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What is the functional relation between PWM duty cycle for LED and its luminous intensity (candelas)? Linear or not?

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You can assume that duty cycle and amount of light emitted are linearly dependent. Usually the rise/fall times of the PWM signal and the LED's brightness are very small compared to the "on" and "off" times so that one may approximate them with an ideal rectangular wave form.

As John U said, however, in reality this may depend on the quality of your PWM signal, the impedances of power lines, etc.

Besides, one may note that many LEDs can be driven at a higher-than-normal current when used in pulsed operation. This can be used to get, for instance, 10% of the light output with 5% duty cycle. (Which then again may seem like 20% of the brightness to the human eye.) The limits for this kind of pulsed operation are usually specified in the datasheet of the LED.

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Unknown until you measure it for your circuit.

LED's are non-linear, and quite variable. If you're lucky there will be some curves / data in the datasheet for your LED, which will be 50% marketing lies, but may give some approximate idea.

Second part: Your PWM / driver circuit may not be entirely linear in power delivered, even if the duty cycle of the micro pin changes perfectly (and there are micros / programming traps that can cause small unintended non-linearities in these things).

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In theory, and looking at systems as lumped matters, one should be able to predict the intensity of the light based on the specific PWM.In other words ratio between on-state and off-state defines resulting intensity of the LED.And that is only a relative idea.

And as the circuits get more complicated, It gets harder and harder to determine exactly how the d.c would affect the intensity. In most cases you can only determine which d.c gives you higher light intensity. But to find a relation between PWM d.c and candelas is not easy nor precise and varies from one circuit to another.

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