Welcome to the boards. I'll take your several questions in order:
I would like to know what formula you are using to convert cd to lm.
1cd = 1 lm/sr, but more specifically, the calculator is simply
lm = cd * 2 * pi * 1 - cos(theta/2) , where theta is the beam angle.
The term 2 * pi * 1 - cos(theta/2) is nothing more than the solid angle subtended by the beam (in steradians, effectively dimensionless):
solid angle = A / r^2 = 2 * pi * h / r = 2* pi * (1- cos( theta/2))
If you'd like a reference, allow me to cite
I'd like to know which is not precise.
The worst problem is that you don't really have specs on how the luminous intensity measurement was taken. The formula I use assumes it represents an average value of the beam that falls within the beam angle. It's entirely possible that a manufacturer should choose to sample more narrowly than this in order to obtain a higher published spec. If you think that isn't likely, you are less cynical than I.
The real point is that my formula is a direct consequence of geometry--error will arise from imprecision in the published spec that violates the assumptions inherent in the calculation.
Also, why don't you already display the formula that you use?
I guess it seemed like a trivial geometry problem to me, and not really the most significant caveat I could issue. It may be worth stating that the formula assumes the beam is a circular cone, and that the luminous intensity represents an average over the published beam width spec.
It's critical for those trying to convert conventional lighting to LEDs to have exact formulas.
I never presumed to serve the professional lighting community. I was under the impression their literature dealt with this sort of calculation as a matter of course.
+++
Okay, to take a step back from that, I should probably note that I wrote the calculator precisely because I was annoyed at websites suggesting there was a constant conversion factor between lumens and candelas (I've seen 4*pi stated), or even more alarming,
Google's suggested conversion.
If you dig hard enough (including offline) there are excellent resources for information on radiometry, photometry, illumination engineering, colorimetry and the related fields. What's available online is spotty. (I'm looking for the chromaticity coordinates of the Planckian locus, for example, and while I can find dozens of instances of it plotted on a CIE chromaticity chart, I can't uncover a formula in terms of x and y. I'm confident this is answered in a book, but I haven't yet obtained the correct book.) With this site I aimed to bridge the gap between enthusiasts or hobbyists and the large body of technical information on lighting.
As to the differences in the formulas, if I read yours correctly, it seems it assumes the beam impinges on a flat surface, mine assumes it impinges on a spherical one. I'll note again that for theta = 2*pi, my formula returns 4*pi, the entire area of a sphere.
[edit: minor typos]
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