AcademyInductance
Academy
LC Oscillations
Level 1 - Physics topic page in Inductance.
Principle
An ideal LC circuit swaps energy between electric and magnetic fields sinusoidally.
Notation
\(L\)
inductance
\(\mathrm{H}\)
\(C\)
capacitance
\(\mathrm{F}\)
\(q(t)\)
capacitor charge
\(\mathrm{C}\)
\(I(t)\)
circuit current
\(\mathrm{A}\)
\(\omega_0\)
natural angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(T\)
oscillation period
\(\mathrm{s}\)
Method
Derivation 1: Loop equation
In an ideal LC circuit, the capacitor voltage and inductor voltage sum to zero.
Loop equation
\[L\frac{dI}{dt}+\frac{q}{C}=0\]
Current-charge relation
\[I=\frac{dq}{dt}\]
Oscillator equation
\[\frac{d^2q}{dt^2}+\frac{1}{LC}q=0\]
Derivation 2: Natural frequency
The charge obeys the same mathematical form as simple harmonic motion.
Angular frequency
\[\omega_0=\frac{1}{\sqrt{LC}}\]
Period
\[T=2\pi\sqrt{LC}\]
Charge solution
\[q(t)=Q\cos(\omega_0t+\phi)\]
Derivation 3: Energy exchange
Energy moves between capacitor electric field and inductor magnetic field.
Capacitor energy
\[U_E=\frac{q^2}{2C}\]
Inductor energy
\[U_B=\frac{1}{2}LI^2\]
Total energy
\[U=\frac{q^2}{2C}+\frac{1}{2}LI^2\]
Rules
Natural frequency
\[\displaystyle \omega_0=\frac{1}{\sqrt{LC}}\]
Period
\[T=2\pi\sqrt{LC}\]
Energy conservation
\[\displaystyle \frac{q^2}{2C}+\frac12LI^2=\text{constant}\]
Maximum current
\[\displaystyle I_{\max}=\frac{Q}{\sqrt{LC}}\]
Examples
Question
An LC circuit has
\[L=0.20\,\mathrm{H}\]
and \[C=50\,\mu\mathrm{F}\]
Find \(\omega_0\).Answer
\[\omega_0=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{(0.20)(50\times10^{-6})}}=316\,\mathrm{rad\,s^{-1}}\]
Checks
- Ideal LC oscillations require no resistance.
- Current is largest when capacitor charge is zero.
- Capacitor energy is largest when current is zero.