AcademyInductance
Academy
LRC Series Circuits
Level 1 - Physics topic page in Inductance.
Principle
A series LRC circuit combines oscillation, damping, and resistance-controlled energy loss.
Notation
\(R\)
series resistance
\(\mathrm{\Omega}\)
\(L\)
inductance
\(\mathrm{H}\)
\(C\)
capacitance
\(\mathrm{F}\)
\(q(t)\)
capacitor charge
\(\mathrm{C}\)
\(I(t)\)
series current
\(\mathrm{A}\)
\(\omega_0\)
undamped natural angular frequency
\(\mathrm{rad\,s^{-1}}\)
Method
Derivation 1: Free series equation
For a source-free series LRC circuit, the inductor voltage, resistor drop, and capacitor voltage sum to zero.
Loop equation
\[L\frac{dI}{dt}+RI+\frac{q}{C}=0\]
Current relation
\[I=\frac{dq}{dt}\]
Charge equation
\[L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{q}{C}=0\]
Derivation 2: Damping
Resistance removes electromagnetic energy and damps the oscillation.
Undamped frequency
\[\omega_0=\frac{1}{\sqrt{LC}}\]
Damping coefficient
\[\gamma=\frac{R}{2L}\]
Underdamped frequency
\[\omega_d=\sqrt{\omega_0^2-\gamma^2}\]
Energy loss rate
\[\frac{dU}{dt}=-I^2R\]
Derivation 3: Regimes
The circuit oscillates only if damping is weak enough.
Underdamped
\[R<2\sqrt{\frac{L}{C}}\]
Critical damping
\[R=2\sqrt{\frac{L}{C}}\]
Overdamped
\[R>2\sqrt{\frac{L}{C}}\]
Rules
Series LRC equation
\[\displaystyle L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{q}{C}=0\]
Natural frequency
\[\displaystyle \omega_0=\frac{1}{\sqrt{LC}}\]
Damping coefficient
\[\displaystyle \gamma=\frac{R}{2L}\]
Damped frequency
\[\displaystyle \omega_d=\sqrt{\frac{1}{LC}-\frac{R^2}{4L^2}}\]
Examples
Question
For
\[L=0.50\,\mathrm{H}\]
\[C=20\,\mu\mathrm{F}\]
and \[R=20\,\Omega\]
find \(\omega_0\).Answer
\[\omega_0=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{(0.50)(20\times10^{-6})}}=316\,\mathrm{rad\,s^{-1}}\]
Checks
- Resistance reduces total electromagnetic energy.
- Weak damping gives decaying oscillations; strong damping gives no oscillation.
- The AC-driven LRC resonance model is handled separately in the AC circuits section.