Having described two-terminal and multi-terminal circuit elements in the “first part” of this book, we have hence discussed the laws of elements. We will now study, in this chapter and the next, KCL/KVL based circuit theoretic techniques that allow us to analyze circuits of varying degrees of “complexity” (a term we make precise in Chap. 4). We will study these techniques by following the classical approach: discussing static (resistive) networks in this chapter and then dynamic (inductive, capacitive, and memristive) networks in Chap. 4. Such a division is not accidental: in terms of circuit variables, dynamic networks usually involve differential equations, unlike static networks. Hence in this chapter we will study simpler resistive networks. We will first discuss the fundamental concept of operating points. Next, we will expand upon graph theoretic concepts and then discuss two of the most important techniques: nodal and tableau analysis. We will conclude the chapter by discussing some general properties of linear resistive networks (superposition, Thévenin-Norton theorems) and nonlinear resistive networks (strict passivity, strict monotonicity).
Typical v − i plot ((0, 0) is in the lower-left corner) of a negative resistance device, the 1N3716 tunnel-diode. Notice the effects of parasitics are visible in the form of hysteresis in the negative resistance region.
A self-loop contains precisely one node and one branch, they are not loops according to Definition 1.6 (of a loop).
4.
If a digraph is not connected, there are two simple solutions to the problem: one approach would be to treat each graph separately. In this case, each part would have its own incidence matrix and ground node. The other approach would be to use a hinged graph, as described in Sect. 3.2.1. We will use both approaches in this book.
5.
Of course, all circuit elements are ideal. We will, nevertheless, occasionally throw in the word “ideal” to remind the reader that “nonphysical” answers (e.g., the Schmitt trigger VTC) are quite possible and even expected. When they do occur, the culprit is not the theory, but the model. Such situations can only be rectified by returning to the drawing board to come up with a more detailed circuit model. Again, in the case of the Schmitt trigger, we will account for physical parasitics to explain the observed behavior.
6.
We will however be able to use tableau analysis from Sect. 3.5 to analyze circuits with any resistive element.
7.
Recall both v and i must be eliminated in node analysis, leaving e as the only variable.
8.
The reader may wish to scan Example 3.5.1 after each step in order to get familiarized first with the notations used in writing the tableau equation.
9.
The “uniqueness” is of course relative to a particular choice of element and node numbers.
10.
Recall that a linear resistive circuit may contain, in addition to two-terminal resistors and independent sources, any multi-terminal or multi-port linear resistors (for example, ideal transformers, gyrators, and all four types of linear-dependent sources).
11.
Compare this description to the definition of superposition from Exercise 1.9.
12.
If you are unfamiliar with the QUCS component notation for the passive sign convention, please be sure to go through the introductory QUCS video online (refer to lab 1).
References
Chua, L.O.: Introduction to Nonlinear Network Theory. McGraw-Hill, New York (1969) (out of print)
