Affine Multiports#

Definition: Affinely Linear Multiport

An \(n\)-port is affine or affinely linear if its I-V characteristic \(\mathcal{F}\) is an \(n\)-dimensional affine subspace of \(\mathbb{R}^{2n}\).

Representations#

Theorem: Existence of an Affine Implicit Representation

Let \(\mathcal{F}\) be the I-V characteristic of an \(n\)-port.

If \(\mathcal{F}\) is affine, then \(\mathcal{F}\) has an implicit representation

\[f\left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix}\right) = \boldsymbol{0} \qquad \iff \qquad \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} \in \mathcal{F},\]

where \(f: \mathbb{R}^{2n} \to \mathbb{R}^{n}\) is an affine function.

Specifically, there exist a surjective linear function \(f_{\text{lin}}: \mathbb{R}^{2n} \to \mathbb{R}^{n}\) and some \(\boldsymbol{c} \in \mathbb{R}^n\) such that

\[f\left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix}\right) = f_{\text{lin}}\left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix}\right) + \boldsymbol{c}.\]

In other words:

\[f_{\text{lin}}\left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix}\right) + \boldsymbol{c} = \boldsymbol{0} \iff \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} \in \mathcal{F}\]

Notation: Matrix Representation of \(f\)

Since \(f_{\text{lin}}\) is a linear transformation, it can be represented as a matrix \(F\) with \(n\) rows and \(2n\) columns:

\[F \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} + \boldsymbol{c} = 0 \iff \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\]

We can, however, split \(F\) exactly into two \(n\times n\)-matrices \(\boldsymbol{A}\) and \(\boldsymbol{B}\), where \(\boldsymbol{A}\) holds the first \(n\) columns of \(F\) and \(\boldsymbol{B}\) holds the last \(n\) columns of \(F\):

\[F = \begin{bmatrix} \boldsymbol{A} & \boldsymbol{B} \end{bmatrix} \qquad \boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{n\times n}\]

If we do this, we get an alternative formulation for the implicit representation:

\[\boldsymbol{A}\boldsymbol{v} + \boldsymbol{B}\boldsymbol{i} = \boldsymbol{e} \qquad \boldsymbol{e} = -\boldsymbol{c}\]

Sometimes, we also use \(\boldsymbol{M}\) and \(\boldsymbol{N}\) for \(\boldsymbol{A}\) and \(\boldsymbol{B}\), respectively.

Proof

TODO

Theorem: Parametric Representation of Affine Multiports

Let \(\mathcal{F}\) be an affine \(n\)-port, let \(\begin{bmatrix}\boldsymbol{v}_0 \\ \boldsymbol{i_0}\end{bmatrix}, \begin{bmatrix}\boldsymbol{v}^{(1)} \\ \boldsymbol{i^{(1)}}\end{bmatrix}, \dotsc, \begin{bmatrix}\boldsymbol{v}^{(n)} \\ \boldsymbol{i^{(n)}}\end{bmatrix} \in \mathcal{F}\), let \(\Delta\boldsymbol{v}^{(k)} = \boldsymbol{v}^{(k)} - \boldsymbol{v}_0\) and let \(\Delta \boldsymbol{i}^{(k)} = \boldsymbol{i}^{(k)} - \boldsymbol{i}_0\).

If \(\begin{bmatrix}\Delta\boldsymbol{v}^{(1)} \\ \Delta \boldsymbol{i}^{(1)}\end{bmatrix}, \dotsc, \begin{bmatrix}\Delta\boldsymbol{v}^{(n)} \\ \Delta \boldsymbol{i}^{(n)}\end{bmatrix}\) are linearly independent, then

\[\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix}\vert & \vert & \vert \\ \Delta\boldsymbol{v}^{(1)} & \cdots & \Delta\boldsymbol{v}^{(n)} \\ \vert & \vert & \vert \\ \vert & \vert & \vert \\ \Delta \boldsymbol{i}^{(1)} & \cdots & \Delta \boldsymbol{i}^{(n)}\\ \vert & \vert & \vert\end{bmatrix}\boldsymbol{\lambda} + \begin{bmatrix}\boldsymbol{v}_0 \\ \boldsymbol{i_0}\end{bmatrix}\]

is a parametric representation of \(\mathcal{F}\).

Notation

We often denote the aforementioned matrix as \(\begin{bmatrix} \boldsymbol{V} \\ \boldsymbol{I} \end{bmatrix}\), where:

\[\boldsymbol{V} = \begin{bmatrix}\vert & \vert & \vert \\ \Delta \boldsymbol{v}^{(1)} & \cdots & \Delta \boldsymbol{v}^{(n)} \\ \vert & \vert & \vert\end{bmatrix} \qquad \boldsymbol{I} = \begin{bmatrix}\vert & \vert & \vert \\ \Delta \boldsymbol{i}^{(1)} & \cdots & \Delta \boldsymbol{i}^{(n)} \\ \vert & \vert & \vert \end{bmatrix}\]

Proof

TODO

Linearization#

Definition: Linearization

Suppose we have an \(n\)-port with I-V characteristic \(\mathcal{F}\) and a point \((\boldsymbol{v}_0, \boldsymbol{i}_0) \in \mathcal{F}\).

Linearization is the process of finding an affine multiport whose I-V characteristic \(\mathcal{F}_{\text{affine}}\) resembles \(\mathcal{F}\) as much as possible around the point \((\boldsymbol{v}_0, \boldsymbol{i}_0)\).

Theorem: Linearization via Implicit Representations

Suppose we have a non-linear \(n\)-port with I-V characteristic \(\mathcal{F}\) and a point \((\boldsymbol{v}_0, \boldsymbol{i}_0) \in \mathcal{F}\).

If \(\mathcal{F}\) has an implicit representation

\[f\left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix}\right) = \boldsymbol{0},\]

then the I-V characteristic \(\mathcal{F}_{\text{affine}}\) of the \(n\)-port which best approximates \(\mathcal{F}\) around \((\boldsymbol{v}_0, \boldsymbol{i}_0)\) has an implicit representation which can be obtained using the Jacobian of \(f\):

\[\left.\mathbf{J}_f\right\vert_{(\boldsymbol{v}_0, \boldsymbol{i}_0)}\left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} -\begin{bmatrix}\boldsymbol{v}_0 \\ \boldsymbol{i}_0\end{bmatrix}\right) = \boldsymbol{0}\]

Notation

The Jacobian \(\mathbf{J}_f\) has \(n\) rows and \(2n\) columns which means that we can split it into two \(n\times n\) matrices, denoted as \(\frac{\partial f}{\partial \boldsymbol{v}}\) and \(\frac{\partial f}{\partial \boldsymbol{i}}\):

\[\mathbf{J}_f = \begin{bmatrix}\frac{\partial f}{\partial \boldsymbol{v}} & \frac{\partial f}{\partial \boldsymbol{i}}\end{bmatrix}\]

The above equation can thus be rewritten in the following form:

\[\left.\frac{\partial f}{\partial \boldsymbol{v}}\right\vert_{(\boldsymbol{v}_0, \boldsymbol{i}_0)}(\boldsymbol{v} - \boldsymbol{v}_0) + \left.\frac{\partial f}{\partial \boldsymbol{i}}\right\vert_{(\boldsymbol{v}_0, \boldsymbol{i}_0)}(\boldsymbol{i} - \boldsymbol{i}_0) = \boldsymbol{0}\]

Proof

TODO

Theorem: Linearization via Explicit Representations

Suppose we have an \(n\)-port with I-V characteristic \(\mathcal{F}\) and a point \(\begin{bmatrix}\boldsymbol{v}_0 \\ \boldsymbol{i}_0 \end{bmatrix} \in \mathcal{F}\).

If \(\mathcal{F}\) has an admittance representation \(\boldsymbol{i} = G(\boldsymbol{v})\), then the I-V characteristic \(\mathcal{F}_{\text{affine}}\) of the \(n\)-port which best approximates \(\mathcal{F}\) around \((\boldsymbol{v}_0, \boldsymbol{i}_0)\) has an admittance representation which can be obtained using \(G\)'s Jacobian:

\[\boldsymbol{i}= J_G(\boldsymbol{v}_0) (\boldsymbol{v} - \boldsymbol{v}_0) + \boldsymbol{i}_0 \]

If \(\mathcal{F}\) has an impedance representation \(\boldsymbol{v} = R(\boldsymbol{i})\), then the I-V characteristic \(\mathcal{F}_{\text{affine}}\) of the \(n\)-port which best approximates \(\mathcal{F}\) around \((\boldsymbol{v}_0, \boldsymbol{i}_0)\) has an impedance representation which can be obtained using \(R\)'s Jacobian:

\[\boldsymbol{v} = J_R(\boldsymbol{i}_0) (\boldsymbol{i} - \boldsymbol{i}_0) + \boldsymbol{v}_0\]

Proof

TODO