# Phase-shift oscillator

Phase-shift oscillator

A phase-shift oscillator is a simple sine wave electronic oscillator. It contains an inverting amplifier, and a feedback filter which 'shifts' the phase by 180 degrees at the oscillation frequency.

The filter must be designed so that at frequencies above and below the oscillation frequency the signal is shifted by either more or less than 180 degrees. This results in constructive superposition for signals at the oscillation frequencies, and destructive superposition for all other frequencies.

The most common way of achieving this kind of filter is using three cascaded resistor-capacitor filters, which produce no phase shift at one end of the frequency scale, and a phase shift of 270 degrees at the other end. At the oscillation frequency each filter produces a phase shift of 60 degrees and the whole filter circuit produces a phase shift of 180 degrees.

Op-Amp implementation

One of the simplest implementations for this type of oscillator uses an operational amplifier (op-amp), three capacitors and four resistors, as shown in the diagram.

The mathematics for calculating the oscillation frequency and oscillation criterion for this circuit are surprisingly complex, due to each R-C stage loading the previous ones. The calculations are greatly simplified by setting all the resistors (except the negative feedback resistor) and all the capacitors to the same values.In the diagram, if R1 = R2 = R3 = "R", and C1 = C2 = C3 = "C", then:

$f_\left\{Oscillation\right\}=frac\left\{1\right\}\left\{2pi RCsqrt\left\{6$

and the oscillation criterion is:

$R_\left\{feedback\right\}=29cdot R$

Without the simplification of all the resistors and capacitors having the same value, the calculations become more complex:

$f_\left\{Oscillation\right\}=frac\left\{1\right\}\left\{2pisqrt\left\{R_2R_3\left(C_1C_2+C_1C_3+C_2C_3\right)+R_1R_3\left(C_1C_2+C_1C_3\right)+R_1R_2C_1C_2$

Oscillation criterion:$R_\left\{feedback\right\}= 2\left(R_1+R_2+R_3\right)+frac\left\{2R_1R_3\right\}\left\{R_2\right\}+frac\left\{C_2R_2+C_2R_3+C_3R_3\right\}\left\{C_1\right\}+frac\left\{2C_1R_1+C_1R_2+C_3R_3\right\}\left\{C_2\right\}+frac\left\{2C_1R_1+2C_2R_1+C_1R_2+C_2R_2+C_2R_3\right\}\left\{C_3\right\}+frac\left\{C_1R_1^2+C_3R_1R_3\right\}\left\{C_2R_2\right\}+frac\left\{C_2R_1R_3+C_1R_1^2\right\}\left\{C_3R_2\right\}+frac\left\{C_1R_1^2+C_1R_1R_2+C_2R_1R_2\right\}\left\{C_3R_3\right\}$

Another version of this circuit can be made by putting an op-amp buffer between each R-C stage which simplifies the calculations.

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