Page 429 - Electromagnetics

and remembering that r r for r in the far zone, we can use the leading terms of a
binomial expansion of the square root to get

2
2(ˆ r · r ) r 2(ˆ r · r ) ˆ r · r

R = r 1 − + ≈ r 1 − ≈ r 1 −
r r r r

≈ r − ˆ r · r . (6.19)
Thus the Green’s function may be approximated as
e − jkr
jkˆ r·r
G(r|r ; ω) ≈ e . (6.20)
4πr
Here we have kept the approximation (6.19) intact in the phase of G but have used
1/R ≈ 1/r in the amplitude of G. We must keep a more accurate approximation for

the phase since k(ˆ r · r ) may be an appreciable fraction of a radian. We thus have the
far-zone approximation for the vector potential
e − jkr jkˆ r·r
˜
˜ i

A e (r,ω) ≈ ˜µ(ω) J (r ,ω)e dV ,
4πr V
which we may use in computing (6.18).
Let us summarize the expressions for computing the far-zone fields:

˜
ˆ ˜
ˆ ˜
E(r,ω) =− jω θA eθ (r,ω) + φA eφ (r,ω) , (6.21)
˜
ˆ r × E(r,ω)
˜
H(r,ω) = , (6.22)
η
e − jkr
˜
A e (r,ω) = ˜ µ(ω)˜ a e (θ,φ,ω), (6.23)
4πr

˜ i

˜ a e (θ, φ, ω) = J (r ,ω)e jkˆ r·r dV . (6.24)
V
Here ˜ a e is called the directional weighting function. This function is independent of r
and describes the angular variation, or pattern, of the fields.
˜ ˜
In the far zone E, H, ˆ r are mutually orthogonal. Because of this, and because the fields
vary as e − jkr /r, the electromagnetic field in the far zone takes the form of a spherical
TEM wave, which is consistent with the Sommerfeld radiation condition.
Power radiated by time-harmonic sources in unbounded space. In § 5.2.1 we
defined the power radiated by a time-harmonic source in unbounded space as the total
time-average power passing through a sphere of very large radius. We found that for a
Hertzian dipole the radiated power could be computed from the far-zone fields through
2π π

2
P av = lim S av · ˆ rr sin θ dθ dφ
r→∞
0 0
where
1
ˇ
ˇ ∗
S av = Re E × H
2
is the time-average Poynting vector. By superposition this holds for any localized source.
Assuming a lossless medium and using phasor notation to describe the time-harmonic

© 2001 by CRC Press LLC