Page 322 - Maxwell House

302 Chapter 6

Here = / is the cut-off frequency. The remarkable fact is that the equity (6.23) is applied

to any TE- or TM-mode in WR of any shape with perfectly conductive walls. The only
difference is the value of critical wavelength. Therefore, the propagation constant for any mode
with < 1 or > is always less than the same constant in free space, i.e. < . It means
⁄

that any wave mode in WR acquires the lesser phase shift than a plane wave in free space at
the same distance . In other words, the wave front in WR should move faster than in free space,
i.e. faster than speed of light. the The explanation of this fact follows from drawing in Figure
6.4.4a. The reflected plane wave moves along the path AB with the speed of light c while its
phase front is perpendicular to the direction of propagation. Meanwhile, the same front shift
measured along the z-axis is greater. In other words, the speed of wave mode in WR exceeds
⁄
⁄
the speed of light and equal to = sin = �1 − ( )⁄ ⁄ 2 > as < 1 and tends to

infinity as → 1. It is not surprising because the phase velocity characterizes the rate of
⁄

phase change only and can be any value including negative. The laws of physics prohibits only
the wave energy to travel faster than light. So look at the diagram depicted in Figure 6.4.4b.
The reflected plane wave transfers its energy from point F to point D with the speed of light
while in the WR the same energy shifts from point F to point G only. If so, the speed of energy
2
⁄
⁄
transferred in WR is = ∙ sin = �1 − ( ) < and trends to zero as → 1. By

the way, we have a quite curious relationship between these two speeds
2
∙ = (6.24)

Now we can define the wavelength Λ in WR as the distance between successive peaks/crests of
a wave meaning that Λ = 2. Therefore,
Λ = 2 = 2 = �1 − ( ) (6.25)
2
⁄
⁄
⁄
⁄

Evidently, Λ > and tends to infinity as → for the propagating modes. The physical proof

follows from Figure 6.4.4a if the vectors c and are replaced with and Λ, respectively. Note

that it follows from (6.24), (6.23) and (6.25) that cos → 0 or → 90°, → and Λ →
as → 0 ( → ∞). It means that any TE- or TM-mode in waveguide of any shape with
⁄
⁄

perfectly conductive wall behaves more like TEM-mode when ≫ 1. Therefore, in
⁄

oversized waveguide ( ≫ ) or at very high frequencies ( ≫ ) in a fixed size waveguide

the transversal components { , } → 0 and the modal field pattern moves closer to TEM-

mode.
Now let us check how large and heavy can be real WRs. For example, the standard WR-2300
designated for frequency band 320 – 490 MHz, i.e. probably low for WR mass applications,
has imposing inside dimensions 0.5842 x 0.2921 m (!) and wall thickness 31.75 mm to be robust
and self-supporting. One meter of such aluminum WR weights 15 kg (!) (50 kg (!!) if copper).
The tiniest copper WR manufactured for frequency band 220 – 325 GHz has dimensions
1.6256x1.1938 mm, wall thickness 0.762 mm and weights 38 g/m. It is not bad if to forget
about Ohmic loss. There is a straightforward but costly way to reduce WR dimensions and
partially weight by replacing the air inside WR with robust and rigid dielectric that diminishes
the wavelength of incident and reflected plane waves to / √ . Accordingly, the WR

dimensions lessens in the same proportion. Besides, the thick metallic walls to get WR
toughness are not needed any more. Due to the skin effect (see Chapter 4) we can get low Ohmic
loss plating on the dielectric surface just several microns of highly conductive metal like copper