**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 90Â° 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_{fÂ }wave can be imagined to be sinusoidally distributed and acting on a uniform air-gap. It can, therefore, be represented as space vector FÌ…_{f}. As the rotor rotates, FÌ…_{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 FÌ…_{f}Â and FÌ…_{ar}Â at the time instant when current (generating) in phase a is maximum positive and is lagging EÌ…_{f}Â (excitation emf due to FÌ…_{f}) by angle Ïˆ. As angle Ïˆ varies, the permeance offered to FÌ…_{ar ,Â }the armature reaction mmf, varies because of a change in its spatial position relative to the d-axis. Consequently FÌ…_{ar} produces Î¦Ì…Ì…_{ar}Â (armature reaction flux/pole) whose magnitude varies with angle Ïˆ (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 FÌ…_{arÂ }into vectors FÌ…_{d}Â along the d-axis and FÌ…_{q}Â 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 FÌ…_{f}Â and 90Â° behind it is the q-axis along EÌ…_{f}. The components of IÌ…_{a}Â and FÌ…_{ar}Â along d- and q-axis are shown in the figure from which it is easily observed, that FÌ…_{d}Â is produced by IÌ…_{d}, the d-axis component of IÌ…_{a} at 90Â° to EÌ…_{f}Â and FÌ…_{q}Â is produced by IÌ…_{q , }the q-axis component of IÌ…_{a}, in phase with EÌ…_{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}Â 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 Î¦Ì…_{d}Â and Î¦Ì…_{q}**Â **are also drawn in Fig. 8.53. The resultant armature reaction flux phasor Î¦Ì…_{ar}Â is now no longer in phase with FÌ…_{ar}Â or IÌ…_{a}Â because P_{d} > P_{q} in Eqs (8.63) and (8.64). In fact (Î¦Ì…_{ar}Â lags or leads IÌ…_{a}Â depending upon the relative magnitudes of the d-axis, q-axis permeances.

The emfs induced by Î¦_{d}Â and Î¦_{q}Â are given by

where

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

The resultant emf induced in the machine is then

Substituting for Î¦Ì…_{d}Â and Î¦Ì…_{q}Â 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 Î´Â 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 Ïˆ=Î¦+Î´ is not known for a given V_{t}, I_{a} and Î¦. The location of E_{fÂ }being unknown, I_{d}Â 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 90Â° to the current phasor IÌ…_{a}Â and CB is drawn atÂ 90Â° to EÌ…_{f} . Now

The phasor EÌ…_{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 EÌ…â€².