Low Percentage PWM LED Brightness (Equation?)

Question

I have a 5 V embedded circuit where a 10 kΩ pot controls the duty cycle of an LED. I noticed that when the duty cycle is only 0.39% (or 1/255 bits), the LED is still perceptibly bright. I'm looking at the PWM signal on an oscilloscope and can see that the +5V fraction takes up only 65 us of a 16.4 ms period.

1. What is the expected brightness of a conventional LED at duty cycles less than 1%? Feels like it should be completely off.

2. Can anyone derive the equation relating brightness to duty cycle in this case?

Also checkout this article which gets close to the answer: https://codeinsecurity.wordpress.com/2023/07/17/the-problem-with-driving-leds-with-pwm/

"However, if you linearly sweep the duty cycle from 0-100%, you won’t see a smooth linear increase in brightness. This is because our eyes are far more sensitive to changes in brightness in the dark than in bright light; the evolutionary reasons for this are fairly obvious. If you graph the PWM duty cycle versus the normalized perceptual brightness, you’ll get a curve that looks something like this [logarithmic curve, no equation given]. At just 5% duty cycle, the perceptual brightness of the LED already exceeds 20% of the maximum."
(this leads to a gamma correction technique)

EDIT: This is a 5 V circuit, typical breadboard LED in series with a 1 kΩ resistor. The 5 V is PWM at variable duty cycle. What happens to the LED over over the full range of an 8 bit duty cycle? At what point is the LED perceptibly off? Apparently not unless the duty cycle is exactly zero.

Answer 1

I'll try to give a helpful answer instead of telling you that this is a question of human perception and very difficult to answer correctly.

Gamma correction is exactly what you need and want. Basically, your LED PWM output sets the light intensity, but what you actually want to set is perceived brightness ("luma").

These follow a power law; but the actual exponent is somewhat subjective (see https://en.wikipedia.org/wiki/Gamma_correction#Power_law_for_video_display).

The good thing is you can just keep things simple and use 2 as gamma factor, which is in the right ballpark and makes everything easy to calculate, so:

led_PWM = brightness ^ 2 = brightness * brightness

Answering your question with this:

1. A PWM value of 1% will still give you 10% perceived brightness! (1% = 10% * 10%)

2. If you want to do this well, decent PWM resolution is required (e.g. 16bit). Because if you have only 8bit PWM resolution, your very smallest "not-off" value is already going to be ~6% "bright" (with 16bit PWM, you get a ~0.4% first increment, which is MUCH better). Here is how to calculate a 16bit PWM output from a 8bit brightness value using integer arithmetic (in C):

   uint16_t gammaCorrectU8ToU16(uint8_t desiredBrightness)
   {
       // 66051 is (0xffff / (0xff * 0xff) << 16), rounded up
       return (uint32_t)desiredBrightness * (uint32_t)desiredBrightness * 66051 >> 16;
   }
   

Answer 2

You will have no trouble seeing that LED going at 0.01% duty cycle in a dark-ish room. To turn the LED off, the PWM setting must be zero. That’s all there’s to it.

Many LEDs, if pulsed at say 50% of continuous current rating, will be visible with pulses as short as 50ns. Microcontrollers don’t normally have enough resolution to go that low :)

The LED itself is quite linear. Our eyes on the other hand have a dynamic range of what. 6 orders of magnitude or more? So yeah, 8-bit PWM is not gonna make a LED appear dark. Maybe in sunlight :)

Answer 3

Human vision has a logarithmic response, and will be more sensitive to low intensities. PWM type drive also triggers another vision path: persistence. So the eye will see the LED pulses and stretch them out, making the perceived light brighter.

Answer 4

LEDs are linear with duty cycle so long as your on time is relatively long relative to the rise time of the driver/LED combination. This will depend to some extent on how you switch it, but probably you will get good linearity down to tens of microseconds, perhaps further.

At very low duty cycles you will find that it starts to deviate from linear because the diode will take time to turn on each cycle.