Consider a sinusoidal oscillator consisting of an amplifier having a frequency-independent gain A (where A is positive) and a second-order bandpass filter with a pole frequency ω0, a pole Q denoted Q, and a positive center-frequency gain K.
a) Find the frequency of oscillation, and the condition that A and K must satisfy for sustained oscillation.

Sinusoidal oscillator frequency of oscillation is given below.

Explanation:

The criterion for a stable oscillator is given in the equation

l A(jw)β(jw) l ≥ 1

In this task A represents the gain of the amplifier , and

β represents gain/attenuation of the second-order bandpass filter.

This sinusoidal oscillation is a special edge case where the product is equal to one.

So the condition is A-K=1

to obtain the sustained oscillations at the desired frequency of oscillations, the product of the voltage gain A and the feedback gain β must be one or greater than one. In this case, the amplifier gain A must be 3. Hence, to satisfy the product condition, feedback gain β must be 1/3.