PIERS Proceedings, Suzhou, China, September 12–16, 2011
368
Generalized Representation of Sideband Radiation Power
Calculation in Arbitrarily Distributed Time-modulated Planar
and Linear Arrays
E. Aksoy and E. Afacan
Department of Electrical & Electronics Engineering, Gazi University, Maltepe, Ankara, Turkey
Abstract— In this study, grid independent general representation of formulation of sideband
radiation (SR) power calculation in time-modulated planar and linear arrays is aimed. It is shown
that both distinct formulations can be written in one form which provides an expectation about
conformal case.
1. INTRODUCTION
Recently a useful formulation which gives the total radiated power associated with the harmonic
frequencies of a TMLA is published by Brégains et al. [1]. After this work, Poli et al. applied
this same idea to rectangular grid planar arrays [2]. Both of these formulations provide calculation
simplicity when the harmonic radiation is concerned but they are still in distinct forms.
In this study, generalization of the formulation originally given in [2] which is constructed on
the idea given in [1] to a grid independent form for linear cases is aimed while trying to keep the
original notation as much as possible.
2. FORMULATION
Consider a planar array consisting of total N elements whose elements are on the x-y plane but not
on a canonical grid. The array factor of this array may be written as
X
In gn (t) ejkxn sin θ cos φ ejkyn sin θ sin φ ,
(1)
AF (θ, φ, t) = ejω0 t
hni
where xn and yn represent apsis and ordinate values in standard Cartesian coordinate system of
corresponding element, respectively. k is the wave number and In represents the element excitations.
gn (t) is a periodic function. Since gn (t) is periodic it can be expanded to Fourier Series which is:
∞
X
gn (t) =
Gnh ejhωp t ,
(2)
h=−∞
where
Gnh
1
=
Tp
Z
Tp /2
−Tp /2
gn (t) e−jhωp t dt.
(3)
As in [2] it is also assumed that:
(
gn (t) =
1, 0 < |t| ≤ t̃2n
.
0, otherwise
According to [1] and [2] the total power radiated on harmonics is given by:
Z
Z
∞
1 2π π X
PSR =
|µh (θ, φ)|2 sin θdθdφ
2 0
0
(4)
(5)
h=−∞
h6=0
where µh (θ, φ) for planar arrays can be written as:
X
µh (θ, φ) =
In ejkxn sin θ cos φ ejkyn sin θ sin φ
hni
From relation |µh (θ, φ)|2 = µh (θ, φ) µ∗h (θ, φ), |µh (θ, φ)|2 becomes:
(6)
Progress In Electromagnetics Research Symposium Proceedings, Suzhou, China, Sept. 12–16, 2011
|µh (θ, φ)|2 =

∗
X
In ejkxn sin θ cos φ ejkyn sin θ sin φ 
In ejkxn sin θ cos φ ejkyn sin θ sin φ 
X
hni
2
|µh (θ, φ)| =
369
hni
X
2
(|In | Gnh ) +
N
−1
X
(7)
< {Im In∗ } Gnh Gmh ejk(xm −xn ) sin θ cos φ ejk(ym −yn ) sin θ sin φ
m,n=0
m6=n
hni
where < {·} represents the real part of a complex number, so that PSR becomes:
PSR =
where
F1 (θ, φ) =
1
[F1 (θ, φ) + F2 (θ, φ)]
2
∞ X
X
Z
2
(|In | Gnh )
h=−∞ hni
h6=0
2π
Z
(8)
π
sin θdθdφ
0
(9)
0
and
F2 (θ, φ) =
∞
X
N
−1
X
< {Im In∗ } Gnh Gmh
h=−∞ m,n=0
h6=0 m6=n
Z 2πZ
0
π
ejk(xm −xn ) sin θ cos φ ejk(ym −yn ) sin θ sin φ sin θdθdφ.
0
(10)
Since Eq. (4) produces real results for Gnh ,Gmh and from [1], the infinite summation of Gnh Gmh
becomes:
∞
X
Gnh Gmh = τmnM inV al − τm τn ,
(11)
h=−∞
h6=0
so that F1 (θ, φ) reduces to:
F1 (θ, φ) = 4π
X
|In |2 [τn (1 − τn )] .
(12)
hni
For F2 (θ, φ) the integral inside summation must be solved. To do this (like in [2]) by using:
µ
µ ¶¶
p
b
−1
2
2
a cos (x) + b sin (x) = a + b cos x − tan
,
(13)
a
complex exponentials in integrals becomes:
“
“
””
√
n
jk sin θ (xm −xn )2 +(ym −yn )2 cos φ−tan−1 xym −y
−xn
m
Fθ = e
.
(14)
So integrals in Eq. (10) may be written as:


Zπ Z2π
“
“
””
√
2
2
−1 ym −yn
 ejk sin θ (xm −xn ) +(ym −yn ) cos φ−tan xm −xn dφ sin θdθ
Fθ =
0
Zπ
=
¶
µ
q
2
2
2πJ0 k sin θ (xm − xn ) + (ym − yn ) sin θdθ.
(15)
0
q
Letting
0
(xm − xn )2 + (ym − yn )2 = C Eq. (15) may be written as:
Zπ/2
2π
−π/2
Z0
J0 (kC cos θ) cos θdθ = 2π
−π/2
Zπ/2
J0 (kC cos θ) cos θdθ + 2π
J0 (kC cos θ) cos θdθ
0
(16)
PIERS Proceedings, Suzhou, China, September 12–16, 2011
370
By letting θ = −θ and since J0 (x) is even Eq. (16) reduces to:
Zπ/2
2π
Zπ/2
J0 (kC cos θ) cos θdθ = 4π
J0 (kC cos θ) cos θdθ.
(17)
0
−π/2
Again by letting u = π/2 − θ Eq. (17) becomes
Zπ/2
Zπ
2π
J0 (kC sin θ) sin θdθ = 2π
0
Zπ/2
J0 (kC cos θ) cos θdθ = 4π
J0 (kC sin θ) sin θdθ.
(18)
0
−π/2
Using the relation [3]
z v+1
Jµ+v+1 (z) = v
2 Γ (v + 1)
Zπ/2
Jµ (z sin θ) sinµ+1 θ cos2v+1 θdθ,
(19)
0
and by letting µ = 0 and v = −1/2 Eq. (17) reduces to:
¡ ¢
Zπ/2
J1/2 (kC) Γ 21
sin (kC)
√
4π
J0 (kC sin θ) sin θdθ = 4π
= 4πj0 (kC) = 4π
.
kC
2kC
(20)
0
Substituting Eq. (20) and Eq. (11) into Eq. (10) F2 (θ, φ) becomes:
µ q
¶
2
2
sin k (xm − xn ) + (ym − yn )
N
−1
X
∗
q
F2 (θ, φ) = 4π
< {Im In } [τmnM inV al − τm τn ]
.
m,n=0
k (xm − xn )2 + (ym − yn )2
(21)
m6=n
Combining Eq. (12) and Eq. (21) PSR becomes:
X
|In |2 [τn (1 − τn )]
PSR = 2π
hni
µ q
¶
2
2
sin k (xm −xn ) +(ym −yn )
N
−1
X
q
+2π
|Im || In | cos (βm − βn )[τmnM inV al −τm τn ]
.(22)
m,n=0
k (xm −xn )2 +(ym −yn )2
m6=n
Integrating Eq. (30) in [1] gives the same result for linear arrays (note that the domain of second
summation is not the same with the case given here).
PSR = 2π
X
|In |2 [τn (1−τn )]+2π
hni
N
−1
X
|Im In | cos (βm −βn ) [τmnM inV al −τm τn ]
m,n=0
m6=n
sin (k (zm −zn ))
(23)
k (zm −zn )
It can be easily seen from Eq. (22) and Eq.
q (23) that both equations are in the same form
q and can
2
2
be represented as one equation. Both (xm − xn ) + (ym − yn ) and (zm − zn ) = (zm − zn )2
gives the Euclidian distance and βm − βn gives the phase difference between distinct elements in
standard Cartesian coordinate system and can be represented as dmn and ∆βmn , respectively. So
that both formulations may be written in one form which is:
PSR = 2π
X
hni
|In |2 [τn (1 − τn )] + 2π
N
−1
X
m,n=0
m6=n
³
´
sin (kdmn )
. (24)
|Im || In | cos ∆βmn [τmnM inV al − τm τn ]
kdmn
Progress In Electromagnetics Research Symposium Proceedings, Suzhou, China, Sept. 12–16, 2011
371
It must be noted that these formulations are valid if there exist only one pulse per period which
are symmetric around t = 0. If there exist more than one pulse in each period Eq. (24) must be
extended to a new general form and if relative positions of pulses are not symmetric around t = 0
total loss seems to be not equal to the symmetric cases.
3. CONCLUSION
In this paper a general representation of total power loss calculation for planar and linear arrays
and some comments about power loss calculations are presented. It is shown that both distinct
formulations given in published works can be combined and written in one form. This form also
provides an expectation about the conformal case.
REFERENCES
1. Brégains, J. C., J. Fondevila-Gómez, G. Franceschetti, and F. Ares, “Signal radiation and
power losses of time-modulated arrays,” IEEE Trans. Antennas Propag., Vol. 56, No. 6, 1799–
1804, 2008.
2. Poli, L., P. Rocca, L. Manica, and A. Massa, “Time modulated planar arrays — Analysis
and optimisation of the sideband radiations,” IET Microw. Antennas Propag., Vol. 4, No. 9,
1165–1171, 2010.
3. Watson, G. N., “A treatise on the theory of Bessel functions,” Cambridge University Press,
London, 1922.