论文标题
隐式线性代数和基本电路理论
Implicit Linear Algebra and Basic Circuit Theory
论文作者
论文摘要
在本文中，我们使用“隐式线性代数”（ILA）得出了电路理论的一些基本结果。这种方法具有简单性和一般性的优势。隐式线性代数在[1]中概述。我们用$ \ mathcal {f} _s $表示所有向量的空间，以及仅包含$ s $上的零向量的空间，$ \ mathbf {0} _s。 $ \ Mathcal {V} _s $中的矢量的点产品为零。 ILA的基本操作是向量空间之间的链接操作（'匹配的组合）$ \ Mathcal {V} _ {sp}，\ Mathcal {V} _ {pq} $（被视为c p，p \ p \ p \ p \ p \ p \ p \ c $ $ s $ s $ s，q，p，p p p p p p p， $ \ MATHCAL {V} _ {sp} \ leftrightArrow \ Mathcal {V} _ {pq} \ equiv \ {（f_s，h_q）： \ Mathcal {V} _ {pq} \}，$和另一个（由$ \ Mathcal {v} _ {sp} \ rightleftharpoons \ Mathcal \ Mathcal {V} _ {V} _ {v} _ {pq} _ {pq} \ equiv \ equiv \ __s， \ Mathcal {V} _ {sp}，（-g_p，h_q）\ in \ Mathcal {v} _ {pq} \}。$ iLA的基本结果是隐式反转定理（该定理$ \ MATHCAL {V} _ {sp} \ leftrightArrow（\ Mathcal {V} _ {sp} \ leftrightArrow \ Mathcal \ Mathcal {V} _S _S）= \ Mathcal {V} \ Mathbf {0} _p \ subseteq \ Mathcal {V} _s \ subseteq \ Mathcal {V} _ {sp} \ LeftrightArrow \ Mathcal \ Mathcal {f} _s $）和含义duality theorem（sp n of $（哪些$） \ Mathcal {V} _ {pq}）^{\ perp} =（\ Mathcal {v} _ {sp}^{\ perp} \ rightleftharpoons \ Mathcal \ Mathcal {v}通过使用ILA介绍Thevenin-Norton定理的概括来说明这一点，在该概述中，我们使用伴随的多端终止通过Gyrator和非常通用的最大功率传递定理来计算多层端口行为，这表明在相邻多层终止过程中通过理想的功率变压器出现的端口条件对应于相对的功率传递。
In this paper we derive some basic results of circuit theory using `Implicit Linear Algebra' (ILA). This approach has the advantage of simplicity and generality. Implicit linear algebra is outlined in [1]. We denote the space of all vectors on $S$ by $\mathcal{F}_S$ and the space containing only the zero vector on $S$ by $\mathbf{0}_S.$ The dual $\mathcal{V}_S^{\perp}$ of a vector space $\mathcal{V}_S$ is the collection of all vectors whose dot product with vectors in $\mathcal{V}_S$ is zero. The basic operation of ILA is a linking operation ('matched composition`) between vector spaces $\mathcal{V}_{SP},\mathcal{V}_{PQ}$ (regarded as collections of row vectors on column sets $S\cup P, P\cup Q,$ respectively with $S,P,Q$ disjoint) defined by $\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ}\equiv \{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (g_P,h_Q) \in \mathcal{V}_{PQ}\},$ and another ('skewed composition`) defined by $\mathcal{V}_{SP}\rightleftharpoons \mathcal{V}_{PQ}\equiv \{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (-g_P,h_Q) \in \mathcal{V}_{PQ}\}.$ The basic results of ILA are the Implicit Inversion Theorem (which states that $\mathcal{V}_{SP}\leftrightarrow(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_S)= \mathcal{V}_S,$ iff $\mathcal{V}_{SP}\leftrightarrow \mathbf{0}_P\subseteq \mathcal{V}_S\subseteq \mathcal{V}_{SP}\leftrightarrow\mathcal{F}_S$) and Implicit Duality Theorem (which states that $(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ})^{\perp}= (\mathcal{V}_{SP}^{\perp}\rightleftharpoons \mathcal{V}_{PQ}^{\perp}$). We show that the operations and results of ILA are useful in understanding basic circuit theory. We illustrate this by using ILA to present a generalization of Thevenin-Norton theorem where we compute multiport behaviour using adjoint multiport termination through a gyrator and a very general version of maximum power transfer theorem, which states that the port conditions that appear, during adjoint multiport termination through an ideal transformer, correspond to maximum power transfer.
