Question 4.AE.6: Synchronous reactances are defined for fundamental frequency......
Synchronous reactances are defined for fundamental frequency. Subjecting the voltage and current to a Fourier series yields the following data: line-to-neutral terminal voltage V_{(l-n)}, phase current I_{(l-n)}, instead of the induced voltage E the no-load voltage V_{(l-n)_noload}≈E can be used for a permanent-magnet machine if saturation is not dominant. The displacement power factor angle φ and the torque angle δ can be determined from an oscilloscope recording and from a stroboscope, respectively. The ohmic resistance R can be measured as a function of the machine temperature. The relationship between these quantities is given by the phasor diagram of Fig. E4.6.1. Calculate the synchronous reactances X_q\ and\ X_d.
From the phasor diagram of Fig. E4.6.1, the following relations for the synchronous reactances X_q\ and\ X_d are derived:
\begin{gathered} X_q=\frac{V_{(1-n)} \sin \delta+R I_{(1-n)} \sin \delta \cos \varphi+R I_{(l-n)} \cos \delta \sin \varphi}{I_{(l-n)} \cos \varphi \cos \delta-I_{(1-n)} \sin \varphi \sin \delta}\quad (E4.6-1) \\ X_d=\frac{E-V_{(l-n)} \cos \delta+R I_{(1-n)} \cos (\delta+\varphi)}{I_{(1-n)} \cos \left(90^{\circ}-\varphi-\delta\right)}\quad (E4.6-2) \end{gathered}An application of these relations is given in [44].