In a balanced three-phase system, which expression correctly gives real power P in terms of V_LL, I, and cosφ?

In a balanced three-phase system, which expression correctly gives real power P in terms of V_LL, I, and cosφ?

• A. P = √3 V_LL I cosφ
• B. P = 3 V_LL I cosφ
• C. P = √3 V_LL I sinφ
• D. P = V_LL I cosφ

Answer: (A)

Real power in a balanced three-phase system is the sum of the real power in each phase. Each phase contributes Pph = Vph Iph cosφ, so Ptotal = 3 Vph Iph cosφ. To express this with line quantities, use the standard relationships between phase and line values in a balanced network, which leads to P = √3 VLL I cosφ. Here VLL is the line-to-line voltage, I is the line current, and cosφ is the power factor (the angle between line voltage and line current). The √3 factor arises from converting from phase quantities to line quantities in a three-phase system. This form directly gives the real power in terms of VLL, I, and cosφ. Other expressions would either omit the √3, replace cosφ with sinφ, or otherwise misrepresent the relationship between line and phase quantities for a balanced load.

Real power in a balanced three-phase system is the sum of the real power in each phase. Each phase contributes Pph = Vph Iph cosφ, so Ptotal = 3 Vph Iph cosφ. To express this with line quantities, use the standard relationships between phase and line values in a balanced network, which leads to P = √3 VLL I cosφ. Here VLL is the line-to-line voltage, I is the line current, and cosφ is the power factor (the angle between line voltage and line current). The √3 factor arises from converting from phase quantities to line quantities in a three-phase system. This form directly gives the real power in terms of VLL, I, and cosφ. Other expressions would either omit the √3, replace cosφ with sinφ, or otherwise misrepresent the relationship between line and phase quantities for a balanced load.