Phase-shift oscillator

A phase-shift oscillator is a simple electronic oscillator circuit. It consists of an inverting amplifier element such as a transistor or op amp, with its output fed back into its input through an filter consisting of a network of resistors and capacitors. The feedback network 'shifts' the phase of the amplifier output by 180 degrees at the oscillation frequency, to give positive feedback. Phase shift oscillators are mostly used at lower frequencies, often in the audio frequency range as audio oscillators.

The filter produces a phase shift that increases with frequency. It must have a maximum phase shift of considerably greater than 180 degrees at high frequencies, so that the phase shift at the desired oscillation frequency is 180 degrees.

The most common way of achieving this kind of filter is using three identical cascaded resistor-capacitor filters, which together produce a phase shift of zero at low frequencies, and 270° at high frequencies. At the oscillation frequency each filter produces a phase shift of 60° so the whole filter circuit produces a phase shift of 180°.

Op-amp implementation

A simple example of a phase-shift oscillator

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:

$f_\mathrm{oscillation}=\frac{1}{2\pi RC\sqrt{6}}$

and the oscillation criterion is:

$R_\mathrm{fb}=29 \cdot R$

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

$f_\mathrm{oscillation}=\frac{1}{2\pi\sqrt{R_2R_3(C_1C_2+C_1C_3+C_2C_3)+R_1R_3(C_1C_2+C_1C_3)+R_1R_2C_1C_2}}$

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

A version of this circuit can be made by putting an op-amp buffer between each R-C stage which simplifies the calculations. Тhe voltage gain of the inverting channel is always unity.

When the oscillation frequency is high enough to be near the amplifier's cutoff frequency, the amplifier will contribute significant phase shift itself, which will add to the phase shift of the feedback network. Therefore the circuit will oscillate at a frequency at which the phase shift of the feedback filter is less than 180 degrees.