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4. Mathematical conventions

   1. General

The basic conventions made in the mathematical procedures described in this standard are:

1. the luminaire is regarded as a point source;

2. light reflected from the surrounds and inter-reflected light is disregarded;

3. obstruction to the light from luminaires by trees and other objects is disregarded;

4. the atmospheric absorption is zero;

5. the road surface is flat and level and has uniform reflecting properties over the area considered;

6. the evaluation in /-tables and r-tables shall be obtained by linear interpolation.

In case of continuous lines of luminaires, generally at low mounting height, it is advisable to check whether the distance between the optical centre of each luminaire to the nearest point of the grid of calculation is greater than or equal to five times the length of the luminous area of a single luminaire. If this is not the case it might be necessary to simulate near-field photometry by fragmenting the luminaire into virtual point light sources of the same light distribution as the entire luminaire. The luminous flux of each virtual light source is an equal proportion of the total luminous flux for the luminaire.

The calculation results shall be presented in the form and with at least the number of digits given in the tables of requirements of EN 13201-2, shown in Table 2.

Table 2 — Number of decimal digits of the lighting requirements

4. Photometric data

   1. General

Photometric data for the light distribution of the luminaires used in the lighting installation are needed for calculating the lighting quality characteristics in this standard. These data are in the form of an intensity table (/-table) which gives the distribution of luminous intensity emitted by the luminaire in all relevant directions. When luminance calculations are to be made, photometric data for the light reflecting properties of the road surface are required in the form of an r-table.

Interpolation is needed in using both these tables to enable values to be estimated for directions between the tabulated angles.

For calculations made in accordance with this standard, an intensity table (/-table) that describes the behaviour of the luminaire with the required accuracy by the aim of calculation shall be used. This 1- table shall be prepared in accordance with EN 13032-1. The coordinate system used for road lighting luminaires is the C-planes system, shown in Figure 1. For floodlight installations, the intensity distribution measured in the B-planes system may be accepted if the calculation program can transfer the intensity values in the C-planes system. In Figure 1, the luminaire is shown at its tilt during measurement.

Luminous intensity shall be expressed in candelas.

The luminous flux used in calculation shall be declared in the calculation report.

Unless specific conditions are mentioned in the calculation report, the luminous flux used shall be that of the light source mentioned in the data sheet of the luminaire.

If the luminous intensity table is given in candelas per kilolumen (cd-klnr¹), its values shall be converted in candelas, considering the luminous flux of all the light sources in the luminaire.

Key

1. luminaire at tilt during measurement

2. longitudinal direction

3. vertical direction

4. direction of luminous intensity

Figure 1 — Orientation of С, у coordinate system in relation to longitudinal direction of
carriageway

Maximum angular intervals stipulated in this standard have been selected to give acceptable levels of interpolation accuracy.

In the (C, y) system of coordinates, luminous intensities shall be provided at the angular intervals stated below.

For all luminaires the angular intervals in vertical planes (y) shall at most be 2,5° from 0° to 180°. In azimuth the intervals shall be varied according to the symmetry of the light distribution from the luminaire as follows:

1. luminaires with no symmetry: the intervals shall at most be 5°, starting at 0°, when the luminaire is at its tilt during measurement, and ending at 355°;

2. luminaires with nominal symmetry about the C = 270° - 90° plane: the intervals shall at most be 5°, starting at 270°, when the luminaire is at its tilt during measurement, and ending at 90°;

3. luminaires with nominal symmetry about the C = 270° - 90° and C = 0°- 180° planes: the intervals shall at most be 5°, starting at 0°, when the luminaire is at its tilt during measurement, and ending at 90°;

4. luminaires with nominally the same light distribution in all C-planes: only one representative set of measurements in a vertical (C-plane) is needed.

Where standards for specific luminaire typologies exist and prescribe improved angular intervals these shall be applied.

The angular intervals stated above shall be reduced in case of a great gradient variation of consecutive luminous intensities.

NOTE In that case, it is the role of photometric laboratories to provide the /-table with relevant reduced angular intervals defined from the angles included in the photometric file.

5.2.2 Linear interpolation in the /-table

To estimate the luminous intensity /(C, y) in the direction (C, y), it is necessary to interpolate between four values of luminous intensity lying closest to the direction, see Figure 2 and Figure 3.

G i*1

Figure 2 — Angles required for linear interpolation of luminous intensity

Figure 3 — Angles required for linear interpolation of luminous intensity (from Figure 2 but showing intensity on z-axis in perspective)

For this purpose, the following formulae or mathematically equivalent formulae shall be used: Interpolation on C angles

Xj)~ 4c_m, /j) _ C — C_m

AOn /j) Оп+1~СгП where

/(Cm, /j) indicates the intensity in column number m and row number j of the /-table, and so on for the other similar symbols;

C is the azimuth, measured about the first photometric axis;

у is the vertical angle measured from the first photometric axis;

j, m, m+1 are integers indicating the number of the column or row in the /-table.

From which:

/(C, _Zj)= /(c_m, _Zj)+ ■ (/(C_m+1, /,)-/(c_m, _Zj)) (4)

^cm+1

Similarly:

^m+l'/j+lJ-^Cm'/j+l) ^Cm+1-^C

m

From which:

/j+i)-7(c_m, /j+i)+ ■ (4c_m+1, /j₊i) /(c_m,yj._ti))

^cm+1 ^cm

At last, interpolation on y:

^/(C</)-/(C/_j) _ /-/j

/(c,_Zj+1)-/(c,_Zj)

From which, finally:

/(c, _z) = i(c, _Zj)+-^{Z Zj} ■ (/(c, _Zj+1)-z(c, _Zj))

I

(6)

(7)

(8)

n these formulae interpolation is first carried out in the C half planes, and then in the у cones. If desired this procedure can be reversed (that is, the interpolation is first carried out in the у cones followed by the C half planes) and the same result obtained.

5.3 The r-table

5.3.1 The r-table format

Road surface reflection data shall be expressed in terms of the reduced luminance coefficient at the angular intervals and in the directions given in Table 3 for the angles (3 and £ indicated in Figure 4.

Generally in r-tables the values are given multiplied by the factor 10⁴. In this case, for calculation purpose, they shall be divided by 10⁴.

Table 3 gives the minimum number of angular directions at which the reduced luminance coefficient shall be specified for luminaires placed at heights, above the road surface, higher than 2 m.

For luminaires of the lighting installation placed at heights, above the road surface, less than or equal to 2 m, Annex В suggests the extended set of angular directions for rvalues.

Key

H mounting height of the luminaire

P observed point

PN normal at P to the road surface

Q photometric centre of the luminaire

QT vertical passing through the photometric centre of the luminaire

ST longitudinal direction

Oh geometrical projection of the observer’s eye to the ground

/andy scalar components of the vector TP (evaluation of tan <p)

P angle between the oriented traces of vertical planes in the horizontal plane of the road surface:

• vertical plane passing through the point of observation and containing P

• vertical plane containing P and passing through the luminaire.

c angle of light incidence at P

a angle of observation

• p installation azimuth

1. luminaire

2. light path

3. observer (0 is the position of the eye of the observer]

Figure 4 — Angular relationships for luminaire at tilt during measurement,
observer, and point of observationTable 3 — Angular intervals and directions to be used in collecting road surface reflection data