# Evaluation of the discrete complex-image method for a NEC-like moment-method solution Page: 5 of 12

0 10
Co / Re(k.2)
C1 /
/C2
/I

b)

.A

4-

C1
-4-

C - - s Re kp)

5 jk2 10 15

-5-
-10 -
-15-

IkV
1

, IM(k:2)

Fig. 2. Contours for evaluation of the Sommerfeld integrals: a) contours in the k2 plane, b) contours in the kp
plane, with branch cuts from k, and k2.
The remainders of the potentials after subtracting the quasistatic terms are

/0o e-ikz2(z+z')
v22 = 2 Rv(kz2) kz2 Jo(kpp)k dkp
foo e-ikz2(z+')
u22 = 2 Ru(kz2) Jo(kpp)k dkp

kz2 _ 1
Skkzk i+ kz2 k? +k3'

Ru(kz2)= kz2 -1.
kz + kz2 2.

The extracted quasistatic terms can be combined with the free space Green's function for
the image by applying the Sommerfeld identity,

o eikz2(z+z') e- jk2R1
0 kz2 Jo(kp)kp dkp = R1 '

(4)

The equations for the electric field components are then

Ev_- jwl"to 82
P 47rk2 apaz
= -jwIpo (02
z 47rk2 (az
EH_- -jWIeRo (82
47rk 8p
EH jWIiJ (18a
S4irk2 paP
EH- -jwlo8
z 47rk2 apaz

(G22 +

k1 + k2 G21

+kJ 22+ kG21) +S
+ k2 \G22 + 21 o#+ 3H
( - k2G_
+ G22- k2 G21) cosin + SP
+k G22 - k2 + G21 si 0 Sp o4
\22 k2 + k4

(5a)
(5b)
(5c)
(5d)
(5e)

where the subscripts on E or S indicate the cylindrical component of the field and the superscript
indicates a vertical electric dipole (V) or horizontal dipole on along the x axis (H). G22 and G21

3

a)

Im(kp)

where

(2a)
(2b)
(3)

-k2!
z

I
I

)+ Sp

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