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

t

yyg

xk

xl

k

l

x

xfygxfJxy)(x,J )(,

C casecos

B case2

sin

A case

)(),()(

0

00

Solution for the co-polar electric field

m

m

n

nymnxmnymnxmn yjkxjkkgkfzJ

dxdyxz)y(x,J )exp()(~)(

~)(

4, 0

2

yyymn

xxxmn d

nkk

d

mkk

2,

200

00000000 sinsin,cossin kkkk yx

),,(20

2 zyxJAkA xxx

Floquet theorem

Aj

Hj

E

00

11

m

m

n

nzmnymnxmnymnxmn

zmn

xmnx zjkyjkxjkkgkf

k

kkJ

dxdyz)y(x,E )exp()(~)(

~2,

220

00

2

Scan impedance (comparison)

20

cellunit

*

20

cellunit

*

*00

),()0,,(

I

dxdyyxJzyxE

I

dsJE

II

PZ

xx

S

Reaction integral

m

mzmn

n

nymnxmn

zmn

xmn

yxS hjkkkP

kk

kk

kDDtZ )2exp(1),(

/

/122

0

20

2

20

2

2

Scan impedance

2sinc

2sinc

4),(

2

20

tklkkltkkP ymnxmn

ymnxmn

2sinc

4

)(sinc

4

)(sinc

16),( 00

2

30

2 tklkklkkkltkkP ymnxmnxmn

ymnxmn

2sinc

2cosc

2),(

20

tklkktkkP ymnxmn

ymnxmn

0for

0for 222

020

22

2220

2220

ymnxmnymnxmn

ymnxmnymnxmnzmn

kkkkkkj

kkkkkkk

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

xdxxGxIkxjk

ZxEx )()()( 2

2

20

xxx

xx

xxx dkxjkkIxI

kGkkZ

kEjkkI )exp()(

~)(,

)(~

1

2

)(~

)(~

220

2)(

~),()( 0

0

VkExVxE xxx

gkVkEgxVxE x

xxx sinc2

)(~

),2/()( 00

gkJV

kExgVxE xxxx 002/122

0 2)(

~,)(/)(

Analytical solution to Pocklington integral equation

aXxaKk x ,/

0 0

220

)()cos(1

)(~

,)(

~)(

)cos()(~

)( dXXaGKXKGdKKGKka

KXKE

Z

jkaXI x

22)2(00 ),()(

8

1)(

~KkaqqHqJ

jKG

•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.

This integral is to be found numerically

Integration contour

)()()(

122)2(

022

022 KkaHKkaJKka

(Miller)

Test of the MATLAB’s solution

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

Miller [23], orig. Eq. (1):constant gap field; outer current;

script circ2.m

1.92 + j0.951.92 + j0.95

2.72 + j2.092.72 + j2.09

3.40 + j3.253.40 + j3.25

Miller [23], orig. Eq. (1):square-root gap field [16],[25]; outer

current; script circ2.m

1.92 + j0.961.92 + j0.97

2.72 + j2.152.72 + j2.17

3.40 + j3.373.40 + j3.41

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.

A passing remark: dipole antenna as a half wave transmission line resonator

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 time-harmonic 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…"

The transmission line is the infinite wire with the second conductor at infinity

Array solution

The feed voltage, gap voltage, and the current for the m-th array element are given by

,...2,1,0),exp( 00 mmdjkVV xxm

,...2,1,0),2/)12(exp(' 00 mmdjkVV xxm

,...2,1,0),exp()( 0 mxjkxxII xmm

00000000 sinsin,cossin kkkk yx

where )(xI is a periodic function of x with the period xd .

Here we assume equal gaps and feeds (for simplicity only)

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

mxxxxxx mdkkjkF

VkE ))(exp()(

2)(

~0

0

)(,sinc,1)( 0 gkJgk

kF xx

x

m

mxxx

xxx dx

mkkkF

d

VkE

2)()(

~0

0

Integrals go away!

Impulse train or Dirac comb

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)(xI S

)(xI S

)2/()/()( 00 xSS dxIVVxI

Periodic component; feeds are shorted out

)2/(')()( 000 xSS dxIVxIVxIV

Express one in terms of another

Total periodic component by superposition

)0(

)2/(' 00

S

xS

I

dIVV 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)

61.0/,08.0,2/ agkad x

Discuss the feed – the center

Solution for a planar array

xxmmm

m

m m

mmm

mm

xxS

d

mkaKKkaq

KF

qHqJq

CCX

d

amj

adZ

kaVXI

2,

,)(

)()(1,

2exp

)/(

4)(

0222

)2(00

2

0

0

Solution for the linear array:

22)2(0

2)2(0

0)2(

0

/)/2(/1

)exp()(

andahqHandqH

djnkqH

ymym

n

nyym

Solution for the planar array:

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

04.0ka

08.0ka

5.0/ ag

%13

%1.7

%0.5

ANSOFT

ANSOFT

ANSOFT

H

D

E

E

E

E

%5.5

%5.3

%5.2

CST

CST

CST

H

D

E

E

E

E

%13

%11

%5.6

ANSOFT

ANSOFT

ANSOFT

H

D

E

E

E

E

%14

%12

%8.4

CST

CST

CST

H

D

E

E

E

E

0.1/ ag

%10

%5.3

%8.3

ANSOFT

ANSOFT

ANSOFT

H

D

E

E

E

E

%2.7

%4.2

%7.1

CST

CST

CST

H

D

E

E

E

E

%6.8

%8.4

%5.5

ANSOFT

ANSOFT

ANSOFT

H

D

E

E

E

E

%10

%7.3

%7.1

CST

CST

CST

H

D

E

E

E

E

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) endend

Limitations of the analytical model: no angular current symmetry

Possible extensions and applications of the analytical model

• Array of strip dipoles •A loaded infinite dipole array (capacitive, or inductive, or resistive loading between the dipole ends)

• An infinite dipole array with an (infinite) array taper (Gaussian amplitude taper, Gaussian phase correction, a load taper, etc.)

• A lossy ground plane , dielectric layers, etc.

Straightforward

Rather challenging, but perhaps important

Challenging

Straightforward

Array of strip dipoles

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.

A predefined transcendental profile

Test (array of /2 dipoles)

a) b)

c) 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) endend

A loaded infinite dipole array (capacitive or inductive loading)

)0(

)2/(' 00

S

xS

I

dIVV

Load

000

)0(

)2/('

ZV

I

dIVV

S

xS

One line in a MATLAB code!

)2/(')()( 000 xSS dxIVxIVxIV Total periodic component by superposition

Unloaded array Loaded array

L

x

Z

VdxI 0

2

0

2

xd

xI

Test (a /2 array)

Lumped LCR boundary

Lumped LCR boundary

Dipole feed

Scan impedance at zenith

C Scan resistance, Ω Scan reactance, Ω

0.0005pF 201.2206.5

-1.5+3.7

0.005pF 242.5249.6

+19.6+26.4

0.05pF 627.0659.9

-9.0-4.5

0.5pF 730.2749.7

-611.5-640.5

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.

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.

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 endend

Four nested loops; total about1,100 data points

A 4:1 non-scanning array (array5.m)

2 3 4 5 6 7 8 9 10

x 109

-19

-18

-17

-16

-15

-14

-13

-12

-11

-10

frequency, Hz

scan

ret

urn

loss

, dB

R=225; C=2.1e-013; h=0.012

A 2:1 scanning array (array6.m)

0 10 20 30 40 50 60-30

-25

-20

-15

-10

-5

0

scan angle , deg

scan

ret

urn

loss

, dB

R=425; C=1.35e-013; h=0.0105

About 20,000 data points

Array taper: preliminary remarks

20

0 1 x

VV

The quadratic amplitude taper leads to a

single sum of the 1D integrals for the 2D array

n

n

nnk )exp(cos 2The 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.