Revisited generalized substitution theorem and its consequences for circuit analysis

The Substitution Theorem (ST) is generally perceived as a mere theoretical curiosity. In this paper, a formerly derived Generalized ST (GST) is carefully revised, which leads to both a Weak Revisited GST (RGST) and a Strong RGST (characterized by noticeably relaxed hypotheses with respect to the GST). Then, despite the common opinion about the ST, such RGSTs are showed to be powerful analytical tools to generalize, make rigorous and rigorously prove several classic results of Circuit Theory, namely: the Substitution Theorem for Multiterminal Circuits (STMC), the Source-Shift Theorem (SST), the Thévenin-Norton Theorem (TNT), the Miller Theorem (MT) alongside its Dual (DMT), and the Augmentation Principle (AP). More specifically, the STMC is extended to an arbitrary set of sources, possibly including nullors. The SST is rigorously derived and possible related ambiguities are removed. Also, all possible hybrid forms of the TNT for multiports are individuated and a precise operative procedure for calculating the relevant entities is provided for all cases. Furthermore, the MT and the DMT are extended to an arbitrary number of variables and to multiports. As to the AP, the constraint regarding the linearity of the augmenting resistors is removed. Finally, thoroughly worked examples are given in which the aforementioned noteworthy consequences of the RGSTs are proved to be efficient tools for analysis by inspection of linear and nonlinear circuits. Among the other things, systematic pencil-and-paper procedures for DC-point and input-output (or driving-point) characteristic calculation in nonlinear networks are derived and applied to circuits with considerably complex topology.