Linear Two-Ports#

A two-port is strictly linear if its I-V characteristic is a two-dimensional subspace of \(\mathbb{R}^4\).

Representations#

Info: Implicit Representation

A strictly linear two-port has an implicit representation

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

where \(\boldsymbol{M} \in \mathbb{R}^{2\times 2}\) and \(\boldsymbol{N} \in \mathbb{R}^{2 \times 2}\).

Info: Parametric Representation

A strictly linear two-port has a parametric representation

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

where \(\boldsymbol{V} \in \mathbb{R}^{2\times 2}\) and \(\boldsymbol{I} \in \mathbb{R}^{2 \times 2}\).

Theorem: Conversions between Implicit and Explicit Representations

Let \(\mathcal{F}\) be a two-port with the following implicit representation:

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

where

\[ \boldsymbol{M} = \begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_1 & \boldsymbol{m}_2 \\ \vert & \vert \end{bmatrix} \qquad \boldsymbol{N} = \begin{bmatrix}\vert & \vert \\ \boldsymbol{n}_1 & \boldsymbol{n}_2 \\ \vert & \vert \end{bmatrix}. \]

The explicit representations of \(\mathcal{F}\) are given below, provided the respective matrix inverses exist:

\(\boldsymbol{G}\)	\(\boldsymbol{R}\)	\(\boldsymbol{H}\)	\(\boldsymbol{H}'\)	\(\boldsymbol{T}\)	\(\boldsymbol{T}'\)	\(\boldsymbol{T}'\) defined via \(\begin{bmatrix} V_2 \\ I_2 \end{bmatrix} = T^{-1}\left(\begin{bmatrix}V_1 \\ -I_1\end{bmatrix}\right)\)
\(-\boldsymbol{N}^{-1}\boldsymbol{M}\)	\(-\boldsymbol{M}^{-1}\boldsymbol{N}\)	\(-\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_1 & \boldsymbol{n}_2 \\ \vert & \vert\end{bmatrix}^{-1}\begin{bmatrix}\vert & \vert \\ \boldsymbol{n}_1 & \boldsymbol{m}_2 \\ \vert & \vert\end{bmatrix}\)	\(-\begin{bmatrix}\vert & \vert \\ \boldsymbol{n}_1 & \boldsymbol{m}_2 \\ \vert & \vert\end{bmatrix}^{-1}\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_1 & \boldsymbol{n}_2 \\ \vert & \vert\end{bmatrix}\)	\(-\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_1 & \boldsymbol{n}_1 \\ \vert & \vert\end{bmatrix}^{-1}\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_2 & -\boldsymbol{n}_2 \\ \vert & \vert\end{bmatrix}\)	\(-\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_2 & -\boldsymbol{n}_2 \\ \vert & \vert\end{bmatrix}^{-1}\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_1 & \boldsymbol{n}_1 \\ \vert & \vert\end{bmatrix}\)	\(-\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_2 & \boldsymbol{n}_2 \\ \vert & \vert\end{bmatrix}^{-1}\begin{bmatrix}\vert & \vert \\ \boldsymbol{m}_1 & -\boldsymbol{n}_1 \\ \vert & \vert\end{bmatrix}\)

Proof

TODO

Theorem: Conversions between Explicit Representations

If all six explicit representations exist and are linear, then their standard matrix representations are related as follows:

	\(R\)	\(G\)	\(H\)	\(H'\)	\(T\)	\(T'\)	\(T'\) defined via \(\begin{bmatrix} V_2 \\ I_2 \end{bmatrix} = T'\left(\begin{bmatrix}V_1 \\ -I_1\end{bmatrix}\right)\)
\(R\)	\(\displaystyle\begin{bmatrix}r_{11} & r_{12} \\ r_{21} & r_{22}\end{bmatrix}\)	\(\displaystyle\frac{1}{\det G} \begin{bmatrix}g_{22} & -g_{12} \\ -g_{21} & g_{11}\end{bmatrix}\)	\(\displaystyle\frac{1}{h_{22}}\begin{bmatrix}\det H & h_{12} \\ -h_{21} & 1\end{bmatrix}\)	\(\displaystyle\frac{1}{h_{11}'}\begin{bmatrix}1 & -h_{12}' \\ h_{21}' & \det H'\end{bmatrix}\)	\(\displaystyle\frac{1}{t_{21}}\begin{bmatrix}t_{11} & \det T \\ 1 & t_{22}\end{bmatrix}\)	TODO	\(\displaystyle\frac{1}{t_{21}'}\begin{bmatrix}t_{22}' & 1 \\ \det T' & t_{11}'\end{bmatrix}\)
\(G\)	\(\displaystyle\frac{1}{\det R} \begin{bmatrix}r_{22} & -r_{12} \\ -r_{21} & r_{11}\end{bmatrix}\)	\(\displaystyle\begin{bmatrix}g_{11} & g_{12} \\ g_{21} & g_{22}\end{bmatrix}\)	\(\displaystyle\frac{1}{h_{11}} \begin{bmatrix}1 & -h_{12} \\ h_{21} & \det H\end{bmatrix}\)	\(\displaystyle\frac{1}{h_{22}'} \begin{bmatrix}\det H' & h_{12}' \\ -h_{21}' & 1\end{bmatrix}\)	\(\displaystyle\frac{1}{t_{12}}\begin{bmatrix}t_{22} & -\det T \\ -1 & t_{11}\end{bmatrix}\)	TODO	\(\displaystyle\frac{1}{t_{12}'}\begin{bmatrix}t_{11}' & -1 \\ -\det T' & t_{22}'\end{bmatrix}\)
\(H\)	\(\displaystyle\frac{1}{r_{22}}\begin{bmatrix}\det R & r_{12} \\ -r_{21} & 1\end{bmatrix}\)	\(\displaystyle\frac{1}{g_{11}}\begin{bmatrix}1 & -g_{12} \\ g_{21} & \det G\end{bmatrix}\)	\(\displaystyle\begin{bmatrix}h_{11} & h_{12} \\ h_{21} & h_{22}\end{bmatrix}\)	\(\displaystyle\frac{1}{\det H'}\begin{bmatrix}h_{22}' & -h_{12}' \\ -h_{21}' & h_{11}'\end{bmatrix}\)	\(\displaystyle\frac{1}{t_{22}}\begin{bmatrix}t_{12} & \det T \\ -1 & t_{21}\end{bmatrix}\)	TODO	\(\displaystyle \frac{1}{t_{11}'}\begin{bmatrix}t_{12}' & 1 \\ -\det T' & t_{21}'\end{bmatrix}\)
\(H'\)	\(\displaystyle\frac{1}{r_{11}}\begin{bmatrix}1 & -r_{12} \\ r_{21} & \det R\end{bmatrix}\)	\(\displaystyle\frac{1}{g_{22}}\begin{bmatrix}\det G & g_{12} \\ -g_{21} & 1\end{bmatrix}\)	\(\displaystyle\frac{1}{\det H} \begin{bmatrix}h_{22} & -h_{12} \\ -h_{21} & h_{11}\end{bmatrix}\)	\(\displaystyle\begin{bmatrix}h_{11}' & h_{12}' \\ h_{21}' & h_{22}'\end{bmatrix}\)	\(\displaystyle\frac{1}{t_{11}}\begin{bmatrix}t_{21} & -\det T \\ 1 & t_{12}\end{bmatrix}\)	TODO	\(\displaystyle \frac{1}{t_{22}'} \begin{bmatrix}t_{21}' & -1 \\ \det T' & t_{12}'\end{bmatrix}\)
\(T\)	\(\displaystyle\frac{1}{r_{21}}\begin{bmatrix}r_{11} & \det R \\ 1 & r_{22}\end{bmatrix}\)	\(\displaystyle\frac{1}{g_{21}}\begin{bmatrix}-g_{22} & -1 \\ -\det G & -g_{11}\end{bmatrix}\)	\(\displaystyle\frac{1}{h_{21}}\begin{bmatrix}-\det H & -h_{11} \\ -h_{22} & -1\end{bmatrix}\)	\(\displaystyle\frac{1}{h_{21}'}\begin{bmatrix}1 & h_{22}' \\ h_{11}' & \det H'\end{bmatrix}\)	\(\displaystyle\begin{bmatrix}t_{11} & t_{12} \\ t_{21} & t_{22}\end{bmatrix}\)	\(\displaystyle\frac{1}{\det T'}\begin{bmatrix}t_{22}' & -t_{12}' \\ -t_{21}' & t_{11}'\end{bmatrix}\)	\(\displaystyle \frac{1}{\det T'} \begin{bmatrix}t_{22}' & t_{12}' \\ t_{21}' & t_{11}'\end{bmatrix}\)
\(T'\)	TODO	TODO	TODO	TODO	\(\displaystyle\frac{1}{\det T}\begin{bmatrix}t_{22} & -t_{12} \\ -t_{21} & t_{11}\end{bmatrix}\)	\(\displaystyle\begin{bmatrix}t_{11}' & t_{12}' \\ t_{21}' & t_{22}'\end{bmatrix}\)	TODO
\(T'\) defined via \(\begin{bmatrix} V_2 \\ I_2 \end{bmatrix} = T'\left(\begin{bmatrix}V_1 \\ -I_1\end{bmatrix}\right)\)	\(\displaystyle \frac{1}{r_{12}}\begin{bmatrix}r_{22} & \det R \\ 1 & r_{11}\end{bmatrix}\)	\(\displaystyle \frac{1}{g_{12}} \begin{bmatrix}-g_{11} & -1 \\ -\det G & -g_{22}\end{bmatrix}\)	\(\displaystyle \frac{1}{h_{12}} \begin{bmatrix}1 & h_{11} \\ h_{22} & \det H\end{bmatrix}\)	\(\displaystyle \frac{1}{h_{12}'} \begin{bmatrix}- \det H' & -h_{22}' \\ -h_{11}' & -1\end{bmatrix}\)	\(\displaystyle \frac{1}{\det T} \begin{bmatrix}t_{22} & t_{12} \\ t_{21} & t_{11}\end{bmatrix}\)	TODO	\(\displaystyle\begin{bmatrix}t_{11}' & t_{12}' \\ t_{21}' & t_{22}'\end{bmatrix}\)

Proof

TODO

Reciprocity#

Theorem: Reciprocity Based on Elements

If a two-port does not contain any elements other than linear resistors, transformers, capacitors and inductors, then it is reciprocal.

Proof

TODO

Theorem: Ratios of Reciprocal Two-Ports

Let \(\mathcal{F}\) be a strictly linear two-port.

If \(\mathcal{F}\) is reciprocal, then we have the following properties:

The ratio of \(v_1\) to \(v_2\) whenever \(i_1 = 0\) is the same as the negative of the ratio of \(i_2\) to \(i_1\) whenever \(v_2 = 0\).

\[ \left.\frac{v_1}{v_2}\right\vert_{i_1 = 0} = \left.-\frac{i_2}{i_1}\right\vert_{v_2 = 0} \]

The ratio of \(v_2\) to \(v_1\) whenever \(i_2 = 0\) is the same as the negative of the ratio of \(i_1\) to \(i_2\) whenever \(v_1 = 0\).

\[ \left.\frac{v_2}{v_1}\right\vert_{i_2 = 0} = \left. -\frac{i_1}{i_2}\right\vert_{v_1 = 0} \]

Proof

TODO

Theorem: Reciprocity via Transmission Representation

If \(\mathcal{F}\) has a forwards transmission matrix \(\boldsymbol{T}\), then it is reciprocal if and only if the determinant of \(\boldsymbol{T}\) is \(1\):

\[ \det \boldsymbol{T} = 1 \]

Proof

TODO

Symmetry#

Theorem: Symmetry via Admittance Representation

A strictly linear two-port with admittance representation

\[ \boldsymbol{i} = \boldsymbol{G} \boldsymbol{v} \]

is symmetrical if and only if

\[ \boldsymbol{G} = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\boldsymbol{G}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \]

i.e. \(\boldsymbol{G}\) remains the same if you swap the elements on its diagonals.

Proof

TODO

Theorem: Symmetry via Impedance Representation

A strictly linear two-port with impedance representation

\[ \boldsymbol{v} = \boldsymbol{R} \boldsymbol{i} \]

is symmetrical if and only if

\[ \boldsymbol{R} = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\boldsymbol{R}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \]

i.e. \(\boldsymbol{R}\) remains the same if you swap the elements on its diagonals.

Proof

TODO

Theorem: Symmetry via Transmission Representations

A strictly linear two-port with existing forwards transmission representation

\[ \begin{bmatrix} v_1 \\ i_1 \end{bmatrix} = \boldsymbol{T} \begin{bmatrix}v_2 \\ -i_2\end{bmatrix} \]

and backwards transmission representation

\[ \begin{bmatrix} v_2 \\ i_2 \end{bmatrix} = \boldsymbol{T}' \begin{bmatrix}v_1 \\ -i_1\end{bmatrix} \]

is symmetrical if and only if

\[ \boldsymbol{T} = \boldsymbol{T}'. \]

Proof

TODO

Theorem: Symmetry \(\implies\) Reciprocity

If a strictly linear two-port two-port is symmetrical, then it is also reciprocal.

Proof

TODO