L06 - Advanced Calculations

1. Overview (KM)

Spectroradiometry

2. Photometry (PR)

Distribution photometry

Photometric Integrators

3. Calculation of Direct Component (PR)

Point sources

Linear sources

Regular areas sources

Irregular area sources

4. Calculation/prediction of reflected and inter-reflected component (GD)

Irregular and regular arrays of luminaires

Domed and coffered ceilings

5. Road lighting (PR)

Reflectance tables

Luminance calculations

Uniformity calculation

Small target visibility

6. Glare (GD)

Glare systems (TM10, CIE, UGR)

7. Field Measurement of Light (PR)

Illuminance and luminance measurement in the field

8. Design for seeing (KM)

Cubic illuminance

L06 (1). Measurement Techniques - Spectroradiometry

Overview (L&L pp. 67-8)

Spectroradiometry is the measurement and analysis of the spectral distributions of lamp sources. The measurements are made using a spectroradiometer. This consists of a monochromator, a photometer and some means of converting the output readings into relative spectral power distributions

The function of the monochromator is to disperse the homogeneous radiation from the light source into a spectrum with provision for isolating discrete bands of energy with known bandwidth. The components of a monochromator are

· Entrance slit

· Collimating device

· Dispersing element – splits up incoming light into its spectral distribution

· Exit slit

The monochromator scans over the spectrum produced by the dispersing element. A typical monochromator has 0.05nm ± 0.2nm resolution, and can scan 22.5 nm/s. A common means of performing spectral power distribution measurements is to take readings of the spectrum at 5nm intervals with a instrument bandwidth of 5nm over the wavelength range required. This is a compromise giving reasonable resolution over the spectrum while keeping down the number of readings.

For accurate results, it is necessary to irradiate the optical system and the photocell uniformly. This can be achieved by irradiating a pressed barium sulphate plate (a nominally perfect diffuser), and mounting the plate so that the entrance slot is uniformly irradiated. The detector is a silicon photodiode with a known spectral response curve.

Diffraction Gratings

In the M300E monochromator the dispersing device is a diffraction grating. This has the advantage over prisms of linear dispersion, but they suffer from the problem of overlapping orders; however this can be overcome with filtering. In addition, gratings are capable of better resolution of the spectra than are prisms.

Light as wave and a particle

Light is electromagnetic radiation, and thus can be regarded as either a wave of electromagnetic energy, or a stream of photons. The main observations in support of light as a wave motion are interference and diffraction effects.

A diffraction grating consists of many narrow equally spaced lines ruled onto glass or metal. Diffraction occurs at each slit and in the same direction each slit creates a similar diffraction pattern.

Let d be the separation between two lines in the grating. Then wavelets from two corresponding points P and Q on two adjacent slits will have a path difference of d sin q. We want to find out where interference between these two adjacent wavelets will be constructive. These wavelets will superimpose if the path difference is a multiple of the wavelength l of the incident light, i.e.

When n = 0, q = 0, and constructive interference appears in line with the incident beam direction. This is known as the zeroth order fringe. It is the brightest fringe.

For a given wavelength of light, l, a bright fringe appears at an angle q where sin q = nl/d i.e. the angle of interference is proportion to the wavelength. If white light is incident on the grating each colour will produce its maximum interference at a different angle, and a spectrum will be formed. When n = 0, all wavelengths form maxima at 0 degrees, and a white maxima will be formed.

When n = 1, the first order spectrum is produced. All colours of light are produced with violet being deflected least. By measuring the spectral power at different angular positions, we can determine the relative spectral power of diffferent parts of the spectrum.

Higher order spectra can be produced, and they tend to spread out the spectrum over a large angular range, allowing more precise measurements to be produced. Unfortunately, these orders can overlap. Filters can be used to overcome these overlapping orders.

Calibration

Calibration is carried out by taking measurements on a suitable lamp of known spectral power distribution under the same conditions as the test lamp. The CL2 absolute spectral irradiance standard lamp is such a lamp. It has the following properties,

· 12V 100W

· Grit blasted

· Aged at 8.3A for 150W (???)

· Tested by a test house, who return the following properties

· Spectral irradiance curve

· Chromaticity co-ordinates

· Illuminance at 0.5m

· Correlated colour temperature

For each point in the spectrum the ratio of the measured current from the detector for the test lamp and the calibrated source is multiplied by the known spectral power distribution of the standard source. This gives the spectral power distribution of the unknown lamp at this point.

The spectral power distribution is the first step in calculating a number of important parameters of a lamp. The relative spectral power distribution is suitable for calculating the chromaticity co-ordinates and the CIE colour rendering index R_a.

Calculation of chromaticity co-ordinates

In order to calculate the chromaticity co-ordinates, we need to calculate the tri-stimulus co-ordinates X, Y, Z. We obtain these from the spectral power distribution f_l using the following equations,

where

Integration limits include the whole visible spectrum. The chromaticity co-ordinates can be calculated from

; ;

where x + y + z =1.

L06 (2). Measurement Techniques – Photometry

Distribution photometry

General principles

The unit of intensity is the Candela and is one lumen per steradian. For photometric data of luminaires, we usually work with candelas per thousand lamp lumens. Intensity is measured indirectly – the photometer measures illuminance, and the inverse square law is used to measure intensity, since the distance is known (I = d ²E). As photometry is dependant on the the inverse square law, it is important that the length of the luminaire is short relative to the path length to the cell (ratio of path length to length of longest side should be greater than 1:5)

Goniophotometers

A goniophotometer is used to measure the intensity distribution of a luminaire. It uses a photocell which has a spectral sensitivity which corresponds to the V(l) curve. We use calibrated lamps, which have been run for 100 hours to allow them time to stabilize.

The specification for all the equipment used to photometer luminaires is laid down in BS 5489 to ensure the accuracy of results. BS 5225 gives details of measurement techniques for obtaining photometric data for luminaires, including laboratory conditions, and procedures for simple luminance and illuminance measures.

Whenever using photometric information the test reference number should be noted. This allows tracking back to the original luminaire and test conditions if a installed scheme fails to work properly.

BS 5225 lays down the following requirements for tolerances on goniophotometers,

· Axis of rotation to the vertical – ±0.5⁰

· Axis of rotation of the arm moving the mirror system or photocell to the horizontal – ±0.5⁰

· The two axes should intersect at the effective centre of the photocell – 10mm

· Angles indicated correctly - ±0.5⁰

· Electricity supply

· Ample current handling capability

· The harmonic content should be below 3%

· To hold voltage to better than values in requirement tables below

· Frequency stability to be introduced by European legislation

· Air temperature and movement

· Measured to ±0.5⁰

· The air around the luminaire to be draught free

· Temperature controlled to values in requirements table below

Requirements Table

Lamp type in luminaire	Filament Lamp	Fluorescent Lamp	Other Discharge
Voltage	Rated Volts ±0.2%	Rated Volts ±0.5%	Rated Volts ±0.5%
Temperature	25⁰C ±5⁰C	25⁰C ±1⁰C	25⁰C ±2⁰C

In summary, the characteristics required for the goniophotometer to produce accurate results are,

· Minimize stray light by light baffles

· Control air temperature

· Avoid errors due to

· Non-linear photocell

· Voltage fluctuations

· Mirror sagging

· Mirror reflectance

The goniophotometer maintains the luminaire in the desired attitude. You measure in the position of luminaire use. To achieve the desired optical path length, mirrors are often used; the path should be at least five times the largest dimension of the light fitting. The mirrors should be optically flat, and have a uniform surface finish.

Angular systems

Indoor and street lighting luminaires are generally measured in their normal operating position and the co-ordinate system should take account of this by having the reference point directly below the luminaire. The C-gamma system of photometric angles is the most common and used for all interior and most exterior ones too. The angle C represents the plane in azimuth, and g represents the plane in elevation. The axis of the system passes through the centre of the luminaire. g are measured from the nadir.

There are two other angular systems B-Beta and A-Alpha. The B-beta system is sometimes used for floodlights and road lanterns. The A-alpha system is used for automobile headlights. Converting from one system to another is a simple matter of geometry.

Floodlights, projectors, automobile headlights and all other concentrating beams have to be tested at much greater distances than indoor luminaires. The detector should be far enough away so that it ‘sees’ the whole of the reflector flashed with light. For normal floodlighting, as used in stadium lighting, 33m is sufficient, but by international agreement automobile headlamps are tested at 25m. One consequence of the long path length is that the photocell has to be fixed, and the luminaire rotated.

Measurement of light

The illuminance meter used to measure light should meet BS 667 – Illuminance meters, although cosine correction is not needed, since the cell is always normal to the light source. It is important that the photopic response of the cell is correct, since the luminaire may colour the light of the lamp, changing its spectral distribution.

Calculation of flux from intensity

Intensity is flux per unit solid angle, so by multiplying intensity values by the solid angles over which they are valid flux may be calculated. The solid angle between two angles of elevation measured from the nadir, q₁ and q₂, is . From this it is easy to calculate the flux for each intensity value, and thus the LOR of the luminaire.

If a sphere is divided up into a series of zones of solid equal angle then the calculation of total flux becomes easier. Russell angles are chosen so that they are in the centre of solid zones. The formula to calculate the Russell angles is,

where n is the number of angles in the set and k is an integer between 0 and N/2 –1.

Calibration of results

There are two methods of calibration,

· One can scan the bare lamp on the goniophotometer, calculate its LOR and work out a scale factor by which to multiply the results of the luminaire scan

· The LOR of the luminaire may be measured in an integrator and then when the total flux of the luminaire is calculated, a factor can be used to make the two agree.

Photometric Integrators

An integrator is used to compare the flux of lamps and to measure the LORs of luminaires.

Operating principles

The inner surface is finished in a matt white paint with a surface reflectance factor of between 75% and 85%. The light from the source is bounced around the sphere and is substantially evenly distributed over the surface. In a sphere it can be shown that the indirect light reaching any point is independent of the position of that point. Hence the inter-reflected illuminance is constant over the whole surface.

Components of an integrator

· Integrator should be spherical

· Possible to use other shape, for example a cube

· If a non-spherical shape is used, checks should be made to ensure uniform reflection properties

· Sphere diameter should be at least ten times the size of any compact lamps, 2 times the length of linear lamps and 1.5 times the maximum dimension of any luminaire used.

· Translucent window should illuminate the photocell, and it should be flat to the edge of the sphere.

· The material of this window should has transmission properties which are independent of the angle if incident light.

· The cell should closely follow the spectral response of the CIE photopic curve, in order to deal correctly with coloured light sources.

· A screen is provided to stop light falling directly on the photocell head.

· Screen size should be no bigger than needed

· Additional screen provided above auxiliary lamp

· A GLS is used as the auxilary lamp

· It is used to assess the self-absorbtion properties of the test lamp.

· The surface finish should be both uniform and achromatic

· Integrators should be repainted regularly as dirt accumulates preferentially on the lower surfaces.

Calculation of LOR for a luminaire

where

A – luminaire in and on; aux lamp off

B – luminaire in and off; aux lamp on

C – luminaire in and on; aux lamp off

D – luminaire in and off; aux lamp on

L06 (5). Road lighting

Lighting for different road classifications

Main roads

On main roads, the needs of the driver are most important. The task of the driver is to locate obstructions in the road ahead. To do this the lighting needs to be bright enough to make small objects with small contrast visible. Over the range 0.5 cd/m²to 2 cd/m², it is estimated that an increase in luminance of 1 cd/m² leads to a 35% decrease in road accidents.

Thus it is luminance that is important to the driver so main road lighting is based on the calculation of luminance. Illuminance is not a good measure of road lighting as the luminance pattern is dependent on the road surface as well as the lighting. The brightness of the area surrounding the road is also important; if it is too bright then it will raise the drivers adaptation level, if it is too dark then it will be hard to see pedestrians off the road who may be waiting to cross.

Minor roads

The lighting on minor roads is aimed much more at the needs of pedestrians. The main objectives in lighting minor roads are

· Detection of obstacles

· Facial recognition

· Visual orientation

· Comfort

For detection of obstacles by pedestrians it has been found that average illuminances in the range 3 to 10 lux is needed with a minimum illuminance value of 1 lux. Facial recognition is closely correlated with semi-cylindrical illuminance. Whilst S-C illuminance is not used in the British Standard, the levels of horizontal illuminance are such that in general requirements for facial recognition are met.

Visual orientation implies the ability to read the names on the road signs and the numbers on the houses. This aspect is not considered in the British Standard although some engineers try to place lanterns so that the road signs are illuminated.

Comfort is again largely controlled by the restriction of glare. As pedestrians move more slowly through an area they do not suffer from disability glare to the same extent. Discomfort glare may be a problem so the intensities of lanterns near the horizontal must be limited and luminaires should not be mounted at eye height.

Measures used in road lighting

Disability glare

Light scatter in the eye causes Disability Glare. This causes extra light to fall on the image of the object. The veiling luminance reduces the contrast that objects have against their background as it contributes to both the object and background luminance; C_EFF = L_B/ L_B + L_V

Consider an object that is just visible in the absence of glare. When glare is introduced it will no longer be visible. To make this object visible again, it will be necessary to increase the contrast. The percentage by which the contrast must be increased to make the object just visible is known as the Threshold Increment (TI). The CIE has arrived at a standard way to calculate the TI that is valid for luminances in the range 0.5 to 5.0 cd/m^-2,

Visual comfort

Visual comfort is of course related to average illuminance. However it is also related to the longitudinal uniformity of the lighting (U_L),

Discomfort glare

This is due to high luminance sources in the visual field. In most installations if the disability glare requirements are met, then there is no problem with discomfort glare.

Overall uniformity

As the adaptation state of the driver’s vision is set by the average value of the road luminance it is important that the darkest point on the road is controlled or else objects in some parts of the road will not be visible. The measure used to control the uniformity of road luminance is overall uniformity (U_O),

Surround ratio

Lighting the area either side of the road is important in allowing drivers to see pedestrians and other road users who may be about to cross the road. The lighting in a zone 5m either side of the road should be bright enough so that pedestrians can be seen, but not so bright as to change the adaptation state of the driver.

The surround ratio (S_R) is defined as the ratio of the average illuminance on a 5m strip adjacent to the road compared with the average illuminance on road. Note that for motorways and other roads where pedestrians are excluded it is not necessary to use the surround ratios.

Visual guidance

A row of street lanterns can provide useful visual cues to the path of the road by day and night. Care should be taken in siting columns at junctions and on bends so that a false impression of the road layout is not given. Indicators such as changing from low to high pressure sodium at junctions as a visual cue is also useful.

Characterization of road surfaces

It is common to characterize road surfaces by a number of parameters,

· The bulk reflection factor q⁰

· The specularity factor S₁,

where r(0,2) is the reduced luminance co-efficient for b=0 and tan g =2 and r(0,2) is the reduced luminance co-efficient for b=0 and tan g =2.

Road have categorizations, with the most common road type in the UK, asphalt, being categorized C2.

Luminance calculation

The luminance of the road is given by where q is a function of a, b and g,

· a - angle from eye to position of observation along road (typically 1⁰)

· b - angle from normal to road at position of measurement to observer

· g - angle from lantern head to position of measurement

Road reflections are normally stored as R-tables where . This makes the luminance calculation much easier as the cos³ is embodied within the table. The table is indexed by b and tan g (where tan g =1 is position below 1^st lantern, etc.)

Veiling luminance calculation

Veiling luminance is calculated as

where q is the angle between where the eye is looking and the direction of the location of the lantern. In most lighting standards the following is used for the calculation,

· Driver one quarter across the road

· Driver 1.5m high

· Driver looking down at 1⁰

· First lantern used in calculation is at 20⁰

· In the BS it is required to include the contribution from all lanterns within 500m of the observer, however the calculation in any row of lanterns may be terminated when the contribution of a given lantern is below 2% of the total.

British Standard requirements

There are two standards used namely,

· BS5489 Part 2 – Traffic Routes

· BS5489 Part 10 – Motorways

The lighting requirements in part 2 are,

Category	Average Illuminance	Overall Uniformity	Longitudinal Uniformity	Examples
2/1	1.5	0.4	0.7	High speed roads. Dual carriageway roads
2/2	1.0	0.4	0.5	Important rural and urban traffic routes. Radial roads
2/3	0.5	0.4	0.5	Connecting, less important roads. Residential major access roads

In addition all roads must have a surround ratio of at least 0.5. TI is limited to 15% for high speed roads and in rural areas, and 30% in all other cases.

For motorways the requirements are,

· Luminance – 2.0 cd/m^-2

· Overall Uniformity – 0.4

· Longitudinal Uniformity – 0.7

· Threshold Increment - 10%

· Luminance on hard shoulder – 0.5 cd/m^-2

L06 (6). Glare

Glare Index Calculation(CIBSE pp 172, 200)

The CIBSE glare index system for the evaluation of discomfort glare is described in CIBSE TM10.

The glare index system can be applied to a wide range of conventional luminaires, but it does have some limitations. It cannot be applied to large area light sources such as luminous ceilings, since the basic formula in the invalid. It cannot be applied to coffered ceilings and similar large cut-off luminaires. Also it may underestimate the discomfort glare for some ceilings-mounted luminaires, especially those in which the luminaire intensity distribution is such that the luminance of ceiling adjacent to the luminaire is greater than the luminance of the luminaire itself. Glare indices are specified for particular building types. These are set three apart, since this is the minimum difference being necessary for a change in discomfort glare sensation to occur.

Basic Glare Index Formula

CIBSE uses the glare formula,

where

· L_b = Luminance of background as seen by observer

· L_j = luminance of luminaire j as seen by the observer

· w_j = solid angle of luminaire j as seen by the observer

· p_j = position index of luminaire n as seen by observer

This method normally requires a computer program, but has the advantage that the formula can be applied to individual, randomly spaced or regular arrays of luminaires for any specified direction. The second method, calculating glare indexes based on photometric data provided by the manufacturer, is sufficiently accurate for most purposes, and is easy to use.

Uncorrected Glare Indices, with correction factors

This method is based upon the following assumptions;

· The luminaires are at a SHR of 1.0

· The luminaires are at a height of 2.0m above eye level

· The total light output of the lamps in the luminaire is 1000 lumens

· the observer is located at the mid-point of a wall, with horizontal line-of-sight towards the centre of the opposite wall

· the eye level is taken as 1.2m above floor level.

Correction terms can be applied to the uncorrected glare index to allow for change in mounting height and lamp output per luminaire. Currently there is no correction for other spacing to height ratios.

Uncorrected Glare Indices are tabulated according to room dimensions and reflectances. We specify the room dimension in terms of multiples of the mounting height above eye level. The x direction is always perpendicular to the line of sight, and the y direction is always parallel to the line of sight. The worst glare conditions will occur for viewing from the centre of wither the long wall or the short wall. Interchanging x and y will allow for this. When the glare index has been found (interpolation may be needed) it must be corrected for

· mounting height above 1.2m eye level

· total lamp luminous flux per luminaire if this differs from 1000 lumens

· extra correction terms if the published UGI table covers a variety of luminaire sizes or lamp types

These correction terms are added, or subtracted, from the initial glare index to give the final glare index of the installation. The height correction and total luminous flux correction may be calculated as follows;

· Height correction term - where H is the height above eye level(m).

· Total lamp luminous flux correction term - where F is the luminous flux per lamp(lumens) and n is the number of lamps per luminaire

Factors in detemining glare index

· Bare batten can be acceptable if the surface are high enough reflectance

· Crosswise viewing increases glare, except for CAT luminaires

· Glare is worse if ceiling height is lower, as the position index is stronger.

· There is a lower glare index for louvred fixtures; lower in the transverse angle, and this is the principal angle of viewing

· Glare is increased if the wattage of the lamps is increased, also if the number of lamps increases, since the luminous area increases.

L06 (7). Field Measurement of Light

Illuminance meters

The performance requirements for illuminance meters are set out in BS 667:1996. The standard defines two types of meter,

· Type L – high accuracy, used for laboratory measurements

· Type F – some accuracy has been sacrificed in order to make the instruments portable.

The standard considers a number of potential causes for errors, and set tolerances on them,

· Calibration uncertainty

· Non-linearity

· Spectral correction factor

· Intra-red response

· Ultra-violet response

· Cosine correction (unless marked as uncorrected) – It is important that light coming away at angles away from the norm are given the correct weighting according to the cosine formula used in the calculation of plane illuminance. This is checked by illuminating the photometer head with a small source, and then rotating the head, checking the angle of rotation and noting differences from expected

· Fatigue

· Temperature change

· Range change

A meter which just meets the requirements of this standard would have a best measurement capability of ±4% (type L) or ±6% (type F) when used on any of its calibrated ranges.

Luminance meters

Luminance meters have a similar set of errors to illuminance meters,

· Calibration uncertainty

· Linearity

· Spectral Correction

· Infra-red response

· Ultra-violet response

· Fatigue

· Temperature change/K

· Directional Response – Luminance meters should have a uniform response across their designed field of view

· Effect from surrounding field – A luminance meter should not respond to light outside its field of measurement. This can be tested by using a gloss trap which is slightly larger than acceptance error, and exposing the meter to a large uniform field of luminance

· Errors of focus – this may be due to changes in the light transmitting properties of the optical system of the meter as the focus is changed

· Range change

A meter which just meets the requirements of this standard would have a best measurement capability of ±5% (type L) or ±7% (type F) when used on any of its calibrated ranges.

Interior measurements

Section 5.3 of the CIBSE Code gives guidance on field measurements or interior lighting. However, since it was published in 1994, it refers to the previous version of BS 667, published in 1968.

Care needs to be taken not only with measuring the light, but measuring and noting the conditions at the time of the test. Things that should be noted are,

· The state and age of any lighting system

· The state of daylight

· The amount of furnishing and people in the space

· The surface finishes of the room

To measure the average illuminance, it is necessary to measure the lighting at several places on the working plane. The minimum number of points that may be measured is dependent on the Room Index,

· Below 1 – 9 points

· 1 -> 2 – 16 points

· 2 -> 3 – 25 points

· 3 and up – 36 points

These values are for SHR of up to 1.5:1. We lay these points out in a regular array.

Road measurements

With road measurement it is important to ensure there is no shadow caused by the person taking the reading. With illuminance measurements it is often to dark to read the display on the meter, so a meter with a hold facility is a good idea so the meter can be read with the use of a torch.

For luminance measures, the following are important,

· Ensure the luminance meter is located in the correct observer position

· Measure and record the geometry of the installation

· Mark out the grid of points to be measured with moveable markers so that the marker may be removed with the meter has been aimed, but before the measurement has been taken

· Check the supply voltage to the lighting, and not the age and condition of the lamps

· Note condition and state of the road surface particularly if it is wet or dry

The meter used should have a 20’ arc of less angle of acceptance in the horizontal direction and smaller than 2’ arc in the vertical direction.

L06 (8). Design for seeing – Cubic Illuminance

Overview

Cubic Illuminance (Cuttle, LR&T 1997) specifies the spatial distribution of illuminance about a point in terms of the illuminances of the six faces of a cube centered at the point. Cuttle proposes it as the basis for a system of applied photometry. The reasoning underlying this proposal is that procedures for both predicting and measuring the six cubic illuminances at a point are practicable. Also, when the cubic illuminance is specified, either by calculation or measurement, a variety of indices relating to the spatial distribution of illumination about the point may be readily derived.

The main advantage of cubic illuminance is that it moves illuminating engineering into the third dimension. It goes beyond horizontal illuminance on the working plane, and allows for the designer to extract information, e.g.,

· Balance of illuminance on horizontal & vertical surfaces

· The spatial distribution of illuminance, and how it affects 3-d objects

· Variation of illuminance, as viewer position moves

The vector approach

Definition: Cubic illuminance is the specification of the directional distribution of incident luminous flux at a point in space in terms of pairs of opposed planar illuminances normal to three mutually perpendicular axes intersecting at the point. A typical specification of cubic illuminance comprises six illuminances related to the surfaces of a small cube centered at the measurement point with the surfaces of the cube aligned in accord with the principle dimensional axes f the surrounding space.

Vector geometry is used, since it represents a useful shorthand notation for deal with the three-dimensional aspects of cubic illuminance. We can consider illuminance to be a vector E, and indeed this concept has been around since the work of Mehmke(1898). The concept of the illumination solid aids visualization of the three-dimensional illuminance distribution around a point. The point may be on the surface of the solid or contained within it, and the distance in any direction from the point to the surface of the solid is proportional to the planar illuminance at the point normal to that direction.

Measurement

Single Cell photometry

The most straightforward was to measure cubic illuminance is to mount a small solid cube at the measurement point, and to measure successively the illuminance on each face. Rowlands & Lowe used this technique, but it is tedious, and difficult to do.

Automatic single cell photometers require rotation of the cell or that a fixed cell is exposed to all the different directions by a mirror, or switching fibre optics.

Six-cell photometry

The simplest version consists of six cells mounted on the face of a cube, and Simons has developed such a device. Refinements such as a laser to target onto walls and automatic collation of information from a laptop are possible. The aim is to produce a device that is as easy to use as a simple photometer, while allowing accurate 3-d information to be gathered.

Calculation

Direct component

We can state the vectorial form of the point source equation,

where is the unit vector in the direction of S at the point of interest P, and is the unit vector normal to the plane of the surface at P. To find the direct component, we sum all the illuminances falling on the plane.

Indirect component

We can determine the inter-reflected component by using radiant flux transfer theory, or radiosity. Usually this accuracy is not needed, and we can use the mean room surface illuminance, which is equal to the mean room surface exitance,

where, FRF is the first reflected flux, .

Derived measures

The illuminance solid contains a vector component, where on any axis the component of the illumination vector equals the illuminance difference in opposite directions. In this way, the vector component represents all the asymmetric components of the illuminance distribution. The symmetric component is obtained by subtracting the vector component.

The illuminance of a surface of any orientation at the reference point is the sum of the illuminances due to the vector and symmetric component. This is a remarkable finding that illuminance distribution can be examined in terms of these two components, and it provides the basis for estimating a variety of indices that relate to characteristics of spatial illumination.

Illuminance equations

Planar Illuminance,

Scalar illuminance,

Hemispherical Illuminance,

Cylindrical Illuminance,

Semi-cylindrical Illuminance,