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