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In a crystal radio a diode is used as a detector. How does it work? If diodes rectify current (clip off the bottom) why doesn't it just distort the signal. How does it pull the audio signal out of the carrier frequency?
Diode detector is used to demodulate AM (Amplitude Modulated) signal. The AM signal is a symmetrical signal having similar information on both sides of the X-axis. So at the demodulator end you can use either side of the signal to reproduce the original signal. The diode does the job of removing any one side (either negative half or positive half).
The capacitor acts as short for AC signal depending on its impedance. The carrier of AM signal is of very high frequency and the capacitor in demodulator is so chosen that it reacts heavily to the carrier signal and eliminates it (by grounding it) thereby producing only the relatively low frequency modulating signal at the output.
The impedance of a capacitor is Xc=1/(2*PI*f*C)
f=Frequency of input signal (carrier frequency in our case)
C=Value of capacitor
Ideally Xc should be zero for desired carrier frequency but it is generally not possible so a very low value like 1R or 10R or so is chosen and value of C is calculated.