Page 504 - Engineering Electromagnetics, 8th Edition

486 ENGINEERING ELECTROMAGNETICS

a and b, along with the material properties, r and µ r , will determine the number of
√
modes that will propagate. For the typical case in which µ r = 1, using n = r ,
√
and identifying the speed of light, c = 1/ µ 0 0 ,we may re-write (100) in a manner
consistent with Eq. (41):

c mπ 2 pπ 2 1/2
ω Cmp = + (101)
n a b
This would lead to an expression for the cutoff wavelength, λ Cmp ,ina manner con-
sistent with Eq. (43):

2πc m 2 2 −1/2
p
λ Cmp = = 2n + (102)
ω Cmp a b
λ Cmp is the free space wavelength at cutoff. If measured in the medium that fills the
waveguide, the cutoff wavelength would be given by Eq. (102) divided by n.
Now, in a manner consistent with Eq. (44), Eq. (99) becomes

2πn λ
β mp = 1 − (103)
λ λ Cmp
where λ is the free space wavelength. As we saw before, a TE mp or TM mp mode can
propagate if its operating wavelength, λ,is less than λ Cmp .

13.5.5 Special Cases: TE m0 and TE 0p Modes

The most important mode in the rectangular guide is the one that can propagate by
itself. As we know, this will be the mode that has the lowest cutoff frequency (or the
highest cutoff wavelength), so that over a certain range of frequencies, this mode will
be above cutoff, while all others are below cutoff. By inspecting Eq. (101), and noting
that a > b, the lowest cutoff frequency will occur for the mode in which m = 1 and
p = 0, which will be the TE 10 mode (remember that a TM 10 mode does not exist, as
can be shown in (91)). It turns out that this mode, and those of the same general type,
are of the same form as those of the parallel-plate structure.
The specific fields for the TE m0 family of modes are obtained from (96a) through
(96e) by setting p = 0, which means, using (86) and (90), that
mπ

(104)
=
p=0
κ m = κ mp a
and κ p = 0. Under these conditions, the only surviving field components in (91)
will be E ys , H xs , and H zs .Itis convenient to define the field equations in terms of
an electric field amplitude, E 0 , which is composed of all the amplitude terms in Eq.
(96e). Specifically, define
κ m ωµ
E 0 =− jωµ 2 A =− j A (105)
κ m0 κ m