A single arm Archimedean spiral printed on a grounded dielectric substrate is analyzed using the method of moments. Piecewise sinusoidal subdomain basis and test functions are used over curved segments that exactly follow the spiral curvature. Results for the input impedance obtained using the curved segmentation approach on MATLAB are compared with those obtained after simulating the model on FEKO. A comparison with published results shows that the curved segment model requires fewer segments and is therefore significantly more computationally efficient than the linear segmentation model.

1. 1
Ashik Imran Akbar Basha and Piyush Kashyap
Abstract— A single arm Archimedean spiral printed on a
grounded dielectric substrate is analyzed using the method of
moments. Piecewise sinusoidal subdomain basis and test functions
are used over curved segments that exactly follow the spiral
curvature. Results for the input impedance obtained using the
curved segmentation approach on MATLAB are compared with
those obtained after simulating the model on FEKO. A comparison
with published results shows that the curved segment model
requires fewer segments and is therefore significantly more
computationally efficient than the linear segmentation model.
Index Terms—Archimedean spiral antenna, curved segment
model, method of moments, piecewise sinusoid
I. INTRODUCTION
METHOD OF MOMENTS modelling of curved wire geometries
using piecewise linear segments tend to require greater
segmentation than similarly sized linear geometries with a
corresponding increase in the number of basis and test
functions. It is possible that the number of segments required to
accurately model the curvature may well exceed that required
for a satisfactory description of the current distribution along
the geometry. This leads to large impedance matrices which
require more memory and CPU processing time. The use of
piecewise curved segments allows to accurately model the
curved geometry in fewer segments. Hence, fewer basis
functions are required and the procedure is therefore
computationally more efficient.
In this paper, the curved segmentation approach is used to
analyze an Archimedean spiral printed on a grounded substrate.
Piecewise subdomain basis and test functions spanning two
segments are used over curved segments. Galerkin’s method is
applied to the Pocklington’s equation for printed wires. The
Green’s functions are expressed as a superposition of
eigenfunctions in Sommerfeld-type integrals.
II. THEORY
A printed spiral considered here is shown in Fig.1. It is
assumed that the radius of the wire is very small compared to
the free space wavelength, thus allowing the thin wire
approximation to be used. The current on the wire will not have
circumferential symmetry. However, it has been shown [1] that
for a thin wire only the axial current component is significant
in determining the radiation and impedance characteristics of
the structure.
The spiral antenna model considered in this paper comprises
a vertical linear segmented feed wire from a source at the
ground plane through the dielectric to the surface, where it is
directly connected to the printed curved segmented wire of the
spiral. A linear section can sometimes be present.
The authors are graduate students with the Electrical Engineering
Department, University of Colorado, Boulder, CO 80309 USA.
It is assumed that the lateral extent of the substrate is infinite
and the eigenmode field solutions used in the paper have been
obtained by satisfying boundary conditions at the dielectric
interface [3].
Fig. 1. Top view of printed spiral model
Fig. 2. Side vew of printed spiral model
If a spiral is Archimedean then
 a 0
(1)
where a is the spiral constant, 0 is the length of feed segment
of an arm and
  yxa ˆsinˆcos0   (2)
The unit tangent vector to the spiral arm can then be expressed
as[4]




d
d
d
d
l /ˆ  (3)
Such that
      
 2
0
2
00
ˆcossinˆsincosˆ


aa
yaaxaa
l



(4)
Note that in this paper, the linear section of the spiral arm is not
considered ie. 00  . The spiral arm is fed at the start of the
curved section by a vertical feed monopole that passes through
Method of Moment analysis of a printed single
arm spiral antenna using curved segments

2. 2
the substrate. The following sections will describe the
procedure to derive the Electric field due to the feed and the
curved spiral.
A. Vertical Feed Monopole
The general electric field integral equation (EFIE) can be
written as
    dvJdvJkE .2
(5)
As illustrated in Fig 2. , the vertical monopole is directed in the
positive z axis. Therefore, only a z directed filamentary current
 zIz  is considered. This assumption reduces the EFIE to the
following for the radial electric field component in free space
above a grounded substrate
 


 zdzI
zr
E
v
r )(
2
(6)
The Green’s function v
 is expressed as a summation of
eigenmodes and obtained by satisfying the boundary conditions
for the electric and magnetic fields at the dielectric interface
[5,6].




0
)(
0 )()cosh(2
tm
hz
e
v
f
d
erJzK

  (7)
)sinh()cosh( hhf eeertm   (8)
where 22
k  , 22
kre   and  04/jK  .
The radial component of the electric field due to the nth
current
expansion is required and can be written as




0
2
)(
1 )(2
tm
hz
nr
f
d
erJPKE

  (9)
where z=h for evaluating the field on the dielectric-air interface
and
   
zdeezIP zz
nn
ee 
)(
2
1 (10)
Piecewise sinusoidal current expansion pulses spanning two
segments are considered and can be represented as
  
 
  
  1
1
1
1
sin
sin
sin
sin














nn
d
ndn
z
nn
d
ndn
z
zzz
zk
zzkI
I
zzz
zk
zzkI
I
(11)
The electric field tangential to a point on the curved segment on
the spiral arm due to a vertical monopole feed current pulse
directed towards positive z axis can therefore be obtained from
equations (4) and (9) by using the operation lEr
ˆ such that
     
 2
0
2
0 sincos


aa
aa
EE r
v
l


 (12)
As a general expression, the cylindrical coordinate frame of the
vertical wire  zr ,, is different from that of the spiral arm
 z,, for an offset feed point. However, if the vertical wire
connects to the center of the printed spiral then r and  
B. Curved spiral segment
A current with amplitude lI  along the spiral arm can be
represented in Cartesian coordinates from equation (4)
 
 
 
 2
0
2
0
2
0
2
0
cossin
sincos












aa
aa
II
aa
aa
II
ly
lx
(13)
where the primes indicate coordinates of the source currents.
Using the electric field integral equation (EFIE) from equation
(5), the field components due to a curved spiral segment can be
written as
    ld
y
G
I
x
G
I
i
GIkE yxii













  
2
(14)
where G and  Green’s functions and are represented as
superposition of eigenmodes as shown below [5]
 



 
d
D
eJKG
e
hz
d



0
)(
0 )(2 (15)
   



0
)(
0 )()1(2 


 
d
DD
eJK
me
hz
dr
(16)
where    hD eee  coth ,    hD eerm  tanh and
i = x or y and z= h to evaluate the fields on the dielectric surface.
On replacing the Cartesian current components with the
components along the spiral contour, equation (14) can be
written as
    ld
y
G
l
y
I
x
G
l
x
I
i
GiIkE llli

















  
ˆ.2
(17)
     
l
G
y
G
l
y
I
x
G
l
x
l













(18)
Using equations (13), (17) and (18) the x and y directed electric
fields can be obtained.
 
 
  ld
l
G
xaa
aa
GkIE lx
















  
2
0
2
02 sincos


 
 
  ld
l
G
yaa
aa
GIkIE lly
















  
2
0
2
02 cossin


(19)
The electric field tangential to a point on the spiral arm due to
a source current on the spiral arm can be similarly obtained
from equations (3, 18) as lEE yx
ˆ

3. 3
 
 
 
 
yx
c
l E
aa
aa
E
aa
aa
E
2
0
2
0
2
0
2
0 cossinsincos










(20)
Piecewise sinusoidal current basis functions are used as the
expansion functions:
  
 
  
  1
1
1
1
sin
sin
sin
sin














nn
nn
l
nn
nn
l
lll
lk
llkI
I
lll
lk
llkI
I
(21)
and noting that
    ldG
l
I
GIld
l
G
I l
ll








 )( (22)
Subsequently, equation (19) can be expressed as
 
     
         
     
ldG
llkllk
lk
kI
E
aaaa
aaaa
n
n
n
n
n
n
nc
l

















 
2
0
22
0
2
sin2cos00
2
1
1
1
1
2
sinsin
sin


 
        ld
l
G
llkllk
lk
kI
n
n
n
n
n
n
n 


















1
1
1
1 coscos
sin
(23)
Using the following simplification to change the variable of
integration for G and 
   
l
G
l
G d
d 





 

(24)
Where the differentials of the Green’s functions with respect to
d can be obtained from equation (14) and (15)
 






d
D
eJK
G
e
hz
d
d






0
2
)(
1 )(2 (25)
   





0
2
)(
1 )()1(2 





d
DD
eJK
me
hz
dr
d
(26)
 
      yyxaxxya
aal
d
d








sin)cos
1
2
0
2
(27)
C. Impedance matrix
The impedance matrix associating a source current pulse n to
a test sinusoid m along the spiral arm can be obtained as
 
      dl
I
E
llkllk
lk
Z
n
a
l
m
m
m
m
m
mmn
















1
1
1
1 sinsin
sin
1
(28)
where,
a = v when the source current pulse is on the vertical feed wire
from equation (11)
a = c when the source current pulse is along a curved spiral
segment (11)
Fig. 3. Input impedance of Archimedean spiral using curved segment model
implemented on MATLAB
Since the variable of integration has changed throughout due to
the curved segmentation approach, a relation between
landl ,,   is needed and can be obtained as follows
  

daal
m
m  
0
2
0
2
(29)
To reduce the number of impedance matrix elements
evaluated, concept of reciprocity can be used, whereby an
impedance element corresponding to a given source segment on
the vertical feed tested on a curved segment is the same as a
source segment on the curved segment tested on the vertical
feed. There also exists vertical to vertical element coupling.
Classical methods used to find the solution of a thin wire dipole
were used to estimate this type of coupling [1]. The moment
method is then completed by solving the impedance matrix
equations for current distribution. A delta gap voltage excitation
source is used between the vertical feed wire and the ground
plane. In this case, the voltage column matrix at the feed test
point is set to unity while all other elements are set to zero.
III. ANALYSIS AND RESULTS
The method of moments as outlined above is implemented in
MATLAB to solve for the input impedance of the spiral
antenna. The frequency range is from 2 to 2.5 GHz. The spiral
antenna parameters are chosen as follows: wire radius = 0.05
cm, flare rate = 0.229 rad/cm, εr = 2.52, 9.9m . Nine curved
segments over the spiral arm and six vertical linear segments
over the vertical feed are considered in the model. It should be
noted that a very low number of curved segments is required
here for estimating parameters with reasonable accuracy. It can
be seen from published results that over twice as many linear

4. 4
segments are required to achieve comparable results [7].
A major difficulty that we faced in this method was the
calculation of the Green’s functions in MATLAB. Since Bessel
functions are highly oscillatory and the integration was over
infinite domain, these numerical integrations were extremely
hard to perform directly. MATLAB’s integral function
prompted several warnings when these calculations were
Fig. 4. FEKO® model of a printed Archimedean spiral antenna
carried out. Errors as high as the order of 108
occurred. Hence,
an adaptive quadrature method with Clenshaw-Curtis
quadrature rules was used to better deal with the singularities.
The integrand had to be transformed to a finite domain by
parametrization and the above mentioned quadrature method
was applied with reasonable accuracy. There also were
singularities through d when the curved segment and test
segment were the same. They were resolved by taking d to
be the thickness of the wire. The input impedance using curved
segmentation from MATLAB is shown in Fig.3.
IV. VALIDATION
The method is validated by simulating the Archimedean
spiral in FEKO (a Method of Moments solver). In accordance
with the EFIE equations used for the numerical solution in the
previous section, the lateral extent of the dielectric and ground
Fig. 5. Input Impedance of Archimedean spiral using FEKO
plane are implemented as infinite in the FEKO model as shown
in Fig. 4. A wire port is defined at the intersection of the feed
wire and ground plane with a voltage source set for 1V and a 50
Ohms port impedance. This simulates the delta gap voltage
feed. The wire radius is set to 0.05 cm and the dielectric
constant is set to 2.52. The input impedance plot as obtained
from simulation in FEKO is shown in Fig. 5. It can be seen that
the input impedances obtained from MATLAB and FEKO over
2.0 GHz to 2.5 GHz are fairly consistent with each other, for
the same geometrical parameters and dielectric permittivity of
the substrate.
V. CONCLUSION
A single arm Archimedean spiral antenna printed on a
grounded dielectric has been analyzed using a method of
moment solution of Pocklington’s equation for printed wires,
using piecewise sinusoidal basis and test functions over curved
segments. The difficulties of implementing it in MATLAB are
discussed. The model was solved for input impedance in
MATLAB and validated in FEKO. Reasonably good
agreements were obtained between the input impedances
obtained from MATLAB and FEKO. The curved segment
technique implemented in this paper uses only 9 sinusoidal
pulses over the spiral arm. If the linear segment approach is
followed, it is shown that the segment length should be about
60/0 [8], thus requiring almost 89 current pulses [3]. The
curved segmentation approach for analysis of printed spirals is
therefore significantly more computationally efficient and
consumes less storage and processing time. This advantage can
be exploited in the analysis of more complex geometries
involving curved contours.
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1982.
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144, no. 4, Aug. 1997.
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