Page 333 - Electromagnetics

c 1/2
where Z TE M = ( ˜µ/ ) is the complex wave impedance. Since we can also write
˜
ˆ r × E(r,θ,ω)
˜
H(r,θ,ω) = ,
Z TE M
the field is TEM to the r-direction, which is the direction of wave propagation as shown
below.
The wave nature of the field is easilyidentified byconsidering the fields in the phasor
domain. Letting ω → ˇω and setting k = β − jα in the exponential function we find that
e − jβr
ˇ
ˆ ˇ
−αr
E(r,θ) = θE 0 e
r sin θ
ˇ
where E 0 = E 0 e jξ E . The time-domain representation maybe found using (4.126):
e −αr
ˆ E
E(r,θ, t) = θE 0 cos( ˇωt − βr + ξ ). (4.383)
r sin θ
We can identifya surface of constant phase as a locus of points obeying
E
ˇ ωt − βr + ξ = C P (4.384)
where C P is some constant. These surfaces, which are spheres centered on the origin, are
called spherical wavefronts. Note that surfaces of constant amplitude as determined by

e −αr
= C A ,
r
where C A is some constant, are also spheres.
The cosine term in (4.383) represents a traveling wave with spherical wavefronts that
propagate outward as time progresses. Attenuation is caused bythe factor e −αr .By
differentiation we find that the phase velocityis
v p = ˇω/β.

The wavelength is given by λ = 2π/β.
Our solution is not appropriate for unbounded space since the fields have a singularity
at θ = 0. To exclude the z-axis we add conducting cones as mentioned on page 105. This
results in a biconical structure that can be used as a transmission line or antenna.
To compute the power carried bya spherical wave, we use (4.381) and (4.382) to obtain
the time-average Poynting flux

1 1 1 E 2 0 −2αr
ˇ ˆ
ˇ ∗ ˆ
S av = Re{E θ θ × H φ}= ˆ r Re 2 e .
φ
2 2 Z ∗ r sin θ
2
TE M
2
The power flux is radial and has densityinverselyproportional to r . The time-average
power carried bythe wave through a spherical surface at r sandwiched between the cones
at θ 1 and θ 2 is
1 1 2 −2αr 2π θ 2 dθ 1 2 −2αr
P av (r) = Re E e dφ = π F Re E e
0
0
2 Z ∗ sin θ Z ∗
TE M 0 θ 1 TE M
where

tan(θ 2 /2)
F = ln . (4.385)
tan(θ 1 /2)
© 2001 by CRC Press LLC