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On Thevenin-Norton and Maximum power transfer theorems

H. Narayanan [email protected] Hariharan Narayanan [email protected] 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

(iP1vP2)=(g11h12h21r22)​(vP1iP2)+(JP1EP2),\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 PP into P1,P2P_{1},P_{2} is known.

In this paper we show how to handle multiports which have port behaviour equations of the general form B​vP−Q​iP=s,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 2​n,2n, where nn 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,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∗Z^{*} of Z,Z, provided Z+Z∗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 1:11:1 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 1:11:1 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 B​vP−Q​iP​"=s,Bv_{P}-Qi_{P"}=s, rather than with those of the kind vP−Z​iP​"=E.v_{P}-Zi_{P"}=E. Instead of working with the adjoint (conjugate transpose) of a matrix Z,Z, we have to work with the adjoint (see Subsection 3.1) of the solution space of B​vP−Q​iP​"=0.Bv_{P}-Qi_{P"}=0. In order for our technique to be effective, we have to show that if B​vP−Q​iP​"=s,Bv_{P}-Qi_{P"}=s, defines the port behaviour of 𝒩P{\cal N}_{P} then the adjoint of the solution space of B​vP−Q​iP​"=0,Bv_{P}-Qi_{P"}=0, is the port behaviour of the multiport 𝒩Pa​d​j{\cal N}_{P}^{adj} which is the adjoint of 𝒩P{\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 𝒩P,{\cal N}_{P}, using controlled sources, norators and nullators, which has any given representation at the ports with nonvoid solution. Suppose the rigid multiport 𝒩P{\cal N}_{P} has the behaviour (see formal definition in Section 3) B​vP−Q​iP​"=s.Bv_{P}-Qi_{P"}=s. The number of equations may range from 0 to 2​n,2n, where n=|P|.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.

Refer to caption
Figure 1: Computation of port behaviour of a multiport

Briefly, the steps in our method are as follows:
1. Build the adjoint multiport 𝒩P~a​d​j.{\cal N}^{adj}_{\tilde{P}}. This is done by retaining the same graph as 𝒩P{\cal N}_{P} but replacing each device by its adjoint (see Example 7). The name of the set of internal edges SS is changed to S~\tilde{S} and that of the port edges PP is changed to P~.\tilde{P}.
2. Between every pair of corresponding ports Pj,P~j,P_{j},\tilde{P}_{j}, insert a source accompanied 1:11:1 gyrator. (For simplicity, in Figure 1 we have taken 𝒩P{\cal N}_{P} to be a 11-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 (vPp,iP​"p)(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 PP will span the source free port behaviour 𝒱^P​P​".\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 𝒜^P​P​":≡(vPp,−iP​"p)+𝒱^P​P​".\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 𝒩P{\cal N}_{P} by its adjoint 𝒩P~a​d​j{\cal N}^{adj}_{\tilde{P}} through a 1:11:1 ideal transformer (vP=vP~;iP=−iP~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.)

(B−Q)​(−Q∗B∗)​λ=s.\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

((v˘Ps​t​a​t)T|(i˘P​"s​t​a​t)T)=λT​(−Q¯|B¯){((\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 B​vP−Q​iP​"=s,Bv_{P}-Qi_{P"}=s, reduces to I​vP−Z​iP​"=E,Iv_{P}-Zi_{P"}=E, and the solution of Equation 2 (taking Z∗:≡Z^{*}:\equiv conjugate transpose of ZZ) yields

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

This corresponds to −(Z+Z∗)​λ=E,((v˘Ps​t​a​t)T|(i˘P​"s​t​a​t)T)=λT​(−Z¯|I),-(Z+Z^{*})\lambda=E,((\breve{v}^{stat}_{P})^{T}|(\breve{i}^{stat}_{P"})^{T})=\lambda^{T}(-\overline{Z}|I), i.e., to (Z+Z∗)​(−i˘P​"s​t​a​t)=E.(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 𝒇\boldsymbol{f} on a finite set XX over 𝔽\mathbb{F} is a function f:X→𝔽f:X\rightarrow\mathbb{F} where 𝔽\mathbb{F} is either the real field ℜ\Re or the complex field ℂ.\mathbb{C}.

The sets on which vectors are defined are always finite. When a vector xx figures in an equation, we use the convention that xx denotes a column vector and xTx^{T} denotes a row vector such as in ‘A​x=b,xT​AT=bTAx=b,x^{T}A^{T}=b^{T}’. Let fYf_{Y} be a vector on YY and let X⊆YX\subseteq Y. The restriction fY|Xf_{Y}|_{X} of fYf_{Y} to XX is defined as follows:
fY|X:≡gX, where gX(e)=fY(e),e∈X.f_{Y}|_{X}:\equiv g_{X},\textrm{ where }g_{X}(e)=f_{Y}(e),e\in X.

When ff is on XX over 𝔽\mathbb{F}, λ∈𝔽,\lambda\in\mathbb{F}, then the scalar multiplication 𝝀​𝒇\boldsymbol{\lambda f} of ff is on XX and is defined by (λf)(e):≡λ[f(e)](\lambda f)(e):\equiv\lambda[f(e)], e∈Xe\in X. When ff is on XX and gg on YY and both are over 𝔽\mathbb{F}, we define 𝒇+𝒈\boldsymbol{f+g} on X∪YX\cup Y by
(f+g)(e):≡f(e)+g(e),e∈X∩Y,(f+g)(e):≡f(e),e∈X∖Y,(f+g)(e):≡g(e),e∈Y∖X.(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−YX-Y in place of X∖Y.X\setminus Y.)

When f,gf,g are on XX over ℂ,\mathbb{C}, the inner product ⟨f,g⟩\langle f,g\rangle of ff and gg is defined by ⟨f,g⟩:≡∑e∈Xf(e)g​(e)¯,\langle f,g\rangle:\equiv\sum_{e\in X}f(e)\overline{g(e)}, g​(e)¯\overline{g(e)} being the complex conjugate of g​(e).g(e). If the field is ℝ,\mathbb{R}, the inner product would reduce to the dot product ⟨f,g⟩:≡∑e∈Xf(e)g(e).\langle f,g\rangle:\equiv\sum_{e\in X}f(e){g(e)}.

When X,Y,X,Y, are disjoint, fX+gYf_{X}+g_{Y} is written as (𝒇𝑿,𝒈𝒀),\boldsymbol{(f_{X},g_{Y})}, The disjoint union of XX and Y.Y. is denoted by 𝑿⊎𝒀.\boldsymbol{X\uplus Y}. A vector fX⊎Yf_{X\uplus Y} on X⊎YX\uplus Y is written as 𝒇𝑿​𝒀.\boldsymbol{f_{XY}}.

We say ff, gg are orthogonal (orthogonal) iff ⟨f,g⟩\langle f,g\rangle is zero.

An arbitrary collection of vectors on XX is denoted by 𝓚𝑿\boldsymbol{\mathcal{K}_{X}}. When XX, YY are disjoint we usually write 𝒦X​Y\mathcal{K}_{XY} in place of 𝒦X⊎Y\mathcal{K}_{X\uplus Y}. We write 𝒦X​Y:≡𝒦X⊕𝒦Y{\cal K}_{XY}:\equiv{\cal K}_{X}\oplus{\cal K}_{Y} iff 𝒦X​Y:≡{fX​Y:fX​Y=(fX,gY),fX∈𝒦X,gY∈𝒦Y}.{\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 𝒦X⊕𝒦Y{\cal K}_{X}\oplus{\cal K}_{Y} as the direct sum of 𝒦X,𝒦Y.{\cal K}_{X},{\cal K}_{Y}.

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

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

For any collection of vectors 𝒦X,\mathcal{K}_{X}, the collection 𝓚𝑿∗\boldsymbol{\mathcal{K}_{X}^{*}} is defined by 𝒦X∗:≡{gX:⟨fX,gX⟩=0},{\mathcal{K}_{X}^{*}}:\equiv\{g_{X}:\langle f_{X},g_{X}\rangle=0\}, It is clear that 𝒦X∗\mathcal{K}_{X}^{*} is a vector space for any 𝒦X.\mathcal{K}_{X}. When 𝒦X\mathcal{K}_{X} is a vector space 𝒱X,\mbox{$\cal V$}_{X}, and the underlying set XX is finite, it can be shown that (𝒱X∗)∗=𝒱X({\mathcal{V}_{X}^{*}})^{*}=\mathcal{V}_{X} and 𝒱X,𝒱X∗\mathcal{V}_{X},{\mathcal{V}_{X}^{*}} are said to be complementary orthogonal. The symbol 0X0_{X} refers to the zero vector on XX and 𝟎𝑿\boldsymbol{0_{X}} refers to the zero vector space on X.X. The symbol ℱ𝑿\boldsymbol{\mbox{$\cal F$}_{X}} refers to the collection of all vectors on XX over the field in question. It is easily seen, when X,YX,Y are disjoint, and 𝒦X,𝒦Y{\cal K}_{X},{\cal K}_{Y} contain zero vectors, that (𝒦X⊕𝒦Y)∗=𝒦X∗⊕𝒦Y∗.({\cal K}_{X}\oplus{\cal K}_{Y})^{*}={\cal K}_{X}^{*}\oplus{\cal K}_{Y}^{*}.

The adjoint K∗K^{*} of a matrix KK is defined by K∗:≡K¯T.K^{*}:\equiv\overline{K}^{T}.
(Later we define the adjoint 𝒱P​P​"a​d​j\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 𝒱X,\mbox{$\cal V$}_{X}, is called a representative matrix for 𝒱X.\mbox{$\cal V$}_{X}. A representative matrix which can be put in the form (I|K)(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)(I\ |\ K) is a standard representative matrix of 𝒱X,\mbox{$\cal V$}_{X}, it is easy to see that (−K∗|I)(-K^{*}|I) is a standard representative matrix of 𝒱X∗.\mbox{$\cal V$}^{*}_{X}. Therefore we must have

Theorem 1.

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

The collection {(fX,λ​fY):(fX,fY)∈𝒦X​Y}\{(f_{X},\lambda f_{Y}):(f_{X},f_{Y})\in\mathcal{K}_{XY}\} is denoted by 𝓚𝑿​(𝝀​𝒀).\boldsymbol{\mathcal{K}_{X(\lambda Y)}}. When λ=−1\lambda=-1 we would write 𝒦X​(λ​Y){\mathcal{K}_{X(\lambda Y)}} more simply as 𝓚𝑿​(−𝒀).\boldsymbol{\mathcal{K}_{X(-Y)}}. Observe that (𝒦X​(−Y))X​(−Y)=𝒦X​Y.(\mathcal{K}_{X(-Y)})_{X(-Y)}=\mathcal{K}_{XY}. We say sets XX, X​"X" are copies of each other iff they are disjoint and there is a bijection, usually clear from the context, mapping e∈Xe\in X to e​"∈X​"e"\in X". When X,X​"X,X" are copies of each other, the vectors fXf_{X} and fX​"f_{X"} are said to be copies of each other with fX​"(e"):≡fX(e),e∈X.f_{X"}(e"):\equiv f_{X}(e),e\in X. The copy 𝒦X​"{\cal K}_{X"} of 𝒦X{\cal K}_{X} is defined by 𝒦X​":≡{fX​":fX∈𝒦X}.{\cal K}_{X"}:\equiv\{f_{X"}:f_{X}\in{\cal K}_{X}\}. When XX and X​"X" are copies of each other, the notation for interchanging the positions of variables with index sets XX and X​"X" in a collection 𝒦X​X​"​Y\mathcal{K}_{XX"Y} is given by (𝓚𝑿​𝑿​"​𝒀)𝑿​"​𝑿​𝒀\boldsymbol{(\mathcal{K}_{XX"Y})_{X"XY}}, that is
(𝒦X​X​"​Y)X​"​X​Y:≡{(gX,fX​",hY):(fX,gX​",hY)∈𝒦X​X​"​Y,gX being copy of gX​",fX​" being copy of fX}.(\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 𝒜X​X​"\mathcal{A}_{XX"} is said to be proper iff the rank of its vector space translate is |X|=|X​"|.|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 𝒦S​P\mathcal{K}_{SP}, 𝒦P​Q\mathcal{K}_{PQ} be collections of vectors on sets S⊎P,S\uplus P, P⊎Q,P\uplus Q, respectively, where S,P,Q,S,P,Q, are pairwise disjoint. The sum 𝓚𝑺​𝑷+𝓚𝑷​𝑸\boldsymbol{\mathcal{K}_{SP}+\mathcal{K}_{PQ}} of 𝒦S​P\mathcal{K}_{SP}, 𝒦P​Q\mathcal{K}_{PQ} is defined over S⊎P⊎Q,S\uplus P\uplus Q, as follows:
𝒦S​P+𝒦P​Q:≡{(fS,fP,0Q)+(0S,gP,gQ), where (fS,fP)∈𝒦S​P,(gP,gQ)∈𝒦P​Q}.\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, 𝒦S​P+𝒦P​Q:≡(𝒦S​P⊕𝟎Q)+(𝟎S⊕𝒦P​Q).\mathcal{K}_{SP}+\mathcal{K}_{PQ}:\equiv(\mathcal{K}_{SP}\oplus{\mathbf{0}}_{Q})+({\mathbf{0}}_{S}\oplus\mathcal{K}_{PQ}).
The intersection 𝓚𝑺​𝑷∩𝓚𝑷​𝑸\boldsymbol{\mathcal{K}_{SP}\cap\mathcal{K}_{PQ}} of 𝒦S​P\mathcal{K}_{SP}, 𝒦P​Q\mathcal{K}_{PQ} is defined over S⊎P⊎Q,S\uplus P\uplus Q, where S,P,Q,S,P,Q, are pairwise disjoint, as follows: 𝒦S​P∩𝒦P​Q:≡{fS​P​Q:fS​P​Q=(fS,hP,gQ),\mathcal{K}_{SP}\cap\mathcal{K}_{PQ}:\equiv\{f_{SPQ}:f_{SPQ}=(f_{S},h_{P},g_{Q}),  where (fS,hP)∈𝒦S​P,(hP,gQ)∈𝒦P​Q.}.\textrm{ where }(f_{S},h_{P})\in\mathcal{K}_{SP},(h_{P},g_{Q})\in\mathcal{K}_{PQ}.\}.
Thus, 𝒦S​P∩𝒦P​Q:≡(𝒦S​P⊕ℱQ)∩(ℱS⊕𝒦P​Q).\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 𝓚𝑺​𝑷\boldsymbol{\mathcal{K}_{SP}} to SS is defined by 𝓚𝑺​𝑷∘𝑺:≡{fS:(fS,fP)∈𝒦S​P}.\boldsymbol{\mathcal{K}_{SP}\circ S}:\equiv\{f_{S}:(f_{S},f_{P})\in\mathcal{K}_{SP}\}. The contraction of 𝓚𝑺​𝑷\boldsymbol{\mathcal{K}_{SP}} to SS is defined by 𝓚𝑺​𝑷×𝑺:≡{fS:(fS,0P)∈𝒦S​P}.\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 𝓚𝑺​𝑷​𝒁∘𝑺​𝑷\boldsymbol{\mathcal{K}_{SPZ}\circ SP}, 𝓚𝑺​𝑷​𝒁×𝑺​𝑷\boldsymbol{\mathcal{K}_{SPZ}\times SP}, respectively when S,P,Z,S,P,Z, are pairwise disjoint, the collections of vectors 𝓚𝑺​𝑷​𝒁∘(𝑺⊎𝑷)\boldsymbol{\mathcal{K}_{SPZ}\circ(S\uplus P)}, 𝓚𝑺​𝑷​𝒁×(𝑺⊎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​(𝒱S​P)=r​(𝒱S​P∘S)+r​(𝒱S​P×P).r({\cal V}_{SP})=r({\cal V}_{SP}\circ S)+r({\cal V}_{SP}\times P). 2. 𝒱S​P∗∘P=(𝒱S​P×P)∗;\mbox{$\cal V$}_{SP}^{*}\circ P=(\mbox{$\cal V$}_{SP}\times P)^{*}; 𝒱S​P∗×S=(𝒱S​P∘S)∗.\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 𝒢{\cal G} be a graph with edge set S.S. We refer to the space of vectors vS,v_{S}, which satisfy Kirchhoff’s Voltage Law (KVL) of the graph 𝒢,\mathcal{G}, by 𝒱𝒗​(𝓖)\boldsymbol{\mbox{$\cal V$}^{v}(\mathcal{G})} and to the space of vectors iS​",i_{S"}, which satisfy Kirchhoff’s Current Law (KCL) of the graph 𝒢,\mathcal{G}, by 𝒱𝒊​(𝓖).\boldsymbol{\mbox{$\cal V$}^{i}(\mathcal{G})}. (We need to deal with vectors of the kind (v,i).(v,i). To be consistent with our notation for vectors as functions, this is treated as a vector (vS,iS​"),(v_{S},i_{S"}), where S​"S" is a disjoint copy of S.S. Therefore 𝒱𝒗​(𝓖)\boldsymbol{\mbox{$\cal V$}^{v}(\mathcal{G})} is taken to be on set SS and 𝒱𝒊​(𝓖)\boldsymbol{\mbox{$\cal V$}^{i}(\mathcal{G})} is taken to be on a disjoint copy S​".S".)

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

Theorem 3.

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

2.5 Networks and multiports

An electrical network 𝒩,\mathcal{N}, or a ‘network’ in short, is a pair (𝒢,𝒦),(\mathcal{G},\mathcal{K}), where 𝒢:≡(V(𝒢),E(𝒢))\mathcal{G}:\equiv(V({\cal G}),E({\cal G})) is a graph and 𝒦,\mathcal{K}, called the device characteristic of the network, is a collection of pairs of vectors (vS,iS​"),S:≡E(𝒢),(v_{S},i_{S"}),S:\equiv E({\cal G}), where S,S​"S,S" are disjoint copies of S,S,\ vS,iS​"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 𝒦S​S​"{\cal K}_{SS"} is affine, we denote it by 𝒜S​S​"{\cal A}_{SS"} and say the network is linear. If 𝒱S​S​"\mbox{$\cal V$}_{SS"} is the vector space translate of 𝒜S​S​",{\cal A}_{SS"}, we say that 𝒜S​S​"{\cal A}_{SS"} is the source accompanied form of 𝒱S​S​".\mbox{$\cal V$}_{SS"}. An affine space 𝒜S​S​"{\cal A}_{SS"} is said to be proper iff its vector space translate 𝒱S​S​"\mbox{$\cal V$}_{SS"} has dimension |S|=|S​"|.|S|=|S"|.

Let SS denote the set of edges of the graph of the network and let {S1,⋯,Sk}\{S_{1},\cdots,S_{k}\} be a partition of S,S, each block SjS_{j} being an individual device. Let S,S​"S,S" be disjoint copies of S,S, with e,e​"e,e" corresponding to edge e.e. The device characteristic would usually have the form ⨁𝒜Sj​Sj​",\bigoplus{\cal A}_{S_{j}S_{j}"}, defined by (Bj​vSj+Qj​iSj​")=sj,(B_{j}v_{S_{j}}+Q_{j}i_{S_{j}"})=s_{j}, with rows of (Bj|Qj)(B_{j}|Q_{j}) being linearly independent. The vector space translate would have the form ⨁𝒱Sj​Sj​",\bigoplus\mbox{$\cal V$}_{S_{j}S_{j}"}, 𝒱Sj​Sj​"\mbox{$\cal V$}_{S_{j}S_{j}"} being the translate of 𝒜Sj​Sj​".{\cal A}_{S_{j}S_{j}"}. Further, usually the blocks SjS_{j} would have size one or two, even when the network has millions of edges. Therefore, it would be easy to build (⨁𝒱Sj​Sj​")∗=⨁𝒱Sj​Sj​"∗.(\bigoplus\mbox{$\cal V$}_{S_{j}S_{j}"})^{*}=\bigoplus\mbox{$\cal V$}_{S_{j}S_{j}"}^{*}.
We say SjS_{j} is a set of norators iff 𝒜Sj​Sj​":≡ℱSj​Sj​",{\cal A}_{S_{j}S_{j}"}:\equiv\mbox{$\cal F$}_{S_{j}S_{j}"}, i.e., there are no constraints on vSj,iSj​".v_{S_{j}},i_{S_{j}"}.
We say SjS_{j} is a set of nullators iff 𝒜Sj​Sj​":≡𝟎Sj​Sj​",{\cal A}_{S_{j}S_{j}"}:\equiv{\mathbf{0}}_{S_{j}S_{j}"}, i.e., vSj=iSj​"=0.v_{S_{j}}=i_{S_{j}"}=0.
We say 𝒱Sj​S^j​Sj​"​S^j​"\mbox{$\cal V$}_{S_{j}\hat{S}_{j}S_{j}"\hat{S}_{j}"} is a gyrator iff vSj=−R​iS^j​";vS^j=R​iSj​",v_{S_{j}}=-Ri_{\hat{S}_{j}"};v_{\hat{S}_{j}}=Ri_{S_{j}"}, where RR is a positive diagonal matrix. We denote by 𝐠Sj​S^j,\mbox{${\bf g}$}^{S_{j}\hat{S}_{j}}, the gyrator where RR is the identity matrix.

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

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

𝒱S​P∩(𝒱S​P∗)S​"​P​"∩𝒦S​S​"=[𝒱S​P⊕(𝒱S​P∗)S​"​P​"]∩𝒦S​S​".\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 𝒦S​P,𝒦P​Q,{\cal K}_{SP},{\cal K}_{PQ}, be collections of vectors respectively on S⊎P,P⊎Q,S\uplus P,P\uplus Q, with S,P,Q,S,P,Q, being pairwise disjoint.

The matched composition 𝓚𝑺​𝑷↔𝓚𝑷​𝑸\boldsymbol{\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{PQ}} is on S⊎QS\uplus Q and is defined as follows:

𝒦S​P↔𝒦P​Q\displaystyle\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{PQ} :≡{(fS,gQ):(fS,hP)∈𝒦S​P,(hP,gQ)∈𝒦P​Q}.\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 𝓚𝑺​𝑷⇌𝓚𝑷​𝑸\boldsymbol{\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{PQ}} is on S⊎QS\uplus Q and is defined as follows:

(𝒦S​P⇌𝒦P​Q)\displaystyle(\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{PQ}) :≡{(fS,gQ):(fS,hP)∈𝒦S​P,(−hP,gQ)∈𝒦P​Q}.Note that\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}
(𝒦S​P⇌𝒦P​Q)=𝒦S​P↔𝒦(−P)​Q.(\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{PQ})\ \ \ =\ \ \ \mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{(-P)Q}.

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

𝒦S​P↔𝒦P​Q=(𝒦S​P∩𝒦P​Q)∘S​Q;𝒦S​P⇌𝒦P​Q=(𝒦S​P∩𝒦(−P)​Q)∘S​Q;{\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;
𝒦S​P↔𝒦P​Q=(𝒦S​P+𝒦(−P)​Q)×S​Q;𝒦S​P⇌𝒦P​Q=(𝒦S​P+𝒦P​Q)×S​Q.{\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
𝒦S∘(S−T)=𝒦S↔ℱT,𝒦S×(S−T)=𝒦S↔𝟎T,T⊆S.{\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 𝒦S​P\mathcal{K}_{SP}, 𝒦P\mathcal{K}_{P} are vector spaces, observe that (𝒦S​P↔𝒦P)=(𝒦S​P⇌𝒦P).(\mathcal{K}_{SP}\leftrightarrow\mathcal{K}_{P})=(\mathcal{K}_{SP}\rightleftharpoons\mathcal{K}_{P}). When S,P,Z,S,P,Z, are pairwise disjoint, we have
(𝒦S​P​Z↔𝒦S)↔𝒦P=(𝒦S​P​Z↔𝒦P)↔𝒦S=𝒦S​P​Z↔(𝒦S⊕𝒦P).(\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 𝒦S:≡𝟎S,𝒦P:≡ℱP,\mathcal{K}_{S}:\equiv{\mathbf{0}}_{S},\mathcal{K}_{P}:\equiv\mbox{$\cal F$}_{P}, the above reduces to 𝒦S​P​Z×P​Z∘Z=𝒦S​P​Z∘S​Z×Z.\mathcal{K}_{SPZ}\times PZ\circ Z\hskip 2.84544pt=\hskip 2.84544pt\mathcal{K}_{SPZ}\circ SZ\times Z.

The multiport 𝒩P:≡(𝒢S​P,𝒦S​S​")\mathcal{N}_{P}:\equiv({\cal G}_{SP},{\cal K}_{SS"}) would impose a relationship between vP,iP​".v_{P},i_{P"}. This relationship is captured by the multiport behaviour (port behaviour for short) 𝒦˘P​P​"\breve{{\cal K}}_{PP"} at P,P, of 𝒩P,{\cal N}_{P}, defined by
𝒦˘P​P​":≡[([𝒱v(𝒢S​P)⊕(𝒱i(𝒢S​P))S​"​P​"]∩𝒦S​S​")∘PP"]P​(−P​")=([𝒱v(𝒢S​P)⊕(𝒱i(𝒢S​P))S​"​P​"]↔𝒦S​S​")P​(−P​")\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")}
=[𝒱v​(𝒢S​P)⊕(𝒱i​(𝒢S​P))S​"​(−P​")]↔𝒦S​S​".=[\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 𝒩P\mathcal{N}_{P} is affine, its port behaviour 𝒦˘P​P​"\breve{{\cal K}}_{PP"} at PP 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,R,
([𝒱S​P⊕(𝒱S​P∗)S​"​(−P​")]∩𝒦S​S​")∘P​P​"([\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"(-P")}]\cap{\cal K}_{SS"})\circ PP" would be the solution of vP=−R​iP​".v_{P}=-Ri_{P"}. But then 𝒦˘P​P​",\breve{{\cal K}}_{PP"}, as defined, would be the solution of vP=R​iP​".v_{P}=Ri_{P"}.

Let the multiports 𝒩R​P,𝒩P~​Q{\cal N}_{RP},{{\cal N}}_{\tilde{P}Q} be on graphs 𝒢R​S​P,𝒢P~​M​Q{\cal G}_{RSP},{\cal G}_{\tilde{P}MQ} respectively, with the primed and double primed sets obtained from R,S,P,P~,M,Q,R,S,P,\tilde{P},M,Q, being pairwise disjoint, and let them have device characteristics 𝒦S,𝒦M{\cal K}^{S},{{\cal K}}^{M} respectively. Let 𝒦P​P~{\cal K}^{P\tilde{P}} denote a collection of vectors 𝒦P​P~​P​"​P~​"P​P~.{\cal K}^{P\tilde{P}}_{P\tilde{P}P"\tilde{P}"}. The multiport [𝓝𝑹​𝑷⊕𝓝𝑷~​𝑸]∩𝓚𝑷​𝑷~,\boldsymbol{[{\cal N}_{RP}\oplus{{\cal N}}_{\tilde{P}Q}]\cap{\cal K}^{P\tilde{P}}}, with ports R⊎QR\uplus Q obtained by connecting 𝒩R​P,𝒩P~​Q{\cal N}_{RP},{{\cal N}}_{\tilde{P}Q} through 𝒦P​P~{\cal K}^{P\tilde{P}}, is on graph 𝒢R​S​P⊕𝒢P~​M​Q{\cal G}_{RSP}\oplus{\cal G}_{\tilde{P}MQ} (the graph obtained by putting 𝒢R​S​P,𝒢P~​M​Q{\cal G}_{RSP},{\cal G}_{\tilde{P}MQ} together with no common nodes) with device characteristic
𝒦S⊕𝒦M⊕𝒦P​P~.{\cal K}^{S}\oplus{{\cal K}}^{M}\oplus{\cal K}^{P\tilde{P}}. When R,QR,Q are void, [𝒩R​P⊕𝒩P~​Q]∩𝒦P​P~[{\cal N}_{RP}\oplus{{\cal N}}_{\tilde{P}Q}]\cap{\cal K}^{P\tilde{P}} would reduce to [𝒩P⊕𝒩P~]∩𝒦P​P~[{\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 𝒩P{\cal N}_{P} is terminated by 𝒩P~{{\cal N}}_{\tilde{P}} through 𝒦P​P~.{\cal K}^{P\tilde{P}}.
This network is on graph 𝒢S​P⊕𝒢P~​M{\cal G}_{SP}\oplus{\cal G}_{\tilde{P}M} with device characteristic 𝒦S​S​"S⊕𝒦M​M​"M⊕𝒦P​P​"​P~​P~​"P​P~.{\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 𝒜S​P,𝒜P​Q{\cal A}_{SP},{\cal A}_{PQ} be affine spaces on S⊎P,P⊎Q,S\uplus P,P\uplus Q, where S,P,QS,P,Q are pairwise disjoint sets. Let 𝒱S​P,𝒱P​Q{\cal V}_{SP},{\cal V}_{PQ} respectively, be the vector space translates of 𝒜S​P,𝒜P​Q.{\cal A}_{SP},{\cal A}_{PQ}. Let 𝒜S​P↔𝒜P​Q{\cal A}_{SP}\leftrightarrow{\cal A}_{PQ} be nonvoid and let
αS​Q∈𝒜S​P↔𝒜P​Q.\alpha_{SQ}\in{\cal A}_{SP}\leftrightarrow{\cal A}_{PQ}. Then, 𝒜S​P↔𝒜P​Q=αS​Q+(𝒱S​P↔𝒱P​Q).{\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 𝒩P1,𝒩P2{\cal N}^{1}_{P},{{\cal N}}^{2}_{P} be multiports on the same graph 𝒢S​P{\cal G}_{SP} but with device characteristics
𝒜S​S​",𝒱S​S​",{\cal A}_{SS"},{\mbox{$\cal V$}}_{SS"}, respectively where 𝒱S​S​",{\mbox{$\cal V$}}_{SS"}, is the vector space translate of the affine space 𝒜S​S​".{\cal A}_{SS"}. Let 𝒩P1,𝒩P2{\cal N}^{1}_{P},{{\cal N}}^{2}_{P} have port behaviours 𝒜˘P​P​"1,𝒱˘P​P​"2,\breve{{\cal A}}^{1}_{PP"},\breve{\mbox{$\cal V$}}^{2}_{PP"}, respectively. If 𝒜˘P​P​"1≠∅,\breve{{\cal A}}^{1}_{PP"}\neq\emptyset, then
𝒱˘P​P​"2=([𝒱v(𝒢S​P)⊕(𝒱i(𝒢S​P))S​"​P​"]↔𝒱S​S​")P​(−P​"),\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 𝒜˘P​P​"1.\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 𝒱S​P,𝒱P​Q{\cal V}_{SP},{\cal V}_{PQ} be vector spaces respectively on S⊎P,P⊎Q,S\uplus P,P\uplus Q, with S,P,Q,S,P,Q, being pairwise disjoint. We then have, (𝒱S​P↔𝒱P​Q)∗=(𝒱S​P∗⇌𝒱P​Q∗).(\mathcal{V}_{SP}\leftrightarrow\mathcal{V}_{PQ})^{*}\ \hskip 2.84544pt=\hskip 2.84544pt\ (\mathcal{V}_{SP}^{*}\rightleftharpoons\mathcal{V}_{PQ}^{*}). In particular, (𝒱S​P↔𝒱P)∗=𝒱S​P∗↔𝒱P∗.(\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 (𝒱X+𝒱Y)∗=(𝒱X∗∩𝒱Y∗)(\mbox{$\cal V$}_{X}+\mbox{$\cal V$}_{Y})^{*}=(\mbox{$\cal V$}^{*}_{X}\cap\mbox{$\cal V$}_{Y}^{*}) and, using Theorem 1, that (𝒱X∩𝒱Y)∗=(𝒱X∗+𝒱Y∗).(\mbox{$\cal V$}_{X}\cap\mbox{$\cal V$}_{Y})^{*}=(\mbox{$\cal V$}^{*}_{X}+\mbox{$\cal V$}_{Y}^{*}). We have, using Theorem 2, (𝒱S​P↔𝒱P​Q)∗=[(𝒱S​P∩𝒱P​Q)∘SQ]∗=[(𝒱S​P∗+𝒱P​Q∗)×SQ]=(𝒱S​P∗⇌𝒱P​Q∗).({\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 𝒱X,\mbox{$\cal V$}_{X}, we have 𝒱X=𝒱(−X).\mbox{$\cal V$}_{X}=\mbox{$\cal V$}_{(-X)}. Therefore (𝒱S​P↔𝒱P)∗=(𝒱S​P∗⇌𝒱P∗)=𝒱S​P∗↔𝒱P∗.(\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 𝒱Sj​Sj​",𝒱^Sj​Sj​"\mbox{$\cal V$}_{S_{j}S_{j}"},\hat{\mbox{$\cal V$}}_{S_{j}S_{j}"} are orthogonal duals of each other iff 𝒱^Sj​Sj​"=𝒱Sj​Sj​"∗.\hat{\mbox{$\cal V$}}_{S_{j}S_{j}"}=\mbox{$\cal V$}^{*}_{S_{j}S_{j}"}. By Theorem 1,
(𝒱Sj​Sj​"∗)∗=𝒱Sj​Sj​".(\mbox{$\cal V$}^{*}_{S_{j}S_{j}"})^{*}=\mbox{$\cal V$}_{S_{j}S_{j}"}. Let 𝒜Sj​Sj​"{\cal A}_{S_{j}S_{j}"} be an affine space with 𝒱Sj​Sj​"\mbox{$\cal V$}_{S_{j}S_{j}"} as its vector space translate. We say 𝒱^Sj​Sj​"\hat{\mbox{$\cal V$}}_{S_{j}S_{j}"} is the adjoint of 𝒜Sj​Sj​",{\cal A}_{S_{j}S_{j}"}, denoted by 𝒱Sj​Sj​"a​d​j,{\mbox{$\cal V$}}^{adj}_{S_{j}S_{j}"}, iff 𝒱^Sj​Sj​"=(𝒱Sj​Sj​"∗)(−Sj​")​Sj.\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 (⊕𝒜Sj​Sj​")a​d​j=(⊕𝒱Sj​Sj​"a​d​j).(\oplus{\cal A}_{S_{j}S_{j}"})^{adj}=(\oplus\mbox{$\cal V$}_{S_{j}S_{j}"}^{adj}). It is clear that 𝒱Sj​Sj​"=(𝒱^Sj​Sj​"∗)(−Sj​")​Sj=(𝒱Sj​Sj​"a​d​j)a​d​j.{\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 𝒜S​S​"{{\cal A}}_{SS"} be an affine device characteristic. Suppose its vector space translate 𝒱S​S​"{\mbox{$\cal V$}}_{SS"} is defined by (is the solution space of) the hybrid equations

(I0−g11−h120I−h21−r22)​(iS1​"vS2vS1iS2​")=(00),\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 SS is partitioned into S1,S2.S_{1},S_{2}. Then 𝒱S​S​"∗{\mbox{$\cal V$}}^{*}_{SS"} is the complementary orthogonal space defined by

(g11∗h21∗I0h12∗r22∗0I)​(i~S1​"v~S2v~S1i~S2​")=(00),\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 𝒱S​S​"a​d​j:≡(𝒱S​S​"∗)(−S​")​S{\mbox{$\cal V$}}^{adj}_{SS"}:\equiv({\mbox{$\cal V$}}^{*}_{SS"})_{(-S")S} is obtained by replacing the v~\tilde{v} variables by −i^-\hat{i} variables and the i~\tilde{i} variables by v^\hat{v} variables. It can be seen that 𝒱S​S​"a​d​j{\mbox{$\cal V$}}^{adj}_{SS"} is defined by

(g11∗h21∗I0h12∗r22∗0I)​(v^S1−i^S2​"−i^S1​"v^S2)=(00),i.e.,​(−g11∗h21∗I0h12∗−r22∗0I)​(v^S1i^S2​"i^S1​"v^S2)=(00).\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=0v=0,i=0 be a nullator. The adjoint will have its voltage and current unconstrained and is therefore a norator.
2. Let vS=Z​iS​"v_{S}=Zi_{S"} be a (multiport) impedance. The adjoint has the characteristic v^S=Z∗​i^S​".\hat{v}_{S}=Z^{*}\hat{i}_{S"}.
3. Consider the current controlled voltage source (CCVS) and the voltage controlled current source (VCCS) shown below.

(100001−r0)​(v1v2i1i2)=(00);(0010−g001)​(v1v2i1i2)=(00);\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)

(0−r100001)​(i^1i^2v^1v^2)=(00);(100−g0100)​(i^1i^2v^1v^2)=(00).\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,gr,g respectively, but the direction of control which was originally from port 11 to port 22 is now from port 22 to port 1.1.
4. Next consider current controlled current source (CCCS) and the voltage controlled voltage source (VCVS) shown below.

(100000−α1)​(v1v2i1i2)=(00);(0010−β100)​(v1v2i1i2)=(00).\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,

(1000001α)​(i^1i^2v^1v^2)=(00);(00101β00)​(i^1i^2v^1v^2)=(00).\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 11 to port 22 is now from port 22 to port 1.1. 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 𝓝𝑷\boldsymbol{{{\cal N}}_{P}} be on graph 𝒢S​P{\cal G}_{SP} with device characteristic 𝒜S​S​"=xS​S​"+𝒱S​S​"{\cal A}_{SS"}=x_{SS"}+{\mbox{$\cal V$}}_{SS"} on S.S. The multiport 𝓝𝑷𝒉​𝒐​𝒎\boldsymbol{{{\cal N}}^{hom}_{P}} is on graph 𝒢S​P{\cal G}_{SP} but has device characteristic 𝒱S​S​".{\mbox{$\cal V$}}_{SS"}. The adjoint 𝒩Pa​d​j{\cal N}^{adj}_{P} of 𝒩P{{\cal N}}_{P} as well as of 𝒩Ph​o​m{{\cal N}}^{hom}_{P} is on graph 𝒢S​P{\cal G}_{SP} but has device characteristic 𝒱S​S​"a​d​j.{\mbox{$\cal V$}}^{adj}_{SS"}.

Let the solution set of 𝒩P{\cal N}_{P} be [𝒱S​P⊕(𝒱S​P∗)S​"​P​"]∩𝒜S​S​".[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"}]\cap{\cal A}_{SS"}. Then the solution set of 𝒩Ph​o​m{{\cal N}}^{hom}_{P} would be [𝒱S​P⊕(𝒱S​P∗)S​"​P​"]∩𝒱S​S​",[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"}]\cap{\mbox{$\cal V$}}_{SS"}, and that of 𝒩Pa​d​j{\cal N}^{adj}_{P} would be [𝒱S​P⊕(𝒱S​P∗)S​"​P​"]∩𝒱S​S​"a​d​j.[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"}]\cap{\mbox{$\cal V$}}^{adj}_{SS"}.

The port behaviour of 𝒩P{\cal N}_{P} would be [𝒱S​P⊕(𝒱S​P∗)S​"​(−P​")]↔𝒜S​S​",[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"(-P")}]\leftrightarrow{\cal A}_{SS"}, that of 𝒩Ph​o​m{{\cal N}}^{hom}_{P} would be
[𝒱S​P⊕(𝒱S​P∗)S​"​(−P​")]↔𝒱S​S​",[\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"(-P")}]\leftrightarrow{\mbox{$\cal V$}}_{SS"}, and that of 𝒩Pa​d​j{\cal N}^{adj}_{P} would be [𝒱S​P⊕(𝒱S​P∗)S​"​(−P​")]↔𝒱S​S​"a​d​j.[\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, 𝒩Ph​o​m{\cal N}^{hom}_{P} and 𝒩Pa​d​j{\cal N}^{adj}_{P} have adjoint port behaviours.

Theorem 8.

Let 𝒩P1,𝒩P2{\cal N}^{1}_{P},{{\cal N}}^{2}_{P} be multiports on the same graph 𝒢S​P{\cal G}_{SP} but with device characteristics
𝒱S​S​"1,𝒱S​S​"2,\mbox{$\cal V$}^{1}_{SS"},{\mbox{$\cal V$}}^{2}_{SS"}, respectively and port behaviours 𝒱˘P​P​"1,𝒱˘P​P​"2,\breve{\mbox{$\cal V$}}^{1}_{PP"},\breve{\mbox{$\cal V$}}^{2}_{PP"}, respectively. Then if 𝒱S​S​"1,𝒱S​S​"2,\mbox{$\cal V$}^{1}_{SS"},{\mbox{$\cal V$}}^{2}_{SS"}, are adjoints of each other so are 𝒱˘P​P​"1,𝒱˘P​P​"2,\breve{\mbox{$\cal V$}}^{1}_{PP"},\breve{\mbox{$\cal V$}}^{2}_{PP"}, adjoints of each other.

Proof.

Let 𝒱S​P:≡𝒱v(𝒢S​P).\mbox{$\cal V$}_{SP}:\equiv\mbox{$\cal V$}^{v}({\cal G}_{SP}). By Theorem 3, we have ((𝒱v​(𝒢S​P))∗)S​"​P​"=𝒱i​(𝒢S​P).((\mbox{$\cal V$}^{v}({\cal G}_{SP}))^{*})_{S"P"}=\mbox{$\cal V$}^{i}({\cal G}_{SP}).
Therefore (𝒱S​P∗)S​"​P​"=𝒱i​(𝒢S​P).(\mbox{$\cal V$}_{SP}^{*})_{S"P"}=\mbox{$\cal V$}^{i}({\cal G}_{SP}). Let 𝒱S​P​S​"​P​":≡𝒱S​P⊕(𝒱S​P∗)S​"​P​".\mbox{$\cal V$}_{SPS"P"}:\equiv\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}_{SP}^{*})_{S"P"}. It is clear that 𝒱S​P​S​"​P​"a​d​j\mbox{$\cal V$}_{SPS"P"}^{adj}
:≡(𝒱S​P​S​"​P​"∗)(−S​")​(−P​")​S​P=𝒱S​P​S​"​P​".:\equiv(\mbox{$\cal V$}_{SPS"P"}^{*})_{(-S")(-P")SP}=\mbox{$\cal V$}_{SPS"P"}. Let 𝒱P​P​"1:≡𝒱S​P​S​"​P​"↔𝒱S​S​"1\mbox{$\cal V$}^{1}_{PP"}:\equiv\mbox{$\cal V$}_{SPS"P"}\leftrightarrow\mbox{$\cal V$}^{1}_{SS"} and let 𝒱2P​P​":≡𝒱S​P​S​"​P​"↔𝒱S​S​"2.{\mbox{$\cal V$}^{2}}_{PP"}:\equiv{\mbox{$\cal V$}}_{SPS"P"}\leftrightarrow{\mbox{$\cal V$}}^{2}_{SS"}.
We then have, (𝒱P​P​"1)a​d​j:≡(𝒱P​P​"1)(−P​")​P∗=(𝒱S​P​S​"​P​"↔𝒱S​S​")(−P​")​P∗=(𝒱S​P​S​"​P​"∗↔𝒱S​S​"∗)(−P​")​P(\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}
=(𝒱S​P​S​"​P​"∗)S​(−P​")​S​"​P↔𝒱S​S​"∗=(\mbox{$\cal V$}_{SPS"P"}^{*})_{S(-P")S"P}\leftrightarrow\mbox{$\cal V$}_{SS"}^{*} =(𝒱S​P​S​"​P​"∗)(−S​")​(−P​")​S​P↔(𝒱S​S​"∗)(−S​")​S=𝒱S​P​S​"​P​"↔𝒱2S​S​"=𝒱2P​P​".=(\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 𝒱P​P​"1,𝒱P​P​"2\mbox{$\cal V$}^{1}_{PP"},{\mbox{$\cal V$}}^{2}_{PP"} are adjoints and therefore 𝒱˘P​P​"1:≡(𝒱P​P​"1)P​(−P​")\breve{\mbox{$\cal V$}}^{1}_{PP"}:\equiv(\mbox{$\cal V$}^{1}_{PP"})_{P(-P")} and 𝒱˘P​P​"2:≡(𝒱P​P​"2)P​(−P​")\breve{\mbox{$\cal V$}}^{2}_{PP"}:\equiv({\mbox{$\cal V$}}^{2}_{PP"})_{P(-P")} are adjoints. ∎

Corollary 9.

Let 𝒩P:≡(𝒢S​P,𝒜S​S​"),{\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{SS"}), with 𝒱S​S​"\mbox{$\cal V$}_{SS"} as the vector space translate of 𝒜S​S​".{\cal A}_{SS"}. Suppose 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is the (nonvoid) port behaviour of 𝒩P{\cal N}_{P} with 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} as its vector space translate. Then 𝒩Pa​d​j:≡(𝒢S​P,𝒱S​S​"a​d​j),{\cal N}^{adj}_{P}:\equiv({\cal G}_{SP},\mbox{$\cal V$}^{adj}_{SS"}), has the port behaviour 𝒱˘P​P​"a​d​j.\breve{\mbox{$\cal V$}}^{adj}_{PP"}.

Proof.

By Theorem 5, port behaviour of 𝒩Ph​o​m:≡(𝒢S​P,𝒱S​S​"),{\cal N}^{hom}_{P}:\equiv({\cal G}_{SP},\mbox{$\cal V$}_{SS"}), is the vector space translate of 𝒜˘P​P​".\breve{{\cal A}}_{PP"}. By definition, the adjoint of 𝒩P{\cal N}_{P} and 𝒩Ph​o​m{\cal N}^{hom}_{P} are the same. The result now follows from Theorem 8, noting that (𝒱˘P​P​"a​d​j)P​(−P​")(\breve{\mbox{$\cal V$}}^{adj}_{PP"})_{P(-P")} is the adjoint of (𝒜˘P​P​")P​(−P​").(\breve{{\cal A}}_{PP"})_{P(-P")}. ∎

Remark 1.

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

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

𝒱P​P​"a​d​j\mbox{$\cal V$}^{adj}_{PP"} is defined by

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

i.e., by vP=Z∗​iP.v_{P}=Z^{*}i_{P}. Now let every device in the multiport have the form vSj=Zj​iSj​"+Ejv_{S_{j}}=Z_{j}i_{S_{j}"}+E_{j} and, further, at the ports let the behaviour be vP=Z​iP+E.v_{P}=Zi_{P}+E. Theorem 8 implies that if every device is replaced by vSj=Zj∗​iSj​",v_{S_{j}}=Z^{*}_{j}i_{S_{j}"}, at the ports the behaviour would be vP=Z∗​iP.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 B​vP−Q​iP=s.Bv_{P}-Qi_{P}=s.

2. The devices in a multiport have few device ports. So building their adjoints is easy. Therefore 𝒩Pa​d​j{\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 𝒩P.{\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 𝒩P:≡(𝒢S​P,𝒜S′​S​"),{\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{S^{\prime}S"}), where 𝒜S′​S​"=αS′​S​"+𝒱S′​S​".{\cal A}_{S^{\prime}S"}=\alpha_{S^{\prime}S"}+\mbox{$\cal V$}_{S^{\prime}S"}.
The multiport 𝒩P{\cal N}_{P} is said to be rigid iff every multiport 𝒩^P:≡(𝒢S​P,𝒜^S′​S​"),\hat{{\cal N}}_{P}:\equiv({\cal G}_{SP},\hat{{\cal A}}_{S^{\prime}S"}), where 𝒜^S′​S​"=α^S′​S​"+𝒱S′​S​",\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 𝒜A​B,𝒜B{\cal A}_{AB},{\cal A}_{B} be affine spaces on sets A⊎B,B,A\uplus B,B, respectively, A,B,A,B, disjoint. Further, let 𝒜A​B,𝒜B{\cal A}_{AB},{\cal A}_{B} have vector space translates 𝒱A​B,𝒱B,\mbox{$\cal V$}_{AB},\mbox{$\cal V$}_{B}, respectively.
We say the pair {𝒜A​B,𝒜B}\{{\cal A}_{AB},{\cal A}_{B}\} has the full sum property iff 𝒱A​B∘B+𝒱B=ℱB.\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B}=\mbox{$\cal F$}_{B}.
We say the pair {𝒜A​B,𝒜B}\{{\cal A}_{AB},{\cal A}_{B}\} has the zero intersection property iff 𝒱A​B×B∩𝒱B=𝟎B.\mbox{$\cal V$}_{AB}\times B\cap\mbox{$\cal V$}_{B}={\mathbf{0}}_{B}.
We say that the pair {𝒜A​B,𝒜B}\{{\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 {𝒜A​B,𝒜B}\{{\cal A}_{AB},{\cal A}_{B}\} be a rigid pair and let 𝒱A​B,𝒱B\mbox{$\cal V$}_{AB},\mbox{$\cal V$}_{B} be the vector space translates of 𝒜A​B,𝒜B,{\cal A}_{AB},{\cal A}_{B}, respectively. Then

  1. 1.

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

  2. 2.

    The zero intersection property of {𝒜A​B,𝒜B}\{{\cal A}_{AB},{\cal A}_{B}\} is equivalent to the statement that ,
    if fA∈𝒜A​B↔𝒜Bf_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B} and (fA,fB),(fA,fB′)∈𝒜A​B∩𝒜B,(f_{A},f_{B}),(f_{A},f^{\prime}_{B})\in{\cal A}_{AB}\cap{\cal A}_{B}, then fB=fB′.f_{B}=f^{\prime}_{B}.

  3. 3.

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

  4. 4.

    A multiport 𝒩P{\cal N}_{P} is rigid iff 𝒩Ph​o​m{\cal N}^{hom}_{P} and 𝒩Pa​d​j{\cal N}^{adj}_{P} are rigid.

  5. 5.

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

Proof.

1. Let 𝒜^A​B=(αA,αB)+𝒱A​B,𝒜^B=βB+𝒱B.\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, 𝒜^A​B↔𝒜^B\hat{{\cal A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B} is nonvoid iff 𝒜^A​B∩𝒜^B\hat{{\cal A}}_{AB}\cap\hat{{\cal A}}_{B} is nonvoid,
i.e., iff 𝒜^A​B∘B∩𝒜^B\hat{{\cal A}}_{AB}\circ B\ \cap\ \hat{{\cal A}}_{B} is nonvoid,
i.e., iff there exist λB∈𝒱A​B∘B,σB∈𝒱B,\lambda_{B}\in\mbox{$\cal V$}_{AB}\circ B,\sigma_{B}\in\mbox{$\cal V$}_{B}, such that αB+λB=βB+σB,\alpha_{B}+\lambda_{B}=\beta_{B}+\sigma_{B}, i.e., such that αB−βB=σB−λB.\alpha_{B}-\beta_{B}=\sigma_{B}-\lambda_{B}.
Clearly when 𝒱A​B∘B+𝒱B=ℱB,\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B}=\mbox{$\cal F$}_{B}, there exist λB∈𝒱A​B∘B,σB∈𝒱B,\lambda_{B}\in\mbox{$\cal V$}_{AB}\circ B,\sigma_{B}\in\mbox{$\cal V$}_{B}, such that αB−βB=σB−λB\alpha_{B}-\beta_{B}=\sigma_{B}-\lambda_{B} so that 𝒜^A​B↔𝒜^B\hat{{\cal A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B} is nonvoid.
On the other hand, if 𝒱A​B∘B+𝒱B≠ℱB,\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B}\neq\mbox{$\cal F$}_{B}, there exist αB,βB\alpha_{B},\beta_{B} such that αB−βB∉𝒱A​B∘B+𝒱B,\alpha_{B}-\beta_{B}\notin\mbox{$\cal V$}_{AB}\circ B+\mbox{$\cal V$}_{B}, so that 𝒜^A​B∩𝒜^B\hat{{\cal A}}_{AB}\cap\hat{{\cal A}}_{B} and therefore 𝒜^A​B↔𝒜^B\hat{{\cal A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B} is void.
By Theorem 4, if 𝒜^A​B↔𝒜^B\hat{{\cal A}}_{AB}\leftrightarrow\hat{{\cal A}}_{B} is nonvoid, its vector space translate is 𝒱A​B↔𝒱B.\mbox{$\cal V$}_{AB}\leftrightarrow\mbox{$\cal V$}_{B}.

2. Let (𝒱A​B×B)∩𝒱B=𝟎B.(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B}={\mathbf{0}}_{B}. If fA∈𝒜A​B↔𝒜Bf_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B} and (fA,fB),(fA,fB′)∈𝒜A​B∩𝒜B,(f_{A},f_{B}),(f_{A},f^{\prime}_{B})\in{\cal A}_{AB}\cap{\cal A}_{B}, then
(0A,(fB−fB′))∈𝒱A​B,(0_{A},(f_{B}-f^{\prime}_{B}))\in\mbox{$\cal V$}_{AB}, and similarly (fB−fB′)∈𝒱B,(f_{B}-f^{\prime}_{B})\in\mbox{$\cal V$}_{B}, so that (fB−fB′)∈(𝒱A​B×B)∩𝒱B=𝟎B.(f_{B}-f^{\prime}_{B})\in(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B}={\mathbf{0}}_{B}.
Next suppose whenever fA∈𝒜A​B↔𝒜Bf_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B} and (fA,fB),(fA,fB′)∈𝒜A​B∩𝒜B,(f_{A},f_{B}),(f_{A},f^{\prime}_{B})\in{\cal A}_{AB}\cap{\cal A}_{B}, we have fB=fB′.f_{B}=f^{\prime}_{B}. Suppose (𝒱A​B×B)∩𝒱B≠𝟎B.(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B}\neq{\mathbf{0}}_{B}. Let gB∈(𝒱A​B×B)∩𝒱Bg_{B}\in(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B} and let gB≠0B.g_{B}\neq 0_{B}.
If fA∈𝒜A​B↔𝒜B,f_{A}\in{\cal A}_{AB}\leftrightarrow{\cal A}_{B}, then there exists fBf_{B} such that (fA,fB)∈𝒜A​B(f_{A},f_{B})\in{\cal A}_{AB} and fB∈𝒜B.f_{B}\in{\cal A}_{B}. Clearly
(fA,fB)+(0A,gB)=(fA,fB+gB)∈𝒜A​B(f_{A},f_{B})+(0_{A},g_{B})=(f_{A},f_{B}+g_{B})\in{\cal A}_{AB} and (fB+gB)∈𝒜B,(f_{B}+g_{B})\in{\cal A}_{B}, so that we must have (fA,fB+gB)∈𝒜A​B∩𝒜B.(f_{A},f_{B}+g_{B})\in{\cal A}_{AB}\cap{\cal A}_{B}. But this means fB=fB+gB,f_{B}=f_{B}+g_{B}, a contradiction. We conclude that (𝒱A​B×B)∩𝒱B=𝟎B.(\mbox{$\cal V$}_{AB}\times B)\cap\mbox{$\cal V$}_{B}={\mathbf{0}}_{B}.

3. We have, (𝒱A​B∘B+𝒱B)∗=𝒱A​B∗×B∩𝒱B∗=ℱB∗=𝟎B,(\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
(𝒱A​B×B∩𝒱B)∗=𝒱A​B∗∘B+𝒱B∗=𝟎B∗=ℱB.(\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 𝒩P:≡(𝒢S​P,𝒜S​S​"){\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{SS"}) is equivalent to the rigidity of
(𝒱v​(𝒢S​P)⊕(𝒱i​(𝒢S​P))S​"​P​",𝒱S​S​").(\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
((𝒱v​(𝒢S​P)⊕(𝒱i​(𝒢S​P))S​"​P​")∗,𝒱S​S​"∗),((\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 ((𝒱i​(𝒢S​P))S​P⊕(𝒱v​(𝒢S​P))S​"​P​",𝒱S​S​"∗)((\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 𝒩Pa​d​j:≡(𝒱v(𝒢S​P)⊕(𝒱i(𝒢S​P))S​"​P​",(𝒱S​S​"∗)(−S​")​S).{\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 𝒩:≡(𝒢S,𝒜S​S​"),𝒩h​o​m:≡(𝒢S,𝒱S​S​").{\cal N}:\equiv({\cal G}_{S},{\cal A}_{SS"}),{\cal N}^{hom}:\equiv({\cal G}_{S},\mbox{$\cal V$}_{SS"}). We have r​(𝒱v​(𝒢S)⊕(𝒱i​(𝒢S))S​")=|S|r(\mbox{$\cal V$}^{v}({\cal G}_{S})\oplus(\mbox{$\cal V$}^{i}({\cal G}_{S}))_{S"})=|S| (using Theorems 1, 3) and r​(𝒱S​S​")=|S|.r(\mbox{$\cal V$}_{SS"})=|S|. Both 𝒩{\cal N} as well as 𝒩h​o​m{\cal N}^{hom} have the same 2​|S|×2​|S|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​"K,K" be representative matrices of vector spaces 𝒱,𝒱∗,\mbox{$\cal V$},\mbox{$\cal V$}^{*}, respectively. Consider the equation

(KK")​(x)\displaystyle\begin{pmatrix}K\\ K^{"}\end{pmatrix}\begin{pmatrix}x\end{pmatrix} =(s0).\displaystyle=\begin{pmatrix}s\\ 0\end{pmatrix}. (13)

First note that the coefficient matrix, in Equation 13, has number of rows equal to r​(𝒱)+r​(𝒱∗)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​"K,K" are linearly independent, in turn implies that a nonzero vector x,x, lies in the intersection of complementary orthogonal complex vector spaces, i.e., satisfies ⟨x,x⟩=0,\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 𝒜˘P​P​"\breve{{\cal A}}_{PP"} be the solution space of the equation B​vP−Q​iP​"=s,Bv_{P}-Qi_{P"}=s, with linearly independent rows and let 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} be the solution space of the equation B​vP−Q​iP​"=0.Bv_{P}-Qi_{P"}=0. Let the dual multiport behaviour 𝒱˘P​P​"∗\breve{\mbox{$\cal V$}}^{*}_{PP"} be the solution space of equation B"​vP−Q"​iP​"=0,B^{"}v_{P}-Q^{"}i_{P"}=0, where the rows of (B"|−Q")(B^{"}|-Q^{"}) form a basis for the space complementary orthogonal to the row space of (B|−Q).(B|-Q).

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

(B⋮−QB"⋮−Q")​(vPiP​")\displaystyle\begin{pmatrix}B&\vdots&-Q\\ B^{"}&\vdots&-Q^{"}\end{pmatrix}\begin{pmatrix}v_{P}\\ i_{P"}\end{pmatrix} =(s0).\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 𝒩P{\cal N}_{P} by another multiport 𝒩P2{\cal N}^{2}_{{P}} in such a way that the port voltage and current vectors of 𝒩P{\cal N}_{P} satisfy an equation of the kind (14) above. Then the port voltage and current vectors of 𝒩P{\cal N}_{P} would be unique and would uniquely determine the port voltage and current vectors of 𝒩P2.{\cal N}^{2}_{{P}}. If both multiports are rigid, this would also uniquely determine the internal voltage and current vectors of both 𝒩P{\cal N}_{P} and 𝒩P2,{\cal N}^{2}_{{P}}, which means that the network, obtained by terminating 𝒩P{\cal N}_{P} by 𝒩P2,{\cal N}^{2}_{{P}}, has a unique solution. We show below that we can build 𝒩P2{\cal N}^{2}_{{P}} by first building 𝒩P~a​d​j{\cal N}^{adj}_{\tilde{P}} and attaching 1:11:1 gyrators to its ports.

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

Let 𝒱S​S​"{\mbox{$\cal V$}}_{SS"} be the vector space translate of 𝒜S​S​"{{\cal A}}_{SS"} and 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} be that of 𝒜˘P​P​".\breve{{\cal A}}_{PP"}. We note that
(𝒱v​(𝒢S​P)⊕(𝒱i​(𝒢S​P))S​"​P​")↔𝒜S​S​"=(𝒜˘P​P​")P​(−P​").(\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 𝒩P{\cal N}_{P} are equivalent, as far as the variables (vP,iP​")(v_{P},i_{P"}) are concerned, to the first set of equations of Equation 14.
Next, (𝒱v​(𝒢S​P)⊕(𝒱i​(𝒢S​P))S​"​P​")↔𝒱S​S​"=(𝒱˘P​P​")P​(−P​"),(\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 𝒩P~a​d​j\mathcal{N}^{adj}_{\tilde{P}} is 𝒱˘P~​P~​"a​d​j:≡(𝒱˘P​P​"a​d​j)P~​P~​".\breve{\mbox{$\cal V$}}^{adj}_{\tilde{P}\tilde{P}"}:\equiv(\breve{\mbox{$\cal V$}}^{adj}_{PP"})_{\tilde{P}\tilde{P}"}. Further, (𝒱˘P~​P~​"a​d​j)P​"​(−P)=(𝒱˘P~​P~​"a​d​j)(−P​")​P=𝒱˘P​P​"∗.(\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 𝒩P~a​d​j\mathcal{N}^{adj}_{\tilde{P}} together with the constraint vP=−iP~​";iP​"=vP~v_{P}=-i_{\tilde{P}"};i_{P"}=v_{\tilde{P}} (the gyrator 𝐠P​P~\mbox{${\bf g}$}^{{P}\tilde{P}}), are equivalent, as far as the variables (vP,iP​")(v_{P},i_{P"}) are concerned, to the defining equations of 𝒱˘P​P​"∗,\breve{\mbox{$\cal V$}}^{*}_{PP"}, i.e., to the second set of equations of Equation 14.

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

The device characteristic 𝒜S​S​"⊕𝒱S~​S~​"a​d​j⊕𝐠P​P~{\cal A}_{SS"}\oplus{\mbox{$\cal V$}}^{adj}_{\tilde{S}\tilde{S}"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}} is proper because r​(𝒱S​S​"⊕𝒱S~​S~​"a​d​j)=r​(𝒱S​S​")+r​(𝒱S​S​"∗)=2​|S|=|S|+|S~|,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​(𝐠P​P~)=2​|P|=|P⊎P~|,r(\mbox{${\bf g}$}^{{P}\tilde{P}})=2|P|=|P\uplus\tilde{P}|, so that dimension of 𝒱S​S​"⊕𝒱S~​S~​"a​d​j⊕𝐠P​P~\mbox{$\cal V$}_{SS"}\oplus{\mbox{$\cal V$}}^{adj}_{\tilde{S}\tilde{S}"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}} equals |S|+|S~|+|P⊎P~|.|S|+|\tilde{S}|+|P\uplus\tilde{P}|. Since the network [𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~[\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 xP​P​"p:≡(vP,−iP​")∈𝒜˘P​P​".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 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} of 𝒜˘P​P​".\breve{{\cal A}}_{PP"}.
Let 𝐠t​vP​P~\mbox{${\bf g}$}_{tv}^{P\tilde{P}} denote the affine space that is the solution set of the constraints
vej=−ie~j​",ej∈P,j≠t,vet+1=−ie~t​";iei=ve~i​",ei∈P.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 𝐠t​iP​P~\mbox{${\bf g}$}_{ti}^{P\tilde{P}} denote the affine space that is the solution set of the constraints
vei=−ie~i​",ei∈P;iej=ve~j​",ej∈P,j≠t,iet+1=ve~t​".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 [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​vP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{tv}^{{P}\tilde{P}} for each et∈Pe_{t}\in P and [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​iP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{ti}^{{P}\tilde{P}} for each et​"∈P​"e_{t}"\in P" (see Figure 1(b) and 1(c)). (We remind the reader that 𝒩Ph​o​m\mathcal{N}^{hom}_{P} is obtained from 𝒩P{\cal N}_{P} by replacing its device characteristic 𝒜S​S​"{\cal A}_{SS"} by the vector space translate 𝒱S​S​".\mbox{$\cal V$}_{SS"}.)

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

We summarize these steps in the following Algorithm.

Algorithm I

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

Step 1. Build the network 𝒩l​a​r​g​e:≡[𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~{\cal N}^{large}:\equiv[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}^{{P}\tilde{P}} on graph 𝒢S​P⊕𝒢S~​P~{\cal G}_{SP}\oplus{\cal G}_{\tilde{S}\tilde{P}}
with device characteristic 𝒜S​S​"⊕𝒱S~​S~​"a​d​j⊕𝐠P​P~,{{\cal A}}_{SS"}\oplus{\mbox{$\cal V$}}^{adj}_{\tilde{S}\tilde{S}"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}}, where 𝒱S​S​"{\mbox{$\cal V$}}_{SS"} is the vector space translate of 𝒜S​S​"{{\cal A}}_{SS"} and 𝒱S~​S~​"a​d​j:≡(𝒱S​S​"∗)(−S~​")​S~.{\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 𝒩l​a​r​g​e{\cal N}^{large} and restrict it to P⊎P​"P\uplus P" to obtain
(vPp,iP​"p).(v^{p}_{P},i^{p}_{P"}). The vector (vPp,−iP​"p)(v^{p}_{P},-i^{p}_{P"}) belongs to 𝒜˘P​P​".\breve{{\cal A}}_{PP"}.
If no solution exists or if there are non unique solutions output ‘𝒩P{\cal N}_{P} not rigid’ and
STOP.
Step 2. Let 𝒩Ph​o​m\mathcal{N}^{hom}_{P} be obtained by replacing the device characteristic 𝒜S​S​"{{\cal A}}_{SS"} by 𝒱S​S​"{\mbox{$\cal V$}}_{SS"} in 𝒩P.\mathcal{N}_{P}.
For t=1,⋯,|P|,t=1,\cdots,|P|, build and solve [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​vP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{tv}^{{P}\tilde{P}} and restrict it to P⊎P​"P\uplus P" to obtain (vPt​v,iP​"t​v).(v^{tv}_{P},i^{tv}_{P"}). (see Figure 1(b).)

The vector (vPt​v,−iP​"t​v)∈𝒱˘P​P​".(v^{tv}_{P},-i^{tv}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.
For t=1,⋯,|P|,t=1,\cdots,|P|, build and solve [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​iP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{ti}^{{P}\tilde{P}} and restrict it to P⊎P​"P\uplus P" to obtain (vPt​i,iP​"t​i).(v^{ti}_{P},i^{ti}_{P"}). The vector (vPt​i,−iP​"t​i)∈𝒱˘P​P​".(v^{ti}_{P},-i^{ti}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}. (see Figure 1(c).)
Step 3. Let 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} be the span of the vectors (vPt​v,−iP​"t​v),(vPt​i,−iP​"t​i),t=1,⋯,|P|.(v^{tv}_{P},-i^{tv}_{P"}),(v^{ti}_{P},-i^{ti}_{P"}),t=1,\cdots,|P|.
Output 𝒜˘P​P​":≡(vPp,−iP​"p)+𝒱˘P​P​".\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 𝒩P:≡(𝒢S​P,𝒜S​S​"){\cal N}_{P}:\equiv({\cal G}_{SP},{\cal A}_{SS"}) be rigid. Let 𝒜˘P​P​"\breve{{\cal A}}_{PP"} be the port behaviour of 𝒩P{\cal N}_{P} and let 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} be the vector space translate of 𝒜˘P​P​".\breve{{\cal A}}_{PP"}. Then the following hold.

  1. 1.

    The network 𝒩l​a​r​g​e:≡[𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~{\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 [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​vP​P~,et∈P,[𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​iP​P~,e​"t∈P​",[\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⊎P​"P\uplus P" gives a vector (vPt​v,iP​"t​v)(v^{tv}_{P},i^{tv}_{P"}) such that (vPt​v,−iP​"t​v)∈𝒱˘P​P​"(v^{tv}_{P},-i^{tv}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"} or a vector (vPt​i,iP​"t​i)(v^{ti}_{P},i^{ti}_{P"}) such that (vPt​i,−iP​"t​i)∈𝒱˘P​P​".(v^{ti}_{P},-i^{ti}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.

  3. 3.

    The vectors (vPt​v,−iP​"t​v),t=1,⋯,|P|,(vPt​i,−iP​"t​i),t=1,⋯,|P|,(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 𝒱˘P​P​".\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 [𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~,[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}^{{P}\tilde{P}}, or that of [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​vP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{tv}^{{P}\tilde{P}} or that of [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​iP​P~[\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 𝒱S​S​"⊕𝒱S​S​"a​d​j⊕𝐠P​P~,{\mbox{$\cal V$}}_{SS"}\oplus{\mbox{$\cal V$}}^{adj}_{SS"}\oplus\mbox{${\bf g}$}^{{P}\tilde{P}}, we get the device characteristic of [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠P​P~.[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}^{{P}\tilde{P}}. All three networks have the same graph 𝒢S​P⊕𝒢S~​P~.{\cal G}_{SP}\oplus{\cal G}_{\tilde{S}\tilde{P}}.
By part 5 of Theorem 11, a linear network 𝒩{\cal N} with a proper device characteristic, has a unique solution iff 𝒩h​o​m{\cal N}^{hom} has a unique solution. We know that [𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~[\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 [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠P​P~,[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}^{{P}\tilde{P}}, has a unique solution and therefore also [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​vP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{tv}^{{P}\tilde{P}} and [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​iP​P~.[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{ti}^{{P}\tilde{P}}.
The restriction of a solution (v^S,v^P,v~S~,v~P~,i^S​",i^P​",i~S~​",i~P~​")(\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 [𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​vP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{tv}^{{P}\tilde{P}} or of
[𝒩Ph​o​m⊕𝒩P~a​d​j]∩𝐠t​iP​P~[\mathcal{N}^{hom}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{ti}^{{P}\tilde{P}} to S⊎P⊎S​"⊎P​"S\uplus P\uplus S"\uplus P" gives a solution of 𝒩Ph​o​m.\mathcal{N}^{hom}_{P}. Its restriction to P⊎P​"P\uplus P" gives (v^P,−i^P​").(\hat{v}_{P},-\hat{i}_{P"}). By Theorem 5, (v^P,i^P​")∈𝒱˘P​P​".(\hat{v}_{P},\hat{i}_{P"})\in\breve{\mbox{$\cal V$}}_{PP"}.
3. Let 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} be the solution space of B​vP−Q​iP​"=0Bv_{P}-Qi_{P"}=0 and let (𝒱˘P​P​"a​d​j)P~​P~​"=(𝒱˘P​P​"∗)(−P~​")​P~,(\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"​vP~+B"​iP~​"=0,Q^{"}v_{\tilde{P}}+B^{"}i_{\tilde{P}"}=0, where the row spaces of (B|−Q),(B"|−Q")(B|-Q),(B^{"}|-Q^{"}) are complementary orthogonal. A vector (vP,vP~,−iP​",−iP~​")(v_{P},v_{\tilde{P}},-i_{P"},-i_{\tilde{P}"}) being the restriction of a solution of [𝒩P⊕𝒩P~a​d​j]∩𝐠t​vP​P~[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{tv}^{{P}\tilde{P}} to P⊎P​"⊎P~⊎P~​"P\uplus P"\uplus\tilde{P}\uplus\tilde{P}" is equivalent to (vP,vP~,iP​",iP~​")(v_{P},v_{\tilde{P}},i_{P"},i_{\tilde{P}"}) being a solution to the equation

(B−Q0000Q"B"I00I0I−I0)​(vPiP​"vP~iP~​")\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} =(00−It0)\displaystyle=\begin{pmatrix}0\\ 0\\ -I^{t}\\ 0\end{pmatrix} (15)

and a vector (vP,vP~,−iP​",−iP~​")(v_{P},v_{\tilde{P}},-i_{P"},-i_{\tilde{P}"}) being the restriction of a solution of [𝒩P⊕𝒩P~a​d​j]∩𝐠t​iP​P~[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}_{ti}^{{P}\tilde{P}} to P⊎P​"⊎P~⊎P~​"P\uplus P"\uplus\tilde{P}\uplus\tilde{P}" is equivalent to (vP,vP~,iP​",iP~​")(v_{P},v_{\tilde{P}},i_{P"},i_{\tilde{P}"}) being a solution to the equation

(B−Q0000Q"B"I00I0I−I0)​(vPiP​"vP~iP~​")\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} =(000−It),\displaystyle=\begin{pmatrix}0\\ 0\\ 0\\ -I^{t}\end{pmatrix}, (16)

where ItI^{t} denotes the tt​ht^{th} column of a |P|×|P||P|\times|P| identity matrix. In the variables vP,iP​",v_{P},i_{P"}, Equation 15 reduces to

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

and Equation 16 reduces to

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

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

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

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

∎

Remark 2.

If the multiport 𝒩P\mathcal{N}_{P} is not rigid, it may not have a solution and then the port behaviour 𝒜˘P​P​"\breve{{\cal A}}_{PP"} would be void. Even if the multiport has a solution, so that the behaviour 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is nonvoid, the above general procedure of solving [𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~,[\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 𝒩P,𝒩P~a​d​j.\mathcal{N}_{P},\mathcal{N}^{adj}_{\tilde{P}}. Therefore, [𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~[\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 [𝒩P⊕𝒩P~a​d​j]∩𝐠P​P~[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap\mbox{${\bf g}$}^{{P}\tilde{P}} has a unique solution iff 𝒩P{\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 11-port transfers maximum power to a load if the latter has value equal to the adjoint (conjugate transpose) of the Thevenin impedance of the 11-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 𝒩P,{\cal N}_{P}, and not explicitly, as an affine space 𝒜˘P​P​".\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,M, we take M¯\overline{M} to be the conjugate and M∗M^{*} to be the conjugate transpose. Our convention for the sign of power associated with a multiport behaviour is that when (v˘P,i˘P​")∈𝒜˘P​P​",(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"}, the power absorbed by the multiport behaviour 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is ⟨v˘P,i˘P​"⟩+⟨i˘P​",v˘P⟩=v˘PT¯​i˘P​"+i˘P​"T¯​v˘P.\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 12\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

minimize(v˘PT¯​i˘P​"+i˘P​"T¯​v˘P)\displaystyle\mbox{minimize}\ \ (\overline{\breve{v}_{P}^{T}}\breve{i}_{P"}+\overline{\breve{i}_{P"}^{T}}\breve{v}_{P}) (20)
(v˘P,i˘P​")∈𝒜˘P​P​",or equivalently,​B​v˘P−Q​i˘P​"=s.\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 (v˘Ps​t​a​t,i˘P​"s​t​a​t),(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}), is a stationary point for the optimization problem 20 , we have

(i˘P​"s​t​a​t)Tδ​v˘¯P+(v˘Ps​t​a​t)Tδ​i˘¯P​"+(i˘P​"s​t​a​t)¯Tδv˘P+(v˘Ps​t​a​t)¯Tδi˘P​"\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"} =0,\displaystyle=0, (22)

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

((i˘P​"s​t​a​t)T|(v˘Ps​t​a​t)T)−λT​(B¯|−Q¯)=0.{((\breve{i}^{stat}_{P"})^{T}|(\breve{v}^{stat}_{P})^{T})-\lambda^{T}(\overline{B}|-\overline{{Q}})}=0.

Thus the stationarity at (v˘Ps​t​a​t,i˘P​"s​t​a​t),(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}), is equivalent to

((v˘Ps​t​a​t)T|(i˘P​"s​t​a​t)T)−λT​(−Q¯|B¯)=0.{((\breve{v}^{stat}_{P})^{T}|(\breve{i}^{stat}_{P"})^{T})-\lambda^{T}(-\overline{Q}|\overline{{B}})}=0.

Since we must have

B​v˘Ps​t​a​t−Q​i˘P​"s​t​a​t=s,B\breve{v}^{stat}_{P}-Q\breve{i}^{stat}_{P"}=s,

the stationarity condition reduces to

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

The vector space translate, 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} of 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is the solution space of the equation, B​v˘P−Q​i˘P​"=0,B\breve{v}_{P}-Q\breve{i}_{P"}=0, and (𝒱˘P​P​"a​d​j)P​(−P​"):≡(𝒱˘P​P​"∗)P​"​P,(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}:\equiv(\breve{\mbox{$\cal V$}}^{*}_{PP"})_{P"P}, is the row space of (−Q¯|B¯).(-\overline{Q}|\overline{B}).
Thus, the stationarity condition says that (v˘Ps​t​a​t,i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}) belongs to 𝒜˘P​P​"∩(𝒱˘P​P​"a​d​j)P​(−P​").\breve{{\cal A}}_{PP"}\cap(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}.
In the case where the multiport has a Thevenin impedance ZZ and Thevenin voltage E,E, the above stationarity condition reduces to the condition (Z+Z∗)(−i˘P​"s​t​a​t)T)=E(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 λ\lambda vector we get a unique stationary vector (v˘Ps​t​a​t,i˘P​"s​t​a​t).(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}). We next show that the stationarity condition is achieved at the ports of the multiport 𝒩P,{\cal N}_{P}, if we terminate it by 𝒩P~a​d​j{\cal N}^{adj}_{\tilde{P}} through the ideal transformer 𝒯P​P~{\cal T}^{P\tilde{P}} resulting in the network [𝒩P⊕𝒩P~a​d​j]∩𝒯P​P~,[{\cal N}_{P}\oplus{\cal N}^{adj}_{\tilde{P}}]\cap{\cal T}^{P\tilde{P}}, where the ideal transformer 𝒯P​P~{\cal T}^{P\tilde{P}} satisfies the equations vP=vP~;iP=−iP~.v_{P}=v_{\tilde{P}};i_{P}=-i_{\tilde{P}}.

Theorem 13.

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

  1. 1.

    A vector (vP,iP​")(v_{P},i_{P"}) is the restriction of a solution of the network 𝒩l​a​r​g​e:≡[𝒩P⊕𝒩P~a​d​j]∩𝒯P​P~{\cal N}^{large}:\equiv[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal T}^{{P}\tilde{P}} to P⊎P​",P\uplus P", iff (vP,−iP​")∈𝒜˘P​P​"∩(𝒱˘P​P​"a​d​j)P​(−P​").(v_{P},-i_{P"})\in\breve{{\cal A}}_{PP"}\cap(\breve{\mbox{$\cal V$}}_{PP"}^{adj})_{P(-P")}.

  2. 2.

    Let (v˘Ps​t​a​t,i˘P​"s​t​a​t)∈𝒜˘P​P​".(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal A}}_{PP"}. Then (v˘Ps​t​a​t,i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}) satisfies stationarity condition with respect to the power absorbed by (v˘P,i˘P​")∈𝒜˘P​P​"(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"} iff (v˘Ps​t​a​t,i˘P​"s​t​a​t)∈𝒜˘P​P​"∩(𝒱˘P​P​"a​d​j)P​(−P​").(\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 (vP1,−iP​"1)(v^{1}_{P},-i^{1}_{P"}) be the restriction of a solution of the multiport 𝒩P,{\cal N}_{P}, to P⊎P​".P\uplus P". Then (v˘P1,i˘P​"1)(\breve{v}^{1}_{P},\breve{i}^{1}_{P"}) satisfies the stationarity condition with respect to the power absorbed by (v˘P,i˘P​")∈𝒜˘P​P​"(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"} iff (vP1,−iP​"1)(v^{1}_{P},-i^{1}_{P"}) is the restriction of a solution of the network [𝒩P⊕𝒩P~a​d​j]∩𝒯P​P~,[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal T}^{{P}\tilde{P}}, to P⊎P​".P\uplus P".

Proof.

1. The restriction of the set of solutions of 𝒩P\mathcal{N}_{P} on graph 𝒢S​P{\cal G}_{SP} to P⊎P​",P\uplus P", is
[(𝒱S​P⊕(𝒱S​P∗)S​"​P​")∩𝒜S​S​"]∘P​P​".[(\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"})\cap{\cal A}_{SS"}]\circ{PP"}. This is the same as (𝒜˘P​P​")P​(−P​").(\breve{{\cal A}}_{PP"})_{P(-P")}.
The restriction of the set of solutions of 𝒩P~a​d​j\mathcal{N}^{adj}_{\tilde{P}} on the disjoint copy 𝒢S~​P~{\cal G}_{\tilde{S}\tilde{P}} of 𝒢S​P,{\cal G}_{SP}, to P~⊎P~​"\tilde{P}\uplus\tilde{P}" is
[(𝒱S~​P~⊕(𝒱S~​P~∗)S~​"​P~​")∩𝒱S~​S~​"a​d​j]∘P~​P~​".[(\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 (𝒱˘P​P​"a​d​j)P~−P~​".(\breve{\mbox{$\cal V$}}^{adj}_{PP"})_{\tilde{P}-\tilde{P}"}.
The restriction of the set of solutions of 𝒩Pa​d​j\mathcal{N}^{adj}_{P} to P⊎P​"{P}\uplus{P}" is [(𝒱S​P⊕(𝒱S​P∗)S​"​P​")∩𝒱S​S​"a​d​j]∘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}"}.
We thus have, [((𝒱S~​P~⊕(𝒱S~​P~∗)S~​"​P~​")∩𝒱S~​S~​"a​d​j)∩𝒯P​P~]∘P​P​"=[((𝒱S~​P~⊕(𝒱S~​P~∗)S~​"​P~​")∩𝒱S~​S~​"a​d​j)∘P~​P~​"]P​(−P​")[((\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")}
=([(𝒱S​P⊕(𝒱S​P∗)S​"​P​")∩𝒱S​S​"a​d​j]∘P​P​")P​(−P​")=𝒱˘P​P​"a​d​j,=([(\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 𝒯P​P~{\cal T}^{P\tilde{P}} are precisely the ones that satisfy vP=vP~​",iP​"=−iP~​".v_{P}=v_{\tilde{P}"},i_{P"}=-i_{\tilde{P}"}.
The restriction of the set of solutions of [𝒩P⊕𝒩P~a​d​j]∩𝒯P​P~,[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal T}^{{P}\tilde{P}}, to P⊎P​"P\uplus P" is therefore equal to
[[(𝒱S​P⊕(𝒱S​P∗)S​"​P​")∩𝒜S​S​"]∘P​P​"]∩[[(𝒱S~​P~⊕(𝒱S~​P~∗)S~​"​P~​")∩𝒱S~​S~​"a​d​j∩𝒯P​P~]∘P​P​"][[(\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}"}]
=(𝒜˘P​P​")P​(−P​")∩𝒱˘P​P​"a​d​j.=(\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 (vE,iE​")(v_{E},i_{E"}) is given by ⟨vE,iE​"⟩+⟨iE​",vE⟩.\langle v_{E},i_{E"}\rangle+\langle i_{E"},v_{E}\rangle.
A vector space 𝒱S​S​"\mbox{$\cal V$}_{SS"} is passive, iff the power absorbed by (xS,yS​")(x_{S},y_{S"}) is nonnegative, whenever (xS,yS​")∈𝒱S​S​".(x_{S},y_{S"})\in\mbox{$\cal V$}_{SS"}. It is strictly passive iff the power absorbed by every nonzero vector in 𝒱S​S​"\mbox{$\cal V$}_{SS"} is positive. An affine space 𝒜S​S​"{\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 𝒩P{\cal N}_{P} be a multiport on graph 𝒢S​P{\cal G}_{SP} with device characteristic 𝒜S​S​".{\cal A}_{SS"}.

  1. 1.

    If 𝒜S​S​"{\cal A}_{SS"} is passive so is the port behaviour 𝒜˘P​P​"\breve{{\cal A}}_{PP"} of 𝒩P.{\cal N}_{P}.

  2. 2.

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

Proof.

1. We assume that 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is nonvoid. We have,
𝒜˘P​P​"=((𝒱S​P⊕(𝒱S​P∗)S​"​P​")↔𝒜S​S​")P​(−P​"),\breve{{\cal A}}_{PP"}=((\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"})\leftrightarrow{\cal A}_{SS"})_{P(-P")}, where 𝒱S​P:≡𝒱v(𝒢S​P).\mbox{$\cal V$}_{SP}:\equiv\mbox{$\cal V$}^{v}({\cal G}_{SP}).
By Theorem 3, 𝒱S​"​P​"∗=(𝒱i​(𝒢S​P))S​"​P​".\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 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is 𝒱˘P​P​"=((𝒱S​P⊕(𝒱S​P∗)S​"​P​")↔𝒱S​S​")P​(−P​").\breve{\mbox{$\cal V$}}_{PP"}=((\mbox{$\cal V$}_{SP}\oplus(\mbox{$\cal V$}^{*}_{SP})_{S"P"})\leftrightarrow\mbox{$\cal V$}_{SS"})_{P(-P")}.
Let (vP,−iP​")(v_{P},-i_{P"}) belong to 𝒱˘P​P​".\breve{\mbox{$\cal V$}}_{PP"}. Then there exist (vS,vP)∈𝒱S​P(v_{S},v_{P})\in\mbox{$\cal V$}_{SP} and (iS​",iP​")∈(𝒱S​P∗)S​"​P​",(i_{S"},i_{P"})\in(\mbox{$\cal V$}^{*}_{SP})_{S"P"}, such that (vS,iS​")∈𝒱S​S​".(v_{S},i_{S"})\in\mbox{$\cal V$}_{SS"}.
By the orthogonality of 𝒱S​P,𝒱S​P∗,\mbox{$\cal V$}_{SP},\mbox{$\cal V$}^{*}_{SP}, it follows that ⟨(vS,vP),(iS​",iP​")⟩=⟨vS,iS​"⟩+⟨vP,iP​"⟩=0,\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 𝒱S​S​",\mbox{$\cal V$}_{SS"}, it follows that ⟨vS,iS​"⟩+⟨iS​",vS⟩≥0.\langle v_{S},i_{S"}\rangle+\langle i_{S"},v_{S}\rangle\geq 0. Therefore ⟨vP,−iP​"⟩+⟨−iP​",vP⟩≥0.\langle v_{P},-i_{P"}\rangle+\langle-i_{P"},v_{P}\rangle\geq 0.
2. Without loss of generality, we assume that the graph 𝒢S​P{\cal G}_{SP} is connected. If PP contains no cutset or circuit of 𝒢S​P,{\cal G}_{SP}, then SS contains both a tree as well as a cotree of 𝒢S​P.{\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 𝒱v​(𝒢S​P)×P=𝒱S​P×P\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 (𝒱i​(𝒢S​P)×P)P​"=𝒱S​"​P​"∗×P​"(\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 (vP,−iP​")∈𝒱˘P​P​",(vP,−iP​")≠0P​P​".(v_{P},-i_{P"})\in\breve{\mbox{$\cal V$}}_{PP"},(v_{P},-i_{P"})\neq 0_{PP"}. As in part 1 above, there exist (vS,vP)∈𝒱S​P(v_{S},v_{P})\in\mbox{$\cal V$}_{SP} and (iS​",iP​")∈𝒱S​P∗,(i_{S"},i_{P"})\in\mbox{$\cal V$}^{*}_{SP}, such that (vS,iS​")∈𝒱S​S​".(v_{S},i_{S"})\in\mbox{$\cal V$}_{SS"}.
Since 𝒱S​P×P\mbox{$\cal V$}_{SP}\times P and 𝒱S​P∗×P\mbox{$\cal V$}^{*}_{SP}\times P are both zero vector spaces, we must have that (vS,iS​")≠0S​S​"(v_{S},i_{S"})\neq 0_{SS"} so that, by strict passivity of 𝒱S​S​",\mbox{$\cal V$}_{SS"}, we have ⟨vS,iS​"⟩+⟨iS​",vS⟩>0.\langle v_{S},i_{S"}\rangle+\langle i_{S"},v_{S}\rangle>0. Further ⟨(vS,vP),(iS​",iP​")⟩=0,\langle(v_{S},v_{P}),(i_{S"},i_{P"})\rangle=0, so that we can conclude ⟨vP,−iP​"⟩+⟨−iP​",vP⟩>0.\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 (v˘Ps​t​a​t,i˘P​"s​t​a​t)∈𝒜˘P​P​",(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal A}}_{PP"}, satisfy the stationarity condition ((v˘Ps​t​a​t)T,(i˘P​"s​t​a​t)T)=λT​(−Q¯|B¯),((\breve{v}^{stat}_{P})^{T},(\breve{i}^{stat}_{P"})^{T})={\lambda}^{T}(-\overline{Q}|\overline{B}), for some λ.{\lambda}. For any (v˘P,i˘P​")∈𝒜˘P​P​",(\breve{v}_{P},\breve{i}_{P"})\in\breve{{\cal A}}_{PP"}, i.e., such that B​v˘P−Q​i˘P​"=s,{B}\breve{v}_{P}-{Q}\breve{i}_{P"}=s, we can write (v˘P,i˘P​")=(v˘Ps​t​a​t+Δ​v˘P,i˘P​"s​t​a​t+Δ​i˘P​").(\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​Δ​v˘P−Q​Δ​i˘P​"=0{B}\Delta\breve{v}_{P}-{Q}\Delta\breve{i}_{P"}=0 and [B¯(Δ​v˘P)¯−Q¯(Δi˘P​"¯)]=0.[\overline{B}\overline{(\Delta\breve{v}_{P})}-\overline{Q}\overline{(\Delta\breve{i}_{P"}})]=0. Therefore ⟨v˘P,i˘P​"⟩+⟨i˘P​",v˘P⟩=⟨v˘P,i˘P​"⟩+⟨i˘P​",v˘P⟩−λT¯[B(Δv˘P)−Q(Δi˘P​")]−λT[B¯(Δ​v˘P)¯−Q¯(Δi˘P​"¯)]\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"}})]
=⟨v˘Ps​t​a​t,i˘P​"s​t​a​t⟩+⟨i˘P​"s​t​a​t,v˘Ps​t​a​t⟩+(Δ​v˘P)T​i˘P​"s​t​a​t¯+(v˘Ps​t​a​t)T​(Δ​i˘P​")¯+(i˘P​"s​t​a​t)T​(Δ​v˘P)¯+(Δ​i˘P​")T​v˘Ps​t​a​t¯=\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}}
+⟨Δv˘P,Δi˘P​"⟩+⟨Δi˘P​",Δv˘P⟩−λT[B¯(Δ​v˘P)¯−Q¯(Δi˘P​"¯)]−λT¯[B(Δv˘P)−Q(Δi˘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
⟨v˘Ps​t​a​t,i˘P​"s​t​a​t⟩+⟨Δ​v˘P,Δ​i˘P​"⟩+((v˘Ps​t​a​t)T+λT​Q¯)​Δ​i˘P​"¯+((i˘P​"s​t​a​t)T−λT​B¯)​Δ​v˘P¯\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}}
+⟨i˘P​"s​t​a​t,v˘Ps​t​a​t⟩+⟨Δ​i˘P​",Δ​v˘P⟩+((v˘Ps​t​a​t)T¯+λT¯​Q)​Δ​i˘P​"+((i˘P​"s​t​a​t)T¯−λT¯​B)​Δ​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 ((v˘Ps​t​a​t)T,(i˘P​"s​t​a​t)T)=λT​(−Q¯|B¯),((\breve{v}^{stat}_{P})^{T},(\breve{i}^{stat}_{P"})^{T})={\lambda}^{T}(-\overline{Q}|\overline{B}), this expression reduces to
⟨v˘Ps​t​a​t,i˘P​"s​t​a​t⟩+⟨i˘P​"s​t​a​t,v˘Ps​t​a​t⟩+⟨Δ​v˘P,Δ​i˘P​"⟩+⟨Δ​i˘P​",Δ​v˘P⟩.\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, ⟨v˘P,i˘P​"⟩+⟨i˘P​",v˘P⟩\langle\breve{v}_{P},\breve{i}_{P"}\rangle+\langle\breve{i}_{P"},\breve{v}_{P}\rangle
=⟨v˘Ps​t​a​t,i˘P​"s​t​a​t⟩+⟨i˘P​"s​t​a​t,v˘Ps​t​a​t⟩+⟨Δ​v˘P,Δ​i˘P​"⟩+⟨Δ​i˘P​",Δ​v˘P⟩.=\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 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is passive we have ⟨Δ​v˘P,Δ​i˘P​"⟩+⟨Δ​i˘P​",Δ​v˘P⟩≥0,\langle\Delta\breve{v}_{P},\Delta\breve{i}_{P"}\rangle+\langle\Delta\breve{i}_{P"},\Delta\breve{v}_{P}\rangle\geq 0, so that
⟨v˘P,i˘P​"⟩+⟨i˘P​",v˘P⟩≥⟨v˘Ps​t​a​t,i˘P​"s​t​a​t⟩+⟨i˘P​"s​t​a​t,v˘Ps​t​a​t⟩.\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 𝒜˘P​P​",\breve{{\cal A}}_{PP"}, is maximum when (v˘P,i˘P​")=(v˘Ps​t​a​t,i˘P​"s​t​a​t).(\breve{v}_{P},\breve{i}_{P"})=(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}).
If 𝒜˘P​P​"\breve{{\cal A}}_{PP"} is strictly passive we have ⟨v˘P,i˘P​"⟩+⟨i˘P​",v˘P⟩>⟨v˘Ps​t​a​t,i˘P​"s​t​a​t⟩+⟨i˘P​"s​t​a​t,v˘Ps​t​a​t⟩\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 (Δ​v˘P,Δ​i˘P​")≠0,(\Delta\breve{v}_{P},\Delta\breve{i}_{P"})\neq 0, so that (v˘Ps​t​a​t,i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}) is the unique maximum delivery vector in 𝒜˘P​P​".\breve{{\cal A}}_{PP"}.

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

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

Theorem 15.

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

  1. 1.

    Let (v˘Ps​t​a​t,i˘P​"s​t​a​t)∈𝒜˘P​P​".(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"})\in\breve{{\cal A}}_{PP"}. Then (v˘Ps​t​a​t,i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}) satisfies

    m​a​x​i​m​i​z​e⟨v˘P,−i˘P​"⟩+⟨−i˘P​",v˘P⟩,(v˘P,i˘P​")∈𝒜˘P​P​",\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 (v˘Ps​t​a​t,i˘P​"s​t​a​t)∈𝒜˘P​P​"∩(𝒱˘P​P​"a​d​j)P​(−P​").(\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 (v˘Ps​t​a​t,−i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},-\breve{i}^{stat}_{P"}) be the restriction of a solution of the multiport 𝒩P,{\cal N}_{P}, to P⊎P​".P\uplus P". Then (v˘Ps​t​a​t,i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},\breve{i}^{stat}_{P"}) satisfies the optimization condition in Equation 24, iff (v˘Ps​t​a​t,−i˘P​"s​t​a​t)(\breve{v}^{stat}_{P},-\breve{i}^{stat}_{P"}) is the restriction of a solution of the network [𝒩P⊕𝒩P~a​d​j]∩𝒯P​P~,[\mathcal{N}_{P}\oplus\mathcal{N}^{adj}_{\tilde{P}}]\cap{\cal T}^{{P}\tilde{P}}, to P⊎P​".P\uplus P".

Remark 3.

Let 𝒱˘P​P​"\breve{\mbox{$\cal V$}}_{PP"} be strictly passive. If (0P,iP​")(0_{P},i_{P"}) or (vP,0P​")(v_{P},0_{P"}) belongs to 𝒱˘P​P​",\breve{\mbox{$\cal V$}}_{PP"}, its strict passivity is violated. Therefore r​(𝒱˘P​P​"×P)r(\breve{\mbox{$\cal V$}}_{PP"}\times P) as well as r​(𝒱˘P​P​"×P​")r(\breve{\mbox{$\cal V$}}_{PP"}\times P") must be zero and consequently r​(𝒱˘P​P​"∘P​")r(\breve{\mbox{$\cal V$}}_{PP"}\circ P") as well as r​(𝒱˘P​P​"∘P)r(\breve{\mbox{$\cal V$}}_{PP"}\circ P) must equal |P|,|P|, using Theorem 2. Let (CP|EP​")(C_{P}|E_{P"}) be the representative matrix of 𝒱˘P​P​".\breve{\mbox{$\cal V$}}_{PP"}. It is clear that CP,EP​"C_{P},E_{P"} are nonsingular so that by invertible row transformation, we can reduce (CP|EP​")(C_{P}|E_{P"}) to the form (ZT|I)(Z^{T}|I) or (I|YT).(I|Y^{T}). We then have ⟨vP,iP​"⟩+⟨iP​",vP⟩=vPT​iP​"¯+iP​"T​vP¯=iP​"T​ZT​iP​"¯+iP​"T​Z​iP​"¯=⟨iP​",(Z+Z∗)​iP​"⟩.\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∗)(Z+Z^{*}) being positive definite. In this case (Z+Z∗)(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.