A Definition of Passive Linear Networks in Terms of Time and Energy

A definition of passive linear network is made: (a) The network is linear. (b) If currents of any wave form are fed to the terminals of the network, the total energy delivered to the network is not negative. (c) No voltages appear between any pair of terminals before a current is fed to the network. When this definition is applied to two terminal networks, i.e., impedances, a necessary and sufficient condition that a two-terminal network be linear passive is that its impedance function be a positive real function. An analysis of multiterminal networks yields as a necessary and sufficient condition from the foregoing hypotheses that a certain Hermitian quadratic form be positive definite. In the case of three-terminal networks, it reduces to 4R11R22−(R12+R21)2−(X12−X21)2≥0,where Rij+jXij=Zij are the terms of the matrix of impedances of the network. The relation of this formula to similar but not identical formulas of Gewertz and Llewellyn, and other consequences of the condition, are discussed.