1. AC Sinusoidal Waveforms, Peak & RMS Values

Instantaneous voltage and current: V(t) = V0 sin(ωt), I(t) = I0 sin(ωt ± φ), where ω = 2πf (standard household f = 50 Hz in India, ω = 100π ≈ 314 rad/s). Root Mean Square (RMS / Effective / Virtual) values: V_rms = V0 / √2 ≈ 0.707 V0; I_rms = I0 / √2 ≈ 0.707 I0. Standard AC voltmeters and ammeters measure RMS values, not peak values. Peak values: V0 = √2 V_rms ≈ 1.414 V_rms (household 220V AC reaches peak of ±311V). Average value over positive half cycle: V_avg = (2 V0) / π ≈ 0.637 V0; Average over full complete cycle is zero.

2. AC Across Pure R, L, and C Components

Pure Resistor (R): Current and voltage are in identical phase (φ = 0). Opposition: R. Pure Inductor (L): Voltage leads current by 90° (π/2 rad) (or current lags voltage). Inductive Reactance: XL = ω L = 2π f L (increases linearly with frequency; blocks high-frequency AC, acts as short circuit for DC where f = 0). Pure Capacitor (C): Current leads voltage by 90° (π/2 rad). Capacitive Reactance: XC = 1 / (ω C) = 1 / (2π f C) (inversely proportional to frequency; blocks DC where f = 0, XC -> ∞; passes high-frequency AC).

3. Series LCR Circuit & Impedance Triangle

In series connection of R, L, and C: Net Impedance (total opposition Z): Z = √[R² + (XL - XC)²]. Phase Angle φ between voltage and current: tan φ = (XL - XC) / R; cos φ = R / Z. If XL > XC: circuit is inductive, voltage leads current; if XC > XL: circuit is capacitive, current leads voltage; if XL = XC: circuit is purely resistive, voltage and current are in phase.

4. Electrical Resonance & Quality Factor (Q)

Series Resonance occurs when inductive reactance equals capacitive reactance: XL = XC -> ω0 L = 1 / (ω0 C). Resonant Angular Frequency: ω0 = 1 / √(L C); Resonant Frequency: f0 = 1 / [2π √(L C)]. Characteristics at resonance: (1) Impedance is at its absolute minimum: Z_min = R; (2) Current is at its maximum: I_max = V / R; (3) Phase difference is zero (φ = 0, power factor cos φ = 1). Quality Factor (Selectivity / Sharpness of tuning): Q = (ω0 L) / R = 1 / (ω0 C R) = (1 / R) √(L / C) = f0 / Bandwidth.

5. Power in AC Circuits & Power Factor

True (Average) Power dissipated in AC circuit: P_avg = V_rms · I_rms · cos φ = (V0 I0 / 2) cos φ. Power Factor: cos φ = R / Z. Apparent Power = V_rms · I_rms. In purely inductive or capacitive circuit (R = 0, φ = 90°): cos φ = 0, so P_avg = 0. The current flowing with zero energy dissipation is known as "Wattless Current": I_wattless = I_rms sin φ. To maximize power efficiency in industrial distribution, power factor is maintained close to 1 by adding parallel power capacitors.

6. Transformers: Transformation Ratio & Efficiency

Operating on mutual induction between primary coil (Np turns) and secondary coil (Ns turns): Transformation Ratio: k = Vs / Vp = Ns / Np = Ip / Is (for ideal 100% efficient transformer). Step-Up Transformer: Ns > Np -> Vs > Vp and Is < Ip (increases voltage, reduces current to minimize transmission line I²R heating). Step-Down Transformer: Ns < Np -> Vs < Vp and Is > Ip. Efficiency of real transformer: η = (Output Power / Input Power) × 100 = (Vs Is cos φ_s) / (Vp Ip cos φ_p) × 100. Energy losses: Copper I²R heating, Iron Eddy currents (minimized by laminated core), Hysteresis loss, and Flux leakage.