Linear Networks; Tellegen Regimes

Kirchhoff’s laws may fail in nonpermissive, transfinite resistive networks. As a result, a general theory for the electrical behavior of all transfinite networks must be based on a more fundamental principle, namely, a generalization of Tellegen’s equation. For a finite network, Tellegen’s equation asserts that any vector of branch voltages satisfying Kirchhoff’s voltage law and any vector of branch currents satisfying Kirchhoff’s current law are orthogonal [26]. This principle can be extended to transfinite networks, which can be used along with Ohm’s law and a certain restriction on the sources to establish a unique “current-voltage regime”—even though Kirchhoff’s laws may be violated around certain loops and at certain nodes. The theory is thoroughly developed in [33, Chapters 3 and 5] and [34, Chapter 5]. We briefly review it in the next section primarily because a substantial simplification arises from our present restriction to permissive networks. Moreover, it turns out that for permissive networks Kirchhoff’s voltage law will always be satisfied around every finite or transfinite loop. The same is true for Kirchhoff’s current law at every node of any rank, but to show this for transfinite nodes requires some work. This latter task occupies most of this chapter.