Page 219 - Electromagnetics

4.5.3 The energy theorem
A convenient expression for the time-average stored energies (4.60) and (4.61) is found
by manipulating the frequency-domain Maxwell equations. Beginning with the complex
conjugates of the two frequency-domain curl equations for anisotropic media,

∗
˜ ∗
∇× E = jω ˜ ¯µ · H ,
˜ ∗
∗
∇× H = J − jω ˜ ¯ · E ,
˜ ∗
˜ ∗
˜ ∗
we differentiate with respect to frequency:
∗
∂E ∂[ω ˜ ¯µ ] ∗ ∂H
˜ ∗
˜ ∗
˜ ∗
∇× = j · H + jω ˜ ¯µ · , (4.66)
∂ω ∂ω ∂ω
∗
∂H ∂J ˜ ∗ ∂[ω ˜ ¯ ] ∗ ∂E
˜ ∗
˜ ∗
˜ ∗
∇× = − j · E − jω ˜ ¯ · . (4.67)
∂ω ∂ω ∂ω ∂ω
These terms also appear as a part of the expansion
∂H ∂E

˜ ∗
˜ ∗
˜ ˜
∇· E × + × H =
∂ω ∂ω
∂H ∂H ∂E ∂E
˜ ∗
˜ ∗
˜ ∗
˜ ∗
˜
˜
˜
˜
· [∇× E] − E ·∇ × + H ·∇ × − · [∇× H]
∂ω ∂ω ∂ω ∂ω
˜
where we have used (B.44). Substituting from (4.66)–(4.67) and eliminating ∇× E and
˜
∇× H by Maxwell’s equations we have
1 ∂H ∂E
˜ ∗
˜ ∗
˜
˜
∇· E × + × H =
4 ∂ω ∂ω
1 ∗ ∂E ∂E 1 ∗ ∂H ∂H
˜ ∗
˜ ∗
˜ ∗
˜ ∗
˜
˜
˜
˜
j ω E · ˜ ¯ · − · ˜ ¯ · E + j ω H · ˜ ¯µ · − · ˜ ¯µ · H +
4 ∂ω ∂ω 4 ∂ω ∂ω
∗
∗
1 ∂[ω ˜ ¯ ] ∂[ω ˜ ¯µ ] 1 ∂J ˜ ∗ ∂E
˜ ∗
˜
˜
˜
˜
+ j E · · E + H · · H − E · + J · .
˜ ∗
˜ ∗
4 ∂ω ∂ω 4 ∂ω ∂ω
Let us assume that the sources and fields are narrowband, centered on ω 0 , and that ω 0
lies within a transparency range so that within the band the material may be considered
†
†
lossless. Invoking from (4.49) the facts that ˜ ¯ = ˜ ¯ and ˜ ¯µ = ˜ ¯µ , we find that the first two
terms on the right are zero. Integrating over a volume and taking the complex conjugate
of both sides we obtain
˜
˜
1 ∂H ∂E
˜ ∗
˜ ∗
E × + × H · dS =
4 S ∂ω ∂ω
˜
1 ∂[ω ˜ ¯ ] ∂[ω ˜ ¯µ] 1 ∂J ˜ ∂E
˜
˜
˜ ∗
− j E · · E + H · · H dV − E · + J · dV.
˜ ∗
˜ ∗
˜ ∗
4 V ∂ω ∂ω 4 V ∂ω ∂ω
Evaluating each of the terms at ω = ω 0 and using (4.60)–(4.61) we have
˜
˜
1 ∂H ∂E
˜ ∗
E × + × H · dS =
˜ ∗
4 S ∂ω ∂ω ω=ω 0
˜

1 ∂J ˜ ∂E
˜ ∗
˜ ∗
− j [ W e + W m ] − E · + J · dV (4.68)
4 V ∂ω ∂ω ω=ω 0
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