fem termpaper

7/30/2019 FEM Termpaper

http://slidepdf.com/reader/full/fem-termpaper 1/6

Analysis of Waveguides Filled with PeriodicMetamaterial Structures

Submitted as a term paper by Debdeep Sarkar (11104028)

&Saptarshi Ghosh (11104093)

1. Introduction:

Finite element method is popular method used in analysis of several classes of partial differential

equation problems, electromagnetic wave propagation based on Helmholtz’ s equation being one of them. Since it is not always possible to write finite element codes for analysis, people often resort tocommercial FEM packages like COMSOL and Ansoft’s HFSS. But for engineering problems, onevital aspect is understanding of the nuances problem and analyzing reduced problem space in such away that the computational resources are used most efficiently, instead of blindly feeding the entiredomain into some software package input.

In this term-paper we are going to take up the problem of wave propagation in a waveguide filled withsub-wavelength thin wire inclusions. FEMLAB 2.3 (an older version of COMSOL) is used for full-wave analysis of this structure. The importance of this structure lies in the fact that sub-wavelengtharrays of metallic and dielectric unit cells give rise to interesting electromagnetic properties which arenot readily found in nature. The 1D array thin wires show plasma-like resonances in the microwaveX-band, which introduces high-pass characteristics with sharp cutoff characteristics. This prevents theflow of electromagnetic energy below a certain frequency, which is determined by the dimensions andspacing of the thin wires. We also use Ansoft’s HFSS to compare the results.

The subsequent section is devoted in design of rectangular waveguide in FEMLAB, since it is ourbasic structure supporting the wire inclusions. In the next section, the overall response of the thin wirefilled waveguide is studied. Finally some concluding discussions are provided.

2. Rectangular Waveguide in FEMLAB:

The first step is to model a rectangular waveguide in FEMLAB and study its scattering matrixparameters as well as field distributions. The waveguide should ideally be modeled in threedimensions, but our final problem (waveguide filled with thin wires) presents symmetry such that wecan easily model it in two dimensions, hence reducing the meshing requirements by a considerableextent. The waveguide dimensions are chosen as a=25.8mm (along y-axis) and b=12.9mm (along z-axis) . For TE 10 mode, the cut-off frequency is (c/2a) [c=speed of light in vacuum] which comes out as5.8 GHz approximately. We operate the waveguide in the X-band (8 to 12 GHz) where it should haveideally zero reflection and unity transmission assuming the waveguide is filled with lossless medium.The length of the waveguide along x-axis is chosen as L=54.6mm which is the guide wavelength at 8GHz.



Figure-1: Simulation of Waveguide in 2D TE Electromagnetics model (FEMLAB)

The problem-space is reduced to two dimensions (along x-y plane) for FEMLAB based modeling, asthe driving electric field (Ez) in dominant TE 10 mode is independent of z-direction. So we draw arectangle centered at origin (0, 0) and having dimensions L along x-axis, and a along y-axis, torepresent the waveguide geometry in the 2D electromagnetic simulation set-up. The left and rightboundaries (1, 4) are taken as matched ports, and the rest two (2, 3) are selected as PEC (PerfectElectric Conductor). In the sub-domain settings, we select lossless isotropic vacuum as the fillingmedium.

The most vital thing in S-parameter analysis is to calculate the powers at input and output ports,where we have set the impedance matching boundary conditions [1]. For that purpose we need toinsert some excitation, which is done by setting up variables E 0z and β. At the port-1 we select:

√

We define the following coupling variables:

W in: total power inflow integrated over the boundary-1 (Port-1);W flow1 : the power outflow integrated over boundary-1 (Port-1);W flow2 : the power outflow integrated over boundary-4 (Port-2);

W in is evaluated by integrating the following quantity over the boundary-1:



Figure-2: Showing the various power flow directions

Figure-4: (Left) decaying modes in the X-band waveguide for frequency 1.6 GHz; (Right) propagating mode for frequency 8 GHz

Figure-3 S-parameters of the Waveguide only as analysed by

FEMLAB

W flow1 and W flow2 are both integrated values of ⃗ where represents the (averagePoynting vector) over the boundaries 1and 4 respectively. Then the S-parameterscan be calculated as follows:

It should be understoodthat W flow1 is basically thedifference of reflected and

incident powers at theport-1. So for calculatingS11 we need to add W flow1 and W in. The calculated S-parameter curve showszero reflection and perfecttransmission. ( | || | )

When we plot the fields (electric and magnetic) over the waveguide structure, we find that for afrequency 1.6 GHz below cutoff there is no transmission (forward propagation of energy from port-1to port-2). But for 8 GHz the field distributions indicate guiding of energy through the structure whichis in agreement with the physics of waveguide operation.

3. Waveguides Filled with Thin-Wire Inclusions:

As illustrated in [2] an infinite array or mesh of thin cylindrical wires made of metals can work aslow-frequency plasma (plasma at microwave frequency range) if they are excited by incident wave



Figure-5: Alignment of thin wire structures inside waveguide

having E-field polarized along the direction of the wires. For the wire-radius r and period of lattice a,the plasma frequency assuming PEC cylindrical wires is given by:

This is done by effective homogenization of the medium assuming the separation a, to be muchsmaller compared to the wavelength of the incident wave (ideally 0.2 times). The effective medium isanisotropic as the effective dielectric constant along the electric field polarization vector can be givenas:

So we see that for incident wave frequency below the plasma frequency, the wave-vector k

(considering TEM wave)will be imaginary, sinceit is governed by therelation:

On the other hand, forfrequencies above the

plasma frequency thewave vector is real.

For our medium(a=7mm, r=0.5mm) , the plasma frequency is around 10.5 GHz (considering TEM waves). Inside thewaveguide, we place the thin wires along the central axis as shown in Figure-5.

Consequently, the excitation by TE 10 mode E-field is strongest along the direction of one dimensionalwire-array. Instead of an infinite array we take only 5 unit cells, which is sufficient to show themetamaterial response. In FEMLAB the 5 unit-cells are considered to above PEC boundary condition.The frequency sweep based analysis over the X-band frequency range shows the S-parameterresponse curve as shown in Figure-6. The S-parameter results are in excellent agreement with the 3Dsimulation of the same struc ture in Ansoft’s HFSS (v11).

The field distribution analysis (Figure-7) (done using both FEMLAB and HFSS) shows thepropagation characteristics of thin-wire included waveguides above plasma frequency (operatingfrequency=11.5 GHz) as well as the stop-band nature below the plasma frequency (operatingfrequency=9 GHz).



Figure-7: S-parameter for the waveguide with thin wire inclusions (HFSS Results)

Figure-8: Distribution of E-fields over the waveguide at frequency 9 GHz which is less thancutoff frequency around 11 GHz

Figure-6 S-parameter for waveguide with thin wire inclusions (FEMLAB Results)



Figure-9: Distribution of E-fields over the waveguide at frequency 11.5 GHz which is greater than cutoff frequency around 11 GHz

4. Conclusion:

After this study of waveguide filled with thin wire structures using finite element method basedFEMLAB, we can take note of some of the points:

Although the actual 3D geometry was modelled in 2D for symmetry considerations, excellentmatch between the S-parameter results is found for the two simulations.

The cut-off frequency for waveguide filled with periodic thin wires is slightly more than thetheoretical plasma frequency of the thin wires. This is can be attributed to the fact that theanalytical formula is based upon the assumption of TEM wave propagation, whereas insidethe waveguide we have guided dominant mode (TE10) propagation over the frequency rangeof 8-12 GHz.

Since the array is finite, it is not “strictly” metamaterial, although it is sub -wavelength andshows the high pass characteristics expected from plasma.

Both FEMLAB and Ansoft’s HFSS are FEM based commercial software packages. HFSS isdedicated purpose EM simulation tool; but FEMLAB is a software package with provision forsolution of several other partial differential equations present in physics and engineering. Wesee that by proper problem formulation, choice of boundary conditions, selection of materialparameters and use of parametric simulation set-up, we can get simulation results comparableto HFSS from FEMLAB.

Acknowledgement:

We thank our course instructor Dr. N. Gupta for providing an opportunity to do the term-paper on atopic of our wish.

References:

1. FEMLAB Electromagnetics Module guide.2. “Physics and Applications of Negative Refractive Index Metamaterials” by S A Ramakrishna

and T.M.Grzegorczyk, SPIE Press and CRC Press.

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