Every oscillator needs 360° of loop phase shift at the frequency of oscillation. An inverting amplifier gives you 180° for free. The phase-shift oscillator gets the other 180° from three cascaded RC high-pass sections, each contributing 60°. No inductors, no transformers, no crystals — just an op-amp and six passive components.

The classic topology puts three identical RC sections between the op-amp output and its inverting input. Each section is a series capacitor followed by a shunt resistor to ground. At exactly one frequency, the network produces 180° of phase shift and an attenuation of 1/29. To sustain oscillation, the amplifier must therefore have a gain of at least 29 (Barkhausen criterion: loop gain ≥ 1).

The oscillation frequency is:

f = 1 / (2π · RC · √6)

Design example: You want a 1 kHz sine wave for testing an audio circuit. Pick C = 10 nF. Solve for R:

• R = 1 / (2π · f · C · √6) = 1 / (2π · 1000 · 10e-9 · 2.449) ≈ 6.5 kΩ
• Feedback resistor Rf ≥ 29 · R = 188 kΩ. Use a 220 kΩ pot or a 180 kΩ + trimmer.

Practical gotchas:

• Loading between stages matters. The equations assume each RC section sees infinite input impedance. In reality, each stage loads the previous one, which is baked into that 1/29 attenuation figure — do not try to buffer the stages unless you re-derive the math.
• Gain trimming is critical. Set Rf too low and it won't start. Set it too high and the output slams into the rails, producing a distorted square-ish wave. For clean sine output, use a nonlinear feedback element (small incandescent lamp, JFET, or a pair of back-to-back diodes with a series resistor) to soft-limit the amplitude.
• Frequency stability is mediocre — component tolerances directly move f. Below ~10 Hz or above ~100 kHz, op-amp phase shift and slew rate start corrupting the assumptions.

Real-world use: Low-frequency test tone generators, audio synthesizer LFOs, sine-wave sources for THD measurement rigs where a Wien bridge feels like overkill. You'll find phase-shift oscillators in old analog function generators and educational kits precisely because they're cheap and require no reactive matching.

Rule of thumb: Start with Rf ≈ 30·R and trim downward until distortion looks acceptable on a scope. If oscillation refuses to start, increase Rf slightly — you're right at the Barkhausen edge.