Page 439 - Electromagnetics

˜ eq

˜ a h (θ, φ, ω) = J (r ,ω)e jkˆ r·r dS , (6.49)
sm
S
are the directional weighting functions.
To compute the fields from the potentials we must apply the curl operator. So we
must evaluate
e e e
− jkr − jkr
− jkr
∇× V(θ, φ) = ∇× V(θ, φ) +∇ × V(θ, φ).
r r r
The curl of V is proportional to 1/r in spherical coordinates, hence the first term on the
2
right is proportional to 1/r . Since we are interested in the far-zone fields, this term can
be discarded in favor of 1/r-type terms. Using
e 1 + jkr e e

− jkr
− jkr − jkr
∇ =−ˆ r ≈−ˆ r jk , kr 1,
r r r r
we have
e e
− jkr − jkr
∇× V(θ, φ) ≈− jkˆ r × V(θ, φ) .
r r
Using this approximation we also establish
e 2 e 2 e
− jkr − jkr − jkr
∇× ∇× V(θ, φ) ≈−k ˆ r × ˆ r × V(θ, φ) = k V T (θ, φ)
r r r
where V T = V − ˆ r(ˆ r · V) is the vector component of V transverse to the r-direction.
With these formulas we can approximate (6.46) and (6.47) as
jk
˜ ˜ ˜
E(r,ω) =− jωA eT (r,ω) + ˆ r × A h (r,ω), (6.50)
c
˜ (ω)
jk
˜ ˜ ˜
H(r,ω) =− jωA hT (r,ω) − ˆ r × A e (r,ω).
˜ µ(ω)
Note that
jk
˜ ˜ ˜
ˆ r × E =− jωˆ r × A eT + ˆ r × ˆ r × A h .
˜ c
˜
˜
˜
˜
Since ˆ r × A eT = ˆ r × A e and ˆ r × ˆ r × A h =−A hT ,wehave

˜ ˜ jk ˜ ˜
ˆ r × E = η − jωA hT − ˆ r × A e = ηH.
˜ µ
Thus
˜
ˆ r × E
˜
H =
η
and the electromagnetic field in the far zone is a TEM spherical wave, as expected.
Example of fields produced by equivalent sources: an aperture antenna. As
an example of calculating the fields in a bounded region from equivalent sources, let
us find the far-zone field in free space produced by a rectangular waveguide opening
into a perfectly-conducting ground screen of infinite extent as shown in Figure 6.6. For
simplicity assume the waveguide propagates a pure TE 10 mode, and that all higher-order

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