Figure 1 Current distributions obtained using the new MoM and TL theory for an open-circuited microstrip line TABLE 1 CPU Times for Three Different Methods Method Used 1 2 3 Ž. CPU time s 482 141 48 calculates the inner products using the directly numerical integration of the four-dimensional integrals. In the numeri- cal integration, a 16-point Gauss quadrature integration algo- rithm is used. Because we select the rooftop function as the basis and testing functions, the convolution over the Green’s function and the basis functions in the inner products can be transferred over to the basis and testing functions, and the convolution can be performed analytically. Therefore, the four-dimensional integrals are reduced to double integrals. The second method is based on this result, and uses a 16-point Gauss quadrature integration algorithm to calculate the double integrals. The third method is based on complete Ž analytical integration which is possible due to the nature of . the new closed-form Green’s functions described in Section III. Note that all three methods use the new closed-form Green’s functions described in Section II, but only the last method makes the maximum use of their special nature that leads to complete analytical integration. From Table 1, we can see that, due to the MoM technique in Section III that exploits analytical integration to the full extent, the computational speed has been improved by a factor of about 10 compared with method 1 and a factor of about 3 compared with method 2, which is the most popular method at present. V. CONCLUSION A method to develop new closed-form Green’s function for planar multilayer microstrip structures has been presented. We have demonstrated that, due to the special form of these new Green’s functions, it is possible to derive closed-form expressions for all of the MoM matrix elements by perform- Ž . ing all of the required usually four-dimensional integrations analytically. This approach has improved the computational efficiency significantly without compromising the precision. REFERENCES 1. J.R. Mosig and F.E. Gardiol, General integral equation formula- tion for microstrip antennas and scatterers, Proc Inst Elect Eng Ž . 132 1985 , 424432. 2. Y.L. Chow, J.J. Yang, D.G. Fang, and G.E. Howard, A closed-form spatial Green’s function for the thick microstrip substrate, IEEE Ž . Trans Microwave Theory Tech 39 1991 , 588592. 3. L. Alatan, M.I. Aksun, K. Mahadevan, and T. Birand, Analytical evaluation of the MoM matrix elements, IEEE Trans Microwave Ž . Theory Tech 44 1996 , 519525. 4. Y. Ge and K.P. Esselle, A fast and general complex image method for evaluating Sommerfeld integrals, Microwave Opt Technol Lett Ž . 30 2001 , 2426. 5. Y. Ge and K.P. Esselle, New closed-form Green’s functions for microstrip structures: Theory and results, IEEE Trans Microwave Ž . Theory Tech accepted . 6. M.I. Aksun and R. Mittra, Estimation of spurious radiation from microstrip etches using closed-form Green’s functions, IEEE Trans Ž . Microwave Theory Tech 40 1992 , 20632069.  2002 John Wiley & Sons, Inc. SYNTHESIS OF DIELECTRIC HELIX SUPPORTS FOR WIDEBAND TRAVELING-WAVE TUBES Sun-Shin Jung, 1 Young-Do Joo, 1 S. Ghosh, 2 B. N. Basu, 3 and Gun-Sik Park 1 1 Vacuum Electrophysics Laboratory School of Physics Seoul National University Seoul 151-742, Korea 2 Bharat Electronics Bangalore 560 013, India 3 Centre of Research in Microwave Tubes Department of Electronics Engineering Institute of Technology Banaras Hindu University Varanasi 221005, India Recei  ed 27 July 2001 ABSTRACT: An n-dielectric tube prototype structure consisting of a helix surrounded by adjoining dielectric tubes in an o erall en elope was studied by HFSS for flat-to-negati  e dispersion characteristics needed for wideband tra eling-wa  e tubes. The prototype structure obtained was then ‘‘ synthesized’’ to obtain a wedge-shaped n-step staircase dielectric support structure for a helix.  2002 John Wiley & Sons, Inc. Mi- crowave Opt Technol Lett 32: 231235, 2002. Key words: synthesis; helix support; helix; tra eling-wa  e tubes; HFSS DOI 10.1002  mop.10140 I. INTRODUCTION Ž . Wideband helix traveling-wave tubes TWTs continue to be important for their application in electronic warfare systems. Ž . A vane-loaded helical slow-wave structure SWS is usually employed in a wideband TWT. The structure consists of a helix supported by a number of dielectric rodsbars of circu- lar or rectangular cross section in a metal envelope provided   with metal vanes projecting radially inward 1, 4 . Such a vane-loaded structure, as needed for wideband device perfor- mance, can have flat-to-negative dispersion characteristics at reasonably high values of interaction impedance, a quantity measuring the axial electric fields interacting with a beam for  a given RF power propagating down the structure 5 . How- ever, at high frequencies, when the size of the structure decreases, not only does the fabrication of the structure become difficult, but also the close proximity of the metallic part, namely, vanes, to the helix entails the risk of arcing in the tube. An alternative is found in a structure in which the 231 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 32, No. 3, February 5 2002