respectively. Note that all YOSC pass through the A (-1,0) in Figure 4, which corresponds the intersection of the r = 0 circle and x = 0 circle; actually it is point, which defines the short stud position. If p = 1, the open circuit point will on the circle || = 1. However, whatever p is, the open circuit point and the match point are in the line  i = tan 2 -  1  r . The distance between the open circuit point and the match point is 1/p. The other OSC characteristics are similar with the above example of OSC. Finally, OSC can be obtained by superimposing ZOSC and YOSC, as shown in Figure 6. 5. CONCLUSION In this article, we analyzed a novel general nonreciprocal lossy transmission line and constructed a graphical tool called OSC for solving this “real general” transmission line problems. Although OSC is more complicated than the standard Smith chart, it can be used to solve transmission lines problems in most conditions. It is also proved that all kinds of previously introduced Smith chart can be classified as one subclass of OSC. So, OSC is the most gener- alized and integrated Smith chart in some sense. Applications of OSC in the analysis and design of general transmission line models will be further investigated in the future. ACKNOWLEDGMENT The authors thank the instructor Li Shulan of School of Electronic Engineering for helpful discussions and express their gratitude to the financial support of Specialized Research Fund for the Doc- toral Program of China Higher Education (No.20030013010). REFERENCES 1. P.H. Smith, Book review: Electronic applications of the smith chart, Microwave J 39 (1996), 178. 2. D. Torungrueng and C. Thimaporn, A generalized zy smith chart for solving nonreciprocal uniform transmission-line problems, Microwave Opt Technol Lett 40 (2004), 57-61. 3. E. Gago-Ribas, C. Dehesa-Martinez, and M.J. Gonzalez-Morales, Com- plex analysis of the lossy-transmission line theory: A generalized smith chart, Turk J Electr Eng Comput Sci 14 (2006), 173-194. 4. F. Urbani, L. Vegni, and A. Toscano, Generalized smith chart for an exponential tapered nonuniform transmission line, Microwave Opt Technol Lett 14 (1997), 36-39. 5. D. Torrungrueng, C. Thimaporn, and N. Chamnandechakun, An appli- cation of the t-chart for solving problems associated with terminated finite lossless periodic structures, Microwave Opt Technol Lett 47 (2005), 594-597. 6. D. Torrungrueng and C. Thimaporn, Application of the t-chart for solving exponentially tapered lossless nonuniform transmission-line problems, Microwave Opt Technol Lett 45 (2005), 402-406. © 2007 Wiley Periodicals, Inc. A NOVEL EXPRESSION FOR EFFECTIVE RADIUS IN CALCULATING THE RESONANT FREQUENCY OF CIRCULAR MICROSTRIP PATCH ANTENNAS Ali Akdagli Department of Electrical and Electronics Engineering, Faculty of Engineering, Mersin University, 33343, Ciftlikkoy, Mersin, Turkey; Corresponding author: akdagli@mersin.edu.tr Received 3 March 2007 ABSTRACT: This letter presents a novel effective radius expression for determining the resonant frequency of circular microstrip patch antennas. The effective patch radius expression is constructed by us- ing differential evolution algorithm and is valid for both electrically thin and thick circular microstrip antennas (MSAs). It is well suited to be used in the development of fast computer aided design algo- rithms. Numerical results obtained for the resonant frequencies of circular MSAs are in good agreement with the experimental results reported by several scientists. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2395–2398, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop. 22767 Key words: circular microstrip antennas; effective radius; resonant frequency; differential evolution 1. INTRODUCTION A growing interest has been observed in the use of microstrip antennas (MSAs) since they are small, low cost, light weight, reproducibility, reliability, ease of fabrication, practical to employ on vehicles, and easy for integration with microwave integrated circuits or monolithic microwave integrated circuit components. Therefore, numerous works have been devoted to the character- ization of these structures [1–28]. MSAs have been used in various configurations: square, rectangular, circular, trapezoidal, and ellip- tical. A circular microstrip patch resonator can be used either as an antenna or as a component of oscillators and filters in microwave integrated circuits. Since the bandwidth of MSAs around their operating resonant frequencies is very narrow, it is important to develop accurate expressions for the calculation of those resonant frequencies. This is the fact that increasing use of MSAs in the electronic commu- nication market requires simple models to analyze their perfor- mance. This letter attempts to develop a simple closed-form ex- pression for the effective patch radius of a circular MSA (Fig. 1), which can be readily used by the MSA designer without any Figure 6 OSC (superimposed) for  1 = - /8,  2 = /6, p = 2. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com] DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 10, October 2007 2395