IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014 1279
A Circuit Model for the Design of Self-Excited EBGResonator Antennas With Miniaturized Unit Cells
Mehdi Hosseini, Member, IEEE, David M. Klymyshyn, Member, IEEE, and Garth Wells
Abstract—A circuitmodel based onBloch theory is introduced tosimplify analysis and design of antennas composed of thick metalelectromagnetic band-gap (EBG) cells with large intercell couplingcapacitance. The cells are composed of thick metal patches peri-odically deployed on a metal-backed dielectric slab. Two versionsof cells are presented that provide large intercell capacitance, onewith narrow high aspect ratio (HAR) gaps between cells and theother with interdigitated gaps between cells. This large capacitancereduces the antenna resonance and dramatically miniaturizes theEBG cells. Three cascaded unit cells are used to demonstrate theapplicability of the circuit model to characterize the recently in-troduced self-excited EBG resonator antenna. Full-wave numer-ical analysis and experimentation validate the robustness and ac-curacy of the model over large variations in electrical/physical celldimensions.
Index Terms—Antenna miniaturization, Bloch, circuit model,electromagnetic band-gap, high aspect ratio, tall transmission line.
I. INTRODUCTION
A CCURATE circuit models can be powerful tools forcharacterizing challenging electromagnetic (EM) struc-
tures such as microwave antennas. Circuit-based analysis canoffer a good set of initial values for time-consuming com-puter-aided parametric design optimization, while also offeringa quick insight into the relationships between performance andstructural parameters. A good circuit model should be ableto predict both input impedance and radiation properties ofantennas [1]. Although prevalent for passive microwave com-ponents like transmission lines (TLs) [2] and filters [3], suchmodels are less common for radiating structures that are notcompletely bound EM problems and have extra complexities.Most published circuit models are applicable to antennas witha planar configuration [e.g., [4] and [5]] that can be representedusing TL theory. The best examples of model-based antennasare planar leaky-wave antennas [4] that belong to the travelingwave family and exhibit both TL and radiating behaviors.Another similar example is a TL model for a broadband spiral
Manuscript received April 12, 2014; revisedMay 30, 2014; accepted June 22,2014. Date of publication June 27, 2014; date of current version July 17, 2014.(Corresponding author: Mehdi Hosseini,)M. Hosseini and D. M. Klymyshyn are with the Department of Electrical
and Computer Engineering, University of Saskatchewan, Saskatoon, SKS7N 5A9, Canada (e-mail: [email protected]; [email protected];[email protected]).G. Wells is with the Synchrotron Laboratory for Micro and Nano De-
vices, Canadian Light Source, Saskatoon, SK S7N 2V3, Canada (e-mail:[email protected]).Color versions of one or more of the figures in this letter are available online
at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LAWP.2014.2333752
antenna and the circuit miniaturization achieved [5]. Further-more, the very recent study in [6] proposes a broadband circuitapproach to model the impedance and fields of antennas usingcharacteristic eigenmodes. Among these, perhaps one of themost successful contributions of circuit models to antennatechnology has been made to antennas with periodic struc-tures [4], [5], [7], [8], most popular and promising of which aremetamaterial-based antennas [4], specifically electromagneticband-gap (EBG)-based antennas [7], [8]. Periodicity allows thedesigner to focus on the antenna building block, the so-calledunit cell, characterize it separately [9], and then use the resultalong with theories like Floquet–Bloch [7], [10] to describethe performance of the structure. As indicated in [11], modelsare sometimes solely based on lumped elements [2], [9] orbased on TLs and embedded lumped elements [5], [10], [11].Most models for periodic structures only focus on reactivebehaviors [2], [9], [11] and ignore real powers, while some alsoinclude the effects of radiation and loss [5], which is also thegoal of this letter.Specifically, this letter presents a (lumped/TL-based) model
suitable for describing unit cells with high capacitance gaps,which can be realized using narrow, high-aspect-ratio, and/orinterdigitated metal structures. Such unit cells can be appliedto the recently introduced self-excited EBG resonator antenna(SE-EBG-RA) structure. The model is applied here to demon-strate the design of such an antenna, in this case a high-effi-ciency antenna formed of three EBG cells, and fed by a mi-crostripline (MSL). In [7], Bloch theory was proposed to an-alyze such structures but was incomplete, being partially de-pendent on full-wave analysis. In this letter, a complete cir-cuit model is presented for the antenna cell, which provides asimple and accurate closed-form set of equations for predictingthe antenna input impedance. More importantly, in addition tothe basic cell proposed in [7], the model is generalized to in-clude cells with high-capacitance interdigitated gaps.
II. EQUIVALENT CIRCUIT MODEL FOR BASIC UNIT CELLS
In [7], a Bloch-theory-based circuit model was proposedto analyze cascaded thick metal unit cells of the general formshown in Fig. 1. This method, though promising, was practi-cally somewhat limiting since it was dependent on computerfull-wave numerical analysis to extract the cascade networkparameters of the unit cell, which were then fed to the Blochequations. The model presented here retains the overall sim-plicity of the approach, however it removes the requirement forany numerical analysis because the cascade parameters are de-termined directly from the physical parameters. The equivalentcircuit model in Fig. 1 is composed of a gap impedance, ,
1280 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
Fig. 1. (a) Bloch unit cell and circuit model representation for a cascade ofsmall thick metal patches on a PEC-backed substrate. (b) Circuit model for theSE-EBG-RA composed of three EBG cells.
with two -long MSLs on each side. is composed of theparallel combination of gap conductance and gap capac-itance . The Bloch dispersion relation of this symmetricalcell (in matrix, ) can be expressed as [10]
(1)
where is the first cascade parameter of the cell in Fig. 1,is the complex propagation constant along the cell,
is the line impedance of the unloaded line (without periodic gaploading), is the cell length, , and is the wavenumberof the unloaded MSL expressed as
(2)
In (2), is the effective relative permittivity given by (7),and and are free-space permittivity and permeability, re-spectively. If , where is the free-space wavelength, then
and , and and can be replaced bythe first two terms of their Taylor series expansions [10], and
. Thus, (1) is simplified to
(3)
Neglecting the radiation represented by in Fig. 1,. In this case, ( ) and (3) becomes
(4)
The cutoff frequency ( ) can be found by setting :
(5)
where is . can also be found using the followingequation [9], with an accuracy of for :
(6)
where is the free-space impedance. It is noted that in (6), thethickness of metal traces is neglected. As for in (6), [12]provides an equation that includes the effect of for an MSLwith a finite thickness
(7)
where and . An interesting observa-tion from (5) is that is canceled by the same term introducedby in (6). Therefore, under certain conditions, themodel doesnot require a particularly accurate estimation of , which canbe difficult to determine for thick metal structures. In fact, (7),which is most accurate for comparatively small , when ap-plied to themodel, is found to produce good results, even forup to 2–3. The in Fig. 1 is attributed to the coplanar couplingon the top of the adjacent cells ( ) plus the parallel-plate cou-pling between the high aspect ratio (HAR) gap in the middle( ), i.e., . The static approach in [8] gives
as ( )
(8)
Also, is found by the following well-known relation:
(9)
Under this circumstance, the only dependency of onoccurs because is still partially dependent on as seenin (8). However, if , then , which ac-cording to (9) eliminates the dependency of on . Par-allel-plate coupling will typically dominate coplanar couplingfor most relevant thick metal structures, meaning that fields willbe mostly constrained inside the narrow air-filled gaps and havenegligible dependency on the substrate material. To include thegap effect in the model, the gap surface is regarded as a radiatingaperture with a radiation conductance, . In [13], E-fields uni-formly distributed on the open end of a parallel-plate waveguideare considered, and an equivalent impedance is driven using theaperture theory with the assumption that the open gap is sur-rounded by infinite PEC planes on each side. In fact, the fieldcondition on the top surface of tall traces (adjacent patches inFig. 1) roughly satisfies this requirement. Thus, is expressedas
(10)
where is the free-space wavenumber. Although canmodel the antenna radiation, it is still a rough estimate sincethick MSLs can radiate themselves [7]. Also, thick traces cancause the gap apertures to be slightly further spaced from thePEC ground and affect . However, this inaccuracy doesnot weaken the model in predicting the zero-crossing in thereactive part of the input impedance (as in Section IV).
III. CAPACITANCE FOR THE INTERDIGITATED UNIT CELL
The circuit model in Section II can be directly applied to char-acterize the basic unit cell with a high-capacitance narrow HARgap shown in Fig. 2(a). Next, the model is extended to describea modified version of the unit cell that employs interdigitatedgaps to increase the gap capacitance, either in thick metal con-figuration of Fig. 2(b) or a more conventional thin metal config-uration. Considering the geometry and parameters depicted, the
HOSSEINI et al.: CIRCUIT MODEL FOR DESIGN OF SELF-EXCITED EBG RESONATOR ANTENNAS WITH MINIATURIZED UNIT CELLS 1281
TABLE IDIMENSIONS AND CHARACTERISTICS OF THE FOUR SE-EBG-RA SAMPLES BASED ON FOUR DIFFERENT UNIT CELLS. IS WAVELENGTH AT , AND IS THE
DIFFERENCE BETWEEN GIVEN BY THE MODEL AND HFSS, DIVIDED BY THE HFSS VALUE. DIMENSIONS ARE IN MILLIMETERS
Fig. 2. Side and top views of EBG unit cells with a thick metal layer and highgap capacitance. (a) Cell with narrow HAR gap. (b) Cell with interdigitated gap.(c) Detailed dimensions of the interdigits in (b).
cumulative coplanar capacitance can be expressed by the sum-mation of capacitances in -axis-oriented gaps ( ) and also-axis-oriented gaps ( )
(11)
where is the number of the periods in Fig. 2(b) (for the caseshown, ). Here is calculated from (8) when isreplaced by and by , respectively. Similarly,is calculated from (8) when is replaced by and is re-placed by [ , , , and are denoted in Fig. 2(c)].Similar to (8) for which , here and
are two necessary conditions for the coplanar capacitance equa-tion by [8]. Also, the parallel-plate capacitance, , can becalculated from (9) when is replaced by
(12)
It is noted that, similarly, in (10) can be replaced byto find the effective at the interdigited gap. In Section IV,the accuracy of (11) and (12) is demonstrated.
IV. VERIFICATION OF THE CIRCUIT MODEL
This section presents several examples to compare the circuitmodel predictions to results obtained through full-wave simu-lations. To this end, cells are used to structure several exampleSE-EBG-RAs. All antenna examples comprise three EBG cells( ) forming a piece of EBG-woven TL, left open on oneside and fed by an MSL from the other side. The configurationis shown in Fig. 1(b). As seen, in addition to at each gap,there are two more radiating apertures at the feed ( ) and
open ( ) sides of the antenna, very similar to radiating aper-tures of a regular patch antenna [13], [14]. Using the lossy TLtheory [15], and considering Fig. 1(b), the input impedance canbe derived as
(13)
When ( , and for , a half-wavelength antenna like a patch antenna is achieved), van-ishes and becomes resistive, and hence, in a similar fashionto patch antennas [13], [14] . In [14], isfound by Harington’s Equation and applied to patch antennas.The equation is identical to (10), except that is replaced by. Also, as in [7], the Bloch line impedance, , can be foundversus , , and , where is given by (1). , which is thesecond cell cascade parameter, is also found from the theory in[15] versus , , and . In (13), is the total length of the EBGTL. Because the antenna has three cells, , where is de-picted in Fig. 1(a). For this antenna, the zero-crossing ( )can be found either from (1) to (13), i.e., entirely based on thecircuit model, or from the HFSS full-wave analysis. By putting
( ) into (4), this resonance can be expressed bywhere is given by (5) and is
(14)
The radiation from each resonator gap is embedded in thetotal radiation losses of the propagating waves along the an-tenna. The radiation from each gap is included in and ,and the radiation from both sides of the antenna is modeled by
and , as expressed by (13). In [7], the behavior of theantenna body as a radiating EBG TL is described in more de-tails, using the dispersion diagram provided.In Table I, four sample SE-EBG-RAs with considerably dif-
ferent EBG cells are considered to demonstrate the applicabilityand generality of the circuit model. The properties obtainedfrom the model [using the dispersion relation (1), rather thanthe simplified relation (4)] are compared to those obtained fromHFSS full-wave analysis. This approach provides two differentevaluations of the antenna matching frequency ( at which
). In Table I, the difference between the predicted( ) for all samples ranges from 0.8% to 4.1%, implying
a satisfactory agreement for design purposes that is consistentwith or better than other analytical circuit models presented forEBG cells with uncertainties as high as 6.5% [9]. In Table I,samples are chosen to have high aspect-ratio (AR) capacitancegaps, ranging from 20 to 30, where as shown inFig. 1(a). One more verification is to examine the accuracy of
1282 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
Fig. 3. Fabricated thin three-cell antenna with micron-scale gaps and its radi-ation pattern. Metal traces are 4- m-thick plated Ni. Cells are as in Fig. 1with mm; mm, mm, m, ,
.
TABLE IIOF SE-EBG-RAS COMPOSED OF CELLS IN FIG. 2(a) AND (b), GIVEN BY THE
MODEL AND HFSS. AND ARE ALSO CALCULATED BY THE MODEL.IS THE DIFFERENCE BETWEEN BY THE MODEL AND HFSS, DIVIDED
BY THE HFSS VALUE
(11) and (12). Table II uses the interdigitated cell in Fig. 2(b) in athree-cell SE-EBG-RA and compares its found by the circuitmodel and HFSS. As seen, , which is quite low con-sidering the complexity of features added to the cell. For com-parison, of the noninterdigitated cell in Fig. 2(a) is includedas well. As seen, of the interdigitated cell has significantlydropped while both cells have identical physical sizes. It is alsoobserved that interdigitation enhances , causing todominate . Obviously, this trend continues as the order anddepth of interdigitation increases.It can be concluded that the model proves relatively accu-
rate in estimation of resonance (where ), provided that, , , , and . Also, although
(6) does not include the effect of thickness and (7) only roughlyincludes that effect, the accuracy as demonstrated ( )is not drastically affected, especially in the case of these struc-tures where dominates . The model also (to some ex-tent) includes the effect of radiation, and thereby, (13) can esti-mate , but not as accurately as .
V. VERIFICATION THROUGH PROTOTYPING
In this section, the effectiveness of the circuit model in de-sign is demonstrated through experimental measurement of twofabricated antenna prototypes. The first prototype is shown inFig. 3. The antenna features 27- m gaps and is made by elec-troplating 4 m Ni on an alumina substrate. The unit cells areas shown in Fig. 1, with dimensions given in Fig. 3. The inputreflection ( ) of the antenna is shown in Fig. 4(a). Themeasured resonance frequency ( ) from Fig. 4 is 12.43 GHz,compared to an predicted by HFSS simulation of 11.54 GHz.
Fig. 4. Measured, HFSS-simulated, and modeled input reflections for (a) theantenna in Fig. 3, (b) the antenna in Fig. 5. Both -axes show .
Fig. 5. Fabricated three-cell antenna. The metal traces are made of30- m-thick copper. The transformer is 1.94 mm wide and 6.96 mm long.
The circuit model predicted of 12.25 GHz is comparableto the HFSS simulation ( ) and is quite respectableconsidering the high operating frequency ( GHz) of thepartially lumped model. Also, with micron-scale features, evensmall discontinuities, for instance in the launch, can introduceuncertainty at higher frequencies. As seen, the difference be-tween the model and experimental results is less than 1.5%.The second prototype and its input reflection are shown in
Figs. 5 and 4(b), respectively. As shown in Fig. 4 for both pro-totypes, although the resonance frequency is well predicted bythe model, the magnitude of is typically higher thanthe measured or simulated results. As explained after (10), themodel only roughly includes the effect of radiation from thegaps and does not consider the effect of the distance betweenthe gap surface and the ground plane. In practice, this affectsthe radiation resistance, and consequently and . TheHFSS-simulated radiation pattern of this antenna at its reso-nance (4.06 GHz) is shown in Fig. 6 as well. As seen, the re-alized gain at broadside is 2.85 dB.In Fig. 5, the unit cell differs from Fig. 1 in that some metal
sections have been removed (indented), forming a dog-bone-like cell. The circuit model in Section II was developed forrectangular unit cells as shown in Fig. 1. However, with someconsiderations, it can also be effectively applied to the cell inFig. 5(a). As shown in Fig. 5(a), associated with the cellis composed of different capacitances, ,where and can be calculated using (8), consideringthe depicted and in Fig. 5(a) for each coplanar capacitance.Also, of the TL in Fig. 1 varies along the length due to thewidth discontinuities. Therefore, two widths of 3.5 and 9 mmcan be considered, each introducing a different to the model.In Table III, these two cases are separately considered, and the
HOSSEINI et al.: CIRCUIT MODEL FOR DESIGN OF SELF-EXCITED EBG RESONATOR ANTENNAS WITH MINIATURIZED UNIT CELLS 1283
Fig. 6. HFSS-simulated radiation pattern of the second prototype at 4.06 GHzfor 30 38 mm ground plane size.
TABLE IIIPROPERTIES OF THE ANTENNA IN FIG. 5 CALCULATED BY THE MODEL.
IS THE DIFFERENCE BETWEEN GIVEN BY THE MODELAND EXPERIMENT, DIVIDED BY THE EXPERIMENTAL VALUE
associated is calculated. As , for the third case, the av-erage of the trace widths in Case 1 and Case 2 is considered asan estimate. As seen, for Case 3, falls somewhere betweenof the Cases 1 and 2, and its estimate for is the best (
is the least at 2%). Even at the worst case (Case 2), ,which is still tolerable.
VI. CONCLUSION
A TL circuit model is proposed for describing thick radiatingEBG unit cells. To demonstrate its effectiveness, this modelis applied to the design of a recently introduced type of an-tenna, the self-excited EBG resonator antenna. Two kinds ofEBG cells are considered for the antenna, including basic rect-angular-shaped cells with narrow high-capacitance gaps and analternative version with high-capacitance interdigitated gaps. Toensure that the model is reliable, it is also applied to the designof four SE-EBG-RAs composed of cells with various physical
and electrical properties. The antenna resonance found by themodel compares favorably with that predicted by HFSS nu-merical analysis (typically difference). Two differentSE-EBG-RA prototypes are fabricated and measured, one ofwhich features narrow gaps of 27 m. Again, it is observed thatthe model and experimental results are very similar, differing byonly 1.5% and 2%. This demonstrates the ability of the simpleequation-based circuit model for accurately predicting the res-onance frequency of radiating EBG unit cells.
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