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Simulate a Dimmable Lamp

July 29, 2018

How should I select the light bulb wattages in a room to get the most options for setting the light level?

The easy solution is to get a dimmable light fixture and bulb. But doing that would make this post very short and I actually want to solve the problem of how to simulate a dimmable lamp with a small number of non-dimmable lamps.

To reduce complexity, consider the case using two lamps. In an ideal solution, one of the lamps should have a low brightness for a dark light level, the other lamp should have a medium brightness for a medium light level and when both lamps are on they should produce a bright light level. This is however not possible due to how humans sense light.

Humans sense light as changes relative to the current light level. For example, lighting 5 candles in a room with 1 already lit candle makes the room appear brighter. Lighting 5 candles in a room with 100 already lit candles barely makes a difference. This makes the problem harder to solve and the solution will not have evenly spaced brightness levels. I choose the sRGB gamma curve to calculate the apparent brightness from the amount of light from the lamps.

Mathematically this can be stated as an optimization problem, where we try to make the brightness steps as evenly distributed as possible. As it has a small number of parameters, and I did not want to spend time on analysis, I choose to find the solution with genetic optimization. It gives a solution but does not give insight into the problem.

For two lamps I got that their relative amount of light (luminance) should be 17.78% and 82.22%. Turning them on and or off should give an appearance similar to the following image for all combinations of off and on:

The step between light gray and white is less than the other brightness differences. But in order to make it larger, the dark gray would become lighter as well.

Increasing the number of lamps allows for more levels:

Three lamps.
5.76%, 24.54% and 69.70%.

Four lamps.
2.40%, 9.25%, 25.71% and 62.64%.

Five lamps.
1.17%, 3.97%, 10.73%, 25.81% and 58.32%.

I hope you found this interresting. Please contact me for any questions, suggestions or other comments.