Alternating Current

Alternating Current

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AC Voltage

\[V = V_0 \sin(\omega t + \phi)\]

Properties & Key Points:

  • The AC voltage varies sinusoidally with time, typically expressed as a function of amplitude and phase.
  • \( V \): Instantaneous voltage.
  • \( V_0 \): Maximum voltage (peak voltage).
  • \( \omega \): Angular frequency.
  • \( t \): Time.
  • \( \phi \): Phase angle.
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AC Current

\[I = I_0 \sin(\omega t + \phi)\]

Properties & Key Points:

  • The AC current varies sinusoidally with time, similar to AC voltage.
  • \( I \): Instantaneous current.
  • \( I_0 \): Maximum current (peak current).
  • \( \omega \): Angular frequency.
  • \( t \): Time.
  • \( \phi \): Phase angle.
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RMS Value of AC Voltage

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

Properties & Key Points:

  • The RMS (Root Mean Square) value of an AC voltage is a measure of its effective value for producing heat, which is equivalent to the DC voltage that would produce the same heat.
  • \( V_{\text{rms}} \): RMS voltage.
  • \( V_0 \): Peak voltage.
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RMS Value of AC Current

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

Properties & Key Points:

  • The RMS value of an AC current is the effective value of the current in terms of heat production.
  • \( I_{\text{rms}} \): RMS current.
  • \( I_0 \): Peak current.
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Average Value of AC Voltage

\[V_{\text{avg}} = \frac{2 V_0}{\pi}\]

Properties & Key Points:

  • The average value of an AC voltage over a half-cycle is the average of the instantaneous voltage.
  • \( V_{\text{avg}} \): Average value of AC voltage.
  • \( V_0 \): Peak voltage.
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Average Value of AC Current

\[I_{\text{avg}} = \frac{2 I_0}{\pi}\]

Properties & Key Points:

  • The average value of an AC current over a half-cycle is the average of the instantaneous current.
  • \( I_{\text{avg}} \): Average value of AC current.
  • \( I_0 \): Peak current.
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Reactance of an Inductor

\[X_L = \omega L\]

Properties & Key Points:

  • The reactance of an inductor represents the opposition to current flow in an AC circuit due to inductance.
  • \( X_L \): Inductive reactance.
  • \( \omega \): Angular frequency.
  • \( L \): Inductance.
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Reactance of a Capacitor

\[X_C = \frac{1}{\omega C}\]

Properties & Key Points:

  • The reactance of a capacitor represents the opposition to current flow in an AC circuit due to capacitance.
  • \( X_C \): Capacitive reactance.
  • \( \omega \): Angular frequency.
  • \( C \): Capacitance.
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Impedance of an AC Circuit

\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]

Properties & Key Points:

  • The impedance is the total opposition to current flow in an AC circuit, which includes both resistance and reactance.
  • \( Z \): Impedance.
  • \( R \): Resistance.
  • \( X_L \): Inductive reactance.
  • \( X_C \): Capacitive reactance.
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Power in an AC Circuit

\[P = V_{\text{rms}} I_{\text{rms}} \cos \phi\]

Properties & Key Points:

  • The power in an AC circuit is the rate at which energy is consumed, and it depends on the voltage, current, and phase difference.
  • \( P \): Power.
  • \( V_{\text{rms}} \): RMS voltage.
  • \( I_{\text{rms}} \): RMS current.
  • \( \phi \): Phase angle between voltage and current.
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Resonance in an AC Circuit

\[\omega_0 = \frac{1}{\sqrt{LC}}\]

Properties & Key Points:

  • Resonance occurs in an RLC circuit when the inductive reactance and capacitive reactance cancel each other out, causing the impedance to be at its minimum.
  • \( \omega_0 \): Resonant angular frequency.
  • \( L \): Inductance.
  • \( C \): Capacitance.
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Quality Factor (Q Factor)

\[Q = \frac{\omega_0 L}{R}\]

Properties & Key Points:

  • The quality factor is a measure of the sharpness of the resonance in an RLC circuit.
  • \( Q \): Quality factor.
  • \( \omega_0 \): Resonant angular frequency.
  • \( L \): Inductance.
  • \( R \): Resistance.
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Average Power in an RLC Circuit

\[P_{\text{avg}} = \frac{V_0 I_0}{2} \cos \phi\]

Properties & Key Points:

  • The average power in an RLC circuit is the total energy dissipated in one complete cycle.
  • \( P_{\text{avg}} \): Average power.
  • \( V_0 \): Peak voltage.
  • \( I_0 \): Peak current.
  • \( \phi \): Phase angle between voltage and current.
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