A phase-shift oscillator is a 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 fed back to its input through a phase-shift network consisting of resistors and capacitors in a ladder network. 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 often used at audio frequency as audio oscillators. 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. The first integrated circuit was a phase shift oscillator invented by Jack Kilby in 1958. Implementations Bipolar implementation This schematic drawing shows the oscillator using a common-emitter connected bipolar transistor as amplifier. The two resistors R and three capacitors C form the RC phase-shift network which provides feedback from collector to base of the transistor. Resistor Rb provides base bias current. Resistor Rc is the collector load resistor for the collector current. Resistor Rs isolates the circuit from the external load. FET implementation This circuit implements the oscillator with a FET. R1, R2, Rs, and Cs provide bias for the transistor. Note that the topology used for positive feedback is voltage series feedback. Op-amp implementation The implementation of the phase-shift oscillator shown in the diagram uses an operational amplifier (op-amp), three capacitors and four resistors. 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: The feedback resistor required to just sustain oscillation is: The equations are simpler when all the resistors (except the negative feedback resistor) have the same value and all the capacitors have the same value. In the diagram, if and , then: and the oscillation criterion is: As with other feedback oscillators, when the power is applied to the circuit, thermal electrical noise in the circuit or the turn-on transient provides an initial signal to start oscillation. In practice, the feedback resistor