Generally I see efficiency related to the lumens per watt, but what is the actual typical efficiency of LEDs in terms of electrical energy in to optical energy out? What sort of conversions apply?
Electronic – How efficient are LEDs
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I'm not ecstatic about the following picture shown in the TI document: -
I've drawn a big red square rectangle on the top diagram and it seems TI are saying the efficiency is between two limits: -
Minimum = 0.95 x 0.80 x 0.80 = 60.80% Maximum = 0.95 x 0.90 x 0.90 = 76.95%
For the wireless diagram they are saying its efficiency is 60% to the equivalent "wired" point but they are neglecting the lower limit so this has a biased feel about it. The lower limit would be: -
Minimum = 0.60 x 0.95 x 0.89 x 0.89 = 45.15%
Read the small print. They say they are using 80% adapter efficiency yet surely a plastic box that plugs into the AC that produces 5V cannot be assumed to have a significantly worse efficiency to a plastic box that produces 19V.
Your "LED Driver" is most likely bad
The driver is clearly out of spec and mostly likely internally damaged. From you photos it isn't clear how the rest of your lights and power sources are connected, but you may have made an error here (I've even seen people connect the AC power line to the DC output side of the supply).
You have a current source, not a voltage source
If you look at the label you will notice that current is specified precisely (600mA) and voltage is specified as a range (16v-28v). You will also notice that the drawings on the label show a single current loop and specify 7 3-Watt LED's in series.
That you provided this equation indicates that you are confused about the difference between the two types of sources:
$$R=\frac{V_{Driver}-V_{LED} \times 7}{I_{LED}} = \frac{35\ V-3.7\ V \times 7}{0.7\ A} = 13\ \Omega\ $$
In your equation, you cannot know the value of $$V_{Driver}$$ as it is determined by the network (it's somewhere between 16 and 28 volts if an acceptable load is attached). Only the current value is constant in the normal operating condition.
Some background
A voltage source presents a single output voltage, no matter what you connect to it. To make that a true statement it will output any current the load wants up until it is incapable of outputting any more (current-limit or failure). Most people are familiar with this behavior as it is intuitive and commonly encountered.
A current source will attempt to output a constant current no matter the load attached to it. It will do this by adjusting the output voltage until it either can raise it no further (limit of it's upward adjustment range) or lower it no further (it will produce insufficient voltage to operate itself and shutdown).
This works via Ohm's law (V = I R) such that increasing the voltage will increase the current flowing and decreasing the voltage will decrease the current. The system is active and senses it's output current (while adjusting its output voltage) until the output current equals the number printed on the label (in this case 600mA).
If nothing (or too little load) is attached, it will output it's maximum voltage as it keeps trying to increase voltage to get increased current... and vice versa if too much load is attached.
Driving LED's
If your LED's are connected as parallel strings of series lights, they will need to be driven by a voltage source. This configuration is cheaper to design and build, but more difficult to install and subject to greater line losses as the LED strings get bigger (since you need to bring the full voltage of the power supply all the way to the end of the line).
If your LED's are connected in series (one to the next), then the same current that drives one light will drive the next. This configuration is used in most higher-end architectural lighting. You use a current source to ensure that no matter how many lights are on the string, the current output remains the same. The advantages are that you can easily add lights to existing strings without worry. The fact that current flows through all lights ensures that voltage losses in the line are minimal (most efficient power distribution). And, LED light output is proportional to current so a current driven approach best ensures uniformity of light output.
There are some limitations however. Current drivers are more expensive as they are produced in lower quantities and have to be matched to the exact fixtures being used. The fixtures must all be the same so that they have the same light-to-current relationship and will not be too dim (or burn up) under the constant current value applied. Series wiring of light fixtures is inconvenient in some installations.
Best Answer
To make things clear let's define what we are talking about.
There are two terms which are mixed up pretty often:
The luminous efficiency is a dimensionless quantity which is derived from the luminous efficacy. It is simply the quotient of luminous efficacy of the source and maximum possible luminous efficacy of radiation.
This is the value you see more often. It usually has the unit of lumen per watt. And gives the luminous flux per power, which is a useful quantity to see how much light we will get with a given power.
With this we have to be a bit careful as well. Because the power can be the radiant flux of the source or the electrical power. So the former can be called luminous efficacy of radiation, and the latter luminous efficacy of a source or overall luminous efficacy.
Now the problem arises, that we cannot see all colors equally well. And lumens are actually weighted based on the response of our eye:
Public Domain, Link
So with this, you can create some values of upper bounds (based on the redefinition of the unit candela). This would be the luminous efficacy of radiation.
Which are:
For more see here.
If you lower the color rendering index (CRI), you can achieve higher values. But not higher than 683 lm/W.
So how efficient are LEDs?
Here we have values of luminous efficacy of a source.
Well there is a race of efficiency. Cree posted a press release with a laboratory LED of 303 lm/W at 5150K. The CRI was not mentioned, I guess it is lower than 95, but based on the data above that seems like it would have a luminous efficiency of something like 80% to 90%.
Of course your average available LED has less. 100 lm/W would be around 25% to 30% and the new 200 lm/W chips announced recently (as of August 2017) reach 50% to 60%.
Note that the above is for photopic vision (day-vision), things change with scotopic vision, but that's usually not so interesting.
If you really want to get into the guts of it, you'd have to take the spectrum of the LED and find out what the highest theoretical maximum for that spectrum is (based on the weighting curve) and then you can calculate the value.
As each and every LED has a different spectrum it is hard to get this data easily.
I hope I haven't made a mistake here, because I always find the topic a bit confusing no matter how many times I revisit it.