Power in Balanced System

This requires that the analysis be done in the time domain. For a Y - connected load, the phase voltages are

                  Van = √2 Vp coswt ,                         Vbn = √2 Vp cos(wt - 120),                                 Vcn = √2 Vp cos(wt + 120)  

where the factor √2 is important because Vp has been defined as the rms value of the phase voltage. If Zy = Z∠Ɵ, the phase currents lag behid their corresponding phase voltage by Ɵ.

                
  ia = √2Ip cos(wt - Ɵ),                        

  ib = √2Ip cos(wt - Ɵ - 120)                    
  ib = √2Ip cos(wt - Ɵ + 120)

TOTAL INSTANTANEOUS POWER

p = pa + pb + pc = VANia + VBNib + VCNic

    = 2VpIp [ coswt cos(wt - Ɵ) + cos(wt  - 120) cos(wt - Ɵ - 120) + cos(wt + 120) cos(wt - Ɵ + 120) ]

p = 3VpIp cosƟ

INSTANTANEOUS POWER PER PHASE IS:

p = VpIp cosƟ

REACTIVE POWER PER PHASE IS:

Qp = VpIp sinƟ

APPARENT POWER PER PHASE IS:

Sp = VpIp

COMPLEX POWER PER PHASE IS :

Sp = Pp + jQp = Vp Ip*