Page 402 - Electromagnetics

where V is the region of the guide between CS 1 and CS 2 . The right-hand side represents
the difference between the total time-average stored electric and magnetic energies. Thus

2 jω [ W e − W m ] =
1 1 1
ˇ
ˇ
ˇ
ˇ ∗
ˇ ∗
ˇ ∗
−ˆ z · (E × H ) dS + ˆ z · (E × H ) dS − (E × H ) · dS,
2 CS 1 2 CS 2 2 S
cond
where S cond indicates the conducting walls of the guide and ˆ n points into the guide. For
a propagating mode the first two terms on the right-hand side cancel since with no loss
ˇ
ˇ
ˇ
E×H is the same on CS 1 and CS 2 . The third term is zero since (E×H )· ˆ n = (ˆ n×E)·H ,
ˇ ∗
ˇ ∗
ˇ ∗
ˇ
and ˆ n × E = 0 on the waveguide walls. Thus we have
W e = W m
for any section of a lossless waveguide.
We may compute the time-average stored magnetic energy in a section of lossless
waveguide of length l as
l
µ
ˇ
ˇ ∗
W m = H · H dS dz.
4 0 CS
For propagating TM modes we can substitute (5.139) to find
µ 2
ˇ
ˇ
∗
W m /l = (βY e ) (ˆ z ×∇ t ψ e ) · (ˆ z ×∇ t ψ ) dS.
e
4 CS
Using
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ

∗

∗
(ˆ z ×∇ t ψ e ) · (ˆ z ×∇ t ψ ) = ˆ z · ∇ t ψ × (ˆ z ×∇ t ψ e ) =∇ t ψ e ·∇ t ψ e ∗
e
e
we have
µ 2
ˇ
ˇ
∗
W m /l = (βY e ) ∇ t ψ e ·∇ t ψ dS.
e
4 CS
Finally, using (5.153) we have the stored energy per unit length for a propagating TM
mode:
µ 2 2
ˇ ˇ
∗
W m /l = W e /l = (ω ) k c ψ e ψ dS.
e
4 CS
Similarly we may show that for a TE mode
2 2
ˇ ˇ
∗
W e /l = W m /l = (ωµ) k c ψ h ψ dS.
h
4 CS
The details are left as an exercise.
As with plane waves in (4.261) we may describe the velocity of energy transport as the
ratio of the Poynting flux density to the total stored energy density:
S av = w T v e .
For TM modes this energy velocity is
1 ω βk ψ e ψ !
2 ˇ ˇ ∗
2 c e β 2 2
v e = µ = = v 1 − ω /ω ,
c
2 (ω ) k ψ e ψ ωµ
2 2 ˇ ˇ ∗
4 c e
which is identical to the group velocity (5.147). This is also the case for TE modes, for
which
1 2 ˇ ˇ ∗ !
2 ωµβk ψ h ψ h β 2 2
c
v e = = = v 1 − ω /ω .
c

2 2 ˇ ˇ ∗
2 (ωµ) k ψ h ψ ωµ
4 c h
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