Chapter 7: Alternating Current

AC VOLTAGE AND CURRENT

Alternating Current (AC)

  • Definition: Current that changes direction periodically
  • Waveform: Usually sinusoidal
  • Mathematical Expression: i = I₀sin(ωt)
  • I₀: Peak current (amplitude)
  • ω: Angular frequency = 2πf
  • f: Frequency (Hz)
  • T: Time period = 1/f

Alternating Voltage

\[ v = V_0\sin(\omega t) \]

V₀: Peak voltage (amplitude)

RMS (Root Mean Square) Values

\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]
\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]

Physical Significance: AC with RMS value produces same heating effect as equivalent DC

AC CIRCUITS WITH RESISTOR

Pure Resistive Circuit

  • Voltage-Current Relationship: v = iR
  • Phase Relationship: Current in phase with voltage
  • Power: P = Vrms·Irms = I²rms·R = V²rms/R
  • Average Power: Pav = (1/2)V₀I₀ = Vrms·Irms

AC CIRCUITS WITH INDUCTOR

Pure Inductive Circuit

  • Voltage-Current Relationship: v = L·di/dt
  • Phase Relationship: Current lags voltage by 90°
  • Inductive Reactance: XL = ωL
  • Impedance: Z = XL
  • Power: P = Vrms·Irms·cos(90°) = 0
  • Average Power: Pav = 0 (no power dissipation)

AC CIRCUITS WITH CAPACITOR

Pure Capacitive Circuit

  • Voltage-Current Relationship: i = C·dv/dt
  • Phase Relationship: Current leads voltage by 90°
  • Capacitive Reactance: XC = 1/(ωC)
  • Impedance: Z = XC
  • Power: P = Vrms·Irms·cos(90°) = 0
  • Average Power: Pav = 0 (no power dissipation)

LCR SERIES CIRCUIT

Impedance

  • Definition: Total opposition to current flow in AC circuit
  • Formula: Z = √(R² + (XL - XC)²)
  • Phase Angle: φ = tan⁻¹((XL - XC)/R)

Current

  • Amplitude: I₀ = V₀/Z
  • Phase: i = I₀sin(ωt - φ)
  • Phase Relationship: Current lags voltage by angle φ

Voltage Across Components

  • Resistor: vR = I₀R·sin(ωt - φ)
  • Inductor: vL = I₀XL·sin(ωt - φ + 90°)
  • Capacitor: vC = I₀XC·sin(ωt - φ - 90°)

Power

  • Instantaneous Power: p = vi
  • Average Power: Pav = Vrms·Irms·cosφ = I²rms·R
  • Power Factor: cosφ
    • Resistive circuit: cosφ = 1
    • Inductive circuit: 0 ≤ cosφ < 1 (lagging)
    • Capacitive circuit: 0 ≤ cosφ < 1 (leading)

RESONANCE IN LCR CIRCUIT

Resonance Condition

\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
  • Impedance: Z = R (minimum)
  • Current: Maximum at resonance
  • Phase Angle: φ = 0 (current in phase with voltage)

Quality Factor (Q-factor)

  • Definition: Sharpness of resonance
  • Formula: Q = ω₀L/R = 1/(ω₀CR)
  • Bandwidth: Δω = ω₀/Q

LC OSCILLATIONS

Free Oscillations

  • Frequency: ω₀ = 1/√(LC)
  • Charge: q = Q₀cos(ω₀t)
  • Current: i = -ω₀Q₀sin(ω₀t)
  • Energy: Oscillates between electric (capacitor) and magnetic (inductor) forms
  • Total Energy: E = (1/2)LI₀² = (1/2)Q₀²/C

TRANSFORMERS

Working Principle

  • Mutual Induction: Changing current in primary induces EMF in secondary
  • Ideal Transformer: No power loss

Transformer Equation

\[ \frac{V_s}{V_p} = \frac{N_s}{N_p} \]
\[ \frac{I_p}{I_s} = \frac{N_s}{N_p} \]
\[ V_s \cdot I_s = V_p \cdot I_p \]

(conservation of power)

Types

  • Step-up: Vs > Vp (Ns > Np)
  • Step-down: Vs < Vp (Ns < Np)

Efficiency

\[ \eta = \frac{P_s}{P_p} \times 100\% \]
  • Losses: Copper losses (I²R), Iron losses (hysteresis, eddy currents)

POWER TRANSMISSION

AC vs DC Transmission

  • Advantage of AC: Voltage can be easily stepped up/down
  • High Voltage Transmission: Reduces I²R losses
  • Power Grid: Generation → Step-up → Transmission → Step-down → Distribution

KEY FORMULAS

  • AC Current: i = I₀sin(ωt)
  • AC Voltage: v = V₀sin(ωt)
  • RMS Values: Irms = I₀/√2, Vrms = V₀/√2
  • Inductive Reactance: XL = ωL
  • Capacitive Reactance: XC = 1/(ωC)
  • Impedance: Z = √(R² + (XL - XC)²)
  • Phase Angle: φ = tan⁻¹((XL - XC)/R)
  • Average Power: Pav = Vrms·Irms·cosφ
  • Resonance Frequency: ω₀ = 1/√(LC)
  • Transformer Equation: Vs/Vp = Ns/Np = Ip/Is