#### 4.1. The Physical Roots

Up to date power LED devices approximately have the band diagram shown in

Figure 8 (based on [

38,

43]). The active portion of the LED die consists of “bulky” p and n confinement layers, an electron blocking layer (EBL), and a sandwich of quantum barrier layers (QB) and quantum wells (QW). The bandgap in quantum wells defines the wavelength of the emitted light; a wider bandgap of the quantum barrier ensures low absorption of emitted photons and a high efficiency of the charge injection into the quantum wells.

In the stationary (DC) condition, the quantity of charge carriers in a material section is defined at a given temperature by the following factors:

Number of generated electrons/holes and their recombination through different mechanisms within the section. These effects can be thoroughly treated by the collision theory, but a simplified quantity called the generation and recombination rate can be used in the treatment of the statistical mechanics, too.

Injection from the adjacent sections, determined by the band structure near the interface.

The imbalance in these phenomena results in a current through all sections of the structure. The continuity equation from statistical mechanics dictates that this current is the same in all sections of the structure, but the actual carriers can be quite different. The exchange of the charge carriers occurs through generation and recombination effects. In the forward direction, the external driving force (voltage) diminishes the potential barriers in the band structure.

In “bulky” diodes, the diode characteristics in the forward direction can be calculated from the thermionic injection of the electrons confined in one side of the junction to the other similar side.

This injection is driven by the diffusion of charge carriers that have a higher energy than the potential barrier separating them from the adjacent material section. As the distribution of the carriers between energy levels is approximately exponential, when an applied forward voltage, ${V}_{F}$, lowers the barrier, exponential growth in the diffusion current is experienced.

By solving the appropriate equations, one obtains a closed analytical formula under simplified conditions. These conditions are such that in the semiconductor material composed of adjoining ideal depleted and neutral zones, there is no particle generation and recombination in the depleted zone, and particles move until charge neutrality is reached in non-depleted zones.

The result is the well-known Shockley equation. By applying the forward voltage, ${V}_{F}$, on the diode, the forward current, ${I}_{F}$, is obtained as given in Equation (1), and reformatted for the forced forward current, ${I}_{F}$, in Equation (2). In Equation (3), ${I}_{F}\xb7{R}_{S}$ describes the additional voltage drop on the diode. ${R}_{S}$ embodies the physical series resistance, but also some other effects, which do not follow the exponential equation.

The number of the movable particles quickly grows with the temperature. Solving the related equations of solid state physics, we can obtain that besides the thermal voltage,

${V}_{T}$, introduced earlier, the

${I}_{0}$ parameter is also temperature dependent:

where

${T}_{j}$ (with a lowercase

j) denotes the junction temperature in Kelvins, and

${W}_{g}$ is the bandgap energy of the semiconductor introduced earlier. The coefficient,

$G$, in Equation (17) and in further equations lumps appropriate constants that represent different material parameters. The value of

$\mathsf{\vartheta}$ for homojunctions is 3 (see, e.g., [

37] or [

42]). This equation is expressed in the same way as in Equation (5), but is now rewritten for an absolute

${T}_{j}$ junction temperature, instead of a temperature change around

${T}_{ref}$.

When a 3D zone faces a 2D well, a complex injection mechanism can be identified: Thermionic emission for the particles above a potential barrier, tunneling into and over the barrier below the barrier height, etc. These equations can be solved numerically [

43,

44], and the results can be approximated with a value of

$\mathsf{\vartheta}$, as shown in Equation (17), that is significantly higher than 3.

By solving the equations of the charge distribution for several sections of the semiconductor, such as the depletion layer at the pn interface, quasi-neutral layers around the depletion layer, and the quantum wells, several equations can be obtained:

Some dominating effects can be treated with the collision theory, for more detail, see, e.g., in [

38]. In the following paragraphs, these effects are summarized briefly.

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

The recombination rate can be written as $R=A\xb7n$, where $A$ is the recombination probability and $n$ is the electron density. The original Shockley-Read-Hall (SRH) recombination model treats the recombination at deep level traps in the band structure. The energy and momentum of the particle will be transferred to the partner in the recombination.

This kind of recombination produces current constituents of the ${I}_{{F}_{A}}\left({V}_{F}\right)={I}_{{0}_{A}}\xb7\mathrm{exp}\left({V}_{F}/\left(2{V}_{T}\right)\right)$ style (${m}_{A}=2$), and becomes saturated at high particle concentrations. Accordingly, in models of bulky diodes, the depleted region where the particle concentration is low is taken into consideration.

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

This kind of recombination adds ${I}_{{F}_{B}}\left({V}_{F}\right)={I}_{{0}_{B}}\xb7\mathrm{exp}\left({V}_{F}/{V}_{T}\right)$ constituents (${m}_{B}=1$). The energy difference between the electron and the hole will be emitted as a photon.

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

Three particles have to be present. The related recombination rate is $R=C\xb7n\xb7n\xb7p$. The probability of this mechanism increases strongly with the carrier concentration.

The current component is not calculated as a closed analytic formula because at high carrier densities, the assumptions defined above do not apply, and the growth would be steeper than ${I}_{{F}_{C}}\left({V}_{F}\right)={I}_{{0}_{C}}\xb7\mathrm{exp}\left({V}_{F}/{V}_{T}\right)$.

In all models, the Auger constituents can be lumped into the ${I}_{F}\xb7{R}_{S}$ term as they appear at high forward currents.

In the forward direction, the generation of carriers in a section can be neglected.

Further effects with the treatment of statistical mechanics are:

Diffusion of minority carriers at low concentrations, that is, in low numbers compared to the density of the majority carriers. This yields ${I}_{{F}_{d}}\left({V}_{F}\right)={I}_{{0}_{d}}\xb7\mathrm{exp}\left({V}_{F}/{V}_{T}\right)$, corresponding to Equation (1) with $m=1$.

Diffusion of both types of carriers at high concentrations, when the quasi-neutrality principle makes the density of both carriers equal. This yields a current constituent in an ${I}_{{F}_{q}}\left({V}_{F}\right)={I}_{{0}_{q}}\xb7\mathrm{exp}\left({V}_{F}/2{V}_{T}\right)$ form.

We can observe that all effects contributing to the junction current correspond to the general form of Equation (18), where ${m}_{X}$ is 1 or 2.

In the standard treatment of text books, disjoint locations in the structure are defined with dominant effects, which are then added up. As such, they claim the

$\mathrm{ln}\left({I}_{F}\right)$-

$\left({V}_{F}\right)$ chart shown in

Figure 9a, with sharp slope breaks at the change of the current forwarding mechanisms. However, real diodes do not expose this alteration of the slope (

Figure 9b), producing a continuous slope over several orders of magnitude.

At this point, the text books typically make a sloppy step, stating that an average ideality factor, $m$, can be used without too much further examination. A more established statement outlining a step to consider separate ideal depleted and neutral zones, and then to add up the currents belonging to disjoint zones is simple, but not true.

In reality, there is more than a zero charge in the depletion zone, the frontier of it to the p and n side is fuzzy, and charge neutrality is questionable. In the treatment of collision theory, instead of statistical mechanics, proportionality would be experienced among different charge movement mechanisms, and one mechanism would establish a particle surplus in a zone (e.g., injection), which enhances the probability of another mechanism (e.g., recombination). Until one such mechanism gets exhausted (for example, SRH recombination decays when deep level traps are mostly filled), a product of the mechanisms is observed:

where

${I}_{0}={I}_{{0}_{a}}\xb7{I}_{{0}_{b}}\xb7{I}_{{0}_{c}}\xb7\dots $ and

$1/m=1/{m}_{a}+1/{m}_{b}+1/{m}_{c}+\dots $.

#### 4.2. Modeling the Electrical Characteristics

By using the proportionality concept, verified by the measurements on many actual LED samples (with peak wavelengths chosen from different ranges of the visible range),

$m$ appears in Equation (17):

Figure 10 shows the measured

${V}_{F}$ of a royal blue power LED at several

${I}_{F}$ currents, from 10 mA to 1 A at a fixed

${T}_{J}=$ 85 °C. A Shockley type operation can be observed at low currents. At higher currents, the

${R}_{S}$ series resistance represents all losses which do not belong to the basic pn-junction operation.

Instead of just fitting a logarithmic trendline on a few points, the ${R}_{S}$ value in Equation (2) can be optimized such that the ${V}_{R}$ = ${I}_{F}\xb7{R}_{S}$ voltage at each measured ${I}_{F}$ is subtracted from the measured ${V}_{F}$ points (blue **+** markers) and the ${R}_{S}$ value which yields the best logarithmic fit on the ${V}_{F}$–${V}_{R}$ relationship in the 10 mA to 1000 mA range is searched for. Herein, this optimization method is referred to as OPT1.

Applying OPT1 on the measured points, the curve of the ${V}_{Fpn}={V}_{F}-{V}_{R}=C\xb7\mathrm{ln}\left({I}_{F}\right)+D$ format is obtained. The parameters, $m$ and ${I}_{0}$, of Equation (2) can be calculated from $C$ and $D$ as $m=C/{V}_{T}$ and ${I}_{0}=\mathrm{exp}\left(-D/C\right)$, respectively.

Repeating the OPT1 procedure to all measured temperatures allows the temperature dependence of the parameters to be measured. First, the results of the method at various ${T}_{J}$ junction temperatures were demonstrated, on a royal blue LED (Cree’s XPE2 type).

The isothermal ${V}_{F}\left({I}_{F}\right)$ characteristics of the LED were recorded at a number of current values, between 1 mA and 1000 mA, at four different ${T}_{J}$ junction temperatures, namely at ${T}_{J}=$ 30, 50, 70, and 85 °C.

The blue

**x** markers in

Figure 11a show the fitted

${R}_{S}$ series resistance values generated by the

OPT1 method at the given temperatures. The temperature dependence in this range is rather “flat”, so a good fit can be achieved by a second order polynomial approximation. For the model parameter,

$m$, the optimization yields several values around

$m=2.5$ (the blue

**x** markers in

Figure 11b).

Figure 12a presents the

${I}_{0}$ values for the best fit.

In Equation (2), the forward voltage, ${V}_{F}$, depends on $m\xb7\mathrm{ln}\left({I}_{0}\right)$, and accordingly small numeric perturbations in the calculated $m$ values can cause unacceptable scatter in the ${I}_{0}$ values, often several orders of magnitude. Supposing that the distribution among different carrier transport mechanisms does not change much in the temperature range modeled, the $m$ value can be fixed.

A similar optimization algorithm denoted by OPT2 was also defined, where $m$ is fixed, and the best (logarithmic) fit for ${R}_{S}$ is targeted again.

Applying

OPT2 on the investigated XPE2 LED samples with

$m=2.5$ chosen, some accuracy of the fitting is lost, but the wild scatter of

${I}_{0}$ changes to demonstrate a reasonable trend (see the red

**+** markers in

Figure 11a,b, and

Figure 12b).

The trend of

${I}_{0}$ in

Figure 12b can be described by several mathematical formulae. The text books of solid state physics [

37] offer several acceptable concepts.

The minimum expectation is that the formula should be valid in the relatively narrow temperature range of the measurement (${T}_{j}=$ 300 K to 360 K in absolute temperature).

The deduction in the related books can be found for the pn-junction in a bulk semiconductor, where

$\mathsf{\vartheta}=3/m$. Thus, Equation (20) will be written in this case as:

and for a smaller range,

$\Delta {T}_{j}$, the change around

${T}_{j}$ is:

resulting in the following ratio of the saturation currents at a

$\Delta {T}_{j}$ temperature shift:

According to this, the

${I}_{0}$ current coefficient plotted in

Figure 12b can be approximated by:

where

${I}_{01}=$ 6.09 × 10

^{−22} and

$a=$ 0.138 in this particular example. The fit is correct in the given range with an

${R}^{2}$ determination coefficient of 0.9972.

A more ambitious attempt is through the direct use of Equation (20). By dividing the formula with the exponential part

$\mathrm{exp}\left(-{V}_{g}/\left(m\xb7{V}_{T}\right)\right),$ the following equation is obtained:

The

OPT2 optimization process already yields

${I}_{0}$ at several temperatures, but to construct the left side of Equation (25), the values of the

${V}_{g}$ bandgap voltage are needed. As shown earlier by Equations (6) and (7), this can be calculated from the

${\lambda}_{peak}$ peak wavelength values obtained from the LEDs’ measured spectral power distributions. The temperature and current dependence of

${\lambda}_{peak}$ can also be taken into consideration, as presented, e.g., in [

5].

In

Figure 13, this “power of

$\mathsf{\vartheta}$” dependence of a royal blue XPE2 LED is illustrated, for which

$\mathsf{\vartheta}=$ 6.6531 was obtained with an

${R}^{2}$ determination coefficient of 0.9966.

Two optimization steps were conducted on the measured data of several LED devices of different colors (in this particular example: Members of Cree’s XPE2 family).

In

Table 2, the measured

${\lambda}_{peak}$ values of these and the calculated bandgap voltages,

${V}_{g}$, are listed. For the white devices, the peak wavelength of the blue part of their spectra was used.

In

Table 3, the modeling parameters for all examined XPE2 devices are listed. To save space, only the minimum and maximum values of

${R}_{S}$ in the given temperature range are presented. In the tables, a color identification code is assigned to the LEDs, which is used below to denote them.

#### 4.3. Modeling the Light Output Characteristics

In the previous subsection, an explicit model for the description of the LED behavior in the electric domain was created, i.e., the applied forward current,

${I}_{F}$, directly determines the forward voltage,

${V}_{F}$, using closed analytic formulae. Besides providing an improved method to capture the phenomena related to the carrier transport and recombination processes in LEDs’ active region, the use of explicit formulae is a clear advantage compared to the quasi black-box model described in

Section 2, which requires that Equations (10) and (13) are solved iteratively (in the Spice circuit, macro implementations of the models of the circuit simulation algorithm are inherently iterative processes, so this difference between the two models is not relevant. However, in simulation environments, such as that depicted in

Figure 4, avoiding internal iteration within the LED chip model is a clear advantage).

As hinted before, the optical domain is traditionally characterized by a bunch of current and temperature dependent parameters. The internal and external quantum efficiency ($\mathrm{IQE}$ and $\mathrm{EQE}$) together yield the ${\eta}_{e}$ energy conversion efficiency also known as the radiant efficiency or wall plug efficiency (${\eta}_{e}=\mathrm{WPE}=\mathrm{IQE}\xb7\mathrm{EQE}$). Then, ${\eta}_{e}$ leads to the emitted total radiant flux, ${\Phi}_{\mathrm{e}}$ (also known as the emitted optical power), and the efficacy (${\eta}_{V}$) or the luminous efficacy of radiation ($\mathrm{K}$) metrics lead to the emitted total luminous flux, ${\Phi}_{\mathrm{V}}$. The chip level multi-domain LED model has to provide the ${\Phi}_{\mathrm{e}}$ and ${\Phi}_{\mathrm{V}}$ fluxes as output, therefore in the following, their modelling is provided.

Following the pattern of the calculation of the voltage drop of the “radiant diode” described in [

5] (see also

Figure 5), first, a new, virtual efficiency figure is introduced, which is called the radiant voltage:

It will be later used in explicit models of the emitted total fluxes. The radiant voltage, ${V}_{rad}$, is tied to the energy conversion efficiency, ${\eta}_{e}$, because ${\eta}_{e}={\Phi}_{\mathrm{e}}/\left({I}_{F}\xb7{V}_{F}\right)={V}_{rad}/{V}_{F}$.

It must be noted that ${V}_{rad}$ is merely a characterization tool, and ageing effects that degrade the external quantum efficiency, like graying of the lens, can significantly change it.

The measurements provided here prove that in a wide current range, where power LEDs are typically operated, ${V}_{rad}$ can be approximated by a product of two Shockley style functions.

In the previous subsection, it was shown that a constant

$m$ parameter, over more orders of magnitude in current, expresses the proportionality between different charge transport mechanisms. Now, one blend of transport effects feeds the radiant recombination by injecting carriers into the quantum well, while other effects (such as the Auger recombination that enhances electron leakage over the electron blocking layer) deport the carriers before they can recombine. Detailed numeric analysis of the actual material compositions and geometries is broadly treated in the literature, like [

43] or [

44].

By maintaining the concept of proportionality that is underpinned by constant

m in the forward current range of interest, whether the measurements support a:

type formula can be checked. The introduction of the notation,

$z=\mathrm{ln}\left({I}_{F}\right)$,

$c=\mathrm{ln}\left({I}_{0a}\right)$,

$d=\mathrm{ln}\left({I}_{0b}\right)$, allows Equation (27) to be rewritten as follows:

which can be further simplified to:

Figure 14 shows the

${V}_{F}$ and

${V}_{rad}$ values of the royal blue and phosphor converted white LEDs measured at different junction temperatures. It was found that the fitting of a quadratic polynomial with

$A$,

$B$, and

$C$ coefficients corresponding to Equation (29) obtained an excellent match on the 10 mA to 1000 mA region of the

${I}_{F}$ total forward current, with an

${R}^{2}$ determination coefficient of around 0.99.

The

${\eta}_{e}$ energy conversion efficiency can be gained by dividing the

${V}_{rad}$ values with the

${V}_{F}$ forward voltage, which is also shown in the diagrams of

Figure 14. This way, through

${V}_{rad}$, an explicit forward current dependent model for the radiant flux is given:

${\Phi}_{\mathrm{e}}={V}_{rad}\xb7{I}_{F}$.

Figure 14a,b show the quadratic fit on

$\mathrm{ln}\left({I}_{F}\right)$ for the royal blue and white devices, while

Figure 14c shows the same for the red and amber devices.

The charts in

Figure 15 present the temperature related changes of the

$A$,

$B$, and

$C$ parameters used in Equation (29), for all colors. The temperature dependence is rather flat again; it was found that a second order polynomial can be fitted at

${R}^{2}$ > 0.99.

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

To prove that the models defined so far describe the electrical and optical domain in the current and temperature range of interest accurately the Cree XPE2 power LED family was characterized. All measurements were done in a combined thermal transient and optical measurement arrangement [

34] that complies with CIE’s and JEDEC’s recent recommendations and standards on the optical and thermal testing of power LEDs [

32,

39,

45].

Five different color LED types were measured, using multiple numbers of samples from each type. To make the modelling easier, the cold plate control mode was set to regulate towards a fixed

${T}_{J}$ junction temperature as described in [

31]. This ensured that the measured DC characteristics and the radiometric and photometric values corresponded exactly to the thermal boundary of a constant

${T}_{J}$, without further model correcting steps. Current levels were selected between 20 mA and 1 A, and

${T}_{J}$ temperatures ranged from 30 to 85 °C.

In order to demonstrate the predictive power of the modeling concept, the red devices from

Table 2 were selected because their

${\eta}_{e}$ radiant efficiency has the largest temperature and current dependence (

Figure 16b). Still, it is expected that the model extracted from a single median device accurately represents the devices in the measured binning class. (The median device is the LED sample, which is closest in terms of one if its selected parameters to the median value of the distribution of that parameter obtained for the entire LED population)

The electrical model of red LEDs was characterized by just a few parameters in the previous subsection. The ${R}_{S}$ values were approximated by a parabolic function with values around 0.7 Ω, $m$ was set to 1.5, and for the “power of $\mathsf{\vartheta}$” dependence of the ${I}_{0}$ saturation, $\mathsf{\vartheta}=$ 6.59 with an ${R}^{2}$ of 0.99 was obtained.

Selecting the LED of median

${V}_{F}$ value from the bin, the

${I}_{F}$–

${V}_{F}$ plots at four

${T}_{J}$ temperatures were measured as shown in

Figure 16a.

In

Figure 17a, a 3D plot of the relative error of the modeled

${V}_{F}$ forward voltage is shown. The chart compares the calculated forward voltage,

${V}_{F}$, from the Shockley model of the median device to its own measured values from

Figure 16a. The accuracy of the model is within ±0.5% on the whole current and temperature range.

The figure and also the subsequent similar figures show the 20 mA to 700 mA range, in which all ${T}_{J}$ values were reached. The colors correspond to a ±4% range of error in the figures, representing an error of ${V}_{F}$ and a ±8% range showing the error of ${\Phi}_{\mathrm{e}}$.

A “worst case” situation is shown in

Figure 17b. Here, the measured forward voltage of the device which has the lowest

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

In

Figure 18 the modeled

${\Phi}_{\mathrm{e}}$ radiant flux of the “median

${V}_{F}$” red sample to the measured

${\Phi}_{\mathrm{e}}$ of the “median

${V}_{F}$” sample itself (

Figure 18a) and to the measured

${\Phi}_{\mathrm{e}}$ of the “lowest

${V}_{F}$” sample (

Figure 18b) is compared. Again, the self-modeling is accurate to 1%, the scatter between samples is higher, and the

${\Phi}_{\mathrm{e}}$ radiant flux of the least similar sample deviates a few percent from the median device.

The test was also carried out on royal blue and phosphor converted white samples, composed of a similar royal blue chip and a yellow phosphor layer. It was found that the

${V}_{F}$ scatter was very low—obviously the same as in the case of the royal blue devices—and the “median” white LED modeled its own radiant flux within 1% (

Figure 19).

However, the white sample of the lowest

${V}_{F}$ showed up to a −8% to +18% difference of

${\Phi}_{\mathrm{e}}$ in the 30 mA to 1000 mA forward current range. This may have occurred due to the larger scatter in the phosphor quality and also indicates that the binning was performed at a single (

${I}_{F\_bin}$,

${T}_{J\_bin}$) operating point (

Figure 20).

This finding corresponds to the results of the variability analysis of the thermal properties of the blue and white LED samples [

46,

47], where a significantly larger scatter of the properties of the white samples was found.

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

The physics based model differs from the ones hardcoded in all Spice program versions for semiconductor diodes. Therefore, a circuit macro in visual basic was created in which the controlled voltage or current sources directly represent the model equations.

It must be emphasized that this model is based on explicit formulae;

${I}_{F}$ and

${T}_{J}$ yield all output data directly, without internal iterations as implied by the circuit scheme in

Figure 5c.

The

${I}_{F}$–

${V}_{F}$ relationship in this model is coded as Equation (2). The

${R}_{S}$ series resistance has the quadratic temperature dependence of Equation (4), as justified in

Figure 11a. The temperature dependent

${I}_{0}$ saturation current was calculated based on Equation (20).

In all equations, the

${V}_{T}$ thermal voltage and the

${V}_{g}$ bandgap voltage were calculated for the actual

${T}_{j}$ junction temperature (given in Kelvins). In the bandgap voltage formula, the peak wavelength was approximated by a relationship derived from Equations (7) and (8) for values of

${T}_{j}$ close to 300 K:

The emitted total radiant flux (optical power) was calculated with:

As supported by

Figure 15, the modest change in the temperature dependence of

${V}_{rad}$ can be reflected by quadratic tuning equations:

The heating power was calculated again, like in the quasi black-box model, using Equation (9). All the coefficients in these equations are model parameters that must be identified for a given LED type, given now by the systematic methodology of subsequent OPT1 and OPT2 operations.

For the ${\Phi}_{\mathrm{V}}$ luminous flux, several approaches were tested, such as deriving it from ${\Phi}_{\mathrm{e}}$ through the fitting of polynomials or searching for a polynomial on the raw ${I}_{F}$. Finally, it was found that best fidelity over a wide current range can be achieved with a polynomial on $\mathrm{ln}{I}_{F}$, Which was calculated similarly to Equation (32), with a different set of constants.

A Spice implementation of the proposed new model was created. A part of it representing the electrical characteristics by controlled sources is shown in

Figure 21.

The

${T}_{J}$ junction temperature can be computed by inserting the thermal model of the packaged LED into the thermal model of the environment. These models are deeply treated in [

7,

48].

A simple Foster-style equivalent thermal chain representing the device and the outer world is shown in

Figure 22. The conversion from the

${T}_{J}$ junction temperature in Celsius to

${T}_{j}$ in Kelvin yields the TJK “voltage” for the model calculations in

Figure 21.

In

Figure 23, the response of a blue XPE2 LED from Cree is monitored in its thermal environment represented by the simple compact thermal model. First, heating can be observed. The current grows linearly from 10 mA to 1 A in the first 1 μs (the current was multiplied by 10 in order to make it visible in the chart. The curvature corresponds to linear growth in the log-lin scale).

In the further two seconds, the increase of the temperature (25 to 70 °C) and the decay of the forward voltage and radiant flux can be observed. The “bumps” of the curves correspond to the thermal time constants in

Figure 22,

${\tau}_{1}=$ 1 ms,

${\tau}_{2}=$ 100 ms.