## Lamp Lumen Maintenance Factor (LLMF) Calculation

The lamp lumen maintenance factor is determined by using a method for projecting the lumen depreciation over the lifetime of the lamp made from LM-80 data using TM-21, which assumes an exponential decay after the first 1,000 hours of use. This form is expressed using the following equation:

Equation 1:

Φ(t)= β exp(−αt)

Where

**Φ(t)**is the normalized luminous flux output at time t and is equal to RMI/RLUX and also known as the Lamp Lumen Maintenance Factor (LLMF)**RMI**= Required Maintained Illuminance**RLUX**= Reported Lux**β**is a projected initial constant of normalized luminous flux**α**is the decay rate constant

A reference to this equation can be found here.

By using this equation, we can create a projection of lumen depreciation data that can then be plotted onto a chart with a y axis representing RMI/RLUX as a percentage of initial lumen output and the x axis representing time in thousands of hours.

## Dirt Depreciation Factor (DDF) Calculation

Since the α (alpha) and β (beta) values from the LM-80 test data do not factor in other light loss factors (OLF) for the depreciation we use the OLF value calculated in the Maintained Illuminance Table to determine a Dirt Depreciation Factor(DDF). The DDF value also assumes an exponential decay over the life of the lamp and can be calculated using the following equation.

OLF = exp(DDF*RRP) which is equivalent to DDF = ln(OLF 1/RRP)

The extrapolation of the LM-80 data using the β and α values from the LM-80 test data and considering the OLF is then performed using equation 1 but with a modified decay constant (α) of

α = α – DDF

The resulting chart is then plotted.

## Recommended Re-lamping Point (RRP) Calculation

The recommended re-lamping point (RRP) of the lamp is calculated using the following equation:

Equation 2: t = ln(B (RLUX/RMI)) / ( α – DDF )

This equation is equivalent to Equation 1 with the DDF considered and also with t representing the recommended re-lamping point.

Where

- t = recommended re-lamping point (RRP)
- DDF = dirt factor
- RMI is the Required Maintained Illuminance for the area based on table 3.1 of AS/NSZ 1680 1:2006
- RLUX is the average lux reading recorded after the installation
- B is a projected initial constant of normalized luminous flux
- α is the decay rate constant

Below is a proof to demonstrate that equations 1 and 2 are equivalent:

Using Equation 1,

Φ(t) = Be−αt

Where

- Φ(t) is the normalized luminous flux output at time t (this value is a percentage represented as a decimal between 0 to 1)_
- B is a projected initial constant of normalized luminous flux_
- α is the decay rate constant_

Φ(t) = RMI/RLUX

Where

- RMI is the Required Maintained Illuminance for the area based on table 3.1 of AS/NSZ 1680 1:2006
- RLUX is the average lux reading recorded after the installation

This gives

RMI/RLUX = Be−αt

Since e-αt= 1/eαt

RMI/RLUX =B / eαt

Invert both sides

RLUX/RMI =eαt / B

Multiple both sides by B

B (RLUX/RMI) =eαt

Since the natural logarithm function ln(x) is the inverse function of the exponential function exp we can say

αt = ln(B (RLUX/RMI))

Divide both sides by α

t = ln(B (RLUX/RMI)) / α

Factoring in the DDF we now have

t = ln(B (RLUX/RMI)) / ( α – DDF )