Optica Applicata, Vol. XLI, No. 3, 2011 Analysis of unifom illumination system with impefect Lambetian LEDs JIAJIE TAN 1, 2, KECHENG YANG 1*, MIN XIA 1, YING YANG 1 1 Wuhan National Laboatoy fo Optoelectonics, School of Optoelectonics Science and Engineeing, Huazhong Univesity of Science and Technology, Wuhan, China, 430074 2 Depatment of Physics and Electonic Infomation Science, Hengyang Nomal Univesity, Hengyang, Hunan, China, 421008 * Coesponding autho: kcyang@mail.hust.edu.cn This pape offes a novel algoithm to bette design LEDs aays fo illumination systems. Fist, the illuminance distibution of LED aays is studied though theoetical analysis. Second, we pesent the algoithm citeion and steps. And finally, we use compute to simulate and veify this method. The esults show that the illumination unifomity is significantly affected by the spacing of each light in the LED aay. The analysis method pesented hee in this thesis can be usefully applied in LED in the illumination engineeing. Keywods: light emitting diode, unifom illumination, impefect Lambetian distibution. 1. Intoduction With the apid development of solid state lighting technologies, the light-emitting diodes (LEDs) ae designed to geneate 10 120 lumens pe LED with an efficiency that supasses incandescent and fluoescent lamps. So the LED will play a key ole in the futue lighting and widely eplace incandescent and fluoescent lamps in indoo illumination system [1 3]. Though a high-powe LED poduces up to 120 lumens pe device, a single LED still cannot povide sufficient illumination fo indoo illumination system whee an illumination of 300 1500 lux is needed. Seveal LEDs must be mounted on the panels to obtain pactical luminous powe and get a unifom illumination [4, 5]. To get unifom illumination, the optical modeling of the illumination system must be equied. Refeence [6] demonstates how the light souce model is chaacteized by geomety, luminous popeties and intensity distibution. As is known, the intensity distibution type of LED includes Lambetian-type, batwing, and side emitting [7]. In othe wods, thee ae some LEDs with impefect Lambetian souce in the eal wold applications. In the liteatue [3 5, 8, 9] we find designs of illumination systems using
508 J. TAN et al. the Lambetian model o a cosine function of the viewing angle, so they can yield definite maximal condition of unifom illumination, and get definite analytical expessions. Since those light-emitting models cannot be depicted with pecision expession, such as the LUXEON K2 geen, blue, ed and white LED [10], when we use LED aay fo lighting, how to design the LED lighting system is the main pupose of this pape. We analyze the featues of the luminous intensity distibution of LEDs in ode to each unifom illumination and we pesent an algoithm to aange the LEDs in aay. We take geat effots to impove the lighting efficiency and educe the cost of encapsulation and of seconday optical design, and this design is moe pactical and easy-to-assemble. The pape is aanged as follows. The LED illumination pinciple is pesented in Section 2. In Section 3, we pesent the algoithm citeion and the algoithm steps. The simulation esults ae given in Section 4. Finally, in Section 5 we daw ou conclusions. 2. LED illumination pinciple 2.1. Assumption With LED lighting, if the light of the LED has not been assigned, the light on the taget is just a ound spot which poduces glae to human eyes, theefoe, unifom illumination plays an impotant ole in lighting system designs. Unifom illumination depends on the following factos: the LED intensity distibution, the distance between the obseve and the adiation suface, the incident angle, the eflectivity and the shape of taget [4]. Hee we only analyze the illumination of a flat taget. The LED lighting can be catalogued into nea-, mid-, and fa-field accoding to the distance between souce and taget [11, 12]. When the LED illumination ange is about 5 times lage than its maximum size, we think that the ange belongs to fa-field illumination aeas, theefoe the LED light souce hee is assumed as a point souce. This pape always takes fa-field illumination as its eseach object, and the LED aays ae placed in the same plane. 2.2. Pactical model of LED The ideal LED optical model is a pefect Lambetian, which means its intensity is popotional to the cosine of the viewing angle, namely I(θ )=I 0 cos(θ ), I 0 (lm/s) is the intensity of the LED axis, θ is the viewing angle, I(θ ) (lm/s) is the intensity at the viewing angle. Aticles [3 5] used accuately expessed models, thus the luminance of the taget can be expessed as E(, θ )=E 0 ()cos m (θ ), whee θ is the viewing angle, the ode m is elated to θ 1/2, given by m = ln(2)/ln(θ 1/2 ), the m depends on the elative position of the LED emitting egion fom the cuvatue cente of the spheical encapsulant. E(, θ ) (lm/m 2 ) is the illuminance of the taget plane, E 0 () is the illuminance (lm/m 2 ) on the axis at distance fom the LED.
Analysis of unifom illumination system... 509 Relative density [a. u.] 0.2 White LED Relative density [a. u.] 0.2 Red LED 100 50 0 50 100 Angula displacement [deg] 100 50 0 50 100 Angula displacement [deg] Relative density [a. u.] 0.2 Uppe bound Blue LED Relative density [a. u.] 0.2 Lowe bound Blue LED 100 50 0 50 100 Angula displacement [deg] 100 50 0 50 100 Angula displacement [deg] Fig. 1. Spatial luminous patten fo LUXEON K2 LED [10]. In some special scenaios, LED luminous intensity distibution I(θ ) cannot be descibed by the above-mentioned fomula. Such a kind of LED is LUMILED of LUXEON K2 seies shown in Fig. 1 [10]. Hee, the figues, fom left to ight, fom top to bottom ae espectively called model A, model B, model C and model D. We can know that only the angula distibution of model A is appoximately depicted by the equation I(θ )=I 0 cos(θ ), and the othes cannot. Consequently, it is difficult to design an LED aay to get unifom illumination. When we design the layout of LED aay, eithe we use its datasheet povided by the manufactue, o we measue the luminous intensity distibution model by ouselves. Finally, we calculate optimal displacement paametes in the LED aay. 2.3. Analysis of the illumination of the taget To descibe the pocess of this design method moe clealy, fo example, we establish the Catesian coodinates shown in Fig. 2. The LED is set in coodinate oigin O, and the optical axis of the LED is pependicula to plane A. Point P in plane A has coodinates x, y and z. In ode to show the amount of light at any point P, take a small aea da aound point P, and assume its luminous flux is dφ (it has solid angle dω), so
510 J. TAN et al. O X Y θ dω Z X θ Plane A X P(x, y, z) da Fig. 2. Schematic of a single LED illumination. the luminous intensity is I =dφ /dω. The elationship of the solid angle, the illuminant aea and the illuminant distance can be expessed by dω =(da/ 2 )cos(θ ), da is the tiny aea, is the distance fom the light souce to illuminant suface, θ stands fo the angle of incidence. We evaluate the luminous flux by dφ =(I / 2 )cos(θ )da, so the illuminance in aea A is E(, θ )=dφ /da = I cos(θ )/ 2. Then we tanslate E(, θ ) into Catesian coodinates (x, y, z). The illuminance at point P(x, y, z) can then be given by Exyz (,, ) = zi ------------------------------------------------- x 2 + y 2 + z 2 3 2 Fo ou scenaio, the LED aay is fixed on the ceiling plane XOY (ceiling plane), if the i-th LED is located in (x i, y i, 0), the illuminance of the i-th LED is given by (1) E i ( x, y, z) = zi -------------------------------------------------------------------------------- ( x x i ) 2 + ( y y i ) 2 + z 2 3 2 (2) The total illuminance E is given by the sum of the illuminance fo N LEDs E = N i = 1 E i (3) E i ( x, y, z) = N i = 1 zi -------------------------------------------------------------------------------- ( x x i ) 2 + ( y y i ) 2 + z 2 3 2 (4) I is the function of θ, the elation between θ and x, y, z, x i, y i can be expessed by θ = zi acos ------------------------------------------------------------------- ( x x i ) 2 + ( y y i ) 2 + z 2 (5)
Analysis of unifom illumination system... 511 2.4. Layout of LED aay We adopt LUXEON K2 LED fo designing lighting system. The oom size is 5.0 m 5.0 m 3.0 m, and the distance between the ceiling and the taget plane is 2.15 m. LEDs fixed in the ceiling of the oom ae aanged as shown in Fig. 3. We take thee types of configuations, that is, the linea LED aay, the squae LED aay, and the cicula LED aay, as in [4, 5]. Because the linea aay is too simple and impactical, we will not conside this case. Futhemoe, the LED s luminous intensity is isotopic, consequently LED-to-LED has the same spacing. The specific aangement method is given by Fig. 3. In the configuation of the squae LED aay, thee is the same LED-to-LED sepaation d, and the LEDs ae placed along axis x and axis y. In the configuation of the cicula LED aay, the LEDs ae put in the cicumfeence with only one LED in the cente of the cicle, and the adius is. Finally, we discuss how many LEDs should be installed in the ceiling as in Fig. 3. Geneally, illuminance of lights is standadized by ISO (Intenational Oganization fo Standadization). In this standad illuminance of 300 1500 lux is equied fo a wok office, but the typical luminous powe of LUXEON K2 LED (model A) is only 130 lm fom the datasheet. Theefoe, 64 LEDs can povide sufficient illuminance in this system. If we use othe type LED (models B, C, D), we equie 81 LEDs to fom the aay. 3. Algoithm 3.1. Analysis of the LED sepaation of LED aays Befoe exploing the LED lighting aay, we fist study illuminance distibution of two LEDs, and then extend the method to the LED aay. In ode to obtain a moe satisfactoy unifom lighting, we should conside the sepaation between LEDs. In this case, the illuminance is given by the sum of the illuminance fo two LEDs as follows Exyz (,, ) zi ( x 1, y 1, 0) zi ---------------------------------------------------------------------------------- ( x 2, y 2, 0) = + ---------------------------------------------------------------------------------- ( x x 1 ) 2 + ( y y 1 ) 2 + z 2 3 2 ( x x 2 ) 2 + ( y y 2 ) 2 + z 2 3 2 (6) d y a y b d x x LED aay LED aay Fig. 3. Configuation of LED aay: squae (a), cicula (b).
512 J. TAN et al. Nomalized illuminance [a. u.] 0.9 0.7 0.3 d = 2.04 m 0.2 4 2 0 2 4 x [m] Nomalized illuminance [a. u.] 0.9 0.7 0.3 0.2 d = 1.50 m 0.1 4 2 0 2 4 x [m] Fig. 4. Simulation of illumination with d = 2.04 m and d =1.50m. The distance between two LEDs is d = ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2, we can adjust d to get a maximally flat illumination in cental egion based on Spaow s citeion and efeences [4, 5]. To explain it clealy, let us take Fig. 4 fo example. When the sepaation of the LEDs d = 2.04 m, the illumination unifomity of the cental egion is inadequate. If the d continues to incease, the unifomity of illumination becomes even wose. When d = 1.50 m, the illumination of the cental egion has the optimum illumination unifomity. We educe d so that we can each the optimum unifomity of illumination nea the cental egion. Theefoe, we can follow cetain ules and algoithms to find the optimum paamete to achieve optimal lighting. 3.2. Algoithm citeion Now, we ae going to discuss the citeia of the algoithm. We know that LED spatial light intensity distibution is isotopic as shown in Fig. 1, and the sum of illuminance of the taget is a convex function, hence thee must exist an optimal lighting space. So, the LED aay we design compises the squae LED aay and the cicula LED aay. Fist, we detemine the maximum sepaation d max of the squae LED aay and the maximum adius max of the cicula LED aay accoding to the oom size and the numbe of LEDs. Second, we calculate the illuminance of the taget, and divide the oom into 256 256 equal-spaced gids, which we define hee as the gid element. We calculate the gid cente illumination, and use it to epesent the aveage illumination of the element. Algoithm citeia 1: E 1 E ----------------------- 2 5% E (7)
Analysis of unifom illumination system... 513 Hee, E = (E 1 + E 2 )/2, E 1, E 2 ae the illuminances of two neighboing gid elements. If E 1 and E 2 meet Eq. (7), we deem that the two gid elements ae unifom illumination. Algoithm citeia 2 the aea suounded by adjacent gid elements whose illuminance satisfies citeia 1 is the lagest. 3.3. Algoithm steps Unde citeia 3.2, the algoithm steps ae as follows. Step 1. Get data of the LED light intensity (angula displacement) fom eithe the datasheet povided by manufactues o the data measued by instument. Step 2. Because the data obtained in step 1 is limited, we utilize spline intepolation to detemine its spatial distibution. Step 3. Discete the eflecting suface into 256 256 equal inteval gids along the x, y axes. Step 4. To initialize the LED aay paamete o d, let = 0 λ 0, d = d 0 λd 0, 1 λ 1, 0 = max, and d 0 = d max. The initial values 0 and d 0 ae elated to the oom size and the numbe of LEDs. Step 5. Evaluate the illuminance of evey gid based on Eqs. (2) and (3). If the illuminance of the neighboing gid element satisfies Eq. (7), count the total aea as unifom illuminance. Step 6. Compaison of the size of the unifom illumination aea. If the aea is not the lagest, then we change the paamete λ, and epeat step 4, step 5 and step 6, until the aea is the lagest. If the unifom illumination aea does not incease again, then exit. 4. Simulation esult 4.1. LED squae aay Now, we take the squae aay of LEDs fo example. In this case, we tansfom Eq. (4) into (8). The total illuminance fo N N LED aay is given by N N Exyz (,, ) = zi ( x x ij ) 2 + ( y y ij ) 2 + z 2 j = 1 i = 1 3 2 (8) Hee, I is the function of x, y, z, x ij, y ij. N is not consideed hee odd no even numbe. The LED aay is fomed by 8 8. The maximal conditions ae difficult to find out, so we esot to a numeical solution. Fist, we stat with d = d max = 0.714 m, by esticting the oom size and the numbe of LEDs. Then we use the algoithm mentioned in Section 3 to seach the spacing of the LEDs, finally find the optimal
514 J. TAN et al. Nomalized iadiation [%] 0.9 0.7 3 2 1 0 1 Y [m] 2 3 3 2 1 0 1 2 X [m] Fig. 5. LED squae aay illuminance of the taget. 3 2.5 2.0 1.5 Y [m] 0.0 1.5 2.0 2.5 3 2 1 0 1 X [m] 2 3 Fig. 6. LED squae aay contou map of illuminance. Nomalized iadiation [%] 0.9 0.7 0.3 0.1 3 2 1 0 1 Y [m] 2 3 3 2 1 0 1 X [m] 2 3 Fig. 7. LED cicula aay illuminance of taget.
Analysis of unifom illumination system... 515 2.5 2.0 1.5 Y [m] 0.0 1.5 2.0 2.5 2 1 0 1 X [m] 2 3 Fig. 8. LED cicula aay contou map of illuminance. paametes d opt = 0.704 m. Figue 5 shows the 8 8 nomalized illuminance distibution map. Figue 6 shows the illumination contou map of the optimal paamete d opt, and its unifom illumination aea is the lagest. 4.2. LED cicula aay Next, we will discuss the case of the cicula aay which is a popula configuation of LED lamps in pactical application. We install one in the cente, and the othe 63 in the cicle. The illumination is as follows N 1 Exyz (,, ) = zi x i = 1 2πi ------------ cos N 1 2 2πi + y ------------ cos N 1 2 + z 2 3 2 + zi x 2 + y 2 + z 2 3 2 + (9) The adius is initialized with = 2.5 m, and is changed step by step based on the algoithm. When the adius is = 1.454 m, the lagest aea of unifom illumination appeas. The illumination is shown in Fig. 7, and the illumination contou map in Fig. 8. 4.3. Simulation esults of othe models The simulation esults of othe models ae not listed one by one as in Sections 4.1 and 4.2. We list all the optimal simulation paametes in the Table. 5. Conclusions In this pape, we analyze the fa-field LED illumination aays, and popose an optimal algoithm to design a LED aay using the impefect Lambetian LED model.
516 J. TAN et al. T a b l e. Simulation of optimal paametes. Squae aay Cicula aay Type of model The numbe of LEDs d [m] The numbe of LEDs [m] Model A 64 0.704 64 1.454 Model B 81 15 81 1.719 Model C 81 48 81 1.328 Model D 81 0.332 81 1.406 The simulation esults show that the algoithm stuctue is moe easonable, and the esult is bette. Compaed with peviously published studies, the methods in this study focus on esolving LED aay lamps whose optical intensity models ae not pefect Lambetian. In summay, this method is not only applicable to Lambetian, but also applicable to the impefect Lambetian. It is the vey chaacteistic of this algoithm. The algoithm fo a eseach design pesented hee is accuate, but it has the shotcoming of being time-consuming and having lage computations. The next step should educe the computation and computing time. Beside these, thee ae some inteesting LED aays which need moe complex calculations which have not been consideed hee, such as the multi-laye concentic stuctue, in which the aays ae not in the same plane. Although this method has its limitations, it is still an impoved and supeio design method. Acknowledgements The authos would like to thank thei laboatoy colleagues and anonymous eviewes fo thei caeful eview of the pape. Refeences [1] STEIGERWALD D.A., BHAT J.C., COLLINS D., FLETCHER R.M., HOLCOMB M.O., LUDOWISE M.J., MARTIN P.S., RUDAZ S.L., Illumination with solid state lighting technology, IEEE Jounal of Selected Topics in Quantum Electonics 8(2), 2002, pp. 310 320. [2] HOLCOMB M.O., MUELLER-MACH R., MUELLER G.O., COLLINS D., FLETCHER R.M., STEIGERWALD D.A., EBERLE S., LIM Y.K., KRAMES M., The LED lightbulb: ae we thee yet? Pogess and challenges fo solid state illumination, OSA Tends in Optics and Photonics Seies 88, 2003, pp. 240 243. [3] YANG H., BERGMANS J.W.M., SCHENK T.C.W., LINNARTZ J.-P.M.G., RIETMAN R., An analytical model fo the illuminance distibution of a powe LED, Optics Expess 16(26), 2008, pp. 21641 21646. [4] MORENO I., AVENDAÑO-ALEJO M., TZONCHEV R.I., Designing light-emitting diode aays fo unifom nea-field iadiance, Applied Optics 45(10), 2006, pp. 2265 2272. [5] MORENO I., TZONCHEV R.I., Effects on illumination unifomity due to dilution on aays of LEDs, Poceedings of SPIE 5529, 2004, pp. 268 275. [6] JONGEWAARD M.P., Guide to selecting the appopiate type of light souce model, Poceedings of SPIE 4775, 2002, pp. 86 98. [7] MORENO I., SUN C.-C., Modeling the adiation patten of LEDs, Optics Expess 16(3), 2008, pp. 1808 1819.
Analysis of unifom illumination system... 517 [8] BENNAHMIAS M., ARIK E., YU K., VOLOSCHENKO D., CHUA K., PRADHAN R., FORRESTER T., JANNSON T., Modeling of non-lambetian souces in lighting applications, Poceedings of SPIE 6669, 2007, p. 66691A. [9] TRYKA S., Spheical object in adiation field fom a point souce, Optics Expess 12(3), 2004, pp. 512 517. [10] Lumileds. http://www.lumileds.com/pdfs/ds51.pdf. [11] MORENO I., SUN C.-C., IVANOV R., Fa-field condition fo light-emitting diode aays, Applied Optics 48(6), 2009, pp.1190 1197. [12] SUN C.-C., CHIEN W.-T., MORENO I., HSIEH C.-C. LO Y.-C., Analysis of the fa-field egion of LEDs, Optics Expess 17(16), 2009, pp. 13918 13927. Received August 19, 2010 in evised fom Novembe 15, 2010