Structural characterization of hybrid equations with tractability index at most two

In circuit simulation, differential-algebraic equations (DAEs) to be solved numerically are obtained through circuit analysis methods. While the modified nodal analysis (MNA) is the most popular method, the hybrid analysis has been shown to have inherent advantage over MNA in terms of index, which is a measure of the numerical difficulty of DAEs. A recent paper of Iwata, Takamatsu, and Tischendorf (2012) has characterized the network structure that leads to DAEs with tractability index zero and one in the hybrid analysis. As a sequel, this paper gives a necessary condition for the hybrid equations to have tractability index at most two. Moreover, we show that this condition is also sufficient for linear time-invariant circuits if dependent sources satisfy a genericity assumption. Since commonly-used circuits fulfill the condition, it is ensured that the hybrid analysis results in a DAE with tractability index at most two in most cases. By combining our results with the previous one, we obtain criteria for tractability index zero/one/two.