Rlc Circuit Calculator

A resistor, inductor, and capacitor together form a second order system whose defining property is resonance. At f0 = 1 / (2 pi sqrt(L x C)) the inductive and capacitive reactances are equal and cancel, leaving purely resistive behavior; note that f0 depends only on L and C, so resistance sets the sharpness of the response but never its center frequency. The quality factor measures that sharpness as the ratio of energy stored to energy lost per cycle, and its form depends on topology: Q = (1/R) x sqrt(L/C) in series, where lower resistance raises Q, but Q = R x sqrt(C/L) in parallel, where higher resistance raises it. Bandwidth between the half power points is f0 / Q, and the damping ratio is 1 / (2Q), separating underdamped ringing from the critically damped and overdamped regimes. The quantity sqrt(L/C) is the characteristic impedance, and in series resonance the voltage across L or C reaches Q times the source voltage. Realistic figures span an enormous range. A 250 microhenry coil with 100 picofarads resonates near 1.0 MHz in the AM broadcast band; 10 millihenries with 1 microfarad lands at about 1.59 kHz in the audio range. Achievable Q depends on the components: air core inductors typically manage 50 to 200, ferrite cored parts 30 to 150, and quartz crystals behave as equivalent series RLC networks with Q of 10,000 to 100,000. The idealization has clear limits. Winding resistance in the inductor and equivalent series resistance in the capacitor usually dominate the loaded Q, so the calculated value is an upper bound. The symmetric half power approximation, f0 plus or minus BW/2, holds only for Q above roughly 10, and every real inductor has a self resonant frequency set by interwinding capacitance, above which it behaves capacitively.

Electrical resonance was predicted before it was seen. In 1853 William Thomson, later Lord Kelvin, analyzed the discharge of a capacitor through an inductive circuit and showed mathematically that the current would oscillate rather than decay smoothly provided the resistance stayed below a critical value related to twice the square root of L over C. Berend Wilhelm Feddersen supplied the experimental proof around 1859 by photographing spark discharges in a rotating mirror, resolving the single visible flash into a train of alternating oscillations. Heinrich Hertz turned the tuned circuit into an instrument between 1886 and 1888, using resonant loops and dipoles to generate and detect electromagnetic waves and confirm Maxwell's theory. Oliver Lodge demonstrated syntonic tuning in 1889, showing that two circuits adjusted to the same resonance would respond selectively to one another, which established the principle that allows separate stations to share a spectrum. Guglielmo Marconi's British patent 7777 of 1900 covered tuned transmitting and receiving circuits and became the commercial foundation of selective radio. The notation Q for the ratio of reactance to resistance was introduced by engineers at Bell Telephone Laboratories in the 1920s, with the name quality factor attaching later; the practice of characterizing filters by Q, bandwidth, and damping ratio spread from telephone loading coil and network design into general electronics through the middle of the century.