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 differential-algebraic form. The index of a DAE circuit model becomes a standard measure for the analytical and numerical difficulties faced in simulations. 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 system into 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. 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. 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. 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 from a non-linear DAE perspective, the augmented nodal analysis formulation was presented 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 and Bryant, 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 equilibrium points of well-posed circuits. 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. Specifically, we present here characterizations of the DC-solvability problem and 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. ____ Los modelos de semiestados basados en ecuaciones algebraico-diferenciales (DAEs) han sido objeto de una gran atencion desde la decada de los 80, habiendo sido aplicados al analisis y a la simulacion de circuitos electricos no lineales. En particular, modelos como el Analisis Tableau (NTA), el Analisis Nodal Aumentado (ANA) o el Analisis Nodal Modificado (MNA), este ultimo muy habitual en programas de simulacion circuital como SPICE y TITAN, proporcionan un sistema algebraico-diferencial. El indice de una DAE es una medida habitual de las dificultades que conlleva su estudio y simulacion. En terminos generales, el indice se puede entender como el numero de pasos necesarios para transformar la DAE original en dos ecuaciones desacopladas: una algebraica y otra diferencial explicita (ODE). En concreto, los sistemas de indice cero son ODEs explicitas, mientras que para los sistemas de indice unidad se puede encontrar un desacoplo de una forma relativamente directa. Los sistemas con indice superior a uno son mas complicados y se requieren tecnicas especificas para su simulacion. En este sentido, la caracterizacion topologica de las configuraciones circuitales de indice bajo (indice menor de dos) cobra relevancia y es realizada por algunos programas de simulacion circuital. Esta caracterizacion se basa tanto en condiciones analiticas sobre los elementos del circuito como en la presencia o ausencia de ciertas configuraciones en el digrafo circuital. En anteriores trabajos, las hipotesis de pasividad para los dispositivos del circuito han sido claves para simplificar la caracterizacion del indice. Estas hipotesis equivalen a la definicion positiva de las matrices de conductancia y de reactancia incremental y, en circuitos sin acoplos, corresponden a que todas las conductancias y reactancias tengan un valor positivo. La presente tesis introduce nuevos metodos basados en arboles para la caracterizacion del indice en circuitos sin acoplos que incluyen tanto elementos activos como pasivos. Nuestros resultados abordan fundamentalmente las formulaciones aumentadas y modificadas. Mientras que MNA ha sido objeto de diferentes estudios desde una perspectiva algebraico-diferencial, ANA se presenta como un paso intermedio entre MNA y NTA, que preserva las condiciones de indice uno para NTA. En esta tesis, las configuraciones de indice uno para ANA quedan caracterizadas por ciertas condiciones sobre los arboles propios del circuito. Por su parte, las configuraciones de indice uno para MNA se describen en termino de arboles normales. Los arboles propios y normales fueron introducidos por Bashkow y Bryant, respectivamente. El estudio de las matrices que caracterizan el indice, en particular de aquellas que describen configuraciones de indice cero para MNA y de indice uno para ANA, nos conduce al concepto de matriz nodal aumentada en un grafo coloreado. Este tipo de matriz no solamente hace posible el analisis de configuraciones de indice bajo sino que tambien permite abordar otros problemas de la teoria de circuitos, en particular, la condicion de solubilidad en continua para puntos de equilibrio de circuitos bien planteados. En este sentido, la caracterizacion de los arboles propios y normales en un grafo coloreado abstracto es un resultado de interes independiente que nos permite analizar el nucleo de las matrices nodales aumentadas. En esta tesis demostramos que los arboles normales en un grafo coloreado verde y azul quedan definidos por todas las combinaciones posibles de un bosque del subgrafo verde y un arbol del conocido como menor de corte azul (blue-cut minor). Demostramos tambien un resultado similar para grafos de tres colores. Finalmente, introducimos las definiciones de arbol equilibrado y par de arboles regular, que permiten el estudio del rango de las matrices nodales aumentadas para grafos coloreados que incluyen acoplos o ramas controladas. Estas definiciones permiten abordar circuitos que incluyen fuentes de corriente controladas por tension (VCCS), que son habituales en la mayor parte de modelos para circuitos integrados. En concreto, en esta tesis caracterizamos configuraciones de indice uno y la condicion de solubilidad en continua para modelos ANA de circuitos que incluyen fuentes controladas. Las configuraciones de indice cero en MNA se estudian para circuitos que incluyen condensadores acoplados.