This stronger driver is able to drive LEDs by applying a high logic level sourced from output pin of a logic IC belonging to common 74xx logic families (HC, HCT, LS, etc.) or CMOS microcontrollers.
The output pin of IC is simulated by a voltage supply and a switch.
General rule of thumb is that the output pin of the IC has to be able to source at least 1 mA of current when it is set to high logic level.
As a real NPN transistor for the first stage, you can use almost any "jellybean" small-signal one, such as BC54x or 2N3904, which are cheap and widely available.
For the second stage, you have to choose a type with greater current handling capability, such as BC337 or 2N2222.
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