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Time Domain Analysis of a 2-D Phased Array

Antennas for Near Field Focusing Radiations

Shih-Chung TuanDept. of Communication Eng.,

Oriental Institute of Technology

Pan-Chiao 22061,Taiwan

Fo012@mail.oit.edu.tw

Hsi-Tseng ChouDept. of Communication Eng.,

Yuan Ze University

Chung-Li 320, Taiwan

hchou@saturn.yzu.edu.tw

AbstractThis paper presents an analytical, transient analysis of

electromagnetic field radiation from a phased and finite periodic

array of antennas for the near-field focused applications. The

elemental current moments of array are assumed with a

transient impulse inputs for the excitations whose phases are

impressed to radiate near-zone focused fields. The transient field

phenomena for each of the Floquet mode expansion were

analyzed. The solution reduces to the case of far-zone fieldradiation by moving the focus point to the far zone. The analysis

shows that the radiation exhibits an impulse field at the focused

point, and finite pulses at locations away from the focus point.

I. INTRODUCTIONA time domain (TD) analysis to predict the transient

radiation from a phased periodic array of antennas isperformed. In this analysis, the array is excited to radiateelectromagnetic (EM) fields focused in the near zone of arrayaperture [1-3], which has found many applications in the near-field antenna (NFA) applications. In particular, a NFA designsand applications in the frequency domain (FD) have been

investigated and found with feasible implementation strategiesfor the phased array antennas. The TD phenomena areinvestigated in this paper to explore the characteristics ofultrawide band operation. The TD phenomena of Floquetmodes in the quantity of field potentials with a transientimpulse excitation in the current moments are examined. Thecurrent work is very general for array radiation phenomenainterpretation as it provides complete and comprehensiveanalysis.

II. FORMULATIONS OF THE TRANSIENT IMPULSE RESPONSEFOR A PLANAR AND RECTANGULARANTENNA ARRAY

A. Transient Phenomena of an Unit Current MomentA planar, rectangular array of (2 1) (2 1)

x yN N+ +

elements of magnetic current moment, ( ', )dp r t , with periods,

dx and dy, in the x- and y- axes, respectively, is illustrated in

Figure 1. This current moment has a transient behavior by

( ', ) ( ') ( ')dp r t dP r t t = (1)

where ( ) is the Dirac delta function. The nmth element of

the array is located at ' ( , ,0)nm x yr nd md = ( x xN n N ,

)y yN m N . The radiation exhibits a transient behavior of

impulse in TD by

( )1

( , ) ( ')4

nm

nm

nm

rt

cdF r t dP r r

= , (2)

where 'nm nmr r r= with ( , , )r x y z = being the observer.

B. Transient Phenomena of an Array of Phased UnitCurrent Moments

The net potential of a NFA is given by

( , )

,

1( , ) ( , ) ( , )

y x

y x

N sNn m

cnm

m N n N o nm

dF r s A n m e dF r sr

= =

= , (3)

where , ( , , )o nm o x o y or x nd y md z = and ,( , ) o nm on m r r =

witho

r being the focus point. In (3), ( , )A n m is an amplitude

taper to reduce the diffraction effects of a finite array. In TD,

(3) becomes

,

1( , ) ( , )

4

( , )( )

( ')

y x

y x

N N

m N n N

nm

o n m nm

dF r t A n m

r n mt

c dP rr r

= =

=

. (4)

Equation (4) can be expressed in terms of Floquet modes by

using the Poisson sum formula, and becomes4 4

1 1

1 1( , ) ( , ) ( , ) ( , )

4 2

w

pq

q p

dF r t C r t G r t F r t

= = = =

= + + AA

, (5)

where each term is associated corner effects, edge effects and

Floquet mode effects as illustrated in Figure 2. It is noted that

each terms in (5) have been evaluated in a closed-formformulation, and will be presented in the conference.

III. RAADIATION CHARACTERISTICSA. An Integration Contour and its Characteristics

An equal time delay contour exists on the array aperture,

which contributes to the radiation field at time, t as illustrated

in Figure 3(a). This contour is either a hyperbolic or elliptical

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curve depending o the observation time and location. It is

found that this contour, )(tCt , is the intersection of a

hyperbolic surface and 0z= plane as illustrated in Figure 3(b)

as an example, where the two focused points of the hyperbolic

surface are located at 1 ( , , )F x y z= and 2 0 0 0( , , )F x y z= ,

respectively. Thus the hyperbolic surface is formed by the two

focuses at the focus and observer, which results in equal time

delays. Detailed discussion of the integration contour will bepresented in the conference.

B. Solution of ( , )wpqF r tThe solution of ( , )

w

pqF r t can be formulated according to the

integration contour. For an example of an elliptical contour, it

can be expressed

[ ]1, 2,( )1 2

0

( ) ( )( , ) ( ') ( )

2 (2 1)(2 1)

d d d d j p e q etw

pq

x y x y

U t t U t t cABLF r t dP r e J

d d N N

+

= + +

(6)

where ( )U is a step function, A and B are related to the radii

of the elliptical contour with 1, 2,( , )d de e being the location of

center. In (6), 2 2 20 0 0tL ct x y z= + + , ( , )d dq is related to

the mode, and0( )J is referred as the incomplete Bessel

function [4]. This formulation is presented here because it

reduces to the case in the far-field focusing antenna array

because the incomplete Bessel function will reduce to the

ordinary Bessel Function as pointed by in [5].The formulations for the cases of linear and hyperbolic

contours will be presented in the conference.

Figure1: A two dimensional periodic array of currentmoments induced on the array antenna elements to radiate

near-zone focused field ator .

x

y

" 2"=Acorner

" 3"=Acorner

" 4"=Acorner

" 1"=Acorner

" 3"=

" 4"=

edge" 1"=

" 2"=

edge

edge

edge

Figure 2: Illustration of edge column/row and corner elementsused to compute ( , )C r t

Aand ( , )G r t in (10).

x

y

aS1( )tC t

2( )tC t

(a) Integration Contour

(b) A hyperbolic surface

(c) Change of integration contour

Figure 3: The variation of integration contour for phased array

aperture, which is formed by the intersection between theaperture and a hyperbolic surface.

REFERENCES

[1]. M. Bogosanovic, A.G. Williamson, Antenna array with beamfocused in near-field zone, Electronics Letters, vol. 39, no. 9, pp.

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Beam Focused in the Near-Field Zone for Application in Noncontact

Microwave Industrial Inspection, IEEE Trans. InstrumentationMeasurements, Vol. 56, No. 6, pp. 2186-2195, Dec. 2007.

[3]. K. D. Stephan, J.B. Mead, D.M. Pozar, L. Wang, J.A. Pearce, A NearField Focused Microstrip Array for a Radiometric TemperatureSensor, IEEE Trans. Antennas Propag., Vol. 55, No. 4, pp. 1199-

1203, April 2007.

[4]. R. Cicchetti and A. Faraone "Incomplete Hankel and modified Besselfunctions: A class of special functions for electromagnetics", IEEE

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[5]. G. Capolino and L. B. Felsen "Time-domain Green's function for aninfinite sequentially excited periodic planar array of dipoles", IEEE

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