Scan Impedance for An Infinite
Dipole Array: Accurate Theory
Model Versus Numerical Software
Sergey N.Makarov1, Angelo Puzella2, and Vishwanth Iyer1
1Worcester Polytechnic Institute, Worcester, MA 01609
2Raytheon Company, Sudbury, MA 01776
ElectroScience Laboratory, OSU, Columbus, Ohio
November 13th 2008
Outline
• Dipole array model with the sinusoidal current
distribution and its limitations
• Analytical solution of VanKoughnett and Yen
• Extension of the solution to feeds/gaps of finite
widths
• Comparison of the model with numerical simulations
(Ansoft HFSS, CST MWS)
• Possible applications and extensions of the model
Array model with the sinusoidal
current distribution – strip dipoles


J(x, y)  xJ 0 f ( x) g ( y ),
  x
case A
  l 
  
  l

 y
f ( x)   sin  k 0   x   case B , g ( y )   

t
  2
case C
cosk 0 x 


Solution for the co-polar electric field
Floquet theorem
m   n  

 4 2
~
J(x, y, z)  x
J 0 ( z )   f (k xmn ) g~(k ymn ) exp(  jk xmn x  jk ymn y)
dxdy
m   n  
k xmn  k x 0 
2 m
2 n
, k ymn  k y 0 
dx
dy
 Ax  k Ax   J x( x, y, z )
2
2
0
k x 0  k 0 sin  0 cos  0 , k y 0  k 0 sin  0 sin  0

E
1
j 0

 H 
1
j 0

 A
2
m   n   2
k 0  k xmn
~
2 2
E x(x, y, z)  
J0  
f (k xmn ) g~(k ymn ) exp(  jk xmn x  jk ymn y  jk zmn z )
dxdy 0 m n  k zmn
Scan impedance (comparison)
Reaction integral
ZS 
P

I 0 I 0*
 *
  E  J ds
unit cell
I
2
0
*
E
(
x
,
y
,
z

0
)

J
( x, y )dxdy
x
x

  unit cell
2
2
2
2
2
2

  k 0  k xmn  k ymn for k 0  k xmn  k ymn  0
k zmn  
2
2
2
2
2
2

 j k xmn  k ymn  k 0 for k 0  k xmn  k ymn  0
Scan impedance
2 2
ZS  2
t Dx Dy k02
I 02
2
2
1  k xmn
/ k02
P(k xmn , k ymn ) 1  exp(  jkzmn 2h)


k zmn / k0
m   n  
m   n  
 k ymnt 
t l k02
 k xmnl 


P(k xmn , k ymn ) 
sinc
sinc


2
4
 2 
 2 
P(k xmn , k ymn ) 
 k ymn t 
 (k xmn  k 0 )l 
 (k xmn  k 0 )l 


sinc
sinc
sinc




2


4
4
16




 2 
t l 2 k 03
 k ymnt 
t k0
 k xmnl 

P(k xmn , k ymn )  2 cosc
sinc 
2
 2 
 2 
Scan impedance (Refs.)
•R. C. Hansen, Phased Array Antennas, Wiley, New York, 1998.
•A. A. Oliner and R. G. Malech, “Mutual coupling in infinite scanning arrays,” in Microwave
Scanning Antennas, Vol. II, R.C. Hansen, ed., Academic Press, 1966, Chapter 3, pp. 195-335.
•L. Stark, Radiation Impedance of a Dipole in Infinite Arrays, Hughes Aircraft Company
Technical Report No. FL60-230, 1960.
•L. Stark, “Radiation impedance of a dipole in an infinite planar phased array,” Radio Science,
vol. 1, March 1966, pp. 361-377.
Comparison with numerical simulations
(Ansoft HFSS, PML)
Motivation
• Establish a more accurate (full wave) analytical
solution for the infinite dipole array
• Compare this solution with numerical software using
the different solvers and the different terminations
(PML/Floquet port)
• Provide quantitative benchmark results for analytical
and numerical models of the infinite dipole array
Base analytical solution
•A. L. VanKoughnett and J. L. Yen, “Properties of a cylindrical antenna in an
infinite planar or collinear array,” IEEE Trans. Antennas Prop., vol. AP-15, no. 6,
Nov. 1967, pp. 750-757.
The idea is to treat an infinite collinear 1D array as one infinitely long dipole
with multiple feeds. A gap between two array elements is to be considered as
another feed since it possesses some nonzero gap voltage. This voltage is
indeed not necessarily equal to the feed voltage. The solution for a 2D array
is then obtained as a combination of the (coupled) 1D array solutions.
Base analytical solution geometry
Why is the infinite dipole (a true
3D TM to x field)?
Because the analytical full-wave solution is known: the Pocklington integral
equation is solved via Fourier transform in the space domain
Pocklington integral equation
and feed models

Z0  2
2
 2  k   I ( x)G( x  x)dx
E x ( x) 
jk  x
 
 V0
~
E x ( x)  V0 ( x), E x (k x ) 
2
 V0
~
k g
E x ( x)  V0  ( x / 2 g ), E x (k x ) 
sinc  x 
2
  
 V0
~
E x ( x)  V0 /  ( g 2  x 2 ) 1 / 2 , E x (k x ) 
J 0 k x g 
2
~

jk E x (k x )
1
~
~
I (k x ) 
,
I
(
x
)

I
(k x ) exp(  jk x x)dk x
~

2
2
2 Z 0 k  k x G(k x )



Analytical solution to Pocklington
integral equation
k x  K / a, x  aX
This integral is to be found numerically
 ~

E x ( K ) cos( KX )
jka
1
~
I(X ) 
dK , G( K )   cos( KX )aG( X )dX
~
 Z 0 0 (ka) 2  K 2 G( K )
 0
1
~
G(K ) 
J 0 (q) H 0( 2 ) (q), q 
8 j
ka2  K 2
•R. H. Duncan and F. A. Hinchey, "Cylindrical antenna theory," J. Research NBS (D-Radio Prop.), vol. 64D, 1960, pp. 569-584.
•T. T. Wu, "Theory of the dipole antenna and the two-wire transmission line," J. Math. Phys., vol. 2, 1961, pp. 550-574.
•R. H. Duncan, "Theory of the infinite cylindrical antenna including the feedpoint singularity in antenna current," J. Research NBS (D-Radio Prop.), vol. 66D, no.
2, 1962, pp. 181-188.
•Y. M. Chen and J. B. Keller, "Current on and input impedance of a cylindrical antenna," J. Research NBS (D-Radio Prop.), vol. 66D, no. 1, Jan.-Feb. 1962, pp.
15-21.
•R. L. Fante, "On the admittance of the infinite cylindrical antenna," Radio Science, vol. 1, no. 9, Sep. 1966, pp. 1041-1044.
•R. W. P. King and T. T. Wu, "The imperfectly conducting cylindrical transmitting antenna," IEEE Trans. Antennas Prop., vol. AP-14, no. 5, Sep. 1966, pp. 524-534.
•E. K. Miller, "Admittance dependence of the infinite cylindrical antenna upon exciting gap thickness," Radio Science, vol. 2, no. 12, Dec. 1967, pp. 1431-1435.
•R. W. P. King, "The linear antenna - eighty years of progress," Proceedings of the IEEE, vol. 55, no. 1, Jan. 1967, pp. 2-16.
•R. A. Hurd and J. Jacobsen, "Admittance of an infinite cylindrical antenna with realistic gap feed," Electronics Letters, vol. 4, no. 19, Sep. 1968, pp. 420-421.
•L.-C. Shen, T. T. Wu, and R. W. P. King, "A simple formula of current in dipole antennas," IEEE Trans. Antennas Prop., vol. AP-16, no. 5, Sep. 1968, pp. 542-547.
Integration contour
(ka)
2

1
 K 2 J 0 ( ka  K 2 ) H 0( 2) ( ka  K 2 )
2
2
(Miller)
Test of the MATLAB’s solution
Table 1. Antenna input admittance in mS for g/a=0.05 obtained using different models. The
corresponding values obtained in Ref. [25] are marked bold.
Method
ka=0.01
ka=0.04
ka=0.08
Chen and Keller [20]; orig. Eq. (22):
constant gap field; total current;
script circ1.m
1.92 + j1.09
2.72 + j2.66
3.40 + j4.38
Chen and Keller [20]; orig. Eq. (22):
square-root gap field; total current;
script circ1.m
1.92 + j1.12
2.72 + j2.78
3.40 + j4.63
1.92 + j0.95
1.92 + j0.95
2.72 + j2.09
2.72 + j2.09
3.40 + j3.25
3.40 + j3.25
1.92 + j0.96
1.92 + j0.97
2.72 + j2.15
2.72 + j2.17
3.40 + j3.37
3.40 + j3.41
Miller [23], orig. Eq. (1):
constant gap field; outer current;
script circ2.m
Miller [23], orig. Eq. (1):
square-root gap field [16],[25]; outer
current; script circ2.m
A passing remark: dipole antenna as a
half wave transmission line resonator
The transmission line is the
infinite wire with the second
conductor at infinity
According to Shen, Wu, and King [26]: "The
foregoing discussion seems to support the following picture of a dipole antenna. An outgoing
traveling wave of current is generated along the dipole antenna when it is driven by a timeharmonic source. It travels along the two arms of the dipole with a speed almost equal to the
speed of light, and decays slowly…, as a result of radiation. It is reflected at the ends of the
dipole… . After it is reflected, the current wave travels in the opposite direction with the
same speed and decays in the same manner as before. The current distribution on the antenna
is just the result of the superposition of the outgoing current wave and all the reflected waves.
This description of the current along a dipole antenna is analogous to that for a lossless
transmission line…"
Array solution
Here we assume equal gaps and
feeds (for simplicity only)
The feed voltage, gap voltage, and the current for
the m-th array element are given by
Vm  V0 exp(  jk x 0 d x m), m  0,1,2,...
Vm  V0 ' exp(  jk x 0 d x (2m  1) / 2), m  0,1,2,...
I m  I ( x  xm ) exp(  jk x 0 x), m  0,1,2,...
k x 0  k 0 sin  0 cos  0 , k y 0  k 0 sin  0 sin  0
where I (x ) is a periodic function of x with the period d x .
Why is the array solution simpler
than the solution for the infinite
dipole?
First guess: a double sum of the
integrals in the complex plane?
m  
 V0
~
E x (k x ) 
F (k x )  exp(  j (k x 0  k x )d x m)
2
m  
k g
F (k x )  1, sinc  x , J 0 (k x g )
  
m  
 V0
2m 
~

E x (k x ) 
F (k x )    k x  k x 0 

dx
dx

m   
Impulse train or Dirac comb
Integrals go away!
The rest of the solution
remains the same
Centerpiece of the VanKoughnett
and Yen’s model for wide feeds
Periodic component; gaps are shorted out
I S (x)
I S (x)
Periodic component; feeds are shorted out
I S ( x)  (V0 / V0 ) I S ( x  d x / 2)
Express one in terms of another
V0 I ( x)  V0 I S ( x)  V0 ' I S ( x  d x / 2) Total periodic component by superposition
V0 '  V0
I S (d x / 2)
I S (0)
No current in the gap: only if IS is constant
there
Fortunately, the square root feed model is
close to this assumption!
Proof (periodic current component)
d x   / 2, ka  0.08, g / a  0.61
Discuss the feed
– the center
Solution for a planar array
Solution for the linear array:

4kaV0 m  
q m2 J 0 (q m ) H 0( 2 ) (q m )
2m a 
1
IS (X ) 
exp   j
X C m ,

,

Z 0 (d x / a ) m  
dx
Cm
F (K m )



2m
q  ka  K , K m  a k x 0 
dx

2
m
2
2
m



Solution for the planar array:
H
( 2)
0
(q m ) 
n  
 exp(  jnk
n  
y0
dy)
 H ( 2)  q 1  nd / a 2   H ( 2)  q (2h / a) 2  nd / a 2
y
0
y
 0  m

 m


Comparison with
numerical simulations:
E-plane scan (a /2
array)
HFSS – Floquet port;
SCT MWS – Floquet port
Comparison with
numerical simulations:
D-plane scan (a /2
array)
HFSS – Floquet port;
SCT MWS – Floquet port
Comparison with
numerical simulations:
H-plane scan (a /2
array)
HFSS – Floquet port;
SCT MWS – Floquet port
Scan impedance error for all scan
angles/planes (the half-wave array)
Parameter
g / a  0.5
g / a  1.0
ka  0.04
ka  0.08
E EANSOFT  5.0% E ECST  2.5%
E EANSOFT  6.5% E ECST  4.8%
E DANSOFT  7.1% E DCST  3.5%
E DANSOFT  11% E DCST  12%
E HANSOFT  13% E HCST  5.5%
E HANSOFT  13% E HCST  14%
E EANSOFT  3.8% E ECST  1.7%
E EANSOFT  5.5% E ECST  1.7%
E DANSOFT  3.5% E DCST  2.4%
E DANSOFT  4.8% E DCST  3.7%
E HANSOFT  10% E HCST  7.2%
E HANSOFT  8.6% E HCST  10%
MATLAB script array3.m
for ph = 1:length(phi0)
for th = 1:length(theta0)
th
kx0
= k*sin(theta0(th))*cos(phi0(ph)); %
phase progression factor
ky0
= k*sin(theta0(th))*sin(phi0(ph)); %
phase progression factor
Km
= a*(kx0 + 2*M*pi/dx);
%
vector (for outer summation)
temp
= ka^2 - Km.^2;
%
vector (for outer summation)
root
= sign(temp).*sqrt(temp);
%
vector (for outer summation)
BESSELJ = besselj(0, root);
%
vector (for outer summation)
%F
= sinc(Km*gtd/pi);
%
uniform gap field
F
= besselj(0, Km*gtd);
%
square root gap field
root1
= sqrt(1
+ (N*dy/a).^2);
%
vector (for inner summation)
root2
= sqrt((2*h/a)^2 + (N*dy/a).^2);
%
vector (for inner summation)
BESSELH = j*zeros(1, length(M));
for n = 1:length(N)
BESSELH = BESSELH + exp(-j*(N(n))*ky0*dy)*...
(besselh(0, 2, root*root1(n)) - besselh(0, 2, root*root2(n)));
end
Cm
= F./(temp.*BESSELJ.*BESSELH);
for p = 1:length(X)
EXP
= exp(-j*2*pi*M*a/dx*X(p));
Is(p)
= 1e3*(4*ka/(const.eta*dx/a))*sum(EXP.*Cm); % Is in mA/V
end
q = 1; %
0 for tip zero current or 1 for center zero current
V0_prime
= -Is(end-1+q)/Is(P/2+q);
Per
= 1:P/2;
I(Per)
= Is(Per) + V0_prime*Is(Per+P/2);
%
I (per) in mA/V
Per
= P/2+1:P+1;
I(Per)
= Is(Per) + V0_prime*Is(Per-P/2);
%
I (per) in mA/V
Im
= I.*exp(-j*kx0*X*a);
%
Current solution
Zs(ph, th) = 1e3/Im(P/2 + q);
%
Scan impedance (Ohm)
end
end
Limitations of the analytical model:
no angular current symmetry
Possible extensions and
applications of the analytical model
• Array of strip dipoles
Straightforward
•A loaded infinite dipole array (capacitive, or inductive, or
resistive loading between the dipole ends)
Straightforward
• An infinite dipole array with an (infinite) array taper
(Gaussian amplitude taper, Gaussian phase correction, a
load taper, etc.)
Rather challenging, but
perhaps important
• A lossy ground plane , dielectric layers, etc.
Challenging
Array of strip dipoles
A predefined transcendental profile
Green’s function remains the same if w/2 is replaced by a
•C. M. Butler, “A formulation of the finite-length narrow slot or strip equation,” IEEE Trans. Antennas Prop., vol.
AP-30, no. 6, Nov. 1982, pp. 1254-1257.
Test (array of /2 dipoles)
a)
c)
b)
d)
Comparison with
numerical simulations:
E-plane scan (a /2
array)
Comparison with
numerical simulations:
D-plane scan (a /2
array)
Comparison with
numerical simulations:
H-plane scan (a /2
array)
MATLAB script array4.m
for ph = 1:length(phi0)
for th = 1:length(theta0)
th
kx0
= k*sin(theta0(th))*cos(phi0(ph)); %
phase progression factor
ky0
= k*sin(theta0(th))*sin(phi0(ph)); %
phase progression factor
Km
= a*(kx0 + 2*M*pi/dx);
%
vector (for outer summation)
temp
= ka^2 - Km.^2;
%
vector (for outer summation)
root
= sign(temp).*sqrt(temp);
%
vector (for outer summation)
BESSELJ = besselj(0, root);
%
vector (for outer summation)
%F
= sinc(Km*gtd/pi);
%
uniform gap field
F
= besselj(0, Km*gtd);
%
square root gap field
root1
= sqrt(1
+ (N*dy/a).^2);
%
vector (for inner summation)
root2
= sqrt((2*h/a)^2 + (N*dy/a).^2);
%
vector (for inner summation)
BESSELH = j*zeros(1, length(M));
for n = 1:length(N)
BESSELH = BESSELH + exp(-j*(N(n))*ky0*dy)*...
(besselh(0, 2, root*root1(n)) - besselh(0, 2, root*root2(n)));
end
Cm
= F./(temp.*BESSELJ.*BESSELH);
for p = 1:length(X)
EXP
= exp(-j*2*pi*M*a/dx*X(p));
Is(p)
= 1e3*(4*ka/(const.eta*dx/a))*sum(EXP.*Cm); % Is in mA/V
end
q = 1; %
0 for tip zero current or 1 for center zero current
V0_prime
= -Is(end-1+q)/Is(P/2+q);
Per
= 1:P/2;
I(Per)
= Is(Per) + V0_prime*Is(Per+P/2);
%
I (per) in mA/V
Per
= P/2+1:P+1;
I(Per)
= Is(Per) + V0_prime*Is(Per-P/2);
%
I (per) in mA/V
Im
= I.*exp(-j*kx0*X*a);
%
Current solution
Zs(ph, th) = 1e3/Im(P/2 + q);
%
Scan impedance (Ohm)
end
end
A loaded infinite dipole array
(capacitive or inductive loading)
V0 I ( x)  V0 I S ( x)  V0 ' I S ( x  d x / 2)
Total periodic component by superposition
d 
d 
V


I x   x   0 I x   x    0
2 
2 
ZL


Unloaded array
Loaded array
I S (d x / 2)
V0 '  V0
I S (0)
I S (d x / 2)
V0 '  V0
V0
I S (0) 
Z Load
One line in a MATLAB code!
Test (a /2 array)
Lumped LCR boundary
Dipole feed
Lumped LCR boundary
Scan impedance at zenith
Scan impedances at 7 at 10 GHz and for different capacitive end-to-end
loads. Two sets of numbers correspond to Ansoft HFSS data and to the
analytical solution (marked bold), respectively. The analytical solution is
obtained with the script array3.m.
C
Scan resistance, Ω
Scan reactance, Ω
0.0005pF
201.2
206.5
-1.5
+3.7
0.005pF
242.5
249.6
+19.6
+26.4
0.05pF
627.0
659.9
-9.0
-4.5
0.5pF
730.2
749.7
-611.5
-640.5
Scan impedance at different
elevation angles
Optimization of a wideband loaded
strip dipole array (only at zenith –
array5.m)
•Ben A. Munk, "A wide band, low profile array of end-loaded dipoles with dielectric slab compensation," 2006 Antenna Applications
Sym., Allerton Park, Monticello, IL, 2006, pp. 149-165.
Four nested loops; total about1,100 data points
for nh = 1:length(h)
%
first optimization loop
for nr = 1:length(R);
%
second optimization loop
r = R(nr);
nr
for nl = 1:length(C)
%
third optimization loop
c = C(nl);
for m = 1:length(f)
[MAIN BODY - single calculation]
end
temp2 = max(max(max(RL)));
temp3 = max(max(max(RL(:,:,1))));
if (temp2 < temp1) & (temp3 < -10) % at least -10dB RL everywhere
NH
= nh;
NR
= nr;
NL
= nl;
temp1
= temp2;
RL_out
= RL;
end
end
end
end
A 4:1 non-scanning array
(array5.m)
R=225; C=2.1e-013; h=0.012
-10
-11
scan return loss, dB
-12
-13
-14
-15
-16
-17
-18
-19
2
3
4
5
6
7
frequency, Hz
8
9
10
9
x 10
A 2:1 scanning array (array6.m)
About 20,000 data points
R=425; C=1.35e-013; h=0.0105
0
scan return loss, dB
-5
-10
-15
-20
-25
-30
0
10
20
30
scan angle , deg
40
50
60
Array taper: preliminary remarks
V0
V0 
1  x 2
The quadratic amplitude taper leads to a
single sum of the 1D integrals for the 2D
array
n  
 cos nk exp( n
n  
2
)
The Gaussian amplitude taper needs this
sum to be expressed analytically, as a
function of k and 
Conclusions
•For the infinite planar array of dipole antennas, we have extended the
analytical model of VanKoughnett and Yen to the case of a finite feeding
gap and a non-uniform field distribution in the gap.
•As a result, we were able to compare the accurate theory model with the
numerical simulations for infinite dipole antennas arrays using two major
antenna software packages - Ansoft HFSS v. 11 and CST Microwave
Studio 2008.
•To our knowledge, such a quantitative comparison for the dipole array has
been performed for the first time.
•The analytical array model seems to be a useful tool for array optimization
MATLAB scripts are available online:
http://ece.wpi.edu/ant/01MATLABAntennaArray
Acknowledgements
•Authors are thankful to Dr. H. Steyskal for his continuous interest and
support of this work, and for critical comments, and to Dr. R. C. Hansen
for his interest and important comments.
•We would like to thank especially Mr. Robert Helsby of Ansoft
Corporation/ANSYS for his encouragement and patience throughout
different stages of this research.
•We are thankful to Mr. Mark Jones, now with Ansoft
Corporation/ANSYS, for valuable insight, time and resources invested
into this project.
•We are grateful to Mr. Frederick Beihold, CST of America, for numerous
and extensive test and validation results related to the present study,
and useful discussions, and to Dr. David Johns, CST of America, for the
support.
•This paper has been completed when one of the authors (SNM) was on
sabbatical leave with Lawrence Livermore National Laboratory,
Livermore, CA.