Page 432 - Electromagnetics

into (6.24) and carrying out the x and y integrals we have the directional weighting
function

l
˜

˜ a e (θ, φ, ω) = ˆ zI(ω) sin k(l −|z |)e jkz cos θ dz .
−l
Writing the sine functions in terms of exponentials we have
˜
ˆ zI(ω) jkl l jkz (cos θ−1) − jkl l jkz (cos θ+1)

˜ a e (θ, φ, ω) = e e dz − e e dz +
2 j 0 0
0 0

+ e jkl e jkz (cos θ+1) − e − jkl e jkz (cos θ−1) .
−l −l
Carrying out the integrals and simplifying, we obtain
˜
2I(ω) F(θ, kl)
˜ a e (θ,φ,ω) = ˆ z
k sin θ
where
cos(kl cos θ) − cos kl
F(θ, kl) =
sin θ
ˆ
is called the radiation function. Using ˆ z = ˆ r cos θ − θ sin θ we find that
˜
2I(ω)
˜ a eθ (θ,φ,ω) =− F(θ, kl), ˜ a eφ (θ, φ, ω) = 0.
k
Thus we have from (6.23) and (6.21) the electric field
˜
jηI(ω) e − jkr
˜
E(r,ω) = θ ˆ F(θ, kl) (6.31)
2π r
and from (6.22) the magnetic field
˜
j I(ω) e − jkr
˜
ˆ
H(r,ω) = φ F(θ, kl). (6.32)
2π r
We see that the radiation function contains all of the angular dependence of the field
and thus describes the pattern of the dipole. When the dipole is short compared to a
wavelength we may approximate the radiation function as
1 2 1 2
1 − (kl cos θ) − 1 + (kl) 1
2
F(θ, kl 1) ≈ 2 2 = (kl) sin θ. (6.33)
sin θ 2
So a short dipole antenna has the same pattern as a Hertzian dipole, whose far-zone
electric field is (5.93).
We may also calculate the radiated power for time-harmonic fields. The time-average
Poynting vector for the far-zone fields is, from (6.25),
ˇ 2
|I| 2
S av = ˆ rη F (θ, kl),
2 2
8π r
and thus the radiated power is
ˇ 2 π
|I| 2
P av = η F (θ, kl) sin θ dθ.
4π 0

© 2001 by CRC Press LLC