Element interconnection laws
The interconnections among elements constrain across- and through-variable relationships. The first element interconnection law requires the concept of a contour “”: a closed path that does not self-intersect superimposed over the linear graph. The first interconnection law is called the continuity law.
Definition
The sum of the through-variables that flow on into a contour on a linear graph is zero, or, in terms of generalized through-variables \(\mathcal{F}_i\) for \(N\) elements with through variables defined as positive into the contour, \[\begin{aligned} \sum_{i=1}^N \mathcal{F}_i = 0. \end{aligned}\]
Contours can enclose any number of nodes and edges, as illustrated in figure 2.5. Kirchhoff's current law (KCL) is the special case of the continuity law for electronic systems.
The second interconnection law we consider requires the concept of a loop “”: a continuous series of edges that begin and end at the same node, not reusing any edges.1 The second interconnection law is called the compatibility law.
Definition
The sum of the across-variable drops on edges around any closed loop on a linear graph is zero, or, in terms of generalized across variables \(\mathcal{V}_i\) for \(N\) elements in a loop, \[\begin{aligned} \sum_{i=1}^N \mathcal{V}_i = 0. \end{aligned}\]
A loop can be “inner” or “outer,” as shown in figure 2.6. Kirchhoff's voltage law (KVL) is the special case of the compatibility law for electronic systems.
For the system shown, (a) write three unique continuity and three unique compatibility equations. Moreover, (b) write a continuity equation solved for ℱ4 in terms of ℱS and ℱ1. Finally, (c) write a compatibility equation solved for 𝒱5 in terms of 𝒱S, 𝒱3, and 𝒱4.
linear graphs and more
Technically, we need not restrict the definition to series that do not reuse edges for purposes of the compatibility law, but these loops are superfluous and we exclude them here.↩︎
Online Resources for Section 2.3
No online resources.