View source for ECE PhD QE ES1 2008 Problem2 - Rhea

[[Category:ECE]]
[[Category:QE]]
[[Category:PE]]
[[Category:problem solving]]
[[Category:Reference Frame Theory]]

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Answers and Discussions for

[[ECE_PhD_QE_ES1_2008| ECE Ph.D. Qualifying Exam ES-1 August 2008]]
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==Problem 2==

To move from stator phase variables to the weird reference frame, pre-multiply by <math>\mathbf{K}_s^w = \begin{bmatrix} 1 & -t^2 \\ +t^2 & 1 \end{bmatrix}</math>. To do the opposite and move from the weird reference frame to stator phase variables, pre-multiply by <math>\mathbf{K}_w^s = \left(\mathbf{K}_s^w\right)^{-1} = \frac{1}{1 + t^4} \begin{bmatrix} 1 & +t^2 \\ -t^2 & 1 \end{bmatrix}</math>. The inverse matrix is found with the explicit formula the the inverse of a 2x2 matrix and its determinant as well.

The voltage equations in stator phase variables are put in vector form.

<math>\begin{equation}
\vec{v}_{abs} = \mathbf{r}_s \vec{i}_{abs} + \mathit{p}\vec{\lambda}_{abs} = r_s \mathbf{I}_2 \vec{i}_{abs} + \mathit{p}\vec{\lambda}_{abs}
\end{equation}</math>

The transformation proceeds next.

<math>\begin{align}
\vec{v}_{qds}^{w} &= \mathbf{K}_s^w \vec{v}_{abs} \\
\vec{v}_{qds}^{w} &= \mathbf{K}_s^w r_s \mathbf{I}_2 \left(\mathbf{K}_w^s \vec{i}_{qds}^{w}\right) + \mathbf{K}_s^w \mathit{p} \left(\mathbf{K}_w^s \vec{\lambda}_{qds}^{w}\right) \\
\vec{v}_{qds}^{w} &= r_s \cancelto{\mathbf{I}_2}{\mathbf{K}_s^w \mathbf{I}_2 \mathbf{K}_w^s} \vec{i}_{qds}^{w} + \mathbf{K}_s^w \left(\mathit{p}\mathbf{K}_w^s\right) \vec{\lambda}_{qds}^{w} + \cancelto{\mathbf{I}_2}{\mathbf{K}_s^w \mathbf{K}_w^s} \mathit{p}\vec{\lambda}_{qds}^{w}
\end{align}</math>

The Product Rule is applied for <math>\mathit{p} \left(\mathbf{K}_w^s \vec{\lambda}_{qds}^{w}\right)</math> to produce two terms, the latter of which also simplifies due to <math>\mathbf{K}_s^w</math> and <math>\mathbf{K}_w^s</math> being matrix inverses. A detour is needed to simplify the middle term.

<math>\begin{align}
\mathbf{K}_s^w \left(\mathit{p}\mathbf{K}_w^s\right) &= \begin{bmatrix} 1 & -t^2 \\ +t^2 & 1 \end{bmatrix} \mathit{p} \left(\frac{1}{1 + t^4} \begin{bmatrix} 1 & +t^2 \\ -t^2 & 1 \end{bmatrix}\right) \\ 
\mathbf{K}_s^w \left(\mathit{p}\mathbf{K}_w^s\right) &= \begin{bmatrix} 1 & -t^2 \\ +t^2 & 1 \end{bmatrix} \frac{-4t^3}{\left(1 + t^4\right)^2} \begin{bmatrix} 1 & +t^2 \\ -t^2 & 1 \end{bmatrix} + \begin{bmatrix} 1 & -t^2 \\ +t^2 & 1 \end{bmatrix} \frac{1}{1 + t^4} \begin{bmatrix} 0 & +2t \\ -2t & 0 \end{bmatrix} \\
\mathbf{K}_s^w \left(\mathit{p}\mathbf{K}_w^s\right) &= \frac{-4t^3}{\left(1 + t^4\right)^2} \begin{bmatrix} 1 + t^4 & 0 \\ 0 & 1 + t^4 \end{bmatrix} + \frac{1}{1 + t^4} \begin{bmatrix} +2t^3 & +2t \\ -2t & +2t^3 \end{bmatrix} \\
\mathbf{K}_s^w \left(\mathit{p}\mathbf{K}_w^s\right) &= \frac{1}{1 + t^4} \begin{bmatrix} -4t^3 & 0 \\ 0 & -4t^3 \end{bmatrix} + \frac{1}{1 + t^4} \begin{bmatrix} +2t^3 & +2t \\ -2t & +2t^3 \end{bmatrix} \\
\mathbf{K}_s^w \left(\mathit{p}\mathbf{K}_w^s\right) &= \frac{1}{1 + t^4} \begin{bmatrix} -2t^3 & +2t \\ -2t & -2t^3 \end{bmatrix}
\end{align}</math>

The vector voltage equation is finished.

<math>\begin{equation}
\vec{v}_{qds}^{w} = r_s \vec{i}_{qds}^{w} + \frac{1}{1 + t^4} \begin{bmatrix} -2t^3 & +2t \\ -2t & -2t^3 \end{bmatrix} \vec{\lambda}_{qds}^{w} + \mathit{p}\vec{\lambda}_{qds}^{w}
\end{equation}</math>

The weird reference frame voltage equations can be expressed separately.

<math>\begin{equation}
\boxed{\vec{v}_{qs}^w = r_s i_{qs}^w - \frac{2t^3}{1 + t^4} \lambda_{qs}^w + \frac{2t}{1 + t^4} \lambda_{ds}^w + \mathit{p}\lambda_{qs}^w}
\end{equation}</math>

<math>\begin{equation}
\boxed{\vec{v}_{ds}^w = r_s i_{ds}^w - \frac{2t}{1 + t^4} \lambda_{qs}^w - \frac{2t^3}{1 + t^4} \lambda_{ds}^w + \mathit{p}\lambda_{ds}^w}
\end{equation}</math>

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==Discussion==

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