ELC-03

Series RLC resonance

ω0 = 1/√(LC), Q = (1/R)√(L/C).

Reading speed

AC circuitsThomsonIEC 60318

Governing equation

\omega_0=\dfrac{1}{\sqrt{LC}},\quad Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}

$C \in \left[0.01, 100\right]$

where

L: Inductance (mH)
C: Capacitance (µF)
R: Resistance (Ω)
f_0: Resonant frequency (Hz)
Q: Quality factor (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

\omega_0=\dfrac{1}{\sqrt{LC}},\quad Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}

$C \in \left[0.01, 100\right]$

$C = 0.01$ $f_0 = 15915.5$

Along the curve

②

Domain of the variable — blue

C_{\min} = 0.01,\quad C_{\max} = 100

$C \in \left[0.01, 100\right]$

$C = 0.01$ $C = 0.01$

Along the curve

③

Response curve — green

\omega_0=\dfrac{1}{\sqrt{LC}},\quad Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}

$C \in \left[0.01, 100\right]$

$C = 0.01$ $f_0 = 15915.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f_0}{\mathrm{d}C}

$C \in \left[0.01, 100\right]$

$C = 0.639$ $f_0^{\prime} = -11530.2$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-03 — Series RLC resonance) is the form associated with Thomson · IEC 60318. Working symbols: $L$ , $C$ , $R$ $\rightarrow$ $f_0$ , $Q$ . The imaginary part of series impedance vanishes when ωL = 1/(ωC), defining Thomson's resonant frequency.

Purpose

Purpose: compute

f_0

Q

from

L

C

R

in Electrical via

\omega_0=\dfrac{1}{\sqrt{LC}},\quad Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}

ω0 = 1/√(LC), Q = (1/R)√(L/C). Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

L = 10.000\,\mathrm{mH}

C = 1.000\,\mathrm{\mu F}

R = 8.000\,\mathrm{Ω}

, the governing relation

\omega_0=\dfrac{1}{\sqrt{LC}},\quad Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}

yields

f_0 = 1591.549\,\mathrm{Hz}

Q = 12.500\,\mathrm{—}

. Series RLC, high-Q approximation for bandwidth. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inductance L10 mHCapacitance C1 µFResistance R8 Ω

Outputs

Resonant frequency f_01591.549 Hz
Quality factor Q12.500 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Series RLC, high-Q approximation for bandwidth.

The imaginary part of series impedance vanishes when ωL = 1/(ωC), defining Thomson's resonant frequency.

Reading speed

Watch on YouTube