Periodic structures of transmission lines have several applications in microwave technology. Due to complicated equations involved with terminated finite lossless periodic structures, a graphical solution assisting in the analysis and design of these structures is necessary. In this study, the T-chart is employed to solve problems associated with these structures, operating in the passband, readily and effectively, especially for unsymmetrical unit cells associated with periodic structures. Only the periodic structures with passive effective characteristic impedances are considered in this paper. It is found that the T-chart provides accurate solutions, compared to analytical solutions of termi- 594MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. [4]. Using the ABCD matrix technique, it can be shown that the equation of the input impedance at each terminal of terminated finite lossless periodic structures is in the same form as that of CCITLs. Due to complicated equations involved with these structures, a graphical solution assisting in analysis and design of these structures is necessary. In this study, the T-chart developed for CCITLs is employed to solve problems associated with terminated finite lossless periodic structures. It is found that the T-chart depends on the phase angle associated with the effective characteristic impedances of periodically loaded transmission lines. It will be shown that the effective characteristic impedances and the effective propagation constant are dependent on various parameters of each unit cell of the periodic structures in a complicated fashion. Once these effective parameters of these structures are known, the theory of CCITLs and the T-chart can be employed to simplify these problems. This paper is organized as follows. Section 2 presents the theory of terminated finite lossless periodic structures in brief. The theory of CCITLs is discussed in section 3. Section 4 illustrates T-Chart solutions of periodic structures. Finally, conclusions are provided in section 5. THEORY OF TERMINATED FINITE LOSSLESS PERIODIC STRUCTURESThis section provides relevant background information of terminated finite lossless periodic structures for developing its graphical solution. It will be useful when applying the T-chart for solving these problems. Figure 1 illustrates a model of an unsymmetrical unit cell associated with lossless periodic structures. The unit cell consists of a length d of reciprocal lossless uniform transmission line loaded with lossless lumped elements (C 1 , L, C, and L 1 ) across the midpoint of the line. Note that this unit cell is similar to that in [1] (see Fig. 1 in [1]), which exhibits NRI as well as negative group delay. The characteristic impedance and the propagation constant of the unloaded transmission line are defined as Z 1 and k, respectively. Assuming operation in the passband, it is found that at the left terminal of the n th unit cell of terminated finite periodic structures, as shown in Figure 2, the phasor voltage and current can be express...

Synthesis Lectures on Antennas will publish 50-to 100-page publications on topics that include both classic and advanced antenna configurations. Each lecture covers, for that topic, the fundamental principles in a unified manner, develops underlying concepts needed for sequential material, and progresses to the more advanced designs. State-of-the-art advances made in antennas are also included. Computer software, when appropriate and available, is included for computation, visualization and design. The authors selected to write the lectures are leading experts on the subject who have extensive background in the theory, design and measurement of antenna characteristics. The series is designed to meet the demands of 21st century technology and its advancements on antenna analysis, design and measurements for engineers, scientists, technologists and engineering managers in the fields of wireless communication, radiation, propagation, communication, navigation, radar, RF systems, remote sensing, and radio astronomy who require a better understanding of the underlying concepts, designs, advancements and applications of antennas.

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seven attenuator sets with the load resistances shown for the FET parameter extraction. The largest four data sets may then be utilized for the final equation solving. Note that the input powerlevel adjustments follow the rules of maximizing N-harmonic levels with higher-order terms in noise-flow levels for N-order coefficient evaluations. SOLUTION ACCURACY EVALUATIONThe extracted coefficients of the FET, shown in Table 2, use the four data sets selected according to the conversion-ratio rules and the additional graphical/numerical evaluations of solution qualities. Figure 2(a) shows the equation plots for the linear coefficients g m1 and g d1 in the four different attenuator cases. The solution quality is determined by the linear independence of the lines, that is, the slopes, and the intersections to the x-axis-the shifts in vertical or horizontal directions of the lines. Following the determining equation in [1], the line slopes are decided by the input/ output voltage ratio changed by the attenuators, while the righthand-side terms, determining the shifts of the lines, are the load resistances for the linear case and the measured harmonic powers for the higher-order coefficients. In summary, the attenuators affect all the behaviors of the lines.The 2 nd -and 3 rd -order extractions can also provide this kind of solution-quality observation, but more than two unknowns are involved ( g m2 , g m1d1 , and g d2 for the 2 nd order; g m3 , g m2d1 , g m1d2 , and g d3 for the 3 rd order), which the plane representation cannot support. Here, we choose g m2/3 and g d2/3 , as shown in Figures 2(b) and 2(c), where the other coefficients are fixed at the extracted values. All the plots in Figure 2 show that the extractions are qualified and robust.The additional solution-quality representations are given quantitatively by using the equation residual errors of the determining equations for more precise and accurate judgments. Table 3 shows the errors in percentages for the extracted values at the seven attenuator-set equations. The larger errors occur at the sets exactly as indicated by the graphical examples that assist users in choosing suitable data. The numerical mean is convenient for the automatic measurements, while the graphical one provide more physical understandings and interpretations. CONCLUSIONThis paper has presented theoretical, graphical and numerical observations for the solution accuracy of FET nonlinear currentcoefficient extractions used in Volterra series analysis. The selection of the attenuator sets is the most important issue that can be assisted by the conversion ratios of the output/input powers for different harmonic contents. The dependencies on the harmonic powers versus the attenuator sets have been demonstrated by the nonlinear current expressions in order to provide users with a clear mathematical interpretation of this phenomenon. The quality of the solution was evaluated by graphical and quantitative means as well as experiments using a commercial MESFET in order to show the validity and r...

The forward-backward method with a novel spectral acceleration algorithm (FB/NSA) has been shown to be an extremely efficient iterative method of moments (MoM) for the computation of scattering from one-dimensional (1-D) perfect electric conducting (PEC) and impedance rough surfaces [1]. The NSA algorithm is employed to rapidly compute interactions between widely separated points in the conventional FB method and is based on a spectral domain representation of source currents and the associated Green's function. For fixed surface roughness statistics, the computational cost and memory storage of the FB/NSA method are ( ) as the surface size increases, where is the total number of unknowns to be solved. This makes studies of scattering from large surfaces, required in low grazing-angle scattering problems, tractable. In this paper, the FB/NSA method is extended to analyze scattering from two-dimensional (2-D) rough surfaces. The NSA algorithm for this case involves a double spectral integral representation of source currents and the 3-D free-space scalar Green's function. The coupling between two spectral variables makes the problem more challenging, and the efficiency improvements obtained for 2-D surfaces are appreciable but not as dramatic as those for 1-D surfaces. However, the computational efficiency of the FB/NSA method for 2-D rough surfaces remains ( ) as one of the surface dimensions increases. Comparisons of numerical results between the conventional FB method and the FB/NSA method for large-scale PEC rough surfaces show that the latter yields identical results to the former with a reduction of CPU time and only a slight increase in memory storage. In addition, the numerical results of FB/NSA method are in good agreement with experimental data obtained from the University of Washington, Seattle, WA.Index Terms-Forward-backward method, novel spectral acceleration algorithm, tough surfaces.

[1] The novel spectral acceleration (NSA) algorithm has been shown to produce an O(N tot ) efficient iterative method of moments for the computation of radiation/scattering from both one-dimensional (1-D) and two-dimensional large-scale quasi-planar structures, where N tot is the total number of unknowns to be solved. This method accelerates the matrix-vector multiplication in an iterative method of moments solution and divides contributions between points into ''strong'' (exact matrix elements) and ''weak'' (NSA algorithm) regions. The NSA method is based on a spectral representation of the electromagnetic Green's function and appropriate contour deformation, resulting in a fast multipole-like formulation in which contributions from large numbers of points to a single point are evaluated simultaneously. In the standard NSA algorithm the NSA parameters are derived on the basis of the assumption that the outermost possible saddle point, f s,max , along the real axis in the complex angular domain is small. For given height variations of quasi-planar structures, this assumption can be satisfied by adjusting the size of the strong region L s . However, for quasi-planar structures with large height variations, the adjusted size of the strong region is typically large, resulting in significant increases in computational time for the computation of the strong-region contribution and degrading overall efficiency of the NSA algorithm. In addition, for the case of extremely large scale structures, studies based on the physical optics approximation and a flat surface assumption show that the given NSA parameters in the standard NSA algorithm may yield inaccurate results. In this paper, analytical formulas associated with the NSA parameters for an arbitrary value of f s,max are presented, resulting in more flexibility in selecting L s to compromise between the computation of the contributions of the strong and weak regions. In addition, a ''multilevel'' algorithm, decomposing 1-D extremely large scale quasi-planar structures into more than one weak region and appropriately choosing the NSA parameters for each weak region, is incorporated into the original NSA method to improve its accuracy.

The forward-backward method with a novel spectral acceleration algorithm (FB/NSA) has been shown to be a highly efficient O(Ntot) iterative method of moments, where Ntot is the total number of unknowns to be solved, for the computation of electromagnetic (EM) wave scattering from both one-dimensional and two-dimensional (2-D) rough surfaces. The efficiency of the method makes studies of backscattering enhancement from moderately rough impedance surfaces at large incident angles tractable. Variations in the characteristics of backscattering enhancement with incident angle, surface impedance, polarization, and surface statistics are investigated by use of the 2-D FB/NSA method combined with parallel computing techniques. The surfaces considered are Gaussian random processes with an isotropic Gaussian spectrum and root-mean-square surface heights and slopes ranging from 0.5 lambda to lambda and from 0.5 to 1.0, respectively, where lambda is the EM wavelength in free space. Incident angles ranging from normal incidence up to 70 degrees are considered in this study. It is found that backscattering enhancement depends strongly on all parameters of interest. America

Impedance matching is a part of the design process for a microwave system to achieve the maximum power delivered to the load.