Power in Balanced 3 ø  Loads:

Balanced loads, in a 3 ø  system, have identical impedance within each secondary winding. The impedance of each winding in a delta load is display as Z_?, and the impedence within a wye load is display as Z_y.  For either the delta or wye connection, the lines A, B, and C provide a 3ø system of voltages.

Figure: Balanced Loads

Within a balanced delta load, the line voltage (V_L) is equivalent to the phase voltage (V_f), and the line current (I_L) is equal to the square root of three times the phase current (√3I_ø). Given Equation (9-5) is a mathematical representation of V_L in a balanced delta load and Equation (9-6) is a mathematical representation of I_L in a balanced delta load.

V_L = V_ø                                                                                   (9-5)

I_L = √3 I_ø                                                                                 (9-6)

In a balanced wye load, the line voltage (V_L) is equivalent to the square root of three times phase voltage ( √3V_ø ), and line current (I_L) is equal to the phase current (I_ø ). Given Equation (9-7) is a mathematical representation of V_L  in a balanced wye load and Equation (9-8) is a mathematical representation of I_L  in a balanced wye load.

V_L = √3V_ø                                                                                                                                                                                     (9-7)

I_L =I_ø                                                                                                        (9-8)

Since the impedance of every phase of a balanced delta or wye load has equal current and phase power is one third of the total power. Given  Equation (9-10) is the mathematical representation for phase power (P_ø ) within a balanced delta or wye load.

P_ø  = V_ø I_ø  cosθ                                                                     (9-10)

Total  power  (P_T)  is  equal  to  three  times  the  single-phase  power. The given Equation  (9-11)  is  the mathematical representation for total power in a balanced delta or wye load.

P_T  = √3V_ø I_ø  cosθ                                                                (9-11)

Within a delta-connected load, V_L  = V_ø  and I_ø = √3 I_L/3  so:

P_T =     √3 V_L I_L cosθ

Within a wye-connected load, I_L  = I_ø  and V_ø = √3 V_L/3 so:

P_T =     √3 V_L I_L cosθ

As  you  could  see,  the  total  power  formulas  for delta- and wye-connected loads are identical.

Overall apparent power (S_T) in volt-amperes and total reactive power (Q_T) in volt-amperes-reactive are  associated  to  total  real  power  (P_T)  in  watts.

A balanced three-phase load hasthe apparent, real, and reactive powers given through:

Figure:  3   Power Triangle

P_T =     √3 V_T I_L cosθ

S_T =     √3 V_T I_L

Q_T=     √3 V_T I_L sinθ