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Question:

How can the current be amplified with a transistor without introducing other power supplies?

The whole circuit is powered by a low voltage, insufficient current supply (for example, LED current is small, brightness is not enough, after amplification work properly), so how to use triode to enlarge?

Answer:

I want to say this is out of the question! You don't have enough power to drive a load, and you can't do it without other energy. Again, the transistor amplification, in fact, it is to borrow other energy
Our input is small Ib, and the output is large Ic. Is it enlarged? The Ic is equal to the power Vcc minus the triode and the feedback resistor. The voltage drop across the Rc is obviously independent of the input voltage (as you say, the power supply). So your idea is self-evident!
In a common emitter circuit, the simple point is to control the large Ic with a small Ib. The current magnitude of the load is Ic, which is opposite to the phase.

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