Linear Multiports#

Definition: Linear Multiport

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

Theorem: Linearity \(\implies\) Affinity

Every linear multiport is also affine.

Proof

TODO

Theorem: Implicit Representation of Linear Multiports

The I-V characteristic \(\mathcal{F}\) of each linear \(n\)-port has an implicit representation as the kernel of a linear function \(f: \mathbb{R}^{2n} \to \mathbb{R}^{n}\):

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

Proof

TODO

Note: Matrix Representation of \(f\)

Since \(f\) is a linear transformation, it can be represented as a matrix \(F\) with \(n\) rows and \(2n\) columns:

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

We can, however, split \(F\) exactly into two \(n\times n\)-matrices \(M\) and \(N\), where \(M\) holds the first \(n\) columns of \(F\) and \(N\) holds the last \(n\) columns of \(F\):

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

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

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

Theorem: Parametric Representation of linear Multiports

The I-V characteristic \(\mathcal{F}\) of each linear \(n\)-port can be expressed as the image of a linear function \(f: \mathbb{R}^n \to \mathbb{R}^{2n}\):

\[\mathcal{F} = f(\mathbb{R}^n)\]

Warning: Non-Uniqueness

This \(f\) need not be unique.

Proof

TODO

Note: Matrix Representation of \(f\)

If \(\begin{bmatrix}\boldsymbol{v}^{(1)} \\ \boldsymbol{i^{(1)}}\end{bmatrix}, \dotsc, \begin{bmatrix}\boldsymbol{v}^{(n)} \\ \boldsymbol{i^{(n)}}\end{bmatrix}\) are a basis for \(\mathcal{F}\), then the \(2n\times n-\)matrix

\[\begin{bmatrix}\vert & \vert & \vert \\ \boldsymbol{v}^{(1)} & \cdots & \boldsymbol{v}^{(n)} \\ \vert & \vert & \vert \\ \vert & \vert & \vert \\ \boldsymbol{i^{(1)}} & \cdots & \boldsymbol{i^{(n)}} \\ \vert & \vert & \vert\end{bmatrix}\]

is the matrix representation of \(f\):

We thus have:

\[\mathcal{F} = \left\{ \begin{bmatrix}\boldsymbol{V} \\ \boldsymbol{I} \end{bmatrix}\in \mathbb{R}^{2n}: \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} = \begin{bmatrix}\vert & \vert & \vert \\ \boldsymbol{v}^{(1)} & \cdots & \boldsymbol{v}^{(n)} \\ \vert & \vert & \vert \\ \vert & \vert & \vert \\ \boldsymbol{i^{(1)}} & \cdots & \boldsymbol{i^{(n)}} \\ \vert & \vert & \vert\end{bmatrix} \boldsymbol{\lambda}, \boldsymbol{\lambda} \in \mathbb{R}^n\right\}\]

We often denote the aforementioned matrix as \(\begin{bmatrix} \boldsymbol{V} \\ \boldsymbol{I} \end{bmatrix}\), where \(\boldsymbol{V} = \begin{bmatrix}\boldsymbol{v}^{(1)} & \cdots & \boldsymbol{v}^{(n)}\end{bmatrix}\) and \(\boldsymbol{I} = \begin{bmatrix}\boldsymbol{i^{(1)}} & \cdots & \boldsymbol{i^{(n)}}\end{bmatrix}\):

\[\mathcal{F} = \left\{ \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix}\in \mathbb{R}^{2n}: \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} = \begin{bmatrix} \boldsymbol{V} \\ \boldsymbol{I} \end{bmatrix} \boldsymbol{\lambda}, \boldsymbol{\lambda} \in \mathbb{R}^n\right\}\]

Theorem: Admittance Parameters

If a linear multiport \(\mathcal{F}\) is also voltage-controlled, then its admittance representation \(g\) is a linear transformation.

Definition: Admittance Parameters

The standard matrix representation of \(g\) is known as the admittance matrix or conductance matrix of \(\mathcal{F}\) and its components are known as the admittance parameters or conductance parameters of \(\mathcal{F}\).

Notation

The admittance matrix is usually denoted in one of the following ways:

\[Y \qquad \boldsymbol{Y} \qquad G \qquad \boldsymbol{G}\]

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

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

and \(\boldsymbol{N}\) is invertible, then \(\mathcal{F}\)'s admittance matrix \(\boldsymbol{G}\) is

\[\boldsymbol{G} = -\boldsymbol{N}^{-1}\boldsymbol{M}.\]

If \(\mathcal{F}\) has a parametric representation

\[\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix} \boldsymbol{V} \\ \boldsymbol{I} \end{bmatrix} \boldsymbol{\lambda},\]

and \(\boldsymbol{V}\) is invertible, then \(\mathcal{F}\)'s admittance matrix \(\boldsymbol{G}\) is

\[\boldsymbol{G} = \boldsymbol{I}\boldsymbol{V}^{-1}.\]

Proof

TODO

Theorem: Impedance Parameters

If a linear multiport \(\mathcal{F}\) is also current-controlled, then its impedance representation \(r\) is a linear transformation.

Definition: Impedance Parameters

The standard matrix representation of \(r\) is known as the impedance matrix or resistance matrix of \(\mathcal{F}\) and its components are known as the impedance parameters or resistance parameters of \(\mathcal{F}\).

Notation

The impedance matrix is usually denoted in one of the following ways:

\[Z \qquad \boldsymbol{Z} \qquad R \qquad \boldsymbol{R}\]

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

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

and \(\boldsymbol{M}\) is invertible, then \(\mathcal{F}\)'s impedance matrix \(\boldsymbol{R}\) is

\[\boldsymbol{R} = -\boldsymbol{M}^{-1}\boldsymbol{N}.\]

If \(\mathcal{F}\) has a parametric representation

\[\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix} \boldsymbol{V} \\ \boldsymbol{I} \end{bmatrix} \boldsymbol{\lambda},\]

and \(\boldsymbol{I}\) is invertible, then \(\mathcal{F}\)'s impedance matrix \(\boldsymbol{R}\) is

\[\boldsymbol{R} = \boldsymbol{V}\boldsymbol{I}^{-1}.\]

Proof

TODO

Theorem: Inverse Hybrid Representations

If a linear multiport \(\mathcal{F}\) has an inverse hybrid representation

\[\begin{bmatrix}\boldsymbol{i}_a \\ \boldsymbol{v}_b \end{bmatrix}= H'\left( \begin{bmatrix}\boldsymbol{v}_a \\ \boldsymbol{i}_b\end{bmatrix} \right),\]

where

\[\boldsymbol{i}_a = \begin{bmatrix} i_1 \\ \vdots \\ i_m\end{bmatrix} \qquad \boldsymbol{v}_a = \begin{bmatrix} v_1 \\ \vdots \\ v_m\end{bmatrix} \qquad \boldsymbol{i}_b = \begin{bmatrix} i_{m+1} \\ \vdots \\ i_n \end{bmatrix} \qquad \boldsymbol{v}_b = \begin{bmatrix} v_{m+1} \\ \vdots \\ v_n\end{bmatrix}\]

for some \(m \in \{1, \dotsc, n\}\), then \(H'\) is linear:

\[\begin{bmatrix}\boldsymbol{i}_a \\ \boldsymbol{v}_b \end{bmatrix} = \boldsymbol{H}' \begin{bmatrix}\boldsymbol{v}_a \\ \boldsymbol{i}_b\end{bmatrix}\]

We can further divide \(\boldsymbol{H}'\) into four matrices:

\[\boldsymbol{H}' = \begin{bmatrix} \boldsymbol{H}_{a,a}' & \boldsymbol{H}_{a,b}' \\ \boldsymbol{H}_{b,a}' & \boldsymbol{H}_{b,b}' \end{bmatrix}\]

\(\boldsymbol{H}_{a,a}'\) contains entries \(H_{jk}'\) where \(j, k \in \{1, \dotsc, m\}\);
\(\boldsymbol{H}_{a,b}'\) contains entries \(H_{jk}'\) where \(j \in \{1, \dotsc, m\}\) and \(k \in \{m+1, \dotsc, n\}\);
\(\boldsymbol{H}_{b,a}'\) contains entries \(H_{jk}'\) where \(j \in \{m+1, \dotsc, n\}\) and \(k \in \{1, \dotsc, m\}\);
\(\boldsymbol{H}_{b,b}'\) contains entries \(H_{jk}'\) where \(j, k \in \{m+1, \dotsc, n\}\).

We then have:

\[\boldsymbol{v} = \begin{bmatrix} \boldsymbol{v}_a \\ \boldsymbol{v}_b\end{bmatrix} = \begin{bmatrix}\boldsymbol{1} & \boldsymbol{0} \\ \boldsymbol{H}_{b, a}' & \boldsymbol{H}_{b, b}'\end{bmatrix}\begin{bmatrix} \boldsymbol{v}_a \\ \boldsymbol{i}_b \end{bmatrix} \qquad \boldsymbol{i} = \begin{bmatrix} \boldsymbol{i}_a \\ \boldsymbol{i}_b\end{bmatrix} = \begin{bmatrix} \boldsymbol{H}_{a, a}' & \boldsymbol{H}_{a, b}' \\ \boldsymbol{0} & \boldsymbol{1} \end{bmatrix} \begin{bmatrix} \boldsymbol{v}_a \\ \boldsymbol{i}_b \end{bmatrix}\]

Proof

TODO

Power#

Theorem: Losslessness of linear Multiports

Let \(\mathcal{F}\) be a linear multiport.

If \(\mathcal{F}\) has a parametric representation \(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix}\boldsymbol{V} \\ \boldsymbol{I}\end{bmatrix}\boldsymbol{\lambda}\), then it is lossless if and only if \(\boldsymbol{V}^{\mathsf{T}} \boldsymbol{I} + \boldsymbol{I}^{\mathsf{T}}\boldsymbol{V} = \boldsymbol{0}\).

If \(\mathcal{F}\) has an admittance representation \(\boldsymbol{i} = \boldsymbol{Y}\boldsymbol{v}\), then it is lossless if and only if \(\boldsymbol{Y}\) is skew symmetric, i.e. \(\boldsymbol{Y} = -\boldsymbol{Y}^{\mathsf{T}}\).

If \(\mathcal{F}\) has an impedance representation \(\boldsymbol{v} = \boldsymbol{Z}\boldsymbol{i}\), then it is lossless if and only if \(\boldsymbol{Z}\) is skew symmetric, i.e. \(\boldsymbol{Z} = -\boldsymbol{Z}^{\mathsf{T}}\).

Proof

TODO

Theorem: Passivity of linear Multiports

A linear multiport with a parametric representation

\[\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix}\boldsymbol{V} \\ \boldsymbol{I}\end{bmatrix}\boldsymbol{\lambda}\]

is passive if and only if \(\boldsymbol{V}^{\mathsf{T}} \boldsymbol{I} + \boldsymbol{I}^{\mathsf{T}}\boldsymbol{V}\) is positive semi-definite:

\[\boldsymbol{V}^{\mathsf{T}} \boldsymbol{I} + \boldsymbol{I}^{\mathsf{T}}\boldsymbol{V} \succeq \boldsymbol{0}\]

Proof

TODO

Theorem: Activity of linear Multiports

A linear multiport with a parametric representation

\[\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = \begin{bmatrix}\boldsymbol{V} \\ \boldsymbol{I}\end{bmatrix}\boldsymbol{\lambda}\]

is active if and only if \(\boldsymbol{V}^{\mathsf{T}} \boldsymbol{I} + \boldsymbol{I}^{\mathsf{T}}\boldsymbol{V}\) is not positive semi-definite:

\[\boldsymbol{V}^{\mathsf{T}} \boldsymbol{I} + \boldsymbol{I}^{\mathsf{T}}\boldsymbol{V} \not\succeq \boldsymbol{0}\]

Proof

TODO

Duality#

Theorem: Duality of linear Multiports

Let \(\mathcal{F}\) and \(\mathcal{G}\) be linear multiports with the following parametric representations:

\[\begin{aligned}\mathcal{F} &= \left\{ \begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathbb{R}^{2n} : \begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} = \begin{bmatrix}\boldsymbol{V}_{\mathcal{F}} \\ \boldsymbol{I}_{\mathcal{F}} \end{bmatrix} \boldsymbol{\lambda}, \boldsymbol{\lambda} \in \mathbb{R}^{n} \right\} \\ \mathcal{G} &= \left\{ \begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathbb{R}^{2n} : \begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} = \begin{bmatrix}\boldsymbol{V}_{\mathcal{G}} \\ \boldsymbol{I}_{\mathcal{G}} \end{bmatrix} \boldsymbol{\lambda}, \boldsymbol{\lambda} \in \mathbb{R}^{n} \right\} \end{aligned}\]

If \(\mathcal{F}\) and \(\mathcal{G}\) are dual with duality constant \(D\), then

\[\begin{bmatrix}\boldsymbol{V}_{\mathcal{G}} \\ \boldsymbol{I}_{\mathcal{G}}\end{bmatrix} = \begin{bmatrix}D\boldsymbol{I}_{\mathcal{F}} \\ \frac{1}{D}\boldsymbol{V}_{\mathcal{F}}\end{bmatrix}\]

Proof

TODO

Reciprocity#

Definition: Reciprocity

An \(n\)-port is reciprocal if

\[\boldsymbol{v}_a^{\mathsf{T}}\boldsymbol{i}_b = \boldsymbol{v}_b^{\mathsf{T}}\boldsymbol{i}_a\]

for all \((\boldsymbol{v}_a, \boldsymbol{i}_a) \in \mathcal{F}\) and all \((\boldsymbol{v}_b, \boldsymbol{i}_b) \in \mathcal{F}\).

Theorem: Reciprocity \(\implies\) Linearity

If a multiport is reciprocal, then it is also linear.

Proof

TODO

Theorem: Reciprocity via Parametric Representations

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

\[\begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} = \begin{bmatrix}\boldsymbol{V} \\ \boldsymbol{I}\end{bmatrix} \boldsymbol{\lambda},\]

then it is reciprocal if and only if

\[\boldsymbol{V}^{\mathsf{T}}\boldsymbol{I} - \boldsymbol{I}^{\mathsf{T}}\boldsymbol{V} = \boldsymbol{0}.\]

Proof

TODO

Theorem: Reciprocity via Explicit Representations

Let \(\mathcal{F}\) be a linear multiport

If \(\mathcal{F}\) has an admittance matrix \(\boldsymbol{G}\), then it is reciprocal if and only if \(\boldsymbol{G}\) is symmetric:

\[\boldsymbol{G} = \boldsymbol{G}^{\mathsf{T}}\]

If \(\mathcal{F}\) has an impedance matrix \(\boldsymbol{R}\), then it is reciprocal if and only if \(\boldsymbol{R}\) is symmetric:

\[\boldsymbol{R} = \boldsymbol{R}^{\mathsf{T}}\]

Proof

TODO