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Microwave Theory and Techniques, IEEE Transactions on

Issue 2 • Date Feb. 2010

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Displaying Results 1 - 25 of 31
  • Table of contents

    Publication Year: 2010 , Page(s): C1 - C4
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  • IEEE Transactions on Microwave Theory and Techniques publication information

    Publication Year: 2010 , Page(s): C2
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  • A Single-Channel Microstrip Electronic Tracking Antenna Array With Time Sequence Phase Weighting on Sub-Array

    Publication Year: 2010 , Page(s): 253 - 258
    Cited by:  Papers (10)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1565 KB) |  | HTML iconHTML  

    We have designed and tested a novel electronic tracking antenna array that is formed by 2 ?? 2 microstrip sub-arrays. Through time sequence phase weighting on each sub-array, the amplitude and phase on each sub-array can be recovered from the output of the resultant single channel. The amplitude and phase on each array can be used to produce the sum and difference radiation pattern by digital signal processing. In comparison with the monopulse system, the RF comparator is eliminated and the number of the receiver channels is reduced from 3 to 1. A proof-of-concept prototype was fabricated and tested. The measured results confirmed the validity and advantages of the proposed scheme. The procedure of channel correction is given. View full abstract»

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  • Planar Ku -Band 4 ,\times, 4 Nolen Matrix in SIW Technology

    Publication Year: 2010 , Page(s): 259 - 266
    Cited by:  Papers (12)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (789 KB) |  | HTML iconHTML  

    In this paper, a 4 ?? 4 Nolen matrix beam-forming network for multibeam antenna applications is designed and demonstrated at 12.5-GHz center frequency. The structure is implemented using substrate integrated waveguide (SIW) technology for its attractive advantages including compact size, low loss, light weight, and planar form well suitable for high-density integration with other microwave and millimeter-wave planar integrated circuits. SIW cruciform couplers are used as fundamental building blocks for their wide range of coupling factors and their specific topology well adapted to the serial feeding topology of a Nolen matrix. The network performances are investigated over a 500-MHz frequency bandwidth ranging from 12.25 to 12.75 GHz. The matrix definition based on SIW cruciform couplers is similar to its microstrip counterpart in terms of coupling factors and phase delays. The whole network is fabricated. Measured results are in good agreement with the theoretical predictions, thus validating the proposed design concept. Using this matrix with a four radiating elements array antenna enables us to investigate the impact of the proposed matrix on the beam pointing angles versus frequency. View full abstract»

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  • Integrated Active Pulsed Reflector for an Indoor Local Positioning System

    Publication Year: 2010 , Page(s): 267 - 276
    Cited by:  Papers (9)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1721 KB) |  | HTML iconHTML  

    This paper presents an indoor localization system based on a frequency modulated continuous wave radar in the industrial-scientific-medical band at 5.8 GHz. An integrated active pulsed reflector behaves as a backscatter by regenerating the incoming phase with phase coherent startup at a constant frequency. The base station (BS) determines the distance to this reflector with a round-trip time-of-flight measurement. The active pulsed reflector is built around a switchable and tunable oscillator. The circuit has been fully integrated in a 0.18-??m CMOS technology. Outdoor measurements revealed a positioning accuracy of 15 cm, while in a harsh multipath environment with omnidirectional antennas a positioning accuracy of 32.88 cm was measured. The localization system is capable of detecting multiple reflectors at the same time, and no synchronization between BSs is needed. View full abstract»

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  • Analysis and Design of Two Low-Power Ultra-Wideband CMOS Low-Noise Amplifiers With Out-Band Rejection

    Publication Year: 2010 , Page(s): 277 - 286
    Cited by:  Papers (10)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1116 KB) |  | HTML iconHTML  

    Two 3-5-GHz low-power ultra-wideband (UWB) low-noise amplifiers (LNAs) with out-band rejection function using 0.18- ??m CMOS technology are presented. Due to the Federal Communications Commission's stringent power-emission limitation at the transmitter, the received signal power in the UWB system is smaller than those of the close narrowband interferers such as the IEEE 802.11 a/b/g wireless local area network, and the 1.8-GHz digital cellular service/global system for mobile communications. Therefore, we proposed a wideband input network with out-band rejection capability to suppress the out-band properties for our first UWB LNA. Moreover, a feedback structure and dual-band notch filter with low-power active inductors will further attenuate the out-band interferers without deteriorating the input matching bandwidth in the second UWB LNA. The 55/48/45 dB maximum rejections at 1.8/2.4/5.2 GHz, a power gain of 15 dB, and 3.5-dB minimum noise figure can be measured while consuming a dc power of only 5 mW. View full abstract»

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  • Analysis and Design of a CMOS UWB LNA With Dual- RLC -Branch Wideband Input Matching Network

    Publication Year: 2010 , Page(s): 287 - 296
    Cited by:  Papers (6)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1178 KB) |  | HTML iconHTML  

    A wideband low-noise amplifier (LNA) based on the current-reused cascade configuration is proposed. The wideband input-impedance matching was achieved by taking advantage of the resistive shunt-shunt feedback in conjunction with a parallel LC load to make the input network equivalent to two parallel RLC-branches, i.e., a second-order wideband bandpass filter. Besides, both the inductive series- and shunt-peaking techniques are used for bandwidth extension. Theoretical analysis shows that both the frequency response of input matching and noise figure (NF) can be described by second-order functions with quality factors as parameters. The CMOS ultra-wideband LNA dissipates 10.34-mW power and achieves S 11 below -8.6 dB, S 22 below -10 dB, S 12 below -26 dB, flat S 21 of 12.26 ?? 0.63 dB, and flat NF of 4.24 ?? 0.5 dB over the 3.1-10.6-GHz band of interest. Besides, good phase linearity property (group-delay variation is only ??22 ps across the whole band) is also achieved. The analytical, simulated, and measured results agree well with one another. View full abstract»

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  • A New Six-Port Transformer Modeling Methodology Applied to 10-dBm 60-GHz CMOS ASK Modulator Designs

    Publication Year: 2010 , Page(s): 297 - 309
    Cited by:  Papers (11)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1050 KB) |  | HTML iconHTML  

    This paper presents a new broadband equivalent-circuit model for millimeter-wave transformers on silicon. The model includes a center tap on the primary and secondary, and considers coupling between all segments of the windings. A corresponding methodology to analytically extract the model from electromagnetic (EM) simulations is developed. The broadband model is verified by EM simulations and measurements. Two amplitude modulatable power oscillators with high power efficiency are demonstrated using low-loss transformers. One achieves an output power of 10.4 dBm near 57 GHz with a total efficiency of 23.6%. Applying amplitude-shift keying modulation, their maximum data rate exceeds 2 Gb/s. Simulations of these circuits showed the transformer model performs well in time-domain simulations. View full abstract»

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  • A CMOS Class-E Power Amplifier With Voltage Stress Relief and Enhanced Efficiency

    Publication Year: 2010 , Page(s): 310 - 317
    Cited by:  Papers (15)  |  Patents (5)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1041 KB) |  | HTML iconHTML  

    This paper proposes a class-E power amplifier (PA) with double-resonance circuit to reduce voltage stress on CMOS transistors. The voltage waveform applied to the CMOS transistor is shaped by harmonic control and the transistors are relieved from breakdowns. A negative capacitance is also implemented for efficiency enhancement, compensating for surplus capacitance from parasitic components on the drain node. Thus, nominal class-E operation is restored and high efficiency is achieved. We present a cascode differential class-E RF PA that is fabricated using a 0.13-?? m CMOS technology that delivers 31.5-dBm output power with 54% drain efficiency and 51% power-added efficiency at 1.8 GHz. View full abstract»

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  • A Dual-Band Self-Oscillating Mixer for C -Band and X -Band Applications

    Publication Year: 2010 , Page(s): 318 - 323
    Cited by:  Papers (8)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (684 KB) |  | HTML iconHTML  

    A self-oscillating mixer that employs both the fundamental and harmonic signals generated by the oscillator subcircuit in the mixing process is experimentally demonstrated. The resulting circuit is a dual-band down-converting mixer that can operate in C -band from 5.0 to 6.0 GHz, or in X-band from 9.8 to 11.8 GHz. The oscillator uses active superharmonic coupling to enforce the quadrature relationship of the fundamental outputs. Either the fundamental outputs of the oscillator or the second harmonic oscillator output signals that exists at the common-mode nodes are connected to the mixer via a set of complementary switches. The mixer achieves a conversion gain between 5-12 dB in both frequency bands. The output 1-dB compression points for both modes of the mixer are approximately -5 dBm and the output third-order intercept point for C -band and X -band operation are 12 and 13 dBm, respectively. The integrated circuit was fabricated in 0.13-??m CMOS technology and measures 0.525 mm2 including bonding pads. View full abstract»

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  • A Novel Alternating and Outphasing Modulator for Wireless Transmitter

    Publication Year: 2010 , Page(s): 324 - 330
    Cited by:  Papers (3)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (484 KB) |  | HTML iconHTML  

    This paper describes a novel alternating and outphasing modulator for the generation and amplification of a linear modulation signal. The architecture requires a linear modulation signal to be represented as two outphasing signals with a constant envelope, which are alternating or switching at the input of two nonlinear amplifiers to produce a linear modulation signal. A power combiner can be employed to cancel the mixed components due to the switching. This will minimize the requirements of the output filter, and hence, simplified the design. This new modulation architecture is simple, and hence, is suitable for all-digital integration. The measurement results of the wideband code division multiple access signal are presented and compared with a conventional linear amplification with nonlinear components architecture. View full abstract»

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  • Enhanced Plasma Wave Detection of Terahertz Radiation Using Multiple High Electron-Mobility Transistors Connected in Series

    Publication Year: 2010 , Page(s): 331 - 339
    Cited by:  Papers (16)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1090 KB) |  | HTML iconHTML  

    We report on enhanced room-temperature detection of terahertz radiation by several connected field-effect transistors. For this enhanced nonresonant detection, we have designed, fabricated, and tested plasmonic structures consisting of multiple InGaAs/GaAs pseudomorphic high electron-mobility transistors connected in series. Results show a 1.63-THz response that is directly proportional to the number of detecting transistors biased by a direct drain current at the same gate-to-source bias voltages. The responsivity in the saturation regime was found to be 170 V/W with the noise equivalent power in the range of 10-7 W/Hz0.5. The experimental data are in agreement with the detection mechanism based on the rectification of overdamped plasma waves excited by terahertz radiation in the transistor channel. View full abstract»

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  • 60-GHz Repeater Link for an ISDB-T Gap-Filler System Based on Self-Heterodyne Technique Applying an Adaptive Distortion Suppression Technique

    Publication Year: 2010 , Page(s): 340 - 347
    Cited by:  Papers (1)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1931 KB) |  | HTML iconHTML  

    A 60-GHz through-repeater link, which adopts the self-heterodyne transmission technique and a distortion suppression technique, is demonstrated as a solution to cost-effectively develop an Integrated Services Digital Broadcasting for Terrestrial gap-filler system. View full abstract»

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  • A 60-GHz 38-pJ/bit 3.5-Gb/s 90-nm CMOS OOK Digital Radio

    Publication Year: 2010 , Page(s): 348 - 355
    Cited by:  Papers (40)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1728 KB) |  | HTML iconHTML  

    A 60-GHz fully integrated bits-in bits-out on-off keying (OOK) digital radio has been designed in a standard 90-nm CMOS process technology. The transmitter provides 2 dBm of output power at a 3.5-Gb/s data rate while consuming 156 mW of dc power, including the on-chip 60-GHz frequency synthesizer. A pulse-shaping filter has been integrated to support high data rates while maintaining spectral efficiency. The receiver performs direct-conversion noncoherent demodulation at data rates up to 3.5 Gb/s while consuming 108 mW of dc power, for a total average transceiver energy consumption of 38 pJ/bit in time division duplex operation. To the best of the authors' knowledge, this is the lowest energy per bit reported to date in the 60-GHz band for fully integrated single-chip CMOS OOK radios. View full abstract»

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  • Effects of Geometrical Discontinuities on Distributed Passive Intermodulation in Printed Lines

    Publication Year: 2010 , Page(s): 356 - 362
    Cited by:  Papers (1)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1043 KB) |  | HTML iconHTML  

    This paper presents the results of experimental study of passive intermodulation (PIM) generation in microstrip lines with U-shaped and meandered strips, impedance tapers, and strips with the profiled edges. It is shown that the geometrical discontinuities in printed circuits may have a noticeable impact on distributed PIM generation even when their effect is indiscernible in the linear regime measurements. A consistent interpretation of the observed phenomena has been proposed on the basis of the phase synchronism in the four-wave mixing process. The results of this study reveal new features of PIM production important for the design and characterization of low-PIM microstrip circuits. View full abstract»

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  • Unconditional Stability Boundaries of a Three-Port Network

    Publication Year: 2010 , Page(s): 363 - 371
    Cited by:  Papers (2)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (446 KB) |  | HTML iconHTML  

    An analytical approach for acquiring the explicit expressions of the boundaries concerning the unconditional stability regions of a three-port network is presented. This approach begins with the expressions for the input reflection coefficients S 11 ' and S 22 ' at ports 1 and 2 of a three-port network having port 3 terminated with ??3 . |S 11 '| = 1 and |S 22 '| = 1 then give two stability circles in the ??3-plane. One can graphically determine that the tangent point of these two stability circles is on the boundary of the unconditional stability region for the terminating port. The explicit expressions of the boundary for port 3 are then acquired. This procedure may be followed for the other two ports in order to fully characterize the unconditional stability boundaries of a three-port network. In addition, issues pertaining to uniqueness and continuity in the plotting of unconditional stability boundaries are addressed. Using the agilent advanced design system software tool, the derived expressions of the unconditional stability boundary for the terminating port and related extreme phase values of reflection coefficients of the other two ports are implemented. These derived expressions can then enhance the computer-aided capability on the stability analysis of a three-port network and provide useful information for the microwave circuit designer. View full abstract»

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  • Analysis and Synthesis of Double-Sided Parallel-Strip Transitions

    Publication Year: 2010 , Page(s): 372 - 380
    Cited by:  Papers (5)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1122 KB) |  | HTML iconHTML  

    Antenna feeders, mixers, and filters made in double-sided parallel-strip technology usually must be adapted to unbalanced lines like the microstrip structure, needing transitions from asymmetric to symmetric waveguides (baluns). In this paper, we propose a new method for the evaluation of a generic tapered balun based on a conformal-mapping technique and an integral equation. This method, along with the use of an optimization technique such as genetic algorithms, allows for quick evaluation of the return losses of any tapered balun and the synthesizing of specific shapes to achieve desired responses in terms of return losses or impedance values. View full abstract»

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  • High- Q RF-MEMS 4–6-GHz Tunable Evanescent-Mode Cavity Filter

    Publication Year: 2010 , Page(s): 381 - 389
    Cited by:  Papers (31)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1604 KB) |  | HTML iconHTML  

    This paper presents a miniature high-Q tunable evanescent-mode cavity filter using planar capacitive RF microelectromechanical system (MEMS) switch networks and with a frequency coverage of 4.07-5.58 GHz. The two-pole filter, with an internal volume of 1.5 cm3, results in an insertion loss of 4.91-3.18- and a 1-dB bandwidth of 17.8-41.1 MHz, respectively, and an ultimate rejection of > 80 dB. RF-MEMS switches with digital/analog tuning capabilities were used in the tunable networks so as to align the two poles together and result in a near-ideal frequency response. The measured Qu of the filter is 300-500 over the tuning range, which is the best reported Q using RF-MEMS technology. The filter can withstand an acceleration of 55-110 g without affecting its frequency response. The topology can be extended to a multiple-pole design with the use of several RF-MEMS tuning networks inside the evanescent-mode cavity. To our knowledge, these results represent the state-of-the-art in RF-MEMS tunable filters. View full abstract»

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  • Compact Hybrid Resonator With Series and Shunt Resonances Used in Miniaturized Filters and Balun Filters

    Publication Year: 2010 , Page(s): 390 - 402
    Cited by:  Papers (27)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (2444 KB) |  | HTML iconHTML  

    A hybrid resonant circuit is proposed in this paper. The circuit is a combination of a shunt resonant circuit and series resonant circuit. With this combination, lower resonant frequency is achieved as compared to the single shunt and series resonant circuits. As a result, a compact resonator with smaller size can be achieved as compared to the conventional quarter- and half-wave resonators. Besides the size reduction, the proposed resonant circuit is able to introduce a transmission zero to improve the stopband suppression in filter design. Based on this circuit, a very compact interdigital coupled microstrip resonator is proposed in this paper. The resonator achieves a small length of nearly 1/10 guided wavelength (??g), which has a length reduction of 63% as compared to the conventional uniform quarter-wave resonator. By using the proposed resonator, a second-order bandpass filter with a small size of 0.144??g ?? 0.128??g and a fourth-order bandpass filter with a size of 0.217??g ?? 0.1??g are built based on the standard filter synthesis methods. Both good performance and miniaturization are achieved for the proposed filters, and the expected transmission zeros are also observed. In addition to the small filters, the proposed resonator is suitable for miniaturized balun bandpass filters. A novel configuration for a balun bandpass filter is proposed based on the aforementioned resonators. A second-order balun bandpass filter with a size of 0.26??g ?? 0.145??g and afourth-order balun bandpass filter with a size of 0.213??g ?? 0.203??g are reported in this paper. Both balun filters achieve good filtering performance, as well as excellent amplitude and phase imbalances, which are less than 1 dB and 1?? in the passband, respectively. View full abstract»

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  • Radio-Optical Dual-Mode Communication Modules Integrated With Planar Antennas

    Publication Year: 2010 , Page(s): 403 - 410
    Cited by:  Papers (3)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1687 KB) |  | HTML iconHTML  

    This paper presents new results on integrated devices for radio and free-space optical dual-mode communication. Two novel hybrid packaging schemes using two different microwave printed antenna designs are presented for the integration of radio-optical front-end circuits on a planar compact printed circuit board with shared electrical and structural components. Full-wave electromagnetic (EM) simulations are presented for antenna optimization to minimize EM interference between the radio and optical circuits. A hybrid radio-optical package design is developed, prototyped, and experimentally studied using a modified quasi-Yagi antenna with split directors to form pads for opto-electronic device integration. Dual-mode link connectivity is investigated in simulations and experiments. A data rate of 2.5 Gb/s is demonstrated for the optical channel despite 15-20-dB signal coupling between the optical and microwave circuits. View full abstract»

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  • A Generalized Formulation for Permittivity Extraction of Low-to-High-Loss Materials From Transmission Measurement

    Publication Year: 2010 , Page(s): 411 - 418
    Cited by:  Papers (5)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (528 KB) |  | HTML iconHTML  

    We have derived a one-variable metric function for fast computations of the relative complex permittivity of low-to-high-loss materials from S 21 measurement at one frequency. We present how this function can be applied for different applications (e.g., relative complex permittivity measurement of a thin low-loss material). In addition, it can be applied as a measurement tool in broadband applications for samples with substantiate lengths, which demonstrate low-loss property at lower frequency bands and high-loss property at higher frequency bands. The derived expressions can work very well in limited frequency-band applications or for dispersive materials since it is based on point-by-point (or frequency-by-frequency) extraction. We measured the relative complex permittivity of two test samples (a low- and high-loss sample) for validation of the derived expressions. View full abstract»

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  • A New 12-Term Open–Short–Load De-Embedding Method for Accurate On-Wafer Characterization of RF MOSFET Structures

    Publication Year: 2010 , Page(s): 419 - 433
    Cited by:  Papers (4)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1544 KB) |  | HTML iconHTML  

    A new algorithm for open-short-load de-embedding of on-wafer S-parameter measurements is presented. Since in typical on-wafer RF transistor test-structures the imperfect grounding of the internal ports is the dominant source of crosstalk between ports, our proposed open-short-load approach resolving a 12-term error model is equally accurate as the current more general 15-term approach, which requires five dummy structures. To demonstrate this, experimental results obtained for six different increasingly sophisticated on-wafer correction schemes using 2-8 different de-embedding standards and resolving between 8-22 error terms, using S -parameter data taken up to 110 GHz on 65- and 45-nm node MOSFET devices are compared. View full abstract»

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  • On a Method to Reduce Uncertainties in Bulk Property Measurements of Two-Component Composites

    Publication Year: 2010 , Page(s): 434 - 439
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (237 KB) |  | HTML iconHTML  

    For two-component composites, we address the inverse problem of estimating the structural parameters and decrease measurement errors in bulk property measurements. A measurement of the effective permittivity at one frequency gives microstructural information about the composite that is used in cross-property bounds to estimate the effective permittivity at other frequencies. We use this information and inverse bounds on microstructural parameters to tighten error bars on permittivity measurements at microwave frequencies. The method can be used in the design of random and periodic composite materials for a large variety of applications. We apply the method to a composite material used in radar applications. View full abstract»

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  • Extended Through-Short-Delay Technique for the Calibration of Vector Network Analyzers With Nonmating Waveguide Ports

    Publication Year: 2010 , Page(s): 440 - 450
    Cited by:  Papers (1)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (1405 KB) |  | HTML iconHTML  

    The extension of the through-short-delay (TSD) technique to the calibration of two-port vector network analyzers (VNAs) with nonmating waveguide ports is reported. The method retains the well-known high accuracy of the basic TSD technique while it enables to calibrate VNAs using two waveguide ports with different cross sections. Comparisons with the reciprocal-short-open-load technique commonly adopted to calibrate VNAs with nonconnectable ports and with theoretical data are reported. The present method can be adopted as either a one- or a two-tier calibration technique. View full abstract»

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  • Accurate Complex Permittivity Inversion From Measurements of a Sample Partially Filling a Waveguide Aperture

    Publication Year: 2010 , Page(s): 451 - 457
    Cited by:  Papers (6)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (342 KB) |  | HTML iconHTML  

    In this paper, we propose a microwave method, which eliminates the necessity of precise knowledge of sample thickness for accurate permittivity determination of thin materials partially filling the waveguide aperture. The method utilizes propagation constant measurements at two different frequencies for this goal. To facilitate the proposed method for dispersive and nondispersive dielectric materials, we have employed a power series representation of the complex permittivity. We have validated the proposed method from permittivity measurements of prepared thin samples by different methods. We have also noted that the accuracy of the proposed method can be increased by the enhancements in the measurement accuracy of conventional methods, which require complete sample filling into the waveguide aperture. View full abstract»

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The IEEE Transactions on Microwave Theory and Techniques focuses on that part of engineering and theory associated with microwave/millimeter-wave components, devices, circuits, and systems involving the generation, modulation, demodulation, control, transmission, and detection of microwave signals. This includes scientific, technical, and industrial, activities. Microwave theory and techniques relates to electromagnetic waves usually in the frequency region between a few MHz and a THz; other spectral regions and wave types are included within the scope of the Society whenever basic microwave theory and techniques can yield useful results. Generally, this occurs in the theory of wave propagation in structures with dimensions comparable to a wavelength, and in the related techniques for analysis and design..

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