Waveguides - ieeetsu

www.ece.rutgers.edu/∼orfanidi/ewa 277
278 Electromagnetic Waves & Antennas – S. J. Orfanidis – June 21, 2004
Fig. 8.9.1
Reflection model of TE 10 mode.
where the electric field was taken to be polarized in the y direction. These field expressions
become component-wise:
E y = ( E 1 e −jkxx + E ′ 1e jkxx) e −jkzz
H x =− 1 η sin θ( E 1 e −jkxx + E ′ 1e jkxx) e −jkzz
H z = 1 η cos θ( E 1 e −jkxx − E ′ 1e jkxx) e −jkzz (8.9.1)
The boundary condition on the left wall, x = 0, requires that E 1 + E ′ 1 = 0. We may write
therefore, E 1 =−E ′ 1 = jE 0 /2. Then, the above expressions simplify into:
E y = E 0 sin k x xe −jkzz
H x =− 1 η sin θE 0 sin k x xe −jkzz
H z = j η cos θE 0 cos k x xe −jkzz (8.9.2)
The boundary condition on the right wall requires sin k x a = 0, which gives rise to
the same condition as (8.4.4), that is, k c a = nπ.
This model clarifies also the meaning of the group velocity. The plane wave is bouncing
left and right with the speed of light c. However, the component of this velocity in
the z-direction will be v z = c sin θ. This is equal to the group velocity. Indeed, it follows
from Eq. (8.9.3) that:
v z = c sin θ = c 1 − ω2 c
ω = v 2 gr (8.9.5)
Eq. (8.9.3) implies also that at ω = ω c , we have sin θ = 0, or θ = 0, that is, the wave
is bouncing left and right at normal incidence, creating a standing wave, and does not
propagate towards the z-direction. Thus, the transmitted power is zero and this also
implies, through Eq. (8.2.9), that α c will be infinite.
On the other hand, for very large frequencies, ω ≫ ω c , the angle θ will tend to 90 o ,
causing the wave to zoom through guide almost at the speed of light.
8.10 Resonant Cavities
Cavity resonators are metallic enclosures that can trap electromagnetic fields. The
boundary conditions on the cavity walls force the fields to exist only at certain quantized
resonant frequencies. For highly conducting walls, the resonances are extremely sharp,
having a very high Q of the order of 10,000.
Because of their high Q, cavities can be used not only to efficiently store electromagnetic
energy at microwave frequencies, but also to act as precise oscillators and to
perform precise frequency measurements.
Fig. 8.10.1 shows a rectangular cavity with z-length equal to l formed by replacing
the sending and receiving ends of a waveguide by metallic walls. A forward-moving wave
will bounce back and forth from these walls, resulting in a standing-wave pattern along
the z-direction.
√
These are identical to Eq. (8.4.3) provided we identify β with k z and k c with k x ,as
shown in Fig. 8.9.1. It follows from the wavevector triangle in the figure that the angle
of incidence θ will be given by cos θ = k x /k = k c /k, or,
cos θ = ω c
ω ,
sin θ = √
1 − ω2 c
ω 2 (8.9.3)
The ratio of the transverse components, −E y /H x , is the transverse impedance, which
is recognized to be η TE . Indeed, we have:
η TE =− E y
=
η
H x sin θ = √
η
1 − ω2 c
ω 2 (8.9.4)
Fig. 8.10.1
Rectangular cavity resonator (and induced wall currents for the TE n0p mode.)
Because the tangential components of the electric field must vanish at the end-walls,
these walls must coincide with zero crossings of the standing wave, or put differently, an
integral multiple of half-wavelengths must fit along the z-direction, that is, l = pλ g /2 =
pπ/β, orβ = pπ/l, where p is a non-zero integer. For the same reason, the standingwave
patterns along the transverse directions require a = nλ x /2 and b = mλ y /2, or