The sum of instantaneous powers in the three phases in a three-phase system:

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The correct answer is Option B: remains constant. In a balanced three-phase system, the sum of instantaneous powers delivered to the load is time-invariant and equal to the average real power. This constancy is a fundamental property that distinguishes three-phase systems from single-phase systems, where instantaneous power pulsates at twice the supply frequency.

Step-by-Step Solution & Concepts:
In a balanced three-phase system, the sum of instantaneous powers delivered to the load is time-invariant and equal to the average real power. This constancy is a fundamental property that distinguishes three-phase systems from single-phase systems, where instantaneous power pulsates at twice the supply frequency.

Working Principle: The instantaneous power in a single phase $p_i(t) = v_i(t) \cdot i_i(t)$ contains a constant component and a double-frequency oscillation. In a three-phase system, the three phases are displaced by $120^{\circ}$ (or $\frac{2\pi}{3}$ radians). When the individual instantaneous power expressions are summed, the oscillating components cancel each other out due to their phase displacement, leaving only a constant sum of $P = 3 V_{ph} I_{ph} \cos \phi$.

Key Formulas: $P_{total} = v_a i_a + v_b i_b + v_c i_c = 3 V_{ph} I_{ph} \cos \phi$; $p(t) = P(1 - \cos(2 \omega t))$ — instantaneous power in single phase

Key Points: Constant power delivery leads to reduced torque pulsations in three-phase motors.; The cancellation of oscillating power relies on a balanced load and balanced supply.; The total power is independent of time $t$ in a balanced three-phase circuit.

Advantages: Constant torque production in rotating electrical machines.; Reduced mechanical vibrations due to uniform power transfer.

Disadvantages: Requires balanced loading to maintain power constancy.; Higher complexity in instrumentation compared to single-phase systems.

Applications: Industrial motor drives and rotating machines.; High-power transmission grids.

Additional Info: Option A is incorrect because the power is non-zero and positive for a passive load.. Options C and D refer to the frequency of power ripples in single-phase systems, not the total sum in three-phase systems.

Answer: B is correct — the sum of instantaneous powers in a balanced three-phase system remains constant at all times.

Exam Tip: Remember that while single-phase systems have a power ripple of $2\omega$, this is essentially filtered out in three-phase systems, which is the primary reason why three-phase motors run much smoother.

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Option A: is zero

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Option C: thrice the line frequency

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Option D: twice the line frequency

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