INT. J. ELECTRONICS,
2000, VOL. 87, NO. 9, 1083 ± 1094
Finite diŒerence time domain analysis of cylindrical coplanar waveguide circuits NIHAD DIB{} and THOMAS WELLER{ There is a growing interest in using cylindrical transmission line structures in microwave applications. In this paper, the ® nite di erence time domain (FDTD) method is used to characterize several three-dimensional cylindrical coplanar waveguide (CCPW) geometries. Speci® cally, a CCPW series stub and a threesection CCPW ® lter are studied both theoretically and experimentally. Moreover, CCPW gap is characterized and the FDTD results are compared to those obtained using conformal mapping techniques. E ective absorbing boundary conditions are employed to truncate the FDTD mesh at the end walls and the outer radial boundary.
1.
Introduction
Cylindrical multiconductor transmission lines are of interest for many applications, in particular for new types of antennas and their feed lines (Nakatani 1988, Elsherbini et al. 1996, Ho et al. 1997, Tam 1998, Scardelletti et al. 1999). The use of a cylindrical substrate is generally dictated by the physical attributes of the system rather than by choice, since the analysis and fabrication of cylindrical structures are more complicated than for a comparable planar implementation. Needless to say, the design and analysis of passive components on cylindrical substrates is not an easy task, especially when there is no angular symmetry. Until now, the integral equation (IE) technique, solved by the method of moments, has been the most commonly used analysis method in this area (Alexopoulos and Nakatani 1987, Silva et al. 1992, Wong et al. 1993, Chen and Wong 1995 a,b,c, Huang and Wong 1995, Lu and Wong 1995, Tsai and Wong 1995, Chen and Wong 1996, Lu and Wong 1996). In Huang et al. (1993), the frequency domain transmission line matrix (TLM) method has been used to analyse microstrip discontinuities on cylindrical substrates. Recently, a versatile two-dimensional ® nite di erence time domain (FDTD) algorithm has been developed in Dib and Weller (2000) to study the dispersion characteristics of arbitrary cylindrical transmission lines. Moreover, an e cient onedimensional FDTD algorithm has been presented in Chen and Mittra (1997) and Shen et al. (1999) for analysing axisymmetric cylindrical waveguides and optical® bre waveguides. The objective of this paper is to use the three-dimensional (3D) FDTD technique (in cylindrical coordinates) to analyse arbitrary cylindrical transmission line discontinuities. The emphasis will be on studying the newly proposed cylindrical coplanar waveguide (CCPW) (Su and Wong 1996 a,b, Alkan et al. 1998, Dib and Al-Zoubi Received 5 July 1999. Accepted 8 February 2000. { Department of Electrical Engineering, Jordan University of Science and Technology, PO Box 3030, Irbid, Jordan. { Department of Electrical Engineering, University of South Florida, Tampa, FL, USA. } Corresponding author. e-mail:
[email protected] International Journal of Electronics ISSN 0020± 7217 print/ISSN 1362± 3060 online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals
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N. Dib and T . W eller
Figure 1.
Cross section of a cylindrical coplanar waveguide (CCPW).
2000, Dib and Weller 2000) shown in ® gure 1. This line has the potential to be used in antenna and wireless communication applications (Scardelletti et al. 1999). The line parameters, °ef f and Z0 , have been analysed in Dib and Weller (2000) using the two-dimensional FDTD technique. In this paper, several three-dimensional CCPW structures are investigated. Our analysis can accurately model circuits in an open environment due to the use of an absorbing boundary condition to truncate the FDTD mesh at its outer radial boundary. Moreover, the problem of the singularity of the ® elds along the z-axis, which results if the ordinary FDTD equations are directly used to update all ® eld components, is handled by discretizing the integral form of one of Maxwell’s equations. Several examples are included, namely: cylindrical microstrip gap, CCPW gap, a series CCPW stub, and a three-section CCPW band-pass ® lter. The FDTD results for the cylindrical microstrip gap are compared to those published in the literature, while the results for the CCPW gap are compared to those obtained using conformal mapping technique. On the other hand, the FDTD results for the stub and ® lter are compared to measurements. The steps taken to build and measure the CCPW structures under study are also described.
2.
Theoretical formulation
Starting from Maxwell’s equations, in a source-f ree, lossless non-magnetic region, the electric ® eld intensity E and magnetic ® eld intensity H are related as @H 5 £ E ˆ ¡·0 @t @E 5£H ˆ° @t
…1† …2†
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FDT D analysis of CCPW circuits
Expanding the above equations in cylindrical coordinates leads to the equations ¡·0
@Hr 1 @Ez @E¿ ˆ ¡ @t r @¿ @z
…3†
¡·0
@H¿ @Er @Ez ˆ ¡ @t @z @r
…4†
¡·0
@Hz 1 @ @E ˆ …rE¿ † ¡ r @t r @r @¿
…5†
°
@Er 1 @Hz @H ¿ ˆ ¡ @t r @¿ @z
…6†
°
@E¿ @Hr @Hz ˆ ¡ @t @z @r
…7†
°
@Ez 1 @ @H ˆ …rH¿† ¡ r @t r @r @¿
…8†
2.1. FDT D equations Discretizing the six equations (3) ± (8) according to the 3D cell shown in ® gure 2, the following di erence equations are obtained. n‡ 1
n¡1
Hr 2…i; j ;k† ˆ Hr 2…i; j ;k† ‡
‡ n‡ 1
4t Ezn …i;j ¡ 1;k† ¡ Ezn …i;j ;k† ·0 i4¿4r n¡1
H¿ 2…i; j ;k† ˆ H¿ 2…i; j ;k† ¡
‡ n‡ 1
…9†
4t Ern …i;j ;k ‡ 1† ¡ Ern …i;j ;k† 4z ·0
4t Ezn …i;j ;k† ¡ Ezn …i ¡ 1;j ;k† 4r ·0 n¡1
Hz 2…i; j ;k† ˆ Hz 2…i; j ;k† ‡
‡
n n 4t E¿ …i;j ;k ‡ 1† ¡ E¿…i;j ; k† 4z ·0
…10†
4t Ern …i;j ;k† ¡ Ern …i;j ¡ 1;k† ·0 …i ¡ 0:5†4r4¿
4t …i ¡ 1†E¿n …i ¡ 1;j ;k † ¡ iE¿n …i ;j ;k† ·0 …i ¡ 0:5†4r
2
n‡1
n‡ 1
3
…11†
4t 4H¿ 2 …i;j ;k † ¡ H¿ 2…i; j ;k ¡ 1†5 Ern‡1 …i; j ;k† ˆ Ern…i; j ;k† ¡ 4z °…i;j ;k † " n‡1 # n‡1 4t Hz 2 …i;j ‡ 1;k† ¡ H z 2 …i;j ;k† ‡ °…i ;j; k† …i ¡ 0:5†4¿4r
…12†
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N. Dib and T . W eller E¿n‡1
Ezn‡1
" n‡1 # n‡1 4t H r 2 …i;j ;k† ¡ Hr 2 …i;j ;k ¡ 1† …i;j ;k† ˆ …i;j ;k† ‡ 4z °…i;j ;k † " n‡1 # n‡ 1 4t Hz 2 …i ‡ 1;j ; k† ¡ Hz 2…i; j ;k† ¡ 4r °…i; j ;k† E¿n
Ezn
…i;j ;k† ˆ
‡
4t …i;j ;k† ‡ °…i ;j ;k†
n‡1
…13†
n‡ 1
Hr 2 …i;j ;k† ¡ Hr 2…i; j ‡ 1;k† i4r4¿
n‡1
n‡1
…i ‡ 0:5†H¿ 2 …i ‡ 1;j ;k† ¡ …i ¡ 0:5†H¿ 2 …i;j ;k† i4r
…14†
In the above, i goes from 1 to Nr , j goes from 1 to N¿ , and k goes from 1 to Nz , where Nr , N¿ , and Nz are the number of cells in the radial r-direction, ¿-direction and z-direction, respectively. For numerical stability of the di erence equations, the following condition should be imposed on the time step 4t (Chen and Fusco 1994): 1 c4t µ q 2 2 2 = = = …1 4r† ‡ …2 4r4¿† ‡ …1 4z†
…15†
where c is the velocity of light in free space.
2.2. Singularities along the z-axis It can be seen from ® gure 2 that the ® eld components E¿ , Ez and Hr must be evaluated along the z-axis (at r ˆ 0). If equations (9) and (14) are directly used to compute Hr…r ˆ 0† and Ez…r ˆ 0† then singularities will exist. Such singularities must be removed since the ® elds there should be ® nite in both the time and frequency domains.
Figure 2.
A typical 3D-FDTD cell in cylindrical coordinate system.
FDT D analysis of CCPW circuits
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It can be noticed from (10) and (11) that only Ez and E¿ along the z-axis are needed to update the adjacent H¿ and Hz ® elds. Hence, H r along the z-axis is not needed. Moreover, to compute Hz …1; j ;k † using (11), E¿…0; j ;k † at r ˆ 0 is multiplied by a factor …i ¡ 1† which is zero-valued since i ˆ 1. Thus, E¿ along the z-axis is also not needed. Finally, only Ez at r ˆ 0 is needed to update all the other relevant ® elds. To determine Ez , the following integral form of Maxwell’s equation in time domain is used ‡
C
- H ¢ d` ˆ °
…
S
@E ¢ ds @t
…16†
where C is a closed contour around the z-axis and S is the surface bounded by this contour. Using the closest closed path around the z-axis, i.e. the one at a distance of …4r=2†, as our contour C, the following FDTD equation for Ez at r ˆ 0 can be obtained Ezn‡1 …0; j ; k† ˆ Ezn…0; j ;k † ‡
N¿
44t X n‡12 H …1;p;k † °N¿ 4r pˆ1 ¿
…17†
where ° is the permittivity of the medium just around the z-axis. Once Ez along the z-axis is known, the rest of the ® eld components can be computed using (9) ± (14).
2.3. Absorbing boundary conditions To be able to analyse open structures and account for radiation, a suitable absorbing boundary condition (ABC) needs to be used to truncate the mesh at its outer radial boundary. In this paper, the following dispersive boundary condition (DBC) is used (Chen and Fusco 1994) @ 1 @ 1 ‡ ‡ ‡ ¬1 @r v1 @t 2R
@ 1 @ 1 ‡ ‡ ‡ ¬2 Etang ˆ 0 @r v 2 @t 2R
…18†
where Etang is the tangential electric ® eld component at the outer radial boundary r ˆ R, v 1 and v 2 are two di erent wave velocities, and ¬1 and ¬2 are arbitrary scalar constants. In our analysis, we set v 1 ˆ v 2 ˆ c, where c is the speed of light in free space. The ® nite di erence equivalent of the above ABC, with v 1 ˆ v 2 ˆ c and ¬1 ˆ ¬2 ˆ 0, can be written as (Chen and Fusco 1994) n ¡ 2C3 Emn¡1 Emn ˆ 2C1 Emn ¡1 ¡ C12 Em¡2
n¡1 ‡ 2…C2 ‡ C1 C3 † Em¡1 ¡ 2C1 C2 Emn¡1 ¡2 2 n¡2 ¡ C32 Emn¡2 ‡ 2C2 C3 Emn¡2 ¡1 ¡ C2 Em¡2
where
…19†
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N. Dib and T . W eller C1 ˆ ‰ 4c4tr…m ¡ 1† ¡ 44rr…m ¡ 1† ¡ c4r4tŠ=D C2 ˆ ‰ 4c4tr…m ¡ 1† ‡ 44rr…m ¡ 1† ¡ c4r4tŠ=D C3 ˆ ‰ 4c4tr…m † ¡ 44rr…m † ‡ c4r4tŠ=D Emn
D ˆ 4c4tr…m † ‡ 44rr…m † ‡ c4t4r
…20†
Here, is the tangential electric ® eld components, i.e. E¿ and Ez , at the outer radial boundary r ˆ r…m † ˆ Nr 4r. Since 3D structures are of interest, then a suitable ABC should also be used at the front and back walls of the mesh in order to absorb the input Gaussian pulse. In our analysis, the super-absorbing ® rst-order Mur boundary condition is used (Mei and Fang 1992). Such an absorber has been widely used in cartesian FDTD algorithms and found to be very e ective. It should be mentioned that the metal strips are assumed to be of zero thickness and are dealt with by setting the tangential electric ® eld components that lie on the strip to zero at all time steps. In addition, at an interface between two di erent dielectrics, the average value of the two relative dielectric constants is used to update E¿ and Ez in (13) and (14). Moreover, a Gaussian pulse is used as an excitation since it is smoothly varying in time, and its Fourier transform is also a Gaussian function centred at zero frequency. Finally, the scattering parameters are obtained using the procedure outlined in Sheen et al. (1990). 3.
Results and discussion
In order to test the developed cylindrical 3D-FDTD algorithm, a variety of cylindrical microstrip structures have been initially investigated and the results were compared with those obtained using other numerical methods. Speci® cally, the cylindrical microstrip open-end and gap discontinuities studied in Lu and Wong (1995), and Chen and Wong (1995 a,b) using the IE technique were analysed using the FDTD method. The results obtained were in very good agreement with the published IE results. Figure 3 shows the FDTD scattering parameters for a cylindrical microstrip gap discontinuity compared to those published in Chen and Wong (1995 a). There is a very good agreement in S 11 while a slight discrepancy appears in S 21. This might be due to the fact that in Chen and Wong (1995 a), the transverse component of the current on the strips is neglected and basis functions for the longitudinal component only are used in the method of moments solution. Several CCPW lines and discontinuities have been fabricated and measured. The realization of geometries with a small radius of curvature was achieved by patterning the circuits on a 3 mm thick copper-clad Te¯ on substrate using photolithographic techniques. The substrate is mounted to a ¯ at support, and FeCl etchant is used to generate the patterns. After etching, the thin substrate is transferred to a Te¯ on rod or tube of the desired dimensions. SMA-type connectors are then mounted to the ends of the rod, and connected to the transmission line by soldering. Two of the CCPW structures that have been characterized, both theoretically and experimentally, are a series short-end stub printed in the ground plane and a threesection bandpass ® lter, illustrated in ® gures 4 and 5, respectively. The ® lter is comprised of three series open-end stubs (printed in the centre conductor) cascaded in series. It should be noted that planar views are shown in ® gures 4 and 5 only. Actually, these structures wrap around a cylindrical substrate as shown in ® gure 1.
FDT D analysis of CCPW circuits
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Figure 3.
Scattering parameter vs. frequency for a cylindrical microstrip gap discontinuity. R ˆ a=b ˆ 0:8, °r ˆ 9:9, h ˆ b ¡ a ˆ S ˆ 0:635 mm, G ˆ 0:8 h.
Figure 4.
A series short-end stub printed in the ground plane. The centre conductor width is 2.0 mm, all slots and gaps are 0.4 mm, and the stub length (L ) is 14.0 mm.
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Figure 5.
N. Dib and T . W eller
A three-section band-pass ® lter. The centre conductor width is 2.0 mm, all slots and gaps are 0.4 mm, and the stub length (L ) is 14.0 mm.
The theoretical and experimental results for the series stub are given in ® gure 6. Except for the discrepancies around 9 GHz, which are due to calibration error, the agreement is very good. Similar agreement is obtained for the ® lter, as shown in ® gure 7. The FDTD parameters used to model both structures are as follows: 4r ˆ 0.3048 mm, 4z ˆ 0.4 mm, Nr ˆ 41, N¿ ˆ 100, and 4t ˆ 3.1256 £ 10¡14 s. As another example, ® gure 8 shows a CCPW gap which can be also used to build ® lters by cascading several gaps in series. Recently, closed form expressions for the capacitances of the equivalent pi-network of a planar CPW gap have been derived in Gevorgian et al. (1996) using conformal mapping method (CMM). Using the same method, closed form expressions for the capacitances of the equivalent pi-network of the CCPW gap have been also derived in Al-Zoubi and Dib (1999). Assuming that the values of these capacitances are frequency independent, the scattering parameters of the CCPW gap as a function of frequency can be obtained using circuit theory. Figure 9 shows the FDTD results compared to those obtained using CMM for three
Figure 6.
Measured and FDTD S-parameters of a CCPW series short-end stub. °r ˆ 2, a ˆ 0, b ˆ 6:4 mm.
FDT D analysis of CCPW circuits
Figure 7.
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measured and FDTD S-parameters of the three-section CCPW ® lter. °r ˆ 2, a ˆ 0, b ˆ 6:4 mm.
Figure 8.
Cylindrical coplanar waveguide gap discontinuity.
di erent gap widths. The agreement is very good except around 11.5 GHz and 18.5 GHz. These two frequencies were identi® ed to be the cuto frequencies of the T M01 and T M11 modes that are excited in a circular waveguide completely ® lled with °r ˆ 2.0 dielectric material. This could be caused by the longitudinal electric ® eld component that exists in the gap between the two centre conductors. It is interesting to note that the dominant T E11 mode, which has a cuto frequency of 9 GHz, has no e ect on the gap characteristics. It can be also seen that as the gap width G increases, the insertion loss increases (i.e. S 21 decreases). This is expected since the coupling between the two lines decreases as G increases.
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N. Dib and T . W eller
(a)
(b)
(c) Figure 9. Scattering parameters for the CCPW gap compared to results obtained using the conformal mapping method (CMM). W ˆ 0:4 mm, S ˆ 2:0 mm, a ˆ 0:0; b ˆ 7:0 mm, °r ˆ 2:0, (a) G ˆ 0:4 mm; (b) G ˆ 0:8 mm, (c) G ˆ 1:2 mm.
FDT D analysis of CCPW circuits 4.
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Conclusions
A three-dimensional ® nite di erence time domain algorithm in cylindrical coordinates has been developed. Absorbing boundary conditions are used to truncate the mesh at its outer radial boundary, and the front and back walls. The algorithm has been validated by comparing the results for several representative cylindrical structures to published numerical data and the experimental results contained herein. The speci® c focus of this paper has been cylindrical coplanar waveguide geometries. It is shown that CCPW ® lters can be realized easily on cylindrical substrates.
References A LEXOPOULOS, N., and N AKATANI, A., 1987, Cylindrical substrate microstrip line characterization. IEEE Transactions on Microwave T heory and T echniques, 35, 843± 849.
A LKAN, M ., G ORUR, A., and K ARPUZ , C., 1998, Quasistatic analysis of cylindrical coplanar
waveguide with multilayer dielectrics. International Journal of RF and Microwave CAE, 8, 303± 314. A L-Z OUBI, A., and D IB, N ., 1999, CAD model of a gap in cylindrical coplanar waveguide. Electronics L etters, 35, 1857± 1858. C HEN, H ., and WONG, K ., 1995 a, Characterization of cylindrical microstrip gap discontinuities. Microwave and Optical T echnology L etters, 9, 260± 263. C HEN, H ., and WONG, K ., 1995 b, Analysis of microstrip open-end and gap discontinuities on a cylindrical body. IEEE 1995 Antennas and Propagation Society Symposium Digest, 1784± 1787. C HEN, H ., and WONG, K ., 1995 c, Characterization of coupled cylindrical microstrip lines mounted inside a ground cylinder. Microwave and Optical T echnology L etters, 10, 330± 333. C HEN, H ., and WONG, K ., 1996, A study of the transverse current contribution on the characteristics of a wide cylindrical microstrip line. Microwave and Optical Technology L etters, 11, 339± 342. C HEN, Q., and F USCO, V., 1994, Three dimensional cylindrical coordinate ® nite di erence time domain analysis of curved slotline. Proceedings of the Second International Conference on Computations in Electromagnetics, pp. 323± 326. C HEN, Y., and M ITTRA, R ., 1997, A highly e cient ® nite di erence time domain algorithm for analyzing axisymmetric waveguides. Microwave and Optical T echnology L etters, 13, 201± 203. D IB, N., and A L-Z OUBI, A., 2000, Quasi-static analysis of asymmetrical cylindrical coplanar waveguide with ® nite-extent ground. International Journal of Electronics, 87, 185± 189. D IB, N., and WELLER, T., 2000, Two dimensional ® nite di erence time domain analysis of cylindrical transmission lines. International Journal of Electronics, 87, 1065± 1081. E LSHERBINI, A., YING, Y., SMITH, C., and P EREYRA, V., 1996, Finite di erence analysis of cylindrical two conductor microstrip transmission line with truncated dielectrics. Journal of Franklin Institute, 333B, 901± 928. G EVORGIAN, S., D ELENIV, A., M ARTINSSON, T., G ALCHENKO, S., L INNER, P., and VENDIK, I., 1996, CAD model of a gap in coplanar waveguide. International Journal of Microwave and Mm-W ave CAE, 6, 369± 377. H O, C., SHUMAKER, P., SMITH, K ., H UANG, D., L IAO, J., and WANG, Y., 1997, Microstrip-fed cylindrical slot antennas for GPS avionics applications. 1997 IEEE Microwave T heory and T echniques Society Symposium Digest, Denver, CO., USA, pp. 1755± 1758. H UANG, C., and WONG, K ., 1995, Analysis of a slot-coupled cylindrical rectangular microstrip antenna. Microwave and Optical T echnology L etters, 8, 251± 253. H UANG, J., VAHLDIECK, R ., and JIN, H ., 1993, Microstrip discontinuities on cylindrical surfaces. 1993 IEEE Microwave T heory and T echniques Society Symposium Digest, pp. 1299± 1302. LU, J., and WONG, K ., 1995, Equivalent circuit of an inside cylindrical microstrip gap discontinuity. Microwave and Optical Technology L etters, 10, 115± 118.
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L U, J., and WONG, K ., 1996, Analysis of slot-coupled double-sided cylindrical microstrip lines. IEEE T ransactions on Microwave T heory and T echniques, 44, 1167± 1170.
M EI, K ., and FANG, J., 1992, Super-absorbtionÐ A method to improve absorbing boundary conditions. IEEE T ransactions on Antennas and Propagation, 40, 1001± 1010.
N AKATANI, A., 1988, Microstrip circuits and antennas on cylindrical substrates. PhD thesis, University of California, Los Angeles.
SCARDELLETTI, M ., WELLER, T., and D IB, N., 1999, CPW-fed slot antennas on cylindrical substrates. Submitted to IEEE T ransactions on Antennas and Propagation.
SHEEN, D., A LI, S., A BOUZAHRA, M ., and K ONG, J., 1990, Application of the three-dimensional ® nite-di erence time-domain method to the analysis of planar microstrip circuits. IEEE T ransactions on Microwave T heory and T echniques, 38, 849± 857. SHEN, G ., C HEN, Y., and M ITTRA, R ., 1999, A nonuniform FDTD technique for e cient analysis of propagation characteristics of optical-® ber waveguide. IEEE T ransactions on Microwave Theory and T echniques, 47, 345± 349. SILVA, F., F ONSECA, S., S OARES, A ., and G IAROLA, A., 1992, E ect of dielectric overlay in a microstrip line on circular cylindrical surface. IEEE Microwave and Guided W ave L etters, 2, 359± 360. SU, H ., and WONG, K ., 1996 a, Full-wave analysis of the e ective relative permittivity of a coplanar waveguide printed inside a cylindrical substrate. Microwave and Optical T echnology L etters, 12, 94± 97. SU, H ., and WONG, K ., 1996 b, Dispersion characteristics of cylindrical coplanar waveguide. IEEE T ransactions on Microwave T heory and T echniques, 44, 2120± 2122. TAM, W., 1998, Stripline-fed cylindrical slot antenna. International Journal of RF and Microwave CAE, 8, 33± 41. T SAI, R ., and WONG, K ., 1995, Characterization of cylindrical microstriplines mounted inside a ground cylindrical surface. IEEE T ransactions on Microwave Theory and T echniques, 43, 1607± 1610. WONG, K ., C HENG, Y., and R OW, J., 1993, Resonance in a superstrate-loaded cylindrical rectangular microstrip structure. IEEE T ransactions on Microwave T heory and T echniques, 41, 814± 819.
