmetry plane as an artificial boundary. The difference in the light propagation can be best visualized by the different distributions of the radiance at the phosphor surface (and also at the internal surfaces of the layered phosphor model) as illustrated in Figure 26. Thus, this problem can be mitigated to some extent by a modified optical model at the symmetry plane: The cell surfaces in contact with the symmetry plane are also the target surfaces of the rays, and the rays hitting them are reflected on the surface as if coming from the symmetrical other side of the CoB structure such that the total flux of the reflected rays is the same as if the rays would have originated from the missing other half of the structure. This still would result in a somewhat different ray distribution and flux compared to the full model because the rays 'reflected' from symmetry plane carry more flux than the rays passing the same plane in the full structure.

**Figure 25.** Simulated distributions of the temperature rise (model setup: 400 mA driving current, 25 ◦C ambient temperature, light propagation described by model g) presented in Section 5.1, with equal blue and yellow absorption, 90% importance level) (**a**) surface temperatures: full model; (**b**) cross-sectional plots of temperature distributions at the blue chip junctions and at the top surface along the green lines; (**c**) surface temperatures: half model.

**Figure 26.** Cross-sectional plots of the radiance distributions along the green lines of (**a**) Figure 25a for the full model and (**b**) Figure 25c for the half model. (Simulation model setup—400 mA driving current, 25 ◦C ambient temperature, light propagation described by model g) presented in Section 5.1, with equal blue and yellow absorption and 90% importance level).

Further analysis showed, that even though with the optical model modified at the symmetry plane the radiance distribution of the half model became more similar to the radiance distribution of the full model, but in terms of temperature distribution the deviation of phosphor temperature at the top in the half model was found to be 5.5 ◦C higher than in the case of simulating the full structure. Overall, taking advantage of the symmetry of the structure to reduce model complexity (which is a common practice in numerical thermal simulations) is not recommended in this multi-domain simulation problem that involves modeling of light propagation as well.

Figure 26 shows that as we move away from the chip, the radiance decreases. The junction layer breaks this pattern, where the radiance is less than in the phosphor layer immediately above it. The reason for this phenomenon is that in the model, the yellow light is reflected from the surface of the chip and does not reach the junction. In Figure 27, we present simulated radiance and luminance maps at the top of the entire phosphor layer of the CoB device by using the model based on the full geometry of the device structure. (Note, that since in the presented light propagation model the spectral power distribution of the converted light was not resolved, the calculated luminance maps are based on approximated luminous flux values associated with the radiant fluxes carried by the rays, therefore all simulated luminance maps presented here are approximate ones only.)

**Figure 27.** Simulated radiance and luminance distributions on the top of the phosphor (model setup: 400 mA driving current, 25 ◦C ambient temperature, light propagation described by model g) presented in Section 5.1, with equal blue and yellow absorption, 90% importance level): (**a**), radiance map; (**b**) luminance map.

#### **6. Simulation Results, Comparison with Measurements**

The means of comparing the detailed multi-domain simulation results of a CoB device to measured data are very limited. As mentioned in Section 2, with usual LED package level testing tools, only the 'ensemble' characteristics, such as the overall forward voltage and the emitted total radiant or luminous flux can be measured. The junction temperature identified with the help of the JEDEC JESD 51-51 electrical test method for LEDs will also be an average value, without any information about the differences of the individual chip temperatures within a CoB device.

With imaging methods, however, we have some hope to measure properties that we can also obtain by simulations, such as the temperature distribution or the luminance distribution at the top surface of a CoB device, using an infrared camera or an imaging luminance meter (luminance measuring camera). Both measurements are problematic. In the case of infrared thermography, one has to make sure that the emitted light does not introduce false information in the IR image. In the case of luminance measurement cameras, the problem is that such cameras are not designed to characterize high-intensity light sources, when a CoB LED is driven by its nominal forward current, it is so bright that a usual luminance measuring camera gets saturated. Therefore, with such a camera we could measure the luminance distribution of our CoB LED device only at very small forward currents (e.g., 80 mA) where the luminance did not cause the camera to saturate yet. To have a considerable temperature rise, for measurements by an IR camera, the investigated CoB LED was driven by 100 mA forward current. During the measurements the CoB device was attached to a temperature-controlled stage, providing a targeted ambient temperature of 25 ◦C (in practice achieving an actual temperature of 24.4 ◦C).

During the simulations an ambient temperature of 25 ◦C was assumed, the thermal boundary conditions were the following: At the bottom of the ceramics substrate of the CoB structure, we assumed a 25 ◦C constant temperature; while at the other outer surfaces of the device (at the sides and at the top), a heat transfer coefficient of 10 W/m2K was applied (representing heat transfer by natural convection roughly). In Figure 28, we present the transient of the average temperature of the CoB device together with two temperature maps grabbed during the heating up process. The image on the top corresponding to 43.7 ◦C average temperature represents already the thermal quasi-steady-state of the device. This is compared to the simulated surface temperature distribution in Figure 29: The peak temperature and the shape of the temperature distribution are well estimated by the simulation, the difference in the measured and simulated peak temperature is 0.68 ◦C only.

Comparing the measured and simulated luminance maps (Figure 30) we can see that the difference between the measured and simulated maximal luminance is 19% while in the average luminance the difference is 7% (measured—2.721 <sup>×</sup> 106 cd/m2, simulated—2.528 <sup>×</sup> 106 cd/m2). One reason for the higher maximum simulated luminance value is the 90% importance level (meaning that 10% of the radiated power was distributed among the major light paths). We could not apply a higher importance level on the available computers. Another probable reason for the discrepancy is that there may be differences in the actual internal structure of the CoB LED compared to the modeled structure, and the actual reflections may differ from the ideal case used in the model.

Note, that at this stage of the development of our FVM simulation model, it is very hard to judge the accuracy of the simulations from the differences between the measured and simulated luminance maps. On the one hand, even at the low driving currents the CoB LED device was too bright for accurate direct imaging with the luminance measuring camera; on the other hand, due to the lack of resolving the spectral power distribution of the converted light in the optical part of our multi-domain model, the luminance estimated from the radiance is only a very rough approximation. Nevertheless, the properly matching orders of magnitude of the simulated and measured luminance values obtained for the brightest spots and the 19% relative difference between maximal values and 7% difference between average values are promising.

A further result from the multi-domain simulations of the CoB device is the voltage distribution on the chip interconnect metallization layers, see Figure 31. The differences between the discrete values corresponding to the chip locations represent the actual forward voltages of the individual chips, calculated by the instances of the chip-level multi-domain LED model, embedded into our FVM solver. The actual voltage drops are determined by the local temperatures of the chips. Unlike in the case of a real, physical CoB device, in the simulation model, we have access to the individual forward voltage values of the chips, present in the LED array of a CoB device.

**Figure 28.** The measured transient of the average temperature of the CoB device and two thermal images grabbed during the heating up process of a physical sample of the investigated CoB device.

**Figure 29.** Steady-state temperature distribution at the top surface of the investigated CoB device at 100 mA driving current: (**a**) measured; (**b**) simulated.

**Figure 30.** Luminance distribution at 80 mA of driving current and at an ambient temperature of 25 ◦C: (**a**) Results as presented by the software of the luminance measurement camera, (**b**) luminance distribution estimated from the simulated radiance map.

**Figure 31.** Simulated voltage distribution in the metallization layer of the anodes of the blue LED chips (400 mA driving current, 25 ◦C ambient temperature).

The simulation also provides insight into the lateral and vertical distributions of other properties, such as the actual junction temperatures of the individual LED chips, radiance at a given plane within the phosphor parallel to the substrate, phosphor temperature as a function of distance from the blue chip surfaces, as illustrated in Figure 32.

Note that, a similar vertical characteristic was published in [23] where the yellow light continuously increases away from the junction, while in Figure 32c it decreases continuously. The phenomenon is caused by the difference between the two light propagation models: In Reference [23] propagation model (a) was used where blue and yellow light travel together, perpendicular to the chip surface, while Figure 32c corresponds to the light propagation model (g) where the yellow light exits the cells of the phosphor model with equal probability in all directions, so a significant portion of it starts down, then it is reflected from the surface of the chip and travels towards the surface of the phosphor. This means that the yellow light passes through many surfaces in both directions. The consequence of this phenomenon is that there is greater radiance near the chip and dissipation occurs closer to the chip than in the case of applying model (a), resulting in a lower average phosphor temperature.

**Figure 32.** Simulated distributions within the CoB device: (**a**) Junction temperature of the blue LED chips; (**b**) blue radiance at the top of the LED chips; (**c**) vertical distribution of the phosphor temperature, blue radiance and yellow radiance at the location indicated by the green X marker (400 mA driving current, 25 ◦C ambient temperature).

As mentioned before, one cannot compare the simulated internal distributions of the forward voltages, chip junction temperatures and vertical phosphor temperature distributions to measurements, the 'ensemble' characteristics though, such as the overall forward voltage or the emitted total radiant flux can be both measured and calculated from the simulation results. In Table 6, we provide a comparison of these quantities. Table 6 also provides temperature data: The 'ensemble' junction temperature, *TJ*\_*ensemble* as measured in compliance with the JEDEC JESD51-51/51-52 standards and an average junction temperature, *TJ*\_*average*, calculated from the distinct junction temperatures used as input to the instances of the multi-domain chip-level LED compact model. Since these temperature values are obtained in different ways, it does not make sense to calculate any relative temperature error from them.


**Table 6.** Measured and simulated 'ensemble' properties of the investigated CoB LED device (400 mA driving current, 25 ◦C ambient temperature).

#### **7. Summary and Conclusions**

In this paper, we proposed and described a methodology for electrical-thermal-radiometric multi-domain modeling and simulation of white CoB LEDs by the combination of compact and distributed modeling methods.

We have also presented a method for multi-domain light modeling in the phosphor, also considering the local temperature dependence of some phosphor properties, such as the conversion efficiency. We implemented different physics/analytic formulae based light propagation models of various complexities and with a few, added heuristics, offering different trade-offs between the need for computational resources and expected accuracy.

A major bottleneck in every numerical simulation is to provide accurate/realistic input data, especially in terms of material properties. Phosphor layers in LEDs pose special problems in this

regard, since the composition of the applied materials are usually not disclosed to the public, not even general data, such as thermal conductivity or exact absorption/emission spectra and their possible temperature dependence. Therefore, in order to set up realistic models, we created our own phosphor samples and our own phosphor-converted white LEDs that were characterized in details, in order to extract the phosphor properties as input for modeling and also, to serve as simple reference structures with controlled properties for validating our simulation models.

With the developed opto-thermal model for phosphor layers we carried out trial simulations for assumed CoB LED structures with different phosphor thicknesses, and with these models we also compared our different light propagation models. We found that for single-chip white LEDs with thin phosphor layers, even the simplest 1D light propagation model may provide sufficiently accurate results. For a general case of any complex CoB device, however, we found that the complex 3D light propagation model is better suited. We found that certain heuristics can be used to speed up the simulations without compromising the accuracy of the results. This 3D light propagation model was implemented in our FVM based numerical simulation code. With our previously proposed Spice-like chip-level multi-domain LED model included, the electrical behavior of the LED chips is also included in the CoB model, allowing to study different chip-level, package level and phosphor level problems (such as the effect of increased local junction heating, due to thermally degraded die attach layers on the luminous flux output) simultaneously, spatially resolved to chip-level or even to smaller scales.

The use of this detailed, distributed thermo-optical model, combined with the LED chips' compact multi-domain model, was demonstrated through the example of the Lumileds 1202s CoB LED devices. This was among the test samples of the round-robin test of the Delphi4LED project and such was already characterized in great detail by multiple independent LED testing laboratories [43]. Using this example, we found that our present modeling approach provides satisfactory accuracy during multi-domain simulation of CoB devices.

The work reported here is far from complete. An obvious approximation is that the spectral power distribution of the converted light is not yet considered. This issue, though, does not impose any theoretical problem; only the 'yellow ray model', indicated in Figure 17, needs to be multiplied according to the number of spectral ranges to which the detailed emission spectrum of the converted light is resolved. This would be important to model the visual performance (such as the total luminous flux, the surface luminance map, color point/correlated color temperature) of the CoB devices accurately enough. Note, however, that in terms of calculations of the radiant properties, our present model already provides accurate results since in the calculation of the yellow photon number the wavelength dependence of the radiant power of the photon flux is considered.

**Author Contributions:** Conceptualization, A.P., L.P., M.N., and Z.K.; methodology, L.P., M.N., Z.K., G.H.; software, L.P., M.N.; validation, G.H. and J.H.; formal analysis, L.P. and M.N.; data curation, G.H. and J.H.; writing—original draft preparation, L.P., M.N., G.H., J.H.; writing—review and editing, A.P., L.P., Z.K., G.H.; supervision, A.P.; project administration, A.P.; funding acquisition, A.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received funding from the European Union's Horizon 2020 research and innovation program through the H2020 ECSEL project Delphi4LED (grant agreement 692465). Co-financing of the Delphi4LED project by the Hungarian government through the NEMZ\_16-1-2017-0002 grant of the National Research, Development and Innovation Fund is also acknowledged. The work related to phosphor characterization was co-funded by the K 128315 grant of the National Research, Development and Innovation Fund. Final validation tests and writing this paper were supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial Intelligence research area (BME FIKP-MI/SC) and the Nanotechnology research area (BME FIKP-NAT) of the Budapest University of Technology and Economics. The support of the Science Excellence Programs at BME under the grant agreement 341 NKFIH-849-8/2019 and BME NC TKP2020 of the Hungarian National Research, Development and Innovation Office is also acknowledged.

**Acknowledgments:** The help of Z. Sárkány (from Mentor, a Siemens business, Budapest, Hungary) in measuring the thermal conductivity of phosphor samples is acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**



#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Lifetime Modelling Issues of Power Light Emitting Diodes**

#### **János Hegedüs, Gusztáv Hantos and András Poppe \***

Department of Electron Devices, Budapest University of Technology and Economics, 1117 Budapest, Hungary; hegedus@eet.bme.hu (J.H.); hantos@eet.bme.hu (G.H.)

**\*** Correspondence: poppe@eet.bme.hu; Tel.: +36-1-463-2721

Received: 27 May 2020; Accepted: 23 June 2020; Published: 1 July 2020

**Abstract:** The advantages of light emitting diodes (LEDs) over previous light sources and their continuous spread in lighting applications is now indisputable. Still, proper modelling of their lifespan offers additional design possibilities, enhanced reliability, and additional energy-saving opportunities. Accurate and rapid multi-physics system level simulations could be performed in Spice compatible environments, revealing the optical, electrical and even the thermal operating parameters, provided, that the compact thermal model of the prevailing luminaire and the appropriate elapsed lifetime dependent multi-domain models of the applied LEDs are available. The work described in this article takes steps in this direction in by extending an existing multi-domain LED model in order to simulate the major effect of the elapsed operating time of LEDs used. Our approach is based on the LM-80-08 testing method, supplemented by additional specific thermal measurements. A detailed description of the TM-21-11 type extrapolation method is provided in this paper along with an extensive overview of the possible aging models that could be used for practice-oriented LED lifetime estimations.

**Keywords:** power LED measurement and simulation; life testing; reliability testing; LM-80; TM-21; LED lifetime modelling; LED multi-domain modelling; Spice-like modelling of LEDs; lifetime extrapolation and modelling of LEDs

#### **1. Introduction**

The typical failure mode of LEDs, unlike fluorescent and incandescent light sources, is not catastrophic failure. The total luminous flux of solid state light sources (SSL) experiences a continuous decrease with the elapsed operating time. The most commonly applied end-of-life criterion of LEDs is related to this constantly declining nature. The most apparent manifestation of LED aging is the luminous flux decrease, or seen from another perspective, to what extent the initial value of the emitted total luminous flux of an LED package is maintained. (sloppy, every day terminology to denote this property of LEDs is called 'lumen maintenance' that we shall rigorously refrain from using; instead, we shall refer to this LED property as 'luminous flux maintenance'.) The IES LM-80 family of test methods have been developed and used in the SSL industry to measure LEDs' luminous flux maintenance. The experiment part of our work presented here is based partly on the provisions of the IES LM-80-08 test method [1]. Besides luminous flux measurements, measuring the continuously shifting forward voltage may be part of the life tests but it is still not required in LM-80 tests and therefore such measurements or the results reporting is often omitted. In addition, LM-80-08 (like any other common life testing method) is defined at predetermined ambient temperatures and does not consider any change in the dissipated power or the degradation of the heat flow path, i.e., the cooling capability of the LED, though such tests were already proposed as early as 2011 [2]. The maximum allowed depreciation of the total luminous flux depends on the exact field of the application; the end

of the SSL product lifetime is considered to be at the time when its light output deteriorates to the critical value. If an LED-based light source operates with a fixed, constant drive current, its value shall be determined so that the level of illumination remains sufficient even at the critical light output level. This results in higher illumination than necessary during most of the life cycle, as well as a significant amount of extra energy consumption. A smart controlling scheme that keeps the light output at a constant level throughout the lifetime can therefore not only increase the visual comfort of the SSL product but also improves its luminous efficiency [3,4]. Some SSL vendors already provide LED drivers that can be pre-scheduled; by gradually increasing the forward current of the LEDs, effects of the continuous total luminous flux degradation can be compensated. The elapsed lifetime dependent controlling scheme is defined for each configuration that consists of the driver, the applied heatsink and LEDs etc. The available CLO solutions most probably rely on extensive life testing results, however, the exact technical details (above the theory of the approved lifetime testing and extrapolating methods) are not put to public.

The recent industrial trends are continuously pushing product development under digitalization to reduce time-to-market and development cost. This mostly means system level computer aided simulations with the so-called digital twins (computer simulation models) of the real life components, like light sources, optical parts, heatsinks etc. Power LED modelling is still an active research area; a recent European H2020 project on LED characterization and modelling (Delphi4LED) [5] undertook to fulfil the growing industrial needs and aimed to generate the measured-data based digital twins of power LEDs [6–9]. Besides many considerations like round-robin testing [10], product variability analysis [11–13], chip-on-board device modelling [14] etc., one could rise the question: how could the electrical, optical and thermal parameter degradation of the LEDs be modelled? There are several analytical models for different stress conditions and parameter changes, e.g., mechanical stresses of the wire bonds [15], termo-hygro-mechanical stresses inside the package [16], shift of parameters in the Shockley diode equation [17], the course and effects of electro migration [18,19], etc.

Our present work originates also from the Delphi4LED in two ways. On the one hand, the models developed (e.g., [7]) and the test methods (see e.g., [10]) used in that project have also been used in this work. On the other hand, we focused on mainstream LEDs of today's SSL industry (operating in the visible range) that were also the subject of investigation in Delphi4LED and in terms of classifying these devices as 'mid-power' or 'high-power', we use the same terminology that was also been used within Delphi4LED [9]. This also explains why in our study we did not consider novel LED structures or recent LEDs aimed for the display industry, despite the fact that top level publications on these devices also share significant amount of test data [20–23]; rather, we re-used some of our own archived data measured during earlier European collaborative R&D projects such as NANOTHERM [24] and we also used our own test data obtained recently.

Reliability testing and investigation of LEDs has long been a hot topic, just some examples are papers [25–40]. Still, there is a lack of a lifetime-lasting multi-physical digital pair of the already existing and widely used solid state lighting solutions, not to mention the novel devices in the research phase such as the LEDs described in papers [20–23]. This exceeded the goals of the aforementioned H2020 research project but the solution of this issue is of an increasing interest as it offers many new options in certain LED applications. The capability of modelling the parameter degradation under different environmental conditions allows to determine the controlling scheme that results in constant light output (CLO). Furthermore, accurate system level lifetime simulations could give appropriate feedback to the luminaire designers by the means of the operational pn-junction temperatures and the suitability of the cooling assembly. These altogether could provide improved reliability, lower power consumption and higher visual comfort during the whole lifetime of streetlighting luminaires.

Our initial concepts and the summary of some of our test data were provided in our prior conference publications [3,4]. In [3] the concept of stabilizing the total emitted luminous flux of LEDs for their foreseen total expected life-span was presented. In [4] actual test data were provided along with our first attempt to extend one of the multi-domain chip level LED models of the Delphi4LED project [7] with the elapsed LED lifetime. The work described in this article is a comprehensive extension of our theory and test data already presented at the THERMINIC workshops in the previous years [3,4].

#### **2. Total Luminous Flux Maintenance Projections**

Reliability and lifetime testing of electrical components is quite a diverse field of research. Due to the complex use of materials in the LED package, various types of the failure mechanisms are induced by the different ambient stress conditions, such as extremely low or high humidity and temperature or high speed change of these, off/on power switching etc. [28–34]. Depending on the field of application, the industry may require a wide range of various reliability tests from the light source manufacturers; to reduce the total testing time it is a common practice to accelerate the degradation mechanisms by increasing the test conditions. The so-called accelerated life tests are based on the Arrhenius model that is used to predict the aging progress under varying degrees of the environmental stress conditions. It is worth mentioning however, that besides the standard laboratory reliability tests widely used in the SSL industry, there are a couple of academic studies about possible new in-field, in-situ test methods aimed primarily for health monitoring, such as identification of LEDs' junction temperature form their emission spectra [35–37] or from changes of certain diode model parameters [38], or from certain dynamic operating characteristics such as the small-signal impedance, non-zero intercept frequency or the optical modulation bandwidth [39,40].

As our work is strongly related to our prior, SSL industry inspired projects, in terms of the considered test methods we aimed to stay as close to the already standardized methods as possible. These are recommendations, approved testing methods and standards, like the IES LM-80-08 and the JESD 22-A family of standards from JEDEC [41,42]. These documents contain requirements on the measurement devices, and specify the needed test conditions like the temperature and humidity [41], change of the stress conditions with time [42], as well as the needed accuracy level of the performed measurements and the set stress parameters [1] etc.

In LED-based lighting applications the main source of any lifetime approximation is provided by the IES LM-80-08 approved method and the IES TM-21-11 technical memorandum [43]. The work described in this paper is also based on these documents. Therefore, first we would like to provide a detailed insight to show the methodology and the roots of our concept.

#### *2.1. IES LM-80*

The IES LM-80-08 description does not provide detailed instructions on the proper sample size or the sample selection, it only states that the samples under test should adequately represent the overall population. It specifies the necessary case temperatures of 55 ◦C and 85 ◦C while the value of the third testing temperature is left to the choice of the manufacturer. The tolerance of the testing case temperatures is 2 ◦C during the burning time, and the temperature of the surrounding air in the chamber should remain within the ±5 ◦C range, which should be continuously monitored by a thermocouple measurement system. The relative humidity level is prescribed to be under 65%.

Duration of the life test should be documented at least with 0.5% accuracy and also the length of any possible power failure should be considered. The optical measurements have to be measured at least at every 1000 h, while the LEDs should be driven by the aging forward current and the ambient (or heatsink) temperature should be 25 ◦C ± 2 ◦C. The total length of the test should be at least 6000 h but it is preferred to reach the total time of 10,000 h. At each photometric measurement interval chromaticity shift should be measured, as well as any possible catastrophic failure of the samples should be monitored and recorded.

The LM-80 method also gives a recommendation on the format and content of the measurement report generated at the end of the life test. Nevertheless, it does not provide provisions to qualify the LED samples and does not state anything about their lifetimes, it provides a procedure only for the measurement of the total luminous flux maintenance.

In 2015 IES published the LM-80-15 approved method [44] which is the revision of LM-80-08. In the new version there are additional requirements towards the optical and colorimetric measurements, but the prescribed three case temperatures have been reduced to only two and even the minimum test duration of 6000 h has been abolished. Regarding our LED modelling concepts the extra colorimetric measurements are not that necessary at this stage while it is still advantageous to keep the 3 case temperatures, therefore, all of our aging tests were still based on the LM-80-08 document [1].

#### *2.2. IES TM-21-11*

The IES TM-21-11 technical memorandum provides a lifetime estimation method to the measurement results of the LM-80-08 testing. The November/December 2011 issue of LEDs Magazine provides a good overview [45] on the extrapolation method.

The extrapolation technique applied by the TM-21-11 method is based on exponential curve fittings to the measured optical data of the LM-80 test. Each case temperature is approximated individually in between which the Arrhenius-equation may be used for interpolations. The recommended sample size is established by 20 packaged LEDs (either on PCB or without it). 30 or more samples would not considerably improve the estimation capability, but there is an uncertainty of extrapolations based on test results of only 10 LEDs. Within this range the number of the samples also sets the limit to the time projection; below 10 tested LEDs the extrapolation method should not be applied, up to 19 tested pieces the document allows a 5.5 times while from 20 samples it admits a 6 times extrapolation of the total test duration. The amount of the measured data used for the exponential curve fitting depends on the total test duration: collected data of the last 5k hours is taken into account up to 10k hours of aging, above that data of the last half of the test is used.

The end of operating lifetime is then defined according to the LM-80 and the TM-21 results. If the pre-defined light output degradation cannot be reached within the extrapolation limit then the result is the maximum extrapolation time itself marked with a "less-than" sign (e.g., "L70 (10k) > 55,000 h" where the "10k" denotes that the LM-80 test lasted for 10,000 h and the 5.5 times rule is applied). If the life output degradation is reached using the TM-21 estimation then the result is reported with an "equals" sign. If the samples reach the minimum light output level during the LM-80 test then the result equals to the testing time in the general reporting formula (e.g., "L90 (5k) = 5100 h").

#### *2.3. The Degradation Model Used by TM-21-11*

The TM-21-11 technical memorandum applies an exponential curve fitting method to extrapolate the measured data in time and the Arrhenius-equation to interpolate between the three different case temperatures. The idea behind these techniques is well described in chemistry; in the following part we will use some of the basic concepts of reaction kinetics in order to give an analytical description of the LED degradation models. Our intention was to build-up and cover the fundamentals for the derivations in the later sections in the lack of such a textbook. The textbook-like manner not only introduces our efforts but also aims to provide a reliable baseline for researchers aiming to build on our work.

Reaction rate of a first-order reaction depends linearly on the reactant concentration [46]. The differential form of the rate law is:

$$\text{Rate} = -\frac{dc}{dt} = k \cdot c \tag{1}$$

where *c* is the changing reactant concentration, *t* is the elapsed time and *k* is the reaction rate coefficient. The separable differential equation can be solved by rearranging it and integrating both sides of the following equation:

$$\int\_{c\_0}^{c} \frac{1}{c} dc = -\int\_{t\_0}^{t} kdt \tag{2}$$

where *c*<sup>0</sup> is the initial concentration and *t*<sup>0</sup> is the initial time instant. The integration should be performed with the condition of *t*<sup>0</sup> = 0 s. After rearranging the achieved formula the rules of logarithm shall be applied. Finally, the integral form of the rate law is:

$$\frac{c}{c\_0} = \exp(-k \cdot t) \tag{3}$$

Considering the total luminous flux as the decreasing quantity of the homogenous aging process, where the initial value of the regression curve fit to the luminous flux is *c*<sup>0</sup> and *c* is the actual value at time *t*, then the TM-21-11 defined extrapolation of the total luminous flux maintenance curve is obtained:

$$\Phi(t) = \emptyset \cdot \exp(-\alpha \cdot t) \tag{4}$$

where the normalized light output is Φ at the time *t*, β is a fitting parameter and α corresponds to the reaction rate coefficient *k* specified at Equation (1).

Homogeneous chemical processes proceeding in the solid phase involves various aging phenomena in plastic and glass, thermal changes induced transformations, recrystallization, transformation of alloys and metals throughout and following a thermal treatment etc. In the solid state these progresses need a lot more time than in gas or in liquid phase. The reaction rate coefficient (i.e., the speed of these reactions) is an exponential function of the absolute temperature. The exact formula is described by the Arrhenius-Equation:

$$k = A \cdot \exp\left[\frac{-E\_a}{k\_B \cdot T}\right] \tag{5}$$

where *A* is a pre-exponential factor, *Ea* is the activation energy (the energy barrier below which the reaction in question does not proceed), *kB* is Boltzmann's constant and *T* is the absolute temperature in kelvins. Concerning the TM-21 interpolations, values of *A* and *Ea* can be calculated if datasets of two or more temperatures are available.

The Arrhenius-equation is typically used to express the *AF* acceleration factor of the reaction rate coefficient at elevated temperatures:

$$AF = \exp\left[\left(\frac{-E\_a}{k\_B \cdot T}\right) \cdot \left(\frac{1}{T\_2} - \frac{1}{T\_1}\right)\right] \tag{6}$$

where *T*<sup>2</sup> is the elevated temperature.

Upon taking into account the speeding-up effect of higher temperatures, the Arrhenius-equation has various expansions describing the effects of other stress conditions as well, like humidity or other non-thermal stresses. In case of LEDs, with respect to the LM-80 testing the most important non-thermal impact corresponds to the forward current [47]. The forward current dependent reaction rate coefficient can be described as:

$$k = A \cdot \exp\left[\frac{-E\_a}{k\_B \cdot T}\right] \cdot I^n \tag{7}$$

where *I* is the forward current and *n* is the so called life-stressor slope [47]. With the help of Equation (7) one can perform the necessary interpolations between the measurement results belonging to the different case temperatures and forward current values captured during an LM-80 life testing of LEDs.

#### *2.4. Further Possible Degradation Trends*

The exponential decay of the total luminous flux output of an LED corresponds to the first-order reaction rate. The order of a reaction defines the relationship between the reaction rate (or the decay rate, in this case) and the changing concentration of the reactant(s) (or the decreasing total luminous

flux, in this case). Practically, the order of a rate law is the sum of the exponents of the changing parameters. Accordingly, the reaction rates of the zero-, first- and second-order reactions are as follows:

$$\text{Rate} = -\frac{d\mathbf{c}}{dt} = k \cdot \mathbf{c}^0 = k \tag{8}$$

$$\text{Rate} = -\frac{dc}{dt} = k \cdot c^1 = k \cdot c \tag{9}$$

$$\text{Rate} = -\frac{dc}{dt} = k \cdot c^2 \tag{10}$$

For the sake of curiosity, a third-order reaction rate with two concentrations of species looks like:

$$\text{Rate} = -\frac{d\mathbf{c}}{dt} = k \cdot \mathbf{c}\_1 + k \cdot \mathbf{c}\_2^2 \tag{11}$$

where *c*<sup>1</sup> and *c*<sup>2</sup> are the concentrations of the two different species (note that the upper indices indicate the order of the reaction while the lower indices refer to the different species). To get the integral form of the above rate laws, the same mathematical steps should be followed as described in case of Equations (1)–(3).

The reaction in which one chemical species is irreversibly transformed into more than one other species is the so called parallel reaction (see also other more complex reactions in [48]). In this case the rates of the parallel reactions add up. Supposing a zero- and a first-order parallel reaction with the coefficients of *k*<sup>1</sup> and *k*<sup>2</sup> respectively, the decay rate can be written as:

$$\text{Rate} = -\frac{d\mathbf{c}}{dt} = k\_1 + k\_2 \cdot \mathbf{c} \tag{12}$$

The separable differential equation can be solved by rearranging it and integrating both sides of the following equation:

$$\int\_{c\_1}^{c\_2} \frac{1}{k\_1 + k\_2 \cdot c} dc = -\int\_{t\_1}^{t\_2} dt \tag{13}$$

$$\frac{\ln(|k\_1 + k\_2 \cdot c\_2|)}{k\_2} - \frac{\ln(|k\_1 + k\_2 \cdot c\_1|)}{k\_2} = -(t\_2 - t\_1) \tag{14}$$

Rearranging Equation (14) and applying the logarithm rules, we get:

$$\ln\left(\left|\frac{k\_1 + k\_2 \cdot c\_2}{k\_1 + k\_2 \cdot c\_1}\right|\right) = -k\_2 \cdot (t\_2 - t\_1) \tag{15}$$

where we know that all terms are positive. Raising both sides of the equation to the power equal to the base of natural logarithm and further rearranging it we get a final version as follows:

$$\mathbf{c}\_{2} = \left[\mathbf{c}\_{1} + \frac{k\_{1}}{k\_{2}}\right] \cdot \exp\left[-k\_{2} \cdot (t\_{2} - t\_{1})\right] - \frac{k\_{1}}{k\_{2}}\tag{16}$$

where *c*<sup>1</sup> and *c*<sup>2</sup> are the concentration values at the time instants *t*<sup>1</sup> and *t*2, respectively. In case of LEDs, a description with parallel reactions should be appropriate when the root causes of the light output degradation can be separated. That is, for example, if the packaged blue LED, the light conversion phosphor material and the lens are aged and tested separately. These aging modes independently reduce the light output of the LED (by the means of decreasing radiant and light conversion efficiencies and light transmission), therefore, theoretically a parallel reaction model with the individual reaction rate coefficients and modes could be matched with the aging results of the complete white LED.

At the EPA ENERGY STAR Lamp Round Table held in San Diego (CA, USA) in 2011, a wider set of decay rate models describing the total luminous flux maintenance of LEDs was proposed by Miller, of the U.S. National Institute of Standards and Technology (NIST, Gaithersburg, MD, USA)—see Table 1 [49]. Among the different decay models one can find zero-, first- and second-order reaction rates, models that are inversely proportional to the elapsed time, and the parallel mixture of the previously listed ones. Miller also drew attention to the fact that the light output degradation trend of LEDs may change significantly during the operation time; Figure 1a,b indicate two relative total luminous flux maintenance curves where the estimated L70 lifetime from the 10,000-h results is halved or doubles compared to the estimation from the 6000-hour results.

**Table 1.** Aging models of different decay rates with the closed form solution, i.e., the integral form (after [49]).


**Figure 1.** The change in aging trends: extrapolation of the 6k and 10k hours may differ dramatically (based on [49]) (**a**) L70(6k)=60,000 h vs. L70(10k)=30,000 h; (**b**) L70(6k)=30,000 h vs. L70(10k) > 60,000 h.

#### **3. The Pn-Junction Temperature During the LM-80 Test**

The pn-junction temperature of LEDs may change significantly during an LM-80 testing procedure due to the increased electrical power consumption, to the decreased energy conversion efficiencies and even due to the possible degradation of the thermal interfaces. The temperature increase can range from only a few degrees Celsius to as high as 20–30 ◦C. Its exact value depends mostly on the testing forward current, the overall thermal resistance, the zero-hour radiant efficiency and on the luminous flux maintenance value reached during the test. Figure 2 shows a theoretical approximation for a high- (a) and a mid-power (b) LED, as the function of the main causes of the increase, assuming at this point, that the 3 V forward voltage and the thermal resistance remain constant. During the calculations we neglected any effects of the temperature sensitive radiant efficiency; the extremely high temperature dependence in case of red and amber LEDs is well-known and acts as a positive loopback to the junction temperature, further increasing the discussed effect.

**Figure 2.** Theoretical pn-junction temperature increase during an LM-80 test in case of (**a**) a high-power and (**b**) a mid-power LED, as the function of the forward current, the thermal resistance, the zero-hour radiant efficiency and the reached luminous flux decay (assuming a constant thermal resistance and a Figure 3. V forward voltage). Note that the curves on (**a**) and (**b**) are identical implying that the junction temperature increase may affect high- and mid-power LEDs equally, depending on the thermal resistance.

**Figure 3.** Effects of the increase in the forward voltage and in the thermal resistance on the pn-junction temperature increase during an LM-80 test. The initial parameters are: 1 A forward current, 40% radiant efficiency, 15 K/W thermal resistance, 3 V forward voltage. The figures show the junction temperature increase during the test as the function of the end of test thermal resistance and forward voltage. The end of test relative light outputs are (**a**) 95% and (**b**) 80%. The black crosses indicate values corresponding to that of in Figure 2a.

It is obvious that a few degrees Celsius change in the junction temperature is not an issue. In this article we are dealing with the >10 ◦C increases caused by high testing currents, high initial efficiency, poor thermal conductivity, significant increase of the forward current and/or the thermal

resistance [50,51] etc. Figure 3a,b indicate the effects of the increase in the forward voltage and in the thermal resistance. During the calculations we considered a power LED with a test current of 1 A. The initial values of the radiant efficiency, the forward voltage and the thermal resistance were 40%, 3 V and 15 K/W in order and the relative junction temperature increase was calculated as the function of the end of test thermal resistance and forward voltage. The end of test relative light output was regarded to be 95% and 80%. Stated practically, Figure 3a,b are the extensions of Figure 2a; the indicated black crosses in the figures correspond to each other. As an example for the parameter increase, Figure 4 shows the aging related forward voltage shift of a Luxeon Z power LED; all the measurement points in the figure belong to the 80 ◦C junction temperature.

**Figure 4.** Shift of the forward voltage—forward current characteristics during the first 9000 h of a Luxeon Z LED sample (aged at 85 ◦C case temperature and 1 A forward current).

Most of the root causes of LED aging is closely connected to the pn-junction temperature. The temperature on which the die-, the interconnection-, the phosphor- and the lens-related aging processes undergo is practically much closer to the pn-junction temperature than the case- or the soldering point temperature. Therefore, we make an attempt to determine the parameter set of the Arrhenius-equation as the function of the changing junction-temperature. To do that first we need to determine the elapsed test-time dependent function of the junction temperature.

#### *3.1. Analytical Calculation of the Pn-Junction Temperature*

The *TJ* pn-junction temperature can be calculated from the *TA* ambient temperature, the optically corrected real *Rth* thermal resistance and the *Pdis* dissipated power:

$$T\_f = T\_A + R\_{th} \cdot P\_{dis} \tag{17}$$

The dissipated power is the difference of the consumed electrical power and the radiant flux. For simplicity we consider *Rth* to be constant over the time, therefore:

$$T\_I(t) = T\_A + R\_{th} \cdot \left( I\_F \cdot V\_F(t, T\_I, I\_F) - \Phi\_c(t, T\_I, I\_F) \right) \tag{18}$$

where *IF* and *VF* are the forward current and the forward voltage, while Φ*<sup>e</sup>* is the total radiant flux. The latter two parameters depend on the elapsed operation time *t*, and the actual junction-temperature and forward current. We assume an exponential decay model for the radiant flux function over time,

so we apply Equations (4) and (7) and we also take into account the *S*Φ*<sup>e</sup>* temperature sensitivity of the radiant flux. For simplicity we consider *S*Φ*<sup>e</sup>* to be constant over the time. Therefore:

$$\Phi\_{\varepsilon}(t, T\_{\parallel}, I\_{\parallel}) = \Phi\_{\varepsilon 0} \cdot \exp\left[-t \cdot A \Phi\_{\varepsilon} \cdot I\_{\parallel}^{n\Phi\_{\varepsilon}} \cdot \exp\left(\frac{-E\_{\rm d}^{\Phi\_{\varepsilon}}}{k\_{\rm B} \cdot T\_{\parallel}}\right)\right] \cdot \left[1 + S\Phi\_{\varepsilon} \cdot \left(T\_{\parallel} - T\_{ref}\right)\right] \tag{19}$$

where Φ*e*<sup>0</sup> and *Tre f* are the initial radiant flux and the reference operating junction temperature of the LED in the test environment at the zero-hour condition. Next, we assume a zero-order model for the increase of the forward voltage (i.e., a value linearly increasing with time) and we also consider the *SVF* temperature sensitivity of the forward voltage. For simplicity we consider *SVF* to be constant over the time:

$$\mathbf{V}\_{\rm F}(t, T\_{\rm f}, I\_{\rm F}) = \left[\mathbf{V}\_{\rm F0} + t \cdot A\_{\rm V\_{\rm F}} \cdot I\_{\rm F}^{\rm nV\_{\rm F}} \cdot \exp\left(\frac{-E\_{\rm a}^{\rm V\_{\rm F}}}{k\_{\rm B} \cdot T\_{\rm f}}\right)\right] \cdot \left[1 + S\_{\rm V\_{\rm F}} \cdot \left(T\_{\rm f} - T\_{\rm ncf}\right)\right] \tag{20}$$

After all this, the overall elapsed test-time dependent junction temperature can be written as:

$$T\_f(t) = -T\_A + R\_{th} \cdot I\_F \cdot \left[\mathbf{V}\_{F0} + t \cdot A\_{\mathbf{V}\_F} \cdot I\_F^{n\mathbf{V}\_F} \cdot \exp\left(\frac{-\mathbf{E}\_a^{\mathbf{V}\_F}}{k\_B \cdot T\_f}\right)\right] \cdot \left[1 + S\_{\mathbf{V}\_F} \cdot \left(T\_f - T\_{nf}\right)\right]$$

$$-R\_{th} \cdot \Phi\_{e0} \cdot \exp\left[-t \cdot A\_{\Phi\_e} \cdot I\_F^{n\Phi\_e} \cdot \exp\left(\frac{-\mathbf{E}\_a^{\Phi\_e}}{k\_B \cdot T\_f}\right)\right] \tag{21}$$

$$\cdot \left[1 + S\_{\Phi\_e} \cdot \left(T\_f - T\_{nf}\right)\right]$$

which is analytically not solvable by ordinary mathematical methods. At this point we gave up with the analytical attempt and tried a measurement based practical method.

#### *3.2. Determination of the Pn-Junction Temperature by Measurement*

The strong temperature dependencies of LEDs make it necessary to measure their optical, electrical and thermal parameters simultaneously. The JEDEC JESD 51-5x family of standards and the related CIE standards [52–61] provide a multi-domain characterization method, especially for LEDs, which includes the measurement of the pn-junction temperature by the help of thermal transient testing [62–64] and the calibrating process of the temperature sensitive parameter, i.e., the *SVF*. Of course, there are other measurement techniques (e.g., as described in [65,66]) beside the mentioned JEDEC and CIE standards/recommendations, we still work with these; over the past 15 years our research team has made significant contributions to the development of the related instruments. The gained experiences and the opportunity to customize the instrumentation also provide a higher flexibility during our investigations. In addition to these, the JEDEC JESD 51-5x family of standards is also conform with the new CIE 225:2017 recommendation and it is also used by many leading SSL companies—the reason why the Delpi4LED consortium has also chosen these standards as the basis of the LED modelling methodology also cited and described in this article.

We have performed the LM-80-08 based life testing of a mid-power LED sample set at the Department of Electron Devices of the Budapest University of Technology and Economics (see details in Section 5). Beside the optical measurements at room temperature specified by the testing method we also performed thermal transient testing both at 25 ◦C and at the testing case temperature. What we had after the measurements were the forward voltages, the radiant fluxes and the junction temperatures at 25 ◦C and at 55 ◦C ambient temperatures. Figure 5 shows the measured pn-junction temperatures at 300 mA forward current. Also, a linear approximation was applied on the measured results after 340 h of aging.

**Figure 5.** Pn-junction temperatures of the tested mid-power LED (#S11) over the testing time, measured at 300 mA forward current, (**a**) at 25 ◦C and (**b**) at 55 ◦C ambient temperatures.

The *S*Φ*<sup>e</sup>* temperature sensitivity of the radiant flux and the *SVF* temperature sensitivity of the forward voltage can be calculated from the measurement results belonging to different junction temperatures. By having these sensitivity values it is possible to make approximate calculations for the operating parameters at any arbitrary junction temperature value.

#### *3.3. The Transient Testing Based Calculation of the Arrhenius-Equation*

The method that we propose consists of four main steps:


The first step was introduced in the previous subsection (see the listed references).

3.3.2. Determine the Pre-Exponential Factor and the Activation Energy

The second step is based on the LM-80 results and on the recorded junction temperature values during the test. Let us recall the differential form of the rate law (Equation (1)) and insert the Arrhenius-equation in place of the reaction rate coefficient *k* (Equation (5)):

$$\text{Rate} = -\frac{dc}{dt} = c\_l \cdot A \cdot \exp\left[\frac{-E\_d}{k\_B \cdot T/(t)}\right] \tag{22}$$

where *ct* and *TJ*(*t*) are the changing concentration and the junction temperature at the time instance of *t*; *ct* corresponds to Φ*<sup>e</sup>* (or Φ*V*). Equation (22) clearly indicates the relationship between the parameters of the Arrhenius-equation and the reaction rate. Let us denote the reaction rate by *D* (representing the fact that it is the derivative of the *c* over *t* function):

$$D = \frac{dc}{dt} \tag{23}$$

At any two measurement times we can write the above equations with the actual parameters:

$$-D\_{t\_1} = c\_{t\_1} \cdot A \cdot \exp\left[\frac{-E\_d}{k\_B \cdot T\_f(t\_1)}\right] \tag{24}$$

$$-D\_{t\_2} = c\_{t\_2} \cdot A \cdot \exp\left[\frac{-E\_d}{k\_B \cdot T\_f(t\_2)}\right] \tag{25}$$

which is a set of equations with two variables (since *Dti* , *cti* and *TJ*(*ti*) are measured values). After rearranging it we get that:

$$A = -\frac{D\_{t\_1}}{c\_{t\_1}} \cdot \exp\left[\frac{E\_a}{k\_B \cdot T\_f(t\_1)}\right] \tag{26}$$

$$E\_d = \frac{k\_B \cdot T\_I(t\_1) \cdot T\_I(t\_2)}{T\_I(t\_2) - T\_I(t\_1)} \cdot \ln\left(\frac{D\_{t\_2} \cdot c\_{t\_1}}{c\_{t\_2} \cdot D\_{t\_1}}\right) \tag{27}$$

The value of *Dti* can be calculated as the derivative of the experimentally acquired luminous flux maintenance curve (see Equation (4)) at the time instance of *ti*:

$$D\_{t\_i} = \frac{d[\beta \cdot \exp(-\alpha \cdot t\_i)]}{dt} \tag{28}$$

$$D\_{t\_i} = -\boldsymbol{\alpha} \cdot \boldsymbol{\beta} \cdot \exp(-\boldsymbol{\alpha} \cdot t\_i) \tag{29}$$

Although, theoretically Equations (26), (27) and (29) should unambiguously assign the values of *A* and *Ea*, still several orders of magnitudes differences may occur among the obtained results. To dissolve this issue it could be a good practice to calculate *Ea* for each measurement time with an arbitrarily fixed *A* value, then sweep *A* until the smallest difference amongst the calculated *Ea* values is reached.

Even if the junction temperature is continuously increasing during aging, its effects on the slope (i.e., the derivative) of the luminous flux maintenance curve (from which *A* and *Ea* were calculated) are negligible in most cases compared to the aging related changes of the light output parameters. Although, the following example will make it clear, that the temperature sensitivity of the optical parameters can be extremely high for red and amber LEDs. The *S*Φ*<sup>e</sup>* temperature sensitivity of the radiant flux of a red power LED from a well-recognized vendor was measured to be −4.3 mW/ ◦C at 1 A forward current whereas the optical power was found to be 1.1 W with a radiant efficiency of 41%. Mounted on a cooling assembly with a 25 K/W junction-to-ambient thermal resistance and supposing a 0.1 V and 1 K/W increase in the forward voltage and thermal resistance respectively, roughly a 7 ◦C increase occurs in the pn-junction temperature until the time of the 10% light output degradation, causing another thermally induced 2.8% drop in the radiant flux—which is not negligible any more. In such cases Equation (23) should be corrected in the following format:

$$D\_{\rm corr} = \frac{dc}{dt} - S\_{\Phi\_{\rm f}} \cdot \frac{dT\_f}{dt} \tag{30}$$

Another possible compensation method of this effect is to determine the optical flux values that would be emitted at the reference junction temperature and use the gained data set as the new maintenance curve:

$$\left. \Phi\_{\varepsilon}(t) \right|\_{T\_{ref}} = \Phi\_{\varepsilon}(t) \cdot \left[ 1 + S\_{\Phi\_{\varepsilon}} \cdot \left( T\_{f} - T\_{ref} \right) \right] \tag{31}$$

We must note, that it could seem to be a good idea to compare the values achieved by the proposed method with the results of an LM-80 test sequence performed on multiple case temperatures. In fact, the technique discussed here only makes sense if the junction temperature increase during the test is significant, but this also means that the parameters calculated from different aging case temperatures would not be well correlated with the junction temperature, therefore the latter method gives a

completely different result in principle. From this reasoning we can tell that only testing at multiple case temperatures provides the needed data if the junction temperature rise is negligible, but otherwise consistent pn-junction temperature based test results can be reached only if the experiment is supported by accurate junction temperature measurements (e.g., thermal transient testing).

#### 3.3.3. Determine the Luminous Flux Maintenance Curve at a Fixed TJ

Our base concept is that LEDs have a kind of "lifetime budget", which (under nominal operating conditions) is consumed at a rate most dependent on the pn-junction temperature. We model the lifetime budget as a junction temperature, forward current and elapsed operating time dependent efficiency η*<sup>t</sup>* (eta *t*) which is to be multiplied by the zero-hour value of the radiant efficiency η*<sup>e</sup>* or the luminous efficacy η*<sup>v</sup>* to get the prevailing light output parameters. It contains the effects of any aging phenomenon, at this point even including the change of the electrical consumption through the change of the forward voltage.

According to our theory the current value of the budget is not dependent of current value of temperature (in such a way maintaining causality). Also, it does not carry any information on the temperature sensitivity of the parameters, therefore the temperature sensitivity of the optical parameters should be applied when the junction temperature is out of the reference value. If the junction temperature remains constant during the test then the lifetime budget is identical to the luminous flux maintenance curve (described by Equation (4)) normalized to 100%.

To calculate the change of the lifetime budget (or the normalized luminous flux maintenance at the reference *TJ*) we need to recall again the differential form of the rate law. Assuming that the exact time function of the temperature change is known, we need to substitute it into:

$$
\Delta \eta\_t = \int\_{c\_1}^{c\_2} \frac{1}{c} d\mathcal{c} = -\int\_{t\_1}^{t\_2} A \cdot \exp\left[\frac{-E\_d}{k\_B \cdot T\_f(t)}\right] dt \tag{32}
$$

The analytical solution of which is not trivial, even in case of a linearly changing temperature value. If the necessary mathematical tools are not available, then a practical solution could be to discretize the problem that way converting the integration to a sum calculation (a series of additions on very short time intervals). This means that the actual value of *k* can be calculated at every time instance knowing the mean value of the temperature, then the differential form of the rate law can be used with the difference that Δ*c* change is calculated during the short time interval Δ*t*. Adding up *c* and Δ*c* will result in the total value of the next time interval.

It should be emphasized that during this step we calculated the theoretical light output parameters of the LED at the reference junction temperature but accounting for the real temperature data, which is not always equal. For example, the operating junction temperature of an LED sample is 85 ◦C at the 55 ◦C case temperature at the beginning of the test. Let us assume that after 10,000 h the operating temperature is 105 ◦C at the same 55 ◦C case temperature. In this case the value substituted into Equation (32) is the real and continuously increasing value (105 ◦C which is 378.15 K after 10k h) but the radiant or luminous flux provided by Equation (32) is the value the LED would emit at the reference 85 ◦C junction temperature. First it could be confusing but we must not forget that we calculated the pre-exponential factor *A* and the activation energy *Ea* as the function of the prevailing junction temperature (therefore the degradation itself is calculated after the aging *TJ* profile). Still, these values describe only the aging effects and they do not carry any information about the temperature sensitivity of the optical parameters. If they would do so, then the method described for Equation (30) or (31) should be applied.

#### 3.3.4. Calculating the Light Output Parameters at any TJ

In the previous subsection we have determined the lifetime budget as the function of the elapsed lifetime and junction temperature profile in the meantime. The calculations result in the actual radiant flux value that would be emitted at the reference junction temperature, after the elapsed operating lifetime *t*. The next step is to apply the *S*Φ*<sup>e</sup>* temperature sensitivity of the radiant flux if the *TJ* value increases/changes significantly during aging:

$$\Phi\_{\varepsilon}(t, T\_{I}) = \Phi\_{\varepsilon 0} \cdot \eta\_{t}(t) \cdot \left[1 + S\_{\Phi\_{\varepsilon}} \cdot \left(T\_{I} - T\_{ref}\right)\right] \tag{33}$$

where Φ*e*<sup>0</sup> is the zero-hour radiant flux, η*<sup>t</sup>* (eta *t*) is the lifetime budget number at time *t*. The value of *S*Φ*<sup>e</sup>* can be determined at zero-hour and used during the whole lifetime as a constant or it can be re-determined at each control measurement. In practice according to the proposed method thermal transient testing should be performed both at case temperature and at the LM-80 prescribed 25 ◦C ambient temperature. From these measurements a linear approximation of all temperature sensitivity parameters can be determined.

#### *3.4. Case Study*

The procedure suggested in Section 3.3 was performed on the measurement results of the LED sample presented in Section 3.2. The steps taken are as follows:


$$A = -D\_{t\_1} \cdot t\_1 \cdot \exp\left[\frac{E\_d}{k\_B \cdot T\_f(t\_1)}\right] \tag{34}$$

$$E\_{\rm a} = \frac{k\_{\rm B} \cdot T\_{\rm f}(t\_1) \cdot T\_{\rm f}(t\_2)}{T\_{\rm f}(t\_2) - T\_{\rm f}(t\_1)} \cdot \ln\left(\frac{D\_{t\_2} \cdot t\_2}{D\_{t\_1} \cdot t\_1}\right) \tag{35}$$


To represent the appropriateness of the proposed technique the R-square values were determined with respect to the measured data. The R<sup>2</sup> values of the simulation are 0.992 and 0.988 for the 25 ◦C and 55 ◦C measurements while that of the logarithmic approximation are 0.993 and 0.983. These values show that the accuracy of the new aging model and the classical curve fitting method is practically the same.

Figure 6 also indicates that the simulated results and the fitted curves have different curvatures and their separation becomes quite significant after around 5000 h. Obviously, neither approximation of the measured values is more accurate than the other. The main cause of this misfit is probably the low statistical power of the measurement results of one single sample (and perhaps the changing aging rate). The purpose of this short case study was only to demonstrate the potential and feasibility of the theory.

**Figure 6.** Measurement and simulation results of the aged mid-power blue LED; the logarithmic regression fitting parameters to the 85 ◦C values are indicated just as an example.

#### **4. LM-80 Based Lifetime Modelling of Power LEDs**

Once the fitting parameters of the total luminous flux maintenance curves (and the pre-exponential factor(s) along with the activation energy (or energies)) are available, it becomes possible to estimate the in-situ light output parameters of an LED at any time (by Equations (7) and (32)), provided that the prevailing junction temperature and forward current values are always known.

In case of a streetlighting luminaire the forward current is either kept constant or it is controlled by a smart device. In fact, aging of the LED driver may cause aging related deviations in the set current but discussion of such issues is out of the scope of this article.

In-situ measurement of the pn-junction temperatures is not impossible, but it requires specialized laboratory equipment. Another solution is to reveal the operating temperature map of the luminaire can be achieved by system level simulations [67–70] that way shifting the junction temperature measurements to a predetermined point of the luminaire. System level simulations with multi-domain LED models make it possible to tell the operating parameters just by measuring the luminaire case temperature which is then on the field, mostly depends on the actual weather conditions.

#### *4.1. Continuous In-Situ Lifetime Modelling of LEDs*

Generating a time-dependent absolute temperature function from years of weather conditions is not realistic, but in the short period of time (e.g., within an hour) it can be assumed that the temperature changes linearly over the time. This approximation makes it possible to solve Equation (32) analytically, but despite of it, the computing capacity of the intelligent control unit of the luminaire may still be insufficient for the required calculations (or it is not acceptable for the unit to be kept busy by these calculations—also not counting with the power consumption of the CPU). If the value of reaction (or decay) rate coefficient is recalculated periodically at short intervals (e.g., every 5–10 min) and approximating the temperature to be constant in the meantime, then Equation (32) is greatly simplified:

$$c\_2 = c\_1 \cdot \exp\{k(T\_I) \cdot [t\_2 - t\_1] \}\tag{36}$$

where *c* is the actual light output value of the LED and the lower indices 1 and 2 denote the initial and the terminating time in-between the LED aging is calculated.

A case study was carried out to present the capabilities of the theory. For this purpose, the LM-80-08 based measurement results of Luxeon Z LEDs aged in our department were used (see the results in Figure 7a). The natural white high-power LEDs had a junction-to-ambient thermal resistance of 35 K/W. In Figure 7b the continuous red curve indicates the exponential fit to the measurement results (the same as the dashed orange curve in Figure 7a). Further eight theoretical aging curves were added in order to make the necessary calculations feasible, corresponding to *TJ* = 85 ◦C and 50 ◦C and to *IF* = 850 mA and 700 mA.

**Figure 7.** (**a**) Averaged LM-80 measurement results of 9 pieces of Luxeon Z samples and extrapolation until 50k hours (*TJ* = 120 ◦C, *IF* = 1 A); (**b**) Exponential curve fit to the LM-80 measurement set along with the eight assumed aging trends.

Figure 8 shows an illustrative example of the theory, applying Equation (36): the stress conditions are abruptly changed at 30k hours of aging from *TJ* = 120 ◦C and *IF* = 1 A to *TJ* = 50 ◦C, *IF* = 0.85 A.

#### *4.2. Lifetime Modelling of Iso-Flux Operation; a Case Study*

By keeping the total luminous flux of solid-state light sources constant, not only the visual comfort can be improved, but the reliability and energy efficiency of the luminaires could also be greatly increased. The advantage of a design method that compensates for the effects of temperature changes has been presented in previous articles [68–70]. The previously proposed methodology has not yet taken into account the aging of LEDs, which in the long time range may be even more significant than the effects of temperature changes.

The total lifespan of LEDs is typically referred to be in the range of 50–100k hours. This period can be easily converted to lifetime in years for light sources that operate without interruption (e.g., tunnel lighting or lamps at industrial facilities), but in case of streetlighting luminaires the conversion is not that straightforward as the on and off time depends on the length of days that alternate continuously throughout the year. In addition, switching on and off the lights does not necessarily depend on the exact occurrence of the twilights; the operating time measured in years should be considered based on astronomical data, taking into account the changing length of the nights. Table 2 illustrates the typical practical lifetime of a luminaire (assuming a rated LED lifetime of 50k hours) in Hungary.


**Table 2.** Elapsed time till 50k hours of operation of a streetlighting luminaire in years, in Hungary.

A case study was carried out on the real and theoretical LED aging results in order to demonstrate the range of differences between a constant current and a constant luminous flux operation. In order to count with realistic temperature variations the daily temperature values of the past decade of the Hungarian city of Szombathely (47.23512◦ N 16.62191◦ E) available at the Hungarian Meteorological Service, have been used, for which the constant current mode and the constant luminous flux operation was compared. The difference between the length of cold winter nights and warm summer nights was also considered by taking into account the annual change in the time differences between the civil twilight. At this stage we dealt with only one single LED, with the junction-to-ambient thermal resistance of 35 K/W.

The results of the case study are shown in Figures 9–11: the consumed electricity can be considerably decreased, particularly in the first year of the operation which means that a significant portion of the cost of a new luminaire installation is recovered within a relatively short time. Figure 10 clearly indicates, that using the smart controlling scheme the light output can be kept constant against the effects of temperature changes in the short run as well as against the LED aging in the time scale of decades (considering only the LED light sources–any other aging effects form a different issue). Due to the more favorable operating conditions (lower forward current and junction temperature values), the expected product lifetime could also be increase significantly.

**Figure 9.** The forward currents applied during the simulation (the absolute maximum DC forward current of the Luxeon Z LEDs is 1 A).

**Figure 11.** Simulated lifetime budget of a Luxeon Z LED.

In case of the smart control that realizes constant luminous flux output throughout the whole product lifetime, it is obvious that the normalized total luminous flux maintenance percentage can hardly be used to determine the end of the product lifetime. For that purpose we propose the lifetime budget efficiency η*<sup>t</sup>* (eta t) as the new end-of-life metric for adaptively controlled LED luminaires. The final results are summarized in Table 3.


**Table 3.** Comparison of operation with constant forward current (700 mA) and with constant light output (425 mW), simulated with the help of our LED aging theory.

#### **5. Lifetime Modelling Based on Multi-Domain Modelling of the LEDs**

The purpose of our multi-domain LED modelling concept is the simultaneous and combined simulation of the optical, electrical and thermal operation parameters. The proposed modelling technique is continuously revised and enhanced in order to achieve higher accuracy, better industrial applicability and also to add further capabilities to the model. Significant improvements have been achieved during the Delphi4LED H2020 European R&D project and the next step shall be modelling the whole lifetime of LED based light sources by the help of a Spice compatible multi-physics model.

Multi-physics LED models are generated by the isothermal LED characteristics which consist of the electrical and optical data as a function of the forward current, measured at a fixed junction temperature. Such measurements can be performed by the help of the T3Ster and TeraLED [71,72] measuring instrument setup; to speed up the rather time-consuming characterization process the vendor also provides an automated control software to the instrumentation. Then the multi-domain LED model can be generated by a global parameter fitting process; a detailed description on the model and its variants, benefits and various properties is provided in the recently published paper in this journal [7].

Combining the existing multi-domain LED model with the measurement results of an LM-80 based life test is a promising attempt. It means that a complete isothermal characterization of the LEDs should be performed as the testing time passes, which is the main drawback of this technique, compared to the common measurement methods during life testing. Its benefit is, however, that the elapsed operating time dependent model parameters can be revealed.

#### *5.1. Evaluation of Precious Life Testing Results*

A sample set of 30 HP LEDs has already been aged under an LM-80-08 based life test, during which the measurements were also extended by the abovementioned isothermal characterization technique. In the earlier study the samples were exposed to the case temperature of 85 ◦C while they were driven by the forward current of 1 A. The whole test lasted for 8k hours (Figure 12a shows the normalized total radiant and luminous maintenance curves). Detailed isothermal characterization was performed in the first 6k hours on 5 LED samples from which the model parameters were determined as the function of the elapsed time (Figure 12b shows an example) [73].

**Figure 12.** (**a**) Total luminous and radiant flux maintenance curves with their extrapolation [73]; (**b**) Time dependence of the model parameters: change of the saturation currents [73].

In case of all the characterized samples the obtained model parameters showed high similarity, and also the time dependent trends were consistent with each other. Still, a proper elapsed lifetime dependent model could not be generated from these. The difference in the ideality factors were found to be relatively small on a linear scale, but due to the exponential form of the Shockley diode equation even very tiny misfits can cause large errors in the output electrical and optical values. Besides this, there is an inflection point at around 1k–1.5k h of aging which prevents any modelling attempts with a simple time-function of the parameters; the use of complicated complex functions would not necessary describe the real physical aging processes but it would significantly increase the difficulty and the needed time of the global parameter fitting process. It is also obvious that various aging phenomena took place in the early "burn-in" part of the LEDs' lifetime and also their significance were changing with time; the collected data before 2.3k h of aging should be omitted during the model

generation process. The appropriate time dependent model, however, cannot be created from only 3 characterization points.

#### *5.2. Launch of a Targeted LM-80 Based Test Sequence*

Instead of fixing the total testing time in advance, it could be more advantageous to pre-define a targeted total luminous flux depreciation level in order to have a better overview on the time evolution of the time evolution of the modelling parameters. Based on this idea a new LM-80 like test was conducted on 18 mid-power LED samples from a well-recognized vendor. The LED type selection was made according to our previously performed tests and detailed measurements. Testing a set of high-power LEDs would also been interesting; preparation of such tests is already underway.

The samples were exposed to the case temperatures of the specified 55 ◦C and 85 ◦C and at the arbitrarily chosen 70 ◦C. At each case temperature, three different forward currents were applied: 220, 260 and 300 mA, the latter one as the absolute maximum allowed forward current of this LED type (the nominal current value is 150 mA). This means only two samples per each aging condition (case temperature/forward current), which is insufficient for TM-21-11 extrapolations but the purpose of the test was much rather to support our theoretical assumptions than to collect statistical data for industrial applications and needs. During the test we did not only measure the necessary parameters prescribed by the LM-80-08 standard but we also performed a complete isothermal forward current–forward voltage–radiant flux characterization of the samples in a 500 mm integrating sphere. These captured iso I–V–L curves were expected to provide enough input data to reveal the effects of the different aging tendencies.

The tested LEDs arrived on 0.8 mm FR4 stripes that had extended thermal pads both on the top and on the bottom sides (see in Figure 13). The samples were electrically connected in series in the chains of 6 LEDs. In order to make the JEDEC JESD 51 compliant individual measurements of the samples the stripes were chopped between the LEDs. Prior to the LM-80 aging test the samples had been pre-characterized during which the radiant flux was measured to be around 50% at the nominal forward current, however, the real thermal resistance was measured to be around 100–150 K/W; from the integral structure functions it was obvious that the most significant part of it belonged to the LED package itself. The exact causes of the unexpectedly high thermal resistance values were not examined in more depth, but according to our assumptions the pre-bake technological step may have been omitted before soldering, therefore the humidity accumulated in the LED package may have caused delamination of the internal mechanical layers during the reflow process.

**Figure 13.** The sample mid-power LEDs mounted on FR4 strips; the printed circuit boards were provided with increased thermal interfaces for better cooling capabilities.

#### *5.3. Results of the LM-80 Based Test*

The LM-80-based investigation of the mid-power LEDs was ended after the elapsed time of 1735 h because of the high failure rate of the LEDs; till the termination of the test 14 LEDs suffered catastrophic failure and two further pieces had contact failure. The rapid and early failure of the samples was unfortunate but not unexpected; in only a few cases the testing junction temperature of the LEDs was close to the allowed maximum value, but in most cases it was far above that. Despite the fact that the testing case temperatures were specified according to the LM-80-08 description, the lifetime testing still went in an accelerated manner (reaching the L70 level was one of our goals anyway).

Figure 14 compares a faulty and an unaged sample. From the figure it can be clearly seen that one of the root causes of the failure may be traced back to the extremely high temperature around the chip that inflicted carbonization of the encapsulant material, causing discoloration and bubbling. Due to the reduced light transmission of the lens higher amount of the blue light was absorbed that further increased the self-heating effect inside the LED package. The catastrophic failure most probably occurred as a result of a thermal runaway.

**Figure 14.** (**a**) A failed and (**b**) an unburnt sample of the investigated mid-power LEDs.

Figure 15 shows the attained total radiant flux maintenance curves; the case temperatures are indicated by different colors while the different forward current values are marked with different line types. Interestingly, the LEDs exposed to the lowest case temperature not only aged faster, but the trend over the time is also different from the others. Deeper investigations were not conducted to reveal the proper reasons and phenomena, but the main reason for the differences may be the higher RH formed at the lower temperature; this assumption is also supported by a previous study of a moisture resistance test on the same LED type, presented in paper [4].

**Figure 15.** The attained total radiant flux maintenance results of the mid-power blue LEDs, sorted by case temperature and forward current.

All in all unfortunately, the captured LM-80 compliant measurement data altogether did not make it possible to further analyze the temperature and forward current dependence of the aging rate. In Section 3.4 however, an attempt was made to achieve the Arrhenius equation parameters, but in this

exact case there is not enough measurement result of the other two case temperatures to support our theory; the resources required for such investigations far exceed the academic capabilities.

#### *5.4. The Elapsed Lifetime Dependent Multi-Domain LED Model*

Using the obtained measurement results an attempt was made in order to investigate the lifetime modelling possibilities and the extrapolation capabilities; for this purpose the multi-domain models of the still functional #S07 and #S11 samples were created at first. Various functions were tried out during a global parameter fitting process in order to set up the lifetime LED models with the best match to the measured characteristics. For this purpose, rudimentary parameter matching software was also developed, which ran on a mid-range 4-core processor for about 1 week (which was about one-tenth of the total software development time). The obtained Shockley model parameters and the value of the series resistance have significant elapsed time dependence. An improved version of the fitting application was re-run iteratively three times in order to achieve the first attempt of our elapsed lifetime dependent multi-physics LED model; Figure 16 indicates examples of the attained model parameters.

**Figure 16.** Electrical (**a**–**c**) and optical (**d**–**f**) model parameters of #S07.

After defining the new lifetime LED model a simulation test bench was created by which the LM-80 based test data could be compared with the modelled values (see the results in Figure 17a). The average absolute inaccuracy of the simulations is 0.5% while the maximum deviation is 1.2%. Figure 17a also shows that due to the quadratic and logarithmic time functions of the fitting parameters of the Shockley diode equation (see in Figure 16a,b) the model fits very well even in the early burn-in period—at least at the time of the measurements. Otherwise, further simulations with much higher time resolution has shown that the model becomes totally inconsistent between 1 and 100 h of aging (see in Figure 17b). These results strongly highlight the risk of using high-degree parameter matching algorithms. The extrapolation applicability of the new model still had to be tested therefore samples #S07 and #S11 were reinstated to the test chamber. In the meantime, the time functions of the fitting parameters were revised in order to eliminate the anomaly of the early aging time.

**Figure 17.** (**a**) Comparison of the simulated and measured total radiant flux maintenance curves of #S07; (**b**) The simulation results of #S07 with higher time resolution—the discontinuity can be clearly seen.

#### *5.5. The Enhanced Time Functions*

While the test continued on the two samples, we reconsidered the time functions that were previously found to provide the best match. After an extensive "trial and error" type investigation of the possibilities it was decided to apply only such functions that push the results only in the same direction while the rest of the parameters were kept constant. The biggest error in each case occurred in the initial burn-in stage, so we decided not to deal with it in the first round. That way the multi-domain model simplified remarkably: the shift of the forward voltage can be modelled by a linearly increasing series resistance, while the saturation current and the ideality factor of the electrical model are constant values. That way the electrical and the optical degradation of the LED can be modelled completely separately: the time function of the light output decay can be applied for the saturation current of the optical branch while its ideality factor and series resistance remain constant.

Considering the initial 100 h of aging, an additive exponential decay with a very short time constant describes well the forward voltage of the LED. The electrical series resistance therefore is formed this way:

$$R\_{\text{ser\\_cl}}(t) = R\_0 + a \cdot t + b \cdot \left[1 - \exp\left(-t \cdot \tau\_{\text{Rser}}\right)\right] \tag{37}$$

where *t* is the elapsed operation time *R*<sup>0</sup> is the zero-hour electrical series resistance, *a* and *b* are fitting parameters and τ*Rser* is the time constant of the initial exponential deviation.

Regarding the radiant flux, so far no correction function was found to describe the burn-in time. The saturation current of the optical branch is therefore:

$$I\_{0\\_nd}(t) = l\_0 + d \cdot \ln(t) \tag{38}$$

where *I*<sup>0</sup> is the zero hour saturation current of the optical branch and *d* is a fitting parameter. According to this function the model is not applicable if *t* < 1 h and also gives inaccurate results if *t* < 100 h.

#### *5.6. Extrapolation Capabilities of the Model*

At 4340 h of the total aging time the two samples #S07 and #S11 were again unloaded from the aging chamber and were re-measured in our integrating sphere. The obtained measurement results (the radiant flux and the forward voltage values) are shown in Figure 18a–d along with the lifetime extrapolation simulation curves. The simulations were performed by the LED models based on the measurement results obtained up to 1000 h.

**Figure 18.** Simulated and measured total radiant flux maintenance of #S07 (**a**) and sample #S11 (**c**); Measured and simulated time function of the forward voltage of sample #S07 (**b**) and sample #S11 (**d**).

For sample #S11, the estimations acquired from the simulations are quite accurate, while the model of sample #S07 overestimates the extent of changes (see the mismatch values in Table 4). A possible explanation for the error could be the fact that the samples had to be removed from the aging environment for each measurement. Re-fixing the samples with different strengths may affect the value of thermal resistance. This possible reason is consistent with the previous assumptions that the high thermal resistance may have been caused by hygro-mechanical stresses that induced delamination of the mechanical layers in the LED packages.

It has to be noted that at this point the model is only valid for the 300 mA aging circumstances. Regarding the temperature issues, according to our theory the Arrhenius-equation parameters determined in Section 3.4 can be directly inserted into the model of #S11 sample:

$$I\_{0\\_rad}(t) = I\_{0\\_rad} - \int\_{t\_0}^{t} \frac{1}{t} \cdot A \cdot \exp\left[\frac{-E\_a}{k\_B \cdot T\_f(t)}\right] dt\tag{39}$$

by which the saturation current of the optical branch can be calculated at any time *t*. Dealing with discretized time steps of very short intervals and considering the temperature to be constant in the meanwhile we can calculate the *I*0\_*rad* value of the next period:

$$I\_{0\\_rad}(t\_2) = I\_{0\\_rad}(t\_1) - A \cdot \exp\left[\frac{-E\_a}{k\_B \cdot T\_f}\right] \cdot \ln\left(\frac{t\_2}{t\_1}\right) \tag{40}$$

With the same considerations it is possible to sensitize the time-dependent electrical series resistance value to the junction temperature. In case of this LED type it should be based on the zero-order model of the differential rate law.


**Table 4.** The error made by the model compared to the measurement results at 4340 h.

#### *5.7. The Required Measurement and Testing Time*

In case of the LM-80-08 based life testing of the blue mid-power LED set the isothermal characterization process was optimized to the needed time: the measured set of operating points was reduced to the minimally sufficient range while the small size of the LED packages and the relatively short thermal time constants (around 30–60 s) were also advantageous to significantly speed up the measurements. In spite of these facts, the required measurement time per each sample was around 180 min at each control event. Such characterization was performed 87 times during the test which altogether amounts a total measurement time of 260 h; an additional 15% of the original 1735 h of LED aging.

The minimal LM-80-08 and TM-21-11 compliant sample set consists of 90 LEDs as the testing process prescribes three case temperatures and at least three testing currents are necessary for proper interpolations in between while the extrapolation technique may be applied in the presence of at least 10 samples per aging conditions. Supposing a typical high-power LED the full characterization time may take even up to 6–8 h which means a total measurement time of 500–700 h at every control event that (according to LM-80-08) must be performed at every 1000 h. Although the measurements could be fully parallelized, it is still not realistic for academy in terms of the price of the currently available instruments required for the isothermal LED characterization. Therefore, supporting the theory described in this paper by appropriate statistical background remains an opportunity much rather for the major industry partners. Also, reducing the necessary measurement time of the current LED characterization system could also help to put this method into common practice.

#### **6. Conclusions**

In this work the LM-80-08 and TM-21-11 documents were briefly introduced after which an extensive description of the applied decay models were provided. In addition to the models used in the accepted methods, we also presented other aging models and the mathematical basis of their application.

Applying the theoretical basics of the light output degradation of LEDs we have introduced a novel method to determine the pre-exponential factor and the activation energy of the Arrhenius-equation only by measuring a sample set of only one aging case temperature instead of the prescribed three. We have also pointed out, that consistent pn-junction temperature based test results can be reached only if the experiment is supported by accurate junction temperature measurements in cases where the junction temperature rise during the life testing is considerable. Our theory was also supported by the

evaluation of real measurement results. The case study showed that the LM-80 based measurement results of a power LED sample set and our LED aging theory of the "lifetime budget" were applied in a case study based on archive meteorological data. The case study showed that the iso-flux (or constant light output) operation mode has very significant benefits in terms of both electrical consumption and lifetime expectancy.

In order to create the elapsed lifetime dependent multi-domain LED model an LM-80-08-based test was performed on 18 blue mid-power LEDs of a well-recognized vendor. During the test even the isothermal characteristics of the samples were captured. Extremely high operating junction temperature of the samples even at the prescribed case temperatures caused fast and early failure of the LEDs. The speed and the trends of the LED aging also showed significant anomalies: the highest aging rate belonged to the samples of the lowest test temperature and even their total luminous flux maintenance curves follow a logarithmic trend instead of the expected exponential one. In the absence of the sufficient aging data ant due to the experienced anomalies it was not possible to determine the forward current dependent LED aging model but a theory was set up to specify the Arrhenius equation's parameters from only one testing temperature, by the help of thermal transient testing. In this specific case the needed extra measurements of the proposed method added a 15% surplus to the life testing duration which is estimated to be one half or one third of the extra time ordinarily required.

As the first approach of the Spice-compatible LED lifetime multi-physics modelling a mid-power LED was modelled from its captured aging results up to the L(78) level. The created LED model matches the measured values with a misfit less than 1.2%.

Simulations with higher time resolution had shown that the achieved model became inconsistent in the very early burn in period, therefore a new aging model was set up. In the meantime two LED samples were reinstated to the test in order to show extrapolation abilities of the lifetime multi-domain LED model. The model in case of sample #S11 performed over expectations, although, in case of #S07 the extrapolations proved to be fairly inaccurate. The cause of the modelling mismatch may be the fact that at this development and research state the LEDs have to be displaced from the aging chamber to perform the necessary measurement in an integrating sphere—it is still an issue that should be solved by a new, appropriate combination of the characterization and life testing methods.

At the academic level the currently running national R&D project allows resources only to such scale of studies. The theory described in this paper should be supported by testing and measurement results of much higher number of LED samples in order to represent appropriately the whole LED population and also the general aging physics of LEDs. Increasing the throughput of the presently applied LED characterization methods would also be needed. We are currently making efforts to develop new procedures to reduce measurement time and also to set up an international joint project consortium to enhance the statistical background of our theory.

**Author Contributions:** Conceptualization, A.P., G.H. and J.H.; Data curation, G.H. and J.H.; Formal analysis, J.H., Funding acquisition, A.P.; Investigation, G.H. and J.H.; Methodology, G.H. and J.H.; Project administration, A.P.; Software, G.H. and J.H.; Supervision, A.P.; Writing—original draft, J.H.; Writing—review and editing, A.P., G.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the K 128315 grant of the National Research, Development and Innovation Fund and was also supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Artificial Intelligence research area (BME FIKP-MI/SC) and the Nanotechnology research area (BME FIKP-NANO) of the Budapest University of Technology and Economics. The support of the "TKP2020, National Challenges Program" of the National Research Development and Innovation Office (BME NC TKP2020) is also acknowledged.

**Acknowledgments:** Support from HungaroLux Light Ltd. (especially from A. Szalai and T. Szabó) is gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Influence of a Thermal Pad on Selected Parameters of Power LEDs** †

#### **Krzysztof Górecki 1,\*, Przemysław Ptak 1, Tomasz Torzewicz <sup>2</sup> and Marcin Janicki <sup>2</sup>**


Received: 24 June 2020; Accepted: 16 July 2020; Published: 20 July 2020

**Abstract:** This paper is devoted to the analysis of the influence of thermal pads on electric, optical, and thermal parameters of power LEDs. Measurements of parameters, such as thermal resistance, optical efficiency, and optical power, were performed for selected types of power LEDs operating with a thermal pad and without it at different values of the diode forward current and temperature of the cold plate. First, the measurement set-up used in the paper is described in detail. Then, the measurement results obtained for both considered manners of power LED assembly are compared. Some characteristics that illustrate the influence of forward current and temperature of the cold plate on electric, thermal, and optical properties of the tested devices are presented and discussed. It is shown that the use of the thermal pad makes it possible to achieve more advantageous values of operating parameters of the considered semiconductor devices at lower values of their junction temperature, which guarantees an increase in their lifetime.

**Keywords:** power LEDs; thermal pads; thermal resistance; measurements; optical efficiency; self-heating; electronics cooling

#### **1. Introduction**

Power LEDs are today the most important components of solid-state lighting sources [1–3]. Temperature strongly influences properties of all semiconductor devices, including power LEDs, [1,2,4–6]. For a single semiconductor device, the value of its junction temperature depends on both the ambient temperature *Ta* and the excess Δ*T* of the device internal junction temperature, which is caused by the self-heating phenomenon [7–13]. Thus, the device temperature rise depends on power dissipated in a considered semiconductor device and on the efficiency of heat removal characterized by thermal parameters. In typically used compact thermal models of semiconductor devices, at the steady state, thermal resistance can be used for this purpose [14,15].

The thermal resistance between the p–n junction of a power LED and the ambient is used to describe the total influence of all components included in the heat flow path, e.g., the package of the device, the printed circuit board (PCB), and the heatsink, on the heat transfer efficiency [4,16]. Thus, the manner of assembly of a power LED can influence its thermal properties and, consequently, electric and optical properties of this device [17,18].

In our previous papers, [18–20], we investigated the dependences of the electric, optical, and thermal parameters of power LEDs on cooling conditions. In particular, these papers show that the influence of cooling conditions on mentioned parameters could be noticeable, e.g., self-heating could cause a high decrease in luminance of light emitted by these diodes.

Additionally, in references [21,22], it is shown that an increase in the semiconductor device internal temperature causes a visible decrease of its lifetime. Besides, the parameters of mounting process [17,23] and the area of soldering pads [14] also influence the device thermal. Manufacturers of power LEDs are continuously improving the quality of packages for these devices, which are characterized by lower and lower values of junction-to-case thermal resistance *Rthj-c*. The value of this parameter depends, e.g., on physical and chemical processes used during the packaging of these devices [23]. In the assembly of power LEDs, a soldering process is typically used. As previously shown [17,24,25], the composition of the soldering alloy, the type of reflow oven, and the soldering temperature profile in time can influence thermal resistance of a soldering joint between the case of a power LED and the PCB.

Górecki et al in the paper [7] describe a problem of multipath heat transfer between the device package and the ambient, whereas Górecki and Zar˛ebski in the paper [14] the influence of selected factors characterizing, e.g., the assembly process of the tested devices on its thermal resistance are analyzed. In order to improve the efficiency of removal of heat generated in power LEDs, a special terminal of these devices is used that makes it possible to conduct only heat between the junction of this device and the PCB. Of course, in order to use such a terminal a special pad (thermal pad) must be situated on the PCB. Górecki and Ptak in the paper [26] we showed the measurements results illustrating the influence of the area of a thermal pad on thermal resistances of selected power LEDs. In these investigations, we used a custom PCB designed by the authors. Górecki et al in the paper [18], some results of measurements thermal parameters of power LEDs mounted on the MCPCB with soldered and not soldered thermal pad are presented. These measurements were performed for the tested devices operating with free convection cooling.

This paper, which is an extended version of our paper [27], illustrates the influence of the use of a thermal pad on thermal, electric and optical properties of selected power LEDs assembled in different types of packages and situated on typical metal core PCBs (MCPCBs) offered by manufacturers of the tested LEDs. The measurement results presented in this paper were performed for power LEDs situated on the cold plate. The temperature of this cold plate was regulated in a wide range of its value. This paper consists of the following parts. First, the employed measurement method is introduced in detail. Next, the tested devices are described. Finally, the obtained results of investigations are presented and discussed.

#### **2. Measurement Method**

This Section describes the measurement method and the set-up used to obtain characteristics of the tested diodes illustrating an influence of the operating conditions on electric, optical and thermal properties of these devices. All the considered parameters are measured simultaneously with the use of the set-up described below. A view of the considered set-up is shown in Figure 1a. This set-up consists of a transient thermal tester, a water-cooling system with a cold plate, a light-tight chamber, a luxmeter, and a radiometer. The interior of the light-tight chamber with the cold plate are shown in Figure 1b [28]. The cold plate with a tested power LED is placed inside the light-tight chamber. The thermocouple is used to measure the temperature of the cold plate. The block diagram showing the main components of the measurement system is presented in Figure 2.

Tested devices are heated by the transient thermal tester, which is also used to record their dynamic temperature responses, i.e., the variations of the voltage drop across LED junctions in time. When isothermal characteristics are measured, the tested devices, soldered to an MCPCB, are placed on a cold plate. The temperature of the cold plate is stabilized at a preset temperature value by a thermostat forcing liquid flow through the plate. Taking into account that, in the case of power devices, the temperature regulation time might be unacceptably long, the control system response might be accelerated, as demonstrated in [29], owing to the use of Peltier thermoelectric modules

inserted between the plate and the PCB. However, if non-isothermal measurements are to be taken in the natural convection cooling conditions, the MCPCB is placed horizontally in thermally insulating clamps. For optical measurements, devices on the cold plate or in the clamp are placed inside the light-tight box and the flux density of emitted light is measured by a radiometer. The operation of the entire measurement system is controlled by a PC where all measurement data are stored.

**Figure 1.** (**a**) View of the measurement equipment; (**b**) the interior of the light-tight chamber with the cold plate and a power LED [29].

**Figure 2.** Block diagram of the measurement set-up.

Thermal resistance was measured with the commercial T3Ster ® equipment manufactured by the MicReD division of Mentor Graphics (Budapest, Hungary) [30]. This equipment is now the industrial standard for device thermal characterization [15] and it realizes the classical pulse measurement method described e.g. in the papers [9]. In this method the tested device is excited by rectangular current pulses. The frequency of this signal is very low, and the duty cycle is close to 1. The user can select the values of low measurement current *IL* and high heating current *IH*. The voltage drop values of tested diodes are measured at the low (*VL*) and high (*VH*) forward current values.

The device under test was placed on the cold plate, whose temperature value was stabilized by a thermostat. The system for water forced cooling of electronic devices is described in the paper [31]. In order to assure higher thermal conductance, some silicon thermal paste was applied between the cold plate surface and the metal core printed circuit board (MCPCB) on which the tested diodes were mounted. The cold plate temperature *Ta* was regulated during the experiment over the range from 10 ◦C to 90 ◦C.

The measurement equipment is dedicated to thermal parameters of typical semiconductor devices, e.g., p–n diodes, but it does not include instruments making it possible to measure optical parameters of power LEDs. Therefore, it is possible to measure with this system only electric thermal resistance

defined in the JEDEC standard [15]. When only electric thermal resistance is measured, the influence of optical power on the results of measurements is neglected.

In order to measure thermal resistance of the considered type of power LEDs, the measurement equipment shown in Figure 1 was used. The thermal resistance *Rth* is measured using the following formula:

$$R\_{th} = \frac{V\_{LE} - V\_{LB}}{a\_T \cdot \left(V\_H \cdot I\_H - P\_{\rm opt}\right)} \tag{1}$$

where α*<sup>T</sup>* denotes the slope of the thermometric characteristic *VL*(*T*) describing the dependence of the diode forward voltage *VL* on temperature at the forward current equal to *IM*; *VLB* and *VLE* denote the values of the diode forward voltage measured at the current IM when measurements, respectively, start and end; *IH* and *VH* denote forward current and forward voltage of the tested diodes during heating process at the steady state, whereas *Popt* is the optical power emitted by tested device.

The measurements of the surface optical power density were performed using the HD2302 radiometer [32] manufactured by Delta Ohm (Caselle di Selvazzano, Italy). The probe of the radiometer was situated at the distance of 17 cm directly above the light source. The optical power emitted by the investigated diodes was measured using the method presented in [18,33]. This method is based on the measurements of surface density of power of the emitted light by means of the radiometer, the data provided by the diode manufacturer, and the application of some classic geometrical dependencies.

The junction temperature values of the tested diodes at the steady state were measured using the standard pulse electric method [34]. Measurements were performed for different values of LED forward current over a wide range of values of cold plate temperature. The values of diode current *ID* and voltage *VD* as well as their junction temperature *Tj* and the surface power density of the emitted radiation Φ*e* were registered simultaneously.

The electric current-voltage characteristics of the tested LEDs measured at the thermal steady state. The measurements were carried out by measuring thermal resistance. The coordinates of points lying on these characteristics are (*VH, IH*).

#### **3. Tested Devices**

The electric, thermal and optical parameters were measured for two types of power LEDs manufactured by the Cree, Inc. (Durham (North Carolina), USA); XPLAWT-00-0000-000BV50E3 (further on called XPLAWT) and MCE4WT-A2-0000-JE5 (further on called MCE). The values of selected parameters characterising the properties of tested devices are provided in Table 1.


**Table 1.** The values of the operating parameters of the tested LEDs.

For the diode XPLAWT, the nominal dissipated power of these LEDs amounts to 10 W, the maximum forward current is equal to 3 A, and the luminous flux at the current of 1050 mA is 460 lm [35]. In turn, for the MCE diode the nominal dissipated power amounts to 2.8 W, the maximum forward current is equal to 0.7 A, and the luminous flux at the current of 350 mA is 100 lm [36]. The typical thermal resistance between the junction and the soldering point *Rthj-s* provided in the datasheets is equal for the considered LEDs to 2.2 K/W and 3 K/W, respectively. The MCE diode contains four independently operating structures, but in our investigations only one of them is powered.

The measurements were taken for the diodes XPLAWT situated on the MCPCBs presented in Figure 3a. These boards have dimensions of 25 mm x 25 mm and the thickness of 2 mm. The package of one diode was soldered to the thermal pad, also shown in the figure, whereas the other one did not use the thermal pad.

**Figure 3.** (**a**) View of the metal core printed circuit board (MCPCB) with the XPLAWT LED and the view of this LED electrode layout for this LED; (**b**) view of the MCE LED soldered to the MCPCB and the layout of this MCPCB.

On the other hand, the MCE diode is a compound device containing four LED structures, which can be powered independently. In the current investigations only one structure was used. The view of the MCE diode and the layout of the PCB used to assemble this diode are shown in Figure 3b. The dimensions of the MCPCB are 35 mm x 35 mm, and its thickness is equal to 2 mm. The thickness of the dielectric layer is 60 μm. Comparing the MCPCBs used for the assembly of both tested power LEDs, one can observe that their surface areas differ by almost two times. Besides, the shape and the dimensions of thermal pads are also visibly different.

#### **4. Results**

Using the measurement method described in Section 2, selected characteristics illustrating electric, optical, and thermal properties of the tested diodes were obtained for these devices operating with soldered thermal pads and non-soldered thermal pads. The electric properties of the considered devices are described by the non-isothermal current-voltage characteristics. The thermal properties are illustrated by the dependencies of thermal resistance on the diode forward current. Finally, the optical properties of these power LEDs are characterized by dependencies of optical power and luminance on the forward current. Moreover, radiant efficiency of the tested power LEDs was calculated. Selected results of these investigations are shown in Figures 4–13. In these figures, the solid lines denote the measurement results obtained for the diodes with the thermal pad soldered, and the dashed ones - for the diodes operating without the thermal pad soldered.

Figure 4 presents the non-isothermal current voltage characteristics of the XPLAWT diodes obtained at selected values of temperature of the cold plate *Ta*, which is equal to temperature of the MCPCB. As it is visible in the figure, an increase in temperature *Ta* shifts the characteristics to the left. This effect is observed as a result of self-heating, which is more visible for the diode operating without the thermal pad. In this case the increase in the diode internal temperature over the cold plate temperature is higher. For both manners of assembly, the differences in the junction temperature and the forward voltage drop increase with current. On the other hand, an increase in junction temperature at a constant value of forward voltage causes approximately exponential increase of the forward current.

**Figure 4.** Measured DC I-V characteristics of the XPLAWT diodes at selected values of cold plate temperature.

In Figure 4 it is shown that differences in the forward voltage of the diode XPLAWT due to the variation of the device junction temperature *Tj* between diodes operating with a thermal pad and without it are equal even to 150 mV at the current of 2 A. In turn, the differences in values of junction temperature at the same value of current exceed even 60 ◦C. The maximum value of junction temperature of the diode operating without a thermal pad is equal to even 145 ◦C at *Ta* = 90 ◦C and *ID* = 2 A. The temperature coefficient of forward voltage changes with the value of forward current and temperature *Ta*. At the forward current equal to 2 A and temperature *Ta* = 10 ◦C, this coefficient is equal to −2.5 mV/K.

Figure 5 presents for different forward current values the measured diode voltage change Δ*VD* in response to the variation of the cold plate temperature from 90 ◦C to 10 ◦C (Δ*Ta* = −80 ◦C) in the case of MCE diodes operating with the thermal pad and without it. The value of Δ*VD* was obtained as the difference of values of forward voltage of the tested LEDs measured at the same value of their forward current and at both above mentioned values of cold pate temperature. As observed, the value of Δ*VD* is an increasing function of current *ID*. Due to the self-heating phenomenon, the value of the considered voltage change is higher for the diode operating without the thermal pad. The observed differences between values of Δ*VD* obtained for both considered mounting methods attain even 60 mV at the current equal to 0.7 A.

**Figure 5.** Measured MCE diode forward voltage change Δ*VD* in response to the cold plate variation by 80 ◦C in function of the diode forward current.

Figure 6 illustrates the dependence of the forward voltage of the MCE diode on temperature *Ta* at two different forward current values. As observed, an increase in temperature of the cold plate causes a decrease in the diode forward voltage. The observed dependence *VD(Ta)* is non-linear. The values of *VD* voltage obtained for the diode operating with the thermal pad are lower than for the diode operating without the pad, but the differences between these values decrease with the increase of temperature *Ta* and with the decrease in forward current *ID*.

**Figure 6.** Measured dependences of the forward voltage of MCE diodes on the cold plate temperature for different forward current values.

Figure 7 illustrates for different cold plate temperature values the influence of the forward current on the junction temperature of the MCE diodes operating in both types of mounting manner considered here. Obviously, an increase in forward current and the cold plate temperature causes an increase in the junction temperature. It is worth observing that the diode operating with the thermal path has even a 25 ◦C lower value of junction temperature *Tj* than the diode of the same type operating without the thermal pad. The influence of the thermal pad on the junction temperature is the most visible for the lowest value of temperature *Ta*. It can be also observed that an increase in the value of forward current causes a decrease of the difference between values of temperature *Tj* obtained at different values of temperature *Ta*. It is a result of a decreasing thermal resistance value of the tested diodes with an increase in the cold plate temperature.

**Figure 7.** Measured dependences of the junction temperature of the MCE diodes on forward current for different values of the cold plate temperature.

The influence of device mounting manner of the considered diodes on their thermal resistance is illustrated in Figures 8 and 9. In Figure 8, the measured dependences of thermal resistance of the XPLAWT (Figure 8a) and MCE (Figure 8b) diodes on their forward current are shown. The measurements were performed at different values of cold plate temperature. It can be easily

noticed that owing to the use of the thermal pad the diode thermal resistance of XPLAWT diode is effectively reduced. Differences in the value of this parameter exceed even 40% for the low value of forward current (100 mA). At higher values of current these differences decrease. For the current equal to 2 A, the value of thermal resistance decreases by less than 30%. For the LED operating without the thermal pad, this decrease is smaller, and it does not exceed 25%.

**Figure 8.** Measured dependences of thermal resistance on forward current for different cold plate temperature values for: (**a**) the XPLAWT; (**b**) MCE diode.

In practice, the results shown in Figure 8a mean that for the forward current equal to 2 A an excess of the device internal temperature above the ambient temperature is equal to about 90 ◦C for the LED with the thermal pad and above 130 ◦C for the LED without the thermal pad, thus demonstrating its importance for the thermal performance of the device. It is also worth noticing that there are no visible differences between the results of measurements obtained for different cold plate temperature values. This proves that in the considered case the value of thermal resistance results mainly from efficiency of heat conduction between the diode junction and the cold plate. The influence of convection and radiation on the thermal resistance value *Rth* is negligibly small. The rate of heat removal through the cold plate is very high, and it does not depend on the velocity of cooling liquid.

It is visible in Figure 8b that for the MCE diode a strong influence of the cold plate temperature on thermal resistance is observed. In the considered range of temperature variations, the thermal resistance can change by over 30%, and for higher *Ta* values, thermal resistance *Rth* is reduced. For the considered diode operating with the thermal pad, an increasing function describes the dependence *Rth(ID)* for low values of current *ID*. The influence of this current *ID* is very weak for the diode operating with the thermal pad, whereas it is very strong for the diode operating without the thermal pad.

Figure 9 presents the dependence of thermal resistance of the MCE diode on the the cold plate temperature. Looking at Figure 9, it is visible that there exists a certain minimum of the thermal resistance *Rth* at the cold plate temperature value *Ta* equal to about 50 ◦C. It is also easy to observe that for the diode operating without the thermal pad the changes in thermal resistance are much bigger than for the diode operating with the pad.

The last part of the presented experimental results illustrates the influence of the thermal pad on the optical parameters of tested power LEDs. The measured surface power density of light emitted by the XPLAWT diode presented in Figure 10 is an increasing function of the diode forward current *ID*. The increase of the cold plate temperature reduces the emitted light power. Moreover, it is also visible that owing to the use of the thermal pad it is possible to obtain higher values of power density. These differences become more apparent with the increased diode current and exceed even 10%.

**Figure 9.** Measured dependences of thermal resistance of the MCE diode on the cold plate temperature for different forward current values.

**Figure 10.** Measured dependences of the surface power density of light emitted by the XPLAWT diode on the LED forward current for different cold plate temperature values.

Figure 11 shows dependences of a change in the surface power density of the emitted light ΔΦ*<sup>e</sup>* on forward current for the MCE diode. This change was measured while changing temperature of the cold plate over the range from 10 ◦C to 90 ◦C. This change is an increasing function on the diode forward current, and at *ID* = 0.7 A, it attains even 0.7 W/m2. The changes in the value of Φ*<sup>e</sup>* obtained for both the considered kinds of mounting the tested power LEDs do not visibly differ between each other, and they are comparable with the measurement error.

**Figure 11.** Measured dependences of the surface power density change ΔΦ*<sup>e</sup>* of the diodes MCE on forward current.

In turn, Figure 12 illustrates the dependence of the surface power density of the emitted light on temperature of the cold plate at selected values of forward current. The considered dependence Φ*e(Ta)* is a decreasing function. The slope of this dependences is much higher for the higher of the considered values of the diode forward current. This slope is equal to even −0.225 %/K for *ID* = 0.7 A, whereas for *ID* = 0.1 A, this slope is practically equal to zero.

**Figure 12.** Measured dependences of the surface power density Φ*e* of the diodes MCE on temperature of the cold plate.

The computed dependences of the efficiency η*opt* of the conversion of electrical energy into light in the tested LEDs is shown in Figure 13a for the diode XPLAWT and in Figure 13b for the diode MCE. This efficiency is equal to the quotient of the optical power *Popt* and the product of the diode current *ID*, and the diode forward voltage drop across its junction *VD*. The computed efficiency η*opt*, in all cases has a distinct maximum. This maximum is observed at the current equal to about 30 mA for the diode without the thermal pad and around 100 mA for the diode with the thermal pad. Interestingly enough, at lower current values, a higher efficiency of conversion is observed for the diode without the thermal pad. However, within this range of currents, the power of the emitted radiation is relatively small, and junction temperature of the diode is nearly to the cold plate temperature.

**Figure 13.** Measured dependences of radiant efficiency on the forward current for different cold plate temperature values for: (**a**) the XPLAWT; (**b**) MCE diodes.

Within the range of forward current values typical for the investigated diodes, the use of the thermal pad makes it possible to achieve higher efficiency η*opt* and a higher power value of the emitted radiation. The change of the cold plate temperature does not influence in an essential way the relationship between the efficiency values computed for both manners of the diode mounting, since an increase of the cold plate temperature by 80 ◦C causes the decrease of efficiency by only several percent. It is interesting that the relationship between the characteristics obtained for the MCE diodes operating in different mounting conditions and the cold plate temperature are much different for the XPLAWT diode. Additionally, for the diode with the soldered thermal pad important differences in values of the efficiency η*opt* obtained for different *Ta* values are noticeable. These differences exceed even 7%. For both diodes the maximum value of the efficiency is observed at the *ID* current in the range from 20 to 50 mA.

In order to compare the influence of thermal pad on properties of the considered power LEDs, some values of parameters of these diodes are collected in Table 2.

**Table 2.** The values of selected electric, optical and thermal parameters of the tested power LEDs operating with soldered (WTP) or not soldered (NTP) thermal pads.


The change in forward voltage Δ*VD* at maximal value of forward current considered here and the change of the cold plate temperature from 10 ◦C to 90 ◦C has much smaller values for the diode XPLAWT, but the highest value of Δ*VD* voltage is observed for the MCE diode without soldered thermal pads. The value of thermal resistance *Rth* measured for the above mentioned values of forward current *ID* is much smaller for the tested devices with soldered thermal pad. Differences in values of this parameter exceed even 50% for MCE diode. Consequently, an excess in junction temperature Δ*Tj* of the tested diode at selected value of current *ID* caused by self-heating phenomenon is much higher for the diode operating without thermal pad.

Finally, the change in surface power density ΔΦ*<sup>e</sup>* at selected value of the current *ID* at changing temperature of the cold plate by 80 ◦C is not big and it does not exceed 5% for both the diodes. Analyzing presented results of measurements, it can be stated that the use of thermal pads makes it possible to reduce visibly the values of thermal resistance and junction temperature of the considered devices, but changes in values of optical and electric parameters are not significant. Consistent with the classic theory of reliability of semiconductor devices [21] and obtained results of measurements, the lifetime of the power LEDs operating with soldered thermal pad can be three times longer than for the same devices operating without the thermal pad.

#### **5. Conclusions**

This paper presented the results of measurements illustrating the influence of the thermal pad on electric, thermal and optical parameters of selected power LEDs. As demonstrated, these parameters influence one another. For example, an increase in thermal resistance causes a decrease in electrical power consumed by the considered diodes. In turn, an increase in junction temperature causes a decrease in the density of emitted light flux.

Based on these results, it is clearly visible that the thermal pad considerably reduces thermal resistance of the considered diodes. This reduction is more visible for the diode situated on the larger MCPCBs. For the MCE diode, its thermal resistance is a decreasing function of forward current and cold plate temperature. It was demonstrated that due to differences in the thermal resistance of power LEDs mounted in a different manner, the differences in the junction temperature of these diodes at the same value of forward current can attain even 30 ◦C. Owing to the more effective cooling of the LED mounted with the thermal pad soldered, it is possible to attain higher power values of the emitted light. This power could be even by 10% higher than for the XPLAWT diode operating without the thermal pad. The observed difference in the power value is an increasing function of the forward current.

Furthermore, it was shown that using the thermal pad results in the increase in the value of the efficiency of the electrical energy conversion into light, especially for higher values of the diode forward current. It was also shown that the increased temperature over the ambient causes the decrease of the conversion efficiency. It is worth pointing out that the efficiency of the tested XPLAWT diodes for low values of forward current exceeded even 80%. Moreover, it is worth noticing that an increase in temperature of the cold plate causes a visible decrease in the diode forward voltage and in the power density of the emitted light. This decrease is the most visible for high forward current values.

The results of performed investigations could be useful for designers of solid-state light sources or substrates dedicated for their applications. Taking into account presented results it might be possible to significantly improve the efficiency and lifetime of the power LEDs used in light sources.

**Author Contributions:** Conceptualization, K.G. and M.J.; methodology, K.G., P.P., and M.J.; investigation, P.P. and T.T.; writing—original draft preparation, K.G. and M.J.; writing—review and editing, K.G., P.P., and M.J.; visualization, K.G. and P.P.; supervision, K.G. and M.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the program of the Ministry of Science and Higher Education called "Regionalna Inicjatywa Doskonało´sci" in the years 2019–2022, project number 006/RID/2018/19, sum of financing 11 870 000 PLN.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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