A MacAdam ellipsis is a region on a chromaticity diagram which contains all colors which are indistinguishable to the average human eye from the color at the center of the ellipse. The contours of the ellipse represent the just noticeable differences of chromaticity. MacAdam ellipses help to show differences of chromaticity between two light sources and their thresholds can be used to determine chromaticity tolerance of the light emitted by lamps.
by Jerzy Pietrzykowski and Miko Przybyła
Chromaticity of light emitted by light sources is usually defined by chromaticity coordinates, which express the ratio of each of the tristimulus values to their sum. In CIE 1931 XYZ colour space, x, y chromaticity coordinates are calculated as follows:
x = X / (X + Y + Z)
y = Y / (X + Y + Z)
where: X, Y, Z – tristimulus values of a colour stimulus.
The values of chromaticity coordinates can be represented on a plane – we get a flat chart called chromaticity diagram where the points defined by chromaticity coordinates represent the chromaticity of colour stimuli.
|Fig. 1. Chromaticity diagram x,y CIE 1931 (provided by GL Optic).|
In many colorimetry applications determination of the chromaticity of a colour stimulus is not sufficient. It is necessary to determine the difference of chromaticity between the stimuli. Initially, there is a temptation to use the distance between the points of comparable colours to evaluate the difference of chromaticity between the stimuli. However, it turns out that the distance between points on the x, y chromaticity diagram is not the equivalent of the impression of chromaticity difference. This is reflected in the statement that a flat chromaticity diagram does not have the properties of a plane in Euclidean geometry and therefore the distance between the points which represent colour stimuli is not the invariable of the plane. This was confirmed by experiments conducted by MacAdam [1, 2], who proved that the places of geometric points on the chromaticity diagram lying at the threshold distance from a given point represent closed ellipsis shaped curves.
|Fig. 2. MacAdam ellipses on the chromaticity diagram x, y CIE 1931 magnified ten times (provided by GL Optic).|
It can be easily noticed that representation on a flat chromaticity diagram is not uniform. The same threshold values of chromaticity difference correspond to different distances from the central point, dependant on the direction in which we are moving, and on the part of the chromaticity diagram which is analyzed.
Different attempts were made to transform the x, y chromaticity diagram into a uniform chromaticity diagram, i.e. into a two dimensional diagram in which the coordinates are defined in such a way that equal distances represent most accurately equal degrees of chromaticity differentiation for the colour stimuli with the same luminance on the entire diagram. On a diagram like this MacAdam ellipses would be transformed into circles with the same radius r.
|Fig. 3. Uniform chromaticity diagram u’, v’ CIE 1976 (provided by GL Optic).|
Presently, u’, v’ CIE 1976 uniform chromaticity diagram is the best transformation . u’, v’ coordinates of the CIE 1976 chromaticity diagram are calculated from x, y chromaticity coordinates, using the formula:
u’ = 4x / (-2x + 12y + 3)
v’ = 9y / (-2x + 12y + 3)
The differences in the dimensions of MacAdam ellipses on the uniform chromaticity diagram are smaller and in some areas, e.g. next to the blackbody curve, they are almost like circles with almost the same radius. MacAdam ellipses can be used to determine the tolerance and difference of chromaticity. Assuming the threshold distance as the difference unit, which corresponds to the just noticeable differences of chromaticity, we can define the size of permitted changes of chromaticity. Six five-threshold MacAdam ellipses are used in PN-EN 60081:2002/A4:2010  to define chromaticity tolerance of the light of tube fluorescent lamps.
Table 1. Correlated colour temperature Tc and chromaticity coordinates u¢, v¢ and chromaticity coordinates x, y of six five-step MacAdam ellipses used in EN 60081.
|Fig. 4. MacAdam ellipses on the uniform chromaticity diagram u’ v’ CIE 1976, magnified ten times (provided by GL Optic).|
An alternative method, equivalent to the method using MacAdam ellipses, was developed at CIE . In this method u’, v’ CIE 1976 uniform chromaticity diagram is used, in which MacAdam ellipses assume a circle like shape and the radii of the circles close to the blackbody curve are practically the same.
As can be seen in Fig. 5, figures looking like circles are obtained after transformation, and their radius r is equal to 0.0055.
Table 2. Categories of tolerance of rated values of chromaticity coordinates recommended in standards IEC 62717:2014 and IEC 62722-2-1:2014.
Size of MacAdam ellipsis
|Category of chromaticity change|
|> 7-step||7 +||7 +|
Fig. 5. Transformation of five-step MacAdam ellipses adopted in EN 60081 to the uniform chromaticity diagram u’, v’ (provided by GL Optic). 
The equation of a circle with center uc’, vc’ and radius r can be described with equation: (u’– uc’)2 + (v’ – vc’)2 = r2. Since the radii of the circles obtained from five-threshold ellipses are equal to 0.0055, the radius of a one-threshold circle is equal to 0.0011 and, generally, the radius of an n-threshold circle is equal to 0.0011n.
The equation of an n-threshold circle can be described as:
(u’ – uc’)2 + (v’ – vc’)2 = (0,0011n)2
An extremely rapid development of while light emitting LED lamps often precedes work on the methods used to evaluate their functional characteristics. Two international standards, i.e. IEC 62717:2014  and 62722-2-1:20154 , were published at the end of 2014. They will also be adopted as European standards. The standards assume application of four categories of tolerance for rated values of chromaticity coordinates to evaluate LED lamps and LED luminaries. The tolerances are defined using MacAdam ellipses with a specified threshold (see Table 2). However, the standards do not explain whether the same MacAdam ellipses which are used in standard PN-EN 60081:2002 should be applied to LED lamps.
More information is given in standard EN 62612(EN 62612:2013) , which will soon be published. There is a clear reference there to standard  and an indication that MacAdam ellipses adopted for double-capped fluorescent lamps should also be used to determine chromaticity tolerance of LED lamps and LED luminaries.
 MacAdam D.L.: Visual sensitivities to colour differences in daylight. J. Opt. Soc. Am., vol. 32, 247 (1942).
 MacAdam D.L.: Specification of small chromaticity differences. J. Opt. Soc. Am., vol. 33, 18-26 (1943).
 ISO/CIE 11664-5:2015 Colorimetry. Part 5: CIE 1976 L*u*v* colour space and u’, v’ uniform chromaticity scale diagram.
 PN-EN 60081:2002/A4:2010 Świetlówki dwutrzonkowe. Wymagania funkcjonalne.
 CIE TN 001:2014 Technical Note: Chromaticity difference specification for light sources.
 IEC 62717:2014 LED modules for general lighting. Performance requirements.
 IEC 62722-2-1:2014 Luminaire performance. Part 2-1: Particular requirements for LED luminaires.
 PN-EN 62612:2015 Lampy samostatecznikowe LED do ogólnych celów oświetleniowych na napięcie zasilające > 50 V. Wymagania funkcjonalne.