Index analysis of semistate systems without passivity restrictions

In the last decades, there has been an increasing interest on semistate models based on differential-algebraic equations (DAEs) for the analysis and simulation of non-linear electrical circuits. Modelling techniques such as Node Tableau Analysis (NTA), Augmented Nodal Analysis (ANA), or Modified Nodal Analysis (MNA), the latter used e.g. in the circuit simulation programs SPICE and TITAN, set up network equations in differentialalgebraic form[48, 59, 60, 139]. The index of a DAE circuit model becomes a standard measure for the analytical and numerical difficulties faced in simulations [16, 60, 65]. Roughly speaking, the notion of the index can be thought of as the number of steps that are necessary to split the original differential-algebraic systemin to two uncoupled systems: an algebraic one, and an explicit differential one. In particular, index zero systems amount to explicit ODEs, while for index one DAEs the aforementioned splitting can be obtained in a relatively simple manner. Differential-algebraic systems with an index higher than one, usually called higher index systems, are more difficult and specific approaches are necessary for their simulation (see e.g. [65, Chap. VII]). In this direction, the topological characterization of low index (index less than two) circuit configurations has become increasingly important, and it is performed by current circuit simulators [46]. Characterizations of this type do not only place analytical conditions on the circuit devices but they also demand the existence or absence of certain configurations on the circuit digraph, which retains the electrical nature of the circuit elements but not their specific constitutive equations. In previous works, passivity assumptions on circuit devices have been very helpful to simplify the characterization of the index for the resulting models [48, 131, 139]. These assumptions amount to the positive definiteness of the incremental conductance and reactance matrices, this being equivalent to demanding that all conductances and reactances are positive in uncoupled circuits. Restricting the coupling effects allowed in the circuit, the present work introduces novel tree-based methods allowing us to characterize the index of common nodal models in a more general framework, based on algebraic assumptions on certain trees within the network. This tree-based index calculation generalizes previous results, making it possible to characterize the index of uncoupled circuits including both passive and active devices. Our results focus mainly on the augmented nodal analysis and the modified nodal analysis formulations. While modified nodal analysis models have been widely studied froma non-linear DAE perspective [45, 48, 95, 130, 139], the augmented nodal analysis formulation was presented in [131] as an intermediate step between MNA and NTA, preserving the index one conditions of node tableau. In the present work, we employ different types of trees for the characterization of low index configurations in the different models. Index one ANA systems are characterized by certain conditions on the proper trees in the circuit. In turn, index one conditions for MNA are stated in terms of normal trees. Proper and normal trees were introduced by Bashkow [11] and Bryant [21, 22, 23], respectively. A key step in our proofs is the factorization of the matrices describing index one for ANA and MNA, where the Cauchy-Binet formula allows us to split the topological component of the circuit from the characteristics of the devices. The study of the above-mentioned matrices, in particular of those describing index zero for MNA and index one for ANA, leads to the notion of an augmented nodal matrix. In the abstract terms of a coloured digraph, this type of matrix allows us not only to characterize low index configurations but also to analyze other problems in circuit theory, such as the DC-solvability condition for equilibriump oints of well-posed circuits [31, 46, 54, 55, 130, 137, 138]. In this context, the characterization of proper and normal trees in abstract coloured digraphs defines a result of independent interest, which allows us to delve into the kernel of the augmented nodal matrix. Regarding this problem, we prove that the normal trees of a green/blue connected graph are defined by all possible combinations of a forest of the green subgraph and a tree of the so-called blue-cut minor. Similarly, for three-colour connected graphs, we show that normal trees can be characterized in terms of red-cut minors and normal forests of the green/blue subgraph. Finally, in order to study the rank of augmented nodal matrices for problems including couplings or controlled branches, we present the novel notions of a balanced tree and a regular tree pair. Although they are introduced in the simpler and more general context of coloured digraphs, both notions can be directly transposed to a circuit theoretic setting. This allows us to examine networks including coupled capacitors or Voltage-Controlled Current Sources (VCCS), which are present in most integrated circuits [5, 87, 136]. Specifically, we present here characterizations of the DC-solvability problemand index one configurations in ANA models of circuits including controlled sources. Additionally, index zero configurations in MNA models are examined for circuits including coupled capacitors