The compound pair has two transistors of opposite polarity (NPN, PNP) connected together to give the maximum current and power gain possible. The arrangement is different to the Darlington pair, so, for example, circuits using positive feedback become possible. You can construct a wide range of projects using this simple combination from audio amplifiers to light sensors to pulse generators, often having the merit of a low component count. The compound pair highlights many interesting technical aspects of transistor behavior, especially as the first transistor operates in the common emitter mode at a very low current.
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