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• You forgot one of the most important things about "regular" LEDs (the ones without special blinking circuitry or UV-phosphorescent LEDs). That is taking advantage of difference between "Forward Current" (otherwise known as "Max Average Forward Current") and "Peak Forward Current". Some LEDs have "Peak Current" many times greater than "Max Average Forward Current". For example: An LED may have a "Forward Current" of 20ma and a "Peak Current" of 100ma. That means that if you are running it off of DC, then you do your calculations based on 20ma and if you exceed this current on a continuous basis, it will overheat and die. But, with that same LED, you can send a short pulse of 100ma without blowing up the LED, and during that time the LED will be 5 times as bright. If you continue at this current, it will quickly overheat and burn out. What good is that? Well, the human eye judges brightness by PEAK brightness. So if you send a 20% duty-cycle current of 100ma, it will average out over time as an average current of 20ma, but your eye will see a brightness the same as if it were driven continuously at 100ma.

You can observe this in many common devices. Many LED stop-lights on automobiles use this fact. If you scan your eyes quickly back and forth, you can see the blinking. Some traffic lights do this. LED clocks use this fact, along with multiplexing the digits, to save circuitry AND to make brighter numbers. You can tell of they're blinking by using the same eye-scanning technique.

Realize that this does not work with many of the more efficient LEDs which work by exciting a phosphorescent emitter with UV light from an otherwise invisible LED. The phosphorescent coating integrates the UV from the pulsing LED and the eye does not see a blinking source at all. Duty-cycle changes the light output almost linearly in that case. Other LED-based products that have internal ICs would also not work with this duty-cycle trick.

Just for a little more info... The failure mode for exceeding "Peak Current" is that the little wire connecting the top end of the LED die instantly melts. The failure mode for exceeding the "Forward Current" is that the entire device overheats and dies a normal overheating death. A proper duty-cycle allows both limits to be met, and the human eye does what it does and judges the brightness by the peak brightness.

• when calculating the current limiting resistor, what value do I use for the forward voltage? The red LED has Vf typical 2.0 and max 2.25, but the green and blue LED have Vf typical 3.5 and max 5.0.

While the tutorial briefly mentions blinking and color-changing LEDs, this specific issue isn't addressed. The answer is in line with Get the details; consult the datasheet.

The datasheet for 'LED - 5mm Cycling RGB (slow)' includes the figures quoted in the question on page 2 in tables for the RED, GREEN and BLUE colors. However, it also includes a table that is intended for circuit/the entire device, and these are the values you would use to determine how to drive the LED as a whole; after all, the individual LED chips aren't even accessible to you - they're locked up inside the component and the circuit inside is what makes them go. Here, specifically, the table notes a drive characteristic of 50mA@4.5V . Usually this is under a header of Operating Characteristics but - as is the case here - may be labeled differently.
Though this is not actually the same as a forward voltage, and the current draw varies depending on color displayed, it is Close Enough™.

While this question is specific to color-changing LEDs, the drive characteristic often also matters for simple flashing/blinking LEDs. Even though they only have a single color and you would think that it would behave as a regular LED, just that it flashes, the circuitry that causes that flashing can be very susceptible to the drive characteristic. For example, the datasheet for a randomly googled green flashing LED notes a typical drive of 20mA@3.0V. However, just down from that it additionally notes that the flashing frequency may vary - from 1.5Hz at 70mA@14V to 2.5Hz at 6mA@3.0V (note that these frequencies are never stable - two color changing / flashing LEDs will practically always be out of sync).

But what if the datasheet doesn't mention the operating characteristics? For example, the datasheet for the Betlux BL-L516 only has the electrical characteristics for each LED chip, not the entire component.
In this case, the important thing to keep in mind is that you want each LED color to light up. If you have an 'emergency services LED' which flashes between red and blue (yes, they exist and are very neat), you want to make sure that both red and blue are able to light up or else it's just a flashing red LED. This essentially sets a minimum mA@V rule: that of the highest required. In the case of the BL-L516, that means that the 20mA@3.8V (typical) for the green/blue LED chips is the minimum that is required, and you can safely take this as your drive characteristic.
If we go back to the datasheet for the LED originally asked about, we can see that green and blue once again require the most (which is often the case - the longer the wavelength (more toward infrared) the less power is required) with 20mA@3.5V. If you look at the fourth table, it notes that the circuit will happily work using a minimum of 2.0V and a maximum of 5.0V. So using 3.5V as an assumed voltage drop at the very least cannot hurt the LED, and at the very elast will make the internal driving circuit do something. There is, however, a small chance that the circuit itself drops enough power for the blue, or even the green, LED to light up less brightly - in addition to any color fading-frequency characteristic changes. If you're finding this to be the case, try reducing the value of your current limiting resistor (where applicable), but be sure to do so in small steps; follow the flowchart and remember that There is such a thing as too much current.

• The data sheet's "Application Diagram" circuit does show the battery connected to the device without a current limiting resistor. This sort of implies that the "IC" has current limiting capability as well as voltage boosting. Anyone tried it on 2V to see if it works? It would seem that putting a small enough resistor for the blue led would likely overload the red led. If the IC has no current regulation capability.

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