Proceedings of the "2013 International Symposium on Electromagnetic Theory"
23PM3A-02
Modeling of Losses in Microstrip Reflectarrays
based on a Simple Equivalent Circuit Approach
Filippo Costa #1, Agostino Monorchio #2
#
Dipartimento di Ingegneria dell’Informazione, University of Pisa, Via G. Caruso 16, 56122 Pisa, Italy
1
2
filippo.costa@iet.unipi.it
agostino.monorchio@iet.unipi.it
Abstract—Reflection losses of microstrip reflectarray antennas
are analytically addressed by approximating the printed
structure through a simple equivalent transmission line model.
The closed-form expressions of surface impedance (real and
imaginary part) are derived through well justified
approximations. The compact formulas containing all the degree
of freedom of reflectarrays allow efficiently studying the
reflection losses. The dependence of the input impedance on the
capacitance associated with the printed pattern is highlighted,
demonstrating that highly capacitive subwavelength elements are
preferable for minimizing reflection losses.
I. INTRODUCTION
Microstrip reflectarray antennas consist of a grounded quasiperiodic array of printed elements able to compensate the
phase displacement of a non-coherent electromagnetic
excitation generated by a feeder [1], [2]. The focusing or
shaping effect can be achieved by slightly modifying the
shape or size of the printed elements in order to locally vary
the phase of the reflected field. Due to the quasi-periodicity of
the structure, the phase diagram can be calculated under the
infinite periodic array assumption. The periodic version of a
reflectarray antenna is analogous to a high-impedance surface
(HIS) [3], [4]. Reflectarrays, differently from HIS structures,
are usually designed with half wavelength spaced elements
and the use of densely packed unit elements was rarely
proposed [5]. Two key figures of merit in the design of a
reflectarray are the bandwidth and the loss minimization.
Usually, in order to minimize losses, substrates with low loss
component and large thicknesses are employed. The effect of
substrate thicknesses on reflectarray losses has been discussed
and quantified with a methodical analysis [6], [7]. While
increasing substrate thickness does indeed reduce losses and
limit the occurrence of a miss-behaved reflection phase [7],
[8], the use of low-loss leads to high manufacturing costs
which is the reason why reflectarrays are not attractive for the
market as an alternative to reflector antennas [9].
In the last few years a new research trajectory for the design
of low-cost and high-performance reflectarray antennas since
the use of subwavelength elements have been proved [10],
[11] to be a good mean both for dramatically reducing
reflectarray losses. By using theory valid for High-Impedance
Surfaces it is also possible to demonstrate that tightly coupled
subwavelength element are the best choice for enlarging the
operating bandwidth [4], [5], [12], [13].
This paper presents a new comprehensive and simple model
able to describe the reflection properties of reflectarray
antennas arising from combinations of physical or geometrical
parameters such as dielectric properties (εr, tan δ), metal
conductivity (σ), element shape, element periodicity, substrate
thickness. The proposed closed-form approach, based on the
equivalent transmission line representation of the printed
structure, allows to derive a compact expression of the real
part of the input impedance as a function of the
aforementioned degrees of freedom. This new formula may be
a useful tool for designing low-loss and low-cost reflectarrays
but also a good mean to understand the reflection loss
mechanisms of reflectarrays.
II. METHODOLOGY
The layout of the investigated structure and its equivalent
circuit are reported in Fig. 1. The analyzed structure is
basically a subwavelength resonant cavity characterized by an
input impedance approaching to infinite and a reflection phase
crossing zero at the resonance [3], [4]. From a circuital point
of view, the amount of power absorbed by the cavity at the
resonance is determined by the value of the real part of the
input impedance (the imaginary part is close to zero at the
resonance). The magnitude of the reflection coefficient at
normal incidence approximately reads:
Γ ≈
(1)
where ZR represents the input impedance of HIS structure and
ζ0 is the characteristic impedance of free space at normal
incidence. ZR is equal to the parallel connection between the
two complex impedances ZFSS and Zd representing the FSS
impedance and the grounded substrate impedance,
respectively. By deriving an explicit expression of the real
part of ZR, reflection properties of the reflectarray can be
easily extracted.
A. Complex Input impedance of the grounded dielectric
substrate
The grounded substrate input impedance is inductive until the
substrate thickness is lower than a quarter wavelength.
'
''
Assuming the small losses hypothesis verified ( ε r ε r ), the
real and the imaginary part of the input impedance Zd=A+jB
can be written as follows [14]:
668
Copyright 2013 IEICE
Re {Z R } − ζ 0
Re {Z R } + ζ 0
Proceedings of the "2013 International Symposium on Electromagnetic Theory"
A
⎤
'' ⎞
⎛
ζ 0 ⎡ ε r''
'
⎜ k0 d ε r ⎟ 1 + tg 2 k0 d ε r' ⎥
ε
tg
k
d
−
⎢
0
r
'
⎥
⎜
ε r' ⎢⎣ 2ε r
2 ε r' ⎟
)
(
B
⎝
⎠
(
(
))
⎦
(2)
)
(
ζ0 ⎡
tg k0 d ε r' ⎤
⎢
⎥⎦
' ⎣
εr
where k0 is the free space propagation constant and d is the
thickness of the dielectric substrate. The real part of the input
impedance depends both on the real and imaginary part of the
dielectric permittivity while the imaginary part of Zd is almost
unchanged with respect to the lossless case.
B. Complex impedance of a Frequency Selective Surface
printed on a lossy dielectric substrate
The impedance of a loss free FSS can be represented through
a series LC circuit or, more simply, by a single capacitor if the
inductive component is low (e. g. square patch element) [15].
The accuracy of the LC model extends from the DC to the
onset of grating lobes, that is, when the repetition period of
the unit cell approaches one wavelength [15]. When an FSS is
printed on a dielectric substrate, the value of the free space
capacitor has to be multiplied by the effective dielectric
permittivity εeff due the surrounding dielectrics [16]. If the
condition of thick substrate is verified [15], the effective
permittivity simply corresponds to the average of the relative
permittivity of the substrate and the relative permittivity of
free space ε eff = (ε r + 1) 2 = ε r' + 1 2 + j ε r'' 2 . As a
(
1 − ω 2 LC0ε eff
jωC0ε eff
= RD + jX
(3)
Under small losses assumption ( ε r' ε r'' ) and assuming that
the resonance frequency of the HIS is smaller or much smaller
than the FSS resonance frequency, the resistor representing
the loss component of the FSS caused by the surrounding
lossy dielectric, approximately reads:
RD −ε r''
ωC ε r' + 1
(
)
C0thin = C0 −
2 Dε 0
π
4π d
−
⎛
⎞
ln ⎜1 − e D ⎟
⎝
⎠
(5)
where d represents the thickness of the dielectric substrate. As
the dielectric thickness diminishes, the capacitance value
increases due to the capacitor created between the FSS and the
ground plane.
C. Complex input impedance of the reflectarray
Once derived the complex expressions of the impedances ZFSS
(R+jX) and Zd (A+jB) the input impedance of the reflectarray
structure ZR can be derived. After some simple algebra, the
real and imaginary terms of ZR can be expressed as follows:
Re {Z R } =
(4)
If a metallic FSS is considered, ohmic losses should be
included within the FSS impedance through an additional
series resistance RO. In microwave range, the resistor
representing ohmic losses is typically one or two order of
magnitude lower than the dielectric resistor in (3). The two
resistors account for two different physical phenomena. RD
takes into account losses due to strong electric fields in the
parallel plate capacitor formed between the edges periodic
patches being electric field lines concentrated in a lossy
medium. RO represents losses due to currents flowing on
imperfect conductor. The free space capacitance and the
inductance of the FSS can be computed by matching the
(b)
normal incidence full-wave response of the elements in
freestanding configuration [15] or by using closed-form
expressions [16] if a square patch element is employed.
If the periodic surface is printed over a thin dielectric
substrate, the influence of higher-order (evanescent) Floquet
modes reflected by the ground plane cannot be neglected. The
influence of the evanescent modes in the equivalent model can
be taken into account by the following substitution [16]:
)
consequence, the lumped impedance ZFSS of the loss free FSS
printed on a lossy substrate is composed by a real and an
imaginary term:
Z FSS =
(a)
Fig. 1. 3D sketch of the analyzed structure (a) and its equivalent circuit (b). C
is the effective capacitance of the FSS printed on the substrate, L is the FSS
inductance and the resistors take into account dielectric and ohmic losses.
( AR − BX )( A + R ) + ( BR − XA )( B + X )
( A + R )2 + ( B + X )2
Im {Z R } =
(
) (
X A2 + BX + B R 2 + BX
)
(6)
(7)
+ (B + X )
For an ideal lossless structure, the resonance condition is
X = − B causes a pole in the imaginary part of ZR. As a lowloss substrate is employed instead the input impedance at the
resonance is characterized by a very high real part and by a
smooth transition through zero of the imaginary part [16].
As previously pointed out, the amount of losses of the
resonant cavity is determined by the value of the real part of
the input impedance. A very large real part leads to a limited
amount of reflection losses whereas a real part close to the
free space impedance determines high absorption [16]. If the
input impedance decreases below the free space impedance
669
( A + R)
2
2
Proceedings of the "2013 International Symposium on Electromagnetic Theory"
the losses starts to decrease again but the typically missbehaved phase behavior is obtained [7], [8].
Since the highest amount of losses is achieved at the
resonance, it is useful to simplify relation (5) by imposing the
resonance condition X = − B and the inequlity B A, R :
Re
{ }
Z Rres
B2
≅
( A + R)
(8)
By replacing B, A and R with the relations (2), (4), the real
part of the input impedance at the resonance can be explicitly
written [14]. Since the resistance of the FSS, RD, is always
larger than the real part of the input impedance of the
grounded dielectric slab, A, before first resonance of the HIS
structure ω0, relation (8) can be written as:
{ }
Re Z Rres ≈
)⎥⎦
(
(
)
2
ζ 02 ⎡ tan 2 k0 d ε r' ⎤ ωC0 ε r' + 1
⎢⎣
−2ε r'' ε r'
if ω < ω0 (9)
III. CONSIDERATION ON REFLECTION LOSSES
The expression of the real part of ZR in (9) is a function of the
FSS capacitance C0 substrate thickness, real and the imaginary
part of the dielectric permittivity. Simple considerations on
reflection losses are drawn in following paragraphs and the
derived relations are compared against full-wave results
obtained through MoM simulations.
Substrate losses: the increase of dielectric losses, ε’’, leads to
a reduction of the real part of the input impedance towards the
free space impedance with a consequent reflection loss
increase.
From (10) it can be easily observed that the effect of the
substrate thickness reduction dominates on the increase of the
capacitance. As a consequence, the value and the real part of
the input impedance at the resonance is reduced and reflection
losses increases.
Capacitive coupling of FSS element at fixed resonance: If the
shift towards lower frequencies due to capacitive coupling
enhancement is compensated by a reduction of the FSS
periodicity, it is possible to observe the effect of the coupling
at a fixed frequency. While the patch gap is made narrower,
the FSS periodicity is reduced in order to maintain the FSS
reactance fixed, that is, the FSS inductance decreases while
the capacitance increases and the element becomes more
compact [4]. In Fig. 2 the magnitude and phase of the
reflection coefficient are reported at a fixed frequency for
various spacings of the patch array. The enhancement of the
capacitance at a fixed frequency with fixed substrate
parameters, leads to a strong reduction of the reflection losses,
that is, the usual practice of designing reflectarrays with λ/2
spaced elements is not the best configuration for the
bandwidth maximization or for loss minimization. The choice
of closely spaced patches results in a one layer reflectarray
with the widest possible fractional bandwidth. It is also worth
to point out the slope of the reflection phase increases as the
absorption of the structure at the resonance increases [16].
This is an additional drawback of sparse elements
configuration which results in a reflectarray sensitive to
fabrication tolerances.
0
300
Capacitive coupling of FSS element: The enhancement of the
capacitive coupling between the printed elements leads to a
shift towards lower frequencies of the resonance frequency
which means lower electrical thickness of the substrate. Since
thickness reduction and capacitance increase are conflicting
trends, it is useful to remove the dependence on the
capacitance from (9). Considering for simplicity an FSS made
of patch array represented by a capacitance only [16], and
applying the resonance condition ( X = − B ) we get:
{ }
Re Z Rres ≈ −
(ε
'
r
)
+1 ζ0
ε r''
ε r'
(
)
⎡ tan k d ε ' ⎤ if ω < ω
0
0
r ⎥
⎢⎣
⎦
(10)
Reflection coefficient magnitude [dB]
200
-2
100
Phase
-3
0
-4
-100
D=λ0/2.2
-5
Reflection coefficient phase [deg]
Substrate thickness: Relation (9) shows that the increase of
the substrate thickness leads to a rise of the real part of the
input impedance proportional to B2 at the resonance. However,
as the substrate thickness is increased, the resonance is shifted
towards lower frequencies proportionally to the square root of
the substrate inductance ( ω0 ≈ 1 Ls C where Ls=B/ω0). The
electrical increase of the thickness is lower than the physical
one due to the shift of the resonance towards higher
wavelength but the use of a thick substrate leads to higher real
part of the input impedance and therefore lower losses than
thin substrates.
Magnitude
-1
D=λ0/3.75
-200
D=λ0/6.6
-6
3
4
5
6
7
Frequency [GHz]
8
9
Fig. 2 - Magnitude and phase of the reflection coefficient of the HIS
for various spacings of the printed elements. Substrate: Fr4 (4.5j0.088) with thickness of 2 mm. Solid lines: MoM simulations;
dashed lines: TL model.
Losses at fixed frequency as a function of the element
geometry: The focusing effect of reflectarray antennas is
achieved by progressively modifying the size of the printed
elements with the aim of locally varying the reflection phase
coefficient of the surface. Four typical FSS elements, reported
in Fig. 1 are analyzed. Fig. 3 reports reflection coefficient
(magnitude and phase) of the four periodic arrays on the top
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Proceedings of the "2013 International Symposium on Electromagnetic Theory"
of a 2.5 mm grounded FR4 substrate at 8 GHz. The curve are
obtained as a function of the physical dimension L/D. The unit
cell periodicity D is 10 mm. The square patch element is
characterized by the smallest amount of losses since the
capacitance of the element, at the resonance, is the highest
one. Indeed, in accordance to (8), the maximization of the
capacitance at the resonance, keeping the other parameters
fixed, leads to a smallest amount of loss. The capacitances and
the inductances of the analyzed elements, computed by
matching the normal incidence full-wave response of the
elements in freestanding configuration, are displayed in Fig. 4.
0
400
-2
200
-3
100
-4
0
Phase
-5
-100
dogbone
ring
patch
dipole
-6
-7
0.5
REFERENCES
[1]
[2]
[3]
[4]
-200
[5]
0.6
0.7
0.8
0.9
-300
[6]
side lenth (L)/periodicity (D)
Fig. 3 - Reflection coefficient magnitude of three FSS elements on
the top of a grounded 2.5 mm FR4 substrate at 8 GHz as a function
of the size L. Cell periodicity D: 10 mm. Solid lines: MoM
simulations; dashed lines: TL model. Investigated unit cell elements.
0.16
16
dogbone
ring
patch
dipole
0.14
[7]
[8]
[9]
14
[10]
0.12
12
0.1
10
0.08
8
0.06
6
0.04
4
0.02
2
[11]
Inductance [nH]
Capacitance [pF]
IV. CONCLUSIONS
Reflection losses of reflectarray antennas have been modeled
through a simple equivalent circuit approach. Closed-form
expressions of the input impedance (real and imaginary part)
have been derived as a function of electric and geometrical
parameters of the structure. The derived expressions can be
employed as a valid tool for the design low-cost reflectarrays
and for understanding the reflection loss mechanisms but also
to demonstrate that, once chosen a given substrate, highly
capacitive elements allows reflection losses minimization and
fractional bandwidth maximization, at the same time.
300
Magnitude
Reflection coefficient phase [deg]
Reflection coefficient magnitude [dB]
-1
lowest one. It is not a good element even for phase excursion
which is rapid and limited.
[12]
[13]
0
0.5
0.6
0.7
0.8
side lenth (L)/periodicity (D)
0.9
[14]
0
[15]
Fig. 4 – Variation of the capacitance and inductance values of the
[16]
investigated elements.
The dogbone element (capacitive loaded cross) leads to the
highest losses since its capacitance at the resonance is the
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