Math behind

[alesh]

Someone suggested that it would be useful to have a thread covering some math behind all those (not only) LED calculations.

So I'll start with explaining what luminous efficacy of radiation and quantum efficacy (?) of radiation are and how they can be calculated from a "spectrum curve".

Firstly we need to digitize an SPD. In this process the input is an image with the SPD curve and the output is set of corresponding relative power values for each unit of wavelength. That is, typically, 400 values between 380 nm and 780 nm. I’m using a software DigitizeIt for extracting the values.
Once this is done we can import the text data into Excel (or whatever you prefer) and plot an XY chart so we can see whether everything went ok.

^^^ a Cree SPD curve


<< text data imported to excel


^^^ text data plotted to a graph (for control)


LER – luminous efficacy of radiation. Tells us how many lumens each watt of a spectrum produces.
NOTE each watt radiated - not dissipated.
NOTE we’ll need luminous efficiency (luminosity) function in the same format as SPD. Refer to this page.

Now some math.
Let’s call SPD function J(λ) and luminosity function y(λ). Then LER can be calculated as following:

This might look a little strange for some but it’s not that complicated.
683.002 lm/W is a constant.
is the area under the graph (SPD multiplied by luminosity function).

is area under the SPD graph.


Still too much? In the language of Excel it’s pretty simple. Let’s have wavelength in column A, SPD in column B and luminosity function in column C. Column D should be filled with B*C then.


Now sum of B column is area under the SPD graph and sum of D column is area under the SPD*luminosity graph. Meaning LER = 683.002 * sum(D: D) / sum(B:B).

FFS I need to upload more files!
QER is gonna follow.

 

[alesh]

With LER we can convert lumens into watts, calculate efficiency of a light source and how much heat it produces. Watt, however, is not a preferred botanical unit. Micromole of photons (in the 400nm to 700nm range) [µmol] is. To be able to convert radiant power into number of photons we’ll have to calculate quantum efficacy of radiation (QER) for our spectra. This quantity tells us how many photons a spectrum produces per radiated energy. The process is quite similar to LER calculation.

Let’s call SPD function J(λ). There will also be some constants:

Planck constant; h = 6.62606957 × 10^-34 J*s
Avogadro constant; NA = 6.0221413 × 10^23 mol-1
Speed of light; c = 299792458 m/s

For each wavelength, energy per mole of photons can be calculated as:

Then:


or if we’re to use 400-700nm range, the equation would change to:


Constants are pretty clear I guess. Then QER is simply ratio of area under the SPD/Energy per mole graph and the SPD graph.

Again Excel will help us. Using the same sheet, we’ve already have 4 columns filled. Now we can add constants somewhere so we can use them later. Column E is to be filled with SPD/energy per mole ((h*c*Na)/λ).

NOTE since wavelength is in nanometers and we’d like to have our final results in micromoles, the equation is changed to (h*c*Na)/(λ*10^-3). It’s for the sake of correct decimal point.

Then QER is simply:
QER = sum(E:E) / sum(B:B)


Zipped Excel book with all the equations is attached.

 

[alesh]

I'm quite fallible sometimes so if you catch a mistake let me know please!

 

[SupraSPL]

This method gets us from lumens to PAR W and from PAR W to umol/s (PPF). Awesome service to the community, thank you sir!

 

[bicit]

Thank you very much, this is awesome

 

[apoulin]

All aboard the learning train! Easy to follow AND there is an Excel file!
Thank you sir

 

[guod]

https://www.rollitup.org/t/various-led-testing-and-extracting-data.833907/#post-10613324

 

[bicit]

@rollitup can this be stickied?

 

[AquariusPanta]

@rollitup can this be stickied?

There should be a sub-thread that the elders here can have the admins place valuable threads into and not just be limited to four stickies... there's just too much damn information here for a few stickies.

Edit:

Did anyone else have trouble at first working with DigitizeIt? It closed on me unexpectedly a few times. Any tips on getting the lines to light up as a whole? I would get sections of a line or two lines on the graph but never one full line; I'm working with the new Vero 18 SPD chart, which has no other colors besides black and white and the lines are really bumpy.

 

[jtizzle]

@alesh -you are the mfn man. If I could post a praising emoticon I would.


Sent from my iPhone using Tapatalk

 

[EfficientWatt]

@alesh : Nice work !

I missed this when originally posted, or wasn't deep enough yet (!)

This is the 3000K 90cri I beleive .. did you do this for each spectrum ?
If yes, may I ask what were your other worked out LERs ?

 

[alesh]

@alesh : Nice work !

I missed this when originally posted, or wasn't deep enough yet (!)

This is the 3000K 90cri I beleive .. did you do this for each spectrum ?
If yes, may I ask what were your other worked out LERs ?

This particular one really is the 3000K 90CRI spectrum from Cree (a CXA).

But yes I've digitized quite a lot of different spectra.
So far I've got:
Cree - CXx 3000K/80CRI/90CRI, 4000K/80CRI, 5000K/70CRI; XP-E red
Bridgelux - Veros both versions, 3000K/80CRI/Décor(97CRI), 4000K/80CRI, 5000K/70CRI
Seoul Semi - SDW87F1C 3000K/80CRI
Philips - CrispWhite 3000K/90CRI; Rebel deep red; Green power 600W HPS
Osram - Oslon hyper/deep red, Plantastar 600W HPS
Inda Gro Pro-420-PAR
Gavita Pro 1000W UL DE
EYE Hortilux 600W Blue MH

 

[Scotch089]

Omg.. You're sitting on a gold mine!

If you ever find time, start a thread! And tag me.. lol

 

[medicinehuman]

I'm quite fallible sometimes so if you catch a mistake let me know please!

I'll take your word for it.

 

[alesh]

Omg.. You're sitting on a gold mine!

If you ever find time, start a thread! And tag me.. lol

No need for a new thread. I can share them if someone's interested.

 

[robincnn]

I am still trying to understand th entire math. I have these values from COB spreadsheet thread. Please update if you have newer version.

I assume these values related to case temprature and current. Vero SPD says Tc 25 degree C
should it be safe to assume these values are for nominal drive current. 2.1 amp for vero 29.

Cree CXB SPD are at Tj 85 and 2400ma for 36v version and 1200ma for 72 v version

Question: if we drive cob at lower current would we not get higher umol/j than these vales since these vales are based on SPD, which is based on higher current than we use.

 

[Abiqua]

I am still trying to understand th entire math. I have these values from COB spreadsheet thread. Please update if you have newer version.
View attachment 3478536
I assume these values related to case temprature and current. Vero SPD says Tc 25 degree C
should it be safe to assume these values are for nominal drive current. 2.1 amp for vero 29.

Cree CXB SPD are at Tj 85 and 2400ma for 36v version and 1200ma for 72 v version

Question: if we drive cob at lower current would we not get higher umol/j than these vales since these vales are based on SPD, which is based on higher current than we use.

SPD is not based on current, afaik, but rather what the diode would produce at 100% efficiency....Current is just a product of efficiency, which will fall or rise depending on the level of current. That said, it has nothing to do with the actual calculation of the SPD....

 

[alesh]

SPD is not based on current, afaik, but rather what the diode would produce at 100% efficiency....

Correct. These values related to SPD and SPD only. It's true that SPD changes with current and temperature but these changes are small and negligible. Beside that I have no means to count them in.

@robincnn Multiply these values by electrical efficiency and you'll get how many photons a light source actually produces per each joule of dissipated energy.

 

[RMF]

Thanks for the effort in trying to explain all this stuff, but to be honest caused more confusion in my mind...
What is SPD? What is luminosity? What's the difference between efficacy and efficiency?
Where do the constants come from and how did you come up all with those formulas?

Not trying to bash anyone here, just trying to learn... Is there some source of information where I could gather more info
about this? I can understand basic math and physics, but there seems to be a lot of concepts missing (or maybe already covered in other threads?) in order to grasp the concepts laid out here, unless one is already a led guru...

Can someone link me to a few sources, texts, whatever, where the basics are covered?
I mean, what kind of knowledge does one need to have in order to grasp the concepts laid out here?

Thanks, and sorry if I somehow come across as being negative... that's not my intention at all... I just feel I need to start from the very basics and build up a more solid foundation of knowledge about the topics covered here.

 

[SupraSPL]

This thread is probably for those who are somewhat advanced but it is great to focus on the basics and return to the basics often. Sometimes it helps to look at things oversimplified.

SPD means color
LER is the maximum lumens/W for a given color.
QER is the maximum photons/s/W for a given color.
In order to get to those maximums we would have to be using lights with 100% efficiency.
Efficiency means how much light and how much heat we get from the LED
These days good COBs range from 35-70% efficient

Efficacy means how "effective" something is. For example if you tried to grow with only pure blue LED, efficiency could be very high but efficacy would be very low.

Luminosity in this case is referring to how well each wavelength (color) stimulates the perception of brightness to the average human eye. It looks like this:


It helps us figure out the LER. Alesh digitizes the SPD graph for a given color and converts each bit of it with the luminosity data to figure out the LER. So if the CREE data tells us we are getting 150 lumens/W and we know the LER is 325, we know that efficiency is about 46%.

It would be great if CREE would just tell us the LER and QER for each color, and the efficiency at each current. But these COBs were intended for human vision so we have to work backwards from SPD graphs and lumen/W data. Aleshs LER and QER numbers have really helped us get a handle on where we stand and how lights will compare.