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 457
For this problem the solutions are
8
c = 3 × 10 m/s
v
v
P
g
8
= 5.51 × 10 m/s
8
= 1.6 × 10 m/s
We can also write a relationship between all three velocities by combining Eqs.
(20.1) and (20.2), resulting in
c = v
pv
(20.3)
g
In any wave the product of frequency and wavelength is the velocity. Thus, for a TEM
wave in unbounded free space we know that
c = f l
(20.4)
fs
Because the frequency f is fixed by the generator and the waveguide is a linear medium,
only the wavelength can change when the velocity changes. In a microwave
waveguide we can relate phase velocity to wavelength as the wave is propagated in the
waveguide:
v
p
λ × c
=
λ
fs
(20.5)
where v p = phase velocity, in meters per second
c = free-space velocity (3 × 10 8 m/s)
l = wavelength in waveguide, in meters
l fs = wavelength in free space (c/f ), in meters (see Eq. [20.4])
Equation (20.5) can be rearranged to find the wavelength in the waveguide:
v l
fs
l = p (20.6)
c
Example 20.2 A 5.6-GHz microwave signal is propagated in a waveguide. Assume
that the internal angle of incidence to the waveguide surfaces is 42 degrees. Calculate
phase velocity, wavelength in unbounded free space, and wavelength in the waveguide.