Aphase-shift oscillator is alinearelectronic oscillator circuit that produces asine wave output. It consists of aninverting amplifier element such as atransistor orop amp with its outputfed back to its input through aphase-shift network consisting ofresistors andcapacitors in aladder network. The feedback network 'shifts' thephase of the amplifier output by 180 degrees at the oscillation frequency to givepositive feedback.^[1] Phase-shift oscillators are often used ataudio frequency asaudio oscillators.

The filter produces a phase shift that increases withfrequency. It must have a maximum phase shift of more than 180 degrees at high frequencies so the phase shift at the desired oscillation frequency can be 180 degrees. The most common phase-shift network cascades three identical resistor-capacitor stages that produce a phase shift of zero at low frequencies and 270° at high frequencies.

The first integrated circuit was a phase shift oscillator invented by Jack Kilby in 1958.^[2]

This schematic drawing shows the oscillator using acommon-emitter connectedbipolar transistor as an amplifier. The two resistorsR and three capacitorsC form theRC phase-shift network which provides feedback from collector to base of the transistor. ResistorR_b provides base bias current. ResistorR_c is the collector load resistor for the collector current. ResistorR_s isolates the circuit from the external load.^[3]

This circuit implements the oscillator with aFET.R₁,R₂,R_s, andC_s providebias for the transistor. Note that the topology used for positive feedback is voltage series feedback.

The implementation of the phase-shift oscillator shown in the diagram uses anoperational amplifier (op-amp), threecapacitors and fourresistors.

The circuit's modeling equations for the oscillation frequency and oscillation criterion are complicated because each RC stage loads the preceding ones. Assuming an ideal amplifier, with very low output impedance and very high input impedance, the oscillation frequency is:

${\displaystyle f_{\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=\frac{1}{2\pi \sqrt{R_2{R}_3(C_1{C}_2+C_1{C}_3+C_2{C}_3)+R_1{R}_3(C_1{C}_2+C_1{C}_3)+R_1{R}_2{C}_1{C}_2}}}$

The feedback resistor required to just sustain oscillation is:

The equations are simpler when all the resistors (except thenegative feedback resistor) have the same value and all the capacitors have the same value. In the diagram, ifR₁=R₂=R₃=R andC₁=C₂=C₃=C, then:

${\displaystyle f_{\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=\frac{1}{2\pi RC\sqrt{6}}}$

and the oscillation criterion is:

${\displaystyle R_{\mathrm{f}\mathrm{b}}=29\cdot R}$

As with other feedback oscillators, when the power is applied to the circuit, thermalelectrical noise in the circuit or the turn-ontransient provides an initial signal to start oscillation. In practice, the feedback resistor must be a little bit larger so the oscillation will grow in amplitude rather than remain the same (small) amplitude. If the amplifier were ideal, then amplitude would increase without limit, but in practice amplifiers are nonlinear and their instantaneous gain varies. As the amplitude increases, amplifier saturation will decrease the amplifier's average gain. Consequently, the oscillation amplitude will keep increasing until the averageloop gain of the circuit falls to unity; at that point, the amplitude will stabilize.

When the oscillation frequency is high enough to be near the amplifier'scutoff 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.

The single op-amp circuit needs a relatively high gain (about 30) to maintain the oscillation due to the RC sections loading each other.^[4] If each RC segment did not affect the others, a gain of about 8 to 10 would be sufficient for oscillation. An isolated version of the oscillator can be made by inserting an op-amp buffer between each RC stage (this also simplifies the modeling equations).

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