Why the current from your wall socket keeps reversing direction, how resistors, inductors and capacitors each respond differently to it, and how a transformer changes voltage without moving parts.
Unlike the steady DC circuits from earlier chapters, an AC source drives a voltage that reverses direction periodically:
For a pure resistor, current simply follows Ohm's Law at every instant:
Since AC voltage and current constantly change sign, their plain average over a full cycle is zero — not useful for describing "how much" current is really flowing. Instead, we use the root mean square (RMS) value, which represents the DC equivalent that would dissipate the same average power.
To track how current and voltage line up (or don't) across R, L, and C, it helps to represent them as phasors — rotating vectors whose length is the amplitude and whose angle from a reference axis represents the phase.
An inductor resists changes in current (self-inductance, from Chapter 6), which delays the current relative to the driving voltage.
A capacitor charges and discharges to follow the applied voltage, but current must flow before the voltage across it can change — putting current ahead of voltage.
Combine a resistor, inductor, and capacitor in series, and the circuit's overall opposition to current — impedance — depends on how the inductive and capacitive reactances partially cancel each other, since they act in exactly opposite phase directions.
At one particular frequency, the inductive and capacitive reactances exactly cancel (XL = XC), leaving impedance at its absolute minimum — equal to just R — and current at its maximum. This is resonance, the same phenomenon behind tuning a radio to a specific station.
Charge a capacitor, then connect it to an inductor with no resistor at all, and energy oscillates back and forth between the capacitor's electric field and the inductor's magnetic field — with no source driving it, exactly like a frictionless mass on a spring trading kinetic and potential energy.
A transformer uses mutual inductance (Chapter 6) between two coils wound on a shared iron core to step AC voltage up or down, with essentially no moving parts.
Real transformers lose some energy to copper losses (I²R heating in the windings), flux leakage, hysteresis in the core, and eddy currents — the last of which is minimized the same way as in motors: by laminating the core.
A 50 Ω resistor is connected to a 220 V (RMS), 50 Hz AC supply. Find the RMS current and the average power dissipated.
Solution: Irms = Vrms/R = 220/50 = 4.4 A.
Pavg = Vrms Irms = 220 × 4.4 = 968 W.
A series LCR circuit has L = 1 H and C = 1 µF. Find the resonant frequency.
Solution: ωr = 1/√(LC) = 1/√(1 × 10⁻⁶) = 1/(10⁻³) = 1000 rad/s.
fr = ωr/(2π) = 1000/6.283 ≈ 159.2 Hz.