Series RLC Circuit and Resonance

Method & assumptions

1. Sinusoidal steady-state series RLC circuit with ideal linear components and positive total series resistance. Enter RMS source voltage, frequency in Hz, inductance in henries and capacitance in farads.
2. ω = 2πf; XL = ωL; XC = 1/(ωC); X = XL − XC. The special input C = 0 bypasses the capacitor (XC = 0); it does not represent a physical zero-capacitance component. L = 0 omits the inductor.
3. Z = R + jX, |Z| = √(R² + X²), I = V/|Z|, φ = atan2(X,R), PF = R/|Z|. Positive φ means inductive impedance and lagging current; negative φ means capacitive impedance and leading current.
4. P = I²R; Q = I²X; S = VI. Component RMS voltages are IR, IXL and IXC. They add as phasors: V² = VR² + (VL − VC)², not as scalar magnitudes.
5. With both L and C present, f0 = 1/(2π√(LC)). At resonance the net reactance vanishes, but the individual L/C voltages can exceed the source voltage. Finite positive resistance is required; the ideal zero-resistance resonance singularity is excluded.
6. No switching transient, parasitics beyond the supplied series resistance, saturation, frequency-dependent losses, nonlinear load, harmonics or thermal rating calculation. Component and source ratings require a separate assessment.

Sources

• MIT OpenCourseWare Physics II: AC Circuits, series RLC and average power, sections 12.3–12.4 ↗