Computing Lux Level Given A Lamp's Power Spectrum

Quote of the Day

Experience is not what happens to a man; it is what a man does with what happens to him.

— Aldous Huxley


Introduction

Figure 1: Xenon Lamp Irradiance Example (Source).

Figure 1: Xenon Lamp Irradiance Example (Source).

I have two engineering interns working in my group this summer. I love having interns around, but they do require more management oversight. I was an intern at Medtronic when I was at university and found the intern experience wonderful. I owe a great debt of gratitude to Ron Meister and Jim Sinclair for their patient support of a very naive young man. Today, I follow their lead and try to support my profession by helping young folks get started.

I have assigned my interns a couple of simple design tasks – one of the tasks involves performing measurements on that amount ambient light that leaks into our enclosures. This task is going to provide data that will drive the design of a simple ambient light sensor for use within our enclosures to tell if the door has been opened. Open door sensors are a notorious source of false alarms. We used to use mechanical/magnetic sensors, but they proved to be unreliable. We are now looking at using ambient light sensors.

One of the interns asked how to convert our ambient light power measurements into lux (visual units). I thought I would rework a xenon flash lamp example I found on the web using Mathcad. This will give him a tool that he can use in his work here this summer.

Background

Requirements

lumen (symbol: lm)
The lumen unit of luminous flux, a measure of the total quantity of visible light emitted by a source. Luminous flux differs from power (radiant flux) in that radiant flux includes all electromagnetic waves emitted, while luminous flux is weighted according to a model of the human eye's sensitivity to various wavelengths. Lumens are related to lux in that one lux is one lumen per square meter.
lux (symbol: lx)
The lux is one lumen per square meter (lm/m2). There is no single conversion factor between lx and W/m2; there is a different conversion factor for every wavelength, and it is not possible to make a conversion unless one knows the spectral composition of the light.
luminous efficiency
Luminous efficacy is a measure of how well a light source produces visible light. It is the ratio of luminous flux to power. The concept of luminous efficiency is complicated by the fact that the eye has two types of receptors: rods and cones. Rods provide our low light vision and their luminous efficiency is described by the scotopic luminous efficiency curve. Cones are used at normal light levels and their luminous efficiency is described by the photopic luminous efficiency curve. My work here will focus on the normal light level (photopic) region of vision.
Figure 2 shows the photopic luminous efficiency curve as defined by the International Commission on Illumination (CIE).
Figure 2: Photopic Luminious Efficiency Curve (Source).

Figure 2: Photopic Luminous Efficiency Curve (Source).

Figure 2 show us that the eye is most sensitive to green light (555 nm). You can achieve a given visual light level with the least power using green light – all other colors will require more power.

Key Formula

The basic math required here is shown in Equation 1, which is just the sum of all the lamp's visual power weighted by the eye's sensitivity.

Eq. 1 {{E}_{v}}=683.002\cdot \int\limits_{{{{\lambda }_{{\min }}}}}^{{{{\lambda }_{{\max }}}}}{{y(\lambda )\cdot J(\lambda )\cdot d\lambda }}

where

  • Ev is the illuminance (i.e. visual brightness) measured in lux
  • y(λ) is the photopic luminosity function.
  • J(λ) is the spectral irradiance of the light – I think of this as the power spectral density of the light.
  • λmax is longest visual wavelength.
  • λin is the shortest visual wavelength.

For 555 nm light, the amount of illuminance is a simply unit conversion using 683.002 lux per W/m2.

Analysis

My Mathcad source and its PDF are included here.

Photopic Luminosity Capture

Figure 3 shows my capture of the photopic luminosity function using Dagra. The curve-fit luminance function that I will be using is highlighted in green.

Figure 3: Capture of the Photopic Luminosity Function.

Figure 3: Capture of the Photopic Luminosity Function.

Capture of Lamp Irradiance

Figure 4 shows my capture of the xenon lamp's irradiance (i.e. power spectrum). The curve-fit irradiance function that I will be using is highlighted in green.

Figure 4: Capture of Xenon Flash Lamp Irradiance.

Figure 4: Capture of Xenon Flash Lamp Irradiance.

Evaluation of Equation 1

Given the data captures in Figures 3 and 4, the evaluation of Equation 1 is simple. My result of 274 lux is quite close to the listed result of 270 lux. The error is from how I captured the data from the images.

Figure 5: Evaluation of Equation 1 for the Xenon Flash Lamp.

Figure 5: Evaluation of Equation 1 for the Xenon Flash Lamp.

Conclusion

This is just a quick illustration of how a common engineering task is made simple through the use of a computer algebra system. Some days I am stunned at how I work today compared to 30+ years ago. The tools today are so much better than what I had when I started out – handheld calculators and graph paper.

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