TRUNCATION
ERRORS
IN
CYLINDRICAL
NEAR
TO
FAR
FIELD
TRANSFORM.
A
PLANE
WAVE
SYNTHESIS
APPROACH
J.
Romeu,
L.
Jo e.
ABSTACI
Cylind ical
Nea
o
a
ield
an enna
ans o ma ion
is
exp essed
as
a
plane
wa e
syn hesis
p oblem
whe e
a
plane
wa e
is
p oduced
by
a
cylind ical
cu en
dis ibu ion
ha
encloses
he
An enna
Unde
Tes
(AUT).
The
cu en
dis ibu ion
ha
p oduces
he
plane
wa e
is
di ec ly
de i ed
om
he
nea
o
a
ield
ans o ma ion
algo i hm.
IN l
DUClIN
In
nea
o
a
ield
an enna
measu emen s
he
An enna
Unde
Tes
(AUT)
nea
ields
a e
measu ed
on
an
enclosing
su ace
close
o
he
AUT.
The
AUT
a
ields
a e
ound
a e
p ocessing
hese
nea
ields.
Plana ,
cylind ical
and
sphe ical
scanning
su aces
a e
used
in
p ac ice.
In
o de
o
pe o m
he
nea
o
a
ield
ans o ma ion,
he
nea
ields
mus
be
known
on
he
en i e
su ace
ha
encloses
he
AUT.
In
he
plana
and
cylind ical
case,
his
condi ion
implies
ha
he
su ace
mus
sp ead
o
-in ini y.
In
p ac ice
an
e o
is
associa ed
o
he
limi ed
ex en
o
he
measu emen
su ace.
This
e o
is
usually
known
as
unca ion
e o .
The
e ec
o
his
e o
on
he
an enna
a
ield
pa e n
is
among
o he
ac o s
hea ily
dependen
on
he
AUT
cha ac e is ics.
The
o mula ion
o
he
nea
o
a
ield
ans o ma ion
as
a
plane
wa e
syn hesis
p ocess
allows
o
exp ess
he
unca ion
e o
as
a
de e io a ion
o
he
syn hesized
wa e.
In
his
way
he
pe o mance
o
he
nea
ield
ange
can
be
exp essed
independen ly
o
he
AUT.
On
he
o he
hand,
plane
wa e
quali y
is
a
common
igu e
o
me i
in
compac
o
a
ield
anges.
EXPRESSION
OF
THE
CYLINDRICAL
NEAR
TO
FAR
FIELD
TRANSFORMATON
AS
A
PLANLWAVE
SYNIEEIS
All
an enna
pa e n
measu emen
schemes
ind
he
AUT
esponse
o
an
inciden
plane
wa e.
In
nea
ield
measu emen s
he
AUT
esponse
o
a
plane
wa e
is
somehow
ob ained,
al hough
i
does
no
appea
explici ly
in
he
ans o ma ion
algo i hm.
In
cylind ical
an enna
nea
ield
measu emen s,
a
p obe
scans
he
AUT
nea
ields
on
a
cylind ical
su ace
( ig.
1).
Conside
a
ecip ocal
case
whe e
he
p obe
is
emi ing
and
he
AUT
ecei ing.
Le
us
de ine
c
(
,
z
(
,
4
')as
he
open
ol age
measu ed
on
he
AUT
access
po
when
he
p obe
is
placed
a
p
o
,
z
'
and
ed
wi h
an
uni a y
ol age
a
i s
inpu .
The
supe index
(p)
can
be
1
o
2
and
e lec s
he
ac
ha
wo
p obes
a e
equi ed
o
pe o m
he
nea
o
a
ield
ans o ma ion.
Conside
now
ha
a
cylind ical
a ay
o
p obes
is
cons uc ed
and
each
p obe
is
ed
a
i s
inpu
by
a
ol age
gi en
by
w
(
z
X,
').
The
AUT
esponse
o
he
cylind ical
a ay
can
be
ound
as:
Uni e si a
Poli ecnica
de
Ca alunya.
Ap a .
30002,
08080
Ba celona,
SPAIN
659
(p)=
X
(
~,) (P)(z`,
¢`)d`dz
(1)
o
X~~~~~~o
Conside
now
ha
w
(
z
k,
'
;
,
zO
,
o
)
exp esses
he
weigh
ha
has
o
be
gi en
o
he
each
elemen
o
he
cylind ical
a ay
placed
a
poin
po'
'
,
z',
o
p oduce
a
plane
wa e
p opaga ing
in
di ec ion
k
z0
=
k
cos
O
o
and
a
gi en
pola iza ion
e,
whe e
ek
=
O
(2)
ko=ksinOosin O*+ksenOocos o5+kcosOoZ
In
his
case
he
ol age
up)
induced
a
he
AUT
e minals
is
he
AUT
esponse
o
a
plane
wa e
p opaga ing
in
he
k
o
di ec ion
and
ha
gi en
pola iza ion
e.
Fo
a
ecip ocal
an enna
his
ol age
is
he
same
as
he
ampli ude
o
he
adia ed
ield
in
he
same
di ec ion
and
pola iza ion.
The
gene al
p oblem
o
inding
he
weigh s
w
(
z
*
,
-
"
;
k
zc
,
¢
o
)does
no
ha e
a
solu ion
o
a
ini e
leng h
cylinde .
An
i e a ing
app oach
o
ind
weigh
unc ions
ha
p oduce
good
app oxima ions
o
plane
wa es
in
a
ce ain
olume
wi h
ini e
sou ces
is
ound
in
[11.
In
ou
case
we
a e
in e es ed
in
inding
he
weigh s
ha
a e
inhe en ly
implici
in
he
ans o ma ion
algo i hm.
The
cylind ical
nea
o
a
ield
ans o ma ion
is
well
desc ibed
elsewhe e
[2,3],
and
can
be
w i en
in
he
ollowing
way:
[F0
(kzo,
00)
J
.
0zkF)
(2
'z1
d
z
L
~~,
(k20,~~0)
j
-~~
J0J
o
(4K
z')J
wi h
-
=6
3senO0
ejk20z
E
Jn(+0-*')
-an2)
an)'
1
(4)
16)
3sen
0A
(n,
k,O)
j
bjb2)
jb(l
)
and
A
(n,
k
)
a
(l)
k
):b
(2)
k
a
(2)
(-k
()b
(5))(-k
=)
In
hese
exp essions
E.(k0,
.
¢0)
and
E,(kzo
.
0
a e
he
AUT
a
ields
in
di ec ion
k,o
.
0,
and
pola iza ionOand
@
espec i ely.
The ol ages Q
(c)
.,
)and
4c
(2
z
Z
)a e he
measu ed
ol ages
by
each
o
he
p obes
equi ed
o
pe o m
he
ans o ma ion
in
posi ion
,
z'.
And
he
coe icien s
a()
),
b
('),
a
(2),
b
(2)
a e
ela ed
o
he
cylind ical
wa e
expansion
o
he
p obe
adia ed
ields.
No ice
ha
(w
z
;k
,
.
9
o
)
a e
he
weigh s
ha
ha e
o
be
gi en
o
p obes
l
and
2,
in
posi ion
*
z
-
o
p oduce
a
plane
wa e
p opaga ing
in
di ec ion
k
zO'
O.
I
we
conside
ha
p obe
1
is
a
e ical
in ini esimal
dipole
and
p obe
2
a
ho izon al
in ini esimal
dipole,
he
cu en
dis ibu ion
o
p oduce
a
e ical
plane
wa e
is:
JkzoZ
V
I
-ii (#oZ)
1
e(p
-
po)
(6)
2
y
2 '
senOoH
()
(k~00
660
and
o
p oduce
a
ho izon al
plane
wa e
is:
1
jk
j
~~~~~~kzon
(7)
J
(Z4
)=---
kezozzin+
ej(0*
2
2
n
k2
poH
(2)
(klppO)
1
~
b6(p-po)
Hn
2)kpopPo)
P
wi hk
o+k2o
-k2.
Each
one
o
hese
cu en
dis ibu ions
p oduce
he
ollowing
ields
inside
o
he
cylinde
E=
411
e
6(8)
00=cosO0coslo0
+
cosO0sin
Oh-sin0
A
and
E
=
)
We
Eo
;+
(9)
411
=
-sinO*
+
cosOO5
Howe e
when
he
cylinde
leng h
is
ini e
hese
ideal
plane
wa es
de e io a e.
The
esul ing
syn hesized
wa e
esul s
om
conside ing
he
cu en
dis ibu ions
gi en
by
equa ions
6
and
7,
bu
wi h
a
ini e
leng h
along
he
z
axis
FRRR
CRIJEA
In
o de
o
quan i y
he
syn hesized
plane
wa e
e o ,
i
is
con enien
o
de ine
he
ollowing
e o
unc ion
[4]:
_-
I
E
e
(
)
12+2I
He ( )12
(10)
=
E-1( )
12
+Ti2
1
Hdpi )
12
whe e
he
de ia ion
om
he
ideal
plane
wa e
is
de ined
as
e =-E-
(11)
E
e
=E
-
E
pi
He
HHPI
This
e o
unc ion
is
a
scala
unc ion
ha
conside s
he
e o
in
he
elec ical
and
magne ic
ield
in
magni ude
and
phase.
I
is
also
in e es ing
o
de ine
he
oo
mean
squa e
alue
o
he
plane
wa e
e o
unc ion
as:
661
1
~~~~~~~~~~~~~~~~(12)
T ms
S[
IT( )
I2dS](
REE;QLTS
By
using
exp essions
(1)
(2)
i
is
possible
o
compu e
he
syn hesized
plane
wa e
inside
he
measu emen
cylinde .
Fo
his
pu pose
he
cu en
dis ibu ions
ha e
been
disc e ized,
and
he
ield
inside
o
he
cylinde
has
been
compu ed
as
he
supe posi ion
o
he
adia ed
ields
o
a
cylind ical
a ay
o
in ini esimal
dipoles.
Figu e
2
shows
he
elec ical
ield
componen s
along
he
z
axis
when
a
e ical
plane
wa e
is
syn hesized.
Figu e
3
shows
he
plane
wa e
e o
unc ion
along
he
z
axis,
o
e ical
plane
wa e,
o
di e en
inciden
angles.
Finally,
igu e
4
shows
he
ms
alue
o
he
plane
wa e
e o
compu ed
along
he
z
axis
o
di e en
cylinde
leng hs
and
in
unc ion
o
he
inciden
angle.
The
as e isk
shows
he
alidi y
angle
(0
,)
o
he
di e en
cylinde s
de ined
as:
anO
=3
Z
m
-
(13)
2(pO-a)
No ice
ha
o
he
di e en
cylinde
leng hs,
he
ms
alue
o
he
plane
wa e
e o
unc ion
is
app oxima ely
he
same
when
he
plane
wa e
inciden
angle
is
he
alidi y
angle.
On
he
o he
hand,
in
igu e
4
he
op
ho izon al
line
is
he
ms
alue
o
he
plane
wa e
e o
conside ing
only
a
quad a ic
phase
e o
along
he
z
axis.
The
maximum
e o
is
n /
8
as
in
he
usual
a
ield
c i e ia.
CQBCL<UQIQNS
The
cylind ical
nea
o
a
ield
ans o ma ion
has
been
exp essed
as
a
plane
wa e
syn hesis
p ocess.
In
his
way
he
unca ion
e o
due
o
he
ini e
leng h
o
he
measu emen
cylinde
can
be
s udied
as
a
de e io a ion
o
he
syn hesized
plane
wa e.
By
using
an
e o
c i e ia
i
is
possible
o
compa e
a
cylind ical
nea
ield
an enna
measu emen
acili y
wi h
a
compac
o
a
a
ield
ange
on
plane
wa e
quali y
basis.
REEERENCL
[1]
Benne .
J.C,
Schoessow.
E.P.,
'
An enna
nea
ield/ a
ield
ans o ma ion using
plane
wa e
syn hesis
echnique',
IEE
P oc.,
ol.
125,
No.
3,
Ma ch
1978.
[2]
Leach.
W.M.,
Pa is.
D.T.,
'P obe
Compensa ed
Nea -Field
Measu emen s
on
a
Cylinde .'
IEEE
T ans.
An ennas
P opaga .,
ol.
AP-21,
July
1973.
[3]
Bo gio i.
G.V.
'In eg al
Equa ion
Fo mula ion
o
P obe
Co ec ed
Fa -Field
Recons uc ion
om
Measu emen s
on
a
Cylinde ',
ol.
AP-26,
July
1978.
[4]
Hansen.
J.E.,
'Sphe ical
Nea
Field
An enna
Measu emen s',
Pe e
Pe eg inus
L d.,
London
1988,
Cap.7.
662
This
wo k
has
been
suppo ed
by
he
Spanish
Commi ee
o
Scien i ic
and
Technical
Resea ch
(CICYT)
unde
g an
91-1034.
Figu e
1.
Syn hesis
geome y.
PLANE
WAVE
SYNTHESIS
plane
wa e
e o ,
e leal
pola Iza on
Nn
*O
-.
n
,"
a
--
_
e
To"
A
---
Wn0
.,.
ea
60
a
_-..
an
40
A
---
-4
-
2
4
.a/
Figu e
3.
Plane
wa e
e o
unc ion.
Figu e
2.
Syn hesized
elec ical
ield.
PLANE
WAVE
SYNTHESIS
RIMS
alue
o
he
plane
wa e
e o , e ical
pola iza ion
L-W1A
L-UA
L-
IA
L..UA
L-MA
-.-.
-_
Figu e
4.
Rms
alue
o
he
plane
wa e
e o .
663
PLANE
WAVE
SYNTHESIS
Elec ical
ield,
e cal
pola iza lon
VMS
--
a/A
sIdwmeM
mA
ACKNQMaJMGM