cTE01 = ⁄
√
Since on the rectangular waveguide we defined a b then
cTE10 cTE01
TE00 do not exist then the lowest frequency that can propagate on this structure is
c TE10 and is the absolute minimum frequency the structure can support and known as dominant TE10 mode..
III. SRR array loaded rectangular waveguides
Artificially structured metamaterials waveguides has attracted increasing researchers in recent years, however this study will base on waveguide with SRR metamaterials. Backward travelling waves is one of the unique properties exhibited by SRR loaded waveguides but there are other
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metamaterial waveguide structures examined by other authors which also have the ability to create an environment for backward wave propagations [14-17]. Regardless of the challenges encountered during the investigation process, great interest has been devoted to waveguide with SRR metamaterials for the past decade. Many investigations have been performed for different SRR parameters and orientation on rectangular waveguide however all created a backward wave mode thereby supporting backward travelling wave. This does not only support electromagnetic wave propagation but also use for waveguide miniaturization [3, 11]. Some focus on other waveguide structures leading to the realization of backward wave mode below the cutoff frequency.
In this thesis SRR loaded rectangular waveguide proposed by [7] have been presented and it was designed to obtain a pass-band below the waveguide cutoff frequency. The identical SRR inserted on two side walls of the rectangular waveguide was reinvestigated again. The design configuration made it possible for the SRR at its resonance frequency below the waveguide cutoff frequency to exhibit a backward travelling wave. This backward wave mode existing within a narrow pass-band below the waveguide cutoff frequency was created by the simultaneously negative permeability and permittivity of the proposed waveguide structure.
We reinvestigated SRR loaded rectangular waveguide [7] with width and height of 35mm and 16mm respectively. An SRR inclusion was printed on a conventional fiber glass of relative permittivity of 4.5 with thickness 1.55mm. The SRR dimensions are: outer radius = 7.6mm, inner radius = 6.4mm, gap distance = 1.4mm and unit cell 15.6mm and the geometry of the proposed waveguide is shown in figure 2.2 The design configuration structure have been computed using CST simulation and the dispersion diagram have been reproduced again as shown in figure 2.3.A backward wave centered at 1.8GHz at a cutoff frequency of 3.9HGz was observed as expected but the variations of SRR has an influence on the location of the backward wave mode. CST software was used for both the dispersion diagram of figure 2.2 and the dispersion diagram of fig. 2. [7], shown as figure 2.4. Comparing the two dispersion diagrams their dispersion results are in agreement, the location of the backward wave modes of the two dispersion diagrams are centered at 1.8 GHz below waveguide the cutoff frequency. With regards to this phenomenon parametric study has been conducted to reinvestigate the factors affecting the location, presence/absence of backward wave mode on the dispersion diagrams. Not only the SRR various size was considered but the parameters of the overall SRR loaded rectangular waveguide structure was looked into. The duration of the CST simulation was also paramount.
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Figure 2.2 Geometry of the proposed waveguide, (a) waveguide cross section (b) SRR dimensions
Figure 2.3 Dispersion diagram of the proposed waveguide with SRR of identical sizes on both sides.
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Figure 2.4.Dispersion diagram of the proposed waveguide with SRR of identical sizes on both sides (solid lines) [7].
CHAPTER 3
RESULTS AND ANALYSIS I. parametric study
Nowadays recent researchers have design SRR array loaded rectangular waveguide of different orientation on the waveguide structure, however the main concern for these structure design is on how to give suitable range of parameters to meet the design objectives. The existence of a backward wave mode on the dispersion diagram may be due to the inappropriate varying electrical structural and dimensions. Despite several parametric study conducted by other authors
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they has not been able to show design configuration method which can be computed using CST in the shortest period of time.
The objective of this study is to examine SRR loaded rectangular waveguide configuration to determine the influence and the effects of the input parameters on the existence of backward wave modal pass-band below the cutoff frequency. This parametric study extended from the previous author [8, 12] on Multiband SRR Loaded Rectangular Waveguide. Variation of design input parameters have continue to be a key factor affecting the exhibition of backward wave mode in metamaterial waveguide structures, an impact on the design configuration and also influences the position of the modal pass-band on the dispersion diagrams. This particular parametric study carried out here was able to provide the possibilities of improving the duration of the CST simulations.
A range of parameters are chosen as shown on table 3.1, various designed configurations have been generated and simulated. Each parameter variation and dispersion results have been analyze, thereby demonstrating the variations of the design causing the most effect either good or bad on the dispersion diagram. The different parameters investigated in this study include the following:
Height
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There were a total number of 729 computed results which all exhibited backward wave on their dispersion diagrams as expected and their respective cutoff frequencies have been observed.
Each of the nine different heights as indicated in table 3.1 have 81 total dispersion diagrams. It was also noticed that backward travelling wave made a shift to left as the scaling factors increase in size. The bandwidths and the center frequencies vary due to the effects of the starting and ending points of the backward travelling wave. It can now be conclude that no matter how small the height of waveguide would be, if you keep on increasing the size of SRR with different scaling factors in ascending order backward wave will be observed as expected.
The effects of the waveguide height dimensions ranging from 14.5mm to 16mm and the SRR size variations have been presented here as follows waveguide heights: height 16mm, SRR radii:
and 0.55 0.95 , Gap space:
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0.55 0.087rad 0.087rad. Backward waves mode exist on all the dispersion graphs of heights 14.5mm, 15.25mm and 16mm but shifting to right as the height increases. There is an influence on the cutoff frequency location. Increasing the heights and the SRR size caused both the backward waves and cutoff frequencies locations on the dispersion graphs to decrease in values and make a shift to the left side due to the constant increase of inner radius whilst maintaining the outer radius of the SRR. Hence the heights and the SRR size variations affect the locations of the backward waves. Below are some of the computed dispersion diagrams and their various bandwidths, center frequencies and cutoff frequencies.
Table 3.2 Height = 14.5mm
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Figure3.2. Height and SRRs dimension effects on dispersion diagrams showing only outer radius of 3.98mm.
The height of the waveguide is 14.5 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.3.98 6.88, 0.55
, 0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
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Figure3.3 SRRs dimensions and Height effects on dispersion diagrams showing only outer radius of 3.98mm.
The height of the waveguide is 14.5 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.3.98 6.88, 0.55
, 0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
Looking at the above graphs with the same height of 14.5mm, it have been noticed that both the backward wave modes and the cutoff frequency values all shift to the right side as their outer and inner diameters of the SRRs increases. Both the outer and inner diameters of the SRRs rings have be scale by factors [0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95]. The higher the scale factor backward wave modes and their respective cutoff frequencies shift to the left side.
Table3.3 Height = 15.25mm
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3.11 2.97465 7.38 0.0407
3.5252 7.6 0.0396
3.88745 7.6 0.0251
3.19 1.84425 5.345 0.0155
2.05645 5.76 0.0151
2.2714 5.99 0.0092
Figure 3.4 SRRs dimensions and Height effects on dispersion diagrams showing only outer radius of 4.19mm.
The height of the waveguide is 15.25mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.4.19 7.24mm, 0.55
, 0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
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Figure 3.5 SRRs dimensions and Height effects on dispersion diagrams showing only outer radius of 7.24mm.
The height of the waveguide is 15.25mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.4.19 7.24mm, 0.55
, 0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
For the case of height 15.25mm which is bigger than height 14.5mm the same observations have been noticed, both the backward wave modes and the cutoff frequency values all shift to the right side as their outer and inner diameters of the SRRs increases. As applied with heights 14.5mm both the outer and inner diameters of the SRRs rings with height 15.25mm have been also scale by factors [0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95]. Comparing the graphs of heights 14.5mm from figure a. to figure i. with the graphs of height 15.25mm from figure a) to figure i) respectively, more shifting to left with a height 15.25mm would be expected. When the SRRs rings are scale with the same scale factor but different heights the locations and positions of both the backward wave modes and the cutoff frequencies differs.
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Table 3.4 Height = 16mm figures Legend Center
frequencies
Cutoff frequencies
Bandwidth
3.20 2.8394 7.21 0.034
3.3729 7.53 0.039
3.72275 7.53 0.022
3.28 1.5131 4.87 0.0056
1.7611 5.13 0.0058
1.97915 5.28 0.0017
Figure 3.6 SRRs dimensions and Height effects on dispersion diagrams showing only outer radius of 4.4mm.
The height of the waveguide is 16mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.4.4 7.6mm, 0.55
,
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0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
Figure 3.7SRRs dimensions and Height effects on dispersion diagrams showing only outer radius of 7.6mm.
The height of the waveguide is 16mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.4.4 7.6mm, 0.55
, 0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
In this parametric study, the highest height was 16mm and comparing its dispersion diagrams with the dispersion diagrams of heights 14.5mm and 15.25mm, both the backward wave modes and their respective cutoff frequencies decrease towards the left side on the dispersion graphs though their respective SRRs rings have been scale with the same factors.
Waveguide width effects:
Waveguide width effects: The effect of waveguide width( ω = 35mm) variation on the modal dispersion have been investigated on three different heights, 14.5mm, 15.25mm and 16mm.The thickness (d = 1.55mm) of the dielectric slab on both side (2* d) of the waveguide have been maintained. The width between the two identical SRRs19
denoted here (a) centered within the two unit cells on both side (2*a) of the waveguide walls have been reduced to half (a). A parametric study of all the SRRs scale by factors [0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95] on all the three heights have been simulated but the figure below only shows the dispersion diagrams for the three different heights, the outer SRR diameter rings 0.95 times each of the three different heights of the waveguide, and the inner SRR diameter of rings 0.55 times the outer diameters of each height.
A rise in both backward wave modes and the ordinary modal cutoff frequencies have occurred with width a reduction. The backward wave mode on each of the three heights dispersion diagrams rise by shifting to the right side comparing with their dispersion diagrams when width (ω = 35mm) was maintained on all the three heights.
Figure 3.8 Height and Waveguide width effects on dispersion diagrams showing only outer diameter of 0.95*height and inner diameter of 0.55*outer diameter
14.5mm Height 16mm, with width 17.5 mm,
, The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm.0.55
0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
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Dielectric Slab d effects:
The thickness of the two dielectric slabs (2* d) loading on the two side walls of the rectangular waveguide have been studied. The two identical SRR arrays printed on the surface of both the dielectric slabs have been simulated reducing the thickness from 1.55mm to 1mm on both the side walls of the waveguide. Three different rectangular waveguides have investigated with all their SRRs scale by factors [0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95]. All designs of different waveguide heights exhibits backward modes on their dispersion graphs as expected. The figure below only shows their dispersion diagrams, the outer SRR diameter of the three rings 0.95 times each of the three different heights of the waveguide, and the inner SRR diameter of the three rings 0.55 times their outer diameters.Comparing these three reduced dielectric slab of 1mm dispersion diagrams to the dispersion diagrams of 1.55mm dielectric slab, it have observed that both the backward wave modes and the cutoff frequencies shift more to left with 1mm than 1.55mm.
Figure 3.9 Height and Dielectric slab effects on dispersion diagrams showing only outer diameter of 0.95*height and inner diameter of 0.55*outer diameter
14.5mm Height 16mm, with width 35mm,
, The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1mm.0.55
0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
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Relative permittivity effects:
The substrate material with relative permittivity = 4.5 have changed to = 1. This reduction on the on the have affected the appearance of the backward wave modes on the three different heights 14.5mm, 15.25mm and 16mm simulation results. All the SRRs of the three heights scale by ring factors [0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95] have been simulated. The outer SRR diameter rings 0.95 times each of the three different heights and their inner SRR diameter of rings 0.55 times their respective outer diameters all possessed backward wave modes as shown in figure 3.10.Comparing the backward wave modes on each of the three heights dispersion diagrams where the relative permittivity have been changed to 1 with dispersion diagrams with relative permittivity = 4.5, the locations of the backward wave modes changes position at different locations decreasing to the left side and their ordinary cutoff frequencies locations were also influence with a smaller relative permittivity. The dispersion graphs of all the three heights with the rings scale by factor 0.95 and 0.55
Figure 3.10Relative permittivity ( )and Height effects on dispersion diagrams showing only outer diameter of 0.95*height and inner diameter of 0.55*outer diameter
14.5mm Height 16mm, with width 35 mm,
, The relative permittivity of each of the two dielectric side slabs is 1 with thickness 1.55mm.0.55
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0.55 0.087rad 0.087rad, represented in the figure in solid lines, asterisk lines and point lines respectively.
II. CST Simulations
Computer Simulation Technology (CST) is a commercial microwave software tool used in the simulations and design of electromagnetic structures. The problem that usually found in CST design simulation is time consuming during the process and at end the results might not be good as expected. First of all before the start of the simulation process identical SRR loaded rectangular waveguide was drawn and various design configurations were generated.
The simulations of metamaterial element loaded waveguide structures are really timed consuming but previous authors have not paid much attention to this problem. My thesis work investigated a way of improving this problem to address long time consumption. Under the boundary conditions symmetry planes have set in particular directions; all to none field directions which involves all cases of fields, all to electric (electric fields are anti-symmetric whilst the magnetic fields are symmetric) or magnetic field directions (magnetic fields are anti-symmetric whilst the electric fields are anti-symmetric), one of the symmetry planes XZ plane electric or YZ plane magnetic or vice versa. The figure 3.11 below show the directions of symmetry planes under the boundary condition of CST design configuration structures.
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Figure 3.11 Setting of symmetry planes under boundary conditions.
The first approach was setting all symmetry planes XZ and YZ to none fields directions and the simulation process took longer hours (total of 9 hours, 35mins and 55sec) but the modal pass-band appeared on the dispersion graphs. This way of setting the symmetry planes thought exhibited backward wave mode as expected but was unable to produce a fast enough simulation process. In the same design all symmetry planes XZ and YZ were set to electric fields directions the simulation was faster (total of 6 hours, 28mins and 14sec) consequently obtaining modal pass-band below the cutoff frequency. This approach of symmetry planes setting to all electric directions does not only speedy up the time but also exhibit modal pass-band. The other two settings of the symmetry planes was also fast enough but backward wave modes could not appear on the dispersion graphs.
Estimated numerical results
The numerical values (Bandwidth, Center frequency and Cutoff frequency) estimated from the dispersion diagrams due to the effects of various waveguide parameters and configurations are presented. The dispersion diagrams are the results performed by the computer simulations of the structure using CST Microwave Studio. The estimated values are tabulated according to the heights of the waveguide, 14.5mm, 15.25mm and 16mm.
Table 3.5 Height = 14.5mm
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25
26
27
6.19 0.54 2.03 5.82 0.017
6.54 0.57 1.93 5.58 0.017
Figure 3.12Bandwidth versus inner radius. Each of the nine outer radii has nine varying inner radii
Table 3.5 shows height, Bandwidth and SRRs dimensions. The height of the waveguide is 14.5 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm. 3.98mm 6.8mm8, 0.55
0.95, 0.55 0.087rad 0.087rad
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Figure 3.13 Center frequencies inner radius. Each of the nine outer radii has nine varying inner radii
Table 3.5 shows height, Center frequency and SRRs dimensions. The height of the waveguide is 14.5 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm. 3.98mm 6.8mm8, 0.55
0.95, 0.55 0.087rad 0.087rad
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Figure 3.14Cutoff frequencies versus inner radius. Each of the nine outer radii has nine varying inner radii
Table 3.5 shows height, Cutoff frequencies and SRRs dimensions. The height of the waveguide is 14.5 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm. 3.98mm 6.8mm8, 0.55
30
31
32
33
5.4328125 0.47 2.05645 5.76 0.0151
5.795 0.51 2.0164 5.7 0.0151
6.1571875 0.54 1.97325 5.63 0.0155
6.519375 0.57 1.93605 5.58 0.0159
6.8815625 0.60 1.84425 5.345 0.0155
Figure 3.15 Bandwidth versus inner radius. Each of the nine outer radii has nine varying inner radii
Table 3.6 shows height, bandwidth and SRRs dimensions. The height of the waveguide is 15.25 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm. 4.19mm 7.24mm8, 0.55
0.95, 0.55 0.087rad 0.087rad
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Figure 3.16 Center frequencies versus inner radius. Each of the nine outer radii has nine varying inner radii
Table 3.6 shows height, Center frequencies and SRRs dimensions. The height of the waveguide is 15.25 mm and its width is 35 mm. The relative permittivity of each of the two dielectric side slabs is 4.5 with thickness 1.55mm. 4.19mm 7.24mm8, 0.55
0.95, 0.55 0.087rad 0.087rad
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Figure 3.17 Cutoff frequencies versus inner radius. Each of the nine outer radii has nine varying
Figure 3.17 Cutoff frequencies versus inner radius. Each of the nine outer radii has nine varying