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LTC1734 Charger

Also very simple scheme. The charge current is set by the resistor R_prog (for example, if you put a 3 kΩ resistor, the current will be 500 mA).
Chips are usually labeled on the case: LTRG (they can often be found in old phones from Samsung).

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The transistor will suit any pnp in general, as long as it is designed for a given charging current.
There is no charge indicator on this circuit, but the datasheet on the LTC1734 states that pin “4” (Prog) has two functions – setting the current and monitoring the end of the battery charge. For example, a diagram with control of the end of the charge using the comparator LT1716.

In this case, the LT1716 comparator can be replaced with a cheap LM358.