Alternating Current

• 1. Fundamentals of AC

  • Alternating Current (AC): Magnitude changes continuously, direction periodically
  • Peak Value ($I_0 / V_0$): Maximum value of current or voltage
  • Mean / Average Value ($I_{avg}$)
    • Over half cycle: $I_{avg} = \frac{2I_0}{\pi} \approx 0.637 I_0$
    • Over full cycle: Zero
  • RMS Value ($I_{rms} / V_{rms}$)
    • Virtual / Effective value measured by AC meters
    • Formula: $I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$

• 2. AC Circuits with Single Components

  • Purely Resistive Circuit ($R$)
    • Phase Difference ($\phi$): $0^\circ$ (Voltage and Current in phase)
    • Opposition: Resistance ($R$)
  • Purely Inductive Circuit ($L$)
    • Phase Difference ($\phi$): Current lags Voltage by $\pi/2$ ($90^\circ$)
    • Inductive Reactance ($X_L$): $X_L = \omega L = 2\pi f L$
    • Direct Current ($f = 0$): $X_L = 0$ (Offers no opposition to DC)
  • Purely Capacitive Circuit ($C$)
    • Phase Difference ($\phi$): Current leads Voltage by $\pi/2$ ($90^\circ$)
    • Capacitive Reactance ($X_C$): $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$
    • Direct Current ($f = 0$): $X_C = \infty$ (Blocks DC completely)

• 3. Series LCR Circuit

  • Impedance ($Z$): Net opposition offered by L, C, and R
    • Formula: $Z = \sqrt{R^2 + (X_L - X_C)^2}$
  • Phase Angle ($\phi$): $\tan \phi = \frac{X_L - X_C}{R}$
  • Electrical Resonance
    • Condition: $X_L = X_C$
    • Impedance becomes minimum: $Z_{min} = R$
    • Current becomes maximum: $I_{max} = \frac{V_{rms}}{R}$
    • Resonant Frequency ($f_r$): $f_r = \frac{1}{2\pi\sqrt{LC}}$
  • Quality Factor ($Q$-Factor)
    • Measures sharpness of resonance
    • Formula: $Q = \frac{\omega_r L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$

• 4. Power in AC Circuits

  • Average Power ($P_{avg}$): $P_{avg} = V_{rms} I_{rms} \cos\phi$
  • Power Factor ($\cos\phi$)
    • Formula: $\cos\phi = \frac{R}{Z}$
    • Pure $R$: Power factor = 1 (Maximum power dissipation)
    • Pure $L$ or $C$: Power factor = 0 (Zero power dissipation)
  • Wattless Current
    • Current flowing in circuit when average power consumed is Zero
    • Occurs in purely inductive or capacitive circuits ($\phi = 90^\circ$)

• 5. AC Devices

  • AC Generator
    • Principle: Electromagnetic Induction
    • Function: Converts Mechanical Energy $\rightarrow$ Electrical Energy
    • Induced EMF: $e = N B A \omega \sin(\omega t) = e_0 \sin(\omega t)$
  • Transformer
    • Principle: Mutual Induction
    • Transformation Ratio ($k$): $k = \frac{N_s}{N_p} = \frac{V_s}{V_p} = \frac{I_p}{I_s}$
    • Step-Up Transformer: $N_s > N_p$ ($V_s > V_p$, $I_s < I_p$)
    • Step-Down Transformer: $N_s < N_p$ ($V_s < V_p$, $I_s > I_p$)
    • Energy Losses & Prevention:
      • Copper Loss ($I^2R$ heat): Use thick copper wires
      • Eddy Current Loss: Use laminated soft iron core
      • Hysteresis Loss: Use soft iron core (low hysteresis loop area)
      • Flux Leakage: Wind primary and secondary coils over each other