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3. Divide 12V (13V) with 3,6V you get number 3
4. 3,6 V X 3 =10,8 V
3.6 V X 4 =14,4 V
5. You can securely take 4 LEDs in series.
Each line of 4 LEDs give you 20mA X 14.4 V = 0,028 W of needed/used power .
6. Desktop lamps are usually halogen lamp about 10-20W or more, so transformer is already overpowered ...
Sincerelly,
I have 4 LED in series... it means 4 x 3 V = 12V total.
Therefore I have 8 lines at 4 LEDs...paralel...
You can put limen at 4 LEDS how much you want
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Or instead of a flat plexiglass surface, make something with a curve, similar to how a headlight works in your car. It takes one bulb, and because of the concave reflective surface, it projects the light with stronger intensity.
ONLY solution that you put so much LED s to reach power consumption of 80 % of halogen. In my case I have to add 20 strip of 4 LED s, to get similar output and quantity of light.
I've used metal duct repair tape as a reflector in several projects and it worked well.