Complex bases in active CR networks and their relations with the network excitation

Abstract

AbstractA complex basis can be defined for a set X of variables, such as branch currents and voltages (excluding the exciting currents and voltages), of a linear lumped‐constant network in the sinusoidal steady state. If a subset Xβ of X satisfies the condition that any variable in X – Xβ can be expressed as a linear combination of variables in Xβ with constant (not including the frequency) coefficients, then Xβ is called a complex basis of X. This paper derives general graph‐theoretical expressions for complex bases of the set of branch voltages and the set of modified branch voltages (= capacitor voltages x j, and/or resistor voltages) of an active CR network which satisfies certain conditions. The expressions derived have the same form as that in the case of a passive CR network. Second, the independence of the modified branch voltages which are obtained by varying both the phasor‐values (= effective values and phases) and the frequency of the excitation, is discussed. Let Xβ(k) be the vector of complex basis variables obtained by the kth excitation and let Xβ = [Xβ(1) ‐Xβ(k) ‐ Xβ(±) ] be the matrix obtained by A excitation. Then it is shown that there always exist some excitations which make Xβ regular. Finally, excitations which make Xβ regultvr can be actually specified, if the network satisfies certain conditions.Acknowledgements. This research was partly supported by the Grant‐in‐Aid for Scientific Research by the Ministry of Education, Science and Culture of Japan, under Grant General Research (C) 58550232 (1983).