Lecture Notes: Power Analysis in AC Circuits

Introduction to Power Analysis in AC Circuits

• Importance: Power calculation is crucial in network theory.

Instantaneous Power

• Definition: Electric power at any instant of time.
• Measurement Unit: Watts.
• Formula:
  • Instantaneous power (P(t)) is the product of instantaneous voltage (V(t)) across the element and the instantaneous current (I(t)) through the element.

  • P(t) = V(t) * I(t)
    

Example Calculation

• Given:
  • Voltage: V(t) = V_m * cos(ωt + θ_v)
  • Current: I(t) = I_m * cos(ωt + θ_i)

• Calculation of Instantaneous Power:

  P(t) = V_m * I_m * cos(ωt + θ_v) * cos(ωt + θ_i)
  

Trigonometric Identity

• Using the identity for product of cosines:
  • 2 * cos(A) * cos(B) = cos(A - B) + cos(A + B)
  • Therefore,

    cos(A) * cos(B) = 1/2 * cos(A - B) + 1/2 * cos(A + B)
    

• In context:
  • A = ωt + θ_v
  • B = ωt + θ_i
  • Substituting,

    P(t) = V_m * I_m / 2 * [cos(θ_v - θ_i) + cos(2ωt + θ_v + θ_i)]
    

Average Power

• Average power (P_avg) is given by:

  P_avg = V_m * I_m / 2 * cos(θ_v - θ_i)
  

  • Time-independent term.
• Instantaneous power has a frequency of 2ω, twice the frequency of voltage and current (ω).