Added:
You advise 3W load on a 50 W transformer.
I suggested 85-90% efficiency.
Transformer will draw magnetising current when unloaded. This will incur copper losses but will not be the full loaded current.
10W additional = 80% of rated power which sounds much too high.
You have still declined to tell us HOW you are measuring power - you say it jumps by 50 Watts. Measured how & with what. The only certain measure would be a power factor correct Watt meter. ie the house meter would qualify - a plug in watt meter quite possibly wouldn't.
More information is needed.
You mention "halogen mal transformers" - are they 50 Hz iron core transformers or smps (switch mode power supply) converters?
You imply that the LEDs operate from 12V and
imply that they may usually run on DC.
You suggest that they run OK on the 12 VAC from the "transformers" which may be
50 Hz AC or
10's to 100's of KHz AC?
You say "it would appear that you are using 15W/LED lap but knowing how you measure this in as much detail as possible could be useful. Or not :-).
You imply that it is not just the VAC x IAC product as yoyt say that in the output VAC x IAC = 15W but compensating for RMS gives 5W. This raises the question as to what it is you measure and how to get the 15W and how you RMS convert to get 5W. (The factor of 3 does not untuitively drop out of any conversion that comes to mind).
SO if you can fill in all the bgaps, and add a diagram if your word picrure would benefit from it, it may help.
Iron core power supplies may be in the 85% - 90% efficiency range but better is conceivable, and any fool can get less than that, and some do. (A 60Hz designed transformer run at 50 Hz can get nice and toasty. Ask me how I know :-).)
An electronic "transformer" can get just about any efficiency at all. 95%+ is doable and 65% would not be a surprise in some few cases. 80%-90% should be hoped for if not always expected.
Running LED lamps that are "designed" [tm] for DC use on AC may cause problems. At 50 Hz flicker may be an issue. IF the lamps have intelligent and/or active internal regulation to a lower DC voltage or to constant current then conceivably the need to provide a pulse of power input twice per cycle (or quite possible once per cycle if they are polarity dependant ) may cause massive converter overload and may even cause core saturation problems upstream in the 230:12 VAC supply. For extra points they may have placed a reverse diode across the supply input for "protection" and this may be causing a massive current peak every unused reverse half cycle.
SO too many guesses needed, not enough data.
A much fuller description should help.
IF you take the 12 VAC, feed through a full wave bridge and then add a LARGE electrolytic cap so there is well under 1 Volt ripple on the DC, and drive the lamp from that, what power does it draw?
Any web links for transformers, LED lamps, ...?
Best Answer
Halogens are MORE efficient than LEDs in selected parts of the IR spectrum if the selected portion is wide enough. It deep-ends how wide a slice of spectrum you want to consider useful.
For narrow bandwidths LEDs are better.
In whole visible range LED is much better than halogen as halogen has much more IR out than visible out.
Halogen is 100% efficient overall as ALL energy in comes out as light and "heat".
A GOOD modern (narrow band) LED at one wavelength puts out 1/3 to 1/2 of energy-in as wavelength-out. The rest goes to heatsink as widish band heat.
If the heatsink IR out is not useful in your application it is lost to you.
IF the LED heatsink IR is useful to you then it too is 100% efficient.
Odds are you do not want the LED's heatsink IR.
As you narrow the halogen bandwidth you use you use less of total output and "useful efficiency falls". When you drop under 1/3 to 1/2 of total energy in bandwidth you consider useful then LED is more efficient.
eg when using a halogen for optical lighting we consider a bandwidth that contains about 5% to 10% of the total energy out to be useful. So a modern LED is always more efficient for lighting at its design wavelength than a halogen is and more efficient in the whole optical band (all energy summed) than a halogen is.
If we use a halogen bulb to illuminate a typical silicon solar panel we find its efficiency RISES as a portion of the IR output falls in the panel's response range.
An (apparently) excellent reference: A major problem in trying to do useful things in this field is that, as in most areas involving people + healing / health / therapy / well being / feel good ... there is an immense amount of hype, hearsay, suspect claims, bad science and general rubbish to wade through. That is not to say that there are not very real and demonstrable benefits available - just that sorting the (w)heat from the chaff can be difficult.
This reference LED Light Therapy provides 29 pages of comment, reported results, and comments on investigations. It's not perfect, but at a quick glance it looks better than much that can be found. And it will probably address the OP's less than fully specified question better than any answer here can do. While the title suggests it's about LEDs it also deals well with halogen light use.
It may be educational to examine the sites usage of terms such as IR, heat and wavelength. Maybe not :-).