AcademyAlternating Current
Academy
Resonance in AC Circuits
Level 1 - Physics topic page in Alternating Current.
Principle
Series resonance occurs when inductive and capacitive reactances are equal. The circuit impedance is then purely resistive and the current is maximum.
Notation
\(\omega_0\)
resonant angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(f_0\)
resonant frequency
\(\mathrm{Hz}\)
\(L\)
inductance
\(\mathrm{H}\)
\(C\)
capacitance
\(\mathrm{F}\)
\(R\)
series resistance
\(\mathrm{\Omega}\)
\(Q\)
quality factor
\(\Delta\omega\)
bandwidth
\(\mathrm{rad\,s^{-1}}\)
Method
Derivation 1: Set reactances equal
At resonance in a series LRC circuit, \(X_L=X_C\).
Resonance condition
\[\omega_0L=\frac{1}{\omega_0C}\]
Resonant angular frequency
\[\omega_0=\frac{1}{\sqrt{LC}}\]
Resonant frequency
\[f_0=\frac{1}{2\pi\sqrt{LC}}\]
Derivation 2: Minimize impedance
At resonance, the reactive part \(X_L-X_C\) is zero, so \(Z=R\). The current is then as large as the resistance allows.
Impedance at resonance
\[Z_{\mathrm{res}}=R\]
Maximum current
\[I_{\max}=\frac{V_{\mathrm{rms}}}{R}\]
Derivation 3: Measure sharpness
The quality factor describes how sharp the resonance is. A small resistance gives a large \(Q\) and a narrow bandwidth.
Series quality factor
\[Q=\frac{\omega_0L}{R}=\frac{1}{\omega_0CR}\]
Bandwidth estimate
\[\Delta\omega=\frac{\omega_0}{Q}\]
Rules
Resonance condition
\[X_L=X_C\]
Resonant angular frequency
\[\omega_0=\frac{1}{\sqrt{LC}}\]
Resonant frequency
\[f_0=\frac{1}{2\pi\sqrt{LC}}\]
Series quality factor
\[Q=\frac{\omega_0L}{R}\]
Examples
Question
Find \(f_0\) for
\[L=0.20\,\mathrm H\]
and \[C=5.0\,\mu\mathrm F\]
Answer
\[f_0=\frac{1}{2\pi\sqrt{LC}}=\frac{1}{2\pi\sqrt{(0.20)(5.0\times10^{-6})}}=159\,\mathrm{Hz}\]
Checks
- Resonance in a series LRC circuit maximizes current.
- At resonance, source voltage and current are in phase.
- The capacitor and inductor voltages can be large but opposite in phase.
- Increasing resistance broadens the resonance and lowers \(Q\).