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

Abstract

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, let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n =</tex> number of terminal-pairs, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_k =</tex> voltage drop across the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> -th terminal-pair. The mean value of reactive energy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> stored in this network is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T = \frac {1}{2j} \Sum_{k=1}^{n}(\bar{E}_k \frac{d}{d\omega} I_k + \bar{I}_k \frac{d}{d\omega} E_k)</tex> . Theorem II: Suppose that an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -terminal-pair reactance network terminated by resistances is driven by a sinusoidal source. Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_0 =</tex> emf of generator, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S =</tex> voltage reflection coefficient at driving terminal-pair, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_1 =</tex> inner resistance of generator, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_k =</tex> resistance terminating <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> -th terminal-pair, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D_k =</tex> the ratio of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_0</tex> to the voltage measured across the resistance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_k</tex> . Then the mean value of the reactive energy stored in the network is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T = \frac{|E_0|^2}{4R_1} |S|^2 \frac{d}{d\omega} (-\arg S)</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+ \Sum_{k=2}{n} \frac {|E_0|^2}{R_k |D_k|^2 } \frac {d}{d\omega} (\arg D_k)</tex> . Some additional remarks, especially on the special but rather practical forms derived from these two theorems, are described.