Most filters trade frequency content for something — low-pass kills highs, bandpass kills both sides. An all-pass filter passes every frequency at the same amplitude but shifts the phase as a function of frequency. Useful? Extremely. Phase delay is how you align signals, build phase shifters for SSB modulators, equalize group delay in audio crossovers, and synthesize quadrature signals.

The classic first-order all-pass uses a single op-amp with a matched RC pair:

• Non-inverting input fed from input via an RC low-pass to ground (R to input, C to ground at the +input)
• Inverting input wired as unity-gain inverter: equal-value resistors from input to inverting node and from output to inverting node

The transfer function works out to H(jω) = (1 − jωRC) / (1 + jωRC). Magnitude is exactly 1 at every frequency. Phase rolls smoothly from 0° at DC to −180° at infinity, passing through −90° at the center frequency f₀ = 1/(2πRC).

Rule of thumb: at f₀, you get −90° phase shift. Want −45°? Operate at f₀/2.41. Want −135°? Operate at 2.41·f₀. The factor 2.41 comes from tan(67.5°).

Worked example — phase shifter for a lock-in amplifier reference at 1 kHz: Suppose you need to dial the reference phase by 90° relative to your signal. Pick R = 16 kΩ and C = 10 nF. Then f₀ = 1/(2π·16k·10n) ≈ 995 Hz — close enough. Replace R with a 20 kΩ pot in series with a 1 kΩ resistor and you can tune phase from roughly −5° to −175° across the dial. The signal amplitude stays rock-solid the whole time, which is why this beats trying to mix amplitude into phase control with a vector modulator.

Practical gotchas:

• Matched resistors matter. The two feedback resistors must be equal — 0.1% mismatch produces an amplitude ripple of about 0.001 dB but also a small gain error that becomes obvious if you cascade several stages.
• Cascade for steeper phase slopes. Two first-order sections give 360° of total shift; three give 540°. Cascading is how Hilbert-transform networks for SSB radios get built.
• Group delay isn't constant. A first-order all-pass has peak group delay at f₀ equal to 2RC seconds. For a 1 kHz section with RC = 1.6×10⁻⁴, that's 320 µs — non-trivial in audio.
• Op-amp GBW must clear your highest frequency by 10×, or the phase response sags below the textbook curve.

Second-order all-pass sections (using a state-variable or biquad core) let you place a complex pole-zero pair, enabling Bessel-style flat group delay or precision Hilbert transformers across decades of bandwidth.