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How to get 5V at least at 1 Ampere from a very low voltage source, at nearly 1 Volt capable of 10 Amperes; using a power transistor or mosfet and larger wire transformer instead of the usual awg#38 on ferrite bead ?
The problem with trying to boost a very low voltage like that is switching losses.
ie: A 0.5V drop across the switching device in a 10V input supply doesn't effect the efficiency as bad as a 0.5V drop across the switching device in a 1V supply would.
You could possibly use the classic joule thief circuit to supply enough voltage to start up one of the standard switching ICs. Once running, the IC could supply it's own source of higher voltage for itself and driving the FET switch.