Details Analysis of phasor Diagram for Synchronous Gen

Blondel's Two Reaction Theory (Theory of Salient Pole Machine) : part 3  
In the phasor diagram shown in the Fig. 4, the angles Ψ and δ are not known, through Vt,  Ia and Φ values are known. Hence the location of Ef is also unknown. The components of Ia, Id  and Iq  can not be determined which are required to sketch the phasor diagram.
Fig. 4

       Let us find out some geometrical relationships between the various quantities which are involved in the phasor diagram. For this, let us draw the phasor diagram including all the components in detail.
We know from the phasor diagram shown in the Fig. 4 that,
       Id  =    Ia  sin Ψ                                          ............. (4)
       Iq = Ia cos Ψ                                              ..............(5)
       cosΨ = Iq/Ia                                              ...............(6)
       The drop Ia Ra has two components which are,
        Id Rd = drop due to Ra in phase with Id
        Iq Ra = drop due to Ra in phase with Iq
       The Id Xd and Iq Rq can be drawn leading Id  and Iq by 90o respectively. The detail phasor diagram is shown in the Fig. 5.
Fig. 5  Phasor diagram for lagging p.f.

       In the phasor diagram,
      OF =  Ef
      OG =  Vt
      GH =    Id Ra and HA = Iq Ra
      GA = Ia Ra
      AE =  Id Xd and EF = Iq Xa
       Now DAC is drawn perpendicular to the current phasor Ia and CB is drawn perpendicular to AE.
       The triangle ABC is right angle triangle,
       But from equations (6), cosΨ = Iq/Ia
        Thus point C can be located. Hence the direction of Ef is also known.
       Now triangle ODC is also right angle triangle,
       Now OD = OI + ID = Vt cos Φ  + Ia Ra
       and CD = AC + AD = Ia Xq + Vt sinΦ 

       As Ia Xq is known, the angle Ψ can be calculated from equation (10). As Φ is known we can write,
       δ = Ψ - Φ for lagging p.f.

       Hence magnitude of Ef  can be obtained by using equation (11).
Note : In the above relations, Φ is taken positive for lagging p.f. For leading p.f., Φ must be taken negative.

Read :   part 1 , part 2

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