## Abstract

High power light-emitting diodes (HPLEDs) are frequently being operated in a pulsed manner. The research presented here focuses on the optical, electrical and thermal behaviour of a HPLED under pulse width modulation (PWM), and has the following twofold aim. Firstly, investigating the temperature dependence of the HPLED’s efficiency, where it was found that the exact method of operation and the definition of calculation is crucial when making claims. Secondly, we propose a method to simulate the absolute emitted luminous flux of a current driven HPLED under PWM. This is done by making use of experimentally determined characteristic parameters of the HPLED. This has as advantage that no further physical measurements are needed to investigate the HPLEDs behavior under numerous different PWM circumstances.

©2009 Optical Society of America

## 1. Introduction

The potential of LEDs to exceed above conventional light sources, especially with regards to efficiency, have experienced an increase in media attention over the last years [1, 2, 3, 4]. Besides its efficiency increase, the controllability of LEDs compared to present conventional light sources is an important and significant benefit. Some of these controllable features are for example: emission spectrum, colour temperature, polarization, temporal modulation as well as spatial modulation [4]. All these advantages imply accurate determination and therefor measurement, so as to be able to utilize these advantages thoroughly. This is important for industrial manufacturing and research and development. In this paper we focus on the temporal modulation, i.e adjusting the duration and period of the current pulse sent to the device. This is known as pulse width modulation (PWM), where the duty cycle *D*, defined in Eq. (1), is an indication of the mean power send to the device.

where *τ* is the pulse duration and *T* the period of the pulse. Previous studies have been conducted which investigated effects such as spectrum changes, dimming schemes, thermal analysis and time evolution of electroluminescence when operating a HPLED underPWM [5, 6, 7, 8]. In this study we investigate the temperature effects on the efficiency/efficacy of a single commercial monochromatic HPLED, with as goal to be able to simulate the absolute total luminous flux under different PWM operating conditions. The HPLED used for this study is a Lumileds LUXEON III Emitter. Certain definitions which are used in this paper make use of photometric quantities. The reader is referred to [9, 10, 11] for a more thorough description of photometry. Throughout this paper dashed lines are included in all figures. This is done to aid the reader, but have no further physical meaning.

In Section 2 the efficiency of a monochromatic HPLED as function of junction temperature is investigated in detail. Firstly some definitions of efficiency/efficacy are stated. After that the relation between the junction temperature, forward voltage and current is experimentally determined. This is done so that the junction temperature is known at a given moment in time when the HPLED is operated. DifferentPWMmeasurements are performed at different operating currents. Here the luminous flux, spectrum, operating current, forward voltage and thus junction temperature are measured simultaneously. Using the measured luminous flux and the spectrum we calculate the radiant flux. The efficiency is calculated in a number of ways and compared. Section 3 proposes a method towards simulating the total luminous flux of the HPLED under PWM. A start has been made to find a simulation model which is directly linked to absolute measurement. The method makes use of the efficiency characteristics of the HPLED, as well as an 1D-model for the junction temperature. The method is implemented in software and verified with experiment. Section 4 concludes the paper with some final remarks.

## 2. Efficiency of a High Power LED under PWM

#### 2.1. Definition of radiant efficiency, external quantum efficiency and luminous efficacy

Before presenting the measurements we shall first state three definitions of efficiency, based on [12] and [13], to quantify what the energy efficiency is of the HPLED. The first one is known as the *radiant efficiency* and plays an important part in Section 3 when we model the luminous flux from the HPLED. It is defined in Eq. (2).

Here Φ* _{e}* is the radiant flux,

*P*is the electrical power applied to the HPLED,

_{d}*V*forward voltage across the HPLED, and

_{f}*I*the forward current applied to the HPLED. Both Φ

_{f}*and*

_{e}*P*are measured in Watt. The power efficiency is an indication as to how efficient a HPLED converts electrical power to optical power. The second definition of efficiency is known as the

_{d}*external quantum efficiency*, and is stated in Eq. (3).

Here *ν*=*c*λ^{-1}, where c is the speed of light in vacuum and λ the wavelength of the emitted radiant flux. This definition provides one with insight into how efficiently electrons recombine with holes, and are emitted as photons into free space. This definition is only valid for monochromatic HPLEDs, and thus relevant to the research presented in this paper. The third and last definition of efficiency gives one an indication of how efficiently a HPLED emits luminous flux with respect to the input power dissipated by it. It is known as the *luminous efficacy of a source* and is stated in Eq. (4).

Here the nominator is the total luminous flux emitted, and the denominator is the electrical power dissipated by the HPLED. In the analysis that follows we will make use of the above stated definitions to analyze the measured results.

#### 2.2. Relation between current, forward voltage and junction temperature of the HPLED

The junction temperature (*T _{j}*) of a HPLED is an important parameter when investigating its efficiency since the radiant and luminous flux is sensitive to temperature changes [13, 14, 15]. Direct measurement of

*T*is not possible since there is no direct method to access the junction without altering the HPLED. Consequently an indirect method has to be used. Different techniques are known which make use of for example, micro-Raman spectroscopy [16], the shift of the emission peak energy [17], and the change of the diode forward voltage [18]. The latter technique was used in this research, since this method has been shown to provide good results [19, 20, 21]. Recently a similar method than the one which was implemented in this research has been proposed [22], which is slightly different in the manner in which the temperature dependence of the forward voltage is derived and applied. The method which was used during this study makes use of the fact that there is a direct relation between

_{j}*T*and the forward voltage (

_{j}*V*) across a HPLED. To make use of this relation, a series of calibration measurements are performed to determine

_{f}*V*at different

_{f}*T*’s as well as at different operating currents (

_{j}*I*) respectively.

_{f}To determine the relations between *V _{f}* and

*T*for the HPLED under investigation we perform the following measurements. The HPLED is placed inside a temperature controlled oven and is current operated with a commercial laser driver. A short current pulse is sent through the HPLED (

_{j}*D*<0.1%), so that a negligible amount of power is dissipated [18] in the HPLED during the measurement. The forward voltage across the HPLED, as well as the

*I*applied to the HPLED are measured simultaneously. This is repeated for a series of

_{f}*I*’s. The oven temperature is then adjusted and the whole series of measurements are repeated until the full temperature range (23°C to 70°C) of the temperature oven is covered. The results obtained are shown in Fig. 1. What is clearly visible is that

_{f}*V*decreases with increasing temperature when

_{f}*I*is kept constant. Defining

_{f}*A*

_{0}=

*dV*/

_{f}*dT*as the derivative and

*V*

_{0}as the voltage at junction temperature

*T*=0°C, one can fit the voltage as function of the temperature with

_{j}*A*

_{0}and

*V*

_{0}depending on the operating current and is stated in Eq. (5).

Figure 2 shows the dependence of *A*
_{0} and *V*
_{0} on the current *I _{f}*.

#### 2.3. PWM measurements and efficiency calculations

In conducting the PWM measurements we used two measurement facilities. One is a goniometer facility equipped with an illuminance meter as well as a spectroradiometer. With this facility one can measure the illuminance distribution of a light source and perform a numerical integration to obtain the total luminous flux. The limitation of the facility is that it is not at present capable of performing time resolved measurements. To work around this limitation a second measurement facility was used. This facility consisted out of a fast illuminance meter and amplifier, a spectroradiometer and a digital oscilloscope. Using this facility one is able to track pulsed signals with a time resolution smaller than 1ms, but has as limitation that it cannot directly measure the total luminous flux of a light source. A conversion factor was determined with which the measured illuminance was converted to luminous flux. This was done by measuring, under the same operational conditions, the total luminous flux with the goniometer, as well as the illuminance using the fast illuminance meter and oscilloscope. The ratio between the two measurements is the conversion factor. By implementing this conversion method one uses an approximation that the total luminous flux of the source increases or decreases by the same amount as the illuminance as measured in a single direction, i.e. the radiation profile does not change with varying PWM operation.

The temperature model presented in this paper requires the radiant flux of the HPLED under investigation as will become clear in Section 3. We calculated the radiant flux from the luminous flux and spectrum measured. To do this we make use of what is known as the luminous efficacy of radiation (LER) [23] and is stated in Eq. (6).

here *S*(*λ*) is the spectrum of the HPLED, *V*(*λ*) is the spectral luminous efficiency function and *K _{m}*=683

*lmW*

^{-1}, and is known as the maximum spectral luminous efficacy of radiation for photopic vision. [11] Knowing the LER for the HPLED and the luminous flux one can calculate the radiant flux as stated in Eq. (7).

where Φ* _{e}* is the radiant flux in Watt (

*W*) and Φ

*is the total luminous flux measured in lumen (*

_{v}*l*). The PWM experiments conducted consist out of a series of measurements at different currents, periods and duty cycles. It was chosen to perform measurements in the millisecond range. Through the whole of this study the pulse shapes were taken to be rectangular in form. Three different currents, namely 700mA, 1000mA and 1400mA are investigated in all the different measurements. During all the measurements the HPLED is allowed to stabilize for approximately 15 minutes. Measurement series were performed over a whole range of duty cycles at a constant period, namely

_{m}*T*=10ms while

*τ*is changed to adjust

*D*=10%,20%, ⋯,90%. This was done to cover a broad range of effective power sent to the device, and in this manner investigate the influence on the radiant/luminous flux, forward voltage, electrical power, junction temperature and spectrum of the HPLED. Other duty cycles and pulse periods were also investigated to see what the dependencies of the changes in optical output are. The different duty cycles and periods are listed in Table 1.

We first present the investigated spectrum in terms of *K _{eff}* at different operating conditions. The different spectra are needed to calculate the radiant flux using Eq. (7) as well as to determine the dependency on duty cycle and period. Figure 3 depicts the calculated

*K*values at different duty cycles and periods as function of the mean power sent to the device.

_{eff}As can be seen from Fig. 3 the *K _{eff}* values are dependent on the mean power sent to the device and not so much dependent on the period of the pulse. Hence, when calculating the radiant flux we shall take into account the changes of

*K*as function of mean power. Knowing the spectrum influences we continue next with the luminous/radiant flux, voltage and temperature changes during a pulse.

_{eff}In our experiments we measure the luminous flux, forward voltage and current simultaneously, from which we can calculate the electrical power, junction temperature and also radiant flux (making use of the previously determined spectra data). Figure 4 depicts an example of a pulse measured with *τ*=7ms and *T*=10ms, i.e D=70%. Figure 5 is zoomed in on the y-axis. The radiant flux is omitted (due to the luminous flux already included) and the calculated junction temperature is added. One sees that the voltage clearly decreases, and that the temperature increases during a pulse.

This increase as well at the start and end temperatures during a pulse was found to be different for different pulse durations with the same maximum pulse current value. This is due to the fact that more effective power is sent to the HPLED as well as that the HPLED has more or less time to cool down. We operated the HPLED at different duty cycles which resulted in different junctions temperatures. We were then able to investigate the effect of different junction temperatures on the radiant/luminous flux and the efficiency of the HPLED. To investigate the relative temperature dependency of the luminous/radiant flux and electrical power dissipated by the HPLED, we normalized these values to their respective starting values. The relative luminous flux and relative dissipated power are depicted in Fig. 6, and the relative radiant flux and relative dissipated power are depicted in Fig. 7.

From Fig. 6 and Fig. 7 one can clearly see that there is a difference between the decrease in luminous and radiant flux. The luminous flux and the power dissipated by the HPLED decreases with increasing temperature at approximately the same rate, while the radiant flux decreases faster than the dissipated power. This difference is due to the spectrum change occurring at different temperatures. Next we will investigate the temperature dependencies of the HPLED further by investigating its efficiency.

Here we investigate the efficiency/efficacy of the HPLED by using the definitions as have been stated in Section 2.1. We start by calculating the luminous efficacy and radiant efficiency using Eq. (2) and Eq. (4) respectively. The results are depicted in Fig. 8 and Fig. 9.

From Fig. 8 and Fig. 9 one can see that the luminous efficacy is close to constant, even though the temperature increases, but that the radiant efficiency decreases with increasing temperature. This implies for this HPLED that the luminous efficacy is to a good degree, temperature independent, and the radiant efficiency temperature dependent. Taking an average over the luminous efficacy values of each of the currents, we see that the luminous efficacy differs for different currents. It namely decreases with increasing current. This effect is due to what is known as the efficiency droop [24]. To investigate the efficiency of the HPLED further, we compute the external quantum efficiency, as defined in Eq. (3). The results are plotted in Fig. 10 as a function of temperature for the three different operating currents. Here it is clear that this quantity is temperature dependent since the radiant flux decreases with increasing temperature while the operating current remains constant.

In the preceding paragraphs we have analyzed the HPLED efficiency in a number of ways with different results depending on the definition used. One often hears in public literature that “t*he efficiency/efficacy of a HPLED is temperature dependent*”. A reason for this discrepancy could be due to the definition of the efficiency/efficacy used, or the method of operation and measurement. As an example if one operates the HPLED in terms of constant voltage (instead of current), then the efficacy/efficiency does decrease as function of temperature. This is so since when the HPLED heats up the voltage decreases. To compensate for this decrease one needs to increase the current, and thus have a lower efficacy/efficiency which is in agreement with what is shown in Fig. 8. It is thus important to be very clear and precise when comparing and discussing the efficiency/efficacy of a HPLED.

## 3. Modeling the emitted luminous flux of a HPLED under PWM

For commercial measurement of the total luminous flux of a HPLED, there is always a tradeoff between the labor cost and product (measurement information). For PWM measurements it is not possible to measure all the different permutations of pulsed operation, as it is not always foreseeable how the HPLED will be used in the future. It would thus be very useful if one could simulate the total luminous flux of a HPLED under many PWM conditions, by using a limited amount of data obtained from experiment. Here we show such a method to simulate the total luminous flux of a HPLED when operated under PWM for different pulse durations, with the current and pulse duration as input parameters. The total luminous flux emitted by the HPLED, as deduced from Eq. (4), is written in the form stated in Eq. (8).

$$={\eta}_{v}\left({I}_{f}\right)\xb7\left[{A}_{0}\left({I}_{f}\right){T}_{j}+{V}_{0}\left({I}_{f}\right)\right]\xb7{I}_{f}.$$

To use this equation to calculate Φ* _{v}* one needs to known

*η*,

_{v}*A*

_{0},

*V*

_{0},

*T*and

_{j}*I*.

_{f}*I*is an input parameter, and thus known. As was shown in Section 2.3 the efficacy can, to a good approximation, be seen as being only dependent on

_{f}*I*and was previously determined, as were

_{f}*A*

_{0}and

*V*

_{0}. The only unknown parameter is

*T*. So the question then becomes, how can we simulate

_{j}*T*so that we can calculate

_{j}*V*and thus determine Φ

_{f}*of the HPLED at a given point in time? The method of modeling*

_{v}*T*, and as result Φ

_{j}*is discussed next.*

_{v}#### 3.1. A model for the junction temperature of a HPLED as function of time

The accurate modeling of the transient temperature behaviour of a HPLED is an arduous task. Numerous studies [25, 26, 27] have shown that one needs to consider a large amount of parameters if one wants to simulate it very accurately. It is not our intend with this study to derive a complete physical model of the HPLED, and to show how these numerous components influence *T _{j}* respectively. We rather set out to show the feasibility of accurately predicting the luminous flux from a HPLED at various PWM settings. A relatively simple, but sufficiently accurate model, was thus decided upon.

For this study the HPLED is modeled as being composed out of the various components as is shown in Fig. 11.

In Fig. 11 the heat is generated in the active layer and transmitted via conduction to the other components and finally to the surrounding air. To describe the heat flow in the HPLED structure one can construct a differential equation for each one of the components [28]. The transient heat flow from the junction to its neighbouring components is written as stated in Eq. (9).

Here *M _{j}* is the mass of the junction in

*k*,

_{g}*C*is the specific heat of junction with units

_{p, j}*J*

*kg*

^{-1}°

*C*

^{-1}and

*h*is the heat transfer coefficient in units of

*Wm*

^{-2}°

*C*

^{-1}.

*T*is the junction temperature,

_{j}*T*is the temperature of the cap, and

_{c}*T*the temperature of the heat slug. The heat created in the HPLED is written as

_{sl}*P*, measured in watt. A similar equation, without the heat source, can also be constructed for the other components, namely the cap, heat slug, thermal material, and heat sink. Each will have its own specific heat, mass, area and transfer coefficient. One doesn’t usually know these material properties of the different components when performing commercial measurements. There are also numerous types of HPLEDs, each with a different structure. To be able to use a single model, and not to have to explicitly define the material properties, we make the following approximations.

_{heat}• The heat transfer to the cap/lens is seen as negligible compared to the other components (slug, thermal material, heat sink) which have as purpose to conduct heat. The function of the cap is to ensure a certain radiation patten, and not heat conduction. This approximation entails that *Ah _{j c}*(

*T*-

_{j}*T*)≈0 in Eq. (9).

_{c}• Any mass and volume changes are negligible.

• The heat sink and heat slug is assumed to be in perfect heat contact, as well as the thermal contact material which was used. They are thus seen as one entity which transfers heat equally well. This lets us equate the two fluxes as *Ah _{j sl}*(

*T*-

_{j}*T*)≈

_{sl}*Ah*(

_{j hs}*T*-

_{j}*T*).

_{hs}• Lastly it is assumed that one can always measure the heat sink temperature by making contact to it with a calibrated temperature sensor.

Substituting these simplifications into Eq. (9), we obtain the differential equation for *T _{j}* as given in Eq. (10).

Here we divided by *M _{j}C_{p, j}*, and defined two coefficient, namely

*H*and

*K*, which are stated in Eq (11).

*K* can be seen as a coefficient characteristic to the cooling down of the HPLED with units of *s*
^{-1}, while *H* can be seen as a parameter indicating how much energy is needed to result in a certain amount of heat in the junction of the HPLED with units of *J* °*C*
^{-1}. To be able to use Eq. (10) to simulate *T _{j}* of the HPLED, we have to determine the two unknown coefficients which is presented in the following paragraphs.

### 3.1.1. Determining K and H

The K and H parameters are determined by making use of transient temperature data of the HPLED. The *K* parameter is associated with cooling down of the HPLED. To determine this parameter we need to raise the junction temperature of the HPLED, and then allow it to cool down while measuring the junction temperature. At that point no additional heat is added, so the *P _{heat}* term in Eq. (10) becomes zero, and can be written as in Eq. (12).

The analytical solution for this differential equation is then: *T _{j}*(

*t*)=

*T*+(

_{hs}*T*-

_{start}*T*)

_{hs}*e*

^{-Kt}. Here

*T*is the heat sink temperature as measured by using a single contact temperature sensor placed onto the heat sink.

_{hs}*T*is the starting temperature at

_{start}*t*=0 when the HPLED reached a certain temperature and the input electrical power was set to zero. Figure 12 depicts the measured heat sink temperature as function of mean power sent to the device for the three operating currents of interest. One observes that the heat sink temperature is dependent on the mean power sent to the device and not on the operating current, and is thus known for all three the operating currents.

It was not possible to directly measure *T _{j}* while the HPLED was cooling down. This is due to the fact that we use Eq. (5) to calculate

*T*from the measured

_{j}*V*. This calculation can only be done if the HPLED is operational (

_{f}*I*>0), otherwise there is no voltage drop across the HPLED to calculate the temperature from. We thus have to infer the change in

_{f}*T*when no current was sent to the device. For this inference we used different PWM data at different operating currents. We saw that after allowing the HPLED to stabilize (approximately 15 minutes), the HPLED reached a certain quasi-steady state. Both the initial and end temperatures of a pulse reproduced over subsequent pulses. Since the time between the pulses were also measured, the time it took for the HPLED to cool down from one temperature to another was known. The calculated

_{j}*K*-values are depicted in Fig. 13 and are plotted as function of mean power.

The result in Fig. 13 is different than one would initially expect, since we calculated it with *I _{f}*=0. One would in first instance expect it to remain constant as function of power. This apparent electrical power dependency has to do with the heat sink measurement. The heat sink was measured only at one point, but has a non-uniform distribution through the whole heat sink. One can clearly see that at lower power the sensitivity to this temperature non-uniformity is the most. Nevertheless, this does not limit one to be able to perform the appropriate simulations. One has only to consider the correct

*K*value. This variable value of

*K*propagates through to

*H*and is shown next.

To determine *H* we first need to specify *P _{heat}* more precisely.

*P*is seen as the difference between the amount of power entering the system in the form of electricity and the amount of power exiting as radiant flux. This is written in Eq. (13) where we have made use of Eq. (5) and Eq. (2).

_{heat}$$=\left[1-{\eta}_{e}\left({I}_{f}\right)\right]\xb7{I}_{f}\xb7\left[{A}_{0}\left({I}_{f}\right){T}_{j}+{V}_{0}\left({I}_{f}\right)\right].$$

Substituting this into Eq. (10) one obtains the differential equation as written in Eq. (14).

One should remark that we use the radiant efficiency here since we need to calculate the amount of power exiting the system as heat. The radiant efficiency has also been determined previously and can be seen as a given in the simulation model. To determine *H* in Eq. (14) we use data where the HPLED had reached steady state or quasi-steady state, i.e. *dT _{j}*/

*dt*≈0. Substituting

*dT*/

_{j}*dt*=0 into Eq. (14) one obtains Eq. (15).

Figure 14 illustrates the calculated *H*-values at different mean operating powers.

Indeed, as was seen in the previous section when calculating K, H decreases with increasing power. This has as implication that if one now applies this model, one needs to use a specific combination of *K* and *H* as function of input power. The *K* and *H* parameters are also HPLED dependent, and have to be determined again if a different HPLED were to be investigated. In the implementation of this model, Eq. (14) was numerically integrated using the starting temperature equal to 23°*C*, which corresponded to the laboratory temperature where the measurements were conducted. We chose a single combination of *H* and *K* corresponding to the nominal effective power sent to the device. It would also have been possible to calculate the instantaneous power in real time and use that combination of parameters. Thus, the model is not limited to using only one set of *H* and *K* parameters, but can also be adapted during the simulation process. In the next section we implement this model and test it against measured data as to see how well it models the physical PWM measurements.

### 3.2. PWM simulation results

As input to the simulation we use the *I _{f}* measured during experiment,

*T*,

_{hs}*η*,

_{v}*η*as well as the

_{e}*K*and

*H*values previously determined for the specify mean powers. Here we will compare the pulse duration (the time when the HPLED was in the “on”-state), and not the whole pulse train. This is already adequate to validate whether the simulation works and how well it matches experiment. In the results which follow, the relative luminous flux is calculated as written in Eq. (16).

### 3.2.1. Comparison between simulated and measured results

In this section we present the simulation results, and we investigate how well the proposed simulation method models experiment. This was done by comparing simulation results with measurements which were not used when determining the model parameters. We thus investigated if it was possible to simulate the luminous flux and temperature of a pulse which had a different combination of *τ* and *T* than which were used in determining the model parameters. Several different choices of *τ* and *T* were chosen. All choices remained in the millisecond range. An example is shown in Fig. 15, where *τ*=2ms, *T*=5ms, and *I _{f}*=700mA.

Here the measured current, temperature and total luminous flux of the HPLED are plotted. Figure 16 shows the differences between the measured and simulated values. As can be seen from the results the luminous flux is within 1% accurate. Simulation validation of the total luminous flux was also done for the other duty cycle combinations as well as other currents, namely 1000 mA and 1400 mA and also there the difference between measured and simulated luminous flux remained within 1%.

## 4. Conclusion

In this paper we investigated the efficiency characteristics of a HPLED, and its transient behaviour when operated under PWM. We demonstrated the feasibility of a method to simulate the total luminous flux emitted by a monochromatic HPLED by using a limited amount of experimentally determined parameters. The radiant efficiency and luminous efficacy characteristics together with a 1D temperature model of the junction temperature were used to model and simulate the emitted luminous flux of the HPLED. This model showed good agreement with experiment. Further research is needed to improve the generality of the model, and to include different types of HPLEDs. This is the subject for further refinement and research.

## Acknowledgements

The authors would like to thank the temperature- and electrical department of VSL for their help with certain measurement equipment used in this research. Also the insightful discussions from colleagues in the optics group of VSL were highly appreciated. This work was partially supported by the Dutch Ministry of Economic Affairs.

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