For a balanced three-phase system, how is the apparent power S calculated from line voltage V_L and line current I_L?

• A. S = √3 × V_L × I_L
• B. S = V_L × I_L
• C. S = 3 × V_L × I_L
• D. S = √2 × V_L × I_L

Answer: (A)

In a balanced three-phase system, the apparent power comes from combining the three phase powers as a whole. The total apparent power can be written as S = √3 times the line voltage times the line current. This result stays valid whether the winding is wired in star or delta, because V_L and I_L are the line-to-line voltage and the line current, and the relationships between line and phase quantities still yield S = √3 V_L I_L.

A quick way to see it is starting from P = 3 V_ph I_ph cosφ and Q = 3 V_ph I_ph sinφ for a balanced system, then S = √(P^2 + Q^2) = 3 V_ph I_ph. Using V_L = √3 V_ph and I_L = I_ph (star) or V_L = V_ph and I_L = √3 I_ph (delta), you get S = √3 V_L I_L in either case. That’s why the correct expression uses √3 times the line quantities.