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268 PIERS Proceedings, Taipei, March 2528, 2013

Time Domain Transient Analysis of Electromagnetic Field Radiation

for Phased Periodic Array Antennas Applications

Shih-Chung Tuan1 and Hsi-Tseng Chou2

1

Department of Communications Engineering, Oriental Institute of Technology, Pan-Chiao, Taiwan2Department of Communications Engineering, Yuan Ze University, Chung-Li, Taiwan

Abstract The increasing interest in the time domain (TD) analysis of ultra wideband orshort pulse target identification and remote sensing applications has resulted in the developmentof new TD techniques to analyze the antenna radiation, which provides more physically appealinginterpretation of wave phenomena. Most recently the applications have been extended to treatthe problems arising in the near zone of electrically large antennas such as the vital life-detectionsystems and noncontact microwave detection systems, where the objects under detection maylocate in the near zone of antenna. The potential applications of near-field antennas continueto grow dramatically and desire more exploration in the near future. A TD analytic solutionto predict the transient radiation from a phased periodic array of elemental antennas is thusdeveloped. This paper presents an analytical transient analysis of electromagnetic field radiationfrom a phased and finite periodic array of antennas for the near- and far-field focused applications.

1. INTRODUCTION

A TD analytic solution to predict the transient radiation from a phased periodic array of elementalantennas is thus developed. In this analysis, the array excitation phases were impressed to radiateelectromagnetic (EM) fields focused in the near zone of array aperture [13]. Potential and practicalantenna designs in the frequency domain (FD) have been investigated and found with feasibleimplementation strategies for the phased array antennas. The TD phenomena are investigated inthis paper. The developed TD analytical solution is a response to the elemental antennas with atransient impulse input on its current moment. The response to a realistic astigmatic finite-energypulse can be obtained by applying the ordinary convolution theorem to obtain the early-timetransient radiation fields generated by the same antennas.

Due to the sophisticated complexity in the analytical analysis in TD, most of past TD EM analy-sis tends to employ TD numerical techniques, such as finite difference time domain (FDTD) and TDintegral approaches, which provided exact solutions, but suffered from computational inefficiency totreat the radiations problems of electrically large antennas. Thus, it remains attractive to developquasi-analytical TD solutions with simple and closed forms, which have the advantage of providingphysical interpretations of wave behaviors. Examples include the developments of TD uniform geo-metrical theory of diffraction (TD-UTD), physical theory of diffraction (TD-PTD), physical optics(TD-PO) and TD aperture integration (TD-AI) techniques that were obtained by using either adirect inverse Laplace transform or an analytical time transform (ATT) of the corresponding FDformulations. These solutions are limited to the transient analysis of antenna radiation with scat-tering mechanisms such as the reflector antennas, and are not applicable to the current situationof direct antenna radiation from a phased array.

The past works most related to the current one are these in [4, 5] where the TD radiations of

two-dimensional infinite or semi-infinite array of dipoles were analyzed. Sequentially linear phaseimpressions were assumed to produce far-field focused radiation of angularly offset beams. The TDphenomena of Floquet modes in the quantity of field potentials with a transient impulse excitationin the current moments were examined. The current work can be viewed as a generalization as itprovides more complete and comprehensive analysis, and will reduce to previous solutions in [4, 5]when the focal point is moved into the far zone of array. In this generalized analysis, one firstconsiders a two dimensional finite array of current moments with phases impressed to radiatefields focused in the near zone of array aperture, where the focal field point can be arbitrarilyselected. Thus the presented analysis is valid for both near- and far-field focus applications. Alsothe assumption of near-field focusing for the array excitation exhibits many unique wave phenomenathat were not revealed in the previous works [4, 5] since they appear to focus the array with alinear phase impression. These phenomena are very important for its application in the near-fieldcommunications.

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Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 2528, 2013 269

2. FORMULATIONS OF THE TRANSIENT IMPULSE RESPONSE FOR A PLANARAND RECTANGULAR ANTENNA ARRAY

2.1. Transient Phenomena of an Unit Current Moment

A planar, rectangular array of (2Nx+1)(2Ny +1) 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 currentmoment has a transient behavior by

dpr, t

= d P

r t t

(1)

where () is the Dirac delta function. The nmth element of the array is located at rnm =(ndx,mdy, 0)(Nx n Nx, Ny m Ny). The radiation exhibits a transient behaviorof impulse in TD by

dFnm(r, t) = 1

4dP(r)(t rnm

c )

rnm, (2)

where rnm= r r

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

2.2. Transient Phenomena of an Array of Phased Unit Current Moments

The net potential of a NFA is given by

dF(r, s) =

Nym=Ny

Nxn=Nx

A(n,m) 1

ro,nmes

c(n,m)dFnm(r, s), (3)

where ro,nm = (xo ndx, yo mdy, zo) and (n,m) = ro,nm ro with ro 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

dF(r, t) = 1

4

Nym=Ny

Nxn=Nx

A(n,m) t

rnm(n,m)c

ro,nmrnm

dP(r). (4)

Equation (4) can be expressed in terms of Floquet modes by using the Poisson sum formula, and

becomes

dF(r, t) =1

4

4=1

C(r, t) +1

2

4=1

G(r, t) +

q=

p=

Fwpq(r, t), (5)

where each term is associated corner effects, edge effects and Floquet mode effects as illustrated inFigure 2. It is noted that each terms in (5) have been evaluated in a closed-form formulation, andwill be presented in the conference.

Figure 1: A two dimensional periodic array of cur-rent moments induced on the array antenna ele-ments to radiate near-zone focused field at ro.

x

y

" 2"=corner

" 3"=

corner

" 4"=corner

" 1"=corner

" 3" =

" 4" =

edge" 1" =

" 2" =

edge

edge

edge

Figure 2: Illustration of edge column/row and cornerelements used to compute C(r, t) and G(r, t).

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270 PIERS Proceedings, Taipei, March 2528, 2013

3. RADIATION CHARACTERISTICS

3.1. An Integration Contour and Its Characteristics

An equal time delay contour exists on the array aperture, which contributes to the radiation field attime,t as illustrated in Figure 3(a). This contour is either a hyperbolic or elliptical curve dependingo the observation time and location. It is found that this contour, Ct(t), is the intersection of ahyperbolic surface andz = 0 plane as illustrated in Figure 3(b) as an example, where the two focused

points of the hyperbolic surface are located at F1= (x,y,z) andF2= (x0, y0, z0), respectively. Thusthe hyperbolic surface is formed by the two focuses at the focus and observer, which results in equaltime delays. Detailed discussion of the integration contour will be presented in the conference.

3.2. Solution ofFwpq

(r, t)

The solution ofFwpq(r, t) can be formulated according to the integration contour. For an exampleof an elliptical contour, it can be expressed

Fwpq(r, t) = [U(t t1) U(t t2)] cABLt

2dxdy(2Nx+ 1)(2Ny+ 1) dP(r)ej(pde1,d+qde2,d)J0() (6)

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

(e1,d, e2,d) being the location of center. In (6), Lt = ct x20

+ y2

0

+ z2

0

, (pd, qd) is related to the

mode, and J0() is referred as the incomplete Bessel function. This formulation is presented herebecause it reduces to the case in the far-field focusing antenna array because the incomplete Besselfunction will reduce to the ordinary Bessel Function as pointed. The formulations for the cases oflinear and hyperbolic contours will be presented in the conference.

x

y

aS

1( )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 intersectionbetween the aperture and a hyperbolic surface.

(a) (p,q)=(0,0) (b) (p,q)=(1,0) (c) (p,q)=(2,2)

Figure 4: Transient responses of various Floquet modes for an infinite and a finite array of current sourceswith impulse excitations. The periods are 0.1m in both x- and y-dimensions. The focus and observationpoints are at (0, 0, 50 m) and (0, 0, 1 m), respectively.

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