Order of complexity of linear active networks

Abstract
The upper bound on the order of complexity σmax of general linear active networks is found. The result thus completes those already obtained for the RLC networks and for a restricted class of active networks.The result is applicable to networks consisting of independent sources, resistors, capacitors, inductors, gyrators, multiwinding ideal transformers and the four types of controlled sources. The voltage graph (NV) and the current graph (NI) associated with the active network (N) are used in the derivation. A common tree is defined as a set of branches which forms a tree in both NV and NI. The network ‘operator matrix’ includes the set of equations resulting from Kirchhoff's voltage and current laws, applied to NV and NI, respectively, and the element behaviour between the corresponding branches in NV and NI.The order of complexity is obtained by an expansion of the determinant of the operator matrix and can be stated with respect to some particular common trees.In the state-space (variable) approach to network synthesis, the order of complexity represents the minimum number of reactive elements required for the realisation

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