Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Identification and Eradication of “Bad Gamma” from Luminance Calculations

By Raymond Yeager, PE, LC, MBA

Sr. Project Engineer, Holophane Recently, the errata for RP-8-00 was issued that consisted of two parts: Adding fixtures toward the observer and adding six cycles of luminaires beyond the calculation area. The first part was needed to insure that fixtures were in the proximity of the observer and were included in the veiling luminance calculation. Without these additional fixtures, the veiling luminance in certain cases could be unrealistically low. However the second part, the addition of six cycles of luminaires beyond the calculation area, introduced a questionable effect into the luminance calculation. This effect was discovered in working on the bench-mark calculations for the Measurements and Calculations Subcommittee. This paper is written to describe the effect, convince readers that a problem exists, and offer a solution to eliminating the problem. Luminance Calculation The luminance calculation is shown in Equation 1. Equation 1: Luminance at a point on the road surface is the intensity striking the point multiplied by the r-factor obtained from the r-Table for a particular surface. This product is divided by the square of the mounting height and the multiplication factor that is build into the r-Tables. The r-factor obtained from the table is dependent upon two angles: Beta and Gamma. These angles are illustrated in RP-8-00 Figure A1. Beta is the angle having its vertex at the calculation point. One side of the angle is a line from the calculation point to the observer 83-meters away. The second side is a line extended through the calculation point from a luminaire whose intensity is striking the point, i.e., the luminaire whose luminance contribution to the point is being calculated. The angle gamma is the angle of incidence at the point.

2

),(HMF

rIL

××= γβ

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Figure A1 from RP-8-00: Beta and gamma angles used in the determination of the r-factor from the r-Table. (Used with permission IESNA) The r-Table is arranged by columns of beta values and rows of tangent gamma values. While gamma is the angle of incidence, the tangent of gamma is the horizontal distance from the calculation point to the point under the luminaire divided by the mounting height of the luminaire. The r-value at the intersection of the beta and tangent gamma is used in the Equation 1 calculation. Figure 2 shows the r-Table for an asphalt surface. The shaded areas indicate locations of non-zero r-values. The non-shaded area in the table indicates the values of zero. Figure 2 does not show the numerical r-values in the table. The purpose of showing the table is to emphasize where the non-zero (shaded or yellow area) and zero values (non-shaded area) are located. The limits of the table values should also be noted. R-values exist for tangent gamma values between 0 and 12. This corresponds to an angle of incidence of between zero and 85.2-degrees. No data is known for angles of incidence greater than 85.2-degrees where the tangent of gamma would be greater than 12.

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Figure 2. R-Table for Asphalt (R3). The yellow shaded area indicates the presence of non-zero values. Non-shaded area indicate the value of zero. Note the upper limit of the rows is tangent gamma = 12. In order to illustrate the problem associated with the second part of the errata, the bench mark calculation photometry and geometry will be used. The photometric distribution of the fixtures have 1000-cd in the hemisphere below the horizon. Luminaires overhang the road by 2 meters on one side. Along the road nine luminaires are located from negative 50 meters to positive 350 meters at 50 meters intervals. The calculation area is between the luminaires 2 and 3 located at 0 and 50 meters along the road. Note the six cycles of luminaires beyond the calculation are provided per part two of the errata. The geometry is illustrated in Figure 3. The road surface is R3. The calculation area has forty calculation points.

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Figure 3: Bench-mark calculation roadway plan and elevation. One example calculation point will be focused upon in the first part of the discussion to illustrate the result of adding fixtures 6 through 9 per part two of the errata. It is the point at 47.5 meters along the calculation area and 7 meters into the road. The beta and tangent gamma angles for the example calculation point are shown on Table 1. The location of the r-values on the table are illustrated in Figure 4 as red circles with cross marks.

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Table 1: Tangent Gamma and Beta values for each luminaire to the example calculation point. Note that the tangent of gamma for luminaires 6 through 9 exceed the maximum table value for tangent gamma. Luminaire Number

Tangent Gamma

Beta (degrees)

1 9.76 177.1 2 4.87 174.0 3 0.56 63.4 4 5.27 5.4 5 10.26 2.8 6 15.26 1.9 7 20.26 1.4 8 25.25 1.1 9 30.26 0.9

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Figure 4: R-value locations on the r-Table for each of the nine luminaires at the example calculation point. In Figure 4, the first five r-values fall inside the boundary of the r-Table. Their values can be imagined to trace a smooth curve, as illustrated by the red curve, through the table until the r-values are plotted for the last four luminaires. Note they are clumped together in the bottom row and move toward the left. These have tangent gamma values greater than the largest table value. The convention of selecting r-values for angles of incidence that exceed the table is to interpolate the r-values in the last row based upon the beta values. Note the imagined smooth curve is interrupted. R-values abruptly move to the left as the beta angle decreases due to the fixtures increased distance to the example calculation point. Table 2 shows the last rows of the r-Table. Note that as beta decreases, the r-values increase. When these inflated r-values are used in Equation 1 for fixtures 6 through 9, the calculated luminance increases as the fixture becomes more distant from the calculation point. Table 2: Last rows of R3 table. Note the r-values increase as the value of beta decreases, i.e., when fixture moves further away from a calculation point, the beta value decreases.

Beta Tan Gamma

0 2 5 10 15 20

11.0 59 26 7.1 2.6 1.9 1.8 11.5 56 24 6.3 2.4 1.8 0 12.0 53 22 5.6 2.1 1.8 0 Bad Gamma The use of the r-Table when the tangent gamma is greater than 12 introduces questionable results. There is no way of knowing what the r-values should be when the tangent of gamma exceeds 12. For this reason the term “bad gamma” is introduced for luminance calculations that use r-values in the calculation for tangent gamma values greater than 12. Table 3 shows the calculated luminance at the example point from each luminaire and the percentage of total luminance contributed to the luminance at that point. The effects of “bad gamma” cannot readily be seen for luminaire 6, but for the following three luminaires the effect is easy to note. Calculations indicate the luminance of the example calculation point increases as the distance between the calculation point and the luminaire increase. This is highly unlikely to occur in an actual installation. Hopefully at this point, the reader is convinced that using six cycles of luminaires beyond the calculation area results in tangent gamma values greater than the values existing on the r-Table for the example used. Additionally, the reader should agree that the luminance values calculated for tangent values greater than 12 are larger than what would be expected to occur in a physical installation.

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Table 3: Luminance contributed to the example calculation point from each luminaire. Note the unlikely increase in luminance for fixtures 7,8, and 9 which increase in distance in 50 meter increments from the example calculation point. Luminaire Number

Contributed Luminance (Cd/ Sq M)

Percent Luminance Contributed

1 0.0000 0% 2 0.0076 2% 3 0.2169 51% 4 0.0501 12% 5 0.0246 6% 6 0.0239 6% 7 0.0311 7% 8 0.0354 8% 9 0.0383 9%

Since roadway calculations are generally based upon an average of calculation values, attention will now be turned to investigating the effect of using r-values based upon a tangent gamma greater than 12 for the entire calculation area. Average Luminance In the benchmark calculation there are forty points and nine luminaires. In total there are 360 separate calculations of luminance made to determine the average luminance in the calculation area. Of these 360 calculations, 184 are made with bad gamma. Bad gamma first occurs for fixture number 5. Twenty-four of the forty calculations are made with the tangent of gamma greater than 12. All of the 160 remaining calculations for luminaires 6 through 9 have bad gamma. An average of 0.539 Cd/ Sq-M is calculated from the nine luminaires. Table 4 lists the average luminance from each luminaire. The luminaires 6 through 9, all having bad gamma, contribute 0.174 Cd/Sq-M to the average. The 24 bad gamma points from luminaire 5 contribute 0.037 Cd/Sq-M to the average. The portion of the average contributed by bad gamma is 0.211 Cd/Sq M. This amounts to 39% of the average as being questionably high. This is a direct result of using luminaires with bad gamma whose effect is compounded by applying part two of the errata.

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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Table 4. Average luminance from each of the nine fixtures. Luminaire Number

Contributed Average Luminance (Cd/ Sq M)

Percent Luminance Contributed

1 0.0000 0% 2 0.0057 10% 3 0.2069 39% 4 0.0616 12% 5 0.0378 7% 6 0.0402 7% 7 0.0431 8% 8 0.0449 8% 9 0.0462 9%

Discussion Availability of computer power has its drawbacks. One of the disadvantages is the ease in which we can be lured into believing that the more fixtures used in a calculation, the more accurate the calculation result will be. The luminance calculation is an example where this is not the case. The true limiting factor in the luminance calculation is the extent in which the data is presented in the r-Table. This data is limited by a tangent gamma of 12 (85.2-degrees angle of light incident at the calculation pooint). The four r-Tables are based upon many measurements of various road surfaces. The resulting tables are representative of a group of measurements. These four tables are the best information available regarding how dry road surfaces reflect light as a function of the beta and gamma angles. The limits of the tables should not be exceeded. Values used in the luminance calculation from the he r-Table trace a smooth curve through the r-Table until the angle of incidence reaches 85.2-degrees. At that point the smooth curve is interrupted and values used abruptly shifts toward the left as r-values are used from the last row for degreasing beta angles. This practice produces higher luminance values than are reasonable expected for sources greater than 12 mounting heights horizontal distance from the calculation point. While it would be possible to derive equation to predict the r-value for angles of incidence greater than 85.2-degrees, the temptation should be resisted. Such an attempt would violate a cardinal rule for using experimental results: “never extrapolate beyond the area where data is collected.” Recommendation Avoiding bad gamma begins by recognizing how it occurs. To date, the luminance calculation procedure has been based upon the number of luminaire cycles. However the equation for luminance is based upon angles determined from the geometry. The r-Tables themselves are organized in by angles. Photometric data files for sources express intensity as a function of lateral and horizontal angles. Recommendations for the

Identification and Eradication of “Bad Gamma” from Luminance Calculations

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calculation procedure should follow this lead. Instead of specifying a number of luminaire cycles that should be used in the luminance calculation, the test to exclude or include the luminance at a point from a luminaire should be based upon the limits of the angular data contained in the r-Tables. Part two of the errata should be changed to incorporate this rule. Luminance contribution to a point from a luminaire should be included until the angle of incidence to calculation points exceeds 85.2-degrees. After that occurrence, the luminance contribution from a source to a point should not be included in the luminance calculation of that point. Following this new rule, the limits of the r table would not be exceeded. There would be no extrapolation beyond the boundaries of the measured r values. The occurrence of bad gamma would be avoided.