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application brief AB20 3A
replaces AN1149 3A
Advanced Electrical Design Models
Table of Contents
Diode Equation Forward Voltage Model Derivation of Diode Model Calculation of Diode Model Parameters "Worst case" Diode Models Advanced Thermal Modeling Equations Maximum Forward Current Vs. Ambient Temperature Thermally Stabilized Luminous Flux 2 2 2 3 4 4 4
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Diode Equation Forward Voltage Model
Traditionally, the forward current versus forward voltage characteristics of a p n junction diode have been expressed mathematically with the "Diode Equation" below. The diode equation approximately models the low current (> 1 A) performance of an LED emitter. However, at forward currents above a few mA, the ohmic losses must be included to accurately model the forward voltage. Thus, the diode equation becomes:
Where: VF = forward voltage, V IF = forward current, A n = ideality factor, 1 n 2 IO = reverse saturation current, A T = temperature, K k = Boltzmann constant, 1.3805 x e 23 joule/K q = electron charge, 1.602 x e 19 coulomb Note: at room temperature (25 C), kT/q = 0.02569 V. 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: The values for the diode equation model can be calculated by using three test currents ( IF1, IF2, and IF3, such that IF1 < IF2 < IF3). Then, the values of n, IO, and RS would generate an equation that intercepts the forward characteristics of at these points: (IF1, VF1), (IF2, VF2), and (IF3, VF3) such as shown in Figure 3.1A. The equations for n, IO, and RS are shown below: Where: RS = internal series resistance, ohms
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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 AB20 3, Figure 3.8.
Since there is little correlation between the forward voltages at each test condition, there are eight possible worst case 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 RS):
Figure 3.2A. Diode Equation Forward Voltage Model for HPWA-xHOO LED Emitter Shown in Figure 3.8 (Semi-Log Scale).
(IF1, VF1 min), (IF2, VF2 min), (IF3, VF3 min) (n LLL, IO LLL, RS LLL) (IF1, VF1 min), (IF2, VF2 min), (IF3, VF3 max) (n LLH, IO LLH, RS LLH) (IF1, VF1 min), (IF2, VF2 max), (IF3, VF3 min) (n LHL, IO LHL, RS LHL) (IF1, VF1 min), (IF2, VF2 max), (IF3, VF3 max) (n LHH, IO LHH, RS LHH) (IF1, VF1 max), (IF2, VF2 min), (IF3, VF3 min) (n HLL, IO HLL, RS HLL)
Using the values of the nominal forward voltage at the three test currents in Equations #3.3A, #3.4A, and #3.5A would generate the typical diode equation forward voltage model. (IF1, VF1 nom), (IF2, VF2 nom), (IF3, VF3 nom) (n nom, IO nom, RS nom)
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(IF1, VF1 max), (IF2, VF2 min), (IF3, VF3 max) (n HLH, IO HLH, RS HLH) (IF1, VF1 max), (IF2, VF2 max), (IF3, VF3 min) (n HHL, IO HHL, RS HHL) (IF1, VF1 max), (IF2, VF2 max), (IF3, VF3 max) (n HHH, IO HHH, RS HHH)
VF max = VDIODE (IF, n HHH, IO HHH, RS HHH) = VDIODE (IF, n MAX, IO MAX, RS MAX)
For analyzing the operation of an electronic 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 forward current as well as forward current as a function of forward voltage. The difficulty in using the diode equation (with the RS term) is that IF as a function of VF can only be solved through an iterative process. In addition, 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 emitter over temperature.
In most situations, the worst case range of forward current and forward voltage can be estimated with only two permutations of the diode equation model: VF min = VDIODE (IF, n LLL, IO LLL, RS LLL) = VDIODE (IF, n MIN, IO MIN, RS MIN)
Advanced Thermal Modeling Equations
Note that, Equations #3.3 in AB20 3 or #3.6 in AB20 3 can be combined with Equation #3.9 in AB20 3 to derive the maximum DC forward current, IF MAX, versus ambient temperature, TA, and thermal resistance, RJA, shown in Figure 4 of the SuperFlux LED Data Sheet. TJ MAX TA + R JA IF MAX VF MAX TA + R JA IF MAX (VO HH + RS HHIF MAX ) Or written as a standard quadratic equation: Equations #3.7 in AB20 3, #3.8 in AB20 3, and
RJARS HHIF MAX + RJAVO HHIF MAX + TA - TJ MAX 0
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Figure 3.4A shows Equation #3.6A graphed as a function of TA and RJA for an HPWA xH00 LED emitter with a maximum expected forward voltage (i.e. VF = 2.67 V at 70 mA). Values of TJ MAX = 125 C, VO HH = 1.83 V, and RS HH = 12 ohms were used for Figure 3.4A. Note that Figure 3.4A is the same as Figure 4a, "HPWA XX00 Maximum DC Forward Current vs. Ambient Temperature" graph, in the SuperFlux LED Data Sheet.
#3.9 in AB20 3 can be combined together in different ways to model the luminous flux (or luminous intensity) of LED emitters due to the effects of internal self heating (i.e. RJAPD) and ambient temperature. Equation #3.7A models the expected reduction in luminous flux due to internal self heating compared to the
Thus, the positive root solution of IF MAX is equal to:
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instantaneous luminous flux (i.e. at initial turn on) when the LED emitter is driven at a constant forward current at a constant ambient temperature. Equation #3.8A models the thermally stabilized luminous flux at any forward 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 RJA 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 SuperFlux LED Data Sheet. Finally, Equation #3.10A models the thermally stabilized luminous
flux over temperature compared to the thermally stabilized luminous flux at test conditions of IF TEST, VF TEST, and RJA TEST, at 25C. Note for Equations #3.8A, #3.9A, and #3.10A, that for forward currents over 30 mA, m 1.0.
Figure 3.4A. Maximum DC Forward Current versus Ambient Temperature for HPWA-xxOO LED Emitter with Different System Thermal Resistances.
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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 AB20 3 and #3.8 in AB20 3 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 AB20 3, #3.8 in AB20 3, and #3.9 in AB20 3 in order to model various thermal effects on the light output of the emitter. Note that Equation #3.10A models Figure 3 (Normalized Luminous Flux versus Forward Current) on the SuperFlux LED Data Sheet.
Figure 3.5A shows Equation #3.9A graphed as a function of IF and RJA for an HPWA xH00 LED emitter with a nominal forward voltage (i.e., VF = 2.25 V at 70 mA). Values of RJA TEST = 200
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/HPWT xx00 Relative Luminous Flux vs. Forward Current" graph, in the SuperFlux LED Data Sheet.
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Company Information
Lumileds is a world class 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 high flux 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: 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 info@lumileds.com Lumileds Lighting, LLC 370 West Trimble Road San Jose, CA 95131
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. AB20 3A (Nov 2002)
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