Alternating Current

Instantaneous values of AC voltage and current. Vm, Im are peak (maximum) values. ω = 2πf is angular frequency. φv, φi are initial phases. The phase difference φ = φv − φi determines whether current leads or lags voltage.

Root-mean-square value: the equivalent DC value that produces the same average power dissipation. The 230 V / 50 Hz mains supply means Vrms = 230 V, so Vm = 230√2 ≈ 325 V. All AC power calculations use RMS values.

Average value over a full cycle is zero for a symmetric sinusoid. Average over a half cycle (as used in rectifier analysis) is 2Vm/π. Relevant for DC ammeters measuring rectified AC and for computing charge delivered per half cycle.

Form factor relates RMS to average value; peak factor (crest factor) relates peak to RMS. For a sinusoidal waveform, these are fixed constants. Non-sinusoidal waveforms have different values — used in power quality and meter calibration.

Relations between angular frequency ω (rad s⁻¹), frequency f (Hz), and time period T (s). For Indian mains supply: f = 50 Hz, T = 0.02 s, ω = 100π rad s⁻¹. For US mains: f = 60 Hz.

In a pure resistor, current and voltage are in phase (φ = 0). Impedance equals resistance R. Power is dissipated: P = Vrms²/R = Irms²R. Ohm's law holds instantaneously for resistors — no frequency dependence.

Opposition offered by a pure inductor to AC. SI unit: ohm (Ω). XL increases linearly with frequency — inductor blocks high-frequency AC but allows DC (f=0, XL=0). Current lags voltage by 90°. No power is dissipated in a pure inductor.

Current lags voltage by 90° in a pure inductor. Phasor of I is 90° behind V. The back-EMF exactly equals applied voltage at all instants. Average power = VrmsIrms cos(90°) = 0. Energy is stored and returned each half-cycle, never dissipated.

Opposition offered by a pure capacitor to AC. SI unit: ohm (Ω). XC decreases with increasing frequency — capacitor blocks DC (f=0, XC→∞) but passes high-frequency AC. Current leads voltage by 90°. No power is dissipated in a pure capacitor.

Current leads voltage by 90° in a pure capacitor. Phasor of I is 90° ahead of V. The capacitor charges to equal and opposite voltage at each instant. Average power = 0. Mnemonic: CIVIL (Capacitor: I before V; Inductor: V before I).

In an R-L series circuit, voltage leads current by φ. Impedance Z is the phasor sum of R and XL (Pythagorean, not arithmetic). φ > 0: inductive circuit. As f → ∞, Z → XL → ∞. As f → 0, Z → R.

In an R-C series circuit, voltage lags current (or current leads voltage) by φ. φ < 0: capacitive circuit. As f → 0, Z → XC → ∞ (blocks DC). As f → ∞, Z → R. Used in filters and phase-shifting circuits.

General impedance of a series RLC circuit. Phase angle φ: positive (inductive) when XL > XC, negative (capacitive) when XC > XL, zero at resonance. The RLC circuit is the most general AC series circuit — every other combination is a special case.

VR is in phase with I; VL leads I by 90°; VC lags I by 90°. VL and VC are in antiphase. Total voltage V is the phasor sum. Note: VL and VC can individually exceed the source voltage V — voltage magnification at resonance.

At resonance, inductive and capacitive reactances cancel. Z = R (minimum), I = Vm/R (maximum). φ = 0 (unity power factor). Voltage and current are in phase. The resonant frequency f₀ depends only on L and C, not on R.

Q-factor measures sharpness of resonance. High Q: narrow bandwidth, high voltage magnification. At resonance: VL = VC = Q·V (Q times the source voltage). Also: Q = f₀/Δf = ω₀/Δω where Δω is the bandwidth. Q > 10 is considered sharp resonance.

Bandwidth Δω is the frequency range over which power ≥ ½ of maximum (current ≥ Im/√2 — the half-power or −3 dB points). ω₁ and ω₂ are the lower and upper half-power angular frequencies. Narrow bandwidth = high Q = sharp selectivity.

At resonance, voltage across L or C is Q times the supply voltage. For Q = 50 and V = 10 V: VL = VC = 500 V. This voltage magnification can damage capacitors/inductors in high-Q circuits. The two voltages cancel each other (antiphase), leaving only IR across R.

Instantaneous power fluctuates at twice the source frequency. It can be negative (energy returned to source by reactive components). The time-average of this gives true average power. Expanding using product-to-sum: p = ½VmIm[cosφ − cos(2ωt − φ)].

True power is the time-average of instantaneous power. cosφ is the power factor. P = Irms²R shows power is only dissipated in resistance. In a pure inductor or capacitor: φ = 90°, P = 0. In a pure resistor: φ = 0°, P = VrmsIrms (maximum).

Power factor is the cosine of the phase angle between voltage and current. Unity (φ=0) for pure resistance; zero (φ=90°) for pure reactance. Lagging PF: inductive loads. Leading PF: capacitive loads. Low PF causes high current for same power — poor efficiency.

Apparent power S (VA) is the product of RMS voltage and current — what the source supplies. Reactive power Qr (VAR) oscillates between source and reactive elements; does no net work. True power P (W) does real work. Power triangle: S² = P² + Qr².

Current has two components: active (in phase with V, does work: P = V·Iactive) and wattless (90° out of phase, does no work). Pure inductor/capacitor carries only wattless current. Power factor correction (adding capacitors) reduces the wattless component in industrial loads.

For an ideal transformer (k=1 coupling, no losses): voltage ratio equals turns ratio. k < 1: step-down; k > 1: step-up. Based on Faraday's law: both coils experience the same dΦ/dt, so V₁/N₁ = V₂/N₂.

In an ideal transformer, power is conserved: input = output. Current ratio is the inverse of turns ratio. Stepping up voltage steps down current by the same factor — long-distance power transmission uses high voltage (low current) to minimise I²R losses.

Real transformers have copper losses (I²R in windings) and iron/core losses (eddy currents + hysteresis in the core). Eddy currents are minimised by laminating the core. Hysteresis losses are minimised by soft magnetic materials. Typical efficiency: 95–99%.

A transformer reflects the load impedance to the primary side multiplied by (N₁/N₂)². Used for impedance matching in audio amplifiers and RF circuits to maximise power transfer. A step-down transformer makes a high-impedance load appear low-impedance to the source.