Modeling Of Solar Radiation Spectrum Using Light-Emitting Diodes
The task of reproduction of the natural sunlight spectrum is quite topical. The developers of illumination systems for human life support, researchers in the area of physiology, zoology and biotechnology, developers of wireless atmospheric optical communication lines (FSO) are waiting for its solution . Chromaticity coordinates and value of color rendering index are the major characteristics used for the description of radiation spectrum of lighting fixture. In the papers [2, 3] the use of specified characteristics for the synthesis of spectral characteristic of light-emitting diode module was considered. For the lighting fixtures using light-emitting diodes in accordance with the standard , the requirements to permissible deviations of correlated color temperature are established with the specification of the thresholds of deviations of chromaticity coordinates .
In this article, the approximation process of solar radiation spectrum is considered with the aids of light-emitting diodes of 5–7 types: cool white (CW), warm white (WW), ultraviolet (UV), blue, green, yellow and red. The spectral characteristics of light-emitting diodes expressed in tabular and graphic forms were obtained in the laboratory of Lighting Faculty of N.P. Ogarev Mordovia State University. It was accepted to express them in relative units for the convenience of comparison (Fig. 1). The characteristics of used light-emitting diodes are given in the table.
Luminophors with cool white and warm white chromaticity used in the production of light-emitting diodes do not always provide the convenient conditions of illumination. The problem of color rendering improvement can be solved in two ways. Firstly, the needed light rendering can be achieved by the combination of different light-emitting diodes – white and color in one lighting fixture. Secondly, the same goal can be achieved with the use of complex combination of luminiferous mixture providing the radiation in all spectral regions. In order to increase the color rendering index, it is necessary to use the additive radiating in red-yellow spectral region in the mixture. In some cases, the additive providing the fluorescence in green spectral region is required. The share of blue component in the spectrum of light-emitting diodes with luminiferous coating is high in connection with incomplete absorption and transformation of exciting blue radiation of light-emitting diode crystal by luminophor. In some application scopes, the additive providing the radiation in ultraviolet spectral region is required.
Solution of the task aimed at the obtainment of set radiation spectrum for the lighting fixture based on light-emitting diodes can be found as a result of creation of the mathematical model, in which the radiation spectrums of all components of luminiferous mixture or spectrums of many-colored light-emitting diodes can be used in the capacity of source data. When modeling the radiation approximation process, the study of the procedure for color rendering improvement is performed for the fixture, the radiation spectrum of which is analogous to the spectrum of solar radiation.
The spectrum shown in Fig. 2 is accepted as the model of natural sunlight. It has the following spectral characteristics:
Spectrum boundaries: 345–800 nm;
Dimensions of data array: 92;
Maximum of spectral characteristic: 50.1725 relative units;
Wavelength of maximum of spectral characteristic: 480 nm.
Let us accept the following designations of the spectral characteristics of light-emitting diodes: f1 – cool white, f2 – warm white, f3 – ultraviolet, f4 – blue, f5 – green, f6 – yellow, f7 – red. Accordingly, the number of light-emitting diodes of every type can be designated as follows: n1 – cool white, n2 – warm white, n3 – ultraviolet, n4 – blue, n5 – green, n6 – yellow, n7 – red.
The task of reproduction of sunlight spectrum is the task, which is inverse in relation to the synthesis of light-emitting diode module. It was considered earlier by the authors [2, 3].
The following ratio is taken as the basis of modeling of daylight approximation process using the light-emitting diodes during the formation of radiation spectrum of the lighting fixture with light-emitting diode on the basis of set color temperature:
where FeSum is the light flux of the lighting fixture consisting of N (seven) types of light-emitting diodes; mi are the additional operational coefficients.
The spectral characteristics of light-emitting diodes and daylight are represented by the arrays of numerical data in estimations. At the same time, the dimensions of arrays are different, as a rule. Array dimensions were assumed to be equal to 46 for the spectral characteristics of light-emitting diodes, which have been used earlier. Therefore, it is necessary to reduce the dimensions to the same value for the solution of new task. In this case, the array of light-emitting diode spectrum is taken as the basis. Therefore, the task of interpolation of sunlight spectral characteristic was solved preliminarily. Practically, it means the ranking of sunlight spectral characteristic.
After the reduction to the common dimensions of spectral characteristic, we can proceed to the task of mean-square approximation . Taking into account the formula (1) the following expression can be written
where fi are the elements of daylight spectrum array; FeSumi refer to the elements of spectral characteristic of all light-emitting diodes; n refer to the array dimensions.
Then, on the basis of the expression (2) the mean-square error of approximation can be calculated using the commonly accepted expression for its evaluation of mean-square approximation :
where n refer to the array dimensions of spectral characteristic.
Due to the fact that number of light-emitting diodes is variable quantity or has integer value, the extremum conditions obtained from the expression (2) cannot be used. Therefore, it is necessary to use the procedure of search of the number of light-emitting diodes of every type with the following evaluation of mean-square approximation.
It is obvious that for the total search of all types of light-emitting diodes the algorithm with embedded loops with the nesting level of seven must be developed. Assuming the variation of number of light-emitting diodes from zero to some specific value, as a result, for the minimization of the expression (2) we will obtain extremely long-term operation of solution search.
In the model experiment it is accepted to vary the number of light-emitting diodes of every type from 0 to 400. Developed software algorithm reminds of the known optimization method – method of coordinate-wise descent. Number of light-emitting diodes of the relevant type plays the role of every coordinate. Specific user-defined function, in which the number of light-emitting diodes of one type varies and other values remain constant, is used for every type of light-emitting diode. The calculation of mean-square error is performed at every loop iteration in accordance with the formula (3). Then, the set of light-emitting diodes, which gives the lowest error, is selected. Value of error and number of light-emitting diodes of considered type are reintroduced into the main program where the analogous function for the light-emitting diodes of next type is sequentially called for. For the evaluation of approximation of resulting spectrum of light-emitting diodes to the spectrum of sunlight the reduced relative error is used and evaluated in percentage. The reduction was performed by the maximum of sunlight spectral characteristic. Thus, as a result of the variation loop of number of light-emitting diodes the lowest approximation error is calculated. Additionally, the spectral powers of radiation area were calculated by the valuation of the area below the curves of spectral characteristics. As a result of modeling and approximation the following results were obtained: radiation power of spectrum – 14345.1 relative units; radiation power of light-emitting diodes – 11049.7 relative units; reduced relative error of approximation of solar radiation spectrum – 21.4204%. Number of light-emitting diodes of every type in light-emitting diode source: cool white – 206; warm white – 376; ultraviolet – 4; blue – 0; green – 11; yellow – 0; red – 58; total number of light-emitting diodes – 655. Comparative diagram of characteristics is shown in Fig. 3.
In case when ultraviolet and then green light-emitting diodes do not participate in approximation, the relative reduced error increases up to 22.6571% and 24.4948% respectively. In these cases the errors turn out to be higher (comparatively with 21.4204%). Obviously, the additional introduction of light-emitting diodes with the specific color will cause the decrease of approximation error. As numerical experiments showed, the main share in resulting spectrum belongs to cool white and warm white light-emitting diodes.