What is the formula for three-phase apparent power in a balanced system?

What is the formula for three-phase apparent power in a balanced system?

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

Answer: (A)

In a balanced three-phase system, the total apparent power is found by summing the per-phase powers. Each phase has a voltage of V_phase and current I_phase, and if the load is balanced, all three phases contribute equally: S = 3 V_phase I_phase. For a star (Y) connection, the phase voltage is V_phase = V_LL / √3, and the line current equals the phase current, I_phase = I_L. Substituting gives S = 3 (V_LL / √3) I_L = √3 V_LL I_L. So the three-phase apparent power is the product of the line-to-line voltage, the line current, and √3. The other options don’t match this relationship: using 3 × V_LL × I_L would be too large by a factor of √3, √2 × V_LL × I_L introduces an unnecessary √2 factor, and V_LL × I_L alone represents a single-phase power, not the total three-phase power.

In a balanced three-phase system, the total apparent power is found by summing the per-phase powers. Each phase has a voltage of V_phase and current I_phase, and if the load is balanced, all three phases contribute equally: S = 3 V_phase I_phase. For a star (Y) connection, the phase voltage is V_phase = V_LL / √3, and the line current equals the phase current, I_phase = I_L. Substituting gives S = 3 (V_LL / √3) I_L = √3 V_LL I_L. So the three-phase apparent power is the product of the line-to-line voltage, the line current, and √3.

The other options don’t match this relationship: using 3 × V_LL × I_L would be too large by a factor of √3, √2 × V_LL × I_L introduces an unnecessary √2 factor, and V_LL × I_L alone represents a single-phase power, not the total three-phase power.