Learning Objectives
- Understand AC voltage and current and their representation
- Analyse pure resistive, capacitive, and inductive AC circuits
- Study LCR series circuits, impedance, and resonance
- Understand power in AC circuits and power factor
- Learn about transformers and AC generators
Key Concepts
AC Voltage and Current
AC voltage: V = V₀ sin ωt, AC current: I = I₀ sin(ωt + φ)
RMS (Root Mean Square) values: V_rms = V₀/√2, I_rms = I₀/√2
Average value over full cycle: 0. Over half cycle: 2V₀/π = 0.637V₀.
Household AC: 220 V (rms), 50 Hz in India.
AC Through Pure Resistor
V and I are in phase (φ = 0°). I = V₀ sin ωt / R.
AC Through Pure Inductor
Current lags voltage by 90° (π/2). Inductive reactance: X_L = ωL = 2πfL.
I = V₀/(ωL) × sin(ωt - π/2). Average power = 0 (wattless component).
AC Through Pure Capacitor
Current leads voltage by 90° (π/2). Capacitive reactance: X_C = 1/(ωC) = 1/(2πfC).
I = V₀ωC × sin(ωt + π/2). Average power = 0.
LCR Series Circuit
Impedance: Z = √[R² + (X_L - X_C)²] = √[R² + (ωL - 1/ωC)²]
Phase angle: tan φ = (X_L - X_C)/R
If X_L > X_C: circuit is inductive, current lags. If X_C > X_L: circuit is capacitive, current leads.
Current: I₀ = V₀/Z
Resonance
When X_L = X_C: impedance is minimum (Z = R), current is maximum.
Resonant frequency: f₀ = 1/(2π√(LC)), ω₀ = 1/√(LC)
At resonance: V_L = V_C (may exceed source voltage), current and voltage are in phase.
Quality factor: Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C). Higher Q means sharper resonance.
Power in AC Circuits
Instantaneous power: P = VI
Average power: P_avg = V_rms × I_rms × cos φ = (V₀I₀/2) cos φ
Power factor: cos φ = R/Z. Maximum power at resonance (cos φ = 1).
For pure L or C: cos φ = 0 (wattless, power = 0).
Apparent power: S = V_rms × I_rms. True power: P = S cos φ.
Transformer
Device to change AC voltage. Based on mutual induction.
Transformation ratio: V_s/V_p = N_s/N_p = I_p/I_s
Step-up: N_s > N_p (voltage increases, current decreases).
Step-down: N_s < N_p (voltage decreases, current increases).
For ideal transformer: V_pI_p = V_sI_s (power conserved). Efficiency = P_output/P_input × 100%.
Energy losses: copper losses (I²R), iron/eddy current losses, hysteresis loss, flux leakage.
Summary
AC is described by peak, RMS, and average values. In pure R, L, C circuits, the phase relationship between V and I differs. LCR series circuits show impedance and resonance phenomena. At resonance, impedance is minimum and current is maximum. Power in AC circuits depends on the power factor cos φ. Transformers change AC voltage levels using electromagnetic induction.
Important Terms
- RMS Value: Effective value of AC, V_rms = V₀/√2
- Reactance: Opposition to AC by L or C (X_L = ωL, X_C = 1/ωC)
- Impedance: Total opposition to AC, Z = √(R² + (X_L - X_C)²)
- Resonance: X_L = X_C, maximum current, Z = R
- Power Factor: cos φ = R/Z, determines true power
- Transformer: Changes AC voltage using mutual induction
Quick Revision
- V_rms = V₀/√2; I_rms = I₀/√2
- X_L = ωL; X_C = 1/ωC; Z = √(R² + (X_L - X_C)²)
- Resonance: ω₀ = 1/√(LC); Z_min = R; I_max = V/R
- P = V_rms I_rms cos φ; cos φ = R/Z
- Transformer: V_s/V_p = N_s/N_p
- Phase: R (in phase), L (I lags 90°), C (I leads 90°)