Math Question, How Do You Really Figure This Out

Do you have the data sheet?
Possible you are going over tha max standard operating current.
Try a 22R...I doubt you will see the light output difference anyhow.

 

An LED is not a white hot piece of an accurate length and diameter of wire that can be made to draw certain current at a certain voltage (incandescent light bulb). Instead an LED cannot be made accurately so its datasheet shows a range of forward voltage, a minimum, a typical and a maximum. Some expensive LEDs are tested then "binned" into bins, each bin has a certain voltage.

Since an LED has a range of voltage then the resistor should have about 33% if the supply voltage across it. Calculate the maximum current and select the resistor value when the LED has a minimum voltage and calculate the current to be enough when the LED has a maximum voltage.

 

Or you can buy thousands of LEDs, then measure them all and sort them into piles that have similar voltages.
Many Chinese LED products doo dat.

 

You are already splitting hairs...for example your 13.15 ohm resistor comes with a tolerance value of maybe 1-5 %. But as your graph indicates that the Vf increases about .036V as the current increases by 2mA, you won't get a runaway condition unless the resistor value decreases as the current through it increases with a small wattage type. I've seen where heat buildup in the LED can cause thermal runaway in some so you'd have to take that into account if your driving voltage increases too with a higher temperature inside the enclosure. You driving voltage probably has a thermal component to its actual value too. Choose your max current through the LED which gives you a Vf to design to and just use that for your calculations.

 

Good observation. Forget precision. Settle on any one. Use a 13 ohm 5% resistor and let the current settle where it will.

 

When I buy LEDs their forward voltages do not match. But when I found strings of Chinese LEDs (powered from 4.5VDC) they are all in parallel and the forward voltage matched. Even the series-connected outdoors blue LEDs matched pretty well. Even my cheap Chinese flashlight has 24 well-matched white LEDs in parallel.

The string of blue Chinese outdoors LEDs was also thrown away but I rescued it. Of course it didn't work. You know why?
The wires on the LEDs were simply touching the socket, no soldering. Where they touched there was some corrosion. Every LED worked perfectly when I soldered them into my project.

 

If you want to calculate directly what the current will be at the supplied voltage and a particular external resistor, you have to know the logarithmic I-V equation of diode. Each diode has a slightly different equation due to manufacturing inconsistencies. There is a branch of mathematics called "curve fitting" that deduces the equation from a set of data points. As others have pointed out above, it is hardly worth the effort to do this. The load line method as illustrated above by Harold Kapp is a good way to find the current graphically if you have the diode I-V curve available. Or, you can plot the curve yourself. The iterative method you used should converge on the solution asymptotically.

Ratch

 

Did you measure the forward voltage of your LEDs and group them into matched sets?
Then did you calculate a different resistor value for each set?

 

Since you have only 1 LED then you must have measured its forward voltage so that if it is minimum then it does not burn out and if it is maximum then it is not too dim.

Temperature change causes a tiny variance, its wide range of possible forward voltage is a huge variance.

 

In real life it might quickly burn out or not light. The datasheet is supposed to say its minimum and maximum voltages so that you can design your circuit to use it properly.