TABLE 1 Gains and Axial Ratios in Broadside Direction for the Different GNSS Standards GNSS Standards Proposed Antenna GPS GLONASS Galileo L1 L2 L5 G1 G2 E5a E5b L1 Gain @zenith >5.4 >5.8 >3.7 >8.2 >5.2 >3.7 >4.4 >5.4 (dBi) AR @zenith <1.2 <2.3 <2.1 <1.2 <2 <2.2 <2.3 <1.2 (dB) 3. CONCLUSIONS In this article, we have presented a dual-band planar GNSS antenna with a wideband circular polarization. The different stand- ards covered are the GPS, GLONASS, and Galileo. The return loss bandwidth is around 22% in the L1 band and 14% in the other band L2. The AR bandwidth is 20% in the first band and 11.3% in the second one. The radiation pattern shows that the antenna has 6 and 8.4 dBi in the normal direction at the central frequencies 1.23 and 1.6 GHz, respectively. The total dimensions of the antenna are 170  190 mm2 with a thickness of 7.651 mm. REFERENCES 1. S. Gao, K. Clarke, M. Unwin, J. Zackrisson, W.A. Shiroma, J. Akagi, K. Maynard, P. Garner, L. Boccia, G. Amendola, G. Massa, C. Underwood, M. Brenchley, and M.N. Sweeting, Antennas for modern small satellites, IEEE Antennas Propag Mag 51 (2009), 40–56. 2. L. Zheng and S. Gao, Compact dual-band printed square quadrifilar helix antenna for global navigation satellite systems receivers, Microwave Opt Technol Lett 53 (2011), 993–997. 3. C.M. Su and K.L. Wong, A dual-band GPS microstrip antenna, A SIMPLE AND EFFICIENT FINITE DIFFERENCE FREQUENCY DOMAIN METHOD FOR THE ANALYSIS AND DESIGN OF LEAKY-WAVE ANTENNAS Feng Xu School of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China; Corresponding author: feng.xu@njupt.edu.cn Received 25 March 2012 ABSTRACT: The finite difference frequency domain method for the analysis of periodic structures will lead to a large-scale nonsymmetrical generalized eigenvalue problem, especially when the open periodic structures are analyzed. In this article, the method is greatly simplified, because the large-scale sparse matrix technique and the shift-and-invert Arnoldi technique, which must be applied to significantly reduce the memory need and increase simulation speed, are implemented by means of some existed codes, for example, matlab codes. Besides, the difference equations of this method are much simpler than those of the previous method, because the longitudinal field components do not have to be eliminated to approximately transform the generalized eigenvalue problem to a standard eigenvalue problem. As a result, the method proposed here is quite simple and efficient for the analysis of leaky- wave antenna, because a leaky-wave antenna can be regarded as direct geometrical evolution from a guided-wave structure that permits to leak energy along its longitudinal direction. VC 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2814–2817, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27216 Key words: Arnoldi techniques; finite difference frequency domain method; large nonsymmetrical generalized eigenvalue problem; leaky-wave antenna; periodic guided-wave structure Microwave Opt Technol Lett 33 (2002), 238–240. 1. INTRODUCTION 4. A.A. Heidari, M. Heyrani, and M. Nakhkash, A dual-band circu- larly polarized stub loaded microstrip patch antenna for GPS appli- cations, Prog Electromagn Res 92 (2009), 195–208. 5. A.M. El-Tager, M.A. Eleiwa, and M.I. Salama, A circularly polar- ized dual-frequency Square Patch Antenna for TT&C satellite applications, Progress in Electromagnetic Research Symp, Beijing, China, 2009, pp. 523–527. 6. G. Beddeleem, J.M. Ribero, G. Kossiavas, R. Staraj, and E. Fond, Dual-frequency circularly polarized antenna, Microwave Opt Tech- nol Lett 50 (2008), 177–180. 7. S. Maci, G. Biffi Gentili, P. Piazzesi, and C. Salvador, Dual-band slot-loaded patch antenna, IEE Proc Microwave Antennas Propag 142 (1995), 225–232. 8. X.F. Peng, S.S. Zhong, S.Q. Xu, and Q. Wu, Compact dual-band GPS microstrip antenna, Microwave Opt Technol Lett 44 (2005), 58–61. 9. G. Zhao, Y.C. Jiao, X. Yang, C. Lin, and Y. Song, Wideband cir- cularly polarized microstrip antenna using broadband quadrature power splitter based on metamaterial transmission line, Microwave Opt Technol Lett 51 (2009), 1790–1793. 10. I.-H. Lin, M. De Vincentis, C. Caloz, and T. Itoh, Arbitrary dual- band components using composite right/left handed transmission lines, Microwave Theory Tech 52 (2004), 1142–1149. 11. C.H. Tseng and C.L. Chang, A broadband quadrature power split- ter using metamaterial transmission line, IEEE Microwave Wireless Compon Lett 18 (2008), 25–27. 12. C. Collado, A. Grau, and F. De Flaviis, Dual-band planar quadra- ture hybrid with enhanced bandwidth response, Microwave Theory Tech 54 (2006), 180–188. 13. Agilent Technologies Inc., Palo Alto, CA, Nov. 2004. VC 2012 Wiley Periodicals, Inc. The determination of dispersion characteristics of a periodic guided-wave structure is a very important subject in practical RF engineering design. In slow-wave structures, backward-wave oscillators, corrugated antennas, and leaky-wave antennas, peri- odic structures have been measured and investigated by many researchers due to their importance [1–5]. Accurate prediction of the propagation behavior of periodic structures is an essential step in the successful design of these systems. Although a lot of commercial software can be used to ana- lyze the periodic structures, the complex propagation constants cannot be obtained directly, because there is no accurate single periodic structure model. Usually, twice simulations for the same periodic structure but with different lengths should be done. After that, the accurate propagation constants can be obtained from these parameters using the so-called numerical calibration technique. Some commercial software provides sin- gle periodic structure model, for example, the eigenvalue solu- tion (used for resonant cavities) of HFSS. Using this solution and combining the equivalent resonant cavity model [6, 7], in some cases, one can accurately extract the propagation constant. However, when the attenuation constant is large, for example, the case of the leaky-wave antennas, this solution is not accu- rate. Especially, when the phase between two periodic planes is close to 180, the solution will crash down. On the other side, a variety of numerical techniques have been used to analyze periodic guided-wave structures. Among them, the finite difference frequency domain (FDFD) method is one of the most versatile techniques for solving partial differen- tial equations, and it has been used to analyze modal 2814 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 12, December 2012 DOI 10.1002/mop 

characteristics of arbitrarily periodic guided-wave structures [7– 9]. However, even if the simplest FDFD method, the large-scale sparse matrix technique and the shift-and-invert Arnoldi tech- nique should be used to significantly reduce the memory need and increase simulation speed. For most designers and research- ers, it is a burden to learn this kind of complicated mathematics, which they do not need to use it in their research. In this article, the large-scale sparse matrix technique and the shift-and-invert Arnoldi technique are implemented using existed codes (matlab codes are used in this article). As a result, the complicated mathematic transformation has been removed. The remained job for researchers is just to construct the normal three dimension difference model of the space fields. Moreover, in this method, the longitudinal field components are not elimi- nated as it is done in Refs. 8 and 9, the final difference equa- tions are much simpler, and the boundary conditions can be processed easily. The calculation of complex propagation con- stants is direct and convenient when compared with the method of Ref. 7. 2. FORMULATIONS In a periodic guided-wave structure, when the propagation direc- tion is in z-direction, according to the Floquet’s theorem for per- iodic structures [1], the electromagnetic field components can be expressed as, *ðx; y; z; tÞ ¼ e*ðx; y; zÞeczejxt E *ðx; y; zÞeczejxt *ðx; y; z; tÞ ¼ h H (1a) (1b) where c is the propagation constant (c ¼ a þ jb), e(x,y,z), and h(x,y,z) is a periodic function with respect to z. If the periodic length of a periodic guided-wave structure is d, then e*ðx; y; z þ dÞ ¼ e*ðx; y; zÞ *ðx; y; z þ dÞ ¼ h *ðx; y; zÞ h (2a) (2b) When the frequency is input parameter for solving the com- plex propagation constant, by substituting Eq. (1) into the Max- well’s differential equations, one can get, 0 ¼ ðjxl þ rm cex ¼ ðjxl þ rm cey ¼ ðjxl þ rm x Þhx þ @ey @ez @z @y y Þhy þ @ex @ez @z @x z Þhz þ @ex @ey @y @x chy ¼ ðjxex þ rxÞex þ @hy @hz @z @y chx ¼ ðjxey þ ryÞey þ @hx @hz @z @x 0 ¼ ðjxez þ rzÞez þ @hx @hy @x @y (3a) (3b) (3c) (4a) (4b) (4c) where r and e are the electric conductivity and permittivity of the medium, respectively. rm and l are magnetic conductivity and permeability of the medium, respectively. As shown in Figure 1, the periodic boundary conditions are disposed along z direction. For the open periodic problems, the absorbing boundary conditions can be easily set up. If the number of meshes in z direction is p and the space step in z direction is Dz, from Eq. (2), the periodic boundaries can be written as, e0ðx; y; 0Þ ¼ epðx; y; dÞ hpþ1ðx; y; d þ Dz=2Þ ¼ h1ðx; y; Dz=2Þ  The difference equation of (3a) can be regarded as, (5a) (5b)  c 2 ey  l; m; n 1 2       þ ey l; m; n þ 1 2   l; m; n þ 1 2 l; m þ 1 2 ey ez þ 1 Dz 1 Dy  ¼ ðjxl þ rm  ey   x ðl; m; nÞÞhxðl; m; nÞ  l; m; n 1 2 l; m 1 2   ez ð6Þ ; n ; n Other difference equations are similar with Eq. (6). After that, an eigenvalue problem can be obtained, cBx ¼ Ax (7) where x ¼ [hx, hy, hz, ex, ey, ez]T, if the electromagnetic waves propagate along z direction. 2.1. Structures of Matrices From Eqs. (3) and (7), the matrix A is, 2 6666664 A ¼ H D H D H H H A42 A43 D A51 H A53 H A61 A62 H H H H A15 A16 H A24 H A26 D A34 A35 H H H D H D H 3 7777775 (8) where, D is a diagonal matrix, H is zero matrix, and each block matrix Amn is bidiagonal matrix. Obviously, the matrix A is large sparse matrix. The vector x ¼ [hx, hy, hz, ex, ey, ez]T is always disposed by first starting along the periodic direction (z direction). The matrix B is also a large sparse matrix, H H H H Bh H H H H Bh H H H H H H H H H Be H H H H Be H H H H H H H H H H H (9) where, Be or Bh is also a bidiagonal matrix. If the number of the meshes in z direction equals to p. The core matrix of Be and Bh are, . . 0:5 0:5 0 . .. 0 0 . . . . . . .


0:5 0:5 . . . . . . 0 0 . .. 0 0:5 . . . .


. . . . . . . 0 . . . 0 . . . . . . .


. . . 0:5 0 . . 0 . . . .. . . 0 0:5 0:5


. . . . . . . 0 0:5 0:5 0 . .. 0 . . (10a) (10b) pp pp 2 6666664 B ¼ 2 6666664 6666664 2 core ¼ Bh core ¼ Be 3 7777775 3 7777775 7777775 3 DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 12, December 2012 2815 

Figure 1 Yee meshes of single period As a result, it can be concluded that the matrix B is a large nonsymmetrical sparse singular matrix. By means of the Arnoldi technique, the eigenvalues (complex propagation constants) of this generalized eigenvalue problem can be extracted accurately and quickly. 2.2. Sparse Matrix and Arnoldi Techniques From above, matrix A and B are large-scale sparse matrix. The sparse matrix techniques should be used to significantly reduce the memory need. To simplify this process, some existed subco- des can be applied, for example, the subcode ‘‘sparse’’ of mat- lab software. In this case, one just needs to calculation the non- zero values in matrix A and B and record them in a vector. At the same time, one also needs to record their position informa- tion in the matrices (row number and column number) and save them into the two integer vectors. After that, using the ‘‘sparse’’ in the environment of matlab, the matrix A and B without zero elements can be formed. The Arnoldi method is an efficient procedure for approximat- ing a subset of the eigensystem of the large sparse matrices [10]. The Arnoldi method is a generalization of the Lanczos process and reduces to that method when the matrices are sym- metric. To greatly increase the simulation speed, the shift-and- invert Arnoldi technique must be implemented to directly solve the large nonsymmetrical generalized eigenvalue problem [11– 16]. Similarly, one can also use some existed codes, for exam- ple, the code ‘‘sptarn’’ (related to the shift-and-invert Arnoldi technique) of matlab software to bypass this complicated mathe- matic process. At last, a simple and efficient numerical algo- rithm for the analysis of the leaky-wave antennas can be con- structed in the environment of matlab. Figure 3 Comparison of the angles of maximum radiation of the leaky-wave antenna. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com] 3. NUMERICAL RESULTS To validate the proposed method, a dielectric-inset waveguide leaky-wave is analyzed. In Figure 2, a periodic leaky-wave antenna based on a metal-strip-loaded dielectric inset waveguide is illustrated. This type of open periodic waveguide supports the propagation of leaky-wave modes, which can be used to set up leaky-wave antenna. Due to its desirable properties, the leaky- wave antenna is suitable to be used in millimeter waveband. This leaky-wave antenna has been analyzed using the transverse equivalent network method [17] and full-wave spectral-domain method [18]. Obviously, the FDFD method can also be used to analyze this antenna based on the versatility of the FDFD method. As shown in Figure 2, the wide wall of rectangular wave- guide is a, and the narrow wall is b. The relative dielectric con- stant in the waveguide is 9.0. One of the narrow walls is replaced by periodic metal strips. The width of metal strip is w, and periodic length is p. The results of the lowest mode in Ref. 17 are used to compare with the results of the FDFD method. The sizes of one example (Fig. 9 of Ref. 17) are: a ¼ 1.4 mm, b ¼ 1.4 mm, c/a ¼ 2.0, (p w)/p ¼ 0.3. Figure 3 shows the Figure 2 Dielectric-inset waveguide leaky-wave antenna Figure 4 Comparison of the normalized leakage constants of the leaky-wave antenna. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com] 2816 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 12, December 2012 DOI 10.1002/mop 

angle of maximum radiation when frequency changes, which can be calculated according to the equation, 13. M.N. Kooper, H.A. ven der Vorst, S. Poedts, and J.P. Goedbloed, Application of the implicitly updated arnoldi method with a com- plex shift-and-invert strategy in MHD, J Comput Phys 118 (1995), 320–329. (11) 14. R.B. Morgan, On restarting the Arnoldi method for large nonsym- sin hm  b k0 where hm is the angle of maximum radiation, measured from the broadside direction. The backward radiation phenomena can be found from Figure 3. Figure 4 shows the comparison of normal- ized leakage constants. From the figures, the agreement between these results can be observed. 4. CONCLUSIONS technique, A FDFD method combined with the subcodes of matlab is pro- posed to more conveniently analyze the characteristics of the leaky-wave antennas. Using the existed codes related to the large scale sparse matrix technique and the shift-and-invert Arnoldi the complicated mathematic process has been removed from the original FDFD method. As a result, a simple and efficient FDTD method can be constructed for the analysis of the periodic leaky-wave antenna. Besides, the fact that the longitudinal field components are not eliminated in this method makes the construction of the matrices in the model and the implementation of the boundary conditions much eas- ier. The accuracy and efficiency have been verified by the example. REFERENCES metric eigenvalue problems, Math Comput 65 (1996), 1213–1230. 15. D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J Matrix Anal Appl 13 (1992), 357–385. 16. R.B. Lehoucq and D.C. Sorensen, Deflation techniques for an im- plicitly re-started Arnoldi iteration, SIAM J Matrix Anal Appl 17 (1996), 789–821. 17. M. Guglielmi and G. Boccalone, A novel theory for dielectric-inset waveguide leaky-wave antennas, IEEE Trans Antennas Propag 39 (1991), 497–504. 18. J.L. Gomez, F.D. Quesada, and A.A. Melcon, Analysis and design of periodic leaky-wave antennas for millimeter waveband in hybrid waveguide-planar technology, IEEE Trans Antennas Propag 53 (2005), 2834–2842. VC 2012 Wiley Periodicals, Inc. MICROWAVE OSCILLATOR DESIGN USING FOLDED T-TYPE SLOT RESONATOR FOR X-BAND APPLICATIONS Bhanu Shrestha, Ram Krishna Maharjan, and Nam-Young Kim Department of Electronic Engineering, Kwangwoon University, 447-1 Wolgye-dong, Nowon-gu, Seoul 139-701, Korea; Corresponding author: bnu@kw.ac.kr 1. R.E. Collin, Field theory of guided waves, McGraw-Hill, New York, 1960. Received 25 March 2012 2. P.J.B. Clarricoats, M.I. Sobhy, Propagation behavior of periodically loaded waveguides, Proc Inst Elect Eng 115 (1968), 652–661. 3. W. Platte, Microwave measurements of effective dielectric constant of semiconductor waveguides via periodic-structure photoexcita- tion, IEEE Trans Instrum Meas 46 (1997), 717–721. 4. Md.R. Amin, K. Ogura, H. Kitamura, K. Minami, T. Wannabe, Y. Carmel, W. Main, J. Weaver, W.W. Destler, and V.L. Granatstein, Analysis of the electromagnetic waves in an overmoded finite length slow wave structure, IEEE Trans Microwave Theory Tech 43 (1995), 815–822. 5. A. Zeid and H. Baudrand, Electromagnetic scattering by metallic holes and its applications in microwave circuit design, IEEE Trans Microwave Theory Tech 50 (2002), 1198–1206. 6. F. Xu, A. Patrovsky, and K. Wu, Fast simulation of periodic guided-wave structures based on commercial software, Microwave Opt Technol Lett 49 (2007), 2180–2182. 7. F. Xu, K. Wu, and W. Hong, Equivalent resonant cavity model of periodic guided-wave structures and its application in finite differ- ence frequency domain algorithm, IEEE Trans Microwave Theory Tech 55 (2007), 697–702. 8. F. Xu, Y. Zhang, W. Hong, K. Wu, and T.J. Cui, Finite-difference frequency-domain algorithm for modeling guided-wave properties of substrate integrated waveguide, IEEE Trans Microwave Theory Tech 51 (2003), 2221–2227. 9. F. Xu, L. Li, K. Wu, S. Delprat, and M. Chake, Parameter extrac- tion of interdigital slow-wave coplanar waveguide circuits using fi- nite difference frequency domain algorithm, Int J RF Microwave Comput-Aid Eng 18 (2008), 250–259. 10. W.E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalues problem, Quart Appl Math 9 (1951), 17–29. 11. Y. Saad, Numerical methods for large eigenvalue problems, Man- chester University Press in Algorithms and Architectures for Advanced Scientific Computing, 1992. 12. Z. Jia and Y. Zhang, A refined shift-and-invert Arnoldi algorithm for large unsymmetric generalized eigenproblems, Comput Math Appl 44 (2002), 1117–1127. ABSTRACT: In this article, Microwave oscillator was designed, fabricated, and characterized using the folded T-type slot resonator for X-band applications. The miniaturized resonator was designed by terminating the slot lines at both ends of the T-structure. The simulation and experimental results were compared and found a closed agreement between them. The oscillator using the folded T-type slot resonator showed phase noise performances of 105.54 dBc/Hz at a 100 kHz offset from the carrier frequency of 8.5 GHz with an output power of 14.06 dBm. The oscillator exhibited better phase noise characteristic due to the high unloaded Q value of the folded T-type slot resonator, which is considerably higher than that of miniaturized spurline resonators. The figure of merit of the oscillator achieved a 183.34 dBc/Hz. The feasibility of integrating that slot resonator in various monolithic microwave integrated circuit technologies can also be expected. VC 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2817–2821, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27215 Key words: oscillator; slot resonator; microstrip resonator; folded T-type resonator; microwave oscillator 1. INTRODUCTION The phase noise characteristic is very important in a microwave oscillator. The communication system can have a good perform- ance, if the oscillator has an excellent phase noise characteris- tics. Many researchers are trying to obtain a low phase noise using high Q-factor-resonators with different configurations. These systems also demand a low phase noise, low power, and low cost materials. To achieve a low phase noise, a high value resonator-Q factor or a loaded Q is required [1]. There are vari- ous types of resonators used in oscillators that enable a low phase noise and a high Q value, such as spurline resonators [1–6], hairpin resonators, and dielectric resonators. In the case DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 12, December 2012 2817