Wideband Circuit Model for Planar EBG Structures

Abstract

In this paper, we present a comprehensive equivalent circuit model to accurately characterize an important class of electromagnetic bandgap (EBG) structures over a wide range of frequencies. The model is developed based on a combination of lumped elements and transmission lines. The model presented here predicts with high degree of accuracy the dispersion diagram over a wide band of frequencies. Since the circuit model can be simulated using SPICE-like simulation tools, optimization of EBG structures to meet specific engineering criteria can be performed with high efficiency, thus saving significant computation time and memory resources. The model was validated by comparison to full-wave simulation results.

Full text

IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010 169
Wideband Circuit Model for Planar EBG Structures
Baharak Mohajer-Iravani, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE
Abstract—In this paper, we present a comprehensive equivalent
circuit model to accurately characterize an important class of
electromagnetic bandgap (EBG) structures over a wide range of
frequencies. The model is developed based on a combination of
lumped elements and transmission lines. The model presented
here predicts with high degree of accuracy the dispersion diagram
over a wide band of frequencies. Since the circuit model can be
simulated using SPICE-like simulation tools, optimization of EBG
structures to meet specific engineering criteria can be performed
with high efficiency, thus saving significant computation time and
memory resources. The model was validated by comparison to
full-wave simulation results.
Index Terms—Electromagnetic bandgap (EBG) structures,
metamaterials, noise suppression, switching noise.
I. INTRODUCTION
IN THE LAST few years, different methods have been used
to characterize engineered materials including metamate-
rials, electromagnetic bandgap (EBG) structures, and frequency
selective surfaces. Engineered metamaterials are electrically
small resonators and their electrical size is typically smaller
(and sometime much smaller) than over the range of oper-
ating frequencies. These materials may lead to different types
of propagation such as left handed (LH), right handed (RH),
or stopband (bandgap). While metamaterials have traditionally
been considered as any engineered material that gives rise to
left-handed propagation, it also includes the class of engineered
material that is referred to as electromagnetic bandgap (EBG)
structures.
EBG structures have the primary characteristic that propaga-
tion through the structures is inhibited. Characterization of EBG
structures is performed using one of the following methods:
• measurements [1];
• full-wave numerical simulation of the entire structure
based on various methods such as the finite element
method, the finite-difference time-domain method, the
finite integration method, etc., [2], [3];
Manuscript received November 04, 2008; revised March 20, 2009. First pub-
lished October 30, 2009; current version published February 26, 2010. This
work was supported in part by Research in Motion and in part by the National
Science and Engineering Research Council of Canada under the NSERC/RIM
Industrial Research Associate Chair Program and the Discovery Program. This
work was recommended for publication by Associate Editor L.-T. Hwang upon
evaluation of the reviewers comments.
B. Mohajer-Iravani is with the Electrical and Computer Engineering Depart-
ment, University of Maryland, College Park, MD 20742 USA (e-mail: bmo-
hajer@ieee.org).
O. M. Ramahi is with the Electrical and Computer Engineering Department,
University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: oramahi@ece.
uwaterloo.ca).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TADVP.2009.2021156
• dispersion diagram extraction using full-wave numerical
simulation of a single cell [4], [5];
• equivalent circuit modeling based on lumped elements
[6]–[10];
• equivalent circuit modeling based on lumped elements and
transmission lines [11]–[13].
Among these characterization methods, experimental measure-
ments give high accuracy; however, they are costly and time
consuming because of the time it takes to build EBG struc-
tures (either printed circuit boards or chip packages need to be
manufactured first). Full-wave numerical analysis provides ac-
curate results, however, for structures with high variance in their
topological dimensions, simulation time and computer memory
needs can be excessive to render optimization impractical. Other
indirect and efficient numerical analysis techniques were devel-
oped based on the assumption that the actual finite-size struc-
tures behave in a manner similar to an infinite extension of the
same structure [14], [15]. While such assumption is reasonable
when the structure is composed of many cells, it falls short of ac-
curate prediction when only few cells are employed as in cases
where the circuit real-estate is limited [3].
Circuit-based models were developed as an alternative to
time-consuming 3-D full-wave based models. In addition
to their high-efficiency, circuit-based models help in under-
standing how the different operating regions (LH, RH, and
bandgap) are related to the topology and composition of the
engineered EBG structures. However, deriving a model which
predicts accurately the dispersive effects through a wide range
of frequencies is not viable. This is because electromagnetic ef-
fects at higher frequencies are stronger and are more difficult to
model. Therefore, to finalize a specific design, fine tunings and
adjustments based on full-wave analysis and/or experimental
methods become necessary.
In [6], a simple model based on lumped elements for the
mushroom EBG structures was proposed. This model works for
normal wave incidence only at low frequencies where the di-
mensions of the EBG are much smaller than the wavelength
in the host media. Later, a model based on both transmission
line theory and circuit elements was presented in [13]. This
model partially overcoming previous limitation may predict the
bandgap with higher accuracy. In [7], a method was developed
for extracting the parameters of the model for a mushroom-type
EBG embedded in parallel plate waveguide. The method in [7]
uses simple formulas to derive initial values of the lumped el-
ements in the model followed by a numerical algorithm based
on curve fitting the analytically extracted S-parameters to those
extracted using full wave numerical simulations. The technique
reported in [7], while amongst the first efforts for extracting a
circuit model for EBG structures employed for noise mitigation,
was not efficient and robust enough for designing EBG struc-
tures with complex topologies. In [8], a physical-based equiv-
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170 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010
alent circuit model for mushroom EBGs embedded in parallel
plate waveguides was derived. This model gives good predic-
tion of the center frequency of bandgap but fails to predict the
edges of the bandgap. In [11] and [12], transmission line seg-
ments were added to the lumped elements model, leading to
more accurate prediction of the edges of the bandgap as well
as the center frequency. In [9] and [10], basic lumped-elements
models were developed for metamaterials. Those models give
the first LH and RH modes and the existing bandgap in be-
tween. These models give good prediction of the bandgap if the
coupling and the parasitic effects (due to higher order modes)
appear at much higher frequencies than the frequency band of
interest.
In this work, we present a model for an important class of
planar EBG structures that is applicable over a wide range of
frequencies. The structures considered here were recently pro-
posed in [3] and [16] as a robust, highly versatile, and tunable
solution in reducing electromagnetic interference (EMI) in mul-
tilayer printed circuit board (PCB) boards and integrated cir-
cuit (IC) packages. These are simple to construct as they are
composed of square patches connected by meander lines. The
parameters of such topology are easily quantified and when
controlled, can lead to designs that meet specific criteria. In
this work, we develop an advanced equivalent circuit model
based on lumped elements and transmission lines under TEM
mode assumption. More specifically, we show that by consid-
ering enough circuit elements in the modeling of the unit cell of
engineered material, it is possible to predict the performance of
structure at higher frequencies up to the point where the TEM
mode assumption becomes invalid. Therefore, the model pre-
sented here eliminates the drawbacks of earlier circuit-based
models.
This paper is organized as follows. In Section II, we present
the method used to predict the dispersive behavior of infinite
array of EBG structures using the circuit model of a unit cell. In
Sections III–VI, we present the modeling of the different com-
ponents comprising the class of planar EBG structures consid-
ered in this paper. In Section VII, the complete advanced circuit
model is presented along with performance validation. Conclu-
sions are presented in Section VIII.
II. OUTLINE OF MODEL FORMULATION
In this section, we present the method used to predict the dis-
persive behavior of an infinite array of EBG structures using
the circuit model of unit cells, along with the corresponding
full-wave numerical procedure. The actual extraction of the cir-
cuit model will be presented in the next sections.
A 3-D view of the planar EBG structure under study con-
sisting of square patches connected by meander lines is shown
in Fig. 1. Without loss of generality, we develop our model in
the first region of Brillouin zone [17] ( region) where the
wave experiences phase variation in -direction through prop-
agation along an infinite array of EBGs ( ; where
is the propagation constant in the direction and is the
cell size). The phase variation in -direction is ( ;
where is the propagation constant in the direction) where
is an integer number. Notice that the structure is isotropic in
Fig. 1. Planar EBG structure based on patches connected by meander lines.
Fig. 2. Infinite 1-D array of the planar EBG structure with PMC boundaries
used to model the 2-D infinite array in region of Brillouin zone (a) with
and (b) without connecting meanders in -direction.
the and directions. The model developed in this work can be
extended to other regions of Brillouin zone.
By considering the region (in the Brillouin diagram),
we reduce the analysis of the infinite 2-D array to the analysis
of 1-D infinite array in -direction. The effects of the period-
icity in the -direction is accounted for through the use of per-
fect magnetic conducting (PMC) boundaries on both sides of the
1-D array, as shown in Fig. 2(a). The PMC boundary is the only
condition which provides zero phase shifts for all frequencies of
the wave propagating in the -direction. By incorporating this
boundary, we have effectively ignored the side coupling effects
in the -direction. The top view of the unit cell of EBG is shown
in Fig. 3(a). Our objective here is to develop a circuit model for
this type of structure that is valid over a wideband of frequen-
cies. Therefore, the unit cell of the EBG array is reduced to ei-
ther the structure shown in Fig. 3(b) or (c) which corresponds to
an infinite array of 1-D EBGs shown in Fig. 2(b) (the results of
the analysis of these two structures are identical). In this study,
we will provide complete model for Fig. 3(c).
The dispersion diagram of the EBG structure is numerically
extracted using the procedure discussed in [4]. The computa-
tional domain and the boundary setup for extracting the disper-
sion diagram of the unit cell of EBG array are shown in Fig. 3(d).
The pair of periodic boundary condition (PBC) models the pe-
riodicity in the array structure.
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MOHAJER-IRAVANI AND RAMAHI: WIDEBAND CIRCUIT MODEL FOR PLANAR EBG STRUCTURES 171
Fig. 3. Unit cell of the planar EBG structure with meander line as a connecting
bridge. (a) Top view of unit cell of array in Fig. 2(a). (b) and (c) Top view of
unit cell of array in Fig. 2(b). (d) Computational setup used for extracting the
dispersion diagram of EBG structure.
Next, the dispersive behavior of the EBG structure is pre-
dicted through the circuit model of a unit cell and its effective
equivalent transmission line using the ABCD matrix parame-
ters. The infinite array of EBG structure cells is considered as
a transmission line made of engineered materials representing
quasi homogeneous media where the effective propagation con-
stant is and the effective characteristic impedance is .
The ABCD parameters for this transmission line corresponding
to a unit cell (see Fig. 3) are given as
where is the length of the unit cell. The ABCD matrix of aunit
cell of the EBG structure obtained from the equivalent circuit
model is represented as
The frequency independent lumped elements to be used in
the circuit model represent the meander line, the patch, the mu-
tual coupling between the meander line and the patch, and the
step discontinuity (junction) between the patch and the meander
line. The lumped elements will be connected through transmis-
sion lines, which are critical to model the phase shift varia-
tion across the structure. By equating those two ABCD matrices
shown above, the dispersion diagram of the EBG structure can
be extracted.
In the following sections, we introduce the relationship be-
tween the physical parameters of the structure (topological pa-
rameters) and the equivalent lumped elements, and compute
Fig. 4. (a) Top view of patch connected to meander lines from both sides.
(b) Equivalent circuit based on lumped elements.
the parameters of the transfer matrix , , , and .
Throughout this work, we will assume that the metallization
thickness and both metallic and dielectric losses are neglected.
III. MODELS OF PATCH AND PATCH-MEANDER
LINE DISCONTINUITY
The patch and the meander line are implemented using
microstrip technology. Their characteristic impedances are
denoted by and , respectively. is much smaller than
as the width of patch microstrip is much wider than the
width of the meander microstrip line. At low frequencies, the
patch behaves as an electrically short line compared to adjacent
meanders. Therefore, the patch is modeled by a -equivalent
circuit as shown in Fig. 4. The series reactance, , and shunt
susceptance, , in this model are approximated as [18]
(1)
(2)
leading to the capacitance and inductance (see
Fig. 4)
where is the length of square patch, is the guided wave-
length in dielectric substrate, is the speed of light in free space,
and is the effective relative permittivity of dielectric sub-
strate.
The asymmetric step discontinuity between the patch and
the meander as shown in Fig. 5(a) can be modeled through the
T-equivalent circuit shown Fig. 5(b). is the width of patch
and is the width of meander line. If we look to the struc-
ture through the symmetry line of the meander-microstrip junc-
tion, the width of patch is divided into two unequal parts with
widths and (where ). Two new sym-
metric discontinuities are constructed such that the patch width
in the first one has a width as shown in Fig. 5(c) and in the
second discontinuity the patch has the width as shown in
Fig. 5(d). The values of the capacitance and the inductances in
the model of asymmetric discontinuity can be approximated as
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172 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010
Fig. 5. (a) Asymmetrical discontinuity between the patch and the meandered
bridge. (b) Equivalent lumped-element circuit model. (c) and (d) are two new
symmetrical discontinuities introduced for modeling purpose.
the geometric mean of related values for two symmetric cases
as
(3)
where , , and
present the elements modeling
the step discontinuities at the connections shown in
Fig. 5(a), (c), and (d), respectively. The elements modeling the
symmetrical discontinuity in Fig. 5(c) where the width of first
strip is and the width of second strip is are
given by [18], [19]
where the inductance per unit length of the microstrip line
( , 2) is given by and
[18], [19].
and are the characteristic impedance and effective relative
permittivity of the microstrip line with width , respectively,
and is the dielectric thickness in . The capacitance is given
by [18], [19]
is in pF, , , and are in nH. The values of the
equivalent circuit for the configuration in Fig. 5(d) are found
similarly.
Fig. 6. (a) Three-dimensional view of the meander line inductor. Design pa-
rameters are included in the figure. (b) Equivalent circuit model.
For the special asymmetric case where
(meander line is connected to the patch at the edge) same as the
case shown in Fig. 3, the value of the lumped elements can be
approximated by ignoring the effect of the fringing field along
the side (where we have assumed -symmetry). Therefore, the
asymmetric discontinuity can be approximated by a symmet-
rical one where the width of the microstrip line at the junction
is twice that of the original structures ( , ).
IV. MODEL OF MEANDER LINE
A 3-D view of a planar meander line and its design parame-
ters is shown in Fig. 6(a). The meander line which is placed on
top of a metal backed dielectric substrate consists of metallic
arms and metallic bridges connecting arms. The con-
cept of modeling to be discussed in this section is general and
can be extended to any meander line configuration. Without any
loss of generality, we assume that all traces in the meander have
the same parameters of width, , separation gap between ad-
jacent arms, , arm length, , input and output connecting
bridges’ length, , and input and output
arms’ length, . The thickness and rel-
ative permittivity of dielectric substrate are and , respec-
tively. The -equivalent circuit model of meander line is shown
in Fig. 6(b). The inductor is the dominant element in this model
up to the first self resonant frequency. The capacitors model the
parasitic effects. It should be noted that in this model, we have
not taken into account the effect of right angled bends in the me-
ander line. The effect of these bends becomes more pronounced
at frequencies much higher than the range of interest considered
in this work.
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MOHAJER-IRAVANI AND RAMAHI: WIDEBAND CIRCUIT MODEL FOR PLANAR EBG STRUCTURES 173
Fig. 7. Magnetically coupled traces in the meander line. The mutual coupling
between pairs of arms and bridges is marked with solid and dashed lines, re-
spectively.
A. Computation of Inductance
Using Greenhouse formulas [20] and image theory [21], the
total meander line inductance is given by
(4)
where is the inductance of meander pattern. is the mu-
tual inductance between the meander line and its ground image.
Here, it is assumed that the meander line is electrically short.
The elements in circuit model are frequency independent. The
phase shift across the structure which is function of frequency
will be considered in section VI.
The top view of a meander line is shown in Fig. 7. The mag-
netically-coupled arms are marked by the solid lines shown in
Fig. 7. The mutual inductance between one arm and all the mag-
netically coupled arms are sequentially alternating between neg-
ative and positive signs due to direction of current and magnetic
flux lines surrounding the arms. The mutual inductance between
one bridge and all the magnetically coupled bridges are posi-
tive. As a first-order approximation, we have considered only
the coupling between adjacent bridges indicated by dashed line
as shown in Fig. 7. Coupling between other pairs of lines is as-
sumed negligible as there is no overlap along the traces as well
as the increased distance between them. is given by
(5)
where the first sum is due to the self inductance of all sections
in the inductor pattern (total of arms and bridges),
the second sum gives the total mutual inductance between cou-
pled arms, and the third sum gives the total mutual inductance
between adjacent bridges. The self inductance of th section is
given by [22], [23]
Fig. 8. Image of meander line by ground plane. Direction of currents in actual
and image patterns are shown in the figure.
where is the length of the line. All dimensions are in and
is in nH.
The mutual inductance between two parallel arms which are
completely overlapping, as shown in Fig. 7, with effective length
and center to center average distance
is computed by (6) and (7) [20], [23],
[24]
(6)
(7)
All dimensions are in m and is in nH. is the geomet-
rical mean distance between parallel strips.
The mutual coupling between adjacent bridges where the ef-
fective length of strips is equal to and
is approximated using (6) and (7) by [24]
For other configurations of parallel strips with arbitrary shapes,
approximate formulas to compute magnetic mutual coupling
can be found in [24] and [25].
In order to model the effect of the ground plane, we show
in Fig. 8 a meander line and its image. The distance between
the actual meander and its image pattern is equal to twice the
substrate-thickness. The current flowing through the image line
is in opposite direction leading to the total negative mutual in-
ductance represented by , as shown in (4). The mutual in-
ductance between magnetically coupled sections in the physical
line and the image line are computed through similar formulas
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174 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010
Fig. 9. Parasitic capacitances in meander lines.
stated earlier. In those formulas, the geometric mean distance is
approximated by the distance between the centers of the lines.
is computed by
if j is odd
if j is even
(8)
It is to be noted that the mutual inductance between the image
and the physical line is not doubled as the image pattern is not
part of the actual structure.
B. Computation of Parasitic Capacitances
There are two types of parasitic capacitances in the meander
line model, and , as shown in Fig. 6(b). represents
the parasitic capacitance in the meander pattern, which consists
of the different interline capacitances such as , , and
, between adjacent arms, and , and
between adjacent bridges, as shown in Fig. 9. Due to our as-
sumption of zero metallization thickness, , and
are negligible. Also, and between the arms are
much higher than and between the bridges in our
meander designs where we intentionally want to enlarge the
value of inductance per unit area [3], [16]. Therefore, the para-
sitic interline capacitance between two coupled adjacent arms
and is given by
or
where , representing the effective coupled length between
two arms, is approximated by
The total fringing per-unit-length capacitance in the gap be-
tween two arms through air and the dielectric is given by
and , respectively.
In this modeling, we are approximating the parasitic capac-
itance by neglecting coupling between nonadjacent arms and
between the patch and the nonadjacent arms (i.e., only the cou-
pling effect between adjacent strips is considered). Any two
Fig. 10. (a) Uncoupled microstrip line. (b) Symmetric coupled microstrip lines.
(c) Symmetric coupled striplines. The widths of strips and the gaps in between
are equal for all cases. The parasitic capacitances per unit length between the
strips and the ground planes are shown.
adjacent arms in the meander pattern are considered as sym-
metric coupled lines because of the constant strip width. In ad-
dition, the equal gap between the arms gives and
where . Below, we will provide
formulas to compute and . which is the equivalent
capacitance resulting from series of gap capacitances in
the -1 consecutive inter-arm gaps is computed as
(9)
In our case, the expression for is reduced to
considering that all arms are of equal length.
To compute [refer to Fig. 6(b)], we have to consider
bridge-sections and arm-sections separately due to different
coupling mechanism. The per-unit-length capacitance between
the strip and the ground for bridge-sections, which are uncou-
pled microstrip lines, is given by [23].
is result of parallel combination of the three capacitances
shown in Fig. 10(a) [18]
where is the parallel plate per-unit-length capaci-
tance between the strip and the ground plane, given by
. The fringing per-unit-length capaci-
tance between the uncoupled strip and the ground is given
by . and are the
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MOHAJER-IRAVANI AND RAMAHI: WIDEBAND CIRCUIT MODEL FOR PLANAR EBG STRUCTURES 175
relative permittivity and the effective relative permittivity of
the meander microstrip, respectively.
For two symmetrically coupled microstrips shown in
Fig. 10(b), the per-unit-length capacitance between the strip
and the ground is . The even mode capacitance per
unit length is [18], [19] where is the
fringing capacitance between the meander-strip and the ground
in presence of another similar strip. It is given by [19]
where .
The interline mutual capacitance between symmetric lines
discussed earlier within the context of is defined in terms
of the odd and the even mode capacitances. The gap capaci-
tance is given by where is the
odd mode capacitance per unit length. It is defined as
[18], [19] [refer to Fig. 10(b)]. is
the fringing per-unit-length capacitance between two symmet-
rical coupled microstrips through the dielectric gap and is ob-
tained by using a corresponding structure of symmetrically cou-
pled striplines with similar dimensions and twice the substrate
height as shown in Fig. 10(c). is equal to
where and are the per-unit-length even and the odd
mode capacitances of the stripline setup, respectively. In the me-
ander line, these odd and even capacitances for symmetrically
coupled arms are given by [19]
where . and are the elliptic function and
its complement [19], [26], respectively. is [19]
if
if (10)
where . is the fringing per-unit-length ca-
pacitance between two symmetrical coupled microstrip lines
through the air gap, and is given by [19] and [27]
where represents the fringing per-unit-length capaci-
tance between the symmetric coupled coplanar lines when
the dielectric material is air. The width of strips and the sep-
aration between them in coplanar configuration are similar to
the arm-strips in the original meander structure and they are
equal to and , respectively. is given by [19]
where and
. is the fringing per-unit-length capaci-
tance in the surrounding air (not in the air gap between the two
symmetric coupled arm-strips) and is approximated by [27]
where , , , and are the fringing per-unit-length
capacitances in the symmetric coupled microstrips and
striplines with air as a dielectric material .
Fig. 11. (a) Side view of patch and meander-arm microstrips as a pair of asym-
metric coupled microstrip lines. The capacitances per unit length modeling the
coupled lines are shown. (b) Corresponding equivalent circuit model.
The formulas of fringing capacitances have been provided
earlier. The fringing capacitances in the symmetric coupled
striplines are summed as
where .
By using the above mentioned approximation, in three sym-
metric coupled microstrip configuration the parasitic per-unit-
length capacitance between the middle strip and the ground is
given by .
In the next section, the coupling effects between the meander
line and adjacent patches are computed.
V. M ODEL OF COUPLING EFFECT BETWEEN THE PATCH
AND THE MEANDER LINE
In the planar EBG structures under study, the configuration
of a patch and an adjacent arm represents asymmetric coupled
microstrip lines, as shown in Fig. 11(a). All per-unit-length ca-
pacitances characterizing that configuration including the par-
allel plate capacitances and the fringing capacitances are shown
in Fig. 11(a). Fig. 11(b) shows the -equivalent circuit model
for this configuration, where is representing the capac-
itance between the first arm and the ground plane, is the
capacitance between the patch and the ground, and is
the mutual coupling between the patch and the arm. Generally,
the unit cell of this type of designs for EBG structure (patch +
meander line) includes several parallel microstrip lines which
are symmetrically or asymmetrically coupled to each other. To
model these structures accurately appropriate fringing capaci-
tances have to be included in the model.
To this end, we will use the lumped elements as defined in
this section to improve the magnitude value of those elements
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176 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010
introduced in earlier sections. The capacitance is in-
cluded in (refer to Section IV-B) to complete the total value
of the parasitic capacitance between the meander line and the
ground. We can replace in Section III with . Then,
the model approximates the capacitance between the patch and
the ground with higher accuracy due to the consideration of the
fringing capacitances in presence of coupled lines. The lumped
elements in Fig. 11(b) may be approximated by
(11)
(12)
(13)
where is the effective overlapping length between the
patch and the meander arm. For the EBG structures under
study, is equal to . and rep-
resent the parallel plate per-unit-length capacitance for the
patch and the meander-strip, respectively. and
demonstrate the fringing per-unit-length capacitances for the
two coupled symmetric patch-microstrips [28]. Similarly,
and represent fringing per-unit-length capac-
itances for the two coupled symmetric meander-microstrips
[28]. Separation between the two symmetric strips is iden-
tical to the original setup, [Fig. 6(a)]. According to the
expressions provided for meander line model in Section IV,
and . is obtained
from the formula for by replacing with the proper
separation . If in the provided expressions, we replace the
specifications of the meander line with the specifications of the
patch including width ( is replaced by ), characteristic
impedance ( is replaced by ), and relative effective
permittivity ( is replaced by ) then the values of
, , and are easily obtained. , the
per-unit-length capacitance through the dielectric gap is given
by [28] where
is the total fringing per-unit-length capacitance between the two
symmetric coupled meander-strips through the dielectric gap.
That capacitance is computed through the formula provided for
in Section IV by replacing the proper separation gap (
is replaced by ). Similarly, , the total fringing
per-unit-length capacitance between the two symmetric cou-
pled patch-strips through the dielectric gap is computed. The
fringing per-unit-length capacitance through air gap is given
by [28] . is the
total fringing per-unit-length capacitance between the asym-
metric coplanar strips suspended in air. The widths of strips
and the separation between them in coplanar are similar to the
patch-arm microstrips and they are equal to , , and
. is given by [29]
where and is given by
represents the fringing per-unit-length capacitance in
the surrounding air. This capacitance can be approximated by
(14)
Fig. 12. (a) A meander line between two patches. The bridge connecting the
first arm to the patch is shown. (b) Equivalent circuit model of: i) the mutual cou-
pling between the patch and the adjacent meander-arm and ii) the connecting
bridge-trace. (c) Equivalent circuit model of the entire meander line. In this
model, the effect of mutual coupling between the patch and the meander is con-
sidered as interline parasitic capacitance.
where
, , , , , ,
, and represent the fringing per-unit-length
capacitances between the strips and the ground in the symmetric
coupled microstrips and striplines. In these configurations, the
dielectric material is air. The subscripts and refer
to meander-meander and patch-patch coupling, respectively.
These capacitances are computed through the expressions
provided in Section IV by substituting .
The bridge connecting the patch to adjacent arm shown in
Fig. 12(a) is modeled as follows. If the bridge-length, ,isa
considerable portion of the total distance separating the two con-
secutive patches, then we can complete the modeling through
the -equivalent circuit shown in Fig. 12(b). Where, is the
inductance of the bridge-trace and is the capacitance be-
tween the trace and the ground. is the mutual coupling
capacitance between the patch and the meander-arm. However,
if is very small then and may be included in the
total inductance and capacitance modeling the meander line (
and ) discussed in Section IV. In this case, is mod-
eled as part of the interline parasitic capacitances in the meander
pattern as shown in Fig. 12(c).
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MOHAJER-IRAVANI AND RAMAHI: WIDEBAND CIRCUIT MODEL FOR PLANAR EBG STRUCTURES 177
Fig. 13. Advanced equivalent circuit model made of lumped elements and transmission lines modeling the unit cell shown in Fig. 3(c).
VI. MODEL OF ELECTRICAL LENGTH OF PLANAR
EBG STRUCTURE
Due to distributive nature of the inductances and capaci-
tances realized by EBG structures, it is important to include
both lumped elements and transmission lines in the circuit
model. This combination insures the inclusion of the phase
shift due to delay of the field propagating across the structure.
To consider the phase delay across the unit cell of the EBG
structure under study, we include two different types of mi-
crostrip transmission lines in our model consisting of patch-mi-
crostrip and meander-microstrip with total length of and ,
respectively. Where represents the periodicity-
length (or the length of a unit cell of EBG structure), rep-
resents the length of a patch, and represents the length of
separation between the two consecutive patches. Therefore, the
patch-microstrip models the phase difference across the patch
and the meander-microstrip models the phase difference across
the meander line [30] (not along the meander line).
In this study, we adopted similar model as [30] in which the
meander-antenna was modeled as a linear dipole antenna with
inductive loading. For the application involving noise mitiga-
tion in PCB and IC package, the proposed model gives sufficient
levels of accuracy, however, it is possible to make the model
more accurate for higher frequencies by considering the cou-
pling effects vectorially between the arms in the meander line
as in [21]. The ABCD parameters for the microstrip line spec-
ified by the characteristic impedance, the relative effective per-
mittivity, and the line length are available in microwave books
such as [15].
VII. FINAL CIRCUIT MODEL AND MODEL VALIDATION
In Fig. 13, the complete circuit model of a unit cell of the
planar EBG structure under study [Fig. 3(c)] is presented. The
elements along the -direction between the two input and output
ports model the propagation in this direction in an infinite array
of periodic structures. The propagation in -direction is bound
between two PMC boundaries resulting in zero phase shifts (be-
tween the side PMC boundaries) for all frequencies, as dis-
cussed in Section II.
The unit cell in the -direction is consisting of two open-
ended microstrips. The widths of these microstrips are half of
the width of patch and they are located at the ends of the unit cell.
Each of these microstrip lines (from one side up to the middle
of the patch) is modeled by the following elements: 1) the open
end; 2) the microstrip line representing phase delay for wave
propagating in -direction; 3) the inductance of corre-
sponding to the microstrip line with both width and length of
; 4) the capacitance of (which is common be-
tween both the and directions) models the capacitive value
of half of the patch. In Fig. 13, in -direction the characteristic
impedance and the propagation constant of patch-microstrip are
labeled as and , respectively. is the length of patch.
The characteristic impedance and the propagation constant of
meander-microstrip are denoted as and , respectively.
is the distance between two consecutive patches. In the -di-
rection, the characteristic impedance and the propagation con-
stant of half-patch-microstrips are labeled as and .
The final ABCD matrix for this model is simply given by cas-
cading the ABCD matrices of all the subcircuits. The disper-
sion diagram of the structure is finally extracted by equating the
ABCD matrices and , as discussed in Section II.
To validate the circuit model, we consider two separate ge-
ometries and obtain the dispersion diagram using our circuit
model and using the full-wave finite element based simulation
software HFSS [31]. The design parameters of the EBG struc-
tures are: , , ,
, , . The sub-
strate in the first sample has a relative permittivity of
and loss tangent of . For the second example, we
have and . These two examples used
for validation of our circuit model were considered earlier for
noise mitigation in packages [3], [16]. The dispersion/attenua-
tion diagrams are shown in Fig. 14, where . The
corresponding dispersion diagrams extracted by HFSS are also
included. From the results presented in Fig. 14, we observe that
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178 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 1, FEBRUARY 2010
Fig. 14. Dispersion/attenuation diagrams for two planar EBG structures ex-
tracted by two methods: 1) advanced equivalent circuit model and 2) full-wave
numerical simulator HFSS. The parameters of EBGs are as follows:
, , , ,
, . The dielectric is specified as: (a) and
in the first sample and (b) and in
the second sample.
the circuit model predicted the first three propagating modes of
the EBG samples with high accuracy. In fact, strong agreement
between the circuit model and the full-wave simulation results
are observed over a frequency bandwidth extending from very
low frequency up to 5 GHz. What is a strong feature of the cir-
cuit model developed here is that it predicts the edges of the
bandgap with high degree of accuracy, something that earlier
circuit models were not able to achieve.
VIII. CONCLUSION
In this study, we introduced, for the first time, a circuit model
for planar EBG structure, with validity extending over a wider
range of frequencies. The circuit model is implemented as
patches and meander lines as connecting bridges. The circuit
model was developed based on rigorous analysis of the prop-
agation effects of each segment of the structure. As a result,
the developed model predicts the dispersion diagram of EBGs
over a wide frequency range with high accuracy. We validated
the circuit model by obtaining the dispersion diagram for two
different EBG structures reported earlier in [3] and compared
the results to those obtained using full-wave analysis.
EBG structures are electrically small resonators and their
bandgap behavior is typically not readily predictable unless
full-wave based extraction is performed. In the work presented
here, a circuit model was developed for a wide class of EBG
topologies, thus facilitating expedient and efficient extraction of
the bandgap behavior. For example, for the two EBG structures
considered here, it took few seconds to obtain the dispersion di-
agram using the circuit model whereas it took over 4 h to extract
the dispersion diagram using full-wave simulation performed
on a PC with an Intel Pentium 4 processor and 2 GB of RAM.
It is important to note that the circuit model that was developed
here did not require any a priori full-wave based extraction as
was done in previous works (see, for example, [7]).
Once a circuit model is obtained, the effect of different
sources with varying time waveforms can be studied with high
efficiency. Furthermore, the effect of varying the length of
the meander line bridges and/or the effect of varying other
topological parameters can be obtained without resorting to
full-wave simulations.
Finally, we note that the circuit model developed in this work
considered open planar EBG structure. However, the circuit
model can easily be extended to include shielded EBG struc-
tures (i.e., three-layer EBG structures). To include shielding
effects, the circuit model would be developed based on stripline
instead of microstrip line propagation behavior.
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Baharak Mohajer-Iravani (S’03–M’08) received
the B.Sc. degree in electrical engineering from Sharif
University of Technology, Tehran, Iran, in 1998, the
M.Sc. degree (with honors) in electrical engineering
from Amir-Kabir University of Technology (Tehran
Polytechnics), Iran, in 2001, and the M.Sc. and Ph.D.
degrees in electrical and computer engineering from
the University of Maryland, College Park, in 2004
and 2007, respectively.
From 1997 to 2001, she worked on design and
implementation of the circuits and systems. She
was also involved in developing algorithms for speech processing. She was
visiting scholar at University of Waterloo, Waterloo, ON, Canada, from 2006
to 2007. Her research interests include electromagnetic band-gap structures
(EBG), analysis and modeling of microwave and RF devices, high-speed pack-
aging and signal integrity, and electromagnetic compatibility and interference
(EMC/EMI).
Dr. Mohajer-Iravani has been selected for inclusion in 2009 Marquis Who’s
Who in America.
Omar M. Ramahi (F’09) received the B.S. degree in
mathematics and electrical and computer engineering
(summa cum laude) from Oregon State University,
Corvallis, the M.S. and Ph.D. degree in electrical and
computer engineering from the University of Illinois,
Urbana-Champaign.
From 1990 to 1993, he held a visiting fellowship
position at the University of Illinois, Urbana-Cham-
paign. From 1993 to 2000, he worked at Digital
Equipment Corporation (presently, HP), where he
was a member of the Alpha Server Product Devel-
opment Group. In 2000, he joined the faculty of the James Clark School of
Engineering at the University of Maryland at College Park as an Assistant
Professor and later as a tenured Associate Professor. At the University of
Maryland, he was also a faculty member of the CALCE Electronic Prod-
ucts and Systems Center. Presently, he is a Professor in the Electrical and
Computer Engineering Department, University of Waterloo, Waterloo, ON,
Canada. He holds cross appointments with the Department of Mechanical and
Mechatronics Engineering and the Department of Physics and Astronomy. He
has authored and co-authored over 190 journal and conference paper. He is
a co-author of the book EMI/EMC Computational Modeling Handbook, 2nd
Ed. (Springer-Verlag, 2001). He served as a consultant to several companies
and was a co-founder of EMS-PLUS, LLC and Applied Electromagnetic
Technology, LLC.
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