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Tableau Analysis#

Once we have a basis for all instances of Kirchhoff's laws within a given network and we know the characteristics of its elements, we can construct a system of equations which describes the behavior of the network.

Definition: Tableau System

A tableau system of a network is a system of equations which completely describes it.

Algorithm: Tableau Analysis

Tableau analysis is the process of finding a tableau system for a given network:

  1. Construct a network graph.

  2. Find a basis for all KVLs in the network and write it in the form \(\boldsymbol{B}\boldsymbol{v} = \boldsymbol{0}\) .
    - Fundamental loop analysis or mesh-current analysis can be used for this.

  3. Find a basis for all KCLs in the network and write it in the form \(\boldsymbol{A}\boldsymbol{i} = \boldsymbol{0}\).
    - Fundamental cut-set analysis or node-voltage analysis can be used for this.

  4. Combine the implicit representations of all electrical elements into a single \(f(\boldsymbol{v},\boldsymbol{i}) = \boldsymbol{0}\) by using the branch voltage vector and branch current network.

  5. Combine the results from the previous steps to obtain a tableau system:

\[ \left\vert \begin{aligned}\boldsymbol{B}\boldsymbol{v} & = \boldsymbol{0} \\ \boldsymbol{A}\boldsymbol{i} & = \boldsymbol{0} \\ f(\boldsymbol{v},\boldsymbol{i}) & = \boldsymbol{0}\end{aligned}\right. \]
  • The first part can also be written in matrix form:
\[ \left\vert \begin{aligned} \begin{bmatrix} \boldsymbol{B} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A} \end{bmatrix} \begin {bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} & = \boldsymbol{0} \\ f(\boldsymbol{v},\boldsymbol{i}) & = \boldsymbol{0}\end{aligned}\right. \]

Info: Non-Existent Implicit Representation

Theoretically, it might be possible for some elements to not have an implicit representation.

Affine Networks#

If a network contains only affine elements, their characteristics can be combined into a single matrix equation of the following form:

\[ \boldsymbol{M}\boldsymbol{v} + \boldsymbol{N}\boldsymbol{i} = \boldsymbol{e} \]

We can then construct a tableau system of the following form:

\[ \begin{bmatrix} \boldsymbol{B} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A} \\ \boldsymbol{M} & \boldsymbol{N} \end{bmatrix} \begin {bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix}\boldsymbol{0} \\ \boldsymbol{0} \\ \boldsymbol{e}\end{bmatrix} \]

Definition: Tableau Matrix

The matrix

\[ \begin{bmatrix} \boldsymbol{B} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A} \\ \boldsymbol{M} & \boldsymbol{N} \end{bmatrix} \]

is known as a tableau matrix.

Notation

\[ \boldsymbol{T} \]
Example: Tableau Analysis of Affine Networks

Consider the following network:

Tableau Analysis Affine Circuit

We use fundamental loop and cut-set analysis to find a basis for Kirchhoff's laws:

Affine Tableau Analysis Graph

\[ \left\vert \begin{aligned}-v_1 + v_2 + v_5 & = 0 \\ v_2 - v_3 + v_6 & = 0 \\ -v_4 + v_7 & = 0\end{aligned}\right. \implies \begin{bmatrix}-1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_5 \\ v_6 \\ v_7 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \]
\[ \left\vert \begin{aligned}i_1 + i_5 = 0 \\ i_2 - i_5 - i_6 = 0 \\ i_3 + i_6 = 0 \\ i_4 + i_7 = 0 \end{aligned}\right. \implies \begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 & -1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} i_1 \\ i_2 \\ i_3 \\ i_4 \\ i_5 \\ i_6 \\ i_7 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \]

We get the following from the elements:

  • \(v_1 = v_0\) from the voltage source;
  • \(i_5 = 0\) and \(v_5 = 0\) from the nullator;
  • \(v_2 - R_{\text{e}}i_2 = 0\) from the resistor \(R_{\text{e}}\);
  • \(v_3 - Nv_4 = 0\) and \(Ni_3 + i_4 = 0\) from the transformer;
  • \(v_7 - R_{\text{L}}i_7 = 0\) from the resistor \(R_{\text{L}}\).

We can write these in the form

\[ \boldsymbol{M}\boldsymbol{v} + \boldsymbol{N}\boldsymbol{i} = \boldsymbol{0}, \]

where

\[ \begin{aligned} \boldsymbol{M} & = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -N & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \\ \boldsymbol{N} & = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -R_{\text{e}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & N & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -R_{\text{L}} \end{bmatrix} \\ \boldsymbol{e} & = \begin{bmatrix} v_0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{aligned} \]

Combining all results, we get the following tableau system:

TODO

Non-Affine Networks#

For networks which contain both affine and non-affine elements, we can combine the characteristics of the affine elements as before and combine the characteristics of the non-affine elements into \(f_{\text{non-affine}} (\boldsymbol{v}, \boldsymbol{i}) = \boldsymbol{0}\).

We can then construct a tableau system of the following form:

\[ \left\vert \begin{aligned} \begin{bmatrix} \boldsymbol{B} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A} \\ \boldsymbol{M}' & \boldsymbol{N}' \end{bmatrix} \begin {bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} & = \begin{bmatrix}\boldsymbol{0} \\ \boldsymbol{0} \\ \boldsymbol{e}'\end{bmatrix} \\ f_{\text{non-affine}} (\boldsymbol{v}, \boldsymbol{i}) & = \boldsymbol{0} \end{aligned}\right. \]