Two reaction theory; determining Xd and Xq

Two-reaction theory

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

Determination of $X_d$ and $X_q$: slip test

1. Drive the rotor by a prime mover at slightly less than synchronous speed (slip below 1 %), with the field winding open.
2. Apply a reduced balanced three-phase voltage (about 25 % of rated, with the same phase sequence) to the stator terminals.
3. The stator rotating field slowly slides past the rotor poles. The effective reluctance varies between the d-axis and q-axis values, so the armature current and terminal voltage oscillate slowly.
4. Record with an oscillograph (or ammeter and voltmeter readings) the maximum and minimum values:
   • When the field is aligned with the d-axis, the reactance is largest, so the current is minimum and the voltage is maximum.
   • When aligned with the q-axis, the current is maximum and the voltage is minimum.
5. Then

$$X_d = \frac{V_{max}}{I_{min}}, \qquad X_q = \frac{V_{min}}{I_{max}}$$

(per phase values; $R_a$ is small and neglected).

 Test connections
  3-phase reduced V --> stator
  field winding: OPEN (voltmeter across it
                 shows induced emf pulsating)
  prime mover: speed just below Ns

$X_d$ is also found from the open-circuit and short-circuit characteristics ($X_d = \frac{V_{oc}}{I_{sc}}$ at the same field current, unsaturated). $X_q$ by this test is typically 60 to 70 % of $X_d$.