Derivation of the frequency of a phase shift oscillator
Consider the following oscillator circuit: For a circuit to oscillate, it must meet the following criteria: Total loop gain is 1 The total phase shift is a multiple of 2π radians One can see that the op-amp is wired as an inverting amplifier, so it has a phase shift of π radians. Therefore, the phase shift network consisting of capacitors C 1 , C 2 , C 3 and resistors R 1 , R 2 , R 3 must provide the remaining π degrees of phase shift in order for the circuit to meet criterion 2. Let \(f\) be the frequency at which the circuit oscillates. The impedance of the capacitors at frequency \(f\) is \(Z_c=\frac{1}{2\pi jfC}\), where j is the imaginary unit and \(C\) is the capacitance of all the capacitors. Let \(R\) be the resistance of all the resistors except R f . It is also known that given two linear devices with impedances Z 1 and Z 2 (either real or complex impedances): their total impedance when wired in series is \(Z_1+Z_2\) their total impedance when w