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I am trying to use a Joule thief circuit to harvest energy from an almost dead battery, but the references that I found on forums or in conference papers just explain how the Joule thief circuit works, not the configuration and mathematical modelling for achieving the best energy harvesting configuration.

Can anybody explain or give me a reference about mathematical modelling of the Joule thief circuit?

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    \$\begingroup\$ The circuit you used earlier has problems, with respect to your solar cell specifications. You also haven't discussed how you made your measurements regarding the solar cell, as well. The Joule Thief circuit has some varieties, as well. There isn't a "single circuit" to discuss. Instead, it's more like a meandering dialog discussing the problems with the simpler iterations and walking through various iterative ideas about improving it. But it is mainly used for extracting the last bit of energy from a battery and a lot less about solar cell usage. So this is a wide-ranging discussion, really. \$\endgroup\$
    – jonk
    Commented May 24, 2019 at 5:34
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    \$\begingroup\$ That circuit needs a higher voltage to generate 50% efficiency like 2 or 3 V Would you like me to show you? \$\endgroup\$ Commented May 24, 2019 at 6:42

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The Joule thief circuit is a standard Boost converter with a sense winding using a coupled inductor and can be modeled first as a simple Boost converter. You can use an LTspice simulation of an MC34063 boost converter to get started. (LTspice is free, and it runs on a PC).

This question, Efficiency question of # of LEDs for 6 volt 4 cell Joule Thief Flashlight circuit, has the LTspice source code for discrete modeling and simulation of a Joule Thief. Just delete all LED's except one, and change the voltage source to be 1.5V down to whatever still works.

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