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BRNO UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering and Communication Department of Radio Electronics Ing. Peter Kovács DESIGN AND OPTIMIZATION OF ELECTROMAGNETIC BAND GAP STRUCTURES NÁVRH

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BRNO UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering and Communication Department of Radio Electronics Ing. Peter Kovács DESIGN AND OPTIMIZATION OF ELECTROMAGNETIC BAND GAP STRUCTURES NÁVRH A OPTIMALIZACE STRUKTUR S ELEKTROMAGNETICKÝM ZÁDRŽNÝM PÁSMEM Short version of Ph.D. Thesis Discipline: Supervisor: Electronics and communications Ing. Zbyněk Lukeš, Ph.D. 1 KEYWORDS Periodic structures, electromagnetic band gap, computer modeling, evolutionary algorithms. KLÍČOVÁ SLOVA Periodická struktura, elektromagnetické zádržné pásmo, počítačové modelování, evoluční algoritmy. STORAGE PLACE Research department, FEEC BUT, Údolní Brno MÍSTO ULOŽENÍ PRÁCE Vědecké oddělení, FEKT VUT v Brně, Údolní Brno Peter Kovács, CONTENT 1 INTRODUCTION STATE OF THE ART THESIS GOAL ANALYSIS OF ELECTROMAGNETIC BAND GAP STRUCTURES OPTIMIZATION ALGORITHMS DESIGN EXAMPLES AND ANTENNA EXPERIMENTS VERTICAL MONOPOLE ANTENNA ON EBG SURFACE HORIZONTAL WIRE ANTENNA OVER AMC SURFACE CONCLUSION REFERENCES SELECTED PUBLICATIONS OF THE AUTHOR 4 1 INTRODUCTION In telecommunication, electromagnetic waves are most frequently used to transmit information. The communication channel can be realized as wireless one (propagation in free space) or as wired one (waveguides, optical cables, etc.). In all the cases, one of the fundamental prerequisite of a successful connection between two nodes in the communication chain is to transmit the electromagnetic energy into the host medium in the proper shape. In wireless technology, the profile of the radiated energy beam depends on antenna properties. Due to this fact, a specified radiation pattern of the antenna is required for most cases. In order to improve the antenna performance, a novel breakthrough based on so-called metamaterials emerged in the last two decades. Metamaterials are a kind of a material with electromagnetic properties not found in nature. The permittivity and/or permeability of these materials are in a certain frequency range smaller than zero. Depending on the negative value of the permittivity, permeability or both, metamaterials can be classified as epsilon-negative (ENG), mi-negative (MNG) or double-negative (DNG). Conventional materials have permittivity and also permeability larger than zero in all frequency ranges and they are called double-positive (DPS). Materials with negative permittivity and/or permeability can be obtained using periodic structures. In optics and microwave engineering metamaterials are often called as photonic band gap (PBG) or electromagnetic band gap (EBG). Application of EBGs in microwave and antenna technology is very wide; for example superstrates for very narrow beamwidth and high gain [1], artificial magnetic conductors (AMC) for low-profile antenna design [2], special substrates with forbidden frequency band for surface wave propagation [3]. Even if many analytical and approximate approaches were worked out, an exact design of EBGs is possible only by full-wave techniques. Due to the complexity of these structures, stochastic methods are mainly preferred. However, their application in the field of EBGs as special substrates with suppressed electromagnetic wave propagation was explored only superficially up to now. In the work, the attention is focused on the synthesis and optimization of so-called metallo-dielectric electromagnetic band gap (MD-EBG) structures with the forbidden surface wave propagation by means of global optimization algorithms. The developed method is universal (applicable for any type of unit cells) and is based on a full-wave characterization of EBGs and optimization of parameters of the structure to fulfill the requirements. 1.1 STATE OF THE ART Metallic sheets as reflectors or ground planes increase the gain of antennas by 3 db by pre-directing half of the radiation into the opposite direction and partially shield objects located on the other side [4], [5]. Unfortunately, reflectors as good 5 conducting surfaces reverse the phase of impinging electromagnetic waves. Due to that, an antenna needs to be placed at the distance one-quarter wavelength (λ/4) from the reflector to ensure the constructive interference between the incident and reflected waves. This fact disallows the realization of antennas with a very low profile. The second disadvantage of metallic sheets is supporting surface waves. Surface waves are electromagnetic waves bound to the interface of the metal and free space and are propagating from DC up to optical frequencies. If an antenna is placed near a conductive sheet it will radiate plane waves into free space, but it will also generate a current that propagates along the sheet. In the case of an infinitely large metallic plate the presence of surface waves is observable via slight reduction of antenna efficiency. However, the realistic reflector is always finite in size and diffractions of surface waves at the edges result in ripples and deep nulls in the far field radiation pattern and in a worse front-to-back ratio [4], [6]. In order to eliminate the above mentioned disadvantages of a conventional flat metal ground plane (in a narrow frequency band at least), a corrugated plane can be used. The corrugated plane is a kind of a periodic structure that consists of a series of vertical metal plates connected to each other at the bottom. The height of the vertical plates is h = λ/4 and the distance between them (period) D is much smaller than the wavelength λ. Such a structure acts as a short-ended parallel-plate transmission line with an open end at the top. This means that the impedance at the top surface is very high and surface waves are in cutoff. Furthermore, a plane wave polarized with the electric field perpendicular to the ridges will be reflected with no phase reversal, since the effective reflection plane is at the bottom of the slots, λ/4 away [4], [6]. The overall construction of the corrugated plate can be radically simplified by folding up the one-quarter wavelength slots into lumped elements (capacitors and inductors) and distributing them in two dimensions [4]. The four above-mentioned types of materials (DPS, ENG, MNG, DNG) can be realized as metallo-dielectric periodic structures (under the condition that the wavelength is much smaller than the period) as depicted in Fig. 1.1, [7]. The introduced structures are able to suppress transverse magnetic (TM) and/or transverse electric (TE) surface waves in dependence on whether the permittivity and/or permeability is smaller than zero [4], [6], [7]. The DPS material (supports both TM and TE surface waves) can be implemented as a 2D mesh of microstrip lines (a series inductance and a shunt capacitance to the ground plane). For ENG behavior, a shunt inductance is needed in addition to a series inductance and a shunt capacitance (suppression of TM surface waves). A metallic patch on a dielectric substrate represents a series inductance and a series capacitance to neighboring cells and a shunt capacitance to the ground plane its behavior corresponds to MNG (suppression of TE surface waves). A metallic patch on a dielectric substrate with a shorting via acts as a DNG material (suppression both of TM and TE surface waves) due to the series inductance and the capacitance to neighboring cells, and the shunt inductance and the capacitance to the ground plane. This structure is often referred to as mushroom EBG [4]. 6 a) b) c) Fig. 1.1 Possible realizations of metallo-dielectric periodic structures as DPS (a), ENG (b), MNG (c), DNG (d). Left the unit cell, right the equivalent circuit representation. 1.2 THESIS GOAL The design of electromagnetic band gap structures is rather difficult, due to their complex nature. Even if some approximate [4] and analytical [8], [9] methods exist, the only approach valid for any unit cell geometry and frequency ranges has to be based on the full-wave electromagnetic simulation. Unfortunately, only a few introductory texts and tutorials describing the correct setup of the analysis in commercially available software tools (CST MWS, Ansoft HFSS) are provided. Moreover, these texts are often incomplete and unclear. The second difficulty in the design of EBGs is caused by the large factor of uncertainty, how the electromagnetic band gap characteristics of the structure change on the dependence of the unit cell geometry. Without a proper approach, the design of such a structure is based on trial-and-error. Heuristic algorithms as genetic d) 7 algorithms (GA), differential evolution (DE), and particle swarm optimization (PSO) were successfully implemented for a broad range of complex electromagnetic problems. However, their implementation in the design of EBGs with the surface wave band gap can be shown to be a completely new task. Based on the above mentioned facts, the dominant goals of this work can be formulated as follows: Correct simulation setup of EBG unit cell in full-wave software tools CST Microwave Studio and Ansoft HFSS for surface waves dispersion analysis. Fully automated computer design of periodic structures with surface wave band gap. The proposed method should be universal and applicable for all types of EBGs independently on the unit cell geometry. The main objectives are the position of the band gap (the central frequency) and the largest bandwidth. Different design approaches (GA, DE, PSO) should be compared from the viewpoint of the convergence rate. Verification of the proposed design techniques on examples of periodic structures for different applications. 2 ANALYSIS OF ELECTROMAGNETIC BAND GAP STRUCTURES Analysis of electromagnetic band gap structures is based on the Bloch-Floquet theorem [10] which describes the theory of wave propagation in infinite media consisting of the periodic repetition of the unit cell. The unit cell corresponds to the so-called first Brillouin zone which is the smallest polygon defined by the centre axes of vectors connecting the points of a periodic lattice around the origin [11]. The Bloch-Floquet theorem states that each component u k (r) of an electromagnetic wave propagating in the periodic media can be expressed in the form (considering a medium with 2D periodicity) u k j k r ( r) = e v( r), (2.1) where k is the wave vector, r the position vector and v(r) is a periodic function v ( r + p D + q D ) = v( r), (2.2) x for all integers p and q and translation vectors in x and y directions D x and D y, respectively. It means the components of electromagnetic waves for any translation p D x + q D y differ in phase only u k y j k ( p Dx + q Dy ) ( r + p D + q D ) = e u ( r). (2.3) x y According to the above-mentioned, properties of wave propagation in periodic media can be fully described considering only one unit cell and applying periodic boundary conditions at its edges. k 8 The exact position of surface waves pass bands and stop bands (band gaps) of periodic structures can be obtained by the dispersion relation of surface waves along the contour of the irreducible Brillouin zone (see Fig. 2.1). Fig. 2.1 The first Brillouin zone and the irreducible Brillouin zone of a structure with 2D periodicity. In this work, dispersion characterization of surface waves propagating on periodic structures is investigated using CST MWS and Ansoft HFSS. The first of the mentioned uses finite integration technique (FIT) and the second one is based on finite element method (FEM). In both the programs, the unit cell of the structure under investigation has to be drawn with periodic boundary conditions applied in the appropriate directions. The phase shift is changed along the boundary of the irreducible Brillouin zone and frequencies of eigenmodes are obtained in each step. Because of slow-wave behavior of surface waves, dispersion curves are calculated only in the region under the light line. Band gaps occur in frequency intervals, where no dispersion curves in the slow-wave region are present. Fig. 2.2 shows the unit cell setup of a periodic structure with the square lattice from Fig. 2.1 for dispersion analysis in CST MWS and in HFSS. The main difference between the computational models in the considered software tools consists in the following fact. HFSS uses a perfectly matched layer (PML) to represent an infinite air 9 layer above the unit cell. In CST MWS, open boundaries are not allowed in combination with periodic walls, but only perfect electric conductor (PEC) or perfect magnetic conductor (PMC) boundary conditions can be applied. After many computer simulations in both the programs and comparing the obtained results with analytical and experimental considerations, the following rules for the correct surface waves dispersion diagram computation were stated: In CST MWS, an airbox with the height of about twenty times the dielectric slab thickness has to be placed over the unit cell and the PEC boundary condition should be applied instead of the open boundary on the top of the model, see Fig In HFSS, the height of the airbox of circa six times the dielectric slab thickness is sufficient and the PML boundary condition should be applied on the top of the model, see Fig a) b) c) d) Fig. 2.2 Dispersion analysis in CST MWS (up) and in HFSS (down). The unit cell setup on the left, the applied boundary conditions with the irreducible Brillouin zone on the right. 10 3 OPTIMIZATION ALGORITHMS Methods of global optimization are used to find optimal solutions for a given problem in a given feasible space. The global optimization can be applied not only to a final tuning, but also to a rough initial design and progressive improvement. In this thesis, four evolutionary algorithms have been implemented to test the capability of global optimization methods in the design of periodic structures with the surface wave band gap: the genetic algorithm with the single-point crossover (GAs), the genetic algorithm with the multi-point crossover (GAm), the differential evolution (DE) in the basic variant and the particle swarm optimization (PSO). The computational process of the design synthesis is controlled from Matlab in connection with CST MWS for the full-wave dispersion analysis. Matlab and CST cooperate via Visual BASIC for Application (VBA) interface. The synthesis can be divided into the following steps: 1. Initialization of the optimization algorithm, 2. Exporting the values of state variables into text file, 3. Running CST MWS and calling the VBA macro, 4. Updating the CST MWS model using the new values of state variables, 5. Running the simulation, 6. Exporting the results into text file, 7. Reading the text file with the results into Matlab, 8. Investigating the position of the band gap, 9. Assigning fitness, 10. Computing the new values of state variables. Steps 1 to 3 are done in Matlab, steps 4 to 6 are executed by the VBA macro, and finally, the control is given back to Matlab. Reaching the last step, the procedure is repeated beginning with the second one. This chapter is aimed to implement the above-mentioned evolutionary algorithms in the design of the planar EBG unit cell shown in Fig Fig. 3.1 The EBG unit cell under consideration. 11 The period D and the size of the square patch P were selected as state variables to achieve the surface wave band gap in the given frequency interval (the first objective) and the maximum bandwidth (the second objective). The relative permittivity ε r = 6.15 and the height of the dielectric substrate h = mm are considered to be constant. The required center frequency of the band gap of the TE surface waves (occurring between the second and the third dispersion curve) is f c = 5.5 GHz. In all the cases, the fitness (or objective) function F (3.1) is formulated as a two-criterion function with respect to both the band gap position and the maximum bandwidth. The fitness function is going to be minimized. In (3.1), f BG_min is the lower limit, f BG_max the upper limit of the band gap and w 1 and w 2 are weighting coefficients (in this case, w 1 = 1 Hz -2, w 2 = 1). F = w 1 f BG _ max + 2 f BG _ min f c 2 w 2 f BG _ max f c f BG _ min The optimization results are included in Tab The solutions produced by the considered methods are very similar. In all the cases, the optimum values of D and P for the chosen central frequency f c = 5.5 GHz and the maximum obtainable bandwidth are about 15 mm and 12 mm, respectively. The achieved relative bandwidth BW (related to f c = 5.5 GHz) is approximately 21%. Tab. 3.1 Optimization results properties of the planar EBG unit cells designed by different global evolutionary algorithms. GAs GAm DE PSO D [mm] P [mm] CST f' c [GHz] MWS BW [%] HFSS f' c [GHz] BW [%] The optimization algorithms were mutually compared in term of the overall computational time required for the design. The dispersion relation of the planar square EBG unit cell was calculated with the phase step 20 degrees for the first three modes along the irreducible Brillouin zone shown in Fig. 2.1 and Fig Using CST MWS v installed on a PC with the processor Intel Core 2.66 GHz and 8 GB RAM, the average time for completing the dispersion characterization was estimated to 776 seconds. Because of the different setups of the techniques used, measuring the convergence velocity in time is reasonable. For an objective comparison of the methods, the initial population was composed from identical sets of individuals for all the algorithms, and the convergence rates were averaged over 3 realizations of the optimization (Fig. 3.2). Based on the results from Fig. 3.2, the fastest convergence is exhibited by PSO, whereas differences in accuracy of the methods are negligible. (3.1) 12 Fig. 3.2 Comparison of the selected methods of global evolutionary algorithms design of a single-band planar EBG unit cell. 4 DESIGN EXAMPLES AND ANTENNA EXPERIMENTS This chapter is devoted to the experimental verification of the developed design methodology and to the practical exploitation of periodic structures in antenna technology. A conventional mushroom structure with simultaneous EBG and AMC performance at f c = GHz was designed. The theoretical considerations were confirmed by computer models and experiments: suppression of surface waves over EBG structure and in-phase reflection of electromagnetic waves impinging on AMC surface was measured and simulated in full-wave software tools. 4.1 VERTICAL MONOPOLE ANTENNA ON EBG SURFACE In the first experiment, a lambda-quarter monopole antenna was placed perpendicularly on the conventional metal ground plane. Then, the conventional ground was replaced by the EBG structure, see Fig The radiation characteristics of the antennas were measured to confirm the suppression of surface waves excited by the vertical monopole. a) 13 b) Fig. 4.1 Vertical monopole over conventional metal ground plane (a) versus vertical monopole surrounded by electromagnetic band gap surface (b). a) Fig. 4.2 Dispersion diagram of the optimized EBG structure (a), measured transmission curves for TM and TE surface waves on metal and EBG board (b). b) 14 The EBG structure under consideration is a mushroom one with the square unit cell (Fig. 1.1d). Using PSO, the EBG was designed to obtain the full band gap and in-phase reflection for normal wave incidence at GHz. The period of the optimized unit cell is D = 2.22 mm, the patch size P = 2.04 mm and the diameter of via d = 0.40 mm (changed in fabrication process to d = 0.50 mm). The substrate used is the Arlon AD 600 with the height h = mm and the relative permittivity ε r = The calculated dispersion diagram (Fig. 4.2a) predicts the full band gap in the frequency interval 9.00 GHz to GHz. The transmission of electromagnetic waves over the EBG was measured using two identical horn antennas communicating through the surface waves supported by the structure. The results are depicted in Fig. 4.2b. Four measurements were provided and transmission of TM and TE polarized waves over the EBG board and the co

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