Networks with very small and very large parasitics: Natural frequencies and stability

This paper considers a nonlinear time-invariant network N (of order n+h+l) which contains, in addition to the usual elements, h stray elements (stray capacitances and lead inductances) and l sluggish elements (chokes and coupling capacitors). It is proved that the asymptotic stability of any equilibrium point of N is guaranteed once the simplified (i.e., with stray and sluggish elements neglected) linearized network and two other linear networks SHand SLare asymptotically stable. The networks SHand SLare obtained by both a physically intuitive argument and by a rigorous one. It is also proved that if any one of the three linear networks is exponentially unstable, then the equilibrium point of N is unstable. This theory explains the commonly occurring fact that N is unstable even though the simplified linearized network is asymptotically stable. An example illustrates the several possibilities. The asymptotic behavior of the natural frequencies of N (valid in a neighborhood of the equilibrium point) is obtained in the proof. It is shown in Appendix II how the natural modes of the three simple networks are related to those of the given network N.