Resonance and LC oscillations

Tune an old radio and a single knob sweeps a dial until, at one particular setting, a station suddenly leaps out of the hiss, loud and clear. That knob is adjusting a capacitor inside a circuit, and the "leap" is resonance — the moment the circuit's natural rhythm matches the frequency of the incoming signal. The same rhythm shows up even without any driving signal at all: charge a capacitor, connect it to a coil, and the two will trade energy back and forth forever (in the ideal case), oscillating at a frequency fixed only by their own inductance and capacitance. This article ties those two pictures — free LC oscillations and driven series-LCR resonance — together. Both are governed by the same natural angular frequency $\omega_0 = 1/\sqrt{LC}$ , and understanding why is the key to everything from radio tuning to why some circuits respond violently to one frequency and ignore all others.

The core insight

An LC combination has one natural rhythm, $\omega_0=1/\sqrt{LC}$ , at which it oscillates freely when left alone and at which a driven series LCR circuit resonates with maximum current and minimum impedance.

The core idea

Free LC oscillations. Picture an ideal (resistance-free) inductor $L$ connected across a capacitor $C$ that starts fully charged to $Q_0$ . At that instant every joule of energy sits in the capacitor's electric field,

U_E = \frac{1}{2}\frac{Q_0^2}{C}

where: $Q_0$ is the initial charge and $C$ the capacitance. As the capacitor discharges through the coil, current $I$ builds up and energy migrates into the inductor's magnetic field,

U_B = \frac{1}{2}LI^2

where: $L$ is the inductance and $I$ the instantaneous current. When the capacitor is momentarily fully discharged, all the energy has transferred to the magnetic field; the coil's back-EMF then keeps the current flowing in the same direction, recharging the capacitor with reversed polarity. The cycle then reverses and repeats. In an idealised, resistance-free loop this exchange never stops — charge, current and both energy forms oscillate sinusoidally, and total energy $U_E+U_B$ stays constant at every instant (conservation of energy, since no resistor dissipates anything). Any real wire has some resistance, though, so real LC circuits are damped: the oscillation amplitude decays and the energy slowly turns to heat.

The natural frequency. The rate of this back-and-forth exchange is fixed entirely by $L$ and $C$ — nothing else — and is called the natural (or resonant) angular frequency, derived below.

Resonance in a driven series LCR circuit. Now connect a resistor, inductor and capacitor in series to an AC source whose frequency $\omega$ you can vary. The circuit's total opposition to current is the impedance $Z=\sqrt{R^2+(X_L-X_C)^2}$ , where $X_L=\omega L$ and $X_C=1/(\omega C)$ are the inductive and capacitive reactances. Because $X_L$ grows with $\omega$ while $X_C$ shrinks, there is exactly one frequency at which they become equal and cancel each other in the impedance formula — that is resonance. At that frequency $Z$ falls to its minimum possible value, $Z=R$ , so current $I=V_{rms}/Z$ climbs to its maximum, the phase angle between voltage and current becomes $\phi=0$ (the circuit behaves as if it were purely resistive), and the power factor $\cos\phi$ reaches its peak value of 1. Strikingly, this resonant frequency is the same $\omega_0$ that governs free LC oscillations — a driven LCR circuit resonates exactly at the rhythm the bare LC pair would keep on its own.

Figure

Two panels one showing an isolated LC loop with energy sloshing between capacitor and inductor and the other showing current in a driven series LCR circuit peaking sharply at the resonant frequency omega naught

Bandwidth and Q-factor (use, no derivation). Near resonance the current stays large only over a narrow band of frequencies straddling $\omega_0$ ; this width is the bandwidth, usually measured between the points where the power drops to half its peak value. The quality factor (Q-factor) is a dimensionless measure of how sharp that peak is — a higher Q means a narrower bandwidth and a more sharply selective resonance. This sharpness is exactly what a radio tuner exploits: a high-Q LCR circuit responds strongly to one station's frequency while barely responding to its neighbours.

Derivation

Derivation ANatural angular frequency of an ideal LC loop

The goal is to find the single frequency at which an ideal, resistance-free LC loop naturally oscillates, using nothing beyond the defining voltage relations for an inductor and a capacitor and the fact that a sinusoidal function is its own second derivative up to a constant — this frequency then also turns out to be the resonant frequency of a driven series LCR circuit, which is why it deserves a name of its own, $\omega_0$ .

Step 1. Apply Kirchhoff's voltage law around the loop. With no resistor, the voltage across the capacitor must at every instant equal the voltage the inductor develops in opposing the changing current: $\dfrac{Q}{C} = -L\dfrac{dI}{dt}$ , where $Q$ is the instantaneous charge on the capacitor and $I$ is the loop current.

Step 2. Write the current in terms of charge. Since the capacitor discharges to supply the current, $I = -\dfrac{dQ}{dt}$ , so $\dfrac{dI}{dt} = -\dfrac{d^2Q}{dt^2}$ . Substituting into Step 1 gives $\dfrac{Q}{C} = L\dfrac{d^2Q}{dt^2}$ , which rearranges to $\dfrac{d^2Q}{dt^2} = -\dfrac{1}{LC}Q$ .

Step 3. Recognise the form. This is exactly the equation for simple harmonic motion, $\dfrac{d^2Q}{dt^2} = -\omega_0^2 Q$ , whose solution oscillates sinusoidally with angular frequency $\omega_0$ . Matching the coefficient of $Q$ in the two expressions, $\omega_0^2 = \dfrac{1}{LC}$ .

Step 4. Take the square root to get the natural angular frequency,

\omega_0 = \frac{1}{\sqrt{LC}}

This is the rhythm at which charge, current and the two energy forms all oscillate in a free LC loop. Because the same impedance-minimising condition $X_L=X_C$ (that is, $\omega L = 1/(\omega C)$ ) rearranges to this identical expression, a driven series LCR circuit resonates at precisely this frequency too — free oscillation and forced resonance share one root cause, the balance between $L$ and $C$ .

Worked example

An LC circuit has $L=2\ \text{mH}$ and $C=8\ \mu\text{F}$ , and it is later wired into a series LCR circuit with $R=10\ \Omega$ driven by a 100 V RMS AC source of variable frequency. Find (a) the natural angular frequency of the bare LC pair, (b) the corresponding frequency in Hz, and (c) the maximum current the driven LCR circuit can draw.

Step 1. Compute $\omega_0$ from the LC values. $\omega_0 = \dfrac{1}{\sqrt{LC}} = \dfrac{1}{\sqrt{2\times10^{-3}\times8\times10^{-6}}} = \dfrac{1}{\sqrt{1.6\times10^{-8}}}$ . Since $\sqrt{1.6\times10^{-8}} = 1.265\times10^{-4}$ , this gives $\omega_0 = \dfrac{1}{1.265\times10^{-4}} = 7906\ \text{rad/s}$ .

Step 2. Convert to ordinary frequency. $f_0 = \dfrac{\omega_0}{2\pi} = \dfrac{7906}{6.283} = 1258\ \text{Hz} \approx 1.26\ \text{kHz}$ . This is the frequency the source must be tuned to for the driven LCR circuit (built with the same $L$ and $C$ ) to hit resonance.

Step 3. At resonance, $X_L=X_C$ so they cancel in the impedance formula, leaving $Z=\sqrt{R^2+0}=R$ . The current is then at its largest possible value for this circuit, $I_{max}=\dfrac{V_{rms}}{Z}=\dfrac{V_{rms}}{R}$ . With $V_{rms}=100\ \text{V}$ and $R=10\ \Omega$ , $I_{max} = 100/10 = 10\ \text{A}$ , delivered with the circuit behaving as if purely resistive ( $\phi=0$ , power factor 1).

Competitive stretch

Confirm resonance directly at $\omega_0=1000\ \text{rad/s}$ for a second circuit with $L=0.5\ \text{H}$ , $C=2\ \mu\text{F}$ : $X_L=\omega_0L=1000\times0.5=500\ \Omega$ and $X_C=1/(\omega_0C)=1/(1000\times2\times10^{-6})=500\ \Omega$ . Since $X_L=X_C$ , $Z=\sqrt{R^2+(500-500)^2}=R$ , verifying the resonance condition without appealing to the formula for $\omega_0$ at all — a useful cross-check when a problem gives you $\omega$ , $L$ and $C$ separately rather than asking you to compute $\omega_0$ .

Common traps

Thinking resonance means $X_L$ or $X_C$ is individually zero — resonance is where they are equal to each other, not where either vanishes.
Forgetting that at resonance the circuit is purely resistive: phase angle $\phi=0$ , power factor exactly 1, and power delivered is at its maximum, $\bar P_{max}=V_{rms}I_{max}$ .
Assuming LC oscillations continue forever in a real circuit — only an idealised, resistance-free loop sustains undamped oscillation; real wires always carry some resistance, so real oscillations damp out.
Mixing up bandwidth and Q-factor — bandwidth is a frequency range (Hz or rad/s), Q-factor is a dimensionless sharpness ratio; a higher Q means a smaller bandwidth, not a larger one.

Ask the tutor: want to see how this LC energy exchange compares to a mass oscillating on a spring, or work through more resonant-frequency practice problems?

Why does an ideal LC loop keep oscillating forever, but a real one does not?

In the ideal case there is no resistor anywhere in the loop, so there is nothing to convert electrical energy into heat — the total $U_E+U_B$ is perfectly conserved and the exchange between capacitor and inductor repeats indefinitely. Every real wire and coil has some resistance, though, so a little energy is dissipated as heat on every cycle. The oscillation amplitude then shrinks cycle by cycle even though the frequency $\omega_0=1/\sqrt{LC}$ stays essentially the same — this is called a damped oscillation, and it is why a real LC "ringing" circuit eventually falls silent unless something keeps feeding it energy.

Is the resonant frequency of a driven LCR circuit really the same number as the free LC oscillation frequency?

Yes, and that is not a coincidence — both come from the same balance between $L$ and $C$ . For the free loop, solving the loop equation gives $\omega_0=1/\sqrt{LC}$ directly. For the driven circuit, resonance is defined as the frequency where $X_L=X_C$ , that is $\omega L = 1/(\omega C)$ , which rearranges to the identical $\omega=1/\sqrt{LC}$ . Physically, the driven circuit resonates because the source is pushing the LC pair at exactly the rate it "wants" to oscillate on its own, so energy is fed in efficiently every cycle rather than fighting the circuit's natural rhythm.

Why is a high Q-factor useful for tuning a radio?

A radio receiver has to pull out one station's signal from many others crowding nearby frequencies. A high-Q LCR circuit has a very narrow, sharply peaked resonance curve, so it responds strongly only to frequencies extremely close to $\omega_0$ and gives a weak response to anything even slightly off that value. Tuning the dial adjusts $C$ (and hence $\omega_0$ ) until it lines up with the desired station's frequency; because the resonance is narrow, neighbouring stations barely register. A low-Q circuit would respond to a broad range of frequencies at once and mix several stations together.