Capacitors

Capacitor basics

Two conducting plates separated by an insulator (dielectric). When voltage V is applied, charge Q accumulates:

Q = C · V

C is capacitance, units of farads (F). 1 F is huge; typical values are pF, nF, µF, mF.

Parallel-plate capacitor

C = ε₀ · κ · A / d

• ε₀ = 8.854 × 10⁻¹² F/m (permittivity of free space).
• κ = dielectric constant (≥ 1).
• A = plate area (m²).
• d = separation (m).

Capacitance scales with area, dielectric constant, and inversely with separation.

Common dielectrics

Material	κ (relative)	Breakdown E (V/m)
Vacuum	1.000	~3 × 10⁷ (in practice)
Air	1.0006	3 × 10⁶
Paper	3.5	~1 × 10⁷
Polyethylene	2.3	2 × 10⁷
Mica	5-7	~2 × 10⁸
Glass	4-7	~1 × 10⁷
Ceramic (titanate)	10-1000+	1-10 × 10⁶
Tantalum oxide	~25	~10⁸
Water (pure)	~80	—

Energy stored

U = ½ · C · V² = ½ · Q · V = Q² / (2 · C)

For a 100 µF capacitor at 12 V — U = ½ · 100e-6 · 144 = 7.2 mJ.

Supercapacitors (1 F at 5 V) — U = ½ · 1 · 25 = 12.5 J.

Series and parallel

Configuration	C_total	Notes
Series	1/C_total = Σ(1/C_i)	Less than smallest; shares charge
Parallel	C_total = ΣC_i	Sum; shares voltage
Two equal in series	C/2	—
Two equal in parallel	2C	—

(Opposite of resistors — series resistors add; series capacitors are less.)

AC behavior

For sinusoidal voltage V(t) = V_peak·cos(ωt), capacitor current:

I(t) = C · dV/dt = -C · ω · V_peak · sin(ωt)

Magnitude relation — V_peak / I_peak = 1/(ω·C) = X_C (capacitive reactance).

Phase: I leads V by 90° (current peaks before voltage peak).

Frequency	X_C for 1 µF
0 (DC)	∞ (blocks)
60 Hz (line)	2.65 kΩ
1 kHz	159 Ω
1 MHz	0.159 Ω
1 GHz	0.159 mΩ

JavaScript — capacitor calculations

// Charge stored
function chargeStored(C, V) { return C * V; }

// Energy stored
function capEnergy(C, V) { return 0.5 * C * V * V; }

// Parallel-plate capacitance
function parallelPlateC(area_m2, separation_m, kappa = 1) {
  const epsilon_0 = 8.854e-12;
  return epsilon_0 * kappa * area_m2 / separation_m;
}

// 1 cm² plates, 0.1 mm gap, air
console.log(`Air gap C: ${parallelPlateC(1e-4, 1e-4).toExponential(2)} F`);  // 8.85 pF

// Series capacitors
function seriesC(C_array) {
  return 1 / C_array.reduce((sum, C) => sum + 1/C, 0);
}

// Parallel capacitors
function parallelC(C_array) {
  return C_array.reduce((a, b) => a + b, 0);
}

console.log(`100µF + 100µF series: ${(seriesC([100e-6, 100e-6]) * 1e6).toFixed(0)} µF`);  // 50
console.log(`100µF + 100µF parallel: ${(parallelC([100e-6, 100e-6]) * 1e6).toFixed(0)} µF`); // 200

// RC charging
function rcChargeVoltage(V_source, R, C, t) {
  return V_source * (1 - Math.exp(-t / (R * C)));
}

// 12 V, 1 kΩ, 100 µF: τ = 0.1 sec
const tau_rc = 1000 * 100e-6;
console.log(`After τ: ${rcChargeVoltage(12, 1000, 100e-6, tau_rc).toFixed(2)} V`);  // 7.59 (~63%)
console.log(`After 5τ: ${rcChargeVoltage(12, 1000, 100e-6, 5*tau_rc).toFixed(2)} V`);  // 11.92 (~99%)

// Capacitive reactance
function reactanceCap(f_Hz, C) {
  return 1 / (2 * Math.PI * f_Hz * C);
}

console.log(`1 µF at 60 Hz: ${reactanceCap(60, 1e-6).toFixed(0)} Ω`);   // ~2653
console.log(`1 µF at 1 MHz: ${reactanceCap(1e6, 1e-6).toFixed(3)} Ω`);   // 0.16

// Time to charge supercapacitor (1 F at 5V via 1A current source)
function chargeTime(C, V, I_charge) {
  // Q = CV; t = Q/I
  return C * V / I_charge;
}

console.log(`1F to 5V at 1A: ${chargeTime(1, 5, 1)} s`);  // 5 sec

// LC oscillator
function lcResonance(L, C) {
  return 1 / (2 * Math.PI * Math.sqrt(L * C));
}

// 1 µH inductor, 1 nF capacitor
console.log(`L=1µH, C=1nF: f = ${(lcResonance(1e-6, 1e-9) / 1e6).toFixed(2)} MHz`);

// Voltage across capacitor in voltage divider
function capDividerVoltage(V_in, C1, C2) {
  // Series caps; voltage divides inversely as C
  return V_in * C1 / (C1 + C2);
}

Where capacitors matter

• Power supply filtering. Smooth out AC ripple after rectification.
• Decoupling. Place near IC power pins to absorb transient current spikes.
• Timing circuits. RC oscillators, 555 timer, monostable circuits.
• Energy storage. Camera flashes, defibrillators, supercapacitors for vehicles, regenerative braking.
• Filters. Low-pass, high-pass, band-pass filters in audio, RF, signal conditioning.
• Coupling and decoupling. Pass AC signals between stages while blocking DC offsets.
• Tuned circuits. Combined with inductors for resonance — radio tuners, antennas.

Common mistakes

• Confusing series and parallel for capacitors. Opposite of resistors — series capacitors give LESS C; parallel give MORE.
• Polarity errors. Electrolytic caps are polarized; backwards connection often destroys them (sometimes explosively).
• Ignoring breakdown voltage. Each capacitor has a max V; exceed it and dielectric breaks down (cap fails). Use cap rated 2-3× operating V.
• Forgetting leakage current. Real caps have small leakage. Big problem for long-term holding circuits, less so for dynamic ones.
• Using DC analysis on capacitor in AC. AC capacitor passes current proportionally to frequency. DC blocks it.
• Treating capacitor energy ½CV² as work done by source. Source does work CV². Half goes into capacitor; half lost to wire resistance during charging.