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<p><b>New page</b></p><div>A '''phase-shift oscillator''' is a [[linear circuit|linear]] [[electronic oscillator]] circuit that produces a [[sine wave]] output. It consists of an [[inverting amplifier]] element such as a [[transistor]] or [[op amp]] with its output [[feedback|fed back]] to its input through a [[Electronic filter|phase-shift network]] consisting of [[resistor]]s and [[capacitor]]s. The feedback network 'shifts' the [[Phase (waves)|phase]] of the amplifier output by 180 degrees at the oscillation frequency to give [[positive feedback]].<ref>http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/oscphas.html</ref> Phase-shift oscillators are often used at [[audio frequency]] as [[audio oscillator]]s. <br />
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The filter produces a phase shift that increases with [[frequency]]. 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.<br />
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==Implementation==<br />
===Op-amp implementation===<br />
[[Image:RC phase shift oscillator.svg|frame|A simple example of a phase-shift oscillator]]<br />
One of the simplest implementations for this type of oscillator uses an [[operational amplifier]] (op-amp), three [[capacitor]]s and four [[resistor]]s, as shown in the diagram.<br />
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The mathematics for calculating the oscillation frequency and oscillation criterion for this circuit are surprisingly complex, due to each RC 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 ''R''<sub>1</sub>=''R''<sub>2</sub>=''R''<sub>3</sub>=''R'', and ''C''<sub>1</sub>=''C''<sub>2</sub>=''C''<sub>3</sub>=''C'', then:<br />
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<math>f_\mathrm{oscillation}=\frac{1}{2\pi RC\sqrt{6}}</math><br />
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and the oscillation criterion is:<br />
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<math>R_\mathrm{fb}=29 \cdot R</math><br />
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{{Breakafterimages}}<br />
Without the simplification of all the resistors and capacitors having the same values, the calculations become more complex:<br />
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<math>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}}</math><br />
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Oscillation criterion:<br />
<math>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}</math><br />
<math> + \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}</math> <br />
<math> + \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}</math><br />
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In a purely ideal circuit, some sort of non-zero initial condition needs to be injected into the circuit in order to start the oscillation. In practice, the noise present (predominantly thermal noise from the resistors) is sufficient to provide an initial signal to start the oscillation.<br />
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One potential problem with the single op-amp circuit is the high gain required to maintain the oscillation. If it is assumed that each RC segment does not affect the other, a gain of about 8 to 10 will be sufficient to enable oscillation. As mentioned previously, each RC section loads the next section, and a larger gain (about 27 to 30) is required to keep the circuit in oscillation.<ref>{{cite book|last=Mancini|first=Ron|title=Op Amps For Everyone|year=2002|publisher=Texas Instruments|location=Dallas, Texas|pages=15-15,15-16|url=http://www.ti.com/lit/an/slod006b/slod006b.pdf}}</ref> An improved version of this circuit can be made by putting an op-amp buffer between each RC stage. (This also simplifies the calculations.) Тhe voltage gain of the inverting channel is always unity.<br />
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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.<br />
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===FET implementation===<br />
[[File:Circuit Diagram for RC-Phase Shift Oscillator using JFET.png|thumb|Circuit Diagram for RC-Phase Shift Oscillator using JFET]]<br />
The circuit implements the RC phase shift oscillator with FET.<br />
Note that the topology used for positive feedback is voltage series feedback.<br />
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===BJT Implementation===<br />
The circuit implements the RC phase shift oscillator with BJT.<br />
Note that the topology used for positive feedback is voltage shunt feedback.<br />
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{{clear}}<br />
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==References==<br />
{{Reflist}}<br />
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{{Electronic oscillators}}<br />
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{{DEFAULTSORT:Phase-Shift Oscillator}}<br />
[[Category:Oscillators]]<br />
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[[de:Phasenschieber#Phasenschiebergenerator]]</div>en>SmackBot