EMG-06

Series RLC impedance

Z = √(R² + (ωL − 1/(ωC))²).

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AC circuitsSteinmetz

Governing equation

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

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

where

R: Resistance (Ω)
L: Inductance (mH)
C: Capacitance (µF)
f: Frequency (Hz)
Z: Impedance (Ω)
\varphi: Phase (°)

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

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

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

$f = 1$ $Z = 3978.7$

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+\left(\omega L-\dfrac{1}{\omega C}\right)^2}

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

$f = 1$ $Z = 3978.7$

Along the curve

④

Rate of change — amber

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

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

$f = 13.57$ $Z^{\prime} = -152.3$

Along the curve

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-06 — Series RLC impedance) is the form associated with Steinmetz. Working symbols: $R$ , $L$ , $C$ , $f$ $\rightarrow$ $Z$ , $\varphi$ . Phasor addition of resistance and the two reactances yields the magnitude of series impedance, minimum at resonance.

Purpose

Purpose: compute

Z

\varphi

from

R

L

C

f

in Electromagnetism via

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

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

Live realistic example

In symbols

Live case. Given

R = 10.000\,\mathrm{Ω}

L = 25.000\,\mathrm{mH}

C = 40.000\,\mathrm{\mu F}

f = 50.000\,\mathrm{Hz}

, the governing relation

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

yields

Z = 72.417\,\mathrm{Ω}

\varphi = -82.063\,\mathrm{^{\circ}}

. Linear lumped R, L, C driven by a sinusoid. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Resistance R10 ΩInductance L25 mHCapacitance C40 µFFrequency f50 Hz

Outputs

Impedance Z72.417 Ω
Phase \varphi-82.063 °

Reading speed

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Narration of this film

Linear lumped R, L, C driven by a sinusoid.

Phasor addition of resistance and the two reactances yields the magnitude of series impedance, minimum at resonance.

Reading speed

Watch on YouTube