First-Order Bode Phase

Formula

Description

A first-order low-pass filter introduces a phase shift that varies from 0° at DC to −90° at high frequencies, passing through −45° exactly at the cutoff frequency. The phase response is a smooth arctan curve on a logarithmic frequency scale. Phase shift is critical in feedback control systems because excessive phase lag at the unity gain frequency causes oscillation. In audio systems, phase differences between drivers in a crossover network affect the acoustic summing and can create lobes or nulls in the radiation pattern.

Variables

• φ — Phase shift in degrees (negative for low-pass)
• f — Signal frequency (Hz)
• fc — Filter cutoff (−3 dB) frequency (Hz)

Practical Notes

At f = 0.1×fc, phase ≈ −5.7°. At f = fc, phase = −45°. At f = 10×fc, phase ≈ −84.3°. A second-order filter reaches −180° at high frequencies (−90° at fc). Phase margin in feedback systems requires the total loop phase to be less than −180° at the gain crossover frequency. A first-order filter contributes at most −90° of phase, so a single-pole system is inherently stable.