The Joule Thief circuit is a voltage boost circuit, and the supply voltage cannot be more than the LED's forward voltage, which is about 3V for white and Blue LEDs. If you want to run a single white or blue LED from a 9V battery, just connect a 300 or 330 ohm resistor in series with the LED and then connect them to the 9V battery. You can put two LEDs in series along with a 130 or 150 ohm resistor and connect them to the 9V battery. Remember that 20 milliamps current will be drawn and the 9V battery will give maybe 8 hours of full brightness. After that, the LEDs will dim and run for a much longer time at a much reduced brightness. You can increase the resistor to get lower brightness and longer battery life. You could modify the JT circuit to run off the 9V battery. But why would you want to when you can do it simply with a single resistor? But 9V alkaline batteries cost so much more than AA cells, it's really a waste of money to use a 9V when a JT circuit will give you the same results and cost much less for the battery.
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Why? What voltage do you want / need? L
Conceptually, sure. Especially if you only need the kind of output levels a normal Joule Thief gives you.Do I have a design for one? No.