Rlc Series and Parallel Resonance Tool

Physics Electricity and Magnetism • Ac Circuits and Electromagnetic Waves

Written by STEM Calculators Team Published January 18, 2026 Updated February 24, 2026

RLC Series/Parallel Resonance Tool

Finds resonant frequency \(f_{res}=\dfrac{1}{2\pi\sqrt{LC}}\), plus quality factor \(Q=\dfrac{f_{res}}{\text{bandwidth}}\). Includes series/parallel mode, damping factor, and a plot of \(|Z|\) vs frequency showing resonance and \(-3\text{ dB}\) points.

Units: \(\Omega\), H, F, Hz, rad/s. Inputs accept 1e-6, pi, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Inputs

Circuit type:

Inductance \(L\): Capacitance \(C\): Resistance \(R\) (Ω):

Series: series loss \(R\). Parallel: shunt \(R\) (in parallel).

Drive at resonance (I/V “max”)

Drive type:

Amplitude:

Series resonance: for a voltage source, current peaks since \(Z\) is minimal (\(Z\approx R\)).
Parallel resonance: for a current source, voltage peaks since \(|Z|\) is maximal (\(|Z|\approx R\)).

Plot settings

Frequency range:

\(a\):

Lower multiplier for \(f_0\).

\(b\):

Upper multiplier for \(f_0\).

Plot points: Graph text size:

Pan/zoom: drag to pan • mouse wheel/trackpad/pinch to zoom in/out • Reset view restores default.

Ready

Steps

Enter values and click Solve.

Impedance vs frequency

Interactive plot (pan/zoom enabled)

Markers

\(|Z|\) vs \(f\)
resonance \(f_0\)
\(-3\text{ dB}\) points \(f_1,f_2\) (if \(R>0\))

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Theory

Resonance condition

In an ideal RLC circuit, resonance occurs when the inductive and capacitive reactances cancel: \(X_L=X_C\).

\[ X_L=\omega L,\qquad X_C=\frac{1}{\omega C} \] \[ \omega L=\frac{1}{\omega C}\ \Rightarrow\ \omega_0=\frac{1}{\sqrt{LC}},\qquad f_0=\frac{\omega_0}{2\pi}=\frac{1}{2\pi\sqrt{LC}} \]

Series vs parallel resonance behavior

Series RLC: impedance is

\[ Z(\omega)=R+j\left(\omega L-\frac{1}{\omega C}\right) \]

At resonance, the imaginary part is zero, so \(Z(\omega_0)=R\) and \(|Z|\) is minimized. With a voltage source, \(I=V/|Z|\) peaks at resonance.

Parallel RLC: it is often easier to use admittance

\[ Y(\omega)=\frac{1}{R}+j\left(\omega C-\frac{1}{\omega L}\right),\qquad Z(\omega)=\frac{1}{Y(\omega)} \]

At resonance the imaginary part cancels and \(|Z|\) reaches a maximum (ideal model). With a current source, \(V=I|Z|\) peaks at resonance.

Quality factor, bandwidth, and -3 dB frequencies

The quality factor measures how sharp/selective the resonance is: \(Q=\dfrac{f_0}{\Delta f}\), where \(\Delta f\) is the bandwidth between the \(-3\text{ dB}\) points.

Series RLC (series loss \(R\))

\[ Q=\frac{\omega_0 L}{R}=\frac{1}{\omega_0 C R},\qquad \Delta\omega=\frac{R}{L},\qquad \Delta f=\frac{\Delta\omega}{2\pi} \]

Parallel RLC (shunt \(R\))

\[ Q=\omega_0 R C = R\sqrt{\frac{C}{L}},\qquad \Delta\omega=\frac{1}{RC},\qquad \Delta f=\frac{\Delta\omega}{2\pi} \]

Higher \(Q\) means narrower bandwidth — this is the core idea behind “radio tuning” selectivity.

Damping factor (extension)

A standard 2nd-order form uses damping factor \(\alpha\) and damping ratio \(\zeta\): \(\zeta=\alpha/\omega_0\).

Series model

\[ \alpha=\frac{R}{2L},\qquad \zeta=\frac{\alpha}{\omega_0} \]

Parallel model

\[ \alpha=\frac{1}{2RC},\qquad \zeta=\frac{\alpha}{\omega_0} \]

Steps

Impedance vs frequency

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