Resonance and Coupled Circuits: Complete Beginner's Guide

Resonance is the condition in an inductor-capacitor circuit where inductive and capacitive effects exactly cancel, so the circuit responds most strongly at one special frequency. Coupled circuits are two or more coils linked by a shared magnetic field, transferring energy from one coil to another with no wire between them.

Resonance matters because radios tune stations by resonating, and wireless chargers and transformers move power through coupled coils. Together they are the physics behind tuning, filtering and contactless power transfer.

Key Concepts

• Resonant frequency (f0): where inductive reactance equals capacitive reactance, in hertz.
• Series resonance: impedance minimum (equal to R), current maximum; inductor and capacitor voltages can each exceed the supply.
• Parallel resonance: impedance maximum, line current minimum; the circulating LC current can exceed the supply current.
• Quality factor (Q): sharpness of resonance — higher Q, narrower and taller the peak.
• Mutual inductance (M): strength of magnetic linkage between coils, in henries.
• Coefficient of coupling (k): k = M/√(L1·L2), between 0 (none) and 1 (perfect).
• Dot convention: dots mark coil ends of the same instantaneous polarity, fixing the sign of M.

Formulas

f0 = 1/(2π√(LC)) hertz (L in henries, C in farads); ω0 = 1/√(LC) rad/s. At resonance XL = XC = ω0L. Series Q: Q = (1/R)·√(L/C), dimensionless. Bandwidth: BW = f0/Q hertz. Voltage magnification in series resonance: VL = VC = Q·Vs. Mutual inductance: M = k·√(L1·L2) henries; induced voltage v2 = ±M·di1/dt, sign from the dot convention. These assume linear magnetics and sinusoidal steady state.

Worked Example

A series RLC circuit: R = 10 Ω, L = 100 mH, C = 10 µF, Vs = 5 V. f0 = 1/(2π√(0.1 × 10×10⁻⁶)) = 1/(2π × 10⁻³) = 159.15 Hz. Q = (1/10)·√(0.1/10×10⁻⁶) = 0.1 × 100 = 10. Bandwidth = 159.15/10 = 15.9 Hz. At resonance I = Vs/R = 0.5 A, and VC = Q·Vs = 50 V — ten times the supply, the classic resonance surprise. Now couple a second identical coil with k = 0.4: M = 0.4 × 0.1 = 0.04 H. With primary current 0.5 A at ω0 = 1000 rad/s, the open-circuit secondary voltage is V2 = ω0·M·I1 = 1000 × 0.04 × 0.5 = 20 V.

Common Misconceptions

• "At resonance the impedance is zero." Only the reactance cancels; resistance R remains, so series impedance equals R (minimum, not zero).
• "Component voltages can't exceed the supply." In series resonance each reactive voltage is Q times the supply — 50 V across the capacitor from a 5 V source here. This destroys underrated capacitors.
• "Coupled coils need an iron core." Air-core coupling works fine — wireless chargers and radio coils are air-coupled; a core only raises k.

Real-World Applications

Radio and TV tuners select stations with resonant LC circuits. Wireless chargers transfer power through coupled coils at resonance for efficiency. Transformers are tightly coupled coils (k ≈ 0.98+) that change AC voltage. Induction cooktops drive a resonant coil to heat the pan, and RFID tags harvest energy from the reader's coupled field.

Resonance amplifies response at one frequency while magnetic coupling moves energy across a gap — together they tune radios, charge phones wirelessly and power the grid's transformers.