Two reaction model of salient pole machine

Two-reaction model

In a salient-pole machine the air gap is not uniform: it is small along the pole axis (direct axis, d-axis) and large between the poles (quadrature axis, q-axis). So the same mmf produces different flux along the two axes, and the single synchronous reactance of a cylindrical rotor cannot be used. Blondel's two-reaction theory resolves the armature mmf (and the current) into two components, one along each axis, and treats each separately.

• $I_d = I\sin\psi$: direct-axis component, in quadrature with the emf $E$, and acting along the pole axis (magnetising or demagnetising).
• $I_q = I\cos\psi$: quadrature-axis component, in phase with $E$, acting along the inter-polar axis (cross-magnetising).

Here $\psi$ is the angle between $I$ and $E$. Each component produces an armature reaction flux, and so a reactance:

$$X_d = X_{ad} + X_l, \qquad X_q = X_{aq} + X_l$$

with $X_d > X_q$ because the d-axis has the lower reluctance (typically $X_q \approx 0.6\text{--}0.7X_d$).

Phasor equation (generator)

$$\vec E = \vec V + \vec I R_a + j\vec I_d X_d + j\vec I_q X_q$$

          E
         /|
   jIdXd/ |     V at angle delta from E
       /  |
      /   |jIqXq
  ---+----+-----> V
   Iq in phase with E
   Id 90 deg behind/ahead

Load angle and components

From the phasor diagram (neglecting $R_a$), with $\delta$ the angle between $E$ and $V$:

$$I_q = \frac{V\sin\delta}{X_q}, \qquad I_d = \frac{E - V\cos\delta}{X_d}$$

These are used to find the power developed:

$$P = \frac{VE}{X_d}\sin\delta + \frac{V^2}{2}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin2\delta$$

The second term is the reluctance power; it exists even without excitation.

For a cylindrical rotor, $X_d = X_q = X_s$, and the model reduces to the simple synchronous reactance model.