In a balanced three-phase system, which expressions correctly give real, reactive, and apparent power?

In a balanced three-phase system, which expressions correctly give real, reactive, and apparent power?

• A. P = √3 V I cosφ; Q = √3 V I sinφ; S = √3 V I
• B. P = 3 V I cosφ; Q = 3 V I sinφ; S = 3 V I
• C. P = V I cosφ; Q = V I sinφ; S = V I
• D. P = √3 V I sinφ; Q = √3 V I cosφ; S = √3 V I

Answer: (A)

In a balanced three-phase system, real, reactive, and apparent power are tied to the line-to-line voltage, the line current, and the phase angle between voltage and current. The standard relations are P = √3 V_L I cosφ, Q = √3 V_L I sinφ, S = √3 V_L I, where V_L is the line-to-line (line) voltage, I is the line current, and φ is the angle between the voltage and current. This formulation comes from summing the per-phase power in all three phases and using V_L = √3 V_Ph and I = I_Ph in a balanced setup. Real power uses the cosine of the angle, reactive power uses the sine, and apparent power is the product √3 V_L I, independent of φ. If you instead use per-phase values, you’d have P = 3 V_Ph I_Ph cosφ and S = 3 V_Ph I_Ph, which is just another valid form depending on which voltages are labeled V_Ph versus V_L. The expressions that omit the √3 factor or mix up the phase relationships do not describe the balanced three-phase case correctly.

In a balanced three-phase system, real, reactive, and apparent power are tied to the line-to-line voltage, the line current, and the phase angle between voltage and current. The standard relations are P = √3 V_L I cosφ, Q = √3 V_L I sinφ, S = √3 V_L I, where V_L is the line-to-line (line) voltage, I is the line current, and φ is the angle between the voltage and current.

This formulation comes from summing the per-phase power in all three phases and using V_L = √3 V_Ph and I = I_Ph in a balanced setup. Real power uses the cosine of the angle, reactive power uses the sine, and apparent power is the product √3 V_L I, independent of φ.

If you instead use per-phase values, you’d have P = 3 V_Ph I_Ph cosφ and S = 3 V_Ph I_Ph, which is just another valid form depending on which voltages are labeled V_Ph versus V_L. The expressions that omit the √3 factor or mix up the phase relationships do not describe the balanced three-phase case correctly.