Lighting and Circuit Sizing: Method

Lighting and Circuit Sizing

Two of the everyday calculations of a building electrical design under one select: the average and point-by-point illuminance of a room by the lumen (zonal-cavity) method, and the voltage drop, ampacity, overcurrent protection, conduit fill and available fault current of a branch circuit. Both are closed-form methods; the only values a code owns (allowable ampacities, adjustment factors, insulated conductor diameters, recommended illuminances) are entered inputs with sourced defaults, and the page says where the analyst reads them.

The lumen method

The lumen method predicts the average illuminance on a horizontal work plane from the lumens installed: E = N Φ CU LLF / A, with N luminaires of output Φ each, the room area A, the light loss factor LLF (the maintained output as a fraction of the initial, the product of the lamp lumen depreciation, dirt depreciations and any driver factor) and the coefficient of utilization CU, the fraction of the luminaire lumens that reaches the work plane directly or after reflection. The IES zonal-cavity method computes CU from three cavities: the ceiling cavity above the luminaire plane, the room cavity between the luminaires and the work plane, and the floor cavity below the work plane. Each has a cavity ratio 5 h (L + W)/(L W), which measures how much of the flux leaving one end of the cavity reaches the other end.

The ceiling and floor cavities are replaced by equivalent diffuse surfaces at the luminaire and work planes. Their effective reflectances are computed here by flux transfer between the cavity base, its walls and its opening, with Hottel's exact view factor between the two parallel rectangles; the handbook closed form ρ A₀ / (A_s − ρ A_s + ρ A₀), which treats the cavity as an integrating sphere with the opening fraction A₀/A_s, is printed beside it. The two agree within a few percent for shallow cavities; the sphere form runs 10 to 15 percent high for a deep cavity of low reflectance (a 7.5 ft floor cavity with a 20 percent floor), which is why the flux-transfer value is the one carried into the CU.

The luminaire enters through its candela distribution: relative intensities at angles from nadir, scaled so that the downward lumens equal the entered output times the downward fraction (Φ_down = 2π ∫ I(θ) sin θ dθ; the zonal lumens are tabulated). The direct ratio D, the share of the downward flux that reaches the work plane without touching a wall, is integrated from the inverse-square cosine law over the work plane for every luminaire of the uniform array at its own position. This replaces the generic zonal-multiplier table of the handbook with the room's own value. The room cavity is then three surfaces, the ceiling-cavity opening, the work plane and the walls, each of uniform exitance; the upward lumens land on the ceiling plane, D of the downward lumens on the work plane and the rest on the walls, and the exitances solve M_i = ρ_i (E_i,direct + Σ_j F_ij M_j). CU is the work-plane flux per lumen emitted. The test script compares this three-surface model with a patch radiosity of the whole room (ceiling, floor and walls as some 600 to 1,000 patches, the luminaires as point sources with the entered distribution) and finds agreement within 1.5 percent for a recessed cosine down-light and within 3 percent for a suspended direct / indirect luminaire.

The point grid gives the direct illuminance Σ I(θ) cos θ / d² at every cell of the work plane, times the LLF, plus the uniform interreflected component taken from the flux transfer; its average, minimum and maximum give the uniformity ratios (UFC 3-530-01 asks for 2:1 average to minimum in offices). The spacing criterion of the distribution is the spacing-to-mounting-height ratio at which the direct illuminance midway between two luminaires, from the two, equals that straight under one of them from it alone, with the four-luminaire square version beside it; the layout's largest spacing over the mounting height should not exceed the lower of the two (1.29 for a cosine distribution). Lighting power density and efficacy follow from the luminaire watts when entered.

Recommended illuminances are the IES Lighting Handbook's and are not reproduced: the target is entered. The default 30 fc (300 lux) at 2 ft 6 in above the floor is the UFC 3-530-01 criterion for individual and open offices, which also asks for ceilings of at least 85 percent and walls of at least 60 percent reflectance and for the applicable light loss factor in every photometric submittal. The Navy Construction Electrician manual prints the lumen method for a roadway (S = LL MF CU / (fc W), 150 ft for 50,000 lumens, 0.70, 0.43, 2 fc and 50 ft) and the test script reproduces it as the same identity.

The branch circuit

The load current is I = P / (V pf) single-phase or P / (√3 V pf) three-phase, or entered. The conductor area follows from the American Wire Gage definition (diameter 5 × 92^((36 − n)/39) mils, area the square of it in circular mils) or the kcmil designation, and its resistance from the NBS wire tables: ρ₂₀ = 10.371 Ω·cmil/ft for annealed copper of 100 percent IACS conductivity with α₂₀ = 0.00393 per °C (NBS Handbook 100), 17.002 Ω·cmil/ft and 0.00403 per °C for EC-H19 aluminum of 61 percent conductivity (NBS Handbook 109), R = ρ₂₀ (1 + α₂₀ (T − 20)) / cmil per foot at the entered operating temperature (75 °C by default, the usual column of the ampacity tables), with an optional 2 percent lay allowance for stranded conductors. The test script reproduces the handbooks' tables at 0 to 200 °C and the Army FM 5-424 wire characteristics.

The reactance of one conductor is entered (the NEC Chapter 9 Table 9 value for the raceway type) or, when the field is blank, estimated from the spacing: X = 2π f (μ₀/2π) ln(D / (0.7788 r)) per unit length, the two-wire line formula with the geometric mean radius of a solid round conductor. A steel raceway raises the reactance by about a fifth; the estimate suits non-magnetic raceways and cable.

Voltage drop is computed three ways. With the load voltage held at the nominal value and the current lagging by φ, the sending-end voltage is the phasor sum V_s = V + I (cos φ − j sin φ)(R + jX) over the loop (both conductors single-phase; one conductor and the √3 factor three-phase, on the line-to-line voltage), and the exact drop is |V_s| − V. The usual approximation keeps the in-phase part, I (R cos φ + X sin φ) times 2 or √3, which is within a fraction of a percent of the exact value at ordinary angles; the resistive form drops X. The percent drop is referred to the nominal voltage and compared with the entered limit: UFC 3-501-01 sets 3 percent for branch circuits and 2 percent for feeders at rated capacity, FM 5-424 uses 3 percent for branch circuits within 5 percent overall, and the NEC's informational note recommends the same 3 and 5 percent. The smallest catalogue size that meets the limit is reported, and a table and chart give the drop of every size.

Ampacity: the design current is 125 percent of a continuous load (NEC 210.19 and 210.20). The allowable ampacity is a code table value entered at its stated conditions (30 °C ambient, up to three current-carrying conductors); it is corrected for ambient by √((T_c − T_a)/(T_c − 30)), the relation that follows from I²R heating against a fixed temperature rise, and by the entered adjustment factor for more conductors in the raceway. The corrected ampacity must carry the load, and the table ampacity the design current. The breaker is the next standard rating at or above the design current (the NEC 240.6(A) list), checked against the cap entered for the conductor (the small-conductor rule of NEC 240.4(D) limits 12 AWG copper to 20 A). Conduit fill is the sum of the insulated conductor areas over the raceway area against 53, 31 or 40 percent for one, two or more conductors (NEC Chapter 9 Table 1); the insulated diameter and the raceway inside diameter are entered.

Fault current by the point-to-point method: the transformer impedance on its own base is Z_t = (%Z/100) V²/S, split into R and X by the X/R ratio; with an infinite primary the fault current at the secondary terminals is I_fl × 100/%Z, and an entered utility short-circuit MVA adds a source reactance V²/S_sc. At the load end the conductor impedance of the fault loop adds vectorially: I = V_phase / |Z_source + Z_conductor|. The Bussmann form f = Z_c I_sc / V, M = 1/(1 + f), I_end = M I_sc is the same arithmetic with the impedances added as magnitudes (its C constant is the reciprocal of the conductor impedance per unit length) and is printed for comparison. The panel fault current is compared with the breaker's interrupting rating. Line-to-neutral faults on a centre-tapped single-phase transformer, motor contribution and arc impedance are not modelled; the method is the preliminary estimate that IEEE Std 141 describes, not a coordination study.

Limits

The lumen method assumes diffuse surfaces of uniform reflectance, a uniform rectangular array, an empty room and luminaires small compared with their mounting height; partitions, furniture and daylight are outside it. The point grid is direct light plus a uniform interreflected component, which is accurate where the direct component dominates. The circuit sub-tool treats the conductor temperature as entered rather than computed from the load, uses the solid-conductor geometric mean radius for the reactance estimate, and sizes for drop and ampacity only: it does not size the equipment grounding conductor, check the neutral, or apply the harmonic, motor-starting or flicker rules.