arctanRC

arctanRC is a term encountered in discussions of RC circuits and first-order systems, used informally to denote the arctangent of a dimensionless parameter involving resistance and capacitance. It is not a standard mathematical function, but rather a nuisance-free shorthand that appears in analyses where the phase angle of a first-order transfer function is described.

In the common RC low-pass or high-pass transfer function, H(jω) = 1/(1 + jωRC), the magnitude and phase are determined by the complex division. The magnitude is |H(jω)| = 1/√(1 + (ωRC)²), and the phase is φ(ω) = -arctan(ωRC). Thus arctanRC (when written as arctan(ωRC)) represents the phase shift introduced by the RC network at angular frequency ω. Because ω has units of 1/s and RC has units of s, their product ωRC is dimensionless, making the arctan well-defined.

Usage notes and caveats:

- arctanRC is typically written as arctan(ωRC) or tan⁻¹(ωRC) in formal contexts. Some texts or codebases may use arctanRC to highlight the dependence on RC without explicitly inserting ω.

- Taking arctan(RC) alone is not dimensionally meaningful unless RC is nondimensionalized by a reference time or incorporated into a broader, dimensionless parameter.

- The concept is especially relevant when analyzing frequency response and phase lag in first-order systems, or in Bode plot interpretation.

Example: with ω = 100 rad/s and RC = 0.01 s, ωRC = 1, so arctan(ωRC) = arctan(1) ≈ 0.785 rad (≈45°).