Answers and Discussions for
Problem 3
Contemplation
The first step is to ponder the inductance matrix in $ \vec{\lambda}_{abs} = \mathbf{L}_s \vec{i}_{abs} $.
$ \begin{equation} \mathbf{L}_s = \begin{bmatrix} L_{asas} & L_{asbs} \\ L_{bsas} & L_{bsbs} \end{bmatrix} \end{equation} $
Because the self-inductances change with rotor position $ \theta_r $, it is clear that the reluctance machine is salient (a vital operational principle of all reluctance machines). The self-inductances and mutual fit a known form as given. All leakage inductance terms are zero based on that given form.
$ \begin{align} L_{asas} &= \cancel{L_{\ell as}} + L_{Aas} + L_{Bas} \cos\left[2\left(\theta_r - 0\right)\right] = L_{A} + L_{B} \cos\left(2\theta_r\right) \\ L_{bsbs} &= \cancel{L_{\ell bs}} + L_{Abs} + L_{Bbs} \cos\left[2\left(\theta_r - \frac{\pi}{2}\right)\right] = L_{A} - L_{B} \cos\left(2\theta_r\right) \\ L_{asbs} &= L_{bsas} = -L_{B} \sin\left(2\theta_r\right) \end{align} $
Rotor Reference Frame Transformation
To move from stator phase variables to the derived rotor reference frame, pre-multiply by $ \mathbf{K}_s^r =\begin{bmatrix} +\cos(\theta_r) & \sin(\theta_r) \\ \sin(\theta_r) & -\cos(\theta_r) \end{bmatrix} $. To do the opposite and move from the derived rotor reference frame to stator phase variables, pre-multiply by $ \mathbf{K}_r^s = \left(\mathbf{K}_s^r\right)^{-1} = \frac{1}{\cancelto{-1}{-\cos^2(\theta_r) - \sin^2(\theta_r)}} \begin{bmatrix} -\cos(\theta_r) & -\sin(\theta_r) \\ -\sin(\theta_r) & +\cos(\theta_r) \end{bmatrix} = \begin{bmatrix} +\cos(\theta_r) & \sin(\theta_r) \\ \sin(\theta_r) & -\cos(\theta_r) \end{bmatrix} = \mathbf{K}_s^r $. The inverse matrix is found with the explicit formula the the inverse of a 2x2 matrix and its determinant as well. (By coincidence, $ \mathbf{K}_s^r $ is an involutary matrix.)
