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This circuit it´s a simple DC DC converter...only this...not have many "dark" secrets. If You have minor knowledge in any mobile phone, You can see that the LiIon battery voltage for charger are: 4.0 or 4.2VDC maximun, the current charge maybe change during the charge process, but not much.
This circuit can be simulated in ORCAD PSPICE for example in case You need verify some details, I not have much time now for create a schemetic in ORCAD and models for LiIon battery and others devices...if You have time...have a fun!
I can post this circuit here because much EDA brothers have related circuits for communicate with mobiles, but I dont see anything about chargers for this circuits...then I buy one cigarette charger, open it and put the schematic here...
Sorry my brotha! I do not have the exact values, but I suspect that it is between 10 to 100uH because the frequency is not too hight (remember: this is a very cheap charger). I recommend you try different inductors and capacitor values to get one frequency around 25 to 35kHz!