Page 173 - Maxwell House

POYNTING's THEOREM 153

3.4.2 Foster's Reactance Theorem.

Note that the part of the impedance plot shown in Figure 3.4.2c curves around the center of
Smith chart meaning a small reflection from the circuitry. By looking at this impedance plot,
we recognize that the impedance travels clockwise around Smith chart as the frequency is
increased. This observation is not occasional and based on a much deeper concept that is the
part of “Foster’s Reactance Theorem” and another useful tool to check the results of analytical
or computer simulations. The proof of this theorem is based on Maxwell’s equations and can
be considered as the variant of Poynting’s theorem that includes the frequency derivatives of
the electric and magnetic field. For the sake of simplicity, we assume that the complex material
parameters () and () are frequency and time independent.

Applying the chain rule to the second Maxwell’s equation written for some space domain
without internal charge and current sources (passive system) we have

∗
∘ ( x ) = − ( ) = − − ∗
∗ ∗
∗
∗ ∗
∗ 0 0 0 � (3.90)
�
∘
− x =
0
Since the required transformations of (3.90) just slightly deviate from used in Poynting’s
theorem, we go straight to final equation
∗ ∗ ∗ ∗
∗ 2 ∗ 2 ′′ ′′
∘ � x + x � = ( || + || ) + 2 � + � (3.91)
0
0
0
0

To simplify the subsequent considerations, we restrict our study to the practically important
case of small loss
′
′′
′′
≪ , ≪ ′

∗ ∗ � (3.92)

� + � ≪ 2( + )
0 0

Here = || /4 and = || /4 is the density of energy stored in electric and
2
′
′
2

0
0
magnetic fields, respectively. Then (3.47) can be rewritten shorter as
∗ ∗

∘ � x + x � ≈ 4( + ) (3.93)

Before undertaking a general analysis of this equation, it is important to understand the meaning
of inequities (3.92). In general, we can expect that in passive systems the electrical and magnetic
field variations over frequency are continuous and quite smooth. Evidently, that is not true for
the expected high-frequency sensitivity around the narrowly banded resonances. Therefore, the
equity (3.93) correctly describes the field behavior in any spatial domain if the resonance
cavities of high quality are excluded from the study. So we accept this not very limiting
restriction. Then integrating (3.93) over the hole aperture in Figure 3.4.1 and applying the
divergence (Gauss’s) theorem (see Appendix I) we obtain
∗ ∗

∬ � x + x � ∘ ≈ 4( + ) (3.94)