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Simple Boost Converter

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1A at 12V plus 1A at 5V is a total of 17 watts. Your 3.7V battery will need to supply 6 or 7 amperes.

Go home and do some math.
 
I should have given some more details. The battery is a 3.7V 3000mAh lithium ion cell. Only at 5V, we need a 1A supply, which is possible with the efficiencies that switching regulators have. At 12V, we are looking at lower supply currents (around 0.5-0.65A).
 
You can increase output voltage to around 200 to 300 volts if you introduce a ferrite core in your inductor (at the max for the frequency of your timer). You can use a cheap TIP162 or TIP3055 darlington transistor and they do not heat up even with continuous currents of 10 Amps and peak currents of 15 Amps. It is pretty efficient for a boost converter and if you increase the frequency of the timer to 100,000 Hz and use a fast recovery diode of the RUR.... series you can get the voltage to around 50 Kv But you need a fast recovery NPN power transistor for that.
 
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