Page 334 - Electromagnetics

This is independent of r when α = 0. For lossymedia the power decays exponentially
because of Joule heating.
We can write the phasor electric field in terms of the transverse gradient of a scalar
ˇ
potential function !:
e − jkr
ˇ ˆ ˇ ˇ
E(r,θ) = θE 0 =−∇ t !(θ)
r sin θ
where

θ
ˇ
ˇ
!(θ) =−E 0 e − jkr ln tan .
2
By ∇ t we mean the gradient with the r-component excluded. It is easilyverified that
ˇ
1 ∂!(θ) e − jkr
ˇ ˇ ˆ ˇ ˆ ˇ
E(r,θ) =−∇ t !(θ) =−θE 0 = θE 0 .
r ∂θ r sin θ
ˇ
ˇ
Because E and ! are related bythe gradient, we can define a unique potential difference
between the two cones at anyradial position r:
θ 2

ˇ ˇ ˇ ˇ ˇ − jkr
V (r) =− E · dl = !(θ 2 ) − !(θ 1 ) = E 0 Fe ,
θ 1
where F is given in (4.385). The existence of a unique voltage difference is a propertyof
all transmission line structures operated in the TEM mode. We can similarlycompute
the current flowing outward on the cone surfaces. The surface current on the cone at
ˆ ˇ
ˇ
ˇ
ˆ
ˇ
θ = θ 1 is J s = ˆ n × H = θ × φH φ = ˆ rH φ , hence
2π ˇ
ˇ I(r) = J s · ˆ rr sin θdφ = 2π E 0 e − jkr .
ˇ
0 Z TE M
The ratio of voltage to current at anyradius r is the characteristic impedance of the bi-
conical transmission line (or, equivalently, the input impedance of the biconical antenna):
ˇ
V (r) Z TE M
Z = = F.
ˇ I(r) 2π
c
If the material between the cones is lossless (and thus ˜µ = µ and ˜ = are real), this
becomes
η
Z = F
2π
where η = (µ/ ) 1/2 . The frequencyindependence of this quantitymakes biconical anten-
nas (or their approximate representations) useful for broadband applications.
Finally, the time-average power carried by the wave maybe found from
1
1 2 −2αr
ˇ
ˇ ∗
P av (r) = Re V (r)I (r) = π F Re E e .
0
2 Z ∗
TE M
∗
The complex power relationship P = VI is also a propertyof TEM guided-wave struc-
tures.

© 2001 by CRC Press LLC