Undergrad Reducing NxN Matrix to 2x2 w/ Physical Constraints

[waynewec]

TL;DR

Reducing an NxN matrix to a 2x2 by application of physical constraints

Gonna preface by saying I never thought linear algebra would be a class I would regret not taking so much... but in short the goal is to reduce an arbitrary symmetric NxN system using a set of auxiliary constraint relationships, e.g. for a 3x3

<br /> \begin{bmatrix}<br /> V_1\\<br /> V_2\\<br /> V_3\\<br /> \end{bmatrix}<br /> =<br /> \begin{bmatrix}<br /> L_{e11}&amp;L_{e12}&amp;L_{e12}\\<br /> L_{e21}&amp;L_{e22}&amp;L_{e23}\\<br /> L_{e31}&amp;L_{e32}&amp;L_{e33}\\<br /> \end{bmatrix}<br /> \cdot<br /> \begin{bmatrix}<br /> i_1\\<br /> i_2\\<br /> i_3\\<br /> \end{bmatrix}\\<br />
using the following constraints
$V_1=V_2=V_p$
$V_3=V_s$
$i_p=i_1+i_2$
$i_s=i_3$
to end up with an equivalent system with L_s, L_p, and M in terms of the starting L_{eij} matrix
<br /> \begin{bmatrix}<br /> V_p\\<br /> V_s\\<br /> \end{bmatrix}<br /> =<br /> \begin{bmatrix}<br /> L_p&amp;M\\<br /> M&amp;L_s\\<br /> \end{bmatrix}<br /> \cdot<br /> \begin{bmatrix}<br /> i_p\\<br /> i_s\\<br /> \end{bmatrix}<br />
For those interested in the context, this is an application specific usage of the method covered in https://onlinelibrary.wiley.com/doi/full/10.1002/eej.23240 but they glossed a bit over some of the key linear math that I don't understand. Eventually I'll be extending this concept to quite large matrices with more complex auxiliary constraints, but for now I'd appreciate some guidance, and some good resources, to get me goin

 

[Paul Colby]

Have you tried just eliminating variables? $V_1 = V_2$ relates $i_n$ making it possible to express $i_1$ in terms of $i_p$ and $i_s$. Clearly, $i_2 = i_p - i_1$ and $i_3 = i_s$ eliminates $i_2$ and $i_3$.

I get something like,

$V_p = (L_{11}-L_{12})i_1 + L_{12}i_p + L_{13}i_s$
$0 = (L_{11}-L_{12})i_1 + (L_{12}-L_{22})(i_p-i_1) + (L_{13}-L_{23})i_s$
$V_s = (L_{31}-L_{32})i_1 + L_{32}i_p + L_{33}i_s$

Okay, just use the second equation to eliminate $i_1$.

 

[waynewec]

Have you tried just eliminating variables? $V_1 = V_2$ relates $i_n$ making it possible to express $i_1$ in terms of $i_p$ and $i_s$. Clearly, $i_2 = i_p - i_1$ and $i_3 = i_s$ eliminates $i_2$ and $i_3$.

I get something like,

$V_p = (L_{11}-L_{12})i_1 + L_{12}i_p + L_{13}i_s$
$0 = (L_{11}-L_{12})i_1 + (L_{12}-L_{22})(i_p-i_1) + (L_{13}-L_{23})i_s$
$V_s = (L_{31}-L_{32})i_1 + L_{32}i_p + L_{33}i_s$

Okay, just use the second equation to eliminate $i_1$.

Absolutely valid, and an approach I've used, but any changes made to constraints or the order of the input matrix requires extremely tedious manual calculations. I was hoping for a direction that relies on matrix mathematics and could be implemented programmatically. 3x3, not so bad - 9x9 will make me want to kill myself

 

[Paul Colby]

Well, okay. The voltage constrains in matrix form,

$\left(\begin{array}{c} V_1 \\ V_2 \\ V_3\end{array}\right) = \left(\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1\end{array}\right)\left(\begin{array}{c} V_p \\ V_s\end{array}\right)$

The current constraints in matrix form,

$\left(\begin{array}{c} i_p \\ i_s \end{array}\right) = \left(\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{c} i_1 \\ i_2 \\ i_3 \end{array}\right)$

More generaly,

$V = C V_c$

and

$I_c = D I$

Clearly,

$V_c = C^{-1} L D^{-1} I_c$

is the solution. All you need to do is figure out what $C^{-1}$ and $D^{-1}$ really mean.