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Average power in an AC circuit is P = V_rms I_rms cos φ, where cos φ is the power factor — only the resistive component dissipates energy. Resonance occurs when X_L = X_C, i.e., ω₀ = 1/√(LC), giving maximum current for a series RLC circuit and minimum impedance. The quality factor Q = ω₀/Δω measures the sharpness of the resonance peak. Transformers use mutual inductance to step voltage up or down while conserving power (V₁/V₂ = N₁/N₂, I₁/I₂ = N₂/N₁).
Plot impedance |Z| vs. frequency and identify the resonance minimum. Calculate Q for different R values and observe how R broadens the resonance peak. Analyze the transformer equations and explain why high-voltage transmission minimizes resistive losses.
From your study of impedance and reactance, you know that inductors and capacitors present frequency-dependent opposition to current. A key insight carried into AC power is that this opposition — reactance — stores and releases energy rather than consuming it. An inductor builds up a magnetic field on each half-cycle and returns that energy to the circuit; a capacitor charges and then discharges. On average, neither dissipates power. Only resistance absorbs energy irreversibly, converting it to heat. This is the origin of the power factor: in a purely resistive circuit, voltage and current are in phase, and all the apparent power does real work. When reactance shifts current out of phase with voltage, some power sloshes back and forth without being consumed, and the power factor cos φ captures exactly what fraction of the apparent power V_rms I_rms actually does work.
Resonance arises when the inductive and capacitive reactances exactly cancel: X_L = X_C, i.e., ωL = 1/(ωC). Solving for the resonant frequency gives ω₀ = 1/√(LC). At this frequency, the total impedance of a series RLC circuit collapses to just R — the circuit behaves as if the inductor and capacitor aren't there. Current amplitude is maximized, and the power delivered to R is at its peak. Think of it like a swing: push at the natural frequency and the amplitude grows; push at the wrong frequency and energy fights the swing's stored motion. The LC pair is the mechanical analog of the swing's mass and spring.
The quality factor Q = ω₀L/R = ω₀/Δω tells you how sharply tuned the resonance is. High Q means the resonance peak is narrow: only frequencies very close to ω₀ produce large current. Low R (low damping) gives high Q — the circuit is "choosy" about frequency. This is how radio tuners work: by adjusting C, you shift ω₀ to match a particular station's broadcast frequency, while the narrow Q rejects nearby stations. High Q is desirable when selectivity matters; low Q is preferable when you need to pass a broad band of frequencies.
Transformers bring in the power consequences of resonance from a different angle. They exploit mutual inductance between two coils to transfer power while changing voltage and current levels. The ideal transformer conserves power: if V₂ = (N₂/N₁)V₁ steps voltage up, then I₂ = (N₁/N₂)I₁ steps current down proportionally. The reason high-voltage AC transmission is efficient connects directly to your prerequisite knowledge of resistive power dissipation P = I²R: for a given power delivered, transmitting at high voltage means low current, and losses in the transmission line (which have fixed resistance) scale as I². Stepping voltage up by a factor of 10 cuts line losses by a factor of 100. This is why the AC grid operates at hundreds of kilovolts, stepped down locally by distribution transformers before reaching homes.