ELC-10

RLC resonance

ω0 = 1/√(LC). The undamped natural frequency of a series tank.

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CircuitsThomson

Governing equation

\omega_0=1/\sqrt{LC}

$C \in \left[0.1, 500\right]$

where

L: Inductance (H)
C: Capacitance (µF)
R: Resistance (Ω)
\omega_0: Natural ω (rad/s)
Q: Quality (—)

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.

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①

Governing law — red

\omega_0=1/\sqrt{LC}

$C \in \left[0.1, 500\right]$

$C = 0.1$ $\omega_0 = 31622.8$

Along the curve

②

Domain of the variable — blue

C_{\min} = 0.1,\quad C_{\max} = 500

$C \in \left[0.1, 500\right]$

$C = 0.1$ $C = 0.1$

Along the curve

③

Response curve — green

\omega_0=1/\sqrt{LC}

$C \in \left[0.1, 500\right]$

$C = 0.1$ $\omega_0 = 31622.8$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\omega_0}{\mathrm{d}C}

$C \in \left[0.1, 500\right]$

$C = 3.244$ $\omega_0^{\prime} = -4399.8$

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-10 — RLC resonance) is the form associated with Thomson. Working symbols: $L$ , $C$ , $R$ $\rightarrow$ $\omega_0$ , $Q$ . Quality Q = (1/R)√(L/C). Bandwidth Δω = R/L.

Purpose

Purpose: compute

\omega_0

Q

from

L

C

R

in Electrical via

\omega_0=1/\sqrt{LC}

ω0 = 1/√(LC). The undamped natural frequency of a series tank. 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 = 0.010\,\mathrm{H}

C = 10.000\,\mathrm{\mu F}

R = 2.000\,\mathrm{Ω}

, the governing relation

\omega_0=1/\sqrt{LC}

yields

\omega_0 = 3162.28\,\mathrm{rad/s}

Q = 15.81\,\mathrm{—}

. An inductor and a capacitor exchanging energy. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inductance L0.01 HCapacitance C10 µFResistance R2 Ω

Outputs

Natural ω \omega_03162.28 rad/s
Quality Q15.81 —

Reading speed

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Narration of this film

An inductor and a capacitor exchanging energy.

Quality Q = (1/R)√(L/C). Bandwidth Δω = R/L.

Reading speed

Watch on YouTube