RLC Impedance, Resonance & Q-Factor

In 2026, a power-electronics engineer designing a 60 Hz PF-correction cap bank for a 480 V 3Φ motor needs to verify that the resonance fr of the cap-plus-line-impedance loop does not collide with a 5th or 7th harmonic. This calculator gives Z, fr and Q in three keystrokes — and the impedance triangle plus Bode sweep make the harmonic-collision risk visible at a glance.

The foundational analogy was published by William Thomson — later Lord Kelvin — in his 1853 paper On Transient Electric Currents. Thomson noticed that an undamped LC tank oscillates at f = 1/(2π√LC) — exactly the formula every RF engineer uses today. He explicitly drew the analogy to a mass-spring-damper system: L plays the role of mass, 1/C of stiffness, and R of viscous damping. The Q factor, ωL/R, mirrors the mechanical quality of a tuning fork.

Heinrich Hertz turned Thomson's analogy into experiment. Between 1887 and 1888 at Karlsruhe Polytechnic, Hertz built spark-gap LC oscillators with fr ≈ 50 MHz and detector loops with matching self-resonance. His famous 1888 paper Über elektrodynamische Wellen demonstrated that only when the detector loop was tuned to the source did it spark — direct experimental proof of Maxwell's 1873 prediction of electromagnetic waves. The Hertz unit (Hz) was adopted by the 11th CGPM in 1960 in his honour.

James Clerk Maxwell's 1864 A Dynamical Theory of the Electromagnetic Field had already established the j-omega-L and 1/(j-omega-C) reactances mathematically. But it was Charles Proteus Steinmetz at General Electric who, in his 1893 AIEE paper, reduced the working math to the complex-number form every modern engineer uses: Z = R + jX with j = √(−1). Steinmetz's formalism collapsed Thomson's differential equations into Ohm's law for AC and made the impedance triangle on this page a practical design tool overnight.

Hendrik Bode at Bell Labs in 1940 introduced the log-log magnitude plot now called the Bode plot — the second SVG on this page. Bode's 1945 book Network Analysis and Feedback Amplifier Design proved that the slope of |Z(f)| changes by 20 dB/decade at every pole and zero — exactly what the green operating-point dot traces as you sweep frequency across an RLC resonance.

Through the twentieth century RLC analysis became foundational. Marconi's 1901 transatlantic transmission relied on a 25 µH × 4 nF spark-gap tank at fr ≈ 500 kHz. Edwin Armstrong's 1918 superheterodyne receiver used a series-RLC tuned IF strip at 455 kHz still used in AM radios today. The 63.87 MHz Larmor frequency of a 1.5 T MRI scanner's ¹H protons demands a Q ≈ 250 series RLC birdcage coil per IEC 60601-2-33. The IEEE 1459-2010 power-quality standard codifies the unity-power-factor condition φ = 0 — exactly when XL = XC.

What does the answer really mean? An |Z| of 50 Ω in series-RLC at the source frequency means a 1 V source drives 20 mA peak through the network — most of that current circulates between the inductor and capacitor (reactive power) while only the R-leg part does work. A Q of 250 means the −3 dB bandwidth is fr/250 — at 63.87 MHz that is 256 kHz. Sharp peaks let an MRI scanner detect protons; broad peaks let a power filter pass a whole harmonic range. Both fall out of the same Z = √(R² + (XL − XC)²) formula this widget animates.