The principle and basic calculation of CRI

It is well known that color table and color rendering are two important quantities that reflect the color of light sources. Light sources with different spectral power distributions can have the same color table, but the color rendering properties of several light sources with the same color table may be completely different. Therefore, only the combination of color table and color rendering can fully reflect the color characteristics of the light source. Using light sources with different spectral power distributions to illuminate objects will produce different color perceptions. The nature of the light source that determines the color perception of the illuminated object is called color rendering.

1. Basic concepts and calculation formulas
1.1 RGB system
Definition of three primary colors: All colors of light can be formed by mixing certain three kinds of monochromatic light in a certain proportion, but none of these three kinds of monochromatic light can be produced by mixing the other two kinds of light, these three kinds of monochromatic light are called for the three primary colors. In 1931, CIE stipulated that the three primary colors of the RGB system are red (R): 700nm, green (G): 546nm, and blue (B): 435.8nm. In the RGB system, equal-energy white light can be obtained by mixing according to the following formula:

FR : FG : FB =1: 4.5907 : 0.0601 (1-1)

So the color mixing result can be expressed mathematically as

IFI = 1R + 4.5907G + 0.0601B (1-2)

IFI represents the luminous flux after color mixing, and R, G, B are called tristimulus values.
In order to facilitate calculation and more intuitively understand the color characteristics of light sources, the introduction of

These three quantities are called chromaticity coordinates or color coordinates. Because r+g+b=1, as long as the two values in the color coordinates are known, the third one can be obtained, that is, the chromaticity can be represented by a plane diagram, which is the chromaticity diagram. The calculation of the tristimulus value can be calculated by the following form

where P is the spectral power distribution of the light source, and r, g, and b are the 1931 CIE-RGB system standard chromaticity observer spectral tristimulus values, respectively.

1.2 XYZ system
Negative values of primary colors are required to match certain visible spectrum colors in the RGB system, and are inconvenient to use, so the International Commission on Illumination adopted a new color system, the 1931 CIE XYZ system. According to the 1931 CIE RGB system, the system envisages three primary colors (X), (Y), (Z) to represent the original three primary colors (R), (G), (B), XYZ system tristimulus values and RGB system tristimulus values the relationship is as follows

The chromaticity coordinates in the XYZ system are determined by

1.3 CIE1960 Uniform Color Space
In an x-y chromaticity diagram, equal distances of different parts do not represent visually equal chromaticity differences. To overcome this shortcoming, McAdam introduced a new uniform chromaticity u-v chromaticity diagram. The relationship between the uniform chromaticity coordinates u, v and x, y as below:

Since the color adaptation of the light source K to be measured is different from that of the reference illuminant r, the chromaticity coordinates of the light source to be measured must be adjusted to the chromaticity coordinates of the reference illuminant, and this adjustment of color coordinates becomes the adaptive color shift. Calculate the color shift using the following formula:

C, d of the light source to be measured, Cr, dr of the reference illuminator, and Ci, di of each color sample under the light source to be measured are calculated by the following formula:

1.4 Calculation of color difference
To calculate the color difference ΔEi, first convert the chromaticity data into 1964 unified space coordinates, and use the following formula:

In this way, the following formula can be used to calculate the color difference of the same color sample i when the light source to be measured and the reference illuminator are used respectively.

1.5 Color rendering index
The color rendering index Ri of a certain color sample i becomes the special color rendering index, which is calculated by the following formula.

The general color rendering index Ra is calculated by the arithmetic average of 8 special color rendering indexes (i=1, 2, …, 8)

2. Case analysis
Scan a self-ballasted fluorescent lamp with a spectral analysis system to obtain its spectral power distribution. The data are shown in the following table.

Calculated using formula (1-4): R=89.291, G=118.229, B=115.919
Then calculate the tristimulus values in the XYZ system by formula (1-5): X=585.272, Y=639.013, Z=655.166
The chromaticity coordinates of the XYZ system are obtained by formula (1-6): x=0.3115, y=0.3402
Using formula (1-7), the chromaticity data is converted from (X, Y, Z, x, y) values under CIE1931 to 1960 (u, v) coordinates: u=0.1929, v=0.3159

From the measured spectral power distribution and the spectral brightness factor of the test colors 1-8, calculate the chromaticity coordinates of the test colors No. 1-8 under the light source, and obtain the corresponding ui, vi according to (1-7).

Calculate C=2.0506, d=2.0825, and Ci, di from the formula (1-9), and then calculate the color coordinates ui’ and vi’ under the light source after the color adaptation adjustment by the formula (1-8).

Calculate the ‘ * Ui , ‘ * Vi and ‘ * Wi*’ of the color sample under the light source from equation (1-10).

• Calculate the color difference ΔEi of each color sample under the light source and the reference illuminator from the formula (1-11)
• Calculate the special color rendering index Ri of each color sample from (1-12)
• Calculate the average color rendering index Ra=79.9 from (1-13)

3. Solution for testing color rendering index by LISUN
3.1 Option 1 (suitable for laboratory customers or LED factory customers who requiring relatively high test accuracy)
LPCE-2 Integrating Sphere Spectroradiometer LED Testing System is for single LEDs and LED lighting products light measurement. LED’s quality should be tested by checking its photometric, colorimetric and electrical parameters. According to CIE 177, CIE84,  CIE-13.3, IES LM-79-19, Optical-Engineering-49-3-033602, COMMISSION DELEGATED REGULATION (EU) 2019/2015, IESNA LM-63-2 and ANSI-C78.377, it recommends to using an array spectroradiometer with an integrating sphere to test SSL products. The LPCE-2 system is applied with LMS-9000C High Precision CCD Spectroradiometer or LMS-9500C Scientific Grade CCD Spectroradiometer, and A molding integrating sphere with holder base. This sphere is more round and the test result is more accruacy than the traditional integrating sphere.

3.2 Option 2 (suitable for small LED factories or customers with insufficient budget and not required for high precision requirements)
LPCE-3 is a CCD Spectroradiometer Integrating Sphere Compact System for LED Testing. It is suitable for single LED and LED luminaires’ photometric, colorimetric and electrical measurement. The measured data meets the requirements of CIE 177, CIE84,  CIE-13.3, COMMISSION DELEGATED REGULATION (EU) 2019/2015, IES LM-79-19, Optical-Engineering-49-3-033602, IESNA LM-63-2, ANSI-C78.377 and GB standards.

4. Test Report

5. Conclusion
The degree to which the light source presents the natural primary color of the object is the color rendering index of the light source. There is no doubt that the color rendering index is a very important quantity to measure the color characteristics of the light source. At a time when computers are highly popular, the calculation of the color rendering index has been written into the computer program along with the spectrometer, which can be read directly, but it is still necessary to understand the calculation process of the color rendering index.

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Tags：LPCE-2 (LMS-9000) , LPCE-3