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The way the input source is applied to the Class AB amp, it seems that there should be no output that appear. Because when the input is positive half cycle, the diode D1 is reversed biased, there would not be an input to Q1 and Q2 is cutoff. Why is there an output?
Because Those resitors and diodes bias the transistors with some initial current flow. So there is a current flow in transistors - because they are initialy biased and in bias circuit (resistors and diodes) - because diodes are polarized in forward direction. When you apply signal change between the diodes amount of current that flows to/from bases of transistors will change.
the AB amplifier is biased to combine linearity/ sound quality of an A-class amplifier and the efficiency of a B-class amplifier. The biasing is performed by the two diodes and the two resistors.
As long as your input signal is below a certain value, Q1 is still conductive. This value is V_pos,max = Vcc - (R1 • (I_D1 + I_b1) + Vbe). Is the positive input amplitude above this voltage D1 is reversed biased and Q1 is non conductive.
No amplifier uses that extremely simple circuit that is only used in school books.
As the input voltage increases (positive or negative) then the base resistors need to produce more base current but they do the opposite, they reduce the base current. The result is distortion.