Practical_Antenna_Handbook_0071639586

C h a p t e r 2 0 : M i c r o w a v e W a v e g u i d e s a n d A n t e n n a s 459
We can define a general mode equation based on our system of notation:
1
( )
λ C
2
2 2
⎛ m ⎞ ⎛ n ⎞
= ⎜ ⎟ + ⎜ ⎟
⎝ 2 a ⎠ ⎝ 2 b⎠
(20.7)
where l c = longest wavelength that will propagate
a, b = waveguide dimensions (see Fig. 20.2)
m, n = integers defining number of half-wavelengths that will fit in a and b
dimensions, respectively
Evaluating Eq. (20.7) reveals that the longest TE-mode signal that will propagate in
the dominant mode (TE 10 ) is given by
from which we can write an expression for the cutoff frequency:
lc = 2a
(20.8)
f
c
c
=
2a
(20.9)
where f c = lowest frequency that will propagate, in hertz
c = speed of light (3 ×10 8 m/s)
a = wider of the two waveguide dimensions
Example 20.3 A rectangular waveguide has dimensions of 3 × 5 cm. Calculate the TE 10
mode cutoff frequency.
Solution
f
c
c
=
2a
=
(3×
108 m / s
⎛ 1 m ⎞
(2) ⎜5 cm × ⎟
⎝ 100 cm ⎠
×
= 3 10 8
m / s
(2) (0.05 m)
= 3 GHz