Chapter 16: Alternating Current

• Alternating Current (AC)

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  • Current whose magnitude changes continuously with time and direction reverses periodically.
  • Represented by sine or cosine function.
  • $I = I_o \sin(\omega t + \phi)$ or $V = V_o \sin(\omega t + \phi)$.
  • $I_o, V_o$: peak/maximum current/voltage.
  • $\omega$: angular frequency ($2\pi f$).
  • $\phi$: phase angle.

• RMS (Root Mean Square) Values

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  • Effective value of AC, equivalent to DC that produces same heating effect.
  • $I_{rms} = \frac{I_o}{\sqrt{2}} \approx 0.707 I_o$.
  • $V_{rms} = \frac{V_o}{\sqrt{2}} \approx 0.707 V_o$.
  • Most AC meters read RMS values.

• AC through a Resistor (R)

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  • Current and voltage are in phase ($\phi = 0$).
  • $V_R = I_R R$.
  • Power dissipated: $P = I_{rms}^2 R = V_{rms} I_{rms}$.

• AC through an Inductor (L)

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  • Voltage leads current by $90^\circ$ ($\frac{\pi}{2}$ radians).
  • $V_L = I_L X_L$.

  • Inductive Reactance ($X_L$)

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    • Opposition offered by an inductor to the flow of AC.
    • $X_L = \omega L = 2\pi f L$.
    • Unit: Ohm ($\Omega$).
    • $X_L \propto f$ (frequency).
    • For DC ($f=0$), $X_L = 0$ (inductor acts as short circuit).

• AC through a Capacitor (C)

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  • Current leads voltage by $90^\circ$ ($\frac{\pi}{2}$ radians).
  • $V_C = I_C X_C$.

  • Capacitive Reactance ($X_C$)

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    • Opposition offered by a capacitor to the flow of AC.
    • $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$.
    • Unit: Ohm ($\Omega$).
    • $X_C \propto \frac{1}{f}$ (frequency).
    • For DC ($f=0$), $X_C = \infty$ (capacitor acts as open circuit).

• R-C Series Circuit

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  • Resistor and capacitor in series with an AC source.
  • Impedance: $Z = \sqrt{R^2 + X_C^2}$.
  • Phase angle: $\tan\phi = \frac{X_C}{R}$. (Current leads voltage)
  • Current: $I = \frac{V}{Z}$.

• R-L Series Circuit

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  • Resistor and inductor in series with an AC source.
  • Impedance: $Z = \sqrt{R^2 + X_L^2}$.
  • Phase angle: $\tan\phi = \frac{X_L}{R}$. (Voltage leads current)
  • Current: $I = \frac{V}{Z}$.

• R-L-C Series Circuit

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  • Resistor, inductor, and capacitor in series with an AC source.
  • Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$.
  • Phase angle: $\tan\phi = \frac{X_L - X_C}{R}$.
  • Current: $I = \frac{V}{Z}$.

  • Resonance in RLC Circuit

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    • Occurs when $X_L = X_C$.
    • Resonant frequency: $f_r = \frac{1}{2\pi\sqrt{LC}}$.
    • At resonance:
      • Impedance is minimum ($Z = R$).
      • Current is maximum ($I_{max} = \frac{V}{R}$).
      • Circuit behaves purely resistively ($\phi = 0$).
      • Voltage across L and C are equal and opposite, cancelling each other.

  • Quality Factor (Q-factor)

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    • Measure of the sharpness of resonance.
    • $Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r C R} = \frac{1}{R}\sqrt{\frac{L}{C}}$.
    • Higher Q-factor means sharper resonance and better selectivity.

• Power in AC Circuits

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  • Average power: $P_{avg} = V_{rms} I_{rms} \cos\phi$.
  • $\cos\phi$: Power factor.
  • Unit: Watt (W).

  • Power Factor ($\cos\phi$)

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    • Ratio of true power to apparent power.
    • $\cos\phi = \frac{R}{Z}$.
    • For purely resistive circuit: $\phi = 0$, $\cos\phi = 1$ (max power).
    • For purely inductive/capacitive circuit: $\phi = \pm 90^\circ$, $\cos\phi = 0$ (zero power).

  • Wattless Current

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    • Component of current that consumes no power.
    • $I_w = I_{rms} \sin\phi$.
    • Flows when voltage and current are $90^\circ$ out of phase.

• Choke Coil

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  • Inductor used to limit AC current without significant power loss.
  • High inductance, low resistance.
  • Used in fluorescent tubes.

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