PIERS Proceedings, Taipei, March 25–28, 2013
268
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, Taiwan
2
Department of Communications Engineering, Yuan Ze University, Chung-Li, Taiwan
Abstract— The increasing interest in the time domain (TD) analysis of ultra wideband or
short pulse target identification and remote sensing applications has resulted in the development
of new TD techniques to analyze the antenna radiation, which provides more physically appealing
interpretation of wave phenomena. Most recently the applications have been extended to treat
the problems arising in the near zone of electrically large antennas such as the vital life-detection
systems and noncontact microwave detection systems, where the objects under detection may
locate in the near zone of antenna. The potential applications of near-field antennas continue
to grow dramatically and desire more exploration in the near future. A TD analytic solution
to predict the transient radiation from a phased periodic array of elemental antennas is thus
developed. This paper presents an analytical transient analysis of electromagnetic field radiation
from 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 elemental
antennas is thus developed. In this analysis, the array excitation phases were impressed to radiate
electromagnetic (EM) fields focused in the near zone of array aperture [1–3]. Potential and practical
antenna designs in the frequency domain (FD) have been investigated and found with feasible
implementation strategies for the phased array antennas. The TD phenomena are investigated in
this paper. The developed TD analytical solution is a response to the elemental antennas with a
transient impulse input on its current moment. The response to a realistic astigmatic finite-energy
pulse can be obtained by applying the ordinary convolution theorem to obtain the early-time
transient radiation fields generated by the same antennas.
Due to the sophisticated complexity in the analytical analysis in TD, most of past TD EM analysis tends to employ TD numerical techniques, such as finite difference time domain (FDTD) and TD
integral approaches, which provided exact solutions, but suffered from computational inefficiency to
treat the radiations problems of electrically large antennas. Thus, it remains attractive to develop
quasi-analytical TD solutions with simple and closed forms, which have the advantage of providing
physical interpretations of wave behaviors. Examples include the developments of TD uniform geometrical 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 a
direct inverse Laplace transform or an analytical time transform (ATT) of the corresponding FD
formulations. These solutions are limited to the transient analysis of antenna radiation with scattering mechanisms such as the reflector antennas, and are not applicable to the current situation
of 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 phase
impressions were assumed to produce far-field focused radiation of angularly offset beams. The TD
phenomena of Floquet modes in the quantity of field potentials with a transient impulse excitation
in the current moments were examined. The current work can be viewed as a generalization as it
provides 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 first
considers a two dimensional finite array of current moments with phases impressed to radiate
fields focused in the near zone of array aperture, where the focal field point can be arbitrarily
selected. Thus the presented analysis is valid for both near- and far-field focus applications. Also
the assumption of near-field focusing for the array excitation exhibits many unique wave phenomena
that were not revealed in the previous works [4, 5] since they appear to focus the array with a
linear phase impression. These phenomena are very important for its application in the near-field
communications.
Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 25–28, 2013
269
2. FORMULATIONS OF THE TRANSIENT IMPULSE RESPONSE FOR A PLANAR
AND 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, dp̄(r̄0 , 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
¡
¢
¡ ¢ ¡
¢
dp̄ r̄0 , t = dP̄ r̄0 · δ t − t0
(1)
0
where δ(·) is the Dirac delta function. The nmth element of the array is located at r̄nm
=
(ndx , mdy , 0)(−Nx ≤ n ≤ Nx , −Ny ≤ m ≤ Ny ). The radiation exhibits a transient behavior
of impulse in TD by
δ(t − rnm
1
c )
,
(2)
dF̄nm (r̄, t) =
dP̄ (r̄0 )
4π
rnm
0
where r̄nm = r̄ − r̄nm
with r̄ = (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
dF̄ (r̄, s) =
Ny
X
Nx
X
A(n, m)
m=−Ny n=−Nx
1
s
ro,nm
e c φ(n,m) dF̄nm (r̄, s),
(3)
where r̄o,nm = (xo − ndx , yo − mdy , zo ) and φ(n, m) = ro,nm − ro with r̄o 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
³
´

rnm −φ(n,m)
Ny
Nx
δ
t
−
X
X
c
1 
 dP̄ (r̄0 ).
dF̄ (r̄, t) =
A(n, m) ·
(4)
4π
ro,nm rnm
m=−Ny n=−Nx
Equation (4) can be expressed in terms of Floquet modes by using the Poisson sum formula, and
becomes
4
4
∞
∞
X
X
1X
1X
w
C̄` (r̄, t) +
dF̄ (r, t) =
Ḡα (r̄, t) +
F̄pq
(r̄, t),
(5)
4
2
q=−∞ p=−∞
`=1
α=1
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-form formulation, and
will be presented in the conference.
corner
" = 3"
edge
x̂ "α = 1"
edge
"α = 2"
edge
"α = 4"
ŷ
corner
" = 4"
Figure 1: A two dimensional periodic array of current moments induced on the array antenna elements to radiate near-zone focused field at r̄o .
corner
" = 2"
edge
"α = 3"
corner
" = 1"
Figure 2: Illustration of edge column/row and corner
elements used to compute C̄` (r̄, t) and Ḡα (r̄, t).
PIERS Proceedings, Taipei, March 25–28, 2013
270
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 at
time, t as illustrated in Figure 3(a). This contour is either a hyperbolic or elliptical curve depending
o the observation time and location. It is found that this contour, Ct (t), is the intersection of a
hyperbolic surface and z = 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) and F2 = (x0 , y0 , z0 ), 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 be presented in the conference.
w
3.2. Solution of F̄pq
(r̄, t)
w (r̄, t) can be formulated according to the integration contour. For an example
The solution of F̄pq
of an elliptical contour, it can be expressed
w
F̄pq
(r̄, t) = −
[U (t − t1 ) − U (t − t2 )] cABLt
dP̄ (r̄0 )e−j(pd e1,d +qd e2,d ) J˜0 (ξ)
2dx dy (2Nx + 1)(2Ny + 1)
(6)
where U (·) is a step function, A and B are related to p
the radii of the elliptical contour with
(e1,d , e2,d ) being the location of center. In (6), Lt = ct − x20 + y02 + z02 , (pd , qd ) is related to the
mode, and J˜0 (ξ) is referred as the incomplete Bessel function. 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. The formulations for the cases of
linear and hyperbolic contours will be presented in the conference.
Ct (t2 )
x̂
ŷ
Sa
Ct (t1 )
(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 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 sources
with impulse excitations. The periods are 0.1 m in both x- and y-dimensions. The focus and observation
points are at (0, 0, 50 m) and (0, 0, 1 m), respectively.
Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 25–28, 2013
271
4. NUMERICAL EXAMPLES
The array of antennas has a period of 0.1 m in both x- and y-dimensions. The focus point is at
(xo , yo , zo ) = (0, 0, 50 m), which is relatively far away from the array aperture. The observation
point is at (0, 0, 1 m), and is in the near zone. In this examination, one considers the behaviors
of F̄pq (r̄, t), which represents the dominating contributions in the array radiation for the impulse
current distributions for every element.
One first considers the case of an infinite array, and examines the fundamental (p, q) = (0, 0),
(1, 0) and (2, 2) modes, whose results are shown in Figures 4(a)–(c).
5. CONCLUSIONS
This paper presents an analytical transient analysis of electromagnetic field radiation from a phased
and finite periodic array of antennas for the near- and far-field focused applications. The elemental
current moments of array are assumed with a transient impulse input 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 field
radiation 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.
Phenomena of partial cylindrical wave functions have been observed.
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