ELC-37

Series RLC impedance

Z = √(R² + (ωL − 1/ωC)²). Minimum at resonance.

Reading speed

CircuitsSteinmetz

Governing equation

Z=\sqrt{R^2+(\omega L-1/\omega C)^2}

$f \in \left[1, 2000\right]$

where

R: Resistance (Ω)
L: Inductance (mH)
C: Capacitance (µF)
f: Frequency (Hz)
Z: Impedance (Ω)
X: Net reactance (Ω)

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

Z=\sqrt{R^2+(\omega L-1/\omega C)^2}

$f \in \left[1, 2000\right]$

$f = 1$ $Z = 15915.4$

Along the curve

②

Domain of the variable — blue

f_{\min} = 1,\quad f_{\max} = 2000

$f \in \left[1, 2000\right]$

$f = 1$ $f = 1$

Along the curve

③

Response curve — green

Z=\sqrt{R^2+(\omega L-1/\omega C)^2}

$f \in \left[1, 2000\right]$

$f = 1$ $Z = 15915.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Z}{\mathrm{d}f}

$f \in \left[1, 2000\right]$

$f = 13.57$ $Z^{\prime} = -608.9$

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-37 — Series RLC impedance) is the form associated with Steinmetz. Working symbols: $R$ , $L$ , $C$ , $f$ $\rightarrow$ $Z$ , $X$ . Phasor addition of R, jωL and 1/(jωC). The reactive parts cancel at ω0 = 1/√(LC).

Purpose

Purpose: compute

Z

X

from

R

L

C

f

in Electrical via

Z=\sqrt{R^2+(\omega L-1/\omega C)^2}

Z = √(R² + (ωL − 1/ωC)²). Minimum at resonance. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 8.000\,\mathrm{Ω}

L = 20.000\,\mathrm{mH}

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

f = 50.000\,\mathrm{Hz}

, the governing relation

Z=\sqrt{R^2+(\omega L-1/\omega C)^2}

yields

Z = 312.13\,\mathrm{Ω}

X = -312.03\,\mathrm{Ω}

. An RLC string, a Z vs ω curve, a notch. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Resistance R8 ΩInductance L20 mHCapacitance C10 µFFrequency f50 Hz

Outputs

Impedance Z312.13 Ω
Net reactance X-312.03 Ω

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

An RLC string, a Z vs ω curve, a notch.

Phasor addition of R, jωL and 1/(jωC). The reactive parts cancel at ω0 = 1/√(LC).

Reading speed

Watch on YouTube