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AC Circuits

Ch. 13Std 12

Easy Overview

Most of the electrical power we use is alternating current (AC), not direct current (DC). In a DC circuit, current flows steadily in one direction. In an AC circuit, the voltage and current change direction sinusoidally, typically 50 times per second (50 Hz in India). Why do we use AC? Because AC voltage can be easily stepped up or down using transformers, making long-distance power transmission efficient. In an AC circuit, the relationship between voltage and current depends on what elements are present. A resistor behaves the same way for AC as for DC — voltage and current are in phase. But capacitors and inductors behave differently. For an inductor, voltage leads current by 90° (the inductor opposes current changes). For a capacitor, current leads voltage by 90° (the capacitor opposes voltage changes). This phase difference has practical consequences — it affects power delivery and efficiency. The impedance (generalised resistance for AC) of an RLC series circuit is Z = √(R² + (X_L − X_C)²), where X_L = ωL and X_C = 1/(ωC). At a particular frequency called the resonant frequency f₀ = 1/(2π√(LC)), X_L = X_C, the impedance is minimum (just R), and current is maximum. This is resonance — the principle behind tuning a radio to a specific station. Power in AC circuits is not simply VI. The power factor cosφ = R/Z accounts for the phase difference between voltage and current. When voltage and current are out of phase, average power P_avg = V_rms I_rms cosφ. For a purely resistive circuit (cosφ = 1), all power is useful. For purely reactive circuits (cosφ = 0), no net power is consumed — energy oscillates between source and circuit. Understanding AC circuits is essential for power systems, audio electronics, radio communications, and every device that plugs into a wall socket.

AC Voltage and Current — RMS Values

v(t) = V₀ sin(ωt), i(t) = I₀ sin(ωt + φ). RMS: V_rms = V₀/√2, I_rms = I₀/√2. Average power: P_avg = V_rms I_rms cosφ. Household supply: 230 V RMS, 50 Hz, peak ≈ 325 V. RMS is the equivalent DC value that delivers the same power. Average of sin²(ωt) = 1/2.

AC Circuit with Pure Resistor

v = V₀ sinωt, i = v/R = (V₀/R) sinωt. Phase: φ = 0 (in phase). Power: p(t) = V₀I₀ sin²ωt. P_avg = V₀I₀/2 = V_rmsI_rms. Always positive — resistor consumes power. No reactive component.

AC Circuit with Pure Inductor

v = V₀ sinωt. i = (V₀/ωL) sin(ωt − π/2) = (V₀/X_L) sin(ωt − π/2). Inductive reactance X_L = ωL = 2πfL. Current lags voltage by 90°. P_avg = 0 — energy stored in magnetic field half-cycle, returned next half-cycle.

AC Circuit with Pure Capacitor

v = V₀ sinωt. i = ωCV₀ sin(ωt + π/2) = (V₀/X_C) sin(ωt + π/2). Capacitive reactance X_C = 1/(ωC) = 1/(2πfC). Current leads voltage by 90°. P_avg = 0. X_C is large at low frequencies (blocks DC), small at high frequencies (passes AC).

Series LCR Circuit and Impedance

Z = √(R² + (X_L − X_C)²). Phase angle φ = tan⁻¹((X_L − X_C)/R). i(t) = V₀/Z sin(ωt − φ). Current depends on frequency. At resonance: X_L = X_C → Z = R (minimum). Power factor cosφ = R/Z.

Resonance in LCR Circuit

Resonance when ω₀L = 1/(ω₀C) → ω₀ = 1/√(LC), f₀ = 1/(2π√(LC)). At resonance: Z = R (minimum), I₀ = V₀/R (maximum). Circuit is purely resistive. Bandwidth Δω = R/L. Quality factor Q = ω₀L/R = 1/(ω₀RC). High Q → sharp resonance (selective tuning).

Power in AC Circuits and Power Factor

Instantaneous power p(t) = v(t)i(t) = V₀I₀ sin(ωt) sin(ωt − φ). Average power: P_avg = V_rms I_rms cosφ. cosφ = R/Z is power factor. For resistor: cosφ = 1. For inductor/capacitor: cosφ = 0. Most loads have 0 < cosφ < 1. Low power factor wastes power — utilities charge penalties.

Power Factor Correction

Industrial loads (motors) have lagging power factor (inductive). Fixed by adding capacitors in parallel — they supply leading reactive current. Result: cosφ → 1, I decreases (less I²R loss), same real power delivered. Capacitor bank: C = P(tanφ₁ − tanφ₂)/(ωV²).

LC Oscillations

An LC circuit oscillates if charged capacitor is connected to inductor. Charge q(t) = q₀ cos(ω₀t), current i(t) = −ω₀q₀ sin(ω₀t). Energy oscillates between electric (UE = q²/(2C)) and magnetic (UB = ½Li²). Frequency f₀ = 1/(2π√(LC)). Ideal: undamped oscillations. Real: resistance damps oscillations.

Transformer in AC Circuits

V_s/V_p = N_s/N_p. Step-up: N_s > N_p, V_s > V_p. Step-down: N_s < N_p. Power conservation: V_pI_p = V_sI_s (ideal). Real transformers have eddy current losses (laminated core), hysteresis losses, I²R losses in windings. Efficiency η = P_out/P_in × 100%. Modern: η > 98%.

Three-Phase AC

Three voltages 120° apart: V₁ = V₀ sinωt, V₂ = V₀ sin(ωt − 120°), V₃ = V₀ sin(ωt − 240°). Each phase connected via separate wire. Advantages: constant total power, smaller motors, easier to start, uses less copper. Star (Y) and delta (Δ) configurations.

AC Through R, L, C — Phasor Diagrams

Phasor: rotating vector representing sinusoidal quantity. Length = amplitude, initial angle = phase. Rotates at ω. Addition of phasors: vector addition. For R: V and I in phase (along same line). For L: V leads I by 90°. For C: I leads V by 90°. Series LCR: V_R + V_L + V_C = V_supply (phasor sum).

Key Points

  • •v(t) = V₀ sinωt, V_rms = V₀/√2. I_rms = I₀/√2.
  • •R: φ = 0 (in phase). L: I lags V by 90°. C: I leads V by 90°.
  • •X_L = ωL = 2πfL. X_C = 1/(ωC) = 1/(2πfC).
  • •Z = √(R² + (X_L − X_C)²). tanφ = (X_L − X_C)/R.
  • •Resonance: ω₀ = 1/√(LC). f₀ = 1/(2π√(LC)).
  • •At resonance: Z = R, I_max = V/R, φ = 0.
  • •Quality factor Q = ω₀L/R. Bandwidth Δω = R/L.
  • •Average power: P_avg = V_rms I_rms cosφ.
  • •Power factor cosφ = R/Z. PF correction: parallel capacitors.
  • •LC oscillations: f₀ = 1/(2π√(LC)).

Practice Questions

  • Derive expression for impedance of series LCR circuit. Obtain condition for resonance. What is the Q-factor and its significance?
  • Define RMS value of AC. Show that V_rms = V₀/√2. An AC supply 230 V, 50 Hz is applied to a 0.5 H inductor. Find inductive reactance and RMS current.
  • Explain power in AC circuit. Show that P_avg = V_rms I_rms cosφ. What is power factor? A circuit with R = 10 Ω, L = 0.1 H, C = 100 µF at 50 Hz. Find power factor.
  • What is resonance in series LCR circuit? Derive condition for resonance. A series LCR circuit with L = 2 H, C = 50 µF, R = 10 Ω. Find resonant frequency and Q-factor.
  • Explain phasor diagrams for pure R, L, C and series LCR circuit.
  • What is power factor correction? Why is low power factor undesirable? A motor draws 10 A at 230 V, cosφ = 0.6. Calculate capacitor needed to raise pf to 0.9.
  • Explain LC oscillations. Derive expression for frequency. An LC circuit with L = 10 mH, C = 1 µF oscillates. Find frequency and maximum current if max charge = 100 µC.
  • Distinguish between AC and DC. State advantages of AC over DC for power distribution.