Page 503 - Engineering Electromagnetics, 8th Edition

CHAPTER 13 Guided Waves 485

where we define A = A C . Applying Eqs. (79a) through (79d) to (96a) gives the

p
m
transverse field components:
κ m
H xs = jβ mp A sin (κ m x) cos κ p y exp(− jβ mp z) (96b)
κ 2 mp
κ p
H ys = jβ mp A cos (κ m x) sin κ p y exp(− jβ mp z) (96c)
κ 2 mp

κ p
E xs = jωµ A cos (κ m x) sin κ p y exp(− jβ mp z) (96d)
2
κ mp
κ m
E ys =− jωµ A sin (κ m x) cos κ p y exp(− jβ mp z) (96e)
κ 2 mp
These field components pertain to modes designated TE mp .For these modes, either

m or p may be zero, thus allowing the possibility of the important TE m0 or TE 0p
cases, as will be discussed later. Some very good illustrations of TE and TM modes
are presented in Ref. 3.

13.5.4 Cutoff Conditions
The phase constant for a given mode can be expressed using Eq. (81):

2
2
β mp = k − κ mp (97)
Then, using (86), along with (90a) and (90b), we have

mπ 2 pπ 2
2 (98)
β mp = k − −
a b
√
This result can be written in a manner consistent with Eq. (42) by using k = ω µ ,
, appropriate for the rectangular guide.
and defining a radian cutoff frequency, ω c mp
We obtain:

√ ω Cmp 2
β mp = ω µ 1 − (99)
ω
where
1 mπ 2 pπ 2 1/2
ω Cmp = √ + (100)
µ a b
As discussed for the parallel-plate guide, it is again clear from (99) that the operating
frequency, ω, must exceed the cutoff frequency, ω Cmp ,to achieve a real value for
β mp (and thus enabling mode mp to propagate). Equation (100) applies to both TE
and TM modes, and thus some combination of both mode types may be present (or
above cutoff) at a given frequency. It is evident that the choice of guide dimensions,