Design of Metamaterials in HFSS and Extraction of Permittivity and Permeability using NRW Method

Transcription

1 Design of Metamaterials in HFSS and Extraction of Permittivity and Permeability using NRW Method Monika Dhillon, Master of Technology student of Electronics and Communication Engineering of YMCA University of Science and Technology, Faridabad. Pradeep Dimri, Assistant Professor, Department of Electronics and Communication Engineering, YMCAUST Faridabad, Abstract Recently, artificially constructed meta-materials have become of considerable interest, because these materials can exhibit electromagnetic characteristics unlike those of any conventional materials. In this paper, we will introduce NRW method to achieve negative electrical permittivity and negative magnetic permeability in a split ring resonator and thin wire of a metamaterial cell.the electrical permittivity ε and the magnetic permeability µ are the fundamental characteristic quantities which determine the propagation of electromagnetic waves in matter. Keywords Metamaterials (MMs), Electrical permittivity, magnetic permeability, NRW method. 1. Introduction In today s scenario the Demand for compact radiator with sufficiently high bandwidth is rapidly increasing in wireless and space borne applications. Metamaterial Microstrip antennas are a class of antennas which use metamaterials to increase performance of miniaturized (electrically small) antenna systems. The study of electricity and magnetism as independent phenomena can be traced back thousands of years, where scientists throughout the ages have tried to understand the elegance and mystery of these natural phenomena. They are due to the interaction of external electric and magnetic fields with the internal electric and magnetic dipoles of the medium. The permittivity is a measure of the net electric dipole moment per unit volume generated by the material, and the permeability is a measure of the generated magnetic dipole moment per unit volume. These parameters can tell us useful information about a material such as its refractive index, phase velocity, loss, and impedance. 2. MATERIALS BASED ON COEFFICIENTS: 1. Double Positive (DPS), both electrical permittivity and magnetic permeability coefficients are positive. 2. Double Negative (DNG), both electrical permittivity and magnetic permeability coefficients are negative or LHMs. The most popular of MMs are the so-called left-handed (LH) MMs, which are characterized by antiparallel phase and group velocities or, equivalently, negative refractive index. 3. Single Negative (SNG), only one of the coefficients electrical permittivity or magnetic permeability coefficients is negative. 38

2 3. Method for extraction of Ɛ and µ of Metamaterials 1. Free space method 2. Waveguide Method 3. Robust Method 4. NRW Method In this paper, we will discuss the NRW method for parameter extraction ofmetamaterials. 4. Nicolson-Ross-Weir (NRW) Method: To extract effective medium parameters from the normal incidence scattering parameter data, Nicolson- Ross-Weir (NRW) approach, was implemented. While many of the difficulties discussed in the literature are associated with sample sizes and the need for care when they are near a multiple of a half-wavelength, other problems were encountered when trying to apply these techniques to the MTMs under consideration here. The NRW approach begins by introducing the composite terms....(1)...(2) And deriving the following quantities Consequently....(3)..(4).....(5).....(6) The choice of the sign is selected to maintain the expected magnitudes of these terms, i.e., Z 1 and 1 In the MTM cases under consideration, the values of and are highly frequency 39

3 dependent and achieve values near zero and unity. After testing them on numerous MTM cases, the standard extraction expressions were found to be unsatisfactory, particularly in the frequency regions where the permittivity and permeability resonances were expected, i.e., where those values would transition quickly between positive and negative values. The presence of the square root values is particularly difficult in those regions. One cannot anticipate what branches the square root values should lie on without potentially biasing the end results. Using the same process, however, one can derive many other expressions for and Z. Ones that could handle the current MTM cases were sought. For instance, one can obtain the transmission term Z as.(7) Similarly, one can obtain the interface reflection coefficient as:.(8) From Eqns. (2.7) and (2.8) one can obtain the exact expression..(9)... (10) Assuming that the electrical thickness of the MTM slab is not too large and knowing that the complex wavenumber K = ω, one can write Z 1-jkdto obtain the approximate results for the wave impedance and permeability from Eqns. (3.22) and (3.23), respectively.. (11). (12) The permittivity and index of refraction can then be obtained simply as....(13)...(14) The square of the wave impedance can also be obtained as.....(15) Because they avoid the square root issues, these expressions seemed to produce reasonable results for all the cases tested. Notice, however, that the combination of Eqns. (2.11) and (2.12) yields...(16) This expression fails to reproduce the physically expected resonance features even though Eq. (2.11) does. It was found that some of the essential features exhibited by several formulas with 1±V1 in them shared this issue when 0, 1 while 1, 0. These V1 terms average the sharp features associated with and. It was found that even Eq. (2.12) is not as accurate as desired in all cases near the resonances of μ predicted by Eq. (2.11) The resonance features of ℇ r still become smeared. To overcome this difficulty, it was found that when kreald 1.(17) 40

4 ...(18) This expression shows explicitly that ℇ r and should exhibit very similar responses when 0.The extraction formulas were applied to the various MTM cases to see if the resonant permittivity and Permeability behaviours could be identified. Particular consideration was given to the extraction in those frequency regions where DNG behaviour may have occurred. The assumption that kreald 1with the extracted values was tested and was found to be satisfied in all of the cases reported below. Note that the resulting effective permittivity and permeability s are to be interpreted in terms of how the MTM slab acts in reflection and transmission under normal incident plane wave illumination. Also note that other extraction approaches could have been attempted. For instance, one could assume models of ℇ r and and curve fit their coefficients to match the and behaviours. Despite the challenges associated with the NWR approach for these MTM cases, I felt that it was highly desirable to continue to deal with this established approach in order to connect the following results with known behaviours. Moreover, other approaches, e.g., the curve-fitting method, require an assumed medium model and could have led to unwanted biased final conclusions. 5. Results To extract effective medium parameters from the normal incidence scattering parameter data using Nicolson Ross weir (NRW) approach : The unit cell is cubic, with a cell dimension of d=2.5 mm. A 0.25 mm thick substrate of FR4 (Ɛ= 4.4, loss tangent of 0.02) is assumed. A copper SRR and wire are positioned on opposite sides of the substrate,modeling the structures typically produced by lithographic circuit board techniques. The copper thickness is mm.the width of the wire is 0.14 mm, and it runs the length of the unit cell. The outer ring length of the SRR is 2.2 mm and both rings have a linewidth of 41

5 0.2 mm. The gap in each ring is 0.3 mm, and the gap between between the inner and outer rings is 0.15 mm. 6. Conclusion and future work In this paper, the introduction of metamaterials and unique characteristics of these materials, such as backward, negative refractive index was investigated. And some meta-material structures such as the SRR, TW in creating electrical permittivity and magnetic permeability coefficients were examined and also the NRW method used for calculating electrical permittivity and magnetic permeability coefficients of a meta-material cell. 6.1Conclusion: We know that in the propagation of wave through any material, permittivity and permeability play a significant role. And from above work done I conclude the following points. 1. The resonant frequency of any cell depends upon the size of cell. 2. The conductivity also effect the resonant frequency. 42

6 3. Height of the substrate and dielectric constant of the substrate plays significant role in determining the resonant frequency. 6.2Future Work: For further work I will investigate the exact relationship between the cell size and resonant frequency of a metamaterials.also I will investigate the change in resonance frequency with the change in internal structural dimension of a unit cell. 7. References 1. McGinnis, Dave, Measurement of Relative Permittivity and Permeability Using Two Port S- parameter Technique, Pbar Note 585 (www-bdnew.fnal.gov/pbar), April Cotuk, Unit, Scattering from Multi-layered Metamaterials Using Wave Matrices, Naval Postgraduate School Thesis, Sept Chen, Xudong, Grzegorczyk, Tomasz M., Wu, Bae-Ian, Pacheco Jr., Joe, & Kong, Jin Au, Robust Method to Retrieve the Constitutive Effective Parameters of Metamaterials, Phys. Rev. E., 70, , (2004). 4. Zwick, Thomas, Chandrasekhar, Arun, Baks, Christian W., Pfeiffer, Ullrich R., Brebels, Steven, &Gaucher, Brian P., Determination of the Complex Permittivity of Packaging Materials at Millimeter-Wave Frequencies, IEEE Trans. Microwave Theory Tech., 54, No. 3, (2006). 5. Chen, Xudong, Wu, Bae-Ian, Kong, Jin Au, &Grzegorczyk, Tomasz M., Retrieval of the Effective Constitutive Parameters of BianisotropicMetamaterials, Phys. Rev. E., 71, , (2005). 6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett.84, 4184 (2000). 7. R. A. Shelby, D. R. Smith, and S. Schultz, Science292,77 (2001). 8. L. Ran, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, Chin. Sci. Bull.48, 1325 (2003). 9. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B65, (2002). 10. K. S. Yee, Numerical Solution of Initial Boundary Value Problems Involving Maxwell s Equations in Isotropic Media,IEEE Trans. Antennas Propagat., vol. AP-14, pp , May