LED power consumption in joint illumination and communication system

Xiong Deng; Yan Wu; A. M. Khalid; Xi Long; Jean-Paul M. G. Linnartz

doi:10.1364/OE.25.018990

1. Introduction

Light Emitting Diodes (LEDs) are now commonly used for both specialized and general illumination applications. Because the LEDs have many advantages over traditional light sources, such as high luminous efficiency, small size and long lifetime, they are believed to eventually replace conventional incandescent and fluorescent lamps for general lighting in future. At the same time, there has been a growing interest to apply Visible Light Communication (VLC) to LED-based lighting systems. The primary purpose of a Joint Illumination and Communication (JIC) system, e.g. [1–3], is to provide illumination with data communication through VLC as a secondary function. Several challenges occur in designing a JIC system such as to overcome data rate limitations and to minimize power consumption. Many reported works demonstrated new functionalities or developed signal processing techniques to increase data rate [4–6].

VLC is suggested to be a green technology since it modulates the current through existing illumination LEDs and reuses the illumination power for communication [6–8]. Yet, there is a power penalty associated with operating LEDs at rapidly varying power levels. For instance, a Field Effect Transistor (FET) modulator in series with the LED consumes the extra power [9, 10]. Also in the LED itself, extra power loss occurs, due to the convexity of the diode equation [1, 10, 11]. Although the mentioned works addressed the power consumption issue, they mainly focused on the electrical performance of LEDs or drivers.

As the first approximation, the output luminous flux is proportional to the average current (DC-bias) through LED with modulation [1, 9, 12]. However, because of a modest degree of concavity between the luminous flux and the current, even DC-free modulation also reduces the average light output [3]. The effect of a decreased luminous flux induced by Alternating Current (AC) modulation and pulse driving current, compared to a constant driving current was reported in [13–15]. Hence, in a JIC system, one must anticipate that data modulation decreases output luminous flux to some extent [16]. The reduction in light flux can be compensated by an increased current. Otherwise, the low-speed modulation effect in a burst data transmission, where the waiting (no modulation) time is randomly distributed, can lead to perceivable flicker. This paper quantifies the extra power consumption associated with the luminous flux compensation in the LED.

The output luminous flux of LED is determined not only by the current through the LED but also the temperature of the LED device, in particular the junction temperature [17, 18]. These two factors also have effect on the spectrum of the LED and the effectiveness of the phosphor, thereby affecting the colorimetric performance, including the Correlated Color Temperature (CCT), Color Rendering Index (CRI), and Luminous Efficacy of Radiation (LER) [15, 19–21]. In this paper, we further refine the model of LED light output into a system where multiple physical (e.g. electrical, thermal and photonic) mechanisms interact with each other, as shown in Fig. 1. In the quasi-stationary mode, an increasing current will increase the LED output light, but it also increases the forward voltage and thus it leads to increased power dissipation. The power dissipation in the LED will increase the temperature, which in return decreases the light output in terms of decreased efficacy. In the transient mode, the fast data modulation with both up and down current also introduces increased power consumption but with reduced light output.

Fig. 1 Factors affecting the light output of the LED.

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This paper quantifies the LED power consumption, including the extra power loss induced by data transmission. Figure 1 illustrates the interdependencies and refers to the expressions that model the effects which are combined in this paper to express the overall efficacy decrease. In particular, we classify two nonlinear effects with respect to the LED driving current I in this paper. Firstly, for the luminous flux Φ − I nonlinearity, we refer to as ΦINL. Secondly, for the power consumption P − I nonlinearity which was introduced in [10], we refer to as PINL.

2. LED luminous flux versus current nonlinearity (ΦINL)

The perceived output of commercial lighting LEDs is usually described in photometric units, such as the luminous flux in lumen (lm). To evaluate the link budget for communication, the LED output also needs to be expressed in radiometric units such as radiant flux in Watts (W) [22]. Table 1 lists the photometric and radiometric units, where the most relevant units for our purpose are in bold.

Table 1. Photometry and Radiometry

View Table

In the following sections, we express LED luminous flux in terms of the driving current and temperature.

2.1. Luminous flux of LED versus current

The luminous efficacy of the LED source η_l, expressed in lm/W, is defined as the ratio of the luminous flux output Φ_V in lm over the electrical power. Thus

Φ_{V} = η_{l} P_{L E D},

where P_LED = IV is the input electrical power at current I and voltage V. In fact η_l is mostly determined by two factors, namely the Luminous Efficacy of Radiation (LER), denoted as η_LER and Wall-Plug efficiency (WPE), denoted as η_WPE. So,

η_{l} = η_{L E R} η_{W} P_{E} .

2.1.1. Luminous efficacy of radiation (LER)

The LER, defined as the ratio of luminous flux to radiant flux [17], is given by

η_{L E R} = \frac{K_{m} \int_{λ} S_{L E D} (λ) V (λ) d λ}{\int_{λ} S_{L E D} (λ) d λ},

where K_m is a constant with value of 683 and the integrals are taken over the wavelength λ of the perceivable light. As we can see, η_LER is obtained by the relative Spectral Power Density (SPD) S_LED (λ), weighted by the efficiency function V (λ) to take into account the relative sensitivity of eyes. For a given LED with a known and stable (e.g. current independent) spectrum, η_LER becomes a constant. In practice, perceivable chromaticity shift can occur [16], but be avoided by using an active feedback loop [19] and appropriate thermal design [20]. In this paper, the relative SPD of LED is assumed to be fixed. In fact, Guoxing et al. [23] tested the effect of the driving current on the relative SPD and found its effect to be negligible.

2.1.2. Wall-plug efficiency (WPE)

The Wall-Plug efficiency (or radiant efficiency) of the source is defined as the ratio of emitted optical power P_op, over the input electrical power [17], thus

η_{W P E} = \frac{P_{o p}}{P_{L E D}} = \frac{〈 E_{p} 〉}{q V} \frac{N_{p}}{I / q} = \frac{〈 E_{p} 〉}{q V} η_{E Q E},

where 〈E_p〉 is the average energy of a photon, q is the electrical charge and N_p is the number of photons per unit time. The External Quantum Efficiency (EQE) η_EQE is defined as the ratio of the number of photons emitted into free space per second over the number of electrons injected into LED per second, so η_EQE = N_p/(I/q). The EQE can be seen as the combined effect of the Internal Quantum Efficiency (IQE) and the light extraction efficiency, so it is expressed as

η_{E Q E} = η_{I Q E} η_{E x t},

where η_IQE is the ratio of photons generated by electron-hole recombination to the total number of electrons injected into the LED, and η_Ext is defined as the ratio of the photons extracted out into the free space to the photon generated in the Quantum Well (QW). It appeared reasonable to assume that η_Ext remains constant for a varying injection current level [24], so we only consider the efficiency droop in IQE. In GaInN/GaN LEDs, the efficiency droop is known as the gradual decrease of efficiency when the injection current density surpasses a certain value [25].

To calculate the IQE, a generally accepted ABC model is use [24, 26–28]. The carrier recombination can occur either as radiative recombination, during which light is generated or as the non-radiative recombination, without output light. The non-radiative recombination mechanisms include the defect-related Shockley-Read-Hall (SRH) recombination and Auger recombination [26]. For GaInN/GaN LEDs, the recombination rate R is a polynomial function of the carrier concentration m, and it is written as

R = A m + B m^{2} + C m^{3},

where A, B, and C represent the SRH recombination, radiative recombination, and Auger recombination coefficient, respectively [26]. Since only the radiative recombination (according to the B-term) leads to light emission, we get

η_{I Q E} = \frac{B m^{2}}{R} = \frac{B}{A / m + B + C m} .

Based on Eq. (7), an optimum IQE is reached when d_ηIQE(m)/dm = 0, thus at a carrier concentration of $m_{p} = \sqrt{A / C}$ . The corresponding current for the optimum IQE can be calculated through the relation between the current and recombination rate, i.e., I_p = qV_activeR_p, where V_active is the effective active region volume and R_p is calculated from Eq. (6) using m_p. Usually, the optimum IQE is achieved when I_p is below a few mA while beyond this point, the IQE decreases with rising current [25, 27, 29]. In lighting systems where a power LED is used, the operation forward current is up to several hundreds of mA. Thus, within the operation current range of the LED, we assume that η_IQE is a monotonically decreasing polynomial function of current I, viz.,

η_{I Q E} = \sum_{n = 0}^{N} d_{n} I^{n} .

For a typical GaInN/GaN LED [27], the coefficients d_n for a fifth order (N=5) fitting are compared in Fig. 2 with the calculated IQE. The optimum IQE occurs around 1mA and the fitting agrees well with the calculated IQE. The concavity of the LED output light versus current with ABC model is further calculated in Appendix A.

Fig. 2 Internal quantum efficiency versus current.

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Inserting Eqs. (2)–(5) and (8) into (1), we get

Φ_{V} = \frac{〈 E_{p} 〉}{q} η_{L E R} η_{E x t} \sum_{n = 0}^{N} d_{n} I^{n + 1},

which is a nonlinear polynomial function of current I.

2.2. Thermal effect on LED luminous flux

The input power of the LED, P_LED, is partially transmitted optically and partly dissipated as heating, thereby decreasing the quantum efficiency and degrading the light output. To mitigate temperature rises due to heating, a large heat sink can be used, but this would increase cost, weight and size [30]. A typical thermal resistance from the junction to the thermal pad is around R_T = 10°C/W, which causes a temperature increment of ∆T = R_T P_LED (1 − η_WPE)° C. Figure 1 suggests that the thermal effect decreases the forward voltage of LED, typically in the order of ∆V = −0.003V/°C and reduces the output luminous flux in the order of ∆lm = 2.1lm °C [13]. To calculate the output light by taking heating into account, an empirical exponential approximation has been proposed in [31]

Φ_{V}_{_T} \approx Φ_{V} \exp [- k R_{T} P_{L E D} (1 - η_{W} {_{P}}_{E})],

where k is a positive thermal coefficient. However, the time constant associated with the LED heating is relative slow and in the order of one millisecond [31]. In such case, it is reasonable to make a quasi-static approximation and assume that the LED temperature is determined by the rms average current, which simplifies the analysis in the JIC system. Thus in this paper, the thermal effect is only considered in the calculation of the extra power loss when the LED is driven at different illumination levels.

3. Extra power loss due to ΦINL and modulation

Equation (9) and (10) show that the output luminous flux increases non-linearly with the current through the LED. Hence modulation across this concave function leads to a reduced output, but that can be compensated by increasing the input power. In this section we will evaluate the associated extra power loss and illustrate it for a typical commercial LED.

3.1. Luminous flux of a commercial white LED

For a Luxeon Rebel power LED [32], the output luminous flux is shown in Fig. 3 versus the input current. Since higher order terms are negligible, we approximate Eq. (9) by a concave quadratic function

Φ_{V} \approx a I^{2} + b I + c,

where I is in mA. For the Luxeon Rebel, a = −0.000102, b = 0.309 and c = 3.65, with a thermal pad temperature of 25°C maintained.

Fig. 3 Luminous flux output of a typical commercial white LED.

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Figure 3 shows a pronounced difference between a thermal pad that maintains a temperature of 25°C versus without a heat sink. Equation (10) with k = 0.01 is also shown in Fig. 3. Practical heat sinks will perform in the gray area of Fig. 3.

3.2. Extra power for luminous flux compensation

In this section we compensate for the flux loss that we described in the introduction. When a modulation signal s(t) is superimposed upon an average current I, the time-varying current through the LED is expressed as I(t) = [1 + s(t)]I. Within the 3-dB bandwidth of the LED, we can assume the luminous flux instantaneously follows the modulated current I(t). In such case and for a maintained thermal pad temperature of 25°C, the relation between the mean value of luminous flux level E [Φ_V] and average forward current I follows

\begin{array}{l} E [Φ_{V}] = \int_{I} P (I) {a {[1 + s (t)]}^{2} I^{2} + b [1 + s (t)] I + c} d I \\ = Φ_{V} (I) + (2 a I^{2} + b I) E [s (t)] + a I^{2} E [s^{2} (t)], \end{array}

where P(I) is the distribution of the modulated current and Φ_V (I) is calculated from Eq. (11) using I. Since we assumed elsewhere the temperature does not depend on the fluctuation of modulated current, here the temperature is not depend on the current Root-Mean-Square (RMS) value. We have

E [s^{2} (t)] = α_{r m s}^{2}

, where α_rms is the average modulation index which equals the RMS value of s(t), and for DC-balanced signal E [s (t)] = 0. When a binary modulation such as two-level Pulse Amplitude Modulation (2-PAM) is adapted with two current levels [1 − α_m, 1 + α_m]I and equal probability, E [s²(t)] becomes equal to

α_{m}^{2}

.

Due to the nonlinear coefficient a in Eq. (12), any DC-balanced modulation signal such as OFDM signal will lead to decreased luminous flux, which can be only compensated by increasing the (DC) input current. Otherwise, the low-speed modulation effect in the data frame can lead to perceivable flicker. To avoid potential flicker perception and achieve an expected illumination level, the current should be increased from I to some value I_D, as illustrated in Fig. 4. The current increment ∆I = I_D − I reflects the increased power consumption. We can find the desired average current I_D that delivers the expected illumination level, by drawing a horizontal line through the static operation level, i.e., Φ_V(I), and find the crossing with the (timesharing) straight line between the two modulation levels Φ_V [(1 − α_rms)I] and Φ_V [(1 + α_rms)I]. Mathematically, we find I_D by solving a re-written version of Eq. (12) with I_D and Eq. (11) with the current I for unmodulated signals, as

\begin{array}{l} E [Φ_{V}] = a I^{2} + b I + c \\ = a I_{D}^{2} + b I_{D} + c + (2 a I_{D}^{2} + b I_{D}) E [s (t)] + a I_{D}^{2} E [s^{2} (t)] . \end{array}

Fig. 4 LED luminous flux nonlinearity.

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The current increment ∆I for the flux compensation is shown in Fig. 5 for various modulation indexes and flux levels. As we can see, a substantial current increment is needed for communication to maintain the same illumination level. In particular, the extra current could be more than 120mA when the modulation index is around 1 and the expected flux is around 120 lm. Many reported works mainly consider that the same average current would result into the same illumination level, which however is not accurate, since we need to compensate for luminous efficacy nonlinearity. As a rule of thumb, to avoid substantial extra power loss (∆I < 2mA), the product of the illumination level and modulation index should be less than 15.

Fig. 5 Current increment under different illumination level and modulation index.

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Then, the extra power used for flux compensation in LED is calculated by

Δ P_{L E D_f l u x} = P_{L E D} (I_{D}) - P_{L E D} (I),

where P_LED(.) is a monotonically increasing function with the input current, which will be discussed next section.

4. Extra power loss due to PINL and modulation

The current through a LED follows the diode equation

I (t) = I_{s} [\exp (\frac{q V (t)}{n k T}) - 1],

where I_s is the saturation current of the LED, q is the charge of an electron, V (t) is the voltage across the junction, n is the ideality factor (n = 1 to 2), k is Boltzmann constant and T is the temperature in Kelvin. At room temperature kT / q = 26mV. The current I_s highly depends on the LED type, where a typical example is I_s = 4.1 × 10⁻²⁴ with n = 1.4 [33]. Above a few milliamps, the resistance R_L in the LED needs to be considered, and it is typically a few ohms. The total voltage across LED becomes

V (t) = \frac{n k T}{q} \ln [\frac{I (t)}{I_{s}} + 1] + R_{L} I (t) .

In particular, for a constant current level, thus if I(t) = I_D, the power into the junction of LED plus its series resistance becomes

P_{L E D} (I_{D}) = \frac{n k T}{q} I_{D} \ln [\frac{I_{D}}{I_{s}} + 1] + R_{L} I_{D}^{2} .

Clearly, this is a convex function of I_D. Based on Jensen’s inequality, the convex relationship, between the dissipated electrical power and current, leads to the extra power loss when the LED is modulated for data transmission, as shown in Fig. 6, where the extra power for ΦINL is also illustrated.

Fig. 6 Two mechanisms of the extra power loss in the LED.

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In particular, ∆I covers the luminous flux compensation, corresponding to Eq. (14). In this section, the extra power loss due to PINL is quantified, comparing binary modulation (2-PAM) and continuous modulation (OFDM).

4.1. Binary modulation

When binary modulation is adapted with two current levels [1 − α_m, 1 + α_m]I_D and with equal probability, the extra power in the LED is expressed by [10]

\begin{array}{l} Δ P_{L E D_P A M} = \frac{1}{2} \frac{n k T}{q} I_{D} (1 + α_{m}) l n [\frac{I_{D} (1 + α_{m})}{I_{s}} + 1] \\ + \frac{1}{2} \frac{n k T}{q} I_{D} (1 - α_{m}) l n [\frac{I_{D} (1 - α_{m})}{I_{s}} + 1], \\ + \frac{1}{2} R_{L} (1 + α_{m})^{2} I_{D}^{2} + \frac{1}{2} R_{L} (1 - α_{m})^{2} I_{D}^{2} - P_{L E D} (I_{D}) \\ = \frac{n k T}{q} I_{D} \cdot B + R_{L} I_{D}^{2} α_{m}^{2} \end{array}

where we defined the constant

B = \frac{1 - α_{m}}{2} \ln (1 - α_{m}) + \frac{1 + α_{m}}{2} \ln (1 + α_{m}) .

For on-off keying, the factor B = ln 2. Note that, P_LED (I_D) is the power used for illumination when no data transmission is enabled, so we subtracted it in Eq. (18).

4.2. Continuous modulation

To achieve a closed-form expression of the extra power for a continuous, e.g. OFDM signal, the exponential diode equation can be approximated by linear or Taylor series expansion model [34, 35]. In other words, at a given current level I for lighting, we approximate Eq. (16) using the piecewise polynomial linear model

V (t) \approx \frac{n k T}{q} [\ln (\frac{I_{D}}{I_{s}} + 1) - \frac{I_{D}}{I_{D} + I_{s}}] + R_{L E D} i_{O F D M} (t),

where R_LED is the equivalent resistance of LED load, including the LED DC resistance R_L and the differential resistance nkT/(qI_D + I_s) derived from the diode equation.

The V-I curve and its approximation are plotted in Fig. 7. As we can see, the desired current level I_D is the point of tangency between the diode equation and approximation.

Fig. 7 LED V-I curves.

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Similarly for an OFDM transmitter, the extra power loss in the LED becomes

Δ P_{L E D_O F D M} = R_{L E D} α_{r m s}^{2} I_{D}^{2},

which also expresses the signal power.

5. Numerical results

This section presents numerical results for the extra power loss in the LED. We first compare the losses caused by two mechanisms, namely the ΦINL and PINL in the LED. We compare the total extra power loss for different illumination levels and modulation indexes, and we further validate our rule of thumb to avoid substantial extra power loss. At last, the efficacy is discussed.

5.1. Extra power loss of the LED

Due to the ΦINL and data modulation, extra power is lost for luminous flux compensation. It is quantified from Eq. (11)–(14) and (17), as shown in Fig. 8. The numerical results for the loss due to the PINL are also shown in Fig. 8, calculated by Eq. (18) and (21) for 2-PAM and OFDM, respectively. Figure 8 shows that the extra power loss owing to the ΦINL dominates the extra power consumption. As expected, a higher modulation index results in more extra power loss. Moreover, the extra loss due to PINL depends on the modulation waveform, in particular the OFDM needs more extra power than 2-PAM. The waveform-dependent power loss in LED along with that in the modulator results into different overall energy efficiencies for different modulations, as presented in [9].

Fig. 8 Extra power loss of two mechanisms in LED.

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For OFDM signal, the extra power loss for different illumination levels and modulation indexes is shown in Fig. 9. Results for 2-PAM are slightly better than OFDM. Figure 9 shows that the power loss varies from 2 to 640 mW. A system designer should make a tradeoff between the illumination level and the amount of energy for communication (α_rms), according to the extra power budget for data transmission through VLC. In other words, we believe that rather than expressing the Bit Error Rate (BER) for VLC versus the Signal to Noise Ratio (SNR), we preferably benchmark solution for their “extra” power loss, discounting the illumination power that is already available [10].

Fig. 9 Extra power loss in LED with OFDM.

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Higher illumination level tends to consume more extra energy if α_rms keep constant. This is due to the fact that the high illumination level needs high current which leads to the decrease of the efficacy.

In particular, the extra power directly dissipated as heating and thus the LED junction temperature also fluctuates in a burst data transmission. Our proposed design rule with extra ∆I < 2mA would enable the extra power loss to be less than 10 mW for an illumination system that consumes several Watts. This rule is effective to avoid substantial thermal fluctuation that has side effect on the LED lifetime.

5.2. LED efficacy for joint illumination and communication

Since the efficacy is a key performance indicator for the LED systems, we plot in Fig. 10 the efficacy when the data transmission is applied. For low (dimmed) illumination levels at 10 to 120 lumen, the efficacy decreases substantially even without current modulation. This is mainly due to the decrease of the internal quantum efficiency and the heating effect. Moreover, it is interesting to notice that for very low illumination levels, data modulation would have no substantial effect on the efficacy.

Fig. 10 Efficacy with different illumination and modulation index.

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Fig. 11 The first and second derivative of LED light output versus current.

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6. Conclusion

This work showed that extra power was inevitably consumed in an illumination LED when it was used for data transmission. The extra power loss appeared to come from two different mechanisms, namely the output luminous flux nonlinearity (ΦINL) and power consumption nonlinearity (PINL) in the LED. We studied that LED luminous flux was nonlinear with respect to the current due to the decrease of external quantum efficiency and heating effect. The heating effect can be modelled in a quasi-stationary way for high-speed data transmission due to the slow thermal time constant. We found the extra power due to PINL was modulation-dependent, for instance, OFDM consumes more extra power than 2-PAM. The fluctuation of light intensity due to ΦINL is relatively small (a few percent), but it can lead to visible flicker in burst mode transmission and lead to perceivable light output differences when only a fraction of the lamps in a ceiling are used for VLC. The extra power for light compensation and PINL are dissipated as extra heating (up to several degrees), which have an impact on LED life time. Thus, LED thermal design should be optimized by taking into account the modulation in emerging JIC systems.

Appendix A: Concavity of LED output light versus current with ABC model

Within the operation current range, the LED output light can be calculated to be an increasing and concave function of input current I.

Inserting Eqs. (2)–(7) into (1), we get

Φ_{V} (I) = 〈 E_{p} 〉 η_{L E R} η_{E x t} B {[m (I)]}^{2},

where m(I) is a function of current that meets

I = q V_{a c t i v e} (A m + B m^{2} + C m^{3}),

which is a cubic equation follows

C m^{3} + B m^{2} + A m - I / (q V_{a c t i v e}) = 0 .

The first derivative of the Equation (22) is

Φ_{V}^{'} (I) = 2 〈 E_{p} 〉 η_{L E R} η_{E x t} B m (I) m^{'} (I),

and the second derivative of the Equation (22) is

Φ_{V}^{″} (I) = 2 〈 E_{p} 〉 η_{L E R} η_{E x t} B {{[m^{'} (I)]}^{2} + m (I) m^{″} (I)} .

The values of Eq. (25) and (26) determine the monotonicity and convexity of Eq. (1).

Theorem Given a cubic equation, ax³ + bx² + cx + d = 0 with a ≠ 0, we define

D = b^{2} - 3 a c, χ = \frac{9 a b c - 2 b^{3} - 27 a^{2} d}{2 {(\sqrt{D})}^{3}}, κ = \sqrt[3]{χ + \sqrt{χ^{2} - 1}} .

When D ≠ 0, we have three roots as

{\begin{cases} x_{1} = \frac{- b + \sqrt{D} (κ + \frac{1}{κ})}{3 a} \\ x_{2, 3} = \frac{- b + \sqrt{D} (κ + \frac{1}{κ}) c o s \frac{2 π}{3}}{3 a} \pm i \frac{\sqrt{D} (κ - \frac{i}{κ}) s i n \frac{2 π}{3}}{3 a} \end{cases} .

For a typical GaInN/GaN LEDs, we have a = C = 5 × 10⁻²⁹cm⁶s⁻¹, b = B = 10⁻¹⁰cm³s⁻¹, c = A = 10⁷s⁻¹ and d = −I/(q × 3nm × 300µm × 300µm). Based on Eq. (27) and with I ∈ [1 1000]mA, we have D > 0 and χ > 1. Thus, the real value of carrier concentration m can be only expressed by

m (I) = \frac{- b + \sqrt{D} (κ (I) + \frac{1}{κ (I)})}{3 a} .

Thus, the first derivative of the Eq. (29) is

m^{'} (I) = \frac{\sqrt{D}}{3 a} κ^{'} (I) (1 - κ^{- 2} (I)),

and the second derivative of the Eq. (29) is

m^{″} (I) = \frac{\sqrt{D}}{3 a} {κ^{″} (I) (1 - κ^{- 2} (I)) + 2 κ^{- 3} (I) {[κ^{'} (I)]}^{2}} .

Since $κ (I) = \sqrt[3]{χ (I) + \sqrt{χ^{2} (I) - 1}}$ , the first derivative of κ(I) is

κ^{'} (I) = \frac{1}{3} {(χ (I) + \sqrt{χ^{2} (I) - 1})}^{- \frac{2}{3}} {χ^{'} (I) + χ (I) χ^{'} (I) {[χ^{2} (I) - 1]}^{- \frac{1}{2}}} .

and the second derivative of κ(I) is

\begin{array}{l} κ^{″} (I) = - \frac{2}{9} {(χ (I) + \sqrt{χ^{2} (I) - 1})}^{- \frac{5}{3}} {χ^{'} (I) + χ (I) χ^{'} (I) {[χ^{2} (I) - 1]}^{- \frac{1}{2}}}^{2} \\ + \frac{1}{3} {(χ (I) + \sqrt{χ^{2} (I) - 1})}^{- \frac{2}{3}} . \\ {χ^{″} (I) + {[χ^{2} (I) - 1]}^{- \frac{1}{2}} [[χ^{'} {(I)}^{2} + χ (I) χ^{″} (I)]] - {[χ (I) χ^{'} (I)]}^{2} {[χ^{2} (I) - 1]}^{- \frac{3}{2}}} . \end{array}

where χ″(I) = 0 and

χ^{'} (I) = \frac{27 a^{2}}{2 {(\sqrt{D})}^{3}} \frac{1}{q V_{a c t i v e}} .

Inserting Eq. (29)–(34) into Eq. (25)–(26), we have following figures with respect to the current. Thus, the LED output light is an increasing and concave function of input current I.Fig. 11

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Photometric Units		Radiometric Units
Luminous flux	lm	Radiant flux	W
Luminous intensity	cd^† = lm/sr^★	Radiant intensity	W/sr
Illuminance	lux = lm/m²	Irradiance	W/m²
Luminance	cd/m² = lm/sr/m²	Radiance	W/sr/m²

LED power consumption in joint illumination and communication system

Abstract

1. Introduction

2. LED luminous flux versus current nonlinearity (ΦINL)

2.1. Luminous flux of LED versus current

2.1.1. Luminous efficacy of radiation (LER)

2.1.2. Wall-plug efficiency (WPE)

2.2. Thermal effect on LED luminous flux

3. Extra power loss due to ΦINL and modulation

3.1. Luminous flux of a commercial white LED

3.2. Extra power for luminous flux compensation

4. Extra power loss due to PINL and modulation

4.1. Binary modulation

4.2. Continuous modulation

5. Numerical results

5.1. Extra power loss of the LED

5.2. LED efficacy for joint illumination and communication

6. Conclusion

Appendix A: Concavity of LED output light versus current with ABC model

References and links

Cited By

Figures (11)

Tables (1)

Equations (34)

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