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I know this seems like a really basic question, but bear with me. The following circuit diagram is given by one of my old electronics kits:

schematic

simulate this circuit – Schematic created using CircuitLab

The book specifies this is a "light organ" whose sound output at the piezo transducer varies in pitch based on the light input. I understand that the impedance of the photoresistor drops with more light, but how does this oscillator work? What is the role of the BJT in this circuit? (To simplify, one could just replace the photoresistor with a potentiometer or even a fixed resistor.)

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In most basic terms, you need some kind of amplification to make the circuit oscillate. That's the purpose of the transistor--to provide the amplification. The other requirement for oscillation is some kind of feedback from the output to the input. The actual power to the circuit comes from the battery to the tap on that coil. Then, the topmost connection on the coil (which will have the signal induced on it) is fed to the base of the transistor to make it oscillate.

Also of note is that the voltage across the entire coil will be higher than the bottom half (which the transistor is managing) and the higher voltage is more suitable for driving a piezoelectric element.

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Basically this circuit transforms the variations of voltage v(t) of the photoresistor due to the light intensity in variations of current i(t) in the oscillator circuit. This transformation is done by the bipolar transistor, it is its function to amplify the v(t) signal.

The variations of i(t) change the frequency of the circuit oscillator circuit, it causes the different sounds. (example of a RLC circuit frecuency and current dependency )

The BJT must be configured correctly to work into the linear region (non-saturated), then if you change the photoresistor with a potentiometer, it will do the same effect, but consider first the correct range of the potentiometer, other case maybe it won’t work, excess of base current could change the transitor state to the saturated region.

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