Mathematical Formulas for Lighting

See Ushio's online calculator at Lamp Life Calculator- Support.

From http://www.harison-toshiba.com/lampinfo.htm: [Note that this ia a manufacturer of subminiature vacuum lamps.]
Since incandescent lamps may be rerated to suit various design goals, existing standard lamps should be reviewed prior to initiation of a new lamp design. Three basic formulas apply when rerating a lamp, assuming that VA equals application voltage and VD equals design voltage, the following may be applied: (M.S.C.P. represents Mean Spherical Candlepower)




[Note that the first two formulas are very close to the formulas below (3.5~3.4 and 12~13). The differences may be attributed to the different lamp types being considered.]


From Mike Wood, of Mike Wood Consulting.

lumens/LUMENS = (VOLTS/volts)^3.4

life/LIFE = (VOLTS/volts)^13
(i.e. reduce the volts to 90% and the life increases by 393%!)

EFFICIENCY/efficiency = (VOLTS/volts)^1.9

watts/WATTS = (volts/VOLTS)^1.6
(not 'squared' as you would get with a fixed resistance)

coltemp/COLTEMP = (volts/VOLTS)^0.42

See also OSRAM SYLVANIA:Sylvania Automotive Lighting Catalog.


Copied from this post by ship.

LIGHT OUTPUT CALCULATIONS
a dinner candle provides about 12 lumens. A 60-watt Soft White incandescent lamp provides 840 lumens.
Foot-candles = candela / distance in feet * distance in feet
Foot-candles = Lux / 10.764 = lumens/sq. meter;
1 fc=1/10.764 lux
Foot-candles * 10.764 = lumens/sq. meter = lux
Lumens/sq. ft. * 1 = foot-candles; (1fc=1 lumen/ft²)
Lumens/sq. ft. * 10.764 = lumens/sq. meter
Lumens * 0.07958 = spherical candle power
Lumens = Mean Spherical Candlepower x 12.57
Lux * 0.0929 = foot-candles
Lux = candela / (distance in meters * distance in meters)
Lambert * 0.3183 = candela/sq. cm [candela per sq. cm], Lambert * (1/pi) = candles/sq. cm
Lambert * 295.720 = candela/ sq. ft. [candela per sq. ft.]
Lambert * 1 = lumens/sq. cm
Illuminance (lx) = E = Luminous flux falling on area (lm) ÷ Illuminated area (m²)
Illuminance (lx) = E = Luminous intensity (cd) ÷ [distance in meters (m)]²
Luminance (cd/m²) = L = Luminous intensidy (cd) ÷ viewed luminous area (M²)
Luminous efficacy (lm/w) = h = Generated luminous flux (lm) ÷ Electrical power consumed (w)
Efficacy=F(lu)/P(w)
F=Luminous Flux
P=Electrical Power (wattage), in watts

LIGHT BEAM CALCULATIONS:
[See Conventional Photometrics_v7.zip for a down-loadable MS Excel workbook containing most popular fixtures.]
Beam diameter = distance * (2 * tan (beam angle in degrees / 2))
Throw distance = Square root [(horizontal dist. * horizontal dist.) + (vertical dist. * vertical dist.)]
Inverse Square Law: E(fc)=I(cd)/D(ft)². E(fc)=F(lm)/A(ft)²
A=Area in square feet
D=Distance in feet
E=Illumination in footcandles
F=Luminous Flux in Lumens
I=Luminous intensity (Candlepower) in footcandles
Mired Shift Value = (1,000,000/d) - (1,000,000/a); d=Desired Color Temperature (no units), a=Actual Color Temperature (no units)

Fixture Lens Conversion Guide:
For a two-lens system: EFL= (f1*f2)/(f1+f2-d), where EFL=Effective Focal Length, f1=Focal Length of Lens1, f2=Focal Length of Lens2, d=Distance between Lenses.

50° - 3.5Q5 / 360Q-4.5x6.5 (45°)
40° - 3.5Q6 / 360Q-6x9 (37°)
30° - 3.5Q8 / 360Q-6x12 (27°)
20° - 3.5Q10 / 360Q-6x16 (17°)
10° / 12° - 3.5Q12 / 360Q-6x22 [single lens] (9.5°)
5° - None
8x8 (20°)
8x10 (16°)
8x16 (6°)

1Kw PAR64:
ACL (GE 4552-28V, 250W)- Beam: 7°x8°
ACL (GE 4559-28V, 600W)- Beam: 11°x12°
VNSP (FFN)- Beam: 6° x 12° Field: 10° x 24°
NSP (FFP)- Beam: 7° x 14° Field: 14° x 26°
MFL (FFR)- Beam: 12° x 24° Field: 21° x 44°
WFL (FFS)- Beam: 24° x 48° Field: 45° x 71°

POWER CALCULATIONS
Power = Voltage * current (Watts = Volts * Amps)
P(w)=V(v)xI(a)
P=Electrical Power (Wattage) in watts
V=Voltage (EMF), in volts
I=Electrical Current(Amperage), in amps
Current = Power / Voltage (Amps = Watts / Volts)
RMS Volts = 0.707 * Peak Volts RMS Volts = 1.11 * Average Volts
Ohm’s Law: V(v)=I(a) * R(Ω)
V=Voltage (EMF), in volts
I=Electrical Current (Amperage), in amps
R=Resistance, in ohms
Impedance: Z(Ω)=√[R²(Ω)²=X²(Ω²]
Z=Impedance, in ohms
R=Resistance, in ohms
X=Reactance, in ohms
Power Factor: R(Ω)/Z(Ω)
pf=Power Factor
R=Resistance, in ohms
Z=Reactance, in ohms

DC VOLTAGE DROP OF CONDUCTOR (cable) OF L LENGTH
V = voltage drop, I = current
R = resistance of conductor per 1000 feet
L = length of conductor in feet
R for 18awg = 6.51, 16awg = 4.09, 14awg = 2.58
12awg = 1.62, 10awg = 1.02, 8awg = 0.64
V = I * L * (R / 1000) * 1.004

Lamp sizing:
Stage/Studio lamps are often expressed as a number representing the diameter of the lamp in 1/8s of one inch. A PAR64 Lamp has a diameter of 8". A T6 lamp is 3/4". The most common size of fluorescent tube is T8, or 1". The predecessor of the BTN fresnel lamp was the T20, 2.5". A G40 lamp is 5" at its widest diameter, and so on. The most common household lamp is the A19, 2 3/8" diameter.