LC Circuit

A capacitor, an inductor, and a loop

The minimal LC circuit is a loop containing one capacitor of capacitance C and one inductor of inductance L, connected end-to-end. To get the oscillation started, we charge the capacitor first to some voltage V₀, then close the loop.

The capacitor stores energy in its electric field — physically, in the gap between its plates. The energy is U_C = ½ C V² = q²/(2C), where q is the charge on each plate. The inductor stores energy in its magnetic field — physically, in the volume around its windings. The energy is U_L = ½ L I². Both are quadratic in their respective state variables (voltage or current), and both are positive.

The total energy of the loop is conserved (in an ideal circuit with no resistance):

U_total = q²/(2C) + ½ L (dq/dt)² = constant

Differentiating both sides with respect to time and dividing through by dq/dt gives the equation of motion:

L d²q/dt² + q/C = 0

This is the simple harmonic oscillator equation. Its solutions are sinusoids with angular frequency:

ω₀ = 1/√(LC)        f₀ = ω₀/(2π) = 1/(2π√(LC))

The charge oscillates as q(t) = q₀ cos(ω₀t + φ); the current is its derivative, I(t) = −q₀ω₀ sin(ω₀t + φ). The two are 90° out of phase: when charge is maximum, current is zero (capacitor fully charged, no flow); when current is maximum, charge is zero (capacitor fully discharged, all energy in the magnetic field).

Energy bouncing between the two stores

Plug the sinusoidal solutions back into the energies:

U_C(t) = q₀² cos²(ω₀t) / (2C)
U_L(t) = ½ L (q₀ ω₀)² sin²(ω₀t) = q₀² sin²(ω₀t) / (2C)    [using ω₀² = 1/LC]
U_total = q₀²/(2C) × (cos² + sin²) = q₀²/(2C)

The total energy is constant; the individual energies oscillate between zero and the same maximum, 180° out of phase with each other. Twice per cycle the energy is fully in the capacitor (V is at its peak and I = 0), and twice per cycle it is fully in the inductor (I is at its peak and V = 0). The frequency of energy transfer is 2ω₀, while the charge and current oscillate at ω₀.

This is exactly analogous to a mass on a spring oscillating between maximum kinetic energy at the equilibrium point and maximum potential energy at the extremes. The mappings:

Mechanical	Electrical	Quantity
Position x	Charge q	State variable
Velocity v = dx/dt	Current I = dq/dt	Rate of state
Mass m	Inductance L	Inertia
Spring constant k	1/C	Restoring stiffness
Damping b	Resistance R	Loss
Drive force F(t)	EMF source ε(t)	Forcing
Frequency √(k/m)	1/√(LC)	Natural ω₀

Whatever you know about pendulums and springs translates directly into LC behaviour: damped oscillations, driven resonance, beats from coupled oscillators, even amplitude-modulation and parametric amplification — they all carry over.

Worked example: a 5 kHz audio oscillator

Suppose we want an LC tank that resonates at f₀ = 5 kHz. We pick L = 1 mH and solve for C:

f₀ = 1 / (2π √(LC))
5000 = 1 / (2π √(1e-3 × C))
2π × 5000 = 1 / √(1e-3 × C)
31416 = 1 / √(1e-3 × C)
√(1e-3 × C) = 1 / 31416 = 3.183e-5
1e-3 × C = 1.013e-9
C = 1.013e-6 F  ≈  1.0 μF

So an L = 1 mH inductor in series with a C = 1 μF capacitor resonates near 5 kHz. We can verify by plugging back: ω₀ = 1/√(1e-3 × 1e-6) = 1/√1e-9 = 1/3.162e-5 = 31623 rad/s, f₀ = 31623/(2π) = 5033 Hz. Within rounding, exactly the target.

If we initially charge the capacitor to V₀ = 10 V, the stored energy is ½ × 1e-6 × 10² = 5e-5 J = 50 μJ. At the moment all energy is in the inductor, U_L = ½ L I_max² = 50e-6 gives I_max = √(2 × 50e-6 / 1e-3) = √0.1 = 0.316 A. So the peak current is about 316 mA, sloshing back and forth at 5 kHz. The characteristic impedance of the tank is Z₀ = √(L/C) = √(1e-3/1e-6) = √1000 ≈ 31.6 Ω, the ratio of peak voltage to peak current.

For a higher-frequency RF circuit at f₀ = 100 MHz with L = 100 nH, C must be 1/(L(2πf)²) = 1/(1e-7 × (6.28e8)²) = 1/(1e-7 × 3.94e17) = 25 pF. These component values appear all over RF design — every WiFi chipset has dozens of similar tanks for matching networks, filters, and oscillators.

Component values across the frequency spectrum

Application	L	C	f₀
Audio crossover	1 mH	10 μF	1.59 kHz
Subwoofer notch	3.3 mH	22 μF	591 Hz
AM tuning at 1 MHz	250 μH	100 pF	1.0 MHz
FM IF (intermediate frequency)	5 μH	56 pF	9.5 MHz
FM RF tuning at 100 MHz	100 nH	25 pF	100.7 MHz
Wi-Fi 2.4 GHz match	2 nH	2.2 pF	2.4 GHz
Cell phone 5 GHz LO	1 nH	1.0 pF	5.03 GHz
Wireless charging coil	10 μH	100 nF	159 kHz
Induction cooktop	50 μH	1 μF	22.5 kHz

Notice the geometric mean L × C product: at 100 MHz it is 2.5 × 10⁻¹⁸ s²; at 1 MHz it is 2.5 × 10⁻¹⁴; at 1 kHz it is 2.5 × 10⁻⁸. Three orders of magnitude in frequency means six orders in LC product, two of which are absorbed by L and C separately. Designers shrink both as frequency rises until they bottom out at parasitic limits — the inductance of a centimeter of trace or the capacitance of a fingernail-sized SMD component.

Adding resistance: the RLC story

Real circuits always have some resistance R: in the wires, in the inductor's windings, in the dielectric of the capacitor. Adding R turns LC into RLC and the equation of motion becomes:

L d²q/dt² + R dq/dt + q/C = 0

The solutions are exponentially damped sinusoids:

q(t) = q₀ exp(−Rt/2L) cos(ω' t + φ)
ω'  = √(ω₀² − (R/2L)²)
   ≈ ω₀ when R is small

Three regimes:

• Under-damped (R²/(4L²) < 1/(LC), or equivalently R < 2√(L/C)). The circuit oscillates with amplitude that decays exponentially. Q = ω₀L/R counts how many radians the oscillation rings before its amplitude drops by 1/e.
• Critically damped (R = 2√(L/C), Q = 0.5). The circuit returns to zero in the shortest time without oscillating. Used in instrument indicators that should settle quickly without overshoot.
• Over-damped (R > 2√(L/C), Q < 0.5). The decay is purely exponential with two real time constants — no oscillation at all.

The Q factor — quality factor — is the headline performance metric:

Q = ω₀L/R = 1/(ω₀RC) = (1/R)√(L/C)

Q × bandwidth = f₀, so a Q of 100 at f₀ = 1 MHz gives a 10 kHz half-power bandwidth — exactly the AM broadcast channel spacing in the US. Q values for common technologies: discrete RF inductors 50–200; SAW filters in cellphones 1000–10000; quartz crystal resonators 10⁵; NMR rotating frames 10⁶; superconducting microwave cavities 10⁹–10¹¹.

Where LC circuits show up

• AM and FM radio tuning. Every AM radio has an LC tank with variable capacitance (a knob-driven air-gap cap, or a varactor diode) that selects which station's frequency the antenna couples to. AM uses Q ~ 100 at 0.5–1.6 MHz; FM uses Q ~ 50 at 88–108 MHz with bandwidth wide enough for the ±75 kHz deviation.
• Wireless power transfer. Qi wireless chargers transmit at 100–205 kHz through resonant LC pairs. The transmitter coil's L plus a capacitor forms one tank; the receiver coil's L plus a cap forms another. Tuning both to the same f₀ maximizes coupling efficiency, often above 80%. Modern phones tune dynamically as the coils misalign.
• Switching power converter snubbers and matching. Every buck, boost, and resonant LLC converter relies on LC behavior. Resonant LLC topologies operate the LC tank at zero-current or zero-voltage switching points to push efficiency above 95% in 100 W to multi-kW supplies.
• MRI body coils. A bird-cage RF coil resonant at the Larmor frequency (64 MHz at 1.5 T or 128 MHz at 3 T) is a multi-rung LC structure tuned to receive the precessing nuclear magnetisation. Q of 100–200 sets the receiver sensitivity.
• Plasma matching networks. Inductively-coupled and capacitively-coupled plasmas in semiconductor etching reactors use LC pi-networks to match a 13.56 MHz RF generator to the variable plasma impedance. Real-time servo'd capacitors keep the match at SWR < 1.2 even as the etch gas composition changes.

Variants and extensions

• Series LC vs parallel LC. Series LC has minimum impedance at resonance (acts as short to f₀); used as a frequency-selective trap to short out unwanted carriers. Parallel LC has maximum impedance at resonance (acts as an open); used as a tank circuit to develop large voltage from small drive current.
• RLC circuit. Adding resistance R produces the damped harmonic oscillator. R sets the Q factor and the bandwidth of the resonance; large R kills oscillation entirely. The standard textbook circuit beyond ideal LC.
• Coupled LC and band-pass filters. Two LC tanks coupled through a transformer or capacitor produce a two-pole band-pass response. Three or four coupled tanks make sharper responses with flat passband and steep skirts. The basis of every IF strip in classic radios.
• LC oscillator topologies. Hartley (tapped inductor), Colpitts (tapped capacitor), Clapp (capacitive divider with series cap), Pierce (crystal as the L-equivalent). Each adds an active gain element to compensate the R losses and sustain oscillation. Found in microcontroller clocks and RF synthesizers.
• Quantum LC oscillator. Microwave LC resonators on superconducting chips, with L from a Josephson junction's kinetic inductance and C from a parallel-plate capacitor, store quantum states of the EM field at GHz frequencies. The transmon qubit is essentially a non-linear LC oscillator coupled to a measurement resonator.

Common pitfalls

• Forgetting parasitic resistance. Every real inductor has DC resistance and AC core loss; every real capacitor has equivalent series resistance (ESR) and dielectric loss. The "loss" you see in the circuit is the sum of all of these. Tabulated Q values are at a specified frequency and amplitude; out of those conditions the Q can drop by an order of magnitude.
• Treating inductors as ideal at high frequency. Above the self-resonant frequency of the inductor (where its parasitic capacitance resonates with its inductance), the device behaves as a capacitor. Surface-mount inductors typically self-resonate between 100 MHz and 5 GHz; ferrite-core inductors much lower. Always check the SRF in the datasheet.
• Forgetting the factor of 2π. ω₀ = 1/√(LC), but f₀ = ω₀/(2π) = 1/(2π√(LC)). Mixing radians per second with cycles per second by a factor of 6.28 is the most common single mistake in LC calculation.
• Loading the tank with the measurement. Probing a high-Q tank with a 10× scope probe (10 MΩ at DC, but a few pF parallel) detunes and damps it. Use a small coupling capacitor or an active probe with sub-pF input capacitance for accurate Q and frequency measurement.
• Assuming ideal sinusoids in driven circuits. Square-wave or pulse-driven LC tanks ring at their natural frequency on top of the drive. The harmonic content of the drive selects which f₀ the tank really sings at — useful for class-E amplifiers, problematic when the tank is supposed to filter.