CHAPTER 3 Sectorial bandgap
3.5 Measurement results
In this section, the measurement results will be shown. The dimensions of the corrugation we fabricate are as followings: dz = 2mm, g=1.6mm, d = 4.5mm, and εrel
,
groove = 4.3. The structure is shown in Figs. 21, and the dispersion diagram retrieved from ACBC method is in Fig.22. Two k-band (from 18GHz to 26.5GHz) horn antennas are used as the source and receiver, and they are put on the corrugation surface. Since the aperture of the horn antenna is tall from the surface, the absorbers are stuck on the apertures and just leave a thin gap in the bottom in order to be closer as a surface wave. In the beginning, two horn antennas are placed face to face, and theoretically, the energy is 100% through so that the value of S21 should be 0dB. But we want the wave to be more like a surface wave propagating on corrugation surface, the absorbers are added to block most areas of the aperture, so the energy is absorbed. The setting framework described above is shown in Fig. 23, and the section view is in Fig.24.As we can see in Fig. 25 that the reference is about -22dB. Then we put the corrugation under theFigure 20 (b): Transmission coefficient results using Rogers RT6010 as the material of the grooves.
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Figure 21(a): Top view of the fabricated corrugation
Figure 21(b): Side view of the fabricated corrugation
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Figure 22: O→Z part of the dispersion diagram
Figure 23: Measurement framework.
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Figure 25: Measurement results for the existence of the corrugation.
Absorber
Horn aperture
Figure 24: Section view
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horns, and it can be observed that there exists a stopband region between 21GHz to 26GHz, for which this region is also the stopband area in dispersion diagram obtained from ACBC method as shown in Fig. 22. Figure 26 shows the comparison as the corrugation rotates 0 degree and 30 degrees. The trends are similar to the simulation results. The larger angles the corrugation rotates, the start frequency of the stopband will be higher, which means that the stopband bandwidth is smaller.
Figure 26: Comparison for the results as the corrugation rotates 0 degree and 30 degrees.
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IV. Relationship between scattering parameters and dispersion diagram
4-1 Introduction
Most of the time, in order to obtain knowledge of just the width and location in the frequency spectrum of bandgaps of periodic structures, the use of two-port scattering parameters is more direct than the dispersion diagram. Nonetheless both approaches have their own benefits. Scattering parameters can characterize the center frequency, bandwidth, and the attenuation level of the stop band, while dispersion diagram can show the relationship between the frequency and wavenumber, further allowing the retrieval of data such as the phase velocity. For most research in the literature, dispersion diagrams are generated by full wave simulators, and when it comes to the measurement stage, the diagrams are difficult to gauge directly, so the scattering parameters are used to explain the dispersion diagram instead.
But scattering parameters are after all only intermediate results, shedding no insights at all aboupt the wavenumber at that frequency. It will be more convenient if we could know the relationship between the scattering parameters and dispersion diagram, and then we could use the directly measurable S-parameters to generate the measured dispersion diagram.
For a normal periodic structure whose equivalent circuit is not complicated, it is easy to get the expression of the wavenumber by ABCD matrix, which will be briefly introduced later.
But for a more complicated periodic structure, the equivalent circuit is complex, resulting in the difficulty to deal with the matrix in the previous method. Since the equivalent circuit is too complex to analyze, it is a good way to treat this complex unit cell as a uniform material. In other words, the scattering parameters are obtained at first, and then we reconsider the unit cell as a uniform material, deriving the relationship between the scattering parameters and wavenumbers, finally substituting the measured scattering parameters into the relationship. In this section, we will show the process of deriving based on a simple theory, and the accuracy for the transformed dispersion diagram is good.
In [16], Pozar analyzed a periodically loaded transmission line, and its equivalent circuit is shown in Fig.27, where d is the distance between two unit cells, and b is the value of susceptance. Since it can be seen as a cascade of identical two port networks, we can use ABCD matrix to relate the voltages and currents on either side of the unit cell:
1
42 Now consider the phase difference between the nth and (n+1)th terminals
1
exp( )
Substituting Eq. (45.a) and Eq. (45.b) into Eq. (42), we can get1
For nontrivial solutions, the determinant of the matrix in Eq. (46) must vanish, leading to
cosh 2
d A D
(47)
For a lossless periodic structure and symmetric network
c o s d A
(48)Figure 27: Equivalent circuit model of an infinite long periodically loaded transmission line.
jb jb jb jb jb
d
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Since we know the relationship between β and matrix element A in Eq. (48), and also the relationship between matrix element A and scattering parameters in two-port networks is already known, we can easily get the dispersion diagram if we have scattering parameters data.
4-2 Theory
The relationship between the scattering parameters and wavenumbers introduced by Pozar is only suitable for simple cases. For a more complex periodic structure, the equivalent circuit is not easy to retrieve. Even though the equivalent circuit can be retrieved, the susceptance would be too complicated, causing difficulties with the calculation using the abovementioned method. To solve this problem, we can think of it from another perspective, which is by treating the complex unit cell as a uniform material whose effective relative permittivity and permeability are εr and μr, respectively, as shown in Fig.28. The representations of the reflection and transmission coefficients are stated as follow:
11 coefficient in Eq. (49) can be represent as
2 Utilizing the formula of sum of the geometric series, which is
0 Now we assume that both medium in the left-hand side and right-hand side of the material are air, resulting in
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45 Substituting Eq. 14(a) to Eq. 14(d) into Eq. (13), we can obtain
12
Similarly, we can obtain the total transmission coefficient
21(1 23) exp( ) 21(1 23) exp( 3 )( 12 23) ...
Again substituting Eq. 54(a) to Eq. 54(d) into Eq. (57), resulting in
2
Eliminating the exponential terms in Eq. (55) and Eq. (58), we can get
2 2 2
12
S
11 12(S
11S
21 1)S
11 0 (59)
Also we can present Γ12 in terms of the intrinsic impedance:
2 1
Substituting Eq. (60) into Eq. (59) results in
2 2
46 exp(-jβl)=z, we can present refractive index as:
" ' structures, it is subject to the criterion dictating proper choices of the signs of η and n, which are:
'
0
andn
" 0
(65)At last, we can retrieve the effective relative permittivity and permeability easily through Eq.
(61) and Eq. (64)
r
,
rn n
(66)The wavenumbers can be obtained from Eq. (65)
0 r 0 r
k
(67) It should be noticed that the retrieved effective relative permittivity and permeability will have variation with the frequency since the input data of scattering parameters are functions of the frequency. Also it is important to know that they are not the only solutions. The reason is that we can notice in Eq. (64), as different integer m is chosen, different n will be obtained, yielding different results in Eq. (66). That is the reason we need to build the conditions of Eq.(65). To make this method reasonable, it is necessary to use another value of m manually if we find that η or n does not satisfy the conditions in Eq. (65).
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4-3 Verification
In the beginning, the rationalities of the retrieved data in Eq. (67) will be examined. We construct a new homogeneous material whose effective permittivity and permeability are the data retrieved by the above mentioned method, and then inspecting the scattering parameters of the new material. Figure 30 shows the comparison between the corrugation and new material for the scattering parameters. The deep of S21 obtained by retrieving method is shallow, which is caused by the limit when setting a dispersive material in the full-wave simulators, but it is clear that the range of the stopband is similar, which means that characteristic of the new material can indeed be equivalent to the one of corrugation.
Two arbitrary cases will be proved, and the dimensions are as following: d = 8mm, dz=1.8mm, g=1.6mm, and the relative permittivity are 10.2 and 6.15, respectively. Figures 31 show the scattering parameters for these two cases, and Figs. 32 are their corresponding dispersion diagrams in ACBC method and retrieving method. We can see that retrieving method matches to the ACBC method almost perfectly. By using this method, we can get not only just the scattering parameters, but also more information which is like phase velocity at measurement stage in the future.
Figure 30: Comparison between the new material and the corrugation for the S21.
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Figure31 (a): Scattering parameters for the relative permittivity of the grooves as 10.2.
Figure 31 (b): Scattering parameters for the relative permittivity of the grooves as 6.15.
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Figure 32(b): Comparison between the TRT method and retrieving method from scattering parameters as the relative permittivity is 6.15.
Figure 32(a): Comparison between the corrected ACBC method and retrieving method from scattering parameters as the relative permittivity is 10.2.
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V. Conclusion
In this thesis, a new approximation to analyze slow waves on planar corrugated surfaces has been proposed. Based on the ACBC, we can derive the characteristic equation of the corrugation via applying the vector potential method. Also compared to the CST software and TRT, it is shown that for the ACBC method, its accuracy is as good as CST and the speed is as fast as TRT, i.e. extracts the advantages from both.
Meanwhile, a novel idea has also been proposed by applying the characteristic equation obtained by ACBC method, which is called sectorial bandgap. Indeed, for most of the periodic structures which are always designed for wider bandgap for any direction, there are potential applications that the bandgap may be needed just in some directions over a certain frequency range. For example, the corrugations may be used for controlling the direction of signal propagation in microwave circuits. Another potential application is the reduction of the cross polarization of some antennas. In Chapter III, we also present a kind of guideline to show how to design a sectorial bandgap corrugation, which cannot be done for other periodic structures without their own characteristic equations.
At last, the relationship of the scattering parameters and the wavenumbers is introduced.
Due to the complex equivalent circuit of the corrugations, the use of ABCD matrix is not proper here. Instead, an interesting aspect based on [17] is shown, which is to take the corrugation as a new whole structure, and the last dispersion diagram transformed from scattering parameters are matched well to the ACBC method.
The work in this thesis is not the end but a pioneer, since we hope to reach the potential application we mentioned above in the future, and also we wish to provide a more accurate guideline to let the designers reach their standard.
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Reference
[1] R. S. Elliott, “On the theory of corrugated plane surfaces,” IRE Trans. Antennas Propag., pp. 71-81, Apr 1954.
[2] R. W. Hougardy and R. C. Hansen, “Scanning surface wave antennas – oblique surface waves over a corrugated conductor,” IRE Trans. Antennas Propag., pp. 370-376, Oct 1958.
[3] P.-S. Kildal, “Definition of artificially soft and hard surfaces for electromagnetic waves,”
Electronic Letters, vol. 24, no. 3, pp. 168-170, Feb. 1988.
[4] P.-S. Kildal, “Artificially soft and hard surfaces in electromagnetics,” IEEE Transactions
on Antennas & Propagation, vol. 38, no. 10, pp. 1537-1544, Oct. 1990.
[5] J.A.Aas and P.-S. Kildal, “Reduction of forward scattering from struts in reflector antennas,” Proc. 18th
European Microwave Conf., Stockholm, Sept. 1988, pp.494-499
[6] F. Yang and Y. Rahmat-Samii, “Electromagnetic Band Gap Structures in AntennaEngineering,” Cambridge RF and Microwave Engineering Series, Cambridge Univ. Press,
Nov 2008.[7] F. Yang and Y. Rahmat-Samii, ”A low-profile circularly polarized curl antenna over an electromagnetic bandgap (EBG) surface,” Microwave Optical Tech. Lett., vol.31, no. 4, 264-7, November 2001.
[8] A. R. Weily, L.Horvath, K. P. Esselle, B. C. Sanders, and T. S. Bird, “A planar resonator antenna based on a woodpile EBG material,” IEEE Transactions on Antennas &
Propagation, vol.53, no. 1, 216-23,2005.
[9] R. Coccioli, F.R. Yang, K.P. Ma, and T. Itoh, “Aperture-coupled patch antenna on UC-PBG substrates,” IEEE Trans. Microwave Theory Tech, vol. 47, 2131-8, 1999.
[10] T. M. Uusitupa, “Usability studies on approximate corrugation models in scattering analysis,” IEEE Trans. Antennas Propag., vol. AP-54, no. 9, pp. 2486-2496, Sep 2006.
[11] H. A. Kalhor, “Approximate analysis of electromagnetic scattering from corrugating conducting surfaces by surface impedance modeling,” IEEE Trans. Antennas Propag., vol.
AP-25, pp. 721-722, Sep 1977.
[12] P.- S. Kildal, A. Kishk, and Z. Sipus, “Asymptotic boundary conditions for strip-loaded and corrugated surfaces,” Microw. Opt. Technol. Lett., vol. 14, no. 2, pp. 99-101, Feb.
1997.
[13] D.Sievenpiper, L.Zhang, R. F. J. Broas, N.G. Alexopolus, and E.
Yablonovitch, ”High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech., vol.47, 2059-74,1999.
[14] Wei-Zhang, Chang-Hong Liang, Tong-Hao Ding, and Bian Wu, “A novel broadband EBG using multi-via cascaded mushroom-like structure,”2009 Asia-Pacific Microwave Conference, Singapore , Dec.2009.
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[15] C. H. Walter, Traveling Wave Antennas, Chapter 6, pp. 259-260, Peninsular Publishing, 1965.
[16] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998.
[17] A. M. NICOLSON and G.F. Ross, “Measurement of the Intrinsic Properties of Materials by Time-Domain Techniques,” IEEE TRANSSACTIONS ON INSTRUMENTATION AND