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Abstract— This paper presents a generalized analytical
expression for directivity of phased array antennas (PAAs). It
includes important design consideration of PAAs like arbitrary
element type & array geometry, complex excitations, mutual
coupling, scan angle and embedded element pattern. We also
present an approximate expression of directivity, helpful for quick
analysis of large PAAs overcoming computing time constraints of
analytical expressions. It provides a quick PAA design tool for
predicting accurate locations of blind spots in desired scan volume
due to the chosen array geometry. The paper also presents study
on wide scan characteristics of periodic sparse phased array
antennas using the proposed expressions.
Index Terms— sparse array antennas, directivity, scan blindness, wide
scan
I. INTRODUCTION
hased array antennas (PAAs) consists of a radiating
aperture supporting regular or periodic arrangement
(equidistant) of radiating elements (REs). In general, to
avoid occurrence of grating lobes (GLs), the inter-element
separation, d/ ( is the operating wavelength) is kept small
(usually < /2) [1]. Thus, these type of PAAs referred to as
dense arrays. A thumb rule for choosing d is 𝑑/ ≤ 1/(𝑠𝑖𝑛𝜃𝑔 +
𝑠𝑖𝑛𝜃𝑜), where g is the angle at which the first GL shall occur
(usually kept 90o to keep the visible region clear from onset of
GLs) [1] and o, represents maximum desired scan angle.
Hence, for instance in order to achieve a desired scan volume
of ±60o, it is required to have 𝑑/ 0.535. However, these
smaller inter-element separations lead to another problem, i.e.,
stronger mutual coupling between elements, which leads to
scan blindness. The scan blindness implies that at particular
scan angles, input reflection coefficient of an array,|𝑖𝑛| → 1,
leading to total reflection or zero directivity [1]. These locations
referred as blind spots. Sparse phased array antennas (SPAAs)
provides an efficient solution for tackling the problem of GLs
due to array elements’ periodicity and scan blindness
simultaneously [2].
The directivity, D of PAAs, dense or sparse, may serve as a
figure of merit for its characterization and is the focus of the
present paper. Various attempts made in the past for computati-
A. Kedar is with Electronics & Radar Development Establishment (LRDE),
DRDO, Bangalore-560093, India (e-mail: [email protected]). L.P.
Ligthart is em. Prof. (retd.), Delft Univ. of Technology, The Netherlands.
on of D, which presents approximate or exact analytical
expressions for it. Tai [3], Bach [4] and Langston [5] proposed
an approximate expression for D of uniformly spaced linear
arrays. These expressions were valid for only isotropic and
dipole REs. Further, neither of these included effect of the scan
angle, arbitrary array geometry, arbitrary RE type, complex
excitations and mutual coupling. Hansen [6] proposed an
expression for D considering mutual resistance term, array
excitations and SLL. There were some more attempts for
computation of D for arrays with specific element type and
excitations [7-13]. Forman [14] proposed an accurate analytical
expression of D considering effect of arbitrary array geometry,
complex excitation, inter-element spacing and scan angle.
However, it did not consider mutual coupling and restricted
only to REs with 𝑐𝑜𝑠𝑛 patterns. Ligthart [15] proposed a
generalized accurate expression for G including effect of
arbitrary shape, complex excitations, mutual coupling, scan
angle and arbitrary RE type. However, the expressions are
computationally intensive especially in the case of large arrays.
Recently Das [16] has also presented a similar analysis for D.
The expressions proposed are computationally intensive,
restricted to only isotropic REs. The need for accurate
expressions for D and importance of incorporation of embedded
element pattern (EEP) in the expression for directivity is well
explained in references [17-19].
This paper proposes a generalized analytical expression of D
for PAAs, which includes most of the important design
considerations in it such as arbitrary array geometry, arbitrary
complex excitations, mutual coupling, scan angle and EEP. In
general, computation of D using such exact analytical
expressions is time consuming in case of large arrays (number
of elements, N>100). Hence, we also present an approximate
expression for D of PAAs assuming aperture with periodic
array grid of infinite extent, i.e. independent of N. Further,
assumption made is that the array comprises of isotropic REs.
These assumptions do not limit the effectiveness of the
proposed expression in providing an accurate estimate of D.
Further, it helps in assessing the scan performance of the chosen
array grid for PAAs. A comparison with earlier publications
[16, 23-24] made herein the paper validates the proposed
expressions. These expressions are accurate for dense as well
as sparse PAAs and used for studying wide scan characteristics
of SPAAs here with the aid of few case studies. The study
Wide Scanning Characteristics of Sparse Phased
Array Antennas Using an Analytical Expression
for Directivity
Ashutosh Kedar, Senior Member, IEEE and L. P. Ligthart, Fellow, IEEE
P
0018-926X (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2018.2880006, IEEETransactions on Antennas and Propagation
presented in this paper may serve as design considerations for
wide scan phased array antennas.
The remaining paper comprises of various sections; viz.,
Section II presents discussion on the exact expression of
directivity, Section III presents the approximate expression of
directivity, Section IV presents the results and discussion
followed by conclusion, references and appendices.
Fig. 1. Array geometry and its associated coordinate system.
II. EXACT ANALYTICAL EXPRESSION FOR DIRECTIVITY OF A FINITE
SIZED PHASED ARRAY ANTENNA
Fig. 1 shows a planar array with an array boundary, S and its
associated coordinate system. The array consists of REs, N in number,
placed in a quadrant of the xy- plane. An nth element is located at (xn,
yn). The elements are coherently phased to make the array look in a
direction given by (θo, φo). R is the distance of the observation point in
far-field from the origin, O. It is assumed here that the radiation occurs
only in the upper half-space. Directivity, D, of a planar aperture
antenna is defined as [1]:
𝐷(𝜃, 𝜑) = 4𝜋|𝐹(𝜃, 𝜑)𝑚𝑎𝑥|2 𝑃0⁄ (1)
In (1), Po is the total radiating power available from S, expressed as:
𝑃0 = ∫ ∫ |𝐹(𝜃, 𝜑)|2𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜑𝜋
0
2𝜋
0 (2)
In (2), 𝐹(𝜃, 𝜑)is the far-field pattern of the antenna, defined as:
𝐹(𝜃, 𝜑) = ∬ 𝐼(𝑥, 𝑦)𝑒[−𝑖𝑘(𝑥 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑+ 𝑦𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑)] 𝑑𝑥 𝑑𝑦 𝑆
(3)
In (3), 𝐼(𝑥, 𝑦) represents the current distribution over S.
In (1), |𝐹(𝜃, 𝜑)2| = 𝐹(𝜃, 𝜑) ∙ 𝐹(𝜃, 𝜑)∗, where ‘*’ denotes a
complex conjugate operator, where 𝐹(𝜃, 𝜑) is defined in (3) and
𝐹(𝜃, 𝜑)∗ is defined as:
𝐹∗(𝜃, 𝜑) = ∬ 𝐼(,) 𝑒[−𝑖𝑘( 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑+ 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑)] 𝑑 𝑑 𝑆
(4)
In (4), (,)are the coordinates of an arbitrary point on the aperture
plane which plays the same role as (𝑥, 𝑦) coordinates. Therefore,
|𝐹(𝜃, 𝜑)|2 becomes:
|𝐹(𝜃, 𝜑)|2 = 𝐹(𝜃, 𝜑) ∙ 𝐹∗(𝜃, 𝜑) = ∬ 𝐼(𝑥, 𝑦)𝐼∗(𝑥 − , 𝑦 −
) 𝑒[𝑖𝑘{(𝑥−) 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑+(𝑦−) 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑)]𝑑𝑥𝑑𝑦𝑑𝑑 (5)
An inspection of (5) reveals that the exponent represents the phase
dependence of an isotropic radiator in the direction, (,), at a
distance, 𝑑 = √(〖(𝑥 − )〗^2 +〖(𝑦 − )〗^2 ) which can be
written as, 𝑑 = √(^2 + ^2 ). Thus, the exponent is the only factor
having (,) dependence. By taking another isotropic radiator located
at = + , we get two-element array with isotropic radiators
having inter-element separation, d. The mutual impedance between
two isotropic radiators is estimated as [1]:
𝑅0 =𝑠𝑖𝑛𝑘√(𝑥−𝜉)2+(𝑦−𝜂)2
𝑘√(𝑥−𝜉)2+(𝑦−𝜂)2 (6)
Using (5) and (6), (2) gives an expression for 𝑃0 written as:
𝑃0 = 4𝜋∬ ∬ 𝐼(𝑥, 𝑦)𝐼∗(𝜉, 𝜂)𝑠𝑖𝑛𝑘√(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2
𝑘√(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2𝑑𝑥𝑑𝑦𝑑𝜂𝑑𝜉
= 4𝜋∬ 𝜒(𝜉, 𝜂)𝑠𝑖𝑛𝑘√𝜉2+𝜂2
𝑘√𝜉2+𝜂2 (7)
In (7), 𝜒(𝜉, 𝜂) is the convolution of the current distribution on the
aperture given as:
𝜒(𝜉, 𝜂) = ∬ 𝐼(𝑥, 𝑦)𝐼∗(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝑥𝑑𝑦 (8)
The current distribution, I(x,y), is assumed as given below:
𝐼(𝑥, 𝑦) = 𝐼0(𝑥, 𝑦)𝑒−[−𝑖𝑘(𝑥𝑠𝑖𝑛𝜃0𝑐𝑜𝑠𝜑0+𝑦𝑠𝑖𝑛𝜃0𝑠𝑖𝑛𝜑0)] (9)
In (9), the exponential part represents phase of the current
distribution when a beam is scanned at an angle, (𝜃0, 𝜑0). Having
substituted (9) in (1) and after utilization of (7) and (8), an expression
of directivity is derived and given as:
𝐷(𝜃, 𝜑) =|∬𝑠 𝐼0(𝜂,𝜉)𝑑𝜉𝑑𝜂|
2
∬𝑠 𝜒0(𝜉,𝜂)𝑐𝑜𝑠 [𝑘𝑠𝑖𝑛𝜃0(𝜉𝑐𝑜𝑠𝜑0+𝜂𝑠𝑖𝑛𝜑0)]𝑠𝑖𝑛𝑘√𝜉2+𝜂2
𝑘√𝜉2+𝜂2𝑑𝜉𝑑𝜂
(10)
In (10), 𝜒0(𝜉, 𝜂) is the auto correlation function of the current
distribution at the aperture, defined as:
𝜒0 = 𝜒0(𝜉, 𝜂) = ∬𝑠𝐼0(𝑥, 𝑦)𝐼0∗(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝑥𝑑𝑦 (11)
The denominator in (10), accounts for the coupling between two
finitely separated elements (isotropic) (through sinc (·)) and the phase
distribution along the array aperture (through cos(·) terms).
Considering a planar aperture consisting of point sources, the array’s
current distribution may be represented as:
𝐼(𝑥, 𝑦) = ∑ 𝛿(𝑥 − 𝑥𝑛)𝛿(𝑦 − 𝑦𝑛)𝐼𝑛𝑁𝑛=1 (12)
In (12), represents Kronecker’s delta function [21], and 𝐼𝑛 is defined
as:
𝐼𝑛 = 𝐼0𝑛𝑒𝑥𝑝 [−𝑖𝑘𝑠𝑖𝑛𝜃0(𝑥𝑛𝑐𝑜𝑠𝜑0 + 𝑦𝑛𝑠𝑖𝑛𝜑0)] (13)
Thus, (10) is re-written as (see detailed derivation in Appendix A)
𝐷(𝜃, 𝜑) =(∑ 𝐼0𝑚
𝑁𝑚=1 )
2
∑ 𝐼0𝑚𝐼0𝑛∗ 𝑐𝑜𝑠 [𝑘𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0−𝛼𝑛𝑚)]
𝑠𝑖𝑛𝑘𝑟𝑛𝑚𝑘𝑟𝑛𝑚
𝑁𝑚,𝑛=1
(14)
In (14), 𝛼𝑛𝑚 is defined by
𝑐𝑜𝑠𝛼𝑛𝑚 =𝑥𝑛−𝑥𝑚
√(𝑥𝑛−𝑥𝑚)2+(𝑦𝑛−𝑦𝑚)2 (15a)
𝑠𝑖𝑛𝛼𝑛𝑚 =𝑦𝑛−𝑦𝑚
√(𝑥𝑛−𝑥𝑚)2+(𝑦𝑛−𝑦𝑚)2 (15b)
and the distance 𝒓𝒏𝒎 between the elements with indices m and n is
given as:
𝑟𝑛𝑚 = √(𝑥𝑛 − 𝑥𝑚)2 + (𝑦𝑛 − 𝑦𝑚)2 (16)
Hence, (14) represents an exact analytical expression for D of a finite
sized arbitrarily shaped planar array. It is a generalized expression,
which accounts for (a) arbitrary array geometry and array lattice; (b)
mutual coupling (c) arbitrary current distribution on aperture and (d)
antenna look or scan angle. The proposed expression serves as an
efficient mean for characterization of dense as well as sparse PAAs.
The antenna arrays can be classified into different categories based on
the ‘choice of d’ as (a) dense arrays, d/<1; (b) moderately sparse
arrays, 1<d/<2; and (c) highly sparse arrays, d/>2. In the present
paper, the focus is on analysis of periodic or regular SPAAs having
large inter-element separations, d/.
In the case of SPAAs, due to large value of d (~ d/ >1), the mutual
coupling contribution between the elements is negligible and the
expression for D becomes:
𝐷(θ, φ) =(∑ 𝐼𝑜𝑛
𝑁𝑛=1 )
2
∑ 𝐼𝑜𝑚𝐼𝑜𝑛𝑁𝑚,𝑛=1
= 𝑁 (17)
0018-926X (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2018.2880006, IEEETransactions on Antennas and Propagation
Thus, D for SPAAs becomes equal to N in absence of the mutual
coupling. Whereas, in the case of dense array, due to dominance of
mutual coupling, D < N.
The effect of EEP [19-20] can be included in (14) as:
𝐺𝑎(θ, φ) = (1 − |Γ𝑎|2)휀𝑙𝑜𝑠𝑠𝐷(θ, φ) (18)
In (18), Ga is the gain including effect of EEP, Γ𝑎 is active reflection
coefficient [20] and 휀𝑙𝑜𝑠𝑠 is the dissipative loss due to antenna and the
associated transmission network.
A ‘factor of 2’ needs to be multiplied to (14) in case radiation is
assumed only in upper hemisphere. In this case, the maximum D limit
achievable by optimally designed sparse arrays is 𝐷𝑠𝑝 = 2𝑁. The
expression, (14) is time consuming for large arrays (~ N >100). In
order to facilitate quick design aid to PAAs designers to estimate D
and scan performance of PAAs, an approximate analytical expression
is proposed in the next section.
III. APPROXIMATE EXPRESSION FOR DIRECTIVITY OF AN INFINITE
PLANAR PHASED ARRAY ANTENNA
This section presents an approximate expression for D of PAAs,
which provides a quick design tool for its estimation. It considers all
possible interference modes or GLs that may exist in the visible region
due to chosen inter-element separations. The proposed expression
assumes an isophoric and periodic infinite planar PAA and do not
consider finite edge effects. It also considers the effect of mutual
coupling, scan angle and inter-element separations. Further, it helps in
predicting the scan performance of the PAAs by estimating accurately
the angular locations for the onset of blind spots. The total number of
elements in a planar array are assumed as, 𝑁 = 𝑀𝑥 × 𝑀𝑦; Mx and My
are the number of elements along x- and y- axes.
Here onwards, normalized directivity, 𝐷𝑛𝑜𝑟𝑚 is considered for
analysis and drawing various inferences from the study presented in
this paper and is expressed as (see detailed derivation in Appendix B):
𝐷𝑛𝑜𝑟𝑚 =𝐷(θ,φ)
𝑀𝑥𝑀𝑦=
2𝜋𝑑𝑥𝑑𝑦
𝜆2 ∑ [1−(𝜆
𝑑𝑥𝑝−𝑠𝑖𝑛𝜃0)
2−(
𝜆
𝑑𝑦𝑞)
2
]
−1/2
𝑝,𝑞
(19)
The indices, 𝑝 and 𝑞 corresponds to 𝑝𝑞𝑡ℎ GL entering into the real
(visible region) space by the virtue of the choice of dx and dy. A
particular pqth mode contributes towards the summation in
denominator of (19) iff it satisfies the condition given below:
1 − (𝜆
𝑑𝑥𝑝 − 𝑠𝑖𝑛𝜃0)
2− (
𝜆
𝑑𝑥𝑞)
2≥ 0 (20)
Thus, the choice for the number of pq modes is critical while
computing Dnorm to include effects of all possible GLs that may exist
in the visible region. The corresponding expression for the radiated
power is:
𝑃0 = ∑ ∑ 𝑐𝑜𝑠[𝑘𝑑𝑥𝑠𝑖𝑛𝜃0(𝑝1 − 𝑝2)]𝑠𝑖𝑛(𝑘𝑟𝑝1,𝑝2,𝑞1,𝑞2)
𝑘𝑟𝑝1,𝑝2,𝑞1,𝑞2
𝑀𝑦
𝑞1,𝑞2
𝑀𝑥𝑝1,𝑝2
(21)
In (21), 𝑟𝑝1,𝑝2,𝑞1,𝑞2 is given as:
𝑟𝑝1,𝑝2,𝑞1,𝑞2= √𝑑𝑥
2(𝑝1 − 𝑝2)2 + 𝑑𝑦2(𝑞1 − 𝑞2)2 (22)
After summation over two indices (21) becomes (see Appendix B]:
𝑃0 = 𝑀𝑥𝑀𝑦 ∑ ∑ (1 −|𝑝|
𝑀𝑥) (1 −
|𝑞|
𝑀𝑦)
𝑀𝑦−1
𝑞=−(𝑀𝑦−1)
𝑀𝑥−1
𝑝=−(𝑀𝑥−1)
∙
𝑠𝑖𝑛 (𝑘√𝑑𝑥2𝑝2 + 𝑑𝑦
2𝑞2)
𝑘√𝑑𝑥2𝑝2 + 𝑑𝑦
2𝑞2
. (𝑐𝑜𝑠(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃))
(23)
In (23), assuming a large array (𝑀𝑥 → ∞, 𝑀𝑦 → ∞) and considering
the “oscillating” nature of the 𝑐𝑜𝑠(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃), it is assumed that
(1 −|𝑝|
𝑀𝑥) (1 −
|𝑞|
𝑀𝑦) ≈ 1. The use of Poisson’s formula and the
properties of modified Bessel function [21], a simplified expression for
Po is obtained as:
𝑃0 =𝑀𝑥𝑀𝑦𝜋
𝑘𝑑𝑥
∑ ∑ 𝐼0 (𝑘𝑑𝑦𝑚√1 − (𝑐𝑜𝑠𝜃0 −𝑙𝜆
𝑑𝑥))𝑙
∞𝑚=−∞ (24)
Restricted to those values of 𝑙 for which 1 − (𝑐𝑜𝑠𝜃0 −𝑙𝜆
𝑑𝑥) ≥ 0.
In (24), Io denotes modified Bessel’s function of first kind of order
zero [21-22]. The expression of Po may be re-written as:
𝑃0 =𝑀𝑥𝑀𝑦𝜆
2𝜋𝑑𝑥𝑑𝑦
∑ [1 − (𝜆
𝑑𝑥𝑝 − 𝑠𝑖𝑛𝜃0)
2− (
𝜆
𝑑𝑦𝑞)
2
]
−1/2
𝑝,𝑞 (25)
for those values of p and q which satisfies the condition, (20).
Considering ‘dense’ arrays (𝑑𝑥/𝜆 and 𝑑𝑦/𝜆 < 1), (19) becomes:
𝐷𝑛𝑜𝑟𝑚 =2𝜋𝑑𝑥𝑑𝑦
𝜆𝑐𝑜𝑠𝜃0 (26)
which is valid at any angle except for those satisfying the condition
𝑐𝑜𝑠𝜃0 <3
4√2
𝜆
𝐿. (27)
(L denotes length of array). Hence, a conclusion may be drawn from
the above study that in case the distance between the elements in the
equidistant array is small, the directivity varies according the cosine
law, except in the sector where the main beam is near the aperture
plane following (27).
IV. RESULTS AND DISCUSSION
This section presents analyses carried out for PAAs utilizing the
proposed expressions with aid of few case studies.
A. Array Characterization Using Exact Expression (14)
This section presents the analysis of PAAs using exact expression
(14), carried out for linear and planar array configurations. In planar
arrays, square as well as rectangular array geometries are considered.
All the figures presented now onwards consider 𝐷𝑛𝑜𝑟𝑚 (with respect
to Dsp). Here onwards we assume only isotropic REs radiating only in
upper hemisphere.
Fig. 2 shows the variation of 𝐷𝑛𝑜𝑟𝑚 (in dB) vs. scan angle, o, for
different values of d, for a 100-elements linear array along 𝜑𝑐 =0𝑜 plane. It is observed that for d=0.25 (/4) spacing, Dnorm is
maximum in the end-fire direction and is nearly constant in the range
0o-70o. As d increases, Dnorm increases and few dips are observed in the
curves (d=0.7 and 1.2). Further, when d (in ) =0.5, 2 and 3, Dnorm
vs. scan angle is constant.
Fig. 2. Dnorm (in dB) vs. scan angle (in deg) for a linear array.
Fig. 3 shows the variation in 𝐷𝑛𝑜𝑟𝑚 (in dB) vs. o for a 2500-
(50x50) elements square planar array for 𝑑 (𝑖𝑛 𝜆) =0.5, 0.7, 1.2, 2 𝑎𝑛𝑑 3 along 𝜑𝑐=0o
plane. When, 𝑑 = 0.25, 𝐷𝑛𝑜𝑟𝑚
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decreases with the increase in o, up to an angle, g (~60o), after which
it again rises. When, 𝑑 = 0.5, 𝐷𝑛𝑜𝑟𝑚 follows 𝑐𝑜𝑠𝜃 pattern and falls
down to half of its value at some angle, say g (~60o-65o). In the case
of 𝑑 > 1, 𝐷𝑛𝑜𝑟𝑚 remains almost constant with respect to o, except
for few dips in the pattern. These dips in the curves shows the onset of
GLs at specific values of o, representing blind spots due to the
mismatch in array input impedance. However, as d increases further
the depth of dips decreases showing negligible variation in Dnorm,
demonstrating wide scan performance (>80o) for highly sparse PAAs.
Fig. 3. Dnorm (in dB) vs. scan angle for a square planar array.
Fig. 4 shows the variation of Dnorm (in dB) vs. o for a 250- (5 x 50)
elements rectangular PAA. A similar behavior is shown as before
except for less shallow dips (slowly falling edges) compared to the
square array. The analysis is carried out along three -cuts, 0o, 45o and
90o. It is observed that d<1 (dense arrays) the performance is not same
along different -cuts (see curves for d=0.5 and 0.7). However, in
the case of SPAAs (d >1), variation in Dnorm is minimal with respect
to o across the -cuts. This clearly establishes the wide scan behavior
of SPAAs over the hemispherical volume of scan.
(a)
(b)
(c)
Fig. 4. Dnorm (in dB) vs. scan angle for a rectangular planar array.
Fig. 5 shows the variation of D (in dB) vs. d (in steps of 0.05) for
different array sizes at boresight. It shows that 𝐷𝑛𝑜𝑟𝑚increases as N
increases. A strong variation in Dnorm is observed for 𝑑 < 1, but as 𝑑
becomes larger (>1) the value stabilizes and becomes almost
constant. Thus, it establishes the fact that in SPAAs there is least
variation in Dnorm over a wide scan volume.
Table 1 presents comparison of D for a N-elements linear array
[taken from reference [23]] with computed values from (14) showing
a close agreement of results. This validates the accuracy of (14).
Fig. 5. Variation of D (in dB) vs. d for different array sizes.
TABLE 1:
ANALYSIS OF N-ELEMENTS LINEAR ARRAY (D: DIRECTIVITY AT BORESIGHT)
Case N Location
Amplitudes D [23] (dB)
D (dB)
1 14
0.26 0.82 1.36 2.05 2.87 3.72
4.46
0.90 0.78 0.96 0.97 1.00 0.91
0.75
11.6 11.9
2 18
0.28 0.82 1.42 2.04 2.69 3.47
4.32 5.16
5.90
0.91 0.88 0.86 1.00 0.95 0.85
0.84 0.74
0.69
12.7 13.69
3 22
0.24 0.74 1.24 1.75 2.25 2.81
3.36 4.04 4.86 5.70 6.44
0.78 0.95 1.00 0.89 0.83 0.95
0.92 0.93 0.81 0.76 0.73
13.58 14.04
4 26
0.28 0.81 1.37 1.92 2.49 3.08
3.66 4.33 4.98 5.66 6.43 7.25
7.99
0.97 1.00 0.91 0.90 0.90 0.85
0.93 0.90 0.78 0.70 0.73 0.75
0.68
14.30 14.9
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B. Array Characterization Using Approximate Expression (19)
An isophoric 900- (30 x 30) elements PAA having separations, 𝑑𝑥/𝜆
and 𝑑𝑦/𝜆, is considered. The value of Dnorm is computed considering
multiple modes, p and q (satisfying (20)). The variation of Dnorm vs. o,
for different d is shown in Fig. 6, which shows the effect of GLs on the
scan performance of PAAs. The analysis assumes a square array
(d=𝑑𝑥= 𝑑𝑦), although it do not limit the applicability of the proposed
expressions for any arbitrary array arrangement.
Fig. 6 shows that for a dense array (d/=0.5), shown by blue curve,
the variation in Dnorm follows 𝑐𝑜𝑠𝜃𝑜 pattern, and there is a GL onset at
𝜃𝑜 = 90𝑜. Here, two degenerate modes exists at (𝑝, 𝑞) = (0,0) and (1,
0) due to square geometry. There is a 2dB decrease in Dnorm from 1.6
to 0. The expression, (19) simplifies to 𝐷𝑛𝑜𝑟𝑚 = (2𝜋𝑑𝑥𝑑𝑦)𝑐𝑜𝑠𝜃0 𝜆2⁄
for dense array. The condition for validity of the expression of Dnorm
in dense array is 𝑐𝑜𝑠𝜃0 > √𝜆/𝑀𝑥𝑑𝑥 (scanning in x – z plane).
Table 2 lists the various possible 𝑝𝑞 modes’ combinations for
different values of d. It also lists angles of onset for GLs corresponding
these modes. A similar study can be carried out for any other arbitrarily
shaped PAA using these expressions. It is seen that the behavior
observed for variation in Dnorm is similar to that in sub-section A. The
only difference from previous section is that deeper dips (blind spots)
as observed, due to infinite nature of array extent assumed in (19).
Fig. 6. Dnorm curves vs. scan angles for different d.
The curves for Dnorm of SPAAs (d/> 1) highlights the major fact
that the cos law is not at all valid here. SPAAs have distinct
advantageous feature of stabilized Dnorm over wide angular scan sector
due to the decrease in mutual coupling. The only drawback observed
in equidistant SPAAs is a sharp decrease in the directivity at some
specific angles due to several ‘interference’ beams, as is obvious due
to periodicity in array grid. In general, in the case of periodic SPAAs,
due to the number of grating lobes, the energy in the sidelobes get re-
distributed and confined in the regions other than the dips. This leads
to the higher values of directivity as compared to the denser
counterpart at angles other than boresight.
Along with increase in d, correspondingly number of GLs increases,
with respect to various 𝑝𝑞 modes (see Table 2 and Fig.6). It is observed
that for any value of d, the square root in (19) becomes zero at 𝜃𝑜 =90𝑜 resulting in a dip in the computed directivity curves. Further it is
observed that for 𝑑𝑥 > 1 and o< 60o more than 90% of the normalized
directivity is larger than 1, showing non-existence of mutual coupling
in the array. Thus, ultra-wide angle scanning (>60o) is possible by
utilizing large inter–element separations of sparse arrays up to
0~ 85𝑜.
Fig. 7 shows a saw tooth curve illustrating the behavior of, Dnorm vs.
d by varying d/ from 0.5 to 8 in a square planar array at
boresight 𝜃𝑜 = 0𝑜. The value of Dnorm is observed to be zero whenever
𝑑 = 𝑛 (n=1, 2, 3….) showing that the GL at 90o exactly nullifies the
main beam leading to 𝐷𝑛𝑜𝑟𝑚 → 0. For values of 𝑑 = 𝑛/2, Dnorm is
observed almost equal for all values of 𝑛. In the intermediate range
of 𝑑, i.e,𝑛
2< 𝑑 < 𝑛, the value of Dnorm is observed to shoot up but at
the cost of deep nulls in the curves (shown in Fig.6). Thus, the saw-
tooth curve provides a design tool for choosing inter-element
separation, dn, in order to avoid these sharp dips at boresight.
TABLE 2.
VARIOUS POSSIBLE 𝑝𝑞 COMBINATIONS AND CORRESPONDING GL LOCATIONS
FOR DIFFERENT D FOR A SQUARE LATTICE (dx=dy=d).
d (in ) (p, q) o (deg)
0.5 (0,0) and (1,0) 90
1.0 (-1,0), (0,1) and (1,0)
(1,1), (0,0) and (2,0)
0
90
1.2
(-1,0)
(1,1)
(0,1)
(2,0)
(0,0)
9.6
16.3
33.5
41.8
90
1.5
(-1,1)
(-1,0) and (2,0)
(0,1)
(0,0)
(3,0)
4.5
19.5
36
48
90
2.0
(-2,0), (0,2) and (2,0)
(1,2), (-1,0), (1,2) and (3,0)
(0,1)
(2,2)
(0,0), (2,2) and (4,0)
0
30
60
80
90
2.5
(-2,1)
(-1,2)
(3,1)
(-1,1)
(-1,0)
(0,2)
(3,2)
(4,0)
(4,1)
(0,1)
(1,2), (4,2), (0,0) and (5,0)
6.7
11.5
16.5
31
37
37
37
37
43
66
90
3.0
(-3,0), (0,3) and (3,0)
(3,1)
(-2,2)
(-2,1)
(1,3), (4,0) and (-2,0)
(4,1)
(-1,2)
(4,2)
(5,0), (2,3) and (-1,0)
(5,1)
(0,2)
(5,2)
(0,1)
(3,3) and (6,0)
0
3.3
4.5
-15.9
19.5
23
24.3
36.1
41.8
46.5
48.15
67.2
70.5
90
The condition (20) implies the formation of unwanted GLs due to
in-phase field summations, which leads to the “absorption” of a large
percentage of the radiated power (may be 100%). This absorbed power
may lead to the condition, 𝐷𝑛𝑜𝑟𝑚 →0, for infinite PAAs
(𝑀𝑥𝑑𝑥, 𝑀𝑦𝑑𝑦 → ∞), see Fig. 6. A method to reduce these effects is
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usage of specific inter-element separations as will be shown in Fig. 8
and/or choice of aperiodic planar SPAAs [2], which is beyond the
scope of this paper.
Fig. 7. Dnorm with respect to the inter-element separation.
(a)
(b)
(c)
Fig. 8. Dnorm vs. scan angle curves for rectangular arrays.
Fig. 8 illustrates the variation of the Dnorm vs. o for different
combinations of dx and dy in a rectangular array, along different -cuts.
The results are compared with the behavior of a dense array (dx=0.5,
dx=dy). Fig. 8a shows the variation of Dnorm vs. o for 𝑑𝑥 = 0.5𝜆 and
𝑑𝑥 = 1.2𝜆 along 𝜑𝑐 =0o, 45o and 90o planes, computed by modifying
the bracketed term in denominator of (19) as
1 − (𝜆
𝑑𝑥𝑝 − 𝑠𝑖𝑛𝜃0𝑐𝑜𝑠𝜑𝑜)
2− (
𝜆
𝑑𝑥𝑞 − 𝑠𝑖𝑛𝜃0𝑠𝑖𝑛𝜑𝑜)
2 (28)
Thus, Fig.8 demonstrates the applicability of approximate expression
along any -planes. Fig. 8 shows that due to rectangular shape of array,
different scan behavior is observed along 𝜑𝑐 =0o and 90o cuts. Fig. 8b
shows the results corresponding to dx=0.5 and dy=2. It shows
appreciable difference in scan performance of array along different 𝜑𝑐
cuts. It is seen in Fig. 8c, that for particular combination of dx=1.2
and dy=2, a highly sparse array, a wide scan volume for SPAAs is
observed with least variation in Dnorm vs. o. It also shows a wide scan
performance along three -cuts. Thus, an important design
consideration is demonstrated.
A 279-elements concentric ring array, is presented in [24] with
computed directivity of 29.35dB. In [23] Das et.al computes the value
as 29.36dB in 23.72s. The values computed using (14) is 29.357dB,
which is in close agreement to the original estimate. The approximate
expression (19) estimates the value as 29.427dB in 14.6s considering
five GL modes (p=-5:5 and q=-5:5). The results are in agreement and
faster to obtain. Thus, the proposed approximate expression for
directivity provides a quick aid to design a wide scan dense as well as
sparse PAAs.
V. CONCLUSIONS
This paper presents an exact analytical expression for computation
of directivity of phased array antennas. It includes most of the
important design considerations in PAAs like arbitrary array geometry
& element type, arbitrary complex excitations, mutual coupling, scan
angle and embedded element pattern. Further, it presents an
approximate expression of directivity for quick analysis of infinite
PAAs assuming isotropic radiator. Although, the expressions don’t
consider array geometry but it serves as a preliminary design aid for
large arrays. The expressions are validated with available literature.
This helps to check the possible onset of GLs due to chosen inter-
element separations. With the aid of various demonstrations, the wide
scan characteristics of SPAAs, especially highly sparse antenna arrays
is established and shown in the paper.
APPENDIX A
DERIVATION OF (14)
In=Ion (the maximum value of In at (𝜃0, 𝜑0)= (0, 0)) is computed as
|∬𝑠𝐼0(𝜉, 𝜂)𝑑𝜉𝑑𝜂|
2= ∑ 𝐼𝑜𝑚
2𝑁𝑚=1 (A-1)
Referring to the denominator of (10)
∬ 𝜒0(𝜉, 𝜂) cos[𝑘𝑠𝑖𝑛𝜃0(𝜉𝑐𝑜𝑠𝜑0 + 𝜂𝑠𝑖𝑛𝜑0)] 𝑠𝑖𝑛𝑘√𝜉2+𝜂2
𝑘√𝜉2+𝜂2 𝑑𝜉𝑑𝜂 (A-2)
From (10), 𝜒0(𝜉, 𝜂) = ∬𝑠𝐼0(𝑥, 𝑦) 𝐼0∗(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝑥𝑑𝑦 (A-3)
Now for a planar phased array antenna
𝐼𝑜(𝑥, 𝑦) = ∑ 𝐼𝑚 𝑁𝑚=1 𝛿(𝑥 − 𝑥𝑚) 𝛿(𝑦 − 𝑦𝑚) (A-4)
𝐼𝑜(𝑥 − 𝜉, 𝑦 − 𝜂) = ∑ 𝐼𝑛𝑁𝑛=1 𝛿(𝑥 − 𝜉 − 𝜉𝑛) 𝛿(𝑦 − 𝜂 − 𝜂𝑛) (A-5)
𝜒0 = ∑ {𝐼𝑚(𝑥 = 𝑥𝑚, 𝑦 = 𝑦𝑚) ∑ 𝐼𝑛∗(𝜉 = 𝑥𝑚 − 𝜉𝑛, 𝜂 =𝑁
𝑛=1𝑁𝑚=1
𝑦𝑚−𝜂𝑛)} = ∑ {𝐼0𝑚exp [−𝑖𝑘𝑠𝑖𝑛𝜃0(𝑥𝑚𝑐𝑜𝑠𝜑0 + 𝑦𝑚𝑠𝑖𝑛𝜑0)] ∙𝑁𝑚=1
∑ 𝐼𝑜𝑛∗ exp[𝑖𝑘𝑠𝑖𝑛𝜃0((𝑥𝑚 − 𝜉𝑛)𝑐𝑜𝑠𝜑0 + (𝑦𝑚−𝜂𝑛)𝑠𝑖𝑛𝜑0)]𝑁
𝑛=1 }
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= ∑ 𝐼0𝑚𝑁𝑚,𝑛=1 𝐼𝑜𝑛
∗ exp[−𝑖𝑘𝑠𝑖𝑛𝜃0(𝜉𝑛𝑐𝑜𝑠𝜑0 + 𝜂𝑛𝑠𝑖𝑛𝜑0)] =
𝜒0(𝜉𝑛, 𝜂𝑛) ∙ cos [𝑘𝑠𝑖𝑛𝜃0(𝜉𝑛𝑐𝑜𝑠𝜑0 + 𝜂𝑛𝑠𝑖𝑛𝜑0)] (A-6)
Denominator of (10) becomes
∑ 𝐼𝑜𝑚𝐼𝑜𝑛∗
𝑁
𝑚,𝑛=1
cos[𝑘𝑠𝑖𝑛𝜃0{(𝑥𝑚 − 𝜉𝑛)𝑐𝑜𝑠𝜑0 + (𝑦𝑚−𝜂𝑛)𝑠𝑖𝑛𝜑0}]
∙𝑠𝑖𝑛𝑘√(𝑥𝑚 − 𝜉𝑛)2 + (𝑦𝑚−𝜂𝑛)2
𝑘√(𝑥𝑚 − 𝜉𝑛)2 + (𝑦𝑚−𝜂𝑛)2
Putting, 𝜉𝑛 = 𝑥𝑛; 𝜂𝑛 = 𝑦𝑛;
Denominator=
∑ 𝐼𝑜𝑚𝐼𝑜𝑛∗𝑁
𝑚,𝑛=1 . cos[𝑘𝑟𝑛𝑚. 𝑠𝑖𝑛𝜃0{𝑐𝑜𝑠𝛼𝑛𝑚𝑐𝑜𝑠𝜑0 +
𝑠𝑖𝑛𝛼𝑛𝑚𝑠𝑖𝑛𝜑0}] ∙𝑠𝑖𝑛𝑘𝑟𝑛𝑚
𝑘𝑟𝑛𝑚 (A-7)
𝑟𝑛𝑚 = √(𝑥𝑛 − 𝑥𝑚)2 + (𝑦𝑛 − 𝑦𝑚)2;
𝑐𝑜𝑠𝛼𝑛𝑚 =𝑥𝑛−𝑥𝑚
𝑟𝑛𝑚 ; 𝑠𝑖𝑛𝛼𝑛𝑚 =
𝑦𝑛−𝑦𝑚
𝑟𝑛𝑚
Hence, we get the expression
𝐷 =(∑ 𝐼0𝑚
𝑁𝑚=1 )
2
∑ 𝐼0𝑚𝐼0𝑛∗ cos [k𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0−𝛼𝑛𝑚)]
𝑠𝑖𝑛𝑘𝑟𝑛𝑚𝑘𝑟𝑛𝑚
𝑁𝑚,𝑛=1
(A-8)
APPENDIX B
Referring to (10), it is possible to rewrite it as:
𝐷 =(∑ 𝐼0𝑚
𝑁𝑚=1 )
2
∑ 𝐼0𝑚𝐼0𝑛∗ cos [k𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0−𝛼𝑛𝑚)]
𝑠𝑖𝑛𝑘𝑟𝑛𝑚𝑘𝑟𝑛𝑚
𝑁𝑚,𝑛=1
(B-1)
𝐼𝑛 = 𝐼𝑜𝑛 exp(−𝑖𝑘𝑥𝑛𝑠𝑖𝑛𝜃0) (∵ 𝜑0 = 0) (B-2)
𝐼(𝑥, 𝑦) = ∑ 𝐼𝑛𝑁𝑛=1 𝛿(𝑥 − 𝑥𝑛) 𝛿(𝑦 − 𝑦𝑛) (B-3)
∑ 𝐼𝑜𝑚2𝑁
𝑚=1 = 𝑁2(= 𝑀𝑥2𝑀𝑦
2) (B-4)
(B-1) may be re-written as
𝐷 =𝑁2(=𝑀𝑥
2𝑀𝑦2)
∑ cos [𝑘𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠𝛼𝑛𝑚]∙𝑁𝑚,𝑛=1
𝑠𝑖𝑛𝑘𝑟𝑛𝑚𝑘𝑟𝑛𝑚
(B-5)
𝑟𝑛𝑚 = 𝑟𝑚1,𝑚2,𝑛1,𝑛2= √dx
2(𝑚1 − 𝑚2)2 + dy2(𝑛1 − 𝑛2)2 (B-6)
It is known that – 𝑀𝑥 + 1 < 𝑚1 − 𝑚2 < 𝑀𝑥 − 1 and – 𝑀𝑦 + 1 <
𝑛1 − 𝑛2 < 𝑀𝑦 − 1. Introducing new variables 𝑚 = 𝑚1 −
𝑚2 & 𝑛 = 𝑛1 − 𝑛2 , the number of equal 𝑟𝑛𝑚 terms for given 𝑚 & 𝑛
are 𝑀𝑥 ∙ (1 −|𝑚|
𝑀𝑥) and 𝑀𝑦 ∙ (1 −
|𝑛|
𝑀𝑦), respectively.
Hence, denominator of (B-5) becomes
𝑀𝑥 𝑀𝑦 ∑ ∑ (1 −|𝑚|
𝑀𝑥)(1 −
𝑀𝑦−1
𝑛=−(𝑀𝑦−1)𝑀𝑥−1𝑚=−(𝑀𝑥−1)
|𝑛|
𝑀𝑦)
sin (𝑘√𝑑𝑥2𝑚2+𝑑𝑦
2𝑛2)
𝑘√𝑑𝑥2𝑚2+𝑑𝑦
2𝑛2cos (𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃0) (B-7)
For large arrays & considering “oscillating” character of
cos(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃0), it may be assumed that
(1 −|𝑚|
𝑀𝑥) (1 −
|𝑛|
𝑀𝑦) ≈ 1 (B-8)
, denominator in (B-5) becomes
∑ {∑sin(𝑘√𝑑𝑥
2𝑚2+𝑑𝑦2𝑛2)
𝑘√𝑑𝑥2𝑚2+𝑑𝑦
2𝑛2
𝑀𝑦−1
𝑛=−(𝑀𝑦−1)}
𝑀𝑥−1𝑚=−(𝑀𝑥−1) cos(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃0) (B-9)
Let 𝑚𝑑𝑥 = 𝑥 𝑎𝑛𝑑 𝑛𝑑𝑦 = 𝑦 ,⇒ 𝑑𝑥 =𝑚∙𝑑𝑥
𝑀𝑥⇒ Δ𝑚 = 1 =
𝑀𝑥
𝑑𝑥∙ 𝑑𝑥 (B-10a)
Similarly, Δ𝑛 = 1 =𝑀𝑦
𝑑𝑦∙ 𝑑𝑦
⇒ ∑ →𝑀𝑦−1
𝑛=−(𝑀𝑦−1)
𝑀𝑦
𝑑𝑦∙ ∫
sin(𝑘√𝑥2+𝑦2)
𝑘√𝑥2+𝑦2)
∞
−∞ 𝑑𝑦 (−∞ ≤ 𝑦 ≤ ∞) (B-10b)
Thus, the denominator of (B-5) may be re-written as
∑ {𝑀𝑦
𝑑𝑦∙ ∫
sin(𝑘√𝑥2+𝑦2)
𝑘√𝑥2+𝑦2)
∞
−∞}
𝑀𝑥−1𝑚=−(𝑀𝑥−1) cos(𝑘𝑥𝑠𝑖𝑛𝜃0) 𝑑𝑦 =
𝑀𝑥𝑀𝑦
𝑑𝑥𝑑𝑦∫ ∫
sin(𝑘√𝑥2+𝑦2)
𝑘√𝑥2+𝑦2)𝑑𝑦 cos(𝑘𝑥 𝑠𝑖𝑛𝜃0) 𝑑𝑥 (−∞ ≤ 𝑥 ≤
∞
𝑦=−∞
∞
𝑥=−∞
∞) (B-11)
Alternately, the expression in (B-11) may be approximated as given
below (making use of Poisson’s formula & properties of modified
Bessel functions [21]), for those values of l & p integers where √∙= 0.
≈ 𝜆𝑀𝑥𝑀𝑦
𝑘 𝑑𝑥𝑑𝑦
∑ [1 − (𝜆
𝑑𝑥𝑙 − 𝑠𝑖𝑛𝜃0)2 − (
𝜆
𝑑𝑦𝑝)
2
]
−1
2
𝑙,𝑝 (B-12)
The qualifying (𝑙, 𝑝) set gives effect on maximum Dnorm as function of
o.
𝑙 = 0, 𝑝 = 0 ⇒ 𝐷𝑛𝑜𝑟𝑚 =𝐷
𝑀𝑥𝑀𝑦=
2𝜋𝑑𝑥𝑑𝑦
𝜆2 1/𝑐𝑜𝑠𝜃0=
2𝜋𝑑𝑥𝑑𝑦
𝜆2 𝑐𝑜𝑠𝜃0 (B-13)
(B-13) is 𝜃0 dependent and it is valid for dense array, depicting
angular dependence of Dnorm . The condition for Dnorm in dense array
is:
𝑐𝑜𝑠𝜃0 > √𝜆
𝑀𝑥𝑑𝑥 (B-14)
where Mxdx is the length of the array (for scanning in xz plane).
w.r.t. (B-11), Integral 1 in (B-11) becomes:
𝐼𝑛𝑡1 = ∫sin(𝑘√𝑥2+𝑦2)
𝑘√𝑥2+𝑦2𝑑𝑦 =
∞
𝑦=0 ∫sin(𝑘√𝑥2+𝑦2)
𝑘√𝑥2+𝑦2𝑑𝑦
∞
0 (B-15)
In order to solve the integral let
𝑧 = 𝑘√𝑥2 + 𝑦2 ⇒ 𝑑𝑧 =𝑘𝑦
√𝑥2 + 𝑦2𝑑𝑦
𝑘2(𝑥2 + 𝑦2) = 𝑧2 ⇒ 𝑦 = √𝑧2
𝑘2 − 𝑥2 𝑦 > 0; 𝑧 > 𝑘𝑥
𝑑𝑦
𝑘√𝑥2 + 𝑦2=
1
𝑘2𝑦𝑑𝑧 =
1
𝑘√𝑧2 − 𝑘2𝑥2𝑑𝑧
∴ 𝐼𝑛𝑡1 = ∫sin(𝑧)
𝑘√𝑧2−𝑘2𝑥2𝑑𝑧
∞
𝑧=𝑘𝑥 (B-16)
Introducing 𝑡 =𝑧
𝑘𝑥 ⇒ 𝑑𝑡 ∙ 𝑘𝑥 = 𝑑𝑧
∴ 𝐼𝑛𝑡1 = ∫sin 𝑘𝑥𝑡
𝑘√𝑡2 − 1𝑑𝑡
∞
𝑡=1
=2
√𝜋 Γ(1/2)∙
1
𝑘𝐽0(𝑘𝑥) (𝐸𝑞. 9.1.24 [21])
Since Γ (1
2) = √𝜋 ⇒ 𝐼𝑛𝑡1 =
2
π∙
1
𝑘𝐽0(𝑘𝑥) (B-17)
Now
𝐼𝑛𝑡2 = ∫2
π∙
1
𝑘𝐽0(𝑘𝑥)
∞
𝑥=0
cos(𝑘𝑥 𝑠𝑖𝑛𝜃0) 𝑑𝑥 (𝐸𝑞. 11.4.37 [21])
=2
𝜋𝑘
1
√𝑘2 − 𝑘2𝑠𝑖𝑛𝜃0
=2
𝜋𝑘2𝑐𝑜𝑠𝜃0
(B-18)
∴ 𝐷𝑛𝑜𝑟𝑚 =𝑀𝑥
2𝑀𝑦2
𝑀𝑥𝑀𝑦
𝑑𝑥𝑑𝑦∙
2
𝜋𝑘2𝑐𝑜𝑠𝜃0
= 𝑀𝑥 𝑑𝑥𝑀𝑦𝑑𝑦 2𝜋
𝜆2 𝑐𝑜𝑠𝜃0 (B-19)
Comparing expressions (B-19) and (B-13), the dependence on the
number of elements is an extra factor as being shown. Further, we may
write in case that no GL are visible (𝑑𝑥 <𝜆
2𝑎𝑛𝑑 𝑑𝑦 < 𝜆),
𝐼𝑛𝑡2 =2
𝜋𝑘2√1−𝑠𝑖𝑛2𝜃0
(B-20)
On the contrary in the case of visible GLs, for the case of non-dense
array, where 𝑑𝑥 >𝜆
2, we may use the under-mentioned treatment to
assess the effect on the directivity. The array factor in xz-plane [1] may
be written in the alternate form as:
sin(
𝑁
2𝜓)
sin(𝜓
2)
𝑤ℎ𝑒𝑟𝑒 𝜓 = 𝑘𝑑𝑥𝑠𝑖𝑛𝜃 + 𝛽; (B-21)
In (B-21), 𝛽 = 𝑘𝑑𝑥𝑠𝑖𝑛𝜃0 is the phase difference between the
successive elements, and 𝜃0 is the beam-pointing or scan direction, 𝜃0.
Now the GL appear when 𝑘𝑑𝑥
2(𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛𝜃0) =
0, ±𝜋, ±2𝜋, … … , 𝑚𝜋
∴ (𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛𝜃0) = 0, ±𝜆
𝑑𝑥, ±
2𝜆
𝑑𝑥, … … , ±
𝑚𝜆
𝑑𝑥
0018-926X (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2018.2880006, IEEETransactions on Antennas and Propagation
The value above expression in brackets is equal to zero, iff 𝑑𝑥 <𝜆
2,
representing main lobe (desired peak direction). The undesired
“grating lobes” occur for
(𝑠𝑖𝑛𝜃 − (𝑠𝑖𝑛𝜃0 ±𝑚𝜆
𝑑𝑥)) = 0 (B-22)
Using (B-2, B-22), (B-20) becomes
𝐼𝑛𝑡2 =2
𝜋𝑘2 ∑ √1−(𝑚𝜆
𝑑𝑥−𝑠𝑖𝑛𝜃0)2
𝑚
𝑤ℎ𝑒𝑛 𝑑𝑦 < 𝜆 (B-23)
In a similar manner is derived, dependency for 𝑑𝑦 > 𝜆,
considering GLs for 𝑠𝑖𝑛𝜃0 ±𝑚𝜆
𝑑𝑥 in xz plane and ±
𝑛𝜆
𝑑𝑦 in yz plane (no
o dependence, scanning is only in xz plane)
Now, 𝐼𝑛𝑡2 =2
𝜋𝑘2 ∑ √1−(𝑚𝜆
𝑑𝑥−𝑠𝑖𝑛𝜃0)2−(
𝑛𝜆
𝑑𝑦)2
𝑚,𝑛
(B-24)
i.e., summation over all (m, n) values for which
1 − (𝑚𝜆
𝑑𝑥− 𝑠𝑖𝑛𝜃0)2 − (
𝑛𝜆
𝑑𝑦)2 ≥ 0
Hence the expression of normalized directivity becomes
∴ 𝐷𝑛𝑜𝑟𝑚 =𝐷
𝑀𝑥𝑀𝑦
=1
𝑀𝑥𝑀𝑦∙
(𝑀𝑥𝑀𝑦)2
𝑀𝑥𝑀𝑦
𝑑𝑥𝑑𝑦
2
𝜋𝑘2 [1 − (𝜆
𝑑𝑥𝑚 − 𝑠𝑖𝑛𝜃0)
2
− (𝜆
𝑑𝑦𝑛)
2
]
1/2
=2𝜋𝑑𝑥𝑑𝑦
𝜆2 ∑ [1 − (𝜆
𝑑𝑥𝑚 − 𝑠𝑖𝑛𝜃0)
2
− (𝜆
𝑑𝑦𝑛)
2
]
−1/2
𝑚,𝑛
(B-25)
ACKNOWLEDGMENT
The authors would like to thank the reviewers for their comments and
suggestions.
REFERENCES
[1]. C.A. Balanis (ed.), Modern Antenna Handbook, John Wiley & Sons, 2008.
[2]. Special Issue, “Innovative Phased Array Antennas Based on Non-Regular Lattices and Overlapped Subarrays: Part 1,” IEEE Trans. Antennas
Propagat., vol. 62, no. 4, April 2014.
[3]. C. T. Tai, “The Optimum Directivity of Uniformly Spaced Broadside Arrays of Dipoles,” IEEE Trans. Antennas Propagat., vol. 1, pp. 447-454,
1964.
[4]. H. Bach, “Directivity of basic linear arrays,” IEEE Trans. Antennas Propagat., vol. 1, pp. 107-110, 1970.
[5]. L. J. Langston, “Scanned directivity of linear arrays,” IEEE Trans.
Antennas Propagat., vol. 1, pp. 282-284, 1971. [6]. R. C. Hansen, “Comparison of square array directivity formulas,” IEEE
Trans. Antennas Propagat., vol. AP-20, pp. 100–102, Jan. 1972.
[7]. J. P. Daniel, “Directivity of linear microstrip arrays,” Electronic Lett., vol. 23, pp. 897-899, 1987.
[8]. A. S. Jazzi, “Directivity of Chebyshev arrays with arbitrary element
spacing,” Electronic Lett., vol. 31, pp. 772-774, 1995. [9]. R. C. Hansen, “Dolph-Chebyshev array directivity versus spacing,”
Electronic Lett., vol. 32, pp. 1050-1051, 1996.
[10]. D. H. Werner, D. Mulyantini and P. L. Werner, “Closed form representation for directivity of non-uniformly spaced linear arrays with
arbitrary element patterns,” Electronic Lett., vol. 35, pp. 2155-2157, 1999.
[11]. J. D. Mahony, “An Approximate Expression for the Directivity of a Tapered-Cosine Distribution on a Circular Aperture,” IEEE Trans.
Antennas Propagat., Vol. 52, No.2, 2010.
[12]. M. J. Lee, I. Song, S. Yoon, and S. R. Park, “Evaluation of directivity of planar antenna arrays,” IEEE Antennas Propagat. Mag., vol. 42, no.3, pp.
64–67, Jun. 2000.
[13]. A. H. Nuttall and B. A. Cray, "Approximations to directivity for linear,
planar, and volumetric apertures and arrays," IEEE Journal of Oceanic Engineering, vol. 26, no. 3, pp. 383-398, Jul 2001.
[14]. B. J. Forman, “Directivity characteristics of scannable planar arrays,”
IEEE Trans. Antennas Propagat., vol. 1, pp. 245-252, 1972. [15]. L. P. Ligthart, “A new numerical approach to planar phased arrays,” IEEE
AP-S, pp. 453-457, 1975.
[16]. S. Das, D. Mandal, R. Kar and S. P. Ghoshal, "A Generalized Closed Form Expression of Directivity of Arbitrary Planar Antenna Arrays,"
IEEE Trans. Antennas Propagat., vol. 61, no. 7, pp. 3909-3911, July
2013. [17]. R. J. Mailloux, Phased Array Antenna Handbook, Artech House, Second
ed., 2005.
[18]. W. K. Kahn, "Element Efficiency: A Unifying Concept for Array Antennas," IEEE Antennas Propagat. Mag., vol. 49, no. 4, pp. 48-56,
Aug. 2007.
[19]. P. S. Kildal, A. Vosoogh and S. Maci, "Fundamental Directivity Limitations of Dense Array Antennas: A Numerical Study Using
Hannan’s Embedded Element Efficiency," IEEE Antennas Wireless
Propagat. Lett., vol. 15, pp. 766-769, 2016. [20]. D. M. Pozar, “A relation between the active input impedance and the
active element pattern of a phased array,” IEEE Antennas Propagat.
Mag., vol. 51, no. 9, pp. 2486-2489, 2003. [21]. M. Abramowitz and I. A. Stegen, Handbook of mathematical functions
with formula, graphs and mathematical tables, NBS, Washington DC,
1972. [22]. G.N. Watson, A treatise on the theory of Bessel functions / Theory of
Bessel functions, 2nd ed., Cambridge: University Press, 1944. [23]. S. Das. M. Bhattacharya, A. Sen and D. Mandal, “Linear Antenna Array
synthesis with Decreasing Sidelobe and Narrow Beamwidth,” ACEEE Int.
J. on Communications, vol. 3, no. 1, pp. 10-14, Mar. 2012. [24]. R. L. Haupt, "Optimized Element Spacing for Low Sidelobe Concentric
Ring Arrays," IEEE Trans. Antennas Propagat., vol. 56, no. 1, pp. 266-
268, Jan. 2008.
Ashutosh Kedar (M’10–SM’13) was born in Delhi, India on
July 9, 1973. He received his Ph.D., M.Tech. and M.Sc. in Elect., Microwave Elect. and Physics in 2003, 1998 & 1995
respectively, from Univ. of Delhi, India.
He fulfilled Teaching & Research duties at Univ. of Delhi and is currently working as a Radar scientist in LRDE,
Bangalore since 2003. He gives various lectures on radar,
antennas and phased arrays and has published approximately 60 papers. He is reviewer of many international journals like IET, PIERS, ACES, IEEE, etc. He
is listed in Asia’s who’s who.
He received technology group award for 1st indigenous active phased array radar development in country. He received best paper award in ICMARS and
ATMS’18. His research interest includes CEM, phased array antennas & radars.
L.P. Ligthart born in The Netherlands, on September 15, 1946.
He received an Engineer's degree (cum laude) and Ph.D. degree
from Delft Univ. of Technology. He is Fellow of IET, IEEE, and Academician of the Russian Academy of Transport. He
received Honorary Doctorates at MSTUCA in Moscow, Tomsk
State University and MTA Romania. Since 1988, he held a chair on MW transmission, remote sensing, radar and
positioning and navigation at Delft Univ. of Technology. He supervised over
50 PhDs. He founded the IRCTR at Delft Univ. He is founding member of the
EuMA, chaired 1st EuMW in 1998 and initiated EuRAD in 2004. Currently he
is em. prof., elft University, guest professor at Universities in Indonesia and
China, Chairman of CONASENSE, Member BoG of IEEE-AESS. His areas of specialization include antennas & propagation, radar & remote
sensing, satellite, mobile & radio communications. He gives various courses on
radar, remote sensing & antennas and published over 650 papers, various book chapters and books.
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