A quadrature axis is one whose magnetic effect is along the perpendicular to the field pole axis. When the peak of armature mmf coincides with the quadrature axis, the reactance offered to the armature mmf is called quadrature axis reactance.

What is the relation between XD and XQ?

Xd- Direct-axis Synchronous Reactance the ratio of fundamental A.C. armature Voltage to A.C. armature current produced by the direct-axis flux. Xq- Quadrature-axis Synchronous Reactance the ratio of fundamental A.C. armature Voltage to A.C. armature current produced by the quadrature-axis flux.

What are other methods to determine XD and XQ?

The method used to determine X_q and X_d, the direct and quadrature axis reactance is called slip test. Current induced in the damper winding of alternator will produce an appreciable error. Induced voltage in the open circuit field may reach dangerous values.