Class 12 · Chapter 7
Alternating Current
Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.
Alternating Current Short Notes
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Condensed revision points for Alternating Current — for quick recall before exams, not a substitute for the full notes.
Basics of AC
- I = I₀ sin ωt; ω = 2π/T = 2πf. Peak-to-peak = 2I₀.
- AC must have constant amplitude and equal, opposite half-cycles — a wave that never reverses direction isn't AC.
- India: 220 V, 50 Hz. USA: 110 V, 60 Hz.
Average and RMS Values
- I_avg (half cycle) = 2I₀/π ≈ 0.637 I₀; full-cycle average of symmetric AC = 0.
- I_rms = I₀/√2 ≈ 0.707 I₀ — same heating effect as that value of DC.
- Ordering: I₀ > I_rms > I_avg.
- Unmarked/unstated AC values (appliance ratings, meter readings) are always rms.
Phase
- I = I₀ sin(ωt + φ): φ is initial phase (constant); (ωt+φ) is instantaneous phase.
- 'V leads I' = V peaks first. 'I leads V' = I peaks first.
Pure R, L, C Circuits
- R: I in phase with V; I₀ = V₀/R; same behaviour in AC and DC.
- L: I lags V by π/2; X_L = ωL = 2πfL; X_L ∝ f; X_L = 0 at DC.
- C: I leads V by π/2; X_C = 1/(ωC); X_C ∝ 1/f; X_C = ∞ at DC (blocks DC).
- Inductor = low-pass filter; Capacitor = high-pass filter.
Series LR and RC Circuits
- LR: Z_L = √(R²+X_L²); tan φ = X_L/R; emf leads I.
- RC: Z_C = √(R²+X_C²); tan φ = X_C/R; emf lags I.
- Both cases: E = √(V_R² + V_reactive²) — perpendicular phasors, not plain addition.
Series LCR Circuit
- Z = √(R² + (X_L−X_C)²); tan φ = (X_L−X_C)/R.
- X_L > X_C → net inductive, emf leads I. X_C > X_L → net capacitive, emf lags I.
- V_L or V_C alone can exceed source voltage — only possible when both L and C are present.
Resonance
- Condition: X_L = X_C ⟹ ω₀ = 1/√(LC), f₀ = 1/(2π√(LC)).
- At resonance: Z_min = R, I_max = V/R, φ = 0, power factor = 1.
- Series resonant circuit = 'acceptor circuit' (radio/TV tuning).
- Below f₀: capacitive (φ < 0). Above f₀: inductive (φ > 0).
Q-Factor and Bandwidth
- Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C) = f₀/Δf.
- Δf = f₂ − f₁ (half-power frequencies, where net reactance = net resistance).
- ↓R ⟹ ↑Q ⟹ sharper resonance, narrower bandwidth.
- At resonance, V_L = V_C = Q × V (voltage magnification factor = Q).
Power in AC Circuits
- P_avg = V_rms I_rms cos φ (real power). P_apparent = V_rms I_rms.
- Power factor cos φ = R/Z; ranges 0 to 1.
- Pure R: cos φ = 1 (max). Pure L or C: cos φ = 0.
- I_rms cos φ = wattful (active) current; I_rms sin φ = wattless (reactive) current.
Choke Coil
- Large L, very small resistance r, wound on soft iron core.
- Limits AC current with almost zero power loss: cos φ = r/Z ≈ r/(ωL) → 0.
- Works only on AC — an inductor offers zero steady-state opposition to DC.
- A capacitor could do the same job (zero power loss) but is costlier than an equivalent choke.
LC Oscillations
- q = q_m cos ωt; ω = 1/√(LC) — same formula as resonance, undriven case.
- Energy conservation: q_m²/(2C) = ½LI_m².
- Charge and current are π/2 out of phase — charge max ⟺ current zero, and vice versa.
- At T/8, 3T/8, 5T/8...: energy is shared equally between L and C.
- Real circuits have resistance ⟹ damped oscillation (amplitude decays exponentially).
Measuring AC
- Hot-wire (AC) meters: based on heating effect, deflection ∝ I², read I_rms, non-uniform scale.
- Moving-coil (DC) meters read zero in an AC circuit (average of symmetric AC = 0).
- Hot-wire meters work correctly on both AC and DC.
Common Exam Traps
- Don't confuse average value (always taken over a half-cycle) with the full-cycle average (always zero for symmetric AC) — questions often ask for one while implying the other.
- X_L and X_C are reactances, not resistances — they store/return energy and dissipate none, even though they're measured in ohms.
- Resonance needs both L and C together — an RL or RC circuit alone can never resonate, regardless of frequency.
- At resonance, current depends only on R and V — L and C have 'cancelled out' and no longer matter for the current magnitude.
- V_L + V_C is never just added arithmetically with V_R — they're perpendicular (or opposite, for V_L vs V_C) on the phasor diagram.
- A choke's near-zero power loss comes from r ≪ ωL, not from r being exactly zero — real choke coils always have some small resistance.
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