MMIC, with chip size of 1.5  1 mm. 4. MEASUREMENT The measurement is performed by putting the VCO MMIC chip into a test fixture, as shown in Figure 4. A test board is used to supply DC power to the chip and to output the oscillating signal through the SMA connector. There are two capacitors (100 pF and 10 F, respectively) connected in parallel to each DC supply line, which is used to bypass the undesired RF signals. The oscillating curve is displayed on an MS288C spectrum analyzer. The mea- surement results are shown in Figures 5 and 6. The operation condition for this VCO MMIC measurement is: V d = 3.0 V, I ds = 18 mA. Figure 5 shows a typical oscillating signal with oscillating frequency of 25.0232 GHz, and Figure 6 shows the measured oscillating frequency and the associated output-power tuning curves. The designed VCO MMIC has a tuning range of 1.4 GHz at around 25.7 GHz with an output power of about 8 dBm. As mentioned above, the VCO MMIC was measured by insert- ing it into the test fixture shown in Figure 4. Clearly, the measured output power will be less than that released by the chip itself, due to the loss of output microstrip and the connector in the test ficture. On-wafer measurement result shows a loss of about 3 dBm for them; so, the output power released by the chip itself should be about 11 dBm. 5. CONCLUSION An active-biased 26-GHz VCO MMIC, based on a 0.25-m GaAS pHEMT process, has been presented in this paper. Its tuning range is 1.4 GHz with a central frequency of 25.7 GHz, and the output power is 11  1 dBm. The fabricated VCO MMIC chip size is 1.5  1.0 mm. ACKNOWLEDGMENTS This work was financially supported by the Chinese National Hi-Tech “863” Plan (2002AA135270), the Science and Technol- ogy Development Foundation of Shanghai, China (021111122), and the Research and Development Foundation on Applied Mate- rials of Shanghai, China (0204). REFERENCES 1. A. Welthof, H. J. Siweris, et al., A 38/76-GHz automotive radar chip set fabricated by a low-cost pHEMT technology, IEEE MTT-S Int Micro- wave Symp Dig (2002), Seattle, WA 1855–1858. 2. N. Priestly, K. Newsome and I. Dale, A Gunn diode-based surface mount 77-GHz oscillator for automotive applications, IEEE MMT-S Int Microwave Symp Dig (2002), Seattle, WA 1863–1866. 3. M. Camiade, D. Domnesque et al., Full MMIC millimeter-wave front- end for a 76.5-GHz adaptive cruise-control car radar, IEEE MTT-S Int Symp Dig (1999), Anaheim, CA 1489 –1492. 4. X. Zhang et al., Comparison of the phase noise performance of HEMT- and HBT-based oscillators, IEEE Int Symp Dig (1995), Orlando, FL 697– 680. 5. A. Megej and K. Beilenhoff, Active biasing technique for compact wide-band voltage controlled oscillators, IEEE MTT-S Int Symp Dig (2001), Pheonix, AZ 1419 –1422. © 2005 Wiley Periodicals, Inc. ANALYSIS OF ELLIPTICAL WAVEGUIDES BY THE METHOD OF FUNDAMENTAL SOLUTIONS D. L. Young, S. P. Hu, C. W. Chen, C. M. Fan, and K. Murugesan Department of Civil Engineering & Hydrotech Research Institute National Taiwan University Taipei, Taiwan, Republic of China Received 25 August 2004 ABSTRACT: The present work describes the application of the method of fundamental solutions (MFS) for the solution of cutoff wavelengths of ellip- tical waveguides. Since the MFS employs a formulation using boundary values only, the cutoff wavelengths are determined by applying the singular value decomposition (SVD) technique. The use of the MFS to solve the gov- erning (Helmholtz) equation guarantees a solution without singularities, since it does not use discretized points to determine the solution at the inte- rior of the computational domain. The combination of the MFS and SVD techniques has resulted in a simpler and efficient numerical solution proce- dure, as compared to other schemes. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 552–558, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop. 20695 Key words: method of fundamental solutions; singular value decompo- sition; elliptical waveguides; Helmholtz equation 1. INTRODUCTION Elliptical waveguides have been widely applied in many engineer- ing fields. Chu [1] presented the theory of the transmission of electromagnetic waves in hollow conducting pipes of elliptic cross section, and reported numerical results for the cutoff frequency and the attenuation for six waves. It is very crucial to determine the cutoff wavelengths of an elliptical waveguide for the design of waveguides; hence, this field has attracted many researchers in the recent past. Kinzer and Wilson [2] published the first approximate formulae to determine the cutoff frequency of the TE c01 , TM c11 , and TM s11 modes for a given elliptical cross section. Kretzschemar [3] obtained the curves of the cutoff wavelengths for 19 successive modes. Zhang and Shen presented analytical solutions of elliptical waveguides [4], and most of the attention was paid to computing the zeros of the modified Mathieu functions of the first kind. Recently, a numerical analysis of elliptical waveguides using the differential-quadrature method was conducted by Shu [5]. Gold- berg et al. [6] calculated the cutoff wavelengths for the six lowest modes and gave a correction to the field pattern plotted in Chu [1]. Lately, a meshless collocation method with the Wendland radial basis functions was conducted by Jiang et al. [7]. Mesh-dependent methods such as the finite-element method, finite-difference method, and finite-volume method require distrib- uted grid points at the interior of the domain for computing the field variables. Significant computational effort is required for pre-processing the mesh generation in the above methods. In particular, to find numerical solutions for irregular domains in- Figure 6 Oscillating frequency and output-power tuning curves 552 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 6, March 20 2005