Recessed Lighting Layout and Lux Calculator

Fixtures lie at the centres of equal rectangular cells: $x_i=(i-1/2)L/n_x$ and $y_j=(j-1/2)W/n_y$. Wall offsets are half the spacings.

The assumed beam has $I(\theta)=I_0\cos^n\theta$, where $n=\ln(0.5)/\ln[\cos(\beta/2)]$ and $\beta$ is the full half-intensity beam angle. This generalized Lambertian angular model is used in Menéndez and Steendam (2022), pp. 2–4, On the Optimisation of Illumination LEDs for VLP Systems. At 180° the calculator explicitly uses the uniform downward-hemisphere limit, $n=0$, which has no half-maximum boundary.

Integrating over the lower hemisphere gives $\Phi=2\pi I_0/(n+1)$. Thus $I_0=\Phi(n+1)/(2\pi)$, with no artificial beam-edge cutoff. This normalization follows hemisphere integration and agrees with their equation (2); the entered lumens are total downward luminaire flux, including light outside the half-intensity cone.

For mounting distance $h=H-z$ and source-to-point distance $r$, $\cos\theta=h/r$. Inverse-square and receiving-plane cosine laws give

The Australian Government's Basics of Efficient Lighting, second edition (2019), pp. 18–19, explains these laws and addition of source contributions.

The area mean is independent of the display grid. With polar radius $\rho$ about one source, equation (4) of Menéndez and Steendam gives an analytic radial integral:

For a rectangular corner with adjacent edge distances $a,b$, split its azimuth at $\arctan(b/a)$; the boundary is $R=a/\cos\alpha$ in the first part, with $a,b$ exchanged in the other. Sum all four corners and all fixture contributions. The implementation uses the mirror symmetry of the centred grid to reduce this sum.

Composite Simpson integration doubles from 8 to at most 1024 angular intervals. Both of the last two relative changes must be at most $10^{-7}$. The same convergence test is used for automatic sizing and the displayed area mean. An error condition withholds results if integration does not converge or calculated room flux exceeds emitted downward flux. No flux result is clipped to hide an error.

The horizontal-illuminance half-maximum radius is $r_{1/2}=h\sqrt{2^{2/(n+3)}-1}$, which differs from the intensity beam footprint. A warning flags display-grid spacing above one third of this radius, or a display-grid mean differing from the converged mean by more than 1%. Minimum, maximum and coverage remain sampled estimates. $U_0=E_{min}/E_{avg}$ uses the converged mean. Fixed-layout required lumens scale with target/mean.

Automatic sizing works on the long and short room axes, starts from the lumen-budget lower bound and preserves the required count when either axis reaches eight. Each successive candidate has a strictly larger count. It stops at a converged mean at least 3% above target, at an unconverged candidate (results withheld), or at 8 × 8. Exchanging room length and width exchanges the selected fixture axes. This remains a heuristic rather than a minimum-count or uniformity optimization.

Identical fixtures point vertically down, all at one height, in a rectangular room. Each fixture is a point source with a rotationally symmetric cosine-power pattern. All entered flux is emitted downward after the luminaire optics; do not enter bare lamp lumens or apply a second utilization factor. The maintenance factor multiplies every contribution equally.

Only direct artificial light is included. Light outside the room is lost; wall, ceiling and floor reflections are omitted. No daylight, obstructions, interreflection, tilt, asymmetric optics or mixed fixtures are modelled. The calculation plane is horizontal and unobstructed. Numerical extrema and coverage depend on grid resolution; the area mean is integrated separately with a convergence check.

The default dimensions, lumens, aperture and maintenance factor are illustrative design inputs, not a product specification. The beam normalization and layout-sizing heuristic are derived for this tool. HSE target examples are free task guidance checked on 27 October 2026; no paid EN 12464-1 table is reproduced.

For preliminary comparisons of small downward luminaires in rooms up to 20 m per side, with 1–8 fixtures along each side and beam angles from 10° to 180°. Keep the working plane at least 0.1 m below the fixtures. Prefer a mounting distance at least five times the largest luminous aperture dimension (Australian Government manual, p. 19).

The mean, captured flux and required lumens use a separately converged area integral, not the heat-map samples. Results are withheld if the integral fails its convergence test. The display-grid warning refers to the heat map, extrema, uniformity numerator and coverage; increase resolution and compare these sampled values. A smooth-looking heat map is not proof of resolution.

Automatic sizing targets the area mean only. It neither guarantees a minimum illuminance nor optimizes uniformity, and may not reach a demanding target within its 8 × 8 cap. Check the final mean/target ratio. Required lumens assume an unchanged angular pattern when output changes.

IES LM-63 file import, reflections, glare/UGR, emergency-lighting compliance, circuit design, structural clearances and thermal/fire ratings are outside scope. Verify final layouts using manufacturer photometry and measurements. This tool does not establish compliance with EN 12464-1 or local regulations: check against the standard in force. Installation work belongs to a qualified person. Coordinate-table headings and charts explicitly use metres; input and result unit fields support metric/US conversion.