Engineering · Electrical engineering

Impedance of a series RLC circuit

For sinusoidal AC, the series impedance magnitude is the quadratic combination of resistance and net reactance.

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Formula

Z = R^{2} + (ω L - 1/ (ω C))^{2}

Z, R, XL and XC in Ω; ω in rad/s; L in H; C in F

Variables & units

Variables & units
Symbol	Meaning	Unit
$Z$	Magnitude of complex series impedance	Ω
$R$	Ohmic resistance	Ω
$ω$	Angular frequency 2πf	rad/s
$L$	Inductance	H
$C$	Capacitance	F

LaTeX

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

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Derivation & background

For ideal series R, L and C, the complex impedance is R+j(ωL−1/ωC); the displayed expression is its magnitude. It applies in sinusoidal steady state with linear elements. At resonance, inductive and capacitive reactances cancel and Z=R. Component losses and parasitics can change real frequency response.

Exam blueprint

Validity range

Derivation steps

For sinusoidal AC, the series impedance magnitude is the quadratic combination of resistance and net reactance.

1State the model assumptions and list quantities with units.
2Substitute into Z = √[R² + (ωL − 1/(ωC))²]; convert quantities to SI first.
3For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω.

Rearrangements

Base equation for the target quantity

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

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Impedance of a series RLC circuit” and check its unit and magnitude.

For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: Z, R, XL and XC in Ω; ω in rad/s; L in H; C in F.

Exam context

Calculate AC current, phase behavior and frequency response of series RLC networks.

Formula cluster

Electrical engineering

For sinusoidal AC, the series impedance magnitude is the quadratic combination of resistance and net reactance.

T = 2π·√(L·C)

Thomson formula for an LC circuit

Σ I hinein = Σ I hinaus; Σ ΔV = 0

Kirchhoff’s junction and loop rules

P = U·I

Electric Power

Worked example

For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω.

Applications

Calculate AC current, phase behavior and frequency response of series RLC networks.

How do you calculate with Impedance of a series RLC circuit?

Here is how to work through a typical Impedance of a series RLC circuit (Z = √[R² + (ωL − 1/(ωC))²]) task step by step:

1
Task
Work through the numerical example for “Impedance of a series RLC circuit” and check its unit and magnitude.
Solution path
For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω.

Frequently asked questions about Impedance of a series RLC circuit

How do I calculate XL and XC?+

Inductive reactance is XL=ωL and capacitive reactance is XC=1/(ωC). Both are measured in ohms, although they store energy temporarily rather than dissipating it like resistance. Use ω in rad/s and carefully convert millihenry and microfarad. The equation gives impedance magnitude only for ideal series components in sinusoidal steady state. Reactances depend on ω=2πf. To find phase, retain complex form R+j(XL−XC); magnitude alone loses whether the circuit is net inductive or capacitive. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

When does series impedance equal R?+

At resonance, ωL=1/(ωC), so inductive and capacitive reactance cancel in series. Then Z=R and current is ideally in phase with source voltage. Resonant frequency is ω₀=1/√(LC). Losses and parasitics can shift or broaden real resonance. The equation gives impedance magnitude only for ideal series components in sinusoidal steady state. Reactances depend on ω=2πf. To find phase, retain complex form R+j(XL−XC); magnitude alone loses whether the circuit is net inductive or capacitive. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Does Z alone show phase?+

No. Magnitude Z gives only the size of the voltage–current ratio. Phase follows from φ=arctan[(XL−XC)/R] with correct quadrant handling. Positive net reactance is inductive; negative is capacitive. Keep complex impedance for sign and phase analysis. The equation gives impedance magnitude only for ideal series components in sinusoidal steady state. Reactances depend on ω=2πf. To find phase, retain complex form R+j(XL−XC); magnitude alone loses whether the circuit is net inductive or capacitive. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

How do I check the result and units for Impedance of a series RLC circuit?+

Write the quantity and unit from the variable list next to every value: Z, R, XL and XC in Ω; ω in rad/s; L in H; C in F. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The equation gives impedance magnitude only for ideal series components in sinusoidal steady state. Reactances depend on ω=2πf. To find phase, retain complex form R+j(XL−XC); magnitude alone loses whether the circuit is net inductive or capacitive. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

Which model limit matters for Impedance of a series RLC circuit?+

The background states this starting condition: For ideal series R, L and C, the complex impedance is R+j(ωL−1/ωC); the displayed expression is its magnitude.. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The equation gives impedance magnitude only for ideal series components in sinusoidal steady state. Reactances depend on ω=2πf. To find phase, retain complex form R+j(XL−XC); magnitude alone loses whether the circuit is net inductive or capacitive. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]OpenStax, University Physics Volume 2, §14.6 “RLC Series Circuits”.

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Impedance of a series RLC circuit

For sinusoidal AC, the series impedance magnitude is the quadratic combination of resistance and net reactance.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Symbol

Meaning

Unit

Z

Magnitude of complex series impedance

R

Ohmic resistance

ω

Angular frequency 2πf

rad/s

L

Inductance

C

Capacitance

Derivation & background

Exam blueprint

Validity range

Derivation steps

For sinusoidal AC, the series impedance magnitude is the quadratic combination of resistance and net reactance.

1State the model assumptions and list quantities with units.
2Substitute into Z = √[R² + (ωL − 1/(ωC))²]; convert quantities to SI first.
3For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω.

How do you calculate with Impedance of a series RLC circuit?

Here is how to work through a typical Impedance of a series RLC circuit (Z = √[R² + (ωL − 1/(ωC))²]) task step by step:

Task

Work through the numerical example for “Impedance of a series RLC circuit” and check its unit and magnitude.

Solution path

For R=30 Ω, f=50 Hz, L=0.10 H and C=100 μF, XL≈31.4 Ω and XC≈31.8 Ω, so Z≈30.0 Ω.

Frequently asked questions about Impedance of a series RLC circuit

How do I calculate XL and XC?+

When does series impedance equal R?+

Does Z alone show phase?+

How do I check the result and units for Impedance of a series RLC circuit?+

Which model limit matters for Impedance of a series RLC circuit?+

Retain Impedance of a series RLC circuit for exams

Create a curated FSRS exam set for Z = √[R² + (ωL − 1/(ωC))²]: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential