#  On Thevenin-Norton and Maximum power transfer theorems

H. Narayanan<EMAIL_ADDRESS>Hariharan Narayanan
<EMAIL_ADDRESS>Department of Electrical Engineering, Indian
Institute of Technology Bombay  School of Technology and Computer Science,
Tata Institute of Fundamental Research

###### Abstract

In this paper we state and prove complete versions of two basic theorems of
linear circuit theory.

The Thevenin-Norton theorem expresses the port behaviour of a linear multiport
in terms of the zero source and zero port input conditions when the port
behaviour has a hybrid input- output representation of the form

$\displaystyle\begin{pmatrix}i_{P_{1}}\\\
v_{P_{2}}\end{pmatrix}=\begin{pmatrix}g_{11}&h_{12}\\\
h_{21}&r_{22}\end{pmatrix}\begin{pmatrix}v_{P_{1}}\\\
i_{P_{2}}\end{pmatrix}+\begin{pmatrix}J_{P_{1}}\\\ E_{P_{2}}\end{pmatrix},$
(1)

where the partition of $P$ into $P_{1},P_{2}$ is known.

In this paper we show how to handle multiports which have port behaviour
equations of the general form $Bv_{P}-Qi_{P}=s,$ which cannot even be put into
the hybrid form of Equation 1, indeed may have number of equations ranging
from $0$ to $2n,$ where $n$ is the number of ports. We do this through
repeatedly solving with different source inputs, a larger network obtained by
terminating the multiport by its adjoint through a gyrator. The method works
for linear multiports which are consistent for arbitrary internal source
values and further have the property that the port conditions uniquely
determine internal conditions.

The maximum power transfer theorem states that if the multiport has a Thevenin
impedance matrix $Z,$ then the maximum power transfer from the multiport takes
place when we terminate the multiport by another whose Thevenin impedance
matrix is the adjoint (conjugate-transpose) $Z^{*}$ of $Z,$ provided $Z+Z^{*}$
has only positive eigenvalues [1]. The theorem does not handle the case where
the multiport does not have a Thevenin or Norton equivalent.

In this paper we present the most general version of maximum power transfer
theorem possible. This version of the theorem states that ‘stationarity’
(derivative zero condition) of power transfer occurs when the multiport is
terminated by its adjoint, provided the resulting network has a solution. If
this network does not have a solution there is no port condition for which
stationarity holds. This theorem does not require that the multiport has a
hybrid immittance matrix.

###### keywords:

Basic circuits, Implicit duality, Thevenin, Maximum power.

###### MSC:

15A03, 15A04, 94C05, 94C15

††journal: Linear Algebra and its Applications

##  1 Introduction

Note: This paper is a brief and self contained account of the basic circuit
theorems extracted from ‘Implicit Linear Algebra and Basic Circuit Theory’
(ILABC) (arXiv:2005.00838v1 [eess.SY]). ILABC attempts to relate implicit
linear algebra (ILA) and circuit theory and the discussion of the theorems in
that paper is meant to be an application of ILA. The present paper does not
emphasize ILA but only concentrates on Thevenin-Norton and Maximum Power
Transfer Theorems.

In this paper we give complete versions of Thevenin-Norton Theorem and Maximum
Power Transfer Theorem.

Thevenin-Norton Theorem gives a method for computing the port behaviour
(formal definition in Section 3) of a linear multiport which has a hybrid
representation ([11, 4, 9], see [2] for a standard treatment), and for which
we know before hand, which ports should be treated as current or voltage input
kind. The characteristic feature of this approach is the computation of the
port condition for a special port termination (open circuit or short circuit),
and then the computation of the resistance, conductance or hybrid matrix
setting the sources inside to zero. It is clear how to use a standard circuit
simulator for this purpose, if one knew before hand that a hybrid matrix
exists for the network and which are its current and voltage input ports.
However, in general, a hybrid matrix may not even exist for a multiport. A
simple termination such as open or short circuit may result in inconsistency,
so that a standard circuit simulator would give an error message.

We present in this paper a method which works for very general linear
multiports, which we call ‘rigid’ (which may have number of port equations
ranging from $0$ to twice the number of ports). These have nonvoid solution
for arbitrary internal source values and further have a unique internal
solution for a given voltage plus current port condition. These are also the
only kind of multiports which can be handled by standard circuit simulators,
which are built for solving linear circuits with unique solution. Our method
involves terminating the given multiport by its adjoint (see Subsection 3.1)
through a multiport gyrator with identity matrix as the gyrator resistance
matrix. The gyrator has external current or voltage sources attached in such a
way that the original multiport does not see a direct termination by a source
(see Figure 1) and the circuit is solved repeatedly by the simulator. It is to
be noted that the adjoint of a multiport can be built by changing the device
characteristic block by block, usually without serious computation.

We next use these ideas to present the most general form of the maximum power
transfer theorem. We show that the maximum power transfer, if it occurs at
all, corresponds to the port condition that is obtained when we terminate the
multiport by its adjoint through a $1:1$ ideal transformer. When the multiport
is rigid, if the simulator fails to solve for this termination because there
is no solution, it means that power transfer can be unbounded. If it fails to
solve because the solution is non unique, it means that there are an infinity
of port conditions corresponding to stationarity of power transfer.

To summarize, we claim there are two significant contributions in the paper:

  1. 1.

If we terminate a rigid multiport by its adjoint, through a $1:1$ gyrator, the
resulting network always has a unique solution. This holds even if the gyrator
has sources attached (current source in parallel, voltage source in series) to
its ports. Further, every possible port condition of the multiport can be
realized by a suitable distribution of sources to the gyrator ports. This
yields a technique for computing the port behaviour of a rigid multiport,
using a conventional circuit simulator repeatedly.

  2. 2.

To find the maximum power transferred by a rigid multiport, it is enough if we
terminate it by its adjoint and solve the resulting circuit. There is no need
to compute its port behaviour, which is usually a more cumbersome process.

Note on the notation: We have to deal with solution spaces of equations of the
kind $Bv_{P}-Qi_{P"}=s,$ rather than with those of the kind $v_{P}-Zi_{P"}=E.$
Instead of working with the adjoint (conjugate transpose) of a matrix $Z,$ we
have to work with the adjoint (see Subsection 3.1) of the solution space of
$Bv_{P}-Qi_{P"}=0.$ In order for our technique to be effective, we have to
show that if $Bv_{P}-Qi_{P"}=s,$ defines the port behaviour of ${\cal N}_{P}$
then the adjoint of the solution space of $Bv_{P}-Qi_{P"}=0,$ is the port
behaviour of the multiport ${\cal N}_{P}^{adj}$ which is the adjoint of ${\cal
N}_{P}$ (Corollary 9). We therefore have to use notation not usual in circuit
theory, dealing with operations on vector spaces such as sum, intersection,
contraction, restriction, matched composition etc., instead of operations on
matrices.

Example. It is possible to build a rigid (as defined above) multiport ${\cal
N}_{P},$ using controlled sources, norators and nullators, which has any given
representation at the ports with nonvoid solution. Suppose the rigid multiport
${\cal N}_{P}$ has the behaviour (see formal definition in Section 3)
$Bv_{P}-Qi_{P"}=s.$ The number of equations may range from $0$ to $2n,$ where
$n=|P|.$ Neither the Thevenin nor the Norton representation, indeed even a
hybrid representation, need exist. Our method can be used to compute the port
behaviour of this multiport, whereas the usual Thevenin-Norton theorem cannot
handle a case of this level of generality.

Figure 1: Computation of port behaviour of a multiport

Briefly, the steps in our method are as follows:
1\. Build the adjoint multiport ${\cal N}^{adj}_{\tilde{P}}.$ This is done by
retaining the same graph as ${\cal N}_{P}$ but replacing each device by its
adjoint (see Example 7). The name of the set of internal edges $S$ is changed
to $\tilde{S}$ and that of the port edges $P$ is changed to $\tilde{P}.$
2\. Between every pair of corresponding ports $P_{j},\tilde{P}_{j},$ insert a
source accompanied $1:1$ gyrator. (For simplicity, in Figure 1 we have taken
${\cal N}_{P}$ to be a $1$-port.)
3\. Solve the circuit resulting when all the gyrator sources are zero and
internal port sources are active as in Figure 1 (a). Let
$(v^{p}_{P},i^{p}_{P"})$ be the corresponding port condition.
4\. Set internal sources to zero but gyrator accompanied sources active one at
a time as in Figure 1 (b),(c). The resulting set of voltage- current vectors
at $P$ will span the source free port behaviour $\hat{\mbox{$\cal
V$}}_{P{P"}}.$ (We have to change the sign of the current port vector in the
solution to correspond to port behaviour current, which ‘enters’ the
multiport).
5\. The port behaviour $\hat{{\cal
A}}_{P{P"}}:\equiv(v^{p}_{P},-i^{p}_{P"})+\hat{\mbox{$\cal V$}}_{P{P"}}.$

Further, we can also test this port behaviour for condition of maximum power
transfer by terminating ${\cal N}_{P}$ by its adjoint ${\cal
N}^{adj}_{\tilde{P}}$ through a $1:1$ ideal transformer
($v_{P}=v_{\tilde{P}};i_{P}=-i_{\tilde{P}}$) and solving the resulting circuit
by a simulator. Solving this circuit is equivalent to solving Equation 23 of
Section 5.1 rewritten as Equation 2 below. (Note that this equation is not
being explicitly constructed and solved.)

$\displaystyle\begin{pmatrix}B&-Q\end{pmatrix}\begin{pmatrix}-Q^{*}\\\
B^{*}\end{pmatrix}\lambda=s.$ (2)

If this equation has a unique solution, the corresponding port condition

${((\breve{v}^{stat}_{P})^{T}|(\breve{i}^{stat}_{P"})^{T})=\lambda^{T}(-\overline{Q}|\overline{B}})$

is the unique stationarity point. We have to verify whether it is maximum or
minimum by perturbing the port condition around this point. If this equation
has no solution, the power that can be drawn from the multiport is unbounded.
If it has many solutions, there will be an infinity of port conditions
corresponding to stationarity of power transfer. In both these latter cases
our method will only indicate non unique solution and halt.

Finally, note that when the Thevenin equivalent exists, the port equation
$Bv_{P}-Qi_{P"}=s,$ reduces to $Iv_{P}-Zi_{P"}=E,$ and the solution of
Equation 2 (taking $Z^{*}:\equiv$ conjugate transpose of $Z$) yields

$\displaystyle\begin{pmatrix}I&-Z\end{pmatrix}\begin{pmatrix}-Z^{*}\\\
I\end{pmatrix}\lambda=E.$ (3)

This corresponds to
$-(Z+Z^{*})\lambda=E,((\breve{v}^{stat}_{P})^{T}|(\breve{i}^{stat}_{P"})^{T})=\lambda^{T}(-\overline{Z}|I),$
i.e., to $(Z+Z^{*})(-\breve{i}^{stat}_{P"})=E.$ This means that the
stationarity of power transfer occurs when we terminate the multiport by the
adjoint of the Thevenin impedance.

##  2 Preliminaries

The preliminary results and the notation used are from [7].

###  2.1 Vectors

A vector $\boldsymbol{f}$ on a finite set $X$ over $\mathbb{F}$ is a function
$f:X\rightarrow\mathbb{F}$ where $\mathbb{F}$ is either the real field $\Re$
or the complex field $\mathbb{C}.$

The sets on which vectors are defined are always finite. When a vector $x$
figures in an equation, we use the convention that $x$ denotes a column vector
and $x^{T}$ denotes a row vector such as in ‘$Ax=b,x^{T}A^{T}=b^{T}$’. Let
$f_{Y}$ be a vector on $Y$ and let $X\subseteq Y$. The restriction
$f_{Y}|_{X}$ of $f_{Y}$ to $X$ is defined as follows:
$f_{Y}|_{X}:\equiv g_{X},\textrm{ where }g_{X}(e)=f_{Y}(e),e\in X.$

When $f$ is on $X$ over $\mathbb{F}$, $\lambda\in\mathbb{F},$ then the scalar
multiplication $\boldsymbol{\lambda f}$ of $f$ is on $X$ and is defined by
$(\lambda f)(e):\equiv\lambda[f(e)]$, $e\in X$. When $f$ is on $X$ and $g$ on
$Y$ and both are over $\mathbb{F}$, we define $\boldsymbol{f+g}$ on $X\cup Y$
by
$(f+g)(e):\equiv f(e)+g(e),e\in X\cap Y,\ (f+g)(e):\equiv f(e),e\in X\setminus
Y,\ (f+g)(e):\equiv g(e),e\in Y\setminus X.$ (For ease in readability, we will
henceforth use $X-Y$ in place of $X\setminus Y.$)

When $f,g$ are on $X$ over $\mathbb{C},$ the inner product $\langle
f,g\rangle$ of $f$ and $g$ is defined by $\langle f,g\rangle:\equiv\sum_{e\in
X}f(e)\overline{g(e)},$ $\overline{g(e)}$ being the complex conjugate of
$g(e).$ If the field is $\mathbb{R},$ the inner product would reduce to the
dot product $\langle f,g\rangle:\equiv\sum_{e\in X}f(e){g(e)}.$

When $X,Y,$ are disjoint, $f_{X}+g_{Y}$ is written as
$\boldsymbol{(f_{X},g_{Y})},$ The disjoint union of $X$ and $Y.$ is denoted by
$\boldsymbol{X\uplus Y}.$ A vector $f_{X\uplus Y}$ on $X\uplus Y$ is written
as $\boldsymbol{f_{XY}}.$

We say $f$, $g$ are orthogonal (orthogonal) iff $\langle f,g\rangle$ is zero.

An arbitrary collection of vectors on $X$ is denoted by
$\boldsymbol{\mathcal{K}_{X}}$. When $X$, $Y$ are disjoint we usually write
$\mathcal{K}_{XY}$ in place of $\mathcal{K}_{X\uplus Y}$. We write ${\cal
K}_{XY}:\equiv{\cal K}_{X}\oplus{\cal K}_{Y}$ iff ${\cal
K}_{XY}:\equiv\\{f_{XY}:f_{XY}=(f_{X},g_{Y}),f_{X}\in{\cal
K}_{X},g_{Y}\in{\cal K}_{Y}\\}.$ We refer to ${\cal K}_{X}\oplus{\cal K}_{Y}$
as the direct sum of ${\cal K}_{X},{\cal K}_{Y}.$

A collection ${\cal K}_{X}$ is a vector space on $X$ iff it is closed under
addition and scalar multiplication. The notation $\boldsymbol{\mbox{$\cal
V$}_{X}}$ always denotes a vector space on $X.$ For any collection ${\cal
K}_{X},$ $\boldsymbol{span({\cal K}_{X})}$ is the vector space of all linear
combinations of vectors in it. We say ${\cal A}_{X}$ is an affine space on
$X,$ iff it can be expressed as $x_{X}+\mbox{$\cal
V$}_{X}:\equiv\\{y_{X},y_{X}=x_{X}+z_{X},z_{X}\in\mbox{$\cal V$}_{X}\\},$
where $x_{X}$ is a vector and $\mbox{$\cal V$}_{X},$ a vector space on $X.$
The latter is unique for ${\cal A}_{X}$ and is said to be its vector space
translate.

For a vector space $\mbox{$\cal V$}_{X},$ since we take $X$ to be finite, any
maximal independent subset of $\mbox{$\cal V$}_{X}$ has size less than or
equal to $|X|$ and this size can be shown to be unique. A maximal independent
subset of a vector space $\mbox{$\cal V$}_{X}$ is called its basis and its
size is called the dimension or rank of $\mbox{$\cal V$}_{X}$ and denoted by
${\boldsymbol{dim}(\mbox{$\cal V$}_{X})}$ or by ${\boldsymbol{r}(\mbox{$\cal
V$}_{X})}.$ For any collection of vectors ${\cal K}_{X},$ the rank
$\boldsymbol{r}({\cal K}_{X})$ is defined to be $dim(span({\cal K}_{X})).$ The
collection of all linear combinations of the rows of a matrix $A$ is a vector
space that is denoted by $row(A).$

For any collection of vectors $\mathcal{K}_{X},$ the collection
$\boldsymbol{\mathcal{K}_{X}^{*}}$ is defined by
${\mathcal{K}_{X}^{*}}:\equiv\\{g_{X}:\langle f_{X},g_{X}\rangle=0\\},$ It is
clear that $\mathcal{K}_{X}^{*}$ is a vector space for any $\mathcal{K}_{X}.$
When $\mathcal{K}_{X}$ is a vector space $\mbox{$\cal V$}_{X},$ and the
underlying set $X$ is finite, it can be shown that
$({\mathcal{V}_{X}^{*}})^{*}=\mathcal{V}_{X}$ and
$\mathcal{V}_{X},{\mathcal{V}_{X}^{*}}$ are said to be complementary
orthogonal. The symbol $0_{X}$ refers to the zero vector on $X$ and
$\boldsymbol{0_{X}}$ refers to the zero vector space on $X.$ The symbol
$\boldsymbol{\mbox{$\cal F$}_{X}}$ refers to the collection of all vectors on
$X$ over the field in question. It is easily seen, when $X,Y$ are disjoint,
and ${\cal K}_{X},{\cal K}_{Y}$ contain zero vectors, that $({\cal
K}_{X}\oplus{\cal K}_{Y})^{*}={\cal K}_{X}^{*}\oplus{\cal K}_{Y}^{*}.$

The adjoint $K^{*}$ of a matrix $K$ is defined by
$K^{*}:\equiv\overline{K}^{T}.$
(Later we define the adjoint $\mbox{$\cal V$}^{adj}_{PP"}$ of a vector space
which extends this notion (see Subsection 3.1).)

A matrix of full row rank, whose rows generate a vector space $\mbox{$\cal
V$}_{X},$ is called a representative matrix for $\mbox{$\cal V$}_{X}.$ A
representative matrix which can be put in the form $(I\ |\ K)$ after column
permutation, is called a standard representative matrix. It is clear that
every vector space has a standard representative matrix. If $(I\ |\ K)$ is a
standard representative matrix of $\mbox{$\cal V$}_{X},$ it is easy to see
that $(-K^{*}|I)$ is a standard representative matrix of $\mbox{$\cal
V$}^{*}_{X}.$ Therefore we must have

######  Theorem 1.

Let $\mbox{$\cal V$}_{X}$ be a vector space on $X.$ Then $r(\mbox{$\cal
V$}_{X})+r(\mbox{$\cal V$}^{*}_{X})=|X|$ and $((\mbox{$\cal
V$}_{X})^{*})^{*}=\mbox{$\cal V$}_{X}.$

The collection $\\{(f_{X},\lambda f_{Y}):(f_{X},f_{Y})\in\mathcal{K}_{XY}\\}$
is denoted by $\boldsymbol{\mathcal{K}_{X(\lambda Y)}}.$ When $\lambda=-1$ we
would write ${\mathcal{K}_{X(\lambda Y)}}$ more simply as
$\boldsymbol{\mathcal{K}_{X(-Y)}}.$ Observe that
$(\mathcal{K}_{X(-Y)})_{X(-Y)}=\mathcal{K}_{XY}.$ We say sets $X$, $X"$ are
copies of each other iff they are disjoint and there is a bijection, usually
clear from the context, mapping $e\in X$ to $e"\in X"$. When $X,X"$ are copies
of each other, the vectors $f_{X}$ and $f_{X"}$ are said to be copies of each
other with $f_{X"}(e"):\equiv f_{X}(e),e\in X.$ The copy ${\cal K}_{X"}$ of
${\cal K}_{X}$ is defined by ${\cal K}_{X"}:\equiv\\{f_{X"}:f_{X}\in{\cal
K}_{X}\\}.$ When $X$ and $X"$ are copies of each other, the notation for
interchanging the positions of variables with index sets $X$ and $X"$ in a
collection $\mathcal{K}_{XX"Y}$ is given by
$\boldsymbol{(\mathcal{K}_{XX"Y})_{X"XY}}$, that is
$(\mathcal{K}_{XX"Y})_{X"XY}:\equiv\\{(g_{X},f_{X"},h_{Y})\ :\
(f_{X},g_{X"},h_{Y})\in\mathcal{K}_{XX"Y},\ g_{X}\textrm{ being copy of
}g_{X"},\ f_{X"}\textrm{ being copy of }f_{X}\\}.$ An affine space
$\mathcal{A}_{XX"}$ is said to be proper iff the rank of its vector space
translate is $|X|=|X"|.$

###  2.2 Sum and Intersection

Note: This extends the conventional definition of sum and intersection of
vector spaces on the same set.
Let $\mathcal{K}_{SP}$, $\mathcal{K}_{PQ}$ be collections of vectors on sets
$S\uplus P,$ $P\uplus Q,$ respectively, where $S,P,Q,$ are pairwise disjoint.
The sum $\boldsymbol{\mathcal{K}_{SP}+\mathcal{K}_{PQ}}$ of
$\mathcal{K}_{SP}$, $\mathcal{K}_{PQ}$ is defined over $S\uplus P\uplus Q,$ as
follows:
$\mathcal{K}_{SP}+\mathcal{K}_{PQ}:\equiv\\{(f_{S},f_{P},0_{Q})+(0_{S},g_{P},g_{Q}),\textrm{
where }(f_{S},f_{P})\in\mathcal{K}_{SP},(g_{P},g_{Q})\in\mathcal{K}_{PQ}\\}.$
Thus,
$\mathcal{K}_{SP}+\mathcal{K}_{PQ}:\equiv(\mathcal{K}_{SP}\oplus{\mathbf{0}}_{Q})+({\mathbf{0}}_{S}\oplus\mathcal{K}_{PQ}).$
The intersection $\boldsymbol{\mathcal{K}_{SP}\cap\mathcal{K}_{PQ}}$ of
$\mathcal{K}_{SP}$, $\mathcal{K}_{PQ}$ is defined over $S\uplus P\uplus Q,$
where $S,P,Q,$ are pairwise disjoint, as follows:
$\mathcal{K}_{SP}\cap\mathcal{K}_{PQ}:\equiv\\{f_{SPQ}:f_{SPQ}=(f_{S},h_{P},g_{Q}),$
$\textrm{ where
}(f_{S},h_{P})\in\mathcal{K}_{SP},(h_{P},g_{Q})\in\mathcal{K}_{PQ}.\\}.$
Thus,
$\mathcal{K}_{SP}\cap\mathcal{K}_{PQ}:\equiv(\mathcal{K}_{SP}\oplus\mbox{$\cal
F$}_{Q})\cap(\mbox{$\cal F$}_{S}\oplus\mathcal{K}_{PQ}).$

It is immediate from the definition of the operations that sum and
intersection of vector spaces remain vector spaces.

###  2.3 Restriction and contraction

The restriction of $\boldsymbol{\mathcal{K}_{SP}}$ to $S$ is defined by
$\boldsymbol{\mathcal{K}_{SP}\circ
S}:\equiv\\{f_{S}:(f_{S},f_{P})\in\mathcal{K}_{SP}\\}.$ The contraction of
$\boldsymbol{\mathcal{K}_{SP}}$ to $S$ is defined by
$\boldsymbol{\mathcal{K}_{SP}\times
S}:\equiv\\{f_{S}:(f_{S},0_{P})\in\mathcal{K}_{SP}\\}.$ Unless otherwise
stated, the sets on which we perform the contraction operation would have the
zero vector as a member so that the resulting set would be nonvoid. We denote
by $\boldsymbol{\mathcal{K}_{SPZ}\circ SP}$,
$\boldsymbol{\mathcal{K}_{SPZ}\times SP}$, respectively when $S,P,Z,$ are
pairwise disjoint, the collections of vectors
$\boldsymbol{\mathcal{K}_{SPZ}\circ(S\uplus P)}$,
$\boldsymbol{\mathcal{K}_{SPZ}\times(S\uplus P)}.$

It is clear that restriction and contraction of vector spaces are also vector
spaces.

The following is a useful result on ranks of restriction and contraction of
vector spaces and on relating restriction and contraction through
orthogonality [12, 7].

######  Theorem 2.

1\. $r({\cal V}_{SP})=r({\cal V}_{SP}\circ S)+r({\cal V}_{SP}\times P).$ 2\.
$\mbox{$\cal V$}_{SP}^{*}\circ P=(\mbox{$\cal V$}_{SP}\times P)^{*};$
$\mbox{$\cal V$}_{SP}^{*}\times S=(\mbox{$\cal V$}_{SP}\circ S)^{*}.$

###  2.4 KVL and KCL

By a ‘graph’ we mean a ‘directed graph’. Kirchhoff’s Voltage Law (KVL) for a
graph states that the sum of the signed voltages of edges around an oriented
loop is zero - the sign of the voltage of an edge being positive if the edge
orientation agrees with the orientation of the loop and negative if it
opposes. Kirchhoff’s Current Law (KCL) for a graph states that the sum of the
signed currents leaving a node is zero, the sign of the current of an edge
being positive if its positive endpoint is the node in question.

Let ${\cal G}$ be a graph with edge set $S.$ We refer to the space of vectors
$v_{S},$ which satisfy Kirchhoff’s Voltage Law (KVL) of the graph
$\mathcal{G},$ by $\boldsymbol{\mbox{$\cal V$}^{v}(\mathcal{G})}$ and to the
space of vectors $i_{S"},$ which satisfy Kirchhoff’s Current Law (KCL) of the
graph $\mathcal{G},$ by $\boldsymbol{\mbox{$\cal V$}^{i}(\mathcal{G})}.$ (We
need to deal with vectors of the kind $(v,i).$ To be consistent with our
notation for vectors as functions, this is treated as a vector
$(v_{S},i_{S"}),$ where $S"$ is a disjoint copy of $S.$ Therefore
$\boldsymbol{\mbox{$\cal V$}^{v}(\mathcal{G})}$ is taken to be on set $S$ and
$\boldsymbol{\mbox{$\cal V$}^{i}(\mathcal{G})}$ is taken to be on a disjoint
copy $S".$)

The following is a fundamental result on vector spaces associated with graphs.

######  Theorem 3.

Tellegen’s Theorem [10] Let ${\cal G}$ be a graph with edge set S, Then
$\mbox{$\cal V$}^{i}(\mathcal{G})=(\mbox{$\cal
V$}^{v}(\mathcal{G})^{*})_{S"}.$

###  2.5 Networks and multiports

An electrical network $\mathcal{N},$ or a ‘network’ in short, is a pair
$(\mathcal{G},\mathcal{K}),$ where $\mathcal{G}:\equiv(V({\cal G}),E({\cal
G}))$ is a graph and $\mathcal{K},$ called the device characteristic of the
network, is a collection of pairs of vectors $(v_{S},i_{S"}),S:\equiv E({\cal
G}),$ where $S,S"$ are disjoint copies of $S,\ $ $v_{S},i_{S"}$ are real or
complex vectors on the edge set of the graph. In this paper, we deal only with
affine device characteristics and with complex vectors, unless otherwise
stated. When the device characteristic ${\cal K}_{SS"}$ is affine, we denote
it by ${\cal A}_{SS"}$ and say the network is linear. If $\mbox{$\cal
V$}_{SS"}$ is the vector space translate of ${\cal A}_{SS"},$ we say that
${\cal A}_{SS"}$ is the source accompanied form of $\mbox{$\cal V$}_{SS"}.$ An
affine space ${\cal A}_{SS"}$ is said to be proper iff its vector space
translate $\mbox{$\cal V$}_{SS"}$ has dimension $|S|=|S"|.$

Let $S$ denote the set of edges of the graph of the network and let
$\\{S_{1},\cdots,S_{k}\\}$ be a partition of $S,$ each block $S_{j}$ being an
individual device. Let $S,S"$ be disjoint copies of $S,$ with $e,e"$
corresponding to edge $e.$ The device characteristic would usually have the
form $\bigoplus{\cal A}_{S_{j}S_{j}"},$ defined by
$(B_{j}v_{S_{j}}+Q_{j}i_{S_{j}"})=s_{j},$ with rows of $(B_{j}|Q_{j})$ being
linearly independent. The vector space translate would have the form
$\bigoplus\mbox{$\cal V$}_{S_{j}S_{j}"},$ $\mbox{$\cal V$}_{S_{j}S_{j}"}$
being the translate of ${\cal A}_{S_{j}S_{j}"}.$ Further, usually the blocks
$S_{j}$ would have size one or two, even when the network has millions of
edges. Therefore, it would be easy to build $(\bigoplus\mbox{$\cal
V$}_{S_{j}S_{j}"})^{*}=\bigoplus\mbox{$\cal V$}_{S_{j}S_{j}"}^{*}.$
We say $S_{j}$ is a set of norators iff ${\cal
A}_{S_{j}S_{j}"}:\equiv\mbox{$\cal F$}_{S_{j}S_{j}"},$ i.e., there are no
constraints on $v_{S_{j}},i_{S_{j}"}.$
We say $S_{j}$ is a set of nullators iff ${\cal
A}_{S_{j}S_{j}"}:\equiv{\mathbf{0}}_{S_{j}S_{j}"},$ i.e.,
$v_{S_{j}}=i_{S_{j}"}=0.$
We say $\mbox{$\cal V$}_{S_{j}\hat{S}_{j}S_{j}"\hat{S}_{j}"}$ is a gyrator iff
$v_{S_{j}}=-Ri_{\hat{S}_{j}"};v_{\hat{S}_{j}}=Ri_{S_{j}"},$ where $R$ is a
positive diagonal matrix. We denote by $\mbox{${\bf g}$}^{S_{j}\hat{S}_{j}},$
the gyrator where $R$ is the identity matrix.

A solution of $\mathcal{N}:\equiv({\cal G},{\cal K})$ on graph ${\cal
G}:\equiv(V({\cal G}),E({\cal G}))$ is a pair $(v_{S},i_{S"}),S:\equiv E({\cal
G})$ satisfying
$v_{S}\in\mbox{$\cal V$}^{v}(\mathcal{G}),i_{S"}\in\mbox{$\cal
V$}^{i}(\mathcal{G})$ (KVL,KCL) and $(v_{S},i_{S"})\in\mathcal{K}.$ The
KVL,KCL conditions are also called topological constraints. Let $S,S"$ be
disjoint copies of $S,$ let $\mbox{$\cal V$}_{S}:\equiv\mbox{$\cal
V$}^{v}(\mathcal{G}),$ so that, by Theorem 3, $(\mbox{$\cal
V$}^{*}_{S})_{S"}=\mbox{$\cal V$}^{i}(\mathcal{G}),$ and let ${\cal K}_{SS"}$
be the device characteristic of ${\cal N}.$ The set of solutions of
$\mathcal{N}$ may be written as,

$\mbox{$\cal V$}_{S}\cap(\mbox{$\cal V$}^{*}_{S})_{S"}\cap{\cal
K}_{SS"}=[\mbox{$\cal V$}_{S}\oplus(\mbox{$\cal V$}^{*}_{S})_{S"}]\cap{\cal
K}_{SS"}.$

This has the form ‘[Solution set of topological constraints] $\cap$ [Device
characteristics]’.

A multiport $\mathcal{N}_{P}:\equiv({\cal G}_{SP},{\cal K}_{SS"})$ is a
network with some subset $P$ of its edges which are norators, specified as
‘ports’. Let ${\cal N}_{P}$ be on graph ${\cal G}_{SP}$ with device
characteristic ${\cal K}.$ Let $\mbox{$\cal V$}_{SP}:\equiv\mbox{$\cal
V$}^{v}({\cal G}_{SP}),$ so that $(\mbox{$\cal
V$}^{*}_{SP})_{S"P"}=(\mbox{$\cal V$}^{i}({\cal G}_{SP}))_{S"P"},$ and let
${\cal K}_{SS"},$ be the device characteristic on the edge set $S.$ The device
characteristic of a multiport $\mathcal{N}_{P}$ would be ${\cal K}:\equiv{\cal
K}_{SS"}\oplus\mbox{$\cal F$}_{PP"}.$ For simplicity we would refer to ${\cal
K}_{SS"}$ as the device characteristic of ${\cal N}_{P}.$ The multiport is
said to be linear iff its device characteristic is affine. The set of
solutions of $\mathcal{N}_{P}:\equiv({\cal G}_{SP},{\cal K}_{SS"})$ may be
writen, using the extended definition of intersection as

$\mbox{$\cal V$}_{SP}\cap(\mbox{$\cal V$}^{*}_{SP})_{S"P"}\cap{\cal
K}_{SS"}=[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal
V$}^{*}_{SP})_{S"P"}]\cap{\cal K}_{SS"}.$

We say the multiport is consistent iff its set of solutions is nonvoid.

##  3 Matched and Skewed Composition

In this section we introduce an operation between collections of vectors
motivated by the need to capture the relationship between the port voltages
and currents in a multiport.

Let ${\cal K}_{SP},{\cal K}_{PQ},$ be collections of vectors respectively on
$S\uplus P,P\uplus Q,$ with $S,P,Q,$ being pairwise disjoint.

The matched composition
$\boldsymbol{\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{PQ}}$ is on $S\uplus
Q$ and is defined as follows:

$\displaystyle\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{PQ}$
$\displaystyle:\equiv\\{(f_{S},g_{Q}):(f_{S},h_{P})\in{\cal
K}_{SP},(h_{P},g_{Q})\in{\cal K}_{PQ}\\}.$

Matched composition is referred to as matched sum in [7]. It can be regarded
as the generalization of composition of maps to composition of relations [8].

The skewed composition
$\boldsymbol{\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{PQ}}$ is on
$S\uplus Q$ and is defined as follows:

$\displaystyle(\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{PQ})$
$\displaystyle:\equiv\\{(f_{S},g_{Q}):(f_{S},h_{P})\in{\cal
K}_{SP},(-h_{P},g_{Q})\in{\cal K}_{PQ}\\}.\ \mbox{Note that}$
$(\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{PQ})\ \ \ =\ \ \
\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{(-P)Q}.$

When $S$, $Y$ are disjoint, both the matched and skewed composition of ${\cal
K}_{S},{\cal K}_{Y},$ correspond to the direct sum ${\cal K}_{S}\oplus{\cal
K}_{Y}$. It is clear from the definition of matched composition and that of
restriction and contraction, that

${\cal K}_{SP}\leftrightarrow{\cal K}_{PQ}=({\cal K}_{SP}\cap{\cal
K}_{PQ})\circ SQ;\ \ {\cal K}_{SP}\rightleftharpoons{\cal K}_{PQ}\ \ =\ \
({\cal K}_{SP}\cap{\cal K}_{(-P)Q})\circ SQ;$ ${\cal
K}_{SP}\leftrightarrow{\cal K}_{PQ}=({\cal K}_{SP}+{\cal K}_{(-P)Q})\times
SQ;\ \ {\cal K}_{SP}\rightleftharpoons{\cal K}_{PQ}\ \ =\ \ ({\cal
K}_{SP}+{\cal K}_{PQ})\times SQ.$

Further, it can be seen that
${\cal K}_{S}\circ(S-T)={\cal K}_{S}\leftrightarrow\mbox{$\cal F$}_{T},{\cal
K}_{S}\times(S-T)={\cal K}_{S}\leftrightarrow{\mathbf{0}}_{T},T\subseteq S.$
When $\mathcal{K}_{SP}$, $\mathcal{K}_{P}$ are vector spaces, observe that
$(\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{P})=(\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{P}).$
When $S,P,Z,$ are pairwise disjoint, we have
$(\mathcal{K}_{SPZ}\leftrightarrow\mathcal{K}_{S})\leftrightarrow\mathcal{K}_{P}=(\mathcal{K}_{SPZ}\leftrightarrow\mathcal{K}_{P})\leftrightarrow\mathcal{K}_{S}=\mathcal{K}_{SPZ}\leftrightarrow(\mathcal{K}_{S}\oplus\mathcal{K}_{P}).$
When $\mathcal{K}_{S}:\equiv{\mathbf{0}}_{S},\mathcal{K}_{P}:\equiv\mbox{$\cal
F$}_{P},$ the above reduces to $\mathcal{K}_{SPZ}\times PZ\circ Z\hskip
2.84544pt=\hskip 2.84544pt\mathcal{K}_{SPZ}\circ SZ\times Z.$

The multiport $\mathcal{N}_{P}:\equiv({\cal G}_{SP},{\cal K}_{SS"})$ would
impose a relationship between $v_{P},i_{P"}.$ This relationship is captured by
the multiport behaviour (port behaviour for short) $\breve{{\cal K}}_{PP"}$ at
$P,$ of ${\cal N}_{P},$ defined by
$\breve{{\cal K}}_{PP"}:\equiv[([\mbox{$\cal V$}^{v}({\cal
G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal G}_{SP}))_{S"P"}]\cap{\cal
K}_{SS"})\circ PP"]_{P(-P")}=([\mbox{$\cal V$}^{v}({\cal
G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal G}_{SP}))_{S"P"}]\leftrightarrow{\cal
K}_{SS"})_{P(-P")}$
$=[\mbox{$\cal V$}^{v}({\cal G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal
G}_{SP}))_{S"(-P")}]\leftrightarrow{\cal K}_{SS"}.$ When the device
characteristic of $\mathcal{N}_{P}$ is affine, its port behaviour
$\breve{{\cal K}}_{PP"}$ at $P$ would be affine if it were not void.

Note that, if the multiport is a single port edge in parallel with a positive
resistor $R,$
$([\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"(-P")}]\cap{\cal
K}_{SS"})\circ PP"$ would be the solution of $v_{P}=-Ri_{P"}.$ But then
$\breve{{\cal K}}_{PP"},$ as defined, would be the solution of
$v_{P}=Ri_{P"}.$

Let the multiports ${\cal N}_{RP},{{\cal N}}_{\tilde{P}Q}$ be on graphs ${\cal
G}_{RSP},{\cal G}_{\tilde{P}MQ}$ respectively, with the primed and double
primed sets obtained from $R,S,P,\tilde{P},M,Q,$ being pairwise disjoint, and
let them have device characteristics ${\cal K}^{S},{{\cal K}}^{M}$
respectively. Let ${\cal K}^{P\tilde{P}}$ denote a collection of vectors
${\cal K}^{P\tilde{P}}_{P\tilde{P}P"\tilde{P}"}.$ The multiport
$\boldsymbol{[{\cal N}_{RP}\oplus{{\cal N}}_{\tilde{P}Q}]\cap{\cal
K}^{P\tilde{P}}},$ with ports $R\uplus Q$ obtained by connecting ${\cal
N}_{RP},{{\cal N}}_{\tilde{P}Q}$ through ${\cal K}^{P\tilde{P}}$, is on graph
${\cal G}_{RSP}\oplus{\cal G}_{\tilde{P}MQ}$ (the graph obtained by putting
${\cal G}_{RSP},{\cal G}_{\tilde{P}MQ}$ together with no common nodes) with
device characteristic
${\cal K}^{S}\oplus{{\cal K}}^{M}\oplus{\cal K}^{P\tilde{P}}.$ When $R,Q$ are
void, $[{\cal N}_{RP}\oplus{{\cal N}}_{\tilde{P}Q}]\cap{\cal K}^{P\tilde{P}}$
would reduce to $[{\cal N}_{P}\oplus{{\cal N}}_{\tilde{P}}]\cap{\cal
K}^{P\tilde{P}}$ and would be a network without ports. In this case we say the
multiport ${\cal N}_{P}$ is terminated by ${{\cal N}}_{\tilde{P}}$ through
${\cal K}^{P\tilde{P}}.$
This network is on graph ${\cal G}_{SP}\oplus{\cal G}_{\tilde{P}M}$ with
device characteristic ${\cal K}^{S}_{SS"}\oplus{{\cal K}}^{M}_{MM"}\oplus{\cal
K}^{P\tilde{P}}_{PP"\tilde{P}\tilde{P}"}.$

The following result is useful for relating the port behaviour of a multiport
with internal sources to that of the source zero version of the multiport. Its
routine proof is omitted.

######  Theorem 4.

Let ${\cal A}_{SP},{\cal A}_{PQ}$ be affine spaces on $S\uplus P,P\uplus Q,$
where $S,P,Q$ are pairwise disjoint sets. Let ${\cal V}_{SP},{\cal V}_{PQ}$
respectively, be the vector space translates of ${\cal A}_{SP},{\cal A}_{PQ}.$
Let ${\cal A}_{SP}\leftrightarrow{\cal A}_{PQ}$ be nonvoid and let
$\alpha_{SQ}\in{\cal A}_{SP}\leftrightarrow{\cal A}_{PQ}.$ Then, ${\cal
A}_{SP}\leftrightarrow{\cal A}_{PQ}=\alpha_{SQ}+({\cal
V}_{SP}\leftrightarrow{\cal V}_{PQ}).$

The following result is an immediate consequence of Theorem 4.

######  Theorem 5.

Let ${\cal N}^{1}_{P},{{\cal N}}^{2}_{P}$ be multiports on the same graph
${\cal G}_{SP}$ but with device characteristics
${\cal A}_{SS"},{\mbox{$\cal V$}}_{SS"},$ respectively where ${\mbox{$\cal
V$}}_{SS"},$ is the vector space translate of the affine space ${\cal
A}_{SS"}.$ Let ${\cal N}^{1}_{P},{{\cal N}}^{2}_{P}$ have port behaviours
$\breve{{\cal A}}^{1}_{PP"},\breve{\mbox{$\cal V$}}^{2}_{PP"},$ respectively.
If $\breve{{\cal A}}^{1}_{PP"}\neq\emptyset,$ then
$\breve{\mbox{$\cal V$}}^{2}_{PP"}=([\mbox{$\cal V$}^{v}({{\cal
G}_{SP}})\oplus(\mbox{$\cal V$}^{i}({{\cal
G}_{SP}}))_{S"P"}]\leftrightarrow{\mbox{$\cal V$}}_{SS"})_{P(-P")},$ is the
vector space translate of $\breve{{\cal A}}^{1}_{PP"}.$

Implicit Duality Theorem, given below, is a part of network theory folklore.
However, its applications are insufficiently emphasized in the literature.
Proofs and applications may be found in [6, 7, 13, 8]. A version in the
context of Pontryagin Duality is available in [3].

######  Theorem 6.

Implicit Duality Theorem Let ${\cal V}_{SP},{\cal V}_{PQ}$ be vector spaces
respectively on $S\uplus P,P\uplus Q,$ with $S,P,Q,$ being pairwise disjoint.
We then have, $(\mathcal{V}_{SP}\leftrightarrow\mathcal{V}_{PQ})^{*}\ \hskip
2.84544pt=\hskip 2.84544pt\
(\mathcal{V}_{SP}^{*}\rightleftharpoons\mathcal{V}_{PQ}^{*}).$ In particular,
$(\mathcal{V}_{SP}\leftrightarrow\mathcal{V}_{P})^{*}\ \hskip 2.84544pt=\hskip
2.84544pt\ \mathcal{V}_{SP}^{*}\leftrightarrow\mathcal{V}_{P}^{*}.$

###### Proof.

It can be shown that $(\mbox{$\cal V$}_{X}+\mbox{$\cal
V$}_{Y})^{*}=(\mbox{$\cal V$}^{*}_{X}\cap\mbox{$\cal V$}_{Y}^{*})$ and, using
Theorem 1, that $(\mbox{$\cal V$}_{X}\cap\mbox{$\cal V$}_{Y})^{*}=(\mbox{$\cal
V$}^{*}_{X}+\mbox{$\cal V$}_{Y}^{*}).$ We have, using Theorem 2, $({\cal
V}_{SP}\leftrightarrow{\cal V}_{PQ})^{*}=[({\cal V}_{SP}\cap{\cal
V}_{PQ})\circ SQ]^{*}=[({\cal V}_{SP}^{*}+{\cal V}_{PQ}^{*})\times SQ]=({\cal
V}_{SP}^{*}\rightleftharpoons{\cal V}_{PQ}^{*}).$ For any vector space
$\mbox{$\cal V$}_{X},$ we have $\mbox{$\cal V$}_{X}=\mbox{$\cal V$}_{(-X)}.$
Therefore
$(\mathcal{V}_{SP}\leftrightarrow\mathcal{V}_{P})^{*}=(\mathcal{V}_{SP}^{*}\rightleftharpoons\mathcal{V}_{P}^{*})=\mathcal{V}_{SP}^{*}\leftrightarrow\mathcal{V}_{P}^{*}.$
∎

An illustration of the use of Theorem 6 is provided in the next subsection
(Theorem 8).

###  3.1 Adjoint of an affine space and the adjoint multiport

We say $\mbox{$\cal V$}_{S_{j}S_{j}"},\hat{\mbox{$\cal V$}}_{S_{j}S_{j}"}$ are
orthogonal duals of each other iff $\hat{\mbox{$\cal
V$}}_{S_{j}S_{j}"}=\mbox{$\cal V$}^{*}_{S_{j}S_{j}"}.$ By Theorem 1,
$(\mbox{$\cal V$}^{*}_{S_{j}S_{j}"})^{*}=\mbox{$\cal V$}_{S_{j}S_{j}"}.$ Let
${\cal A}_{S_{j}S_{j}"}$ be an affine space with $\mbox{$\cal
V$}_{S_{j}S_{j}"}$ as its vector space translate. We say $\hat{\mbox{$\cal
V$}}_{S_{j}S_{j}"}$ is the adjoint of ${\cal A}_{S_{j}S_{j}"},$ denoted by
${\mbox{$\cal V$}}^{adj}_{S_{j}S_{j}"},$ iff $\hat{\mbox{$\cal
V$}}_{S_{j}S_{j}"}=(\mbox{$\cal V$}^{*}_{S_{j}S_{j}"})_{(-S_{j}")S_{j}}.$ It
is easy to see that $(\oplus{\cal A}_{S_{j}S_{j}"})^{adj}=(\oplus\mbox{$\cal
V$}_{S_{j}S_{j}"}^{adj}).$ It is clear that ${\mbox{$\cal
V$}}_{S_{j}S_{j}"}=(\hat{\mbox{$\cal
V$}}^{*}_{S_{j}S_{j}"})_{(-S_{j}")S_{j}}=({\mbox{$\cal
V$}}^{adj}_{S_{j}S_{j}"})^{adj}.$

######  Example 7.

Let ${{\cal A}}_{SS"}$ be an affine device characteristic. Suppose its vector
space translate ${\mbox{$\cal V$}}_{SS"}$ is defined by (is the solution space
of) the hybrid equations

$\displaystyle\begin{pmatrix}I&0&-g_{11}&-h_{12}\\\
0&I&-h_{21}&-r_{22}\end{pmatrix}\begin{pmatrix}i_{S_{1}"}\\\ v_{S_{2}}\\\
v_{S_{1}}\\\ i_{S_{2}"}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},$ (4)

where $S$ is partitioned into $S_{1},S_{2}.$ Then ${\mbox{$\cal
V$}}^{*}_{SS"}$ is the complementary orthogonal space defined by

$\displaystyle\begin{pmatrix}g^{*}_{11}&h^{*}_{21}&I&0\\\
h^{*}_{12}&r^{*}_{22}&0&I\end{pmatrix}\begin{pmatrix}\ \ \tilde{i}_{S_{1}"}\\\
\tilde{v}_{S_{2}}\\\ \tilde{v}_{S_{1}}\\\
\tilde{i}_{S_{2}"}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},$ (5)

and ${\mbox{$\cal V$}}^{adj}_{SS"}:\equiv({\mbox{$\cal
V$}}^{*}_{SS"})_{(-S")S}$ is obtained by replacing the $\tilde{v}$ variables
by $-\hat{i}$ variables and the $\tilde{i}$ variables by $\hat{v}$ variables.
It can be seen that ${\mbox{$\cal V$}}^{adj}_{SS"}$ is defined by

$\displaystyle\begin{pmatrix}g^{*}_{11}&h^{*}_{21}&I&0\\\
h^{*}_{12}&r^{*}_{22}&0&I\end{pmatrix}\begin{pmatrix}\hat{v}_{S_{1}}\\\
-\hat{i}_{S_{2}"}\\\ -\hat{i}_{S_{1}"}\\\
\hat{v}_{S_{2}}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},\mbox{i.e.,}\begin{pmatrix}-g^{*}_{11}&h^{*}_{21}&I&0\\\
h^{*}_{12}&-r^{*}_{22}&0&I\end{pmatrix}\begin{pmatrix}\hat{v}_{S_{1}}\\\
\hat{i}_{S_{2}"}\\\ \hat{i}_{S_{1}"}\\\
\hat{v}_{S_{2}}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ (6)

The individual devices which are present will usually have very few ports.
Building their adjoints is therefore computationally inexpensive.
1\. Let $v=0,i=0$ be a nullator. The adjoint will have its voltage and current
unconstrained and is therefore a norator.
2\. Let $v_{S}=Zi_{S"}$ be a (multiport) impedance. The adjoint has the
characteristic $\hat{v}_{S}=Z^{*}\hat{i}_{S"}.$
3\. Consider the current controlled voltage source (CCVS) and the voltage
controlled current source (VCCS) shown below.

$\displaystyle\begin{pmatrix}1&0&0&0\\\
0&1&-r&0\end{pmatrix}\begin{pmatrix}v_{1}\\\ v_{2}\\\ i_{1}\\\
i_{2}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix};\begin{pmatrix}0&0&1&0\\\
-g&0&0&1\end{pmatrix}\begin{pmatrix}v_{1}\\\ v_{2}\\\ i_{1}\\\
i_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix};$ (7)

The adjoints are respectively (built by first building the orthogonal dual of
the source zero characteristic, interchanging current and voltage variables
and changing the sign of the current variables)

$\displaystyle\begin{pmatrix}0&-r&1&0\\\
0&0&0&1\end{pmatrix}\begin{pmatrix}\hat{i}_{1}\\\ \hat{i}_{2}\\\
\hat{v}_{1}\\\ \hat{v}_{2}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix};\begin{pmatrix}1&0&0&-g\\\
0&1&0&0\end{pmatrix}\begin{pmatrix}\hat{i}_{1}\\\ \hat{i}_{2}\\\
\hat{v}_{1}\\\ \hat{v}_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$
(8)

Thus the adjoints of CCVS, VCCS remain CCVS,VCCS respectively, with the same
parameters $r,g$ respectively, but the direction of control which was
originally from port $1$ to port $2$ is now from port $2$ to port $1.$
4\. Next consider current controlled current source (CCCS) and the voltage
controlled voltage source (VCVS) shown below.

$\displaystyle\begin{pmatrix}1&0&0&0\\\
0&0&-\alpha&1\end{pmatrix}\begin{pmatrix}v_{1}\\\ v_{2}\\\ i_{1}\\\
i_{2}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix};\begin{pmatrix}0&0&1&0\\\
-\beta&1&0&0\end{pmatrix}\begin{pmatrix}v_{1}\\\ v_{2}\\\ i_{1}\\\
i_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ (9)

The adjoints are respectively,

$\displaystyle\begin{pmatrix}1&0&0&0\\\
0&0&1&\alpha\end{pmatrix}\begin{pmatrix}\hat{i}_{1}\\\ \hat{i}_{2}\\\
\hat{v}_{1}\\\ \hat{v}_{2}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix};\begin{pmatrix}0&0&1&0\\\
1&\beta&0&0\end{pmatrix}\begin{pmatrix}\hat{i}_{1}\\\ \hat{i}_{2}\\\
\hat{v}_{1}\\\ \hat{v}_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$
(10)

Thus the adjoints of CCCS, VCVS are source zero VCVS,CCCS respectively, but
the direction of control which was originally from port $1$ to port $2$ is now
from port $2$ to port $1.$ Also the current gain factor is now the negative of
the voltage gain factor in the CCCS to VCVS case and vice versa in VCVS to
CCCS case.

Let the multiport $\boldsymbol{{{\cal N}}_{P}}$ be on graph ${\cal G}_{SP}$
with device characteristic ${\cal A}_{SS"}=x_{SS"}+{\mbox{$\cal V$}}_{SS"}$ on
$S.$ The multiport $\boldsymbol{{{\cal N}}^{hom}_{P}}$ is on graph ${\cal
G}_{SP}$ but has device characteristic ${\mbox{$\cal V$}}_{SS"}.$ The adjoint
${\cal N}^{adj}_{P}$ of ${{\cal N}}_{P}$ as well as of ${{\cal N}}^{hom}_{P}$
is on graph ${\cal G}_{SP}$ but has device characteristic ${\mbox{$\cal
V$}}^{adj}_{SS"}.$

Let the solution set of ${\cal N}_{P}$ be $[\mbox{$\cal
V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"}]\cap{\cal A}_{SS"}.$ Then the
solution set of ${{\cal N}}^{hom}_{P}$ would be $[\mbox{$\cal
V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"}]\cap{\mbox{$\cal V$}}_{SS"},$
and that of ${\cal N}^{adj}_{P}$ would be $[\mbox{$\cal
V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"}]\cap{\mbox{$\cal
V$}}^{adj}_{SS"}.$

The port behaviour of ${\cal N}_{P}$ would be $[\mbox{$\cal
V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"(-P")}]\leftrightarrow{\cal
A}_{SS"},$ that of ${{\cal N}}^{hom}_{P}$ would be
$[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal
V$}^{*}_{SP})_{S"(-P")}]\leftrightarrow{\mbox{$\cal V$}}_{SS"},$ and that of
${\cal N}^{adj}_{P}$ would be $[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal
V$}^{*}_{SP})_{S"(-P")}]\leftrightarrow{\mbox{$\cal V$}}^{adj}_{SS"}.$

We now have the following basic result on linear multiports [7, 6]. It
essentially states that, ${\cal N}^{hom}_{P}$ and ${\cal N}^{adj}_{P}$ have
adjoint port behaviours.

######  Theorem 8.

Let ${\cal N}^{1}_{P},{{\cal N}}^{2}_{P}$ be multiports on the same graph
${\cal G}_{SP}$ but with device characteristics
$\mbox{$\cal V$}^{1}_{SS"},{\mbox{$\cal V$}}^{2}_{SS"},$ respectively and port
behaviours $\breve{\mbox{$\cal V$}}^{1}_{PP"},\breve{\mbox{$\cal
V$}}^{2}_{PP"},$ respectively. Then if $\mbox{$\cal V$}^{1}_{SS"},{\mbox{$\cal
V$}}^{2}_{SS"},$ are adjoints of each other so are $\breve{\mbox{$\cal
V$}}^{1}_{PP"},\breve{\mbox{$\cal V$}}^{2}_{PP"},$ adjoints of each other.

###### Proof.

Let $\mbox{$\cal V$}_{SP}:\equiv\mbox{$\cal V$}^{v}({\cal G}_{SP}).$ By
Theorem 3, we have $((\mbox{$\cal V$}^{v}({\cal
G}_{SP}))^{*})_{S"P"}=\mbox{$\cal V$}^{i}({\cal G}_{SP}).$
Therefore $(\mbox{$\cal V$}_{SP}^{*})_{S"P"}=\mbox{$\cal V$}^{i}({\cal
G}_{SP}).$ Let $\mbox{$\cal V$}_{SPS"P"}:\equiv\mbox{$\cal
V$}_{SP}\oplus(\mbox{$\cal V$}_{SP}^{*})_{S"P"}.$ It is clear that
$\mbox{$\cal V$}_{SPS"P"}^{adj}$
$:\equiv(\mbox{$\cal V$}_{SPS"P"}^{*})_{(-S")(-P")SP}=\mbox{$\cal
V$}_{SPS"P"}.$ Let $\mbox{$\cal V$}^{1}_{PP"}:\equiv\mbox{$\cal
V$}_{SPS"P"}\leftrightarrow\mbox{$\cal V$}^{1}_{SS"}$ and let ${\mbox{$\cal
V$}^{2}}_{PP"}:\equiv{\mbox{$\cal V$}}_{SPS"P"}\leftrightarrow{\mbox{$\cal
V$}}^{2}_{SS"}.$
We then have, $(\mbox{$\cal V$}^{1}_{PP"})^{adj}:\equiv(\mbox{$\cal
V$}^{1}_{PP"})^{*}_{(-P")P}=(\mbox{$\cal
V$}_{SPS"P"}\leftrightarrow\mbox{$\cal V$}_{SS"})^{*}_{(-P")P}=(\mbox{$\cal
V$}_{SPS"P"}^{*}\leftrightarrow\mbox{$\cal V$}_{SS"}^{*})_{(-P")P}$
$=(\mbox{$\cal V$}_{SPS"P"}^{*})_{S(-P")S"P}\leftrightarrow\mbox{$\cal
V$}_{SS"}^{*}$ $=(\mbox{$\cal
V$}_{SPS"P"}^{*})_{(-S")(-P")SP}\leftrightarrow(\mbox{$\cal
V$}_{SS"}^{*})_{(-S")S}={\mbox{$\cal V$}}_{SPS"P"}\leftrightarrow{\mbox{$\cal
V$}^{2}}_{SS"}={\mbox{$\cal V$}^{2}}_{PP"}.$ Thus $\mbox{$\cal
V$}^{1}_{PP"},{\mbox{$\cal V$}}^{2}_{PP"}$ are adjoints and therefore
$\breve{\mbox{$\cal V$}}^{1}_{PP"}:\equiv(\mbox{$\cal V$}^{1}_{PP"})_{P(-P")}$
and $\breve{\mbox{$\cal V$}}^{2}_{PP"}:\equiv({\mbox{$\cal
V$}}^{2}_{PP"})_{P(-P")}$ are adjoints. ∎

######  Corollary 9.

Let ${\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{SS"}),$ with $\mbox{$\cal
V$}_{SS"}$ as the vector space translate of ${\cal A}_{SS"}.$ Suppose
$\breve{{\cal A}}_{PP"}$ is the (nonvoid) port behaviour of ${\cal N}_{P}$
with $\breve{\mbox{$\cal V$}}_{PP"}$ as its vector space translate. Then
${\cal N}^{adj}_{P}:\equiv({\cal G}_{SP},\mbox{$\cal V$}^{adj}_{SS"}),$ has
the port behaviour $\breve{\mbox{$\cal V$}}^{adj}_{PP"}.$

###### Proof.

By Theorem 5, port behaviour of ${\cal N}^{hom}_{P}:\equiv({\cal
G}_{SP},\mbox{$\cal V$}_{SS"}),$ is the vector space translate of
$\breve{{\cal A}}_{PP"}.$ By definition, the adjoint of ${\cal N}_{P}$ and
${\cal N}^{hom}_{P}$ are the same. The result now follows from Theorem 8,
noting that $(\breve{\mbox{$\cal V$}}^{adj}_{PP"})_{P(-P")}$ is the adjoint of
$(\breve{{\cal A}}_{PP"})_{P(-P")}.$ ∎

######  Remark 1.

1\. Suppose the original port behaviour ${\cal A}_{PP"}$ is defined by
$v_{P}=Zi_{P}+E.$ Its vector space translate $\mbox{$\cal V$}_{PP"}$ is
defined by $v_{P}=Zi_{P}.$ i.e., by

$\displaystyle(I|-Z)\begin{pmatrix}v_{P}\\\ i_{P"}\end{pmatrix}=0.$ (11)

$\mbox{$\cal V$}^{adj}_{PP"}$ is defined by

$\displaystyle(I|-Z^{*})\begin{pmatrix}v_{P}\\\ i_{P"}\end{pmatrix}=0,$ (12)

i.e., by $v_{P}=Z^{*}i_{P}.$ Now let every device in the multiport have the
form $v_{S_{j}}=Z_{j}i_{S_{j}"}+E_{j}$ and, further, at the ports let the
behaviour be $v_{P}=Zi_{P}+E.$ Theorem 8 implies that if every device is
replaced by $v_{S_{j}}=Z^{*}_{j}i_{S_{j}"},$ at the ports the behaviour would
be $v_{P}=Z^{*}i_{P}.$ But, as the theorem indicates, the idea of the adjoint
works well even if we have only a relationship of the kind $Bv_{P}-Qi_{P}=s.$

2\. The devices in a multiport have few device ports. So building their
adjoints is easy. Therefore ${\cal N}^{adj}_{P}$ can be built essentially in
linear time. Thus we can “implicitly” build, essentially in linear time, the
adjoint of the port behaviour of ${\cal N}_{P}.$ We use “implicitly” because
both the port behaviours are available not as explicit equations, but in terms
of multiports. This fact is exploited subsequently to generalize Thevenin-
Norton and maximum power transfer theorems.

##  4 Generalization of Thevenin-Norton Theorem

The characteristic feature of the Thevenin-Norton Theorem is that it is in
terms of repeated solution of networks obtained by suitable termination of a
given multiport. These networks are assumed to have unique solutions. Our
generalization of the Thevenin-Norton theorem is in accordance with this
feature and requires the notion of a rigid multiport. If a multiport is not
rigid, it cannot be a part of a network with unique solution and therefore we
cannot use a conventional simulator to solve a network of which the multiport
is a part. A rigid multiport can permit any nonvoid port behaviour at its
ports. Therefore our generalization of the Thevenin-Norton theorem in terms of
rigid multiports is the best that is possible if conventional circuit
simulators are to be used to compute the port behaviour. Our technique is to
terminate a rigid multiport by its adjoint through a gyrator. We therefore
need to prove that the adjoint of a rigid multiport is rigid. We need the
development in the next subsection for this purpose.

###  4.1 Rigid Multiports

######  Definition 10.

Let multiport ${\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{S^{\prime}S"}),$
where ${\cal A}_{S^{\prime}S"}=\alpha_{S^{\prime}S"}+\mbox{$\cal
V$}_{S^{\prime}S"}.$
The multiport ${\cal N}_{P}$ is said to be rigid  iff every multiport
$\hat{{\cal N}}_{P}:\equiv({\cal G}_{SP},\hat{{\cal A}}_{S^{\prime}S"}),$
where $\hat{{\cal A}}_{S^{\prime}S"}=\hat{\alpha}_{S^{\prime}S"}+\mbox{$\cal
V$}_{S^{\prime}S"},$ has a non void set of solutions and has a unique solution
corresponding to every vector in its multiport behaviour.
Let ${\cal A}_{AB},{\cal A}_{B}$ be affine spaces on sets $A\uplus B,B,$
respectively, $A,B,$ disjoint. Further, let ${\cal A}_{AB},{\cal A}_{B}$ have
vector space translates $\mbox{$\cal V$}_{AB},\mbox{$\cal V$}_{B},$
respectively.
We say the pair $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ has the full sum property
iff $\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B}=\mbox{$\cal F$}_{B}.$
We say the pair $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ has the zero intersection
property iff $\mbox{$\cal V$}_{AB}\times B\cap\mbox{$\cal
V$}_{B}={\mathbf{0}}_{B}.$
We say that the pair $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ is rigid,  iff it has
the full sum property and the zero intersection property.

Multiport rigidity reduces to affine space pair rigidity and the adjoint of a
rigid multiport is also rigid.

######  Theorem 11.

Let $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ be a rigid pair and let $\mbox{$\cal
V$}_{AB},\mbox{$\cal V$}_{B}$ be the vector space translates of ${\cal
A}_{AB},{\cal A}_{B},$ respectively. Then

  1. 1.

The full sum property of $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ is equivalent to
$\hat{{\cal A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B}$ being nonvoid, whenever
$\mbox{$\cal V$}_{AB},\mbox{$\cal V$}_{B}$ are the vector space translates of
$\hat{{\cal A}}_{AB},\hat{{\cal A}}_{B},$ respectively. Further, when the full
sum property holds for $\\{{\cal A}_{AB},{\cal A}_{B}\\},$ $\hat{{\cal
A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B}$ has vector space translate
$\mbox{$\cal V$}_{AB}\leftrightarrow\mbox{$\cal V$}_{B}.$

  2. 2.

The zero intersection property of $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ is
equivalent to the statement that ,
if $f_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B}$ and
$(f_{A},f_{B}),(f_{A},f^{\prime}_{B})\in{\cal A}_{AB}\cap{\cal A}_{B},$ then
$f_{B}=f^{\prime}_{B}.$

  3. 3.

The pair $\\{{\cal A}_{AB},{\cal A}_{B}\\}$ has the zero intersection (full
sum) property iff $\\{\mbox{$\cal V$}^{*}_{AB},\mbox{$\cal V$}^{*}_{B}\\}$ has
the full sum (zero intersection) property. Therefore $\\{{\cal A}_{AB},{\cal
A}_{B}\\}$ is rigid iff $\\{\mbox{$\cal V$}^{*}_{AB},\mbox{$\cal
V$}^{*}_{B}\\}$ is rigid.

  4. 4.

A multiport ${\cal N}_{P}$ is rigid iff ${\cal N}^{hom}_{P}$ and ${\cal
N}^{adj}_{P}$ are rigid.

  5. 5.

A network ${\cal N}:\equiv({\cal G}_{S},{\cal A}_{SS"}),$ where ${\cal
A}_{SS"}$ is proper, has a unique solution iff ${\cal N}^{hom}:\equiv({\cal
G}_{S},\mbox{$\cal V$}_{SS"}),$ where $\mbox{$\cal V$}_{SS"}$ is the vector
space translate of ${\cal A}_{SS"},$ has a unique solution.

###### Proof.

1\. Let $\hat{{\cal A}}_{AB}=(\alpha_{A},\alpha_{B})+\mbox{$\cal
V$}_{AB},\hat{{\cal A}}_{B}=\beta_{B}+\mbox{$\cal V$}_{B}.$
By the definition of matched composition, $\hat{{\cal
A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B}$ is nonvoid iff $\hat{{\cal
A}}_{AB}\cap\hat{{\cal A}}_{B}$ is nonvoid,
i.e., iff $\hat{{\cal A}}_{AB}\circ B\ \cap\ \hat{{\cal A}}_{B}$ is nonvoid,
i.e., iff there exist $\lambda_{B}\in\mbox{$\cal V$}_{AB}\circ
B,\sigma_{B}\in\mbox{$\cal V$}_{B},$ such that
$\alpha_{B}+\lambda_{B}=\beta_{B}+\sigma_{B},$ i.e., such that
$\alpha_{B}-\beta_{B}=\sigma_{B}-\lambda_{B}.$
Clearly when $\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B}=\mbox{$\cal
F$}_{B},$ there exist $\lambda_{B}\in\mbox{$\cal V$}_{AB}\circ
B,\sigma_{B}\in\mbox{$\cal V$}_{B},$ such that
$\alpha_{B}-\beta_{B}=\sigma_{B}-\lambda_{B}$ so that $\hat{{\cal
A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B}$ is nonvoid.
On the other hand, if $\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal
V$}_{B}\neq\mbox{$\cal F$}_{B},$ there exist $\alpha_{B},\beta_{B}$ such that
$\alpha_{B}-\beta_{B}\notin\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B},$
so that $\hat{{\cal A}}_{AB}\cap\hat{{\cal A}}_{B}$ and therefore $\hat{{\cal
A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B}$ is void.
By Theorem 4, if $\hat{{\cal A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B}$ is
nonvoid, its vector space translate is $\mbox{$\cal
V$}_{AB}\leftrightarrow\mbox{$\cal V$}_{B}.$

2\. Let $(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal
V$}_{B}={\mathbf{0}}_{B}.$ If $f_{A}\in{\cal A}_{AB}\leftrightarrow{\cal
A}_{B}$ and $(f_{A},f_{B}),(f_{A},f^{\prime}_{B})\in{\cal A}_{AB}\cap{\cal
A}_{B},$ then
$(0_{A},(f_{B}-f^{\prime}_{B}))\in\mbox{$\cal V$}_{AB},$ and similarly
$(f_{B}-f^{\prime}_{B})\in\mbox{$\cal V$}_{B},$ so that
$(f_{B}-f^{\prime}_{B})\in(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal
V$}_{B}={\mathbf{0}}_{B}.$
Next suppose whenever $f_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B}$ and
$(f_{A},f_{B}),(f_{A},f^{\prime}_{B})\in{\cal A}_{AB}\cap{\cal A}_{B},$ we
have $f_{B}=f^{\prime}_{B}.$ Suppose $(\mbox{$\cal V$}_{AB}\times
B)\cap\mbox{$\cal V$}_{B}\neq{\mathbf{0}}_{B}.$ Let $g_{B}\in(\mbox{$\cal
V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B}$ and let $g_{B}\neq 0_{B}.$
If $f_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B},$ then there exists
$f_{B}$ such that $(f_{A},f_{B})\in{\cal A}_{AB}$ and $f_{B}\in{\cal A}_{B}.$
Clearly
$(f_{A},f_{B})+(0_{A},g_{B})=(f_{A},f_{B}+g_{B})\in{\cal A}_{AB}$ and
$(f_{B}+g_{B})\in{\cal A}_{B},$ so that we must have
$(f_{A},f_{B}+g_{B})\in{\cal A}_{AB}\cap{\cal A}_{B}.$ But this means
$f_{B}=f_{B}+g_{B},$ a contradiction. We conclude that $(\mbox{$\cal
V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B}={\mathbf{0}}_{B}.$

3\. We have, $(\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal
V$}_{B})^{*}=\mbox{$\cal V$}_{AB}^{*}\times B\ \cap\ \mbox{$\cal
V$}_{B}^{*}=\mbox{$\cal F$}^{*}_{B}={\mathbf{0}}_{B},$ and
$(\mbox{$\cal V$}_{AB}\times B\ \cap\ \mbox{$\cal V$}_{B})^{*}=\mbox{$\cal
V$}_{AB}^{*}\circ B+\mbox{$\cal V$}_{B}^{*}={\mathbf{0}}^{*}_{B}=\mbox{$\cal
F$}_{B}.$

4\. By part 1 and 2 above, the rigidity of ${\cal N}_{P}:\equiv({\cal
G}_{SP},{\cal A}_{SS"})$ is equivalent to the rigidity of
$(\mbox{$\cal V$}^{v}({\cal G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal
G}_{SP}))_{S"P"},\mbox{$\cal V$}_{SS"}).$ By part 3, the rigidity of the
latter is equivalent to the rigidity of
$((\mbox{$\cal V$}^{v}({\cal G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal
G}_{SP}))_{S"P"})^{*},\mbox{$\cal V$}^{*}_{SS"}),$ i.e., to the rigidity of
$((\mbox{$\cal V$}^{i}({\cal G}_{SP}))_{SP}\oplus(\mbox{$\cal V$}^{v}({\cal
G}_{SP}))_{S"P"},\mbox{$\cal V$}^{*}_{SS"})$
(using Theorem 3), i.e., to the rigidity of ${\cal
N}_{P}^{adj}:\equiv(\mbox{$\cal V$}^{v}({\cal G}_{SP})\oplus(\mbox{$\cal
V$}^{i}({\cal G}_{SP}))_{S"P"},(\mbox{$\cal V$}^{*}_{SS"})_{(-S")S}).$

5\. Let ${\cal N}:\equiv({\cal G}_{S},{\cal A}_{SS"}),{\cal
N}^{hom}:\equiv({\cal G}_{S},\mbox{$\cal V$}_{SS"}).$ We have $r(\mbox{$\cal
V$}^{v}({\cal G}_{S})\oplus(\mbox{$\cal V$}^{i}({\cal G}_{S}))_{S"})=|S|$
(using Theorems 1, 3) and $r(\mbox{$\cal V$}_{SS"})=|S|.$ Both ${\cal N}$ as
well as ${\cal N}^{hom}$ have the same $2|S|\times 2|S|$ coefficient matrix of
the network equations. The networks have unique solutions iff the matrix is
nonsingular. ∎

###  4.2 Terminating a multiport by its adjoint through a gyrator

A conventional linear circuit simulator can process only linear circuits with
proper device characteristics. Firther, unless the circuit has a unique
solution, the simulator will return an error message. A useful artifice for
processing a multiport through a conventional circuit simulator, is to
terminate it appropriately so that the resulting network, if it has a
solution, has a unique solution. This solution would also contain a solution
to the original multiport. We now describe this technique in detail.

Let $K,K"$ be representative matrices of vector spaces $\mbox{$\cal
V$},\mbox{$\cal V$}^{*},$ respectively. Consider the equation

$\displaystyle\begin{pmatrix}K\\\
K^{"}\end{pmatrix}\begin{pmatrix}x\end{pmatrix}$
$\displaystyle=\begin{pmatrix}s\\\ 0\end{pmatrix}.$ (13)

First note that the coefficient matrix, in Equation 13, has number of rows
equal to $r(\mbox{$\cal V$})+r(\mbox{$\cal V$}^{*})$ (Theorem 1) which is the
number of columns of the matrix. Next, suppose the rows are linearly
dependent. This would imply that a non trivial linear combination of the rows
is the zero vector, which, since the rows of $K,K"$ are linearly independent,
in turn implies that a nonzero vector $x,$ lies in the intersection of
complementary orthogonal complex vector spaces, i.e., satisfies $\langle
x,x\rangle=0,$ a contradiction. Thus, if a matrix is made up of two sets of
rows, which are representative matrices of complementary orthogonal vector
spaces, then it must be nonsingular. Therefore, the coefficient matrix in
Equation 13 is nonsingular.

Let the multiport behaviour $\breve{{\cal A}}_{PP"}$ be the solution space of
the equation $Bv_{P}-Qi_{P"}=s,$ with linearly independent rows and let
$\breve{\mbox{$\cal V$}}_{PP"}$ be the solution space of the equation
$Bv_{P}-Qi_{P"}=0.$ Let the dual multiport behaviour $\breve{\mbox{$\cal
V$}}^{*}_{PP"}$ be the solution space of equation $B^{"}v_{P}-Q^{"}i_{P"}=0,$
where the rows of $(B^{"}|-Q^{"})$ form a basis for the space complementary
orthogonal to the row space of $(B|-Q).$

The constraints of the two multiport behaviours together give the following
equation.

$\displaystyle\begin{pmatrix}B&\vdots&-Q\\\
B^{"}&\vdots&-Q^{"}\end{pmatrix}\begin{pmatrix}v_{P}\\\ i_{P"}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}s\\\ 0\end{pmatrix}.$ (14)

The first and second set of rows of the coefficient matrix of the above
equation are linearly independent and span complementary orthogonal spaces.
Therefore the coefficient matrix is invertible and the equation has a unique
solution. Now suppose we manage to terminate ${\cal N}_{P}$ by another
multiport ${\cal N}^{2}_{{P}}$ in such a way that the port voltage and current
vectors of ${\cal N}_{P}$ satisfy an equation of the kind (14) above. Then the
port voltage and current vectors of ${\cal N}_{P}$ would be unique and would
uniquely determine the port voltage and current vectors of ${\cal
N}^{2}_{{P}}.$ If both multiports are rigid, this would also uniquely
determine the internal voltage and current vectors of both ${\cal N}_{P}$ and
${\cal N}^{2}_{{P}},$ which means that the network, obtained by terminating
${\cal N}_{P}$ by ${\cal N}^{2}_{{P}},$ has a unique solution. We show below
that we can build ${\cal N}^{2}_{{P}}$ by first building ${\cal
N}^{adj}_{\tilde{P}}$ and attaching $1:1$ gyrators to its ports.

Let $\mathcal{N}_{P}$ be a rigid multiport on graph ${\cal G}_{SP}$ and device
characteristic ${{\cal A}}_{SS"},$ and let it have the port behaviour
$\breve{{\cal A}}_{PP"}.$ Let $\mathcal{N}^{adj}_{\tilde{P}}$ be on the copy
${\cal G}_{\tilde{S}\tilde{P}}$ of ${\cal G}_{SP},$ with device characteristic
$({\mbox{$\cal V$}}^{adj}_{SS"})_{\tilde{S}\tilde{S}"}.$ By the definition of
rigidity, the port behaviour $\breve{{\cal A}}_{PP"}$ is nonvoid and by
Theorem 11, $\mathcal{N}^{adj}_{\tilde{P}}$ is also rigid. We will now show
that the network
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ has a unique solution.

Let ${\mbox{$\cal V$}}_{SS"}$ be the vector space translate of ${{\cal
A}}_{SS"}$ and $\breve{\mbox{$\cal V$}}_{PP"}$ be that of $\breve{{\cal
A}}_{PP"}.$ We note that
$(\mbox{$\cal V$}^{v}({\cal G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal
G}_{SP}))_{S"P"})\leftrightarrow{{\cal A}}_{SS"}=(\breve{{\cal
A}}_{PP"})_{P(-P")}.$ Thus the constraints of ${\cal N}_{P}$ are equivalent,
as far as the variables $(v_{P},i_{P"})$ are concerned, to the first set of
equations of Equation 14.
Next, $(\mbox{$\cal V$}^{v}({\cal G}_{SP})\oplus(\mbox{$\cal V$}^{i}({\cal
G}_{SP}))_{S"P"})\leftrightarrow{\mbox{$\cal V$}}_{SS"}=(\breve{\mbox{$\cal
V$}}_{PP"})_{P(-P")},$ using Theorem 4. By Corollary 9, the port behaviour of
$\mathcal{N}^{adj}_{\tilde{P}}$ is $\breve{\mbox{$\cal
V$}}^{adj}_{\tilde{P}\tilde{P}"}:\equiv(\breve{\mbox{$\cal
V$}}^{adj}_{PP"})_{\tilde{P}\tilde{P}"}.$ Further, $(\breve{\mbox{$\cal
V$}}^{adj}_{\tilde{P}\tilde{P}"})_{{P}"(-{P})}=(\breve{\mbox{$\cal
V$}}^{adj}_{\tilde{P}\tilde{P}"})_{(-{P}"){P}}=\breve{\mbox{$\cal
V$}}^{*}_{PP"}.$ Thus the constraints of $\mathcal{N}^{adj}_{\tilde{P}}$
together with the constraint $v_{P}=-i_{\tilde{P}"};i_{P"}=v_{\tilde{P}}$ (the
gyrator $\mbox{${\bf g}$}^{{P}\tilde{P}}$), are equivalent, as far as the
variables $(v_{P},i_{P"})$ are concerned, to the defining equations of
$\breve{\mbox{$\cal V$}}^{*}_{PP"},$ i.e., to the second set of equations of
Equation 14.

Thus, the constraints of
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}},$ are equivalent as far as the variables $(v_{P},i_{P"})$
are concerned, to the constraints of Equation 14. We have seen that this
equation has a unique solution, say $(\hat{v}_{P},-\hat{i}_{P"}).$ Using the
gyrator constraints $v_{P}=-i_{\tilde{P}"};i_{P"}=v_{\tilde{P}},$ we get a
corresponding vector $(\tilde{v}_{\tilde{P}},\tilde{i}_{\tilde{P}"})$ that is
the restriction of a solution of $\mathcal{N}^{adj}_{\tilde{P}}$ to
$\tilde{P}\uplus\tilde{P}".$ Now $(\hat{v}_{P},-\hat{i}_{P"})\in\hat{{\cal
A}}_{PP"}.$ Therefore, $(\hat{v}_{P},\hat{i}_{P"})$ is the restriction of a
solution of ${\cal N}_{P}$ to $P\uplus P".$ By the rigidity of ${\cal N}_{P},$
there is a unique solution
$(\hat{v}_{S},\hat{v}_{P},\hat{i}_{S"},\hat{i}_{P"})$ of ${\cal N}_{P}.$ By
the rigidity of $\mathcal{N}^{adj}_{\tilde{P}},$ there is a unique solution
$(\tilde{v}_{\tilde{S}},\tilde{v}_{\tilde{P}},\tilde{i}_{\tilde{S}"},\tilde{i}_{\tilde{P}"})$
of $\mathcal{N}^{adj}_{\tilde{P}}.$ Thus the vector
$(\hat{v}_{S},\hat{v}_{P},\tilde{v}_{\tilde{S}},\tilde{v}_{\tilde{P}},\hat{i}_{S"},\hat{i}_{P"},\tilde{i}_{\tilde{S}"},\tilde{i}_{\tilde{P}"})$
is the unique solution of
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}.$

The device characteristic ${\cal A}_{SS"}\oplus{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}}$ is
proper because $r(\mbox{$\cal V$}_{SS"}\oplus{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"})=r(\mbox{$\cal V$}_{SS"})+r(\mbox{$\cal
V$}^{*}_{SS"})=2|S|=|S|+|\tilde{S}|,$ $r(\mbox{${\bf
g}$}^{{P}\tilde{P}})=2|P|=|P\uplus\tilde{P}|,$ so that dimension of
$\mbox{$\cal V$}_{SS"}\oplus{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}}$ equals
$|S|+|\tilde{S}|+|P\uplus\tilde{P}|.$ Since the network
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ has proper device characteristic and also a unique
solution, our conventional circuit simulator can process it and obtain its
solution.

We have computed a single vector
$x_{PP"}^{p}:\equiv(v_{P},-i_{P"})\in\breve{{\cal A}}_{PP"}.$ We next consider
the problem of finding a generating set for the vector space translate
$\breve{\mbox{$\cal V$}}_{PP"}$ of $\breve{{\cal A}}_{PP"}.$
Let $\mbox{${\bf g}$}_{tv}^{P\tilde{P}}$ denote the affine space that is the
solution set of the constraints
$v_{e_{j}}=-i_{\tilde{e}_{j}"},e_{j}\in P,j\neq
t,v_{e_{t}}+1=-i_{\tilde{e}_{t}"};i_{e_{i}}=v_{\tilde{e}_{i}"},e_{i}\in P.$
Let $\mbox{${\bf g}$}_{ti}^{P\tilde{P}}$ denote the affine space that is the
solution set of the constraints
$v_{e_{i}}=-i_{\tilde{e}_{i}"},e_{i}\in
P;i_{e_{j}}=v_{\tilde{e}_{j}"},e_{j}\in P,j\neq
t,i_{e_{t}}+1=v_{\tilde{e}_{t}"}.$

Now solve
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}}$ for each $e_{t}\in P$ and
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}}$ for each $e_{t}"\in P"$ (see Figure 1(b) and 1(c)).
(We remind the reader that $\mathcal{N}^{hom}_{P}$ is obtained from ${\cal
N}_{P}$ by replacing its device characteristic ${\cal A}_{SS"}$ by the vector
space translate $\mbox{$\cal V$}_{SS"}.$)

We prove below, in Lemma 12, that each solution yields a vector in
$\breve{\mbox{$\cal V$}}_{PP"}$ and the vectors corresponding to all $e_{t}\in
P,e_{t}"\in P",$ form a generating set for $\breve{\mbox{$\cal V$}}_{PP"}.$

We summarize these steps in the following Algorithm.

###### Algorithm I

Input: A multiport ${\cal N}_{P}$ on ${\cal G}_{SP}$ with affine device
characteristic ${{\cal A}}_{SS"}.$
Output: The port behaviour $\breve{{\cal A}}_{PP"}$ of ${\cal N}_{P}$ if
${\cal N}_{P}$ is rigid.
Otherwise a statement that ${\cal N}_{P}$ is not rigid.

Step 1. Build the network ${\cal
N}^{large}:\equiv[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ on graph ${\cal G}_{SP}\oplus{\cal
G}_{\tilde{S}\tilde{P}}$
with device characteristic ${{\cal A}}_{SS"}\oplus{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}},$ where
${\mbox{$\cal V$}}_{SS"}$ is the vector space translate of ${{\cal A}}_{SS"}$
and ${\mbox{$\cal V$}}^{adj}_{\tilde{S}\tilde{S}"}:\equiv({\mbox{$\cal
V$}}_{SS"}^{*})_{(-\tilde{S}")\tilde{S}}.$ (see Figure 1(a).)
Find the unique solution (if it exists) of ${\cal N}^{large}$ and restrict it
to $P\uplus P"$ to obtain
$(v^{p}_{P},i^{p}_{P"}).$ The vector $(v^{p}_{P},-i^{p}_{P"})$ belongs to
$\breve{{\cal A}}_{PP"}.$
If no solution exists or if there are non unique solutions output ‘${\cal
N}_{P}$ not rigid’ and
STOP.
Step 2. Let $\mathcal{N}^{hom}_{P}$ be obtained by replacing the device
characteristic ${{\cal A}}_{SS"}$ by ${\mbox{$\cal V$}}_{SS"}$ in
$\mathcal{N}_{P}.$
For $t=1,\cdots,|P|,$ build and solve
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}}$ and restrict it to $P\uplus P"$ to obtain
$(v^{tv}_{P},i^{tv}_{P"}).$ (see Figure 1(b).)

The vector $(v^{tv}_{P},-i^{tv}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.$
For $t=1,\cdots,|P|,$ build and solve
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}}$ and restrict it to $P\uplus P"$ to obtain
$(v^{ti}_{P},i^{ti}_{P"}).$ The vector
$(v^{ti}_{P},-i^{ti}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.$ (see Figure
1(c).)
Step 3. Let $\breve{\mbox{$\cal V$}}_{PP"}$ be the span of the vectors
$(v^{tv}_{P},-i^{tv}_{P"}),(v^{ti}_{P},-i^{ti}_{P"}),t=1,\cdots,|P|.$
Output $\breve{{\cal
A}}_{PP"}:\equiv(v^{p}_{P},-i^{p}_{P"})+\breve{\mbox{$\cal V$}}_{PP"}.$
STOP

We complete the justification of Algorithm I in the following lemma.

######  Lemma 12.

Let ${\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{SS"})$ be rigid. Let
$\breve{{\cal A}}_{PP"}$ be the port behaviour of ${\cal N}_{P}$ and let
$\breve{\mbox{$\cal V$}}_{PP"}$ be the vector space translate of $\breve{{\cal
A}}_{PP"}.$ Then the following hold.

  1. 1.

The network ${\cal
N}^{large}:\equiv[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ has a proper device characteristic and has a unique
solution.

  2. 2.

Each of the networks
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}},e_{t}\in
P,[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}},e"_{t}\in P",$ has a unique solution and restriction
of the solution to $P\uplus P"$ gives a vector $(v^{tv}_{P},i^{tv}_{P"})$ such
that $(v^{tv}_{P},-i^{tv}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}$ or a vector
$(v^{ti}_{P},i^{ti}_{P"})$ such that
$(v^{ti}_{P},-i^{ti}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.$

  3. 3.

The vectors $(v^{tv}_{P},-i^{tv}_{P"}),\
t=1,\cdots,|P|,(v^{ti}_{P},-i^{ti}_{P"}),\ t=1,\cdots,|P|,$ form a generating
set for $\breve{\mbox{$\cal V$}}_{PP"}.$

###### Proof.

We only prove parts 2 and 3 since part 1 has already been shown.
2\. If we replace the device characteristic of
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}},$ or that of
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}}$ or that of
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}}$ by its vector space translate, i.e., by
${\mbox{$\cal V$}}_{SS"}\oplus{\mbox{$\cal V$}}^{adj}_{SS"}\oplus\mbox{${\bf
g}$}^{{P}\tilde{P}},$ we get the device characteristic of
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}.$ All three networks have the same graph ${\cal
G}_{SP}\oplus{\cal G}_{\tilde{S}\tilde{P}}.$
By part 5 of Theorem 11, a linear network ${\cal N}$ with a proper device
characteristic, has a unique solution iff ${\cal N}^{hom}$ has a unique
solution. We know that
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ has a proper device characteristic and has a unique
solution. Thus
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}},$ has a unique solution and therefore also
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}}$ and
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}}.$
The restriction of a solution
$(\hat{v}_{S},\hat{v}_{P},\tilde{v}_{\tilde{S}},\tilde{v}_{\tilde{P}},\hat{i}_{S"},\hat{i}_{P"},\tilde{i}_{\tilde{S}"},\tilde{i}_{\tilde{P}"})$
of $[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}}$ or of
$[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}}$ to $S\uplus P\uplus S"\uplus P"$ gives a solution of
$\mathcal{N}^{hom}_{P}.$ Its restriction to $P\uplus P"$ gives
$(\hat{v}_{P},-\hat{i}_{P"}).$ By Theorem 5,
$(\hat{v}_{P},\hat{i}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.$
3\. Let $\breve{\mbox{$\cal V$}}_{PP"}$ be the solution space of
$Bv_{P}-Qi_{P"}=0$ and let $(\breve{\mbox{$\cal
V$}}^{adj}_{PP"})_{\tilde{P}\tilde{P}"}=(\breve{\mbox{$\cal
V$}}^{*}_{PP"})_{(-\tilde{P}")\tilde{P}},$ be the solution space of
$Q^{"}v_{\tilde{P}}+B^{"}i_{\tilde{P}"}=0,$ where the row spaces of
$(B|-Q),(B^{"}|-Q^{"})$ are complementary orthogonal. A vector
$(v_{P},v_{\tilde{P}},-i_{P"},-i_{\tilde{P}"})$ being the restriction of a
solution of
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{tv}^{{P}\tilde{P}}$ to $P\uplus P"\uplus\tilde{P}\uplus\tilde{P}"$ is
equivalent to $(v_{P},v_{\tilde{P}},i_{P"},i_{\tilde{P}"})$ being a solution
to the equation

$\displaystyle\begin{pmatrix}B&-Q&0&0\\\ 0&0&Q^{"}&B^{"}\\\ I&0&0&I\\\
0&I&-I&0\end{pmatrix}\begin{pmatrix}v_{P}\\\ i_{P"}\\\ v_{\tilde{P}}\\\
i_{\tilde{P}"}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}0\\\ 0\\\ -I^{t}\\\
0\end{pmatrix}$ (15)

and a vector $(v_{P},v_{\tilde{P}},-i_{P"},-i_{\tilde{P}"})$ being the
restriction of a solution of
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}_{ti}^{{P}\tilde{P}}$ to $P\uplus P"\uplus\tilde{P}\uplus\tilde{P}"$ is
equivalent to $(v_{P},v_{\tilde{P}},i_{P"},i_{\tilde{P}"})$ being a solution
to the equation

$\displaystyle\begin{pmatrix}B&-Q&0&0\\\ 0&0&Q^{"}&B^{"}\\\ I&0&0&I\\\
0&I&-I&0\end{pmatrix}\begin{pmatrix}v_{P}\\\ i_{P"}\\\ v_{\tilde{P}}\\\
i_{\tilde{P}"}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}0\\\ 0\\\ 0\\\
-I^{t}\end{pmatrix},$ (16)

where $I^{t}$ denotes the $t^{th}$ column of a $|P|\times|P|$ identity matrix.
In the variables $v_{P},i_{P"},$ Equation 15 reduces to

$\displaystyle\begin{pmatrix}B&-Q\\\
B^{"}&-Q^{"}\end{pmatrix}\begin{pmatrix}v_{P}\\\ i_{P"}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0\\\ -B^{"}I^{t}\end{pmatrix},$ (17)

and Equation 16 reduces to

$\displaystyle\begin{pmatrix}B&-Q\\\
B^{"}&-Q^{"}\end{pmatrix}\begin{pmatrix}v_{P}\\\ i_{P"}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0\\\ Q^{"}I^{t}\end{pmatrix}.$ (18)

It is clear that a vector belongs to $\breve{\mbox{$\cal V$}}_{PP"}$ iff it is
a solution of

$\displaystyle\begin{pmatrix}B&-Q\\\
B^{"}&-Q^{"}\end{pmatrix}\begin{pmatrix}v_{P}\\\ i_{P"}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0\\\ x\end{pmatrix},$ (19)

for some vector $x.$ The space of all such $x$ vectors is the column space of
the matrix $(B^{"}|-Q^{"}).$ Noting that, for any matrix $K,$ the product
$KI^{t}$ is the $t^{th}$ column of $K,$ we see that the solutions, for
$t=1,\cdots,|P|,$ of Equations 17 and 18, span $\breve{\mbox{$\cal
V$}}_{PP"}.$

∎

######  Remark 2.

If the multiport $\mathcal{N}_{P}$ is not rigid, it may not have a solution
and then the port behaviour $\breve{{\cal A}}_{PP"}$ would be void. Even if
the multiport has a solution, so that the behaviour $\breve{{\cal A}}_{PP"}$
is nonvoid, the above general procedure of solving
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}},$ will yield non unique internal voltages and currents in
the multiports $\mathcal{N}_{P},\mathcal{N}^{adj}_{\tilde{P}}.$ Therefore,
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ will have non unique solution. (In both the above cases
our conventional circuit simulator would give error messages.) Thus
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf
g}$}^{{P}\tilde{P}}$ has a unique solution iff ${\cal N}_{P}$ is rigid.

##  5 Maximum power transfer for linear multiports

The original version of the maximum power transfer theorem, states that a
linear $1$-port transfers maximum power to a load if the latter has value
equal to the adjoint (conjugate transpose) of the Thevenin impedance of the
$1$-port. In the multiport case the Thevenin equivalent is an impedance matrix
whose adjoint has to be connected to the multiport for maximum power transfer.
These should be regarded as restricted forms of the theorem since they do not
handle the case where the Thevenin equivalent does not exist.

It was recognized early that a convenient way of studying maximum power
transfer is to study the port conditions for which such a transfer occurs [1,
5]. We will use this technique to obtain such port conditions for an affine
multiport behaviour. In the general, not necessarily strictly passive, case,
we can only try to obtain stationarity of power transfer, rather than maximum
power transfer. After obtaining these conditions we show that they are in fact
achieved, if at all, when the multiport is terminated by its adjoint (which is
easy to build), through an ideal transformer. This means that the multiport
behaviour need only be available as the port behaviour of a multiport ${\cal
N}_{P},$ and not explicitly, as an affine space $\breve{{\cal A}}_{PP"}.$

###  5.1 Stationarity of power transfer for linear multiports

We fix some preliminary notation needed for the discussion of the maximum
power transfer theorem. For any matrix $M,$ we take $\overline{M}$ to be the
conjugate and $M^{*}$ to be the conjugate transpose. Our convention for the
sign of power associated with a multiport behaviour is that when
$(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"},$ the power absorbed
by the multiport behaviour $\breve{{\cal A}}_{PP"}$ is
$\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle=\overline{\breve{v}_{P}^{T}}\breve{i}_{P"}+\overline{\breve{i}_{P"}^{T}}\breve{v}_{P}.$
(We omit the scale factor $\frac{1}{2}$ for better readability of the
expressions involved.) The power delivered by it, is therefore the negative of
this quantity.

The maximum power transfer problem is

$\displaystyle\mbox{minimize}\ \
(\overline{\breve{v}_{P}^{T}}\breve{i}_{P"}+\overline{\breve{i}_{P"}^{T}}\breve{v}_{P})$
(20) $\displaystyle(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"},\
\mbox{or equivalently,}\ B\breve{v}_{P}-Q\breve{i}_{P"}=s.$ (21)

If $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}),$ is a stationary point for
the optimization problem 20 , we have

$\displaystyle(\breve{i}^{stat}_{P"})^{T}\overline{\delta\breve{v}}_{P}+(\breve{v}^{stat}_{P})^{T}\overline{\delta\breve{i}}_{P"}+(\overline{\breve{i}^{stat}_{P"})}^{T}{\delta\breve{v}}_{P}+(\overline{\breve{v}^{stat}_{P})}^{T}{\delta\breve{i}}_{P"}$
$\displaystyle=0,$ (22)

for every vector $(\delta\breve{v}_{P},\delta\breve{i}_{P"}),$ such that
$({B}({\delta\breve{v}}_{P})-{{Q}}({\delta\breve{i}}_{P"}))=0.$ Therefore,
Equation 22 is satisfied iff, for some vector $\lambda,$

${((\breve{i}^{stat}_{P"})^{T}|(\breve{v}^{stat}_{P})^{T})-\lambda^{T}(\overline{B}|-\overline{{Q}})}=0.$

Thus the stationarity at $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}),$ is
equivalent to

${((\breve{v}^{stat}_{P})^{T}|(\breve{i}^{stat}_{P"})^{T})-\lambda^{T}(-\overline{Q}|\overline{{B}})}=0.$

Since we must have

$B\breve{v}^{stat}_{P}-Q\breve{i}^{stat}_{P"}=s,$

the stationarity condition reduces to

$\displaystyle\begin{pmatrix}B&-Q\end{pmatrix}\begin{pmatrix}-Q^{*}\\\
B^{*}\end{pmatrix}\lambda=s,$ (23)

The vector space translate, $\breve{\mbox{$\cal V$}}_{PP"}$ of $\breve{{\cal
A}}_{PP"}$ is the solution space of the equation,
$B\breve{v}_{P}-Q\breve{i}_{P"}=0,$ and $(\breve{\mbox{$\cal
V$}}_{PP"}^{adj})_{P(-P")}:\equiv(\breve{\mbox{$\cal V$}}^{*}_{PP"})_{P"P},$
is the row space of $(-\overline{Q}|\overline{B}).$
Thus, the stationarity condition says that
$(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})$ belongs to $\breve{{\cal
A}}_{PP"}\cap(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}.$
In the case where the multiport has a Thevenin impedance $Z$ and Thevenin
voltage $E,$ the above stationarity condition reduces to the condition
$(Z+Z^{*})(-\breve{i}^{stat}_{P"})^{T})=E$ (see example in Section 1).

We note that, even when the multiport is rigid, Equation 23 may have no
solution, in which case we have no stationary vectors for power transfer. If
the equation has a unique solution, using that $\lambda$ vector we get a
unique stationary vector $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}).$ We
next show that the stationarity condition is achieved at the ports of the
multiport ${\cal N}_{P},$ if we terminate it by ${\cal N}^{adj}_{\tilde{P}}$
through the ideal transformer ${\cal T}^{P\tilde{P}}$ resulting in the network
$[{\cal N}_{P}\oplus{\cal N}^{adj}_{\tilde{P}}]\cap{\cal T}^{P\tilde{P}},$
where the ideal transformer ${\cal T}^{P\tilde{P}}$ satisfies the equations
$v_{P}=v_{\tilde{P}};i_{P}=-i_{\tilde{P}}.$

######  Theorem 13.

Let $\mathcal{N}_{P},$ on graph ${\cal G}_{SP}$ and device characteristic
${{\cal A}}_{SS"},$ have the port behaviour $\breve{{\cal A}}_{PP"}.$ Let
${\mbox{$\cal V$}}_{SS"},\breve{\mbox{$\cal V$}}_{PP"}$ be the vector space
translates of ${{\cal A}}_{SS"},\breve{{\cal A}}_{PP"},$ respectively. Let
$\mathcal{N}^{adj}_{\tilde{P}}$ be on the disjoint copy ${\cal
G}_{\tilde{S}\tilde{P}}$ of ${\cal G}_{SP},$ with device characteristic
${\mbox{$\cal V$}}^{adj}_{\tilde{S}\tilde{S}"}:\equiv({\mbox{$\cal
V$}}^{adj}_{SS"})_{\tilde{S}\tilde{S}"}.$

  1. 1.

A vector $(v_{P},i_{P"})$ is the restriction of a solution of the network
${\cal
N}^{large}:\equiv[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal
T}^{{P}\tilde{P}}$ to $P\uplus P",$ iff $(v_{P},-i_{P"})\in\breve{{\cal
A}}_{PP"}\cap(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}.$

  2. 2.

Let $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal A}}_{PP"}.$
Then $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})$ satisfies stationarity
condition with respect to the power absorbed by
$(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"}$ iff
$(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal
A}}_{PP"}\cap(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}.$

  3. 3.

Let $(v^{1}_{P},-i^{1}_{P"})$ be the restriction of a solution of the
multiport ${\cal N}_{P},$ to $P\uplus P".$ Then
$(\breve{v}^{1}_{P},\breve{i}^{1}_{P"})$ satisfies the stationarity condition
with respect to the power absorbed by
$(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"}$ iff
$(v^{1}_{P},-i^{1}_{P"})$ is the restriction of a solution of the network
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal
T}^{{P}\tilde{P}},$ to $P\uplus P".$

###### Proof.

1\. The restriction of the set of solutions of $\mathcal{N}_{P}$ on graph
${\cal G}_{SP}$ to $P\uplus P",$ is
$[(\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"})\cap{\cal
A}_{SS"}]\circ{PP"}.$ This is the same as $(\breve{{\cal A}}_{PP"})_{P(-P")}.$
The restriction of the set of solutions of $\mathcal{N}^{adj}_{\tilde{P}}$ on
the disjoint copy ${\cal G}_{\tilde{S}\tilde{P}}$ of ${\cal G}_{SP},$ to
$\tilde{P}\uplus\tilde{P}"$ is
$[(\mbox{$\cal V$}_{\tilde{S}\tilde{P}}\oplus(\mbox{$\cal
V$}^{*}_{\tilde{S}\tilde{P}})_{\tilde{S}"\tilde{P}"})\cap{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"}]\circ{\tilde{P}\tilde{P}"}.$ This we know, by
Corollary 9, to be the same as $(\breve{\mbox{$\cal
V$}}^{adj}_{PP"})_{\tilde{P}-\tilde{P}"}.$
The restriction of the set of solutions of $\mathcal{N}^{adj}_{P}$ to
${P}\uplus{P}"$ is $[(\mbox{$\cal V$}_{{S}{P}}\oplus(\mbox{$\cal
V$}^{*}_{{S}{P}})_{{S}"{P}"})\cap{\mbox{$\cal
V$}}^{adj}_{{S}{S}"}]\circ{{P}{P}"}.$
We thus have, $[((\mbox{$\cal V$}_{\tilde{S}\tilde{P}}\oplus(\mbox{$\cal
V$}^{*}_{\tilde{S}\tilde{P}})_{\tilde{S}"\tilde{P}"})\cap{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"})\cap{\cal
T}^{P\tilde{P}}]\circ{PP"}=[((\mbox{$\cal
V$}_{\tilde{S}\tilde{P}}\oplus(\mbox{$\cal
V$}^{*}_{\tilde{S}\tilde{P}})_{\tilde{S}"\tilde{P}"})\cap{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"})\circ{\tilde{P}\tilde{P}"}]_{P(-P")}$
$=([(\mbox{$\cal V$}_{{S}{P}}\oplus(\mbox{$\cal
V$}^{*}_{{S}{P}})_{{S}"{P}"})\cap{\mbox{$\cal
V$}}^{adj}_{{S}{S}"}]\circ{{P}{P}"})_{P(-P")}=\breve{\mbox{$\cal
V$}}^{adj}_{PP"},$ since vectors in ${\cal T}^{P\tilde{P}}$ are precisely the
ones that satisfy $v_{P}=v_{\tilde{P}"},i_{P"}=-i_{\tilde{P}"}.$
The restriction of the set of solutions of
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal
T}^{{P}\tilde{P}},$ to $P\uplus P"$ is therefore equal to
$[[(\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"})\cap{\cal
A}_{SS"}]\circ{PP"}]\cap[[(\mbox{$\cal
V$}_{\tilde{S}\tilde{P}}\oplus(\mbox{$\cal
V$}^{*}_{\tilde{S}\tilde{P}})_{\tilde{S}"\tilde{P}"})\cap{\mbox{$\cal
V$}}^{adj}_{\tilde{S}\tilde{S}"}\cap{\cal T}^{P\tilde{P}}]\circ{{P}{P}"}]$
$=(\breve{{\cal A}}_{PP"})_{P(-P")}\cap\breve{\mbox{$\cal V$}}_{PP"}^{adj}.$

Part 2 follows from the discussion preceding the theorem and part 3 follows
from part 1 and part 2. ∎

###  5.2 Maximum Power Transfer Theorem for passive multiports

We show below that the stationarity conditions of the previous subsection
reduce to maximum power transfer conditions when the multiport is passive.

The power absorbed by a vector $(v_{E},i_{E"})$ is given by $\langle
v_{E},i_{E"}\rangle+\langle i_{E"},v_{E}\rangle.$
A vector space $\mbox{$\cal V$}_{SS"}$ is passive, iff the power absorbed by
$(x_{S},y_{S"})$ is nonnegative, whenever $(x_{S},y_{S"})\in\mbox{$\cal
V$}_{SS"}.$ It is strictly passive iff the power absorbed by every nonzero
vector in $\mbox{$\cal V$}_{SS"}$ is positive. An affine space ${\cal
A}_{SS"}$ is (strictly) passive iff its vector space translate is (strictly)
passive. A multiport is (strictly) passive iff its port behaviour is
(strictly) passive.

We now have a routine result which links passivity of the device
characteristic of a multiport to its port behaviour. (A cutset of a graph is a
minimal set of edges which when deleted increases the number of connected
components of the graph, equivalently, in the case of a connected graph, is a
minimal set of edges not contained in any cotree.)

######  Lemma 14.

Let ${\cal N}_{P}$ be a multiport on graph ${\cal G}_{SP}$ with device
characteristic ${\cal A}_{SS"}.$

  1. 1.

If ${\cal A}_{SS"}$ is passive so is the port behaviour $\breve{{\cal
A}}_{PP"}$ of ${\cal N}_{P}.$

  2. 2.

If ${\cal A}_{SS"}$ is strictly passive and $P$ contains no loops or cutsets
of ${\cal G}_{SP},$ then the port behaviour $\breve{{\cal A}}_{PP"}$ of ${\cal
N}_{P}$ is also strictly passive.

###### Proof.

1\. We assume that $\breve{{\cal A}}_{PP"}$ is nonvoid. We have,
$\breve{{\cal A}}_{PP"}=((\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal
V$}^{*}_{SP})_{S"P"})\leftrightarrow{\cal A}_{SS"})_{P(-P")},$ where
$\mbox{$\cal V$}_{SP}:\equiv\mbox{$\cal V$}^{v}({\cal G}_{SP}).$
By Theorem 3, $\mbox{$\cal V$}^{*}_{S"P"}=(\mbox{$\cal V$}^{i}({\cal
G}_{SP}))_{S"P"}.$ By Theorem 4, it follows that the vector space translate of
$\breve{{\cal A}}_{PP"}$ is $\breve{\mbox{$\cal V$}}_{PP"}=((\mbox{$\cal
V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"})\leftrightarrow\mbox{$\cal
V$}_{SS"})_{P(-P")}.$
Let $(v_{P},-i_{P"})$ belong to $\breve{\mbox{$\cal V$}}_{PP"}.$ Then there
exist $(v_{S},v_{P})\in\mbox{$\cal V$}_{SP}$ and
$(i_{S"},i_{P"})\in(\mbox{$\cal V$}^{*}_{SP})_{S"P"},$ such that
$(v_{S},i_{S"})\in\mbox{$\cal V$}_{SS"}.$
By the orthogonality of $\mbox{$\cal V$}_{SP},\mbox{$\cal V$}^{*}_{SP},$ it
follows that $\langle(v_{S},v_{P}),(i_{S"},i_{P"})\rangle=\langle
v_{S},i_{S"}\rangle+\langle v_{P},i_{P"}\rangle=0,$ and
by the passivity of $\mbox{$\cal V$}_{SS"},$ it follows that $\langle
v_{S},i_{S"}\rangle+\langle i_{S"},v_{S}\rangle\geq 0.$ Therefore $\langle
v_{P},-i_{P"}\rangle+\langle-i_{P"},v_{P}\rangle\geq 0.$
2\. Without loss of generality, we assume that the graph ${\cal G}_{SP}$ is
connected. If $P$ contains no cutset or circuit of ${\cal G}_{SP},$ then $S$
contains both a tree as well as a cotree of ${\cal G}_{SP}.$ If the voltages
assigned to the branches of a tree are zero, the branches in its complement
will have zero voltage. Therefore $\mbox{$\cal V$}^{v}({\cal G}_{SP})\times
P=\mbox{$\cal V$}_{SP}\times P$ must necessarily be a zero vector space. If
the currents in the branches of a cotree are zero the branches in its
complement will have zero current. Therefore $(\mbox{$\cal V$}^{i}({\cal
G}_{SP})\times P)_{P"}=\mbox{$\cal V$}^{*}_{S"P"}\times P"$ must necessarily
be a zero vector space.
Now let $(v_{P},-i_{P"})\in\breve{\mbox{$\cal V$}}_{PP"},(v_{P},-i_{P"})\neq
0_{PP"}.$ As in part 1 above, there exist $(v_{S},v_{P})\in\mbox{$\cal
V$}_{SP}$ and $(i_{S"},i_{P"})\in\mbox{$\cal V$}^{*}_{SP},$ such that
$(v_{S},i_{S"})\in\mbox{$\cal V$}_{SS"}.$
Since $\mbox{$\cal V$}_{SP}\times P$ and $\mbox{$\cal V$}^{*}_{SP}\times P$
are both zero vector spaces, we must have that $(v_{S},i_{S"})\neq 0_{SS"}$ so
that, by strict passivity of $\mbox{$\cal V$}_{SS"},$ we have $\langle
v_{S},i_{S"}\rangle+\langle i_{S"},v_{S}\rangle>0.$ Further
$\langle(v_{S},v_{P}),(i_{S"},i_{P"})\rangle=0,$ so that we can conclude
$\langle v_{P},-i_{P"}\rangle+\langle-i_{P"},v_{P}\rangle>0.$ ∎

When a port behaviour is passive or strictly passive, by taking into account
second order terms, we can show that the stationarity condition implies a
maximum power delivery condition.

Let $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal A}}_{PP"},$
satisfy the stationarity condition
$((\breve{v}^{stat}_{P})^{T},(\breve{i}^{stat}_{P"})^{T})={\lambda}^{T}(-\overline{Q}|\overline{B}),$
for some ${\lambda}.$ For any $(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal
A}}_{PP"},$ i.e., such that ${B}\breve{v}_{P}-{Q}\breve{i}_{P"}=s,$ we can
write
$(\breve{v}_{P},\breve{i}_{P"})=(\breve{v}^{stat}_{P}+\Delta\breve{v}_{P},\breve{i}^{stat}_{P"}+\Delta\breve{i}_{P"}).$
We then have ${B}\Delta\breve{v}_{P}-{Q}\Delta\breve{i}_{P"}=0$ and
$[\overline{B}\overline{(\Delta\breve{v}_{P})}-\overline{Q}\overline{(\Delta\breve{i}_{P"}})]=0.$
Therefore
$\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle=\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle-\overline{{{\lambda}}^{T}}[{B}{(\Delta\breve{v}_{P})}-{Q}{(\Delta\breve{i}_{P"}})]-{{\lambda}^{T}}[\overline{B}\overline{(\Delta\breve{v}_{P})}-\overline{Q}\overline{(\Delta\breve{i}_{P"}})]$
$=\langle\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}\rangle+\langle\breve{i}^{stat}_{P"},\breve{v}^{stat}_{P}\rangle+(\Delta\breve{v}_{P})^{T}\overline{\breve{i}^{stat}_{P"}}+(\breve{v}^{stat}_{P})^{T}\overline{(\Delta\breve{i}_{P"})}+(\breve{i}^{stat}_{P"})^{T}\overline{(\Delta\breve{v}_{P})}+(\Delta\breve{i}_{P"})^{T}\overline{\breve{v}^{stat}_{P}}$
$+\langle\Delta\breve{v}_{P},\Delta\breve{i}_{P"}\rangle+\langle\Delta\breve{i}_{P"},\Delta\breve{v}_{P}\rangle-{{\lambda}^{T}}[\overline{B}\overline{(\Delta\breve{v}_{P})}-\overline{Q}\overline{(\Delta\breve{i}_{P"}})]-\overline{{\lambda}^{T}}[{B}{(\Delta\breve{v}_{P})}-{Q}{(\Delta\breve{i}_{P"}})].$
We can rewrite the right side as
$\langle\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}\rangle+\langle\Delta\breve{v}_{P},\Delta\breve{i}_{P"}\rangle+((\breve{v}^{stat}_{P})^{T}+{\lambda}^{T}\overline{Q})\overline{\Delta\breve{i}_{P"}}+((\breve{i}^{stat}_{P"})^{T}-{\lambda}^{T}\overline{B})\overline{\Delta\breve{v}_{P}}$
$+\langle\breve{i}^{stat}_{P"},\breve{v}^{stat}_{P}\rangle+\langle\Delta\breve{i}_{P"},\Delta\breve{v}_{P}\rangle+(\overline{(\breve{v}^{stat}_{P})^{T}}+\overline{{\lambda}^{T}}{Q}){\Delta\breve{i}_{P"}}+(\overline{(\breve{i}^{stat}_{P"})^{T}}-\overline{{\lambda}^{T}}{B}){\Delta\breve{v}_{P}}.$
Applying the condition
$((\breve{v}^{stat}_{P})^{T},(\breve{i}^{stat}_{P"})^{T})={\lambda}^{T}(-\overline{Q}|\overline{B}),$
this expression reduces to
$\langle\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}\rangle+\langle\breve{i}^{stat}_{P"},\breve{v}^{stat}_{P}\rangle+\langle\Delta\breve{v}_{P},\Delta\breve{i}_{P"}\rangle+\langle\Delta\breve{i}_{P"},\Delta\breve{v}_{P}\rangle.$
Therefore,
$\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle$
$=\langle\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}\rangle+\langle\breve{i}^{stat}_{P"},\breve{v}^{stat}_{P}\rangle+\langle\Delta\breve{v}_{P},\Delta\breve{i}_{P"}\rangle+\langle\Delta\breve{i}_{P"},\Delta\breve{v}_{P}\rangle.$

If $\breve{{\cal A}}_{PP"}$ is passive we have
$\langle\Delta\breve{v}_{P},\Delta\breve{i}_{P"}\rangle+\langle\Delta\breve{i}_{P"},\Delta\breve{v}_{P}\rangle\geq
0,$ so that
$\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle\geq\langle\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}\rangle+\langle\breve{i}^{stat}_{P"},\breve{v}^{stat}_{P}\rangle.$
Equivalently, the power delivered by $\breve{{\cal A}}_{PP"},$ is maximum when
$(\breve{v}_{P},\breve{i}_{P"})=(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}).$
If $\breve{{\cal A}}_{PP"}$ is strictly passive we have
$\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle>\langle\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}\rangle+\langle\breve{i}^{stat}_{P"},\breve{v}^{stat}_{P}\rangle$
whenever $(\Delta\breve{v}_{P},\Delta\breve{i}_{P"})\neq 0,$ so that
$(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})$ is the unique maximum delivery
vector in $\breve{{\cal A}}_{PP"}.$

By Lemma 14, if a multiport ${\cal N}_{P}$ is passive, so is its port
behaviour. Therefore, Equation 23 also gives the condition for maximum power
transfer from a passive ${\cal N}_{P}.$

We thus have, from the above discussion, using Theorem 13, the following
result.

######  Theorem 15.

Let $\mathcal{N}_{P},$ on graph ${\cal G}_{SP}$ and with passive device
characteristic ${{\cal A}}_{SS"},$ have the port behaviour $\breve{{\cal
A}}_{PP"}.$ Let ${\mbox{$\cal V$}}_{SS"}$ be the vector space translate of
${{\cal A}}_{SS"}.$ Let $\mathcal{N}^{adj}_{\tilde{P}}$ be on the disjoint
copy ${\cal G}_{\tilde{S}\tilde{P}}$ of ${\cal G}_{SP},$ with device
characteristic ${\mbox{$\cal V$}}^{adj}_{\tilde{S}\tilde{S}"}.$

  1. 1.

Let $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal A}}_{PP"}.$
Then $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})$ satisfies

$\displaystyle maximize\ \
\langle\breve{v}_{P},-\breve{i}_{P"}\rangle+\langle-\breve{i}_{P"},\breve{v}_{P}\rangle,\
\ \ \ \ \ (\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"},$ (24)

iff $(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal
A}}_{PP"}\cap(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}.$

  2. 2.

Let $(\breve{v}^{stat}_{P},-\breve{i}^{stat}_{P"})$ be the restriction of a
solution of the multiport ${\cal N}_{P},$ to $P\uplus P".$ Then
$(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})$ satisfies the optimization
condition in Equation 24, iff $(\breve{v}^{stat}_{P},-\breve{i}^{stat}_{P"})$
is the restriction of a solution of the network
$[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal
T}^{{P}\tilde{P}},$ to $P\uplus P".$

######  Remark 3.

Let $\breve{\mbox{$\cal V$}}_{PP"}$ be strictly passive. If $(0_{P},i_{P"})$
or $(v_{P},0_{P"})$ belongs to $\breve{\mbox{$\cal V$}}_{PP"},$ its strict
passivity is violated. Therefore $r(\breve{\mbox{$\cal V$}}_{PP"}\times P)$ as
well as $r(\breve{\mbox{$\cal V$}}_{PP"}\times P")$ must be zero and
consequently $r(\breve{\mbox{$\cal V$}}_{PP"}\circ P")$ as well as
$r(\breve{\mbox{$\cal V$}}_{PP"}\circ P)$ must equal $|P|,$ using Theorem 2.
Let $(C_{P}|E_{P"})$ be the representative matrix of $\breve{\mbox{$\cal
V$}}_{PP"}.$ It is clear that $C_{P},E_{P"}$ are nonsingular so that by
invertible row transformation, we can reduce $(C_{P}|E_{P"})$ to the form
$(Z^{T}|I)$ or $(I|Y^{T}).$ We then have $\langle v_{P},i_{P"}\rangle+\langle
i_{P"},v_{P}\rangle=v_{P}^{T}\overline{i_{P"}}+i_{P"}^{T}\overline{v_{P}}=i_{P"}^{T}Z^{T}\overline{i_{P"}}+i_{P"}^{T}\overline{Zi_{P"}}=\langle
i_{P"},(Z+Z^{*})i_{P"}\rangle.$ The strict passivity is equivalent to
$(Z+Z^{*})$ being positive definite. In this case $(Z+Z^{*})$ is invertible
and maximum transfer port condition is unique.

##  6 Conclusions

We have given a method for computing the port behaviour of linear multiports
using standard circuit simulators which are freely available. This will work
for linear multiports which have non void solutions for arbitrary internal
source values and further have a unique internal solution corresponding to a
port condition consistent with the port behaviour. The procedure involves
termination by the adjoint multiport through a gyrator. The above procedure
can be regarded as the most general form of Thevenin-Norton theorem according
to the view point that the theorem involves the computation of the unique
solutions of certain circuits which result by some termination of the linear
multiport.

We have used termination by the adjoint multiport through an ideal transformer
to give a condition for maximum power transfer for general linear multiports.
This is the most general form of the Maximum power transfer theorem that is
possible in terms of stationarity of power delivered.

Data availability statement
The authors confirm that there is no associated data for this manuscript.

Acknowledgements
Hariharan Narayanan was partially supported by a Ramanujan Fellowship.

## References

  * [1] C. A. Desoer, The Maximum Power Transfer Theorem for n-Ports, IEEE Transactions on Circuit Theory, (1973) 328–330.
  * [2] C. A. Desoer & E. S. Kuh, Basic circuit theory, McGraw-Hill, New York, 1969.
  * [3] G. D. Forney & M. D. Trott, The Dynamics of Group Codes : Dual Abelian Group Codes and Systems, IEEE Transactions on Information Theory 50 (11) (2004)
  * [4] H. F. Mayer, Ueber das Ersatzschema der Verstärkerröhre [On equivalent circuits for electronic amplifiers], Telegraphen- und Fernsprech-Technik (in German), 15: (1926) 335–337.
  * [5] H. Narayanan, On the maximum power transfer theorem, Int. J. Elect. Enging Educ., 15 (1978) 161–167.
  * [6] H. Narayanan, A Unified Construction of Adjoint Systems and Networks, Circuit Theory and Applications 14 (1986) 263–276.
  * [7] H. Narayanan, Submodular Functions and Electrical Networks, Annals of Discrete Mathematics, vol. 54, North Holland, Amsterdam, 1997.
Open 2nd edition, http://www.ee.iitb.ac.in/$\ \tilde{}$ hn/ , 2009.

  * [8] H. Narayanan, Implicit Linear Algebra and its Applications, arXiv:1609.07991v1 [math.GM] (2016).
  * [9] E. L. Norton, Design of finite networks for uniform frequency characteristic, Bell Laboratories, Technical Report TM26–0–1860 (1926).
  * [10] B. D. H. Tellegen, A general network theorem, with applications, Philips Res. Rept., 7 (1952) 259–269.
  * [11] L. Thevenin, Extension of Ohm’s law to complex electromotive circuits, Annales Telegraphiques, 1883.
  * [12] W. T. Tutte, Lectures on matroids, Journal of Research of the National Bureau of Standards, B69 (1965) 1–48.
  * [13] A. J. van der Schaft, Interconnection and geometry, in The Mathematics of Systems and Control: from Intelligent Control to Behavioral Systems, Editors J.W. Polderman, H.L. Trentelman, Groningen, 203–218, 1999.