application brief AB203A replaces AN11493A Advanced Electrical Design Models Table of Contents Diode Equation Forward Voltage Model 2 Derivation of Diode Model 2 Calculation of Diode Model Parameters 2 “Worstcase” Diode Models 3 Advanced Thermal Modeling Equations 4 Maximum Forward Current Vs. Ambient Temperature 4 ThermallyStabilized Luminous Flux 4 1 Diode Equation Forward Voltage Model Traditionally, the forward current versus forward The diode equation approximately models the voltage characteristics of a pn junction diode low current (> 1 µA) performance of an LED have been expressed mathematically with the emitter. However, at forward currents above a “Diode Equation” below. few mA, the ohmic losses must be included to accurately model the forward voltage. Thus, the diode equation becomes: Where: VF = forward voltage, V Where: IF = forward current, A R′S = internal series resistance, ohms n = ideality factor, 1 ≤ n ≤ 2 IO = reverse saturation current, A The values for the diode equation model can be T = temperature, °K calculated by using three test currents ( IF1, IF2, k = Boltzmann constant, 1.3805 x e23 joule/°K and IF3, such that IF1 < IF2 < IF3). Then, the values q = electron charge, 1.602 x e19 coulomb of n, IO, and R´S would generate an equation Note: at room temperature (25 °C), kT/q = that intercepts the forward characteristics of at 0.02569 V. these points: (IF1, VF1), (IF2, VF2), and (IF3, VF3) such as shown in Figure 3.1A. The equations for n, IO, and R′S are shown below: The reverse saturation current, IO, varies by several orders of magnitude over the automotive temperature range so this effect must be included to properly model the forward characteristics of the LED lamp over temperature. For forward voltage, VF, greater than a few hundred millivolts, the exponential term predominates and the equation can be re written as: 2 Figure 3.1A. Diode Equation Forward Voltage Model for LED Emitter (Semi-Log Scale). Figure 3.3A. Worst-Case Diode Equation Forward Voltage Models for LED Emitters. Note Graph Shows Forward Voltage Variations for LED Emitters from a Single Forward Voltage Category, Tested at IF = 70 mA. Figure 3.2A shows how the diode equation model compares to the forward current versus forward voltage curve shown in AB203, Figure 3.8. Since there is little correlation between the forward voltages at each test condition, there are eight possible worstcase permutations of forward voltage at the three test currents. As shown in Figure 3.3A, these eight combinations of forward voltage can be used with Equations #3.3A, #3.4A, and #3.5A to generate eight different diode equation forward voltage models (n, IO, and R′S): (IF1, VF1 min), (IF2, VF2 min), (IF3, VF3 min) ⇒ Figure 3.2A. Diode Equation Forward Voltage Model for HPWA-xHOO LED Emitter Shown in Figure 3.8 (Semi-Log Scale). (n LLL, IO LLL, R′S LLL) Using the values of the nominal forward voltage (IF1, VF1 min), (IF2, VF2 min), (IF3, VF3 max) ⇒ at the three test currents in Equations #3.3A, (n LLH, IO LLH, R′S LLH) #3.4A, and #3.5A would generate the typical diode equation forward voltage model. (IF1, VF1 min), (IF2, VF2 max), (IF3, VF3 min) ⇒ (n LHL, IO LHL, R′S LHL) (IF1, VF1 nom), (IF2, VF2 nom), (IF3, VF3 nom) ⇒ (n nom, IO nom, R′S nom) (IF1, VF1 min), (IF2, VF2 max), (IF3, VF3 max) ⇒ (n LHH, IO LHH, R′S LHH) (IF1, VF1 max), (IF2, VF2 min), (IF3, VF3 min) ⇒ (n HLL, IO HLL, R′S HLL) 3 (IF1, VF1 max), (IF2, VF2 min), (IF3, VF3 max) ⇒ VF max = VDIODE (IF, n HHH, IO HHH, R′S HHH) (n HLH, IO HLH, R′S HLH) = VDIODE (IF, n MAX, IO MAX, R′S MAX) (IF1, VF1 max), (IF2, VF2 max), (IF3, VF3 min) ⇒ For analyzing the operation of an electronic (n HHL, IO HHL, R′S HHL) circuit, it is convenient to be able to write the electrical forward characteristics of a component both in terms of forward voltage as a function of (IF1, VF1 max), (IF2, VF2 max), (IF3, VF3 max) ⇒ forward current as well as forward current as a (n HHH, IO HHH, R′S HHH) function of forward voltage. The difficulty in using the diode equation (with the R´S term) is that IF as In most situations, the worstcase range of a function of VF can only be solved through an forward current and forward voltage can be iterative process. In addition, the reverse estimated with only two permutations of the saturation current, IO, varies by several orders of diode equation model: magnitude over the automotive temperature range so this effect must be included to properly VF min = VDIODE (IF, n LLL, IO LLL, R′S LLL) model the forward characteristics of the LED = VDIODE (IF, n MIN, IO MIN, R′S MIN) emitter over temperature. Advanced Thermal Modeling Equations Note that, Equations #3.3 in AB203 or #3.6 in Figure 3.4A shows Equation #3.6A graphed as a AB203 can be combined with Equation #3.9 in function of TA and RθJA for an HPWAxH00 LED AB203 to derive the maximum DC forward emitter with a maximum expected forward current, IF MAX, versus ambient temperature, TA, voltage (i.e. VF = 2.67 V at 70 mA). Values of and thermal resistance, RθJA, shown in Figure 4 TJ MAX = 125 °C, VO HH = 1.83 V, and RS HH = 12 of the SuperFlux LED Data Sheet. ohms were used for Figure 3.4A. Note that Figure 3.4A is the same as Figure 4a, “HPWA TJ MAX ≅ TA + R θJA IF MAX VF MAX XX00 Maximum DC Forward Current vs. Ambient ≅ TA + R θJA IF MAX (VO HH + RS HHIF MAX ) Temperature” graph, in the SuperFlux LED Data Sheet. Or written as a standard quadratic equation: Equations #3.7 in AB203, #3.8 in AB203, and RθJARS HHIF MAX + RθJAVO HHIF MAX + TA – TJ MAX ≅ 0 2 #3.9 in AB203 can be combined together in different ways to model the luminous flux (or Thus, the positive root solution of IF MAX is equal luminous intensity) of LED emitters due to the to: effects of internal selfheating (i.e. RθJAPD) and ambient temperature. Equation #3.7A models the expected reduction in luminous flux due to internal selfheating compared to the 4 instantaneous luminous flux (i.e. at initial turn flux over temperature compared to the thermally on) when the LED emitter is driven at a constant stabilized luminous flux at test conditions of IF TEST, forward current at a constant ambient VF TEST, and RθJA TEST, at 25°C. Note for Equations temperature. Equation #3.8A models the #3.8A, #3.9A, and #3.10A, that for forward thermally stabilized luminous flux at any forward currents over 30 mA, m ≈ 1.0. current compared to the instantaneous luminous flux prior to heating at a specified forward current and a constant ambient temperature. Equation #3.9A models the thermally stabilized luminous flux at any forward current compared to the thermally stabilized luminous flux at test conditions of IF TEST, VF TEST, and RθJA TEST at a constant ambient temperature. A good example of an application for Equation #3.9A is the normalized luminous flux versus forward current graph shown in Figure 3 of the Figure 3.4A. Maximum DC Forward Current versus Ambient Temperature for HPWA-xxOO LED Emitter with Different System Thermal Resistances. SuperFlux LED Data Sheet. Finally, Equation #3.10A models the thermally stabilized luminous 5 This section discussed the key concepts of modeling the electrical, optical, and thermal performance of LED signal lights. Equation #3.6A is a combination of Equations #3.3 in AB203 and #3.8 in AB203 that can be used to calculate the maximum forward current as a function of ambient temperature and thermal resistance. Note that this equation models Figure 4 (Maximum DC Forward Current versus Ambient Temperature) on the SuperFlux LED Data Sheet. Equations #3.7A, #3.8A, #3.9A, and #3.10A Figure 3.5A. Thermally Stabilized Luminous Flux versus DC Forward Current for HPWx-xHOO LED Emitter with Different System Thermal Resistances. show different combinations of equations #3.7 in AB203, #3.8 in AB203, and #3.9 in AB203 in order to model various thermal effects on the Figure 3.5A shows Equation #3.9A graphed as light output of the emitter. Note that Equation a function of IF and RθJA for an HPWAxH00 LED #3.10A models Figure 3 (Normalized Luminous emitter with a nominal forward voltage (i.e., VF = Flux versus Forward Current) on the SuperFlux 2.25 V at 70 mA). Values of RθJA TEST = 200 LED Data Sheet. °C/W, m = 1.0, k = –0.0106, VO NOM = 1.802 V, and RS NOM = 6.4 ohms were used for Figure 3.5A. Note that Figure 3.5A is the same as Figure 3, “HPWA/HPWTxx00 Relative Luminous Flux vs. Forward Current” graph, in the SuperFlux LED Data Sheet. 6 Company Information Lumileds is a worldclass supplier of Light Emitting Diodes (LEDs) producing billions of LEDs annually. Lumileds is a fully integrated supplier, producing core LED material in all three base colors (Red, Green, Blue) and White. Lumileds has R&D development centers in San Jose, California and Best, The Netherlands. Production capabilities in San Jose, California and Malaysia. Lumileds is pioneering the highflux LED technology and bridging the gap between solid state LED technology and the lighting world. Lumileds is absolutely dedicated to bringing the best and brightest LED technology to enable new applications and markets in the Lighting world. LUMILEDS www.luxeon.com www.lumileds.com For technical assistance or the location of your nearest Lumileds sales office, call: 2002 Lumileds Lighting. All rights reserved. Lumileds Lighting is a joint venture between Agilent Technologies and Philips Lighting. Luxeon is a trademark of Lumileds Lighting, LLC. Product specifications are subject to change without notice. Publication No. AB203A (Nov 2002) 7 Worldwide: +1 408-435-6044 North America: +1 408 435 6044 Europe: +31 499 339 439 Asia: +65 6248 4759 Fax: 408 435 6855 Email us at [email protected] Lumileds Lighting, LLC 370 West Trimble Road San Jose, CA 95131