Page 405 - Electromagnetics

or TM modes. For instance, if a = b then the cutoff frequency of the TE nm mode is
identical to that of the TE mn mode. If a ≥ b then the TE 10 mode has the lowest cutoff
frequency and is termed the dominant mode in a rectangular guide. There is a finite
band of frequencies in which this is the only mode propagating (although the bandwidth
is small if a ≈ b.)
Calculation of the time-average power carried by propagating TE and TM modes is
left as an exercise.

5.4.4 TE–TM decomposition in spherical coordinates
It is not necessary for the longitudinal direction to be constant to achieve a TE–TM
decomposition. It is possible, for instance, to represent the electromagnetic field in terms
of components either TE or TM to the radial direction of spherical coordinates. This may
be shown using a procedure identical to that used for the longitudinal–transverse decom-
position in rectangular coordinates. We carry out the decomposition in the frequency
domain and leave the time-domain decomposition as an exercise.

TE–TM decomposition in terms of the radial fields. Consider a source-free re-
gion of space filled with a homogeneous, isotropic material described by parameters ˜µ(ω)
c
and ˜ (ω). We substitute the spherical coordinate representation of the curl into Fara-
˜
˜
day’s and Ampere’s laws with source terms J and J m set equal to zero. Equating vector
components we have, in particular,
˜
1 1 ∂E r ∂
˜
˜
− (r E φ ) =− jω ˜µH θ (5.156)
r sin θ ∂φ ∂r
and
˜
1 ∂ ∂ H r c ˜
˜
(r H θ ) − = jω˜ E φ . (5.157)
r ∂r ∂θ
We seek to isolate the transverse components of the fields in terms of the radial compo-
c
nents. Multiplying (5.156) by jω˜ r we get
˜ ∂(r E φ )
˜
c 1 ∂E r c 2 ˜
jω˜ − jω˜ = k r H θ ;
sin θ ∂φ ∂r
next, multiplying (5.157) by r and then differentiating with respect to r we get
˜
2 ˜
∂ 2 ∂ H r c ∂(r E φ )
˜
(r H θ ) − = jω˜ .
∂r 2 ∂θ∂r ∂r
Subtracting these two equations and rearranging, we obtain
∂ 2 c 1 ∂E r ∂ H r

2 ˜ 2 ˜
˜
+ k (r H θ ) = jω˜ + .
∂r 2 sin θ ∂φ ∂r∂θ
This is a one-dimensional wave equation for the product of r with the transverse field
˜
component H θ . Similarly

2 ˜ 2 ˜
∂ 2 c ∂E r 1 ∂ H r
˜
+ k (r H φ ) =− jω˜ + ,
∂r 2 ∂θ sin θ ∂r∂φ

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