Verification, enhancement and mathematical analysis of EBG structure using complex geomsetrical shapes and eigenmode analysis approach
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Abstract
The study presents the enhancement in analysis formulae and its verification for Electromagnetic Band Gap (EBG) structure using the Eigenmode analysis method. Eigenmode analysis of EBG structure is a compact and prolific method for obtaining its performance parameters like surface bandgap (bandwidth) and resonant frequency. The enhancement in mathematical expressions of the gap width, the capacitance, and the bandwidth is carried out using the simple geometrical shapes like square, circle, and hexagon. The verification of the enhanced formulae of the EBG structure is conducted using complex shapes. The theoretical and simulated results agree with each other up to 94% accuracy.
Keywords
Unit cell EBG structure Surface bandgap1 Introduction
In the recent past, microstrip antenna with EBG structure was popular due to its peculiar features like surface wave suppression, bandwidth enhancement, and improvement in the efficiency of antenna [1, 2, 3, 4, 5]. The characterization of EBG structure is obtained using techniques like planewaveexpansion method [6, 7, 8], finitedifference method [9], finiteelement method [10, 11], and transfermatrix method [12, 13]. A variety of other techniques are also used, such as the effective medium theory [14], phasedarray method [15], Eigenmode expansion method [16], array scanning method [17].], and hybrid methods [18, 19], Fullwave simulation, Finite Domain Time Difference (FDTD) method and circuit modelbased method [20]. The fullwave simulation method based on a diffraction grating is developed for a finite number of periodical layers of the EBG structure [21].
Generalizedpenciloffunction (GPOF) algorithm is developed as a postprocessing procedure in the finitedifference timedomain (FDTD) method for EBG structure analysis. The technique reduces the computational burden by reducing an excessive number of time steps [22]. The circuit model is build up using the patches and meander lines as connecting bridges based on the rigorous analysis of the propagation effects for each unit cell of the EBG structure [23].
The circuit model for the hybrid structure consisting of resistance, which represents the permeability losses due to the ferrite layer is proposed. It is also demonstrated that both the inductor and resistor components are frequencydependent [24].
The research of any technology relies on mathematical modeling and reaches upto computational simulation. In this paper, the computational simulation method is used to enhance the basic analysis formulae. The FDTD based Eigenmode analysis is used to simulate the unit cells of various shapes of the EBG structure. These simulated results are put into a mathematical model, and the enhancement of formulae like the gap width, capacitance and surface bandgap are achieved.
The major contributions of this paper are focused on,

Investigation and validation of the expressions of the capacitance and bandwidth for considered geometrical shapes like circle, hexagon, and complex designs like circular fractal shapes where the gapwidth is not constant at every point on the edge of cells.

The analysis is carried out using the Eigenmode analysis approach, and the expressions of bandwidth and capacitance are validated and enhanced.
The paper organization is as follows. In Sect. 2, a model of EBG structure and its mathematical analysis is demonstrated. Section 3 discusses the Eigenmode analysis approach using the simulation setup for EBG analysis. Section 4 focuses on simulations of various shapes, while Sect. 5 elaborates on results, and suggestions regarding the enhancements in the analysis of the bandwidth expressions. Section 6 describes the brief conclusions.
2 EBG structure and its mathematical analysis
The EBG structure’s for the calculation of bandwidth, resonant frequency, and impedance of EBG structure are derived as follows [20, 32, 36].
The equations from (1)–(12) are used to find the performance parameters of the EBG structure like bandwidth, resonant frequency, and the surface impedance. These equations are derived for a squareshaped EBG cell where the gap width ‘g’ is constant between adjacent cells.
3 Eigenmode approach for EBG Structure
The EBG structure is analyzed using fullwave simulation methods like Finite Element Method (FEM) and Method of Moments (MOM). The fullwave simulation methods require huge memory and time for their execution.
FDTDPBC based unit cell analysis method is carried out using two approaches viz. scattering mode analysis and Eigenmode analysis. For the first one, an EBG cell is placed with an airbox of finite size, the sidewalls of the air box are applied with Periodic Boundary Condition (PBC). The upper surface of the air box is applied with a floquet port. The scattering analysis gives a reflection and transmission curves for the selected frequency range, and the surface band is predicted.
For Eigenmode analysis, the EBG cell is placed in the airbox of a height six times more than the height of the substrate. A Perfectly Matched Layer (PML) boundary is applied to terminate the airbox. This airbox is enclosed in another airbox whose sidewalls are applied with PBC along x and y directions. The phase shift along the transverse direction is kept at 0°, while the phase along the direction of propagation is varied from 0° to 180°.
The dispersion curve and the surface band gap for various modes obtained using Eigenmode analysis and a graphical method [39, 40].
The following steps are carried out for analysis of EBG structure using the FDTDPBC method wherein the electromagnetic fields are calculated using the following technique [20].

The electromagnetic (EM) fields in the interior of the EBG cell in free space are updated Yee’s scheme.
 Equations (13) and (14) are used to calculate EM fields on the periodic boundaries of a computational domain.$$E \left( {x = 0,y,z,t} \right) = E \left( {x = p,y,z,t} \right)e^{{jpk_{x } }}$$(13)where the periodicity of EBG cells is p.$$H \left( {x = 0,y,z,t} \right) = H \left( {x = p,y,z,t} \right)e^{{jpk_{x } }}$$(14)
 The elimination of the reflections in the Perfectly Matched Layer (PML) region is possible by setting the material properties as per (15) (16), (17), and (18).$$\overline{\overline{s}} = \left[ {\begin{array}{*{20}c} {s_{x}^{  1} } & 0 & 0 \\ 0 & {s_{x} } & 0 \\ 0 & 0 & {s_{x} } \\ \end{array} } \right].\left[ {\begin{array}{*{20}c} {s_{y} } & 0 & 0 \\ 0 & {s_{y}^{  1} } & 0 \\ 0 & 0 & {s_{y} } \\ \end{array} } \right].\left[ {\begin{array}{*{20}c} {s_{z} } & 0 & 0 \\ 0 & {s_{z} } & 0 \\ 0 & 0 & {s_{z}^{  1} } \\ \end{array} } \right]$$(15)$$s_{x} = 1 + \frac{{\sigma_{x} }}{{j\omega \varepsilon_{0} }}$$(16)$$s_{y} = 1 + \frac{{\sigma_{y} }}{{j\omega \varepsilon_{0} }}$$(17)The following procedure is carried out to implement a constant k_{x} method.$$\left( {s_{z} = 1 + \frac{{\sigma_{z} }}{{j\omega \varepsilon_{0} }}} \right)$$(18)

The propagation constant k_{x}(β) is obtained.

Acquired results are plotted in k_{x}frequency plane.

The Eigen frequencies are identified for the propagation constants in x and y directions.

The extraction of the resonant frequencies of surface waves is undertaken.

The Dispersion is plotted to obtain the surface band gap [20, 41].
In this research work, Eigenmode analysis approach is applied to obtain the bandwidth and the center frequency of the unit EBG cell. This empirical bandwidth is verified using a mathematical approach, and an enhancement in its formulation is suggested for an improvement in results.
4 Simulation of experiments for various shapes of EBG cell
In this section, the simulations are carried out for the basic geometrical shapes using Eigenmode analysis. The simulations are conducted using HighFrequency Simulation Software version 11 (HFSS 11). The EBG cells of the same footprint and gap widths are taken into consideration. The effect of gap width for complex shapes and its expression is enhanced using fractal shapes.
Comparison of bandgaps and center frequencies for three basic shapes
Shape of unit cell  Width of patch w (mm)  Gap width g (mm)  Surface bandgap (GHz)  Center frequency f_{r} (GHz) 

Square  6  1  1.83–3.26  2.545 
Circle  6  1  2.08–4.83  3.455 
Hexagon  6  1  2.11–4.72  3.415 
It is observed from Eqs. (1) and (12) that the gap width plays a vital role in deciding the capacitance of the EBG structure as well as its bandwidth.
Therefore, for accurate analysis of such complex shapes, there is a need to modify the gap width equation. Various gap widths should be considered between the adjacent cells. The gap width for such shapes is selected in such a way that the average gap width is found to be equal to 1 mm for the verification of practical and simulated results.
Two iterations based circular fractal shaped EBG cell is designed having a footprint size of 6 mm and simulated using Eigenmode analysis approach.
Parameters for circular fractal shaped EBG cell
Shape of unit cell of fractal shape  Width of patch w (mm)  Gap width g_{avg} (mm)  Surface bandgap (GHz)  Center frequency f_{r} (GHz) 

Circle  6  1  2.11–4.72  3.415 
Circle with 1st iteration  6  1  1.8–5.21  3.505 
Circle with 2nd iteration  6  1  1.1–6.32  3.71 
Table 2 clearly states that even for the same width of patch and gapwidth, variation in the parameters for various shapes is obtained. The gapwidth is kept as 1 mm in all cases, but for square shape, it was constant for every point on adjacent edges. The gap width for other shapes is not the same throughout the curvature. It is, therefore, needed to enhance the theoretical expression for capacitance as it depends on a gapwidth.
5 Discussions on the results
This section focuses on the need for improvement in bandwidth expression of the EBG structure. The improvement is suggested based on the simulation results of fractal shapes of EBG cells obtained using FDTDPBC analysis method.

Mathematical analysis is carried out considering the squareshaped EBG cell with various patch widths and gap widths.

The corresponding equivalent, capacitance, inductance, and bandwidth are calculated.
Theoretical and practical values of parameters of squareshaped EBG structure
Patch width W (mm)  Height of substrate h (mm)  Gap width G (mm)  Theoretical inductance L (nH)  Theoretical inductance C (pF)  Theoretical Bandwidth BW (Hz)  Bandgap using Eigenmode analysis approach (GHz) 

4  1.6  0.5  2.0112  0.162827  0.294953146  2.9801 
4  1.6  1  2.0112  0.129281  0.331015708  4.4345 
4  1.6  2  2.0112  0.09941  0.377486817  7.1025 
6  1.6  0.5  2.0112  0.275485  0.226760828  1.2043 
6  1.6  1  2.0112  0.222809  0.252144925  1.6559 
6  1.6  2  2.0112  0.000169179  9.150464168  701441.9 
8  1.6  0.5  2.0112  0.0000563929  15.84910154  5062117.6 
8  1.6  1  2.0112  0.000112785  11.20703169  146948.3 
8  1.6  2  2.0112  0.000225568  7.924605745  417133.7 
Theoretical and practical values of parameters of complex shapes of EBG
Shape of unit cell  Width of patch w (mm)  Surface bandgap bandwidth (GHz)  Calculated bandwidth using enhanced formula in GHz 

Square  6  1.48  1.4692 
Hexagon  6  2.75  2.6987 
Circle  6  2.61  2.581 
Fractal circle with 1st iteration  6  3.41  3.395 
Fractal circle with 2nd iteration  6  5.22  5.183 
The strong agreement between the simulated and calculated results (obtained from the enhanced formula of bandwidth) is the achievement of the research. The earlier research has focused on the new techniques of EBG structure analysis. While this paper discusses enhancing the traditional analysis formulae so that the accurate mathematical analysis can be carried out.
6 Conclusion
The modern tools for the analysis of complex designs of microstrip antenna and EBG structure are useful for the prediction of their performance. The Eigenmode based analysis method proves to be efficient for the verification of mathematical analysis and enhancement of expressions used for the analysis. The gap width for some geometrical shapes
like circular and hexagonal unit cells is not uniform at every point between the adjacent cells. Therefore, an improvement in gapwidth calculation based on sampling and averaging method is suggested. This improvement is validated with practical results obtained using FDTDPBC based Eigenmode analysis approach. The equation for gap width is verified for complex shapes based on circular fractal shapes of EBG cells. Further, the analytical formulation of capacitance is enhanced using regression analysis and verified with simulated results. The Eigenmode analysis can be further investigated for exploring different parameters of EBG structure and similar materials like Artificial Magnetic Materials (AMC), High Impedance Surface (HIS), Photonic Band Gap (PBG) materials.
Notes
Funding
There is no funding from any agency for the research work proposed in the Paper.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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