Implicit Linear Algebra and Basic Circuit Theory
Abstract
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 FS and the space containing only the zero vector on S by 0S. The dual VS of a vector space VS is the collection of all vectors whose dot product with vectors in VS is zero. The basic operation of ILA is a linking operation ('matched composition`) between vector spaces VSP,VPQ (regarded as collections of row vectors on column sets S P, P Q, respectively with S,P,Q disjoint) defined by VSP VPQ \(fS,hQ):((fS,gP)∈ VSP, (gP,hQ) ∈ VPQ\, and another ('skewed composition`) defined by VSP VPQ \(fS,hQ):((fS,gP)∈ VSP, (-gP,hQ) ∈ VPQ\. The basic results of ILA are the Implicit Inversion Theorem (which states that VSP(VSP VS)= VS, iff VSP 0P⊂eq VS⊂eq VSPS) and Implicit Duality Theorem (which states that (VSP VPQ)= (VSP VPQ). 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.