Which set expresses real power, reactive power and apparent power for an AC circuit with V, I and phase angle φ?

Which set expresses real power, reactive power and apparent power for an AC circuit with V, I and phase angle φ?

• A. P = VI cos φ; Q = VI sin φ; S = P^2 + Q^2
• B. P = VI cos φ; Q = VI sin φ; S = sqrt(P^2 + Q^2)
• C. P = VI; Q = VI; S = VI
• D. P = V I cos φ; Q = VI sin φ; S = VI

Answer: (B)

Understanding how P, Q, and S relate in an AC circuit shows why these formulas fit together: real power is the portion of power that actually does work, P = VI cos φ, reactive power is the portion that alternates between store and release, Q = VI sin φ, and apparent power combines both as a single magnitude of the complex power, S = sqrt(P^2 + Q^2). This last relation comes from treating power as a vector with components P and Q; its magnitude is the square root of the sum of squares, which is the standard way to define apparent power. If you substitute P and Q from the first two expressions, you get S = sqrt((VI cos φ)^2 + (VI sin φ)^2) = VI, confirming consistency with the usual rms-quantity interpretation. The alternative that lists S as VI directly is numerically correct for rms values but the universally explicit form is S = sqrt(P^2 + Q^2), which is why that choice best expresses all three powers clearly and consistently. The other options fail because they either misuse the relationship (S without a square root) or assign impossible values to P and Q (making P and Q both equal to VI), or omit the proper linkage between S and the P–Q components.

Understanding how P, Q, and S relate in an AC circuit shows why these formulas fit together: real power is the portion of power that actually does work, P = VI cos φ, reactive power is the portion that alternates between store and release, Q = VI sin φ, and apparent power combines both as a single magnitude of the complex power, S = sqrt(P^2 + Q^2). This last relation comes from treating power as a vector with components P and Q; its magnitude is the square root of the sum of squares, which is the standard way to define apparent power. If you substitute P and Q from the first two expressions, you get S = sqrt((VI cos φ)^2 + (VI sin φ)^2) = VI, confirming consistency with the usual rms-quantity interpretation. The alternative that lists S as VI directly is numerically correct for rms values but the universally explicit form is S = sqrt(P^2 + Q^2), which is why that choice best expresses all three powers clearly and consistently. The other options fail because they either misuse the relationship (S without a square root) or assign impossible values to P and Q (making P and Q both equal to VI), or omit the proper linkage between S and the P–Q components.