Alternating Current Class 12 Chapter Notes

Key points

• AC current and voltage vary periodically, usually sinusoidally: I = I0 sin ωt and V = V0 sin ωt. Here I0 and V0 are peak values.
• RMS value is the effective DC equivalent for heating effect: Irms = I0/√2 and Vrms = V0/√2 for sinusoidal AC.
• In pure resistor: voltage and current are in phase; in pure inductor: voltage leads current by π/2; in pure capacitor: current leads voltage by π/2.
• Inductive reactance XL = ωL = 2πfL increases with frequency; capacitive reactance XC = 1/ωC = 1/(2πfC) decreases with frequency.
• For series LCR circuit: Z = √(R² + (XL - XC)²), I0 = V0/Z, tan φ = (XL - XC)/R.
• Average power in AC circuit: Pavg = Vrms Irms cos φ. The factor cos φ is called power factor.
• At resonance in series LCR: XL = XC, Z = R, current is maximum, phase angle φ = 0 and f0 = 1/(2π√LC).
• Transformer works on mutual induction and is designed for AC, not steady DC. Ideal transformer relation: Vs/Vp = Ns/Np = Ip/Is.

Important definitions and terms

Alternating current

Electric current whose magnitude changes continuously with time and direction reverses periodically.

Peak value

Maximum instantaneous value of AC voltage or current, denoted by V0 or I0.

RMS value

Root mean square value of AC; the equivalent DC value producing the same heat in a resistor in the same time.

Reactance

Opposition offered by an inductor or capacitor to AC due to changing current or voltage, measured in ohm.

Impedance

Total effective opposition offered by an AC circuit, combining resistance and reactance; denoted by Z.

Phase difference

Angular difference between two sinusoidally varying quantities such as voltage and current.

Power factor

Cosine of phase angle between voltage and current; it decides how much of apparent power is converted into useful power.

Resonance

Condition in a series LCR circuit when XL = XC, impedance is minimum and current is maximum.

Transformer

Device based on mutual induction used to step up or step down AC voltage.

Core idea: why AC circuits need phasors

In DC circuits, resistance alone decides the current after steady state. In AC circuits, the voltage changes continuously, so inductors and capacitors respond differently. An inductor opposes change in current by producing back emf; hence current builds up late and lags voltage. A capacitor must be charged and discharged repeatedly; current flows first to build charge, so current leads voltage. Because voltage and current may not peak together, ordinary algebra is not enough for combining them. We use phasors, rotating vectors representing sinusoidal quantities. Their projections give instantaneous values, while angles show phase difference. In a series LCR circuit, VR is along current, VL is 90° ahead of current and VC is 90° behind current. The net reactive voltage is VL - VC, giving the impedance triangle and the formulas for Z, tan φ and power factor.

Formula sheet for fast revision

R, L and C in AC circuits: one-glance comparison

How to solve AC numericals quickly

1. Identify the circuit type —
2. Convert given values correctly —
3. Find reactances —
4. Calculate impedance and current —
5. Check phase and power —

Short worked examples

• {"title":"Example 1: RMS and peak voltage","bodyMarkdown":"A household AC supply is 220 V. This is RMS value. Peak value V0 = √2 Vrms = 1.414 × 220 ≈ 311 V. Board trap: do not write 220 V as maximum voltage."}
• {"title":"Example 2: Series LCR impedance","bodyMarkdown":"Given R = 30 Ω, XL = 50 Ω, XC = 10 Ω. Z = √(30² + (50 - 10)²) = √(900 + 1600) = 50 Ω. If Vrms = 100 V, Irms = 100/50 = 2 A."}
• {"title":"Example 3: Resonance frequency","bodyMarkdown":"For L = 0.5 H and C = 2 μF, f0 = 1/(2π√LC). LC = 0.5 × 2 × 10⁻⁶ = 10⁻⁶, so √LC = 10⁻³. Hence f0 ≈ 1/(2π × 10⁻³) ≈ 159 Hz."}

Power in AC circuits: what actually gets consumed?

Instantaneous power is p = vi, but in AC it keeps changing with time. The useful value is average power over one complete cycle. In pure L and pure C circuits, energy is alternately stored and returned to the source, so average power is zero. In circuits with resistance, energy is dissipated as heat, so real power is non-zero. The formula Pavg = Vrms Irms cos φ is central. Here cos φ is the power factor. If φ is large, current may be high but useful power is low. This is why power factor correction is important in practical AC systems. At resonance in a series LCR circuit, φ = 0, cos φ = 1 and the circuit absorbs maximum real power for a given applied voltage.

Exam tips and common traps

• In CBSE numericals, assume given AC voltage like 220 V is RMS unless maximum or peak value is explicitly stated.
• Average value of sinusoidal AC over a full cycle is zero, but RMS value is not zero. Do not confuse these.
• For inductor and capacitor phase: remember ELI and ICE. In L, EMF leads current; in C, current leads EMF.
• At series resonance, impedance is not zero; it is minimum and equal to R. Current is maximum but finite.
• Write units: XL, XC and Z in ohm; L in henry; C in farad; f in hertz; ω in rad s⁻¹.
• In transformer questions, ideal relation uses power conservation, so stepping up voltage steps down current.

Quick revision checks

• Q: What is the RMS value of current i = 10 sin ωt? A: Irms = 10/√2 = 7.07 A.
• Q: In a pure capacitive circuit, does current lead or lag voltage? A: Current leads voltage by π/2 or 90°.
• Q: What is the condition for resonance in a series LCR circuit? A: XL = XC, or ω0L = 1/(ω0C).
• Q: A series LCR circuit has R = 20 Ω and Z = 40 Ω. What is the power factor? A: cos φ = R/Z = 20/40 = 0.5.

Frequently Asked Questions