Alternating Current— Notes, Formulas & Revision

An AC source produces a sinusoidal voltage v(t) = V₀ sin(ωt), where V₀ is the peak amplitude and ω = 2πf is the angular frequency.

Indian mains: 230 V rms, 50 Hz. So V₀ ≈ 325 V and T = 20 ms.

RMS value is the DC-equivalent — the steady DC voltage that would deliver the SAME average power to a resistor.

For a sine wave: V_rms = V₀/√2 ≈ 0.707 V₀.

Phasor representation: V₀ is the length of a vector rotating at ω; its y-projection (or x-projection, by convention) gives the instantaneous value.

The PHASE of an AC source is its angular offset at t = 0 — it sets when the wave 'starts'.

AC waveforms can also be triangular or square, but unless otherwise stated 'AC' means sinusoidal.

When AC voltage v = V₀ sin(ωt) drives a pure resistor R, the current is i = (V₀/R) sin(ωt).

V and I are IN PHASE — they peak, zero, and reverse together. Phase angle φ = 0.

Peak current: I₀ = V₀/R. RMS current: I_rms = V_rms/R = I₀/√2.

Average power: P_avg = V_rms · I_rms = V_rms²/R = I_rms²·R — exactly like a DC formula but with rms values.

Resistors dissipate energy continuously — instantaneous power p(t) = v·i = (V₀²/R) sin²(ωt) ≥ 0.

On a phasor diagram, V and I phasors point in the SAME direction.

Resistors do not store energy; they dissipate it as heat (Joule heating).

A pure inductor opposes changes in current. Driven by v = V₀ sin(ωt), current is i = (V₀/X_L) sin(ωt − π/2).

Current LAGS voltage by exactly 90° (φ = +π/2). At the instant V is at its peak, I is zero and rising.

Inductive reactance: X_L = ωL = 2πfL. Units: Ω. Rises linearly with frequency.

At DC (f = 0): X_L = 0 (inductor acts as a wire). At very high f: X_L → ∞ (inductor BLOCKS AC).

No power is dissipated on average: P_avg = V_rms·I_rms·cos(π/2) = 0. Energy oscillates between source and inductor's magnetic field.

Phasor: I lags V by 90° — current phasor is 90° clockwise from voltage phasor.

Inductors are 'frequency-dependent resistors' — but they STORE energy, they don't dissipate it.

A pure capacitor lets the current jump first, voltage builds up. Driven by v = V₀ sin(ωt), current is i = (V₀/X_C) sin(ωt + π/2).

Current LEADS voltage by 90°. When V is zero (about to rise), I is at its peak.

Capacitive reactance: X_C = 1/(ωC) = 1/(2πfC). Units: Ω. FALLS with frequency.

At DC (f = 0): X_C → ∞ (capacitor BLOCKS DC after charging). At very high f: X_C → 0 (capacitor passes AC like a wire).

Average power: P_avg = 0 (cos(π/2) = 0). Like an inductor, no net energy dissipation.

Phasor: I leads V by 90° — current phasor is 90° counter-clockwise from voltage phasor.

Capacitors store energy in the electric field — energy goes back and forth between source and capacitor.

A series LCR circuit driven by AC has resistor, inductor, and capacitor sharing the SAME current — but with different phase relationships to the source voltage.

Impedance: Z = √(R² + (X_L − X_C)²) — the AC analogue of total resistance.

Peak current: I₀ = V₀/Z. RMS current: I_rms = V_rms/Z.

Phase angle between V and I: φ = arctan((X_L − X_C)/R). Positive φ → V leads I (inductive). Negative → V lags I (capacitive).

Voltage across R is IN PHASE with current; across L leads I by 90°; across C lags I by 90°.

When X_L = X_C (resonance), Z = R (minimum), I is maximum, and circuit behaves purely resistive.

Phasor diagram: V_R along I, V_L at +90° from I, V_C at −90° from I. Net voltage = vector sum.

Average power = V_rms · I_rms · cos φ. The factor cos φ is the POWER FACTOR.

Series LCR resonance: at f₀ = 1/(2π√(LC)), the inductive and capacitive reactances cancel and impedance is minimum (= R).

Current is MAXIMUM at resonance: I_max = V_rms/R, regardless of L or C separately.

At resonance, V and I are IN PHASE — circuit looks purely resistive even though L and C are present.

Quality factor Q = (1/R)√(L/C) = ω₀L/R = 1/(ω₀RC). Higher Q ⇒ sharper resonance, narrower bandwidth.

Bandwidth Δf = f₀/Q = R/(2πL). Wide R (resistance) ⇒ broad resonance peak. Low R ⇒ sharp, selective.

Resonance is the principle of RADIO TUNING — varying C tunes Q-circuit to the broadcast frequency.

V across L and V across C can each be Q × V_source — much larger than the source. This is voltage MAGNIFICATION at resonance.

Parallel (anti-)resonance behaves OPPOSITELY: impedance is MAXIMUM, current minimum (for ideal components).

Instantaneous power in an AC circuit: p(t) = v(t)·i(t). For sinusoidal v and i with phase difference φ: p(t) varies, but ITS AVERAGE matters.

Average power: P_avg = V_rms · I_rms · cos φ. The factor cos φ is called the POWER FACTOR.

Pure R: cos φ = 1 ⇒ full power delivered.

Pure L or C: cos φ = 0 ⇒ no power dissipated (wattless current — energy oscillates but doesn't transfer net).

Mixed circuit: cos φ = R/Z. Higher reactance ⇒ smaller cos φ ⇒ less real power for same current.

Real power (W) vs apparent power (VA): P = V_rms·I_rms·cos φ (real), S = V_rms·I_rms (apparent). Reactive power Q = V_rms·I_rms·sin φ (VAR).

Power factor correction: industries add capacitors in parallel with inductive loads to bring cos φ close to 1. Lowers I for the same P → reduces transmission losses.

An LC circuit (inductor + capacitor, no resistance) oscillates indefinitely once given an initial charge or current — energy bouncing between E-field of C and B-field of L.

Charge on capacitor: q(t) = q₀ cos(ω₀ t). Current: i(t) = −q₀ω₀ sin(ω₀ t).

Angular frequency: ω₀ = 1/√(LC). Period T = 2π√(LC).

Energy: U_E = q²/(2C) in capacitor; U_B = ½LI² in inductor. Total U_tot = q₀²/(2C) is CONSTANT.

Energy oscillates at 2ω₀ — it transfers between C and L twice per cycle.

Direct mechanical analogue: SHM of a mass on a spring. Q ↔ x, I ↔ v, L ↔ m, 1/C ↔ k, energy U_B ↔ KE, U_E ↔ PE.

Adding any resistance turns it into damped LCR oscillation — amplitude decays exponentially.

A transformer is two coils (primary, secondary) magnetically coupled — usually via a laminated iron core to maximise flux linkage.

Ideal-transformer relation: V_s/V_p = N_s/N_p (turns ratio).

If N_s > N_p: step-up transformer (V_s > V_p but I_s < I_p). If N_s < N_p: step-down.

For an ideal (lossless) transformer: V_p·I_p = V_s·I_s (power conservation).

Transformers ONLY work with AC — they need a changing Φ. DC produces no induction.

Real transformers have losses: copper (I²R in windings), iron (eddy + hysteresis in core), flux leakage. Efficiency typically 95-99%.

Power transmission grid uses high-voltage AC (220-765 kV) precisely so transformers can step it up at the source and step it down at the consumer end — minimising I²R losses in transmission lines.

An AC generator converts mechanical rotation into electrical AC by spinning a coil in a uniform magnetic field.

EMF: ε(t) = N·B·A·ω·sin(ωt), where N = turns, B = field, A = coil area, ω = angular speed.

Peak EMF: ε₀ = NBAω. Frequency f = ω/(2π) — directly set by rotation rate.

Flux through the coil: Φ = NBA·cos(ωt). The EMF is −dΦ/dt = NBAω·sin(ωt) — exactly Faraday's law.

Slip rings and brushes feed AC out (without rectification). A commutator (split-ring) would convert it to DC.

Most power plants — thermal, hydro, nuclear, wind — drive a turbine that turns a 3-phase generator. The grid runs at fixed f (50 Hz in India, 60 Hz in US).

Indian power-plant generators run typically at 3000 rpm to produce 50 Hz from a 2-pole machine.

Inductive reactance X_L = ωL = 2πfL — rises LINEARLY with frequency.

Capacitive reactance X_C = 1/(ωC) = 1/(2πfC) — falls HYPERBOLICALLY with frequency.

At low f: X_C is huge (block) and X_L is small (pass) — capacitor blocks DC, inductor lets it through.

At high f: X_L is huge (block) and X_C is small (pass) — inductor blocks high-frequency, capacitor passes it.

The two curves cross at f₀ = 1/(2π√(LC)) — the resonant frequency. At this f, X_L = X_C.

This frequency dependence is the basis of FILTERS: high-pass uses series C, low-pass uses series L, etc.

RC circuits act as low-pass or high-pass depending on which element is in series; RL circuits are dual.

Root-mean-square (rms) value of an AC quantity is the DC equivalent that delivers the SAME average power to a resistor.

For sinusoidal AC: V_rms = V₀/√2 and I_rms = I₀/√2.

Mean voltage over a full sine cycle is ZERO (positive and negative halves cancel) — that's why ⟨V²⟩ is used.

Average power: P_avg = V_rms·I_rms = V_rms²/R = I_rms²·R (matches DC form).

rms is what AC meters display by default. '230 V mains' means 230 V rms.

Average over a half-cycle (magnitude only): |V|_avg = 2V₀/π ≈ 0.637 V₀ — DIFFERENT from rms.

For NON-sinusoidal waveforms: rms must be computed from the specific waveform. For a square wave amplitude V₀: V_rms = V₀ (no factor of √2).

A phasor is a rotating vector representing a sinusoidal quantity — its length is the amplitude, its angle is the phase.

All phasors in a circuit rotate at the SAME angular frequency ω. Relative phase angles between them stay constant.

By convention, we view the phasors in a rotating frame so the current is along the +x axis. Voltage phasors are drawn relative to it.

For R: V_R is in phase with I (along +x).

For L: V_L leads I by +90° (along +y).

For C: V_C lags I by −90° (along −y).

Total source voltage = vector sum of V_R, V_L, V_C. For series LCR: V = √(V_R² + (V_L − V_C)²).

Adding phasors graphically gives both the magnitude (Z·I) and phase (φ = tan⁻¹((X_L−X_C)/R)).