Which of the following expresses the total three-phase real power for a balanced system?

• A. P_total = V_phase I_phase cosφ
• B. P_total = √3 V_LL I_L cosφ
• C. P_total = 3 V_LL I_L cosφ
• D. P_total = V_LL I_L sinφ

Answer: (B)

In a balanced three-phase system, the total real power is the sum of the real power from each phase. For one phase, real power is P_phase = V_phase I_phase cosφ. In a balanced setup, V_phase equals V_LL/√3 and the phase current equals the line current I_L. Multiply by three to get the total: P_total = 3 (V_LL/√3) I_L cosφ = √3 V_LL I_L cosφ. This uses line quantities, which is why the √3 factor appears. This form correctly accounts for all three phases at the same power factor. Using V_LL I_L cosφ omits the √3 relationship between phase and line values, and using 3 V_LL I_L cosφ would overcount by a factor of √3. The sinφ form would represent reactive power, not real power.