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GATE 2026 EE – Question 42

Electric Circuits · Transient response of DC and AC networks, sinusoidal steady-state analysis, resonance, two port networks, balanced three phase circuits, star-delta transformation, complex power and power factor in AC circuits · 2 marks · Multiple choice

An electrical component has voltage drop $v=V_m\sin(\omega t)$, when the current through it is $i=I_m\sin(\omega t-\theta)$.

What is the average power dissipated over a half cycle corresponding to $\omega$?

  1. 0
  2. $V_mI_m\cos\theta$
  3. $\dfrac{V_mI_m}{2}\cos\theta$
  4. $\dfrac{V_mI_m}{4}\cos\theta$

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Show answer and explanation

Correct answer: (C) $\dfrac{V_mI_m}{2}\cos\theta$

Explanation

The instantaneous power is $p=V_mI_m\sin\omega t\sin(\omega t-\theta)=\dfrac{V_mI_m}{2}\left[\cos\theta-\cos(2\omega t-\theta)\right]$. Averaged over a half cycle ($0$ to $\pi/\omega$), the second term integrates to $\int_0^\pi\cos(2x-\theta)dx=0$, so the average is $\dfrac{V_mI_m}{2}\cos\theta$.