I understand the points about calculating the series resistance for an LED and the battery voltage, and about using a small resistor just to be safe even if the battery voltage is "about right" for the LED.
The general advice "do not connect LEDs in parallel" refers to this sort of circuit:
This circuit is more complicated than the others, without offering any benefit. You've already got two R2's to limit the current through the diodes. R1 serves no additional purpose. If you want to lower the current through the whole circuit, simply raise the values of the R2 resistors.
So you should ask yourself what advantage this circuit offers over the other. How is it better, or what can it do that the others can't?
In this case you seem to believe that having low values for R2 is an advantage. Why would it be? You need to limit the current through the diodes, and because of the way that semiconductors work, a resistor in parallel would not reliably do this. The reason for this is that a diode that heats up begins to offer lower resistance, which allows more current to flow through it, which causes it to heat up. You can wind up with thermal runaway.
With diodes in parallel, you can wind up with most of the current flowing through one diode. Since the resistor is valued for twice the desired current per diode (since it's in parallel), the diode can easily be stressed into failure.
With individual resistors, the current is limited for each diode, avoiding this problem. Adding yet another resistor in parallel would offer no advantage that I can see.
I'm always looking for a new idea that will be more productive than its
the last two will work.. and the problem is not some much burn out, it is uneven forward current. One will dominate the other, which ever one has the lowest forward voltage.
LED's work with current, not voltage, and have a forward starting point of voltage before they even start to exert current, which is the reason for the R, a simple way to get sufficient voltage but limiting the current.
Trying to regulate the voltage with out over doing the current rating on the LED is very hard to do, unless you have a current source that is regulated or simply linearly limited like an R.
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LEDs are current drive devices so you will always need to limit the current somehow. You will sometimes meet LEDs driven directly from a battery but then the internal resistance of that battery accounts for the current limiting. Sometimes you see LEDs directly in parallel which may do for LEDs from the same batch as long as they are not driven to their limits. It is nevertheless considered bad practice.
Usually you will not use three resistors if you can do with two of them. If the LEDs differ widely they may also influence each other. Nevertheless there may be some use for this circuit. For instance you may want to spread the dissipated heat.
Doesn't mean it's a good design that should be emulated...
The problem with LEDs is that their resistance isn't constant, it varies with voltage.
There's a point where the resistance of a LED drops off a cliff, allowing enough current through to burn them out. That voltage needed for that is usually very very close to their 'optimum' voltage. If you try to run them anywhere near that voltage without current limiting you're living dangerously.
I've run into places where I'm switching on and off tri color leds from the same supply with a common anode (in my case) then a limiting resistor in the supply as well as the cathodes may come in handy to balance light output among different colors.
The typical small (like, 20 to 50 mA rated) LED has the classic exponential current:voltage curve at lower currents, and gets ohmic at higher currents. Actually, almost all diodes do this. Check the data sheets on specific parts.
allowing enough current through to burn them out.
Not usually true. Again, check the curves on a specific part.
Constant-current drive is better of course, but CV can be usable.