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 V_t,  I_a and Φ values are known. Hence the location of E_f is also unknown. The components of I_a, I_d  and I_q  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,

       I_d  =    I_a  sin Ψ                                          ............. (4)

       I_q = I_a cos Ψ                                              ..............(5)

       cosΨ = I_q/I_a                                              ...............(6)

       The drop I_a R_a has two components which are,

        I_d R_d = drop due to R_a in phase with I_d

        I_q R_a = drop due to R_a in phase with I_q

       The I_d X_d and I_q R_q can be drawn leading I_d  and I_q by 90^o respectively. The detail phasor diagram is shown in the Fig. 5.

Fig. 5  Phasor diagram for lagging p.f.

       In the phasor diagram,

      OF =  E_f

      OG =  V_t

      GH =    I_d R_a and H_A = I_q R_a

      GA = I_a R_a

      AE =  I_d X_d and EF = I_q X_a

       Now DAC is drawn perpendicular to the current phasor I_a and CB is drawn perpendicular to AE.

       The triangle ABC is right angle triangle,

       But from equations (6), cosΨ = I_q/I_a

        Thus point C can be located. Hence the direction of E_f is also known.

       Now triangle ODC is also right angle triangle,

       Now OD = OI + ID = V_t cos Φ  + I_a R_a

       and CD = AC + AD = I_a X_q + V_t sinΦ 

       As I_a X_q 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 E_f  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.