Relations Between Reactive Energy and Group Delay in Lumped-Constant Networks

The following two theorems are proved: Theorem I: A lumped-constant network is supposed to be excited by means of one or more sinusoidal sources with the same frequency. Here we take out all resistive elements and voltage sources from the network. Then we obtain a multiterminal network including only reactive elements. For this network, letn =number of terminal-pairs,E_k =voltage drop across theK-th terminal-pair. The mean value of reactive energyTstored in this network is given byT = 1/2j ₖ₌₁ⁿ(Ē_k d/dω I_k + Ī_k d/dω E_k). Theorem II: Suppose that ann-terminal-pair reactance network terminated by resistances is driven by a sinusoidal source. LetE_0 =emf of generator,S =voltage reflection coefficient at driving terminal-pair,R_1 =inner resistance of generator,R_k =resistance terminatingK-th terminal-pair,D_k =the ratio ofE_0to the voltage measured across the resistanceR_k. Then the mean value of the reactive energy stored in the network is given byT = |E_0|^2/4R_1 |S|^2 d/dω (- S)+ ₖ₌₂{n} |E_0|^2/R_k |D_k|^2 d/dω ( D_k). Some additional remarks, especially on the special but rather practical forms derived from these two theorems, are described.