**Two Reaction Model of Salient Pole Synchronous Machine:**

In a Two Reaction Model of Salient Pole Synchronous Machine, the flux established by a mmf wave is independent of the spatial position of the wave axis with respect to the field pole axis. On the other hand, in a salient-pole machine as shown in the cross-sectional view of Fig. 8.52, the permeance offered to a mmf wave is highest when it is aligned with the field pole axis (called the **direct-axis or d-axis**) and is lowest when it is oriented at 90° to the field pole axis (called the **quadrature axis or q-axis**).

Though the field winding in a Two Reaction Model of Salient Pole Synchronous Machine is of concentrated type, the B-wave produced by it is nearly sinusoidal because of the shaping of pole-shoes (the air-gap is least in the centre of the poles and progressively increases on moving away from the centre). Equivalently, the F_{f }wave can be imagined to be sinusoidally distributed and acting on a uniform air-gap. It can, therefore, be represented as space vector F̅_{f}. As the rotor rotates, F̅_{f} is always oriented along the d-axis and is presented with the d-axis permeance. However in case of armature reaction the permeance presented to it is far higher when it is oriented along the d-axis then it is oriented along the q-axis.

Figure 8.52 shows the relative spatial location of F̅_{f} and F̅_{ar} at the time instant when current (generating) in phase a is maximum positive and is lagging E̅_{f} (excitation emf due to F̅_{f}) by angle ψ. As angle ψ varies, the permeance offered to F̅_{ar , }the armature reaction mmf, varies because of a change in its spatial position relative to the d-axis. Consequently F̅_{ar} produces Φ̅̅_{ar} (armature reaction flux/pole) whose magnitude varies with angle ψ (which has been seen earlier to be related to the load power factor). So long as the magnetic circuit is assumed linear (i.e. superposition holds), this difficulty can be overcome by dividing F̅_{ar }into vectors F̅_{d} along the d-axis and F̅_{q} along the q-axis as shown by dotted lines in Fig. 8.52.

Figure 8.53 shows the phasor diagram corresponding to the vector diagram of Fig. 8.52. Here the d-axis is along F̅_{f} and 90° behind it is the q-axis along E̅_{f}. The components of I̅_{a} and F̅_{ar} along d- and q-axis are shown in the figure from which it is easily observed, that F̅_{d} is produced by I̅_{d}, the d-axis component of I̅_{a} at 90° to E̅_{f} and F̅_{q} is produced by I̅_{q , }the q-axis component of I̅_{a}, in phase with E̅_{f.}

Figure 8.54 shows the relative locations of F_{d}, F_{q} and field poles and the B-waves produced by these. It is easily seen that the B-waves contain strong third-harmonic space waves. For reasons already discussed, one can proceed with the analysis on the basis of space fundamentals of B_{d} and B_{q} while neglecting harmonics.

The flux components/pole produced by the d- and q-axis components of armature reaction mmf are

The flux phasors Φ̅_{d} and Φ̅_{q}** **are also drawn in Fig. 8.53. The resultant armature reaction flux phasor Φ̅_{ar} is now no longer in phase with F̅_{ar} or I̅_{a} because P_{d} > P_{q} in Eqs (8.63) and (8.64). In fact (Φ̅_{ar} lags or leads I̅_{a} depending upon the relative magnitudes of the d-axis, q-axis permeances.

The emfs induced by Φ_{d} and Φ_{q} are given by

where

- K
_{e}= emf constant of armature winding

The resultant emf induced in the machine is then

Substituting for Φ̅_{d} and Φ̅_{q} from Eqs (8.63) and (8.64),

Let

Obviously

It then follows from Eq. (8.67) that

Also for a realistic machine

Combining Eqs (8.71) and (8.72)

Define

It is easily seen that

Equation (8.73) can now be written as

The phasor diagram depicting currents and voltages as per Eq. (8.75) is drawn in Fig. 8.55 in which δ is the angle between the excitation emf E_{f} and the terminal voltage V_{t}.

**Analysis of Phasor Diagram of Salient Pole Synchronous Generator:**

In the phasor diagram of Two Reaction Model of Salient Pole Synchronous Machine Fig. 8.55, the angle ψ=Φ+δ is not known for a given V_{t}, I_{a} and Φ. The location of E_{f }being unknown, I_{d} and l_{q} cannot be found which are needed to draw the phasor diagram. This difficulty is overcome by establishing certain geometric relationships for the phasor diagrams which are drawn in Fig. 8.56(a) for the generating machine and in Fig. 8.56(b) for the motoring machine.

In Fig. 8.56(a) AC is drawn at 90° to the current phasor I̅_{a} and CB is drawn at 90° to E̅_{f} . Now

The phasor E̅_{f} can then be obtained by extending OC by +CD for generating machine and -CD for motoring machine, where CD is given by Eq. (8.80). Let us indicate phasor OC by E̅′.