Steady-State Responses of NRD-Guide LSE/LSM Mode Oscillators in MM-Wave Bands

The analysis methods for the steady-state responses of the mm-wave band NRD-guide negative impedance oscillators based on nonlinear microwave autonomous circuits harmonic balanced method are presented in the paper in details. Firstly, the large-signal nonlinear lumped equivalent circuits of the Gunn diode are studied in mm-wave bands. Then, the performances of two kinds of NRD-guide autonomous circuits, the probe-exciting LSM ₁₁ ^o-mode oscillator and the LSE ₁₁ ^o-mode oscillator, are analyzed by the way of extracting the large-signal dynamic harmonic admittance (conductance) of the Gunn diode or directly configuring the harmonic balance equations for the oscillator. The input impedance of the exciting probe in the oscillator and the performances of the load-pulling and the local stability of the NRD-guide oscillator are also involved.     

The digital predistortion of non-linear frequency modulation of voltage-controlling-oscillator in MM-band

The reason and rectification method of the non-linear frequency modulation of voltage-controlling-oscillator in millimeter-wave band have been presented in this paper. The digital predistortion for the controlling signal of the varactor in VCO possesses high rectification sensitivity, simple circuit and convenient adjustment. The method can be used in mm-wave and microwave VCOs that are fabricated by all kinds' transmission lines, and can eliminate the dispersion of FM characteristic in VCOs that are usually caused by the machining deviation and the inconsistent parameters of semiconductors. The digital predistortion technique has been applied in NRD-guide VCO in Ka-band, which main performances are: f0 = 36.30648GHz, the frequency-stability reaches 2.26×10^–5, the maximum frequency-deviation is 150MHz, the linear-FM error is less than 5%.     

A new type NRD-guide receiving front-end in Ka-band

A new type millimeter-wave receiving frond-end using NRD-guide as internal transmission line is presented in this paper. The receiving frond-end consists of a NRD-guide crossbar balanced mixer, a NRD-guide LSE mode local oscillator and two impedance-converters at two input ports of the mixer circuit. Firstly, the propagation characteristics of some transmission modes often used in NRD-guide are given, and the conditions which these modes are applied as the operating modes are discussed. Secondly, the principle of the balanced mixed which is regarded as the key circuit unit in the receiving frond-end is analysed by using vector Green's function, and the conversion loss of the mixer is computed with an equivalent circuit in Ka-band. And then, from the view of obtaining the optimum performances of the whole receiving frond-end, some methods of designing local oscillator are expound. Some experimental results of the receiving frond-end at Kaband are given in the paper lastly. Because the frond-end have the impedance matching and converting circuits, it possesses better electrical performance. The integrated frond-end which have compact size and excellent shock strength is suitable for varied millimeter-wave system.     

Discontinuity of NRD-Guide and Its Application in Millimeter Wave Circuits

In this paper, the performances of two typical discontinuous structures of the NRD-guide, the traverse metallic rod to disturb an incident wave and the traverse probe to excite required modes, are analyzed and computed by utilizing the theory of modes orthogonality of the NRD-guide. The moment method is used to evaluate the current distribution on the rod and probe and the affection of some higher-order modes is taken into account simultaneously. On these bases, the multitone exciting steady states of two kinds of important circuits that employ the above discontinuous structures, the NRD-guide crossbar balanced mixer and the LSM ₁₁ ⁰ -mode oscillator, are simulated by the harmonic balance technique based on the multi-dimensional discrete Fourier transformation (MDFT-HB method).     

Steady State Responses of NRD-Guide Crossbar Balanced Mixer in MM-Wave Band

The periodic and quasi-periodic steady states of the nonradiative dielectric waveguide (NRD-guide) crossbar single-balanced mixer excited by single tone or dual tones in mm-wave band are deeply researched in the paper. Based on the wideband dispersion characteristics, the mode self-contained property and the modes orthogonality and a series of modes excitation criterions of the NRD-guide, a novel mixer is designed and analyzed. Two important parameters, the local oscillator and incident wave inductive electromotive forces corresponding to the LSE/LSM mode-converter and LSM-mode probe respectively, are calculated firstly by using the wire-antenna theory and the moment method. According to the theorem of band-pass signal sampling, a kind of integrated and general quasi-periodical steady-state responses algorithm for the nonlinear microwave/mm-wave circuits excited by multiple tones, the inter-modulation balance method based on the multi-dimension discrete Fourier transformation (MDFT-IMB), is designed and then successfully applied to calculate the performances of the above NRD-guide mixer.     

The development of nonradiative dielectric waveguide transmitting front-end in Ka-band

The operating theory,design method and experimental results of the nonradiative dielectric waveguide (NRD-guide) transmitting front-end in millimeter wave band are related in details in this paper. As a kind of dielectric waveguide,the NRD-guide possesses fine transmission performances, and it can be used to fabricate the millimeter wave hyterodyne integrated circuits. The two kinds NRD-guide transmitting front-ends substantiated in the paper, the voltage controlling oscillator(VCO) and twin-Gunn-diode power-combiner,have very similar structures to each other and possess compact sizes, fine machinery and electric performances.For the VCO front-end in Ka-band, the frequency modulation band is greater than 150MHz, the output power is larger than 20mW.For the power-combiner, the output power is larger than 40mW,the combination efficiency is better than 90%,the frequency stabilization reaches 1.68×10^–5. Combined with the receiving front-end reported before,the transmitter and receiver can be composed to a dielectric waveguide R/T module to be applied in some millimeter wave sub-systems.     

Essentially nonlinear piezoelectric shunt circuits applied to mistuned bladed disks

An essentially nonlinear piezoelectric shunt circuit is proposed for the practical realization of nonlinear energy sink, and then applied to a mistuned bladed disk for blade vibration reduction. First, the global dynamics of a single degree-of-freedom linear mechanical oscillator, coupled to an essentially nonlinear shunted piezoelectric attachment, is studied. Under certain conditions, the nonlinear targeted energy transfer, i.e. a fast, passive energy transfer from the mechanical oscillator to the nonlinear attachment is observed. A numerical method, referred to as the variable-coefficient harmonic balance method, is developed to calculate quasi-periodic responses arising in the electromechanical system under harmonic forcing. Characterized by the nonexistence of a resonance frequency, the essentially nonlinear shunt circuit is able to work robustly over a broad frequency band with a smaller inductance requirement compared with the linear resonant shunt circuit.     

Theoretical Study of Millimeter Wave Harmonic Oscillator Stabilized with a Transmission Cavity

An equivalent circuit model of millimeter wave second harmonic oscillator stabilized with a transmission cavity has been proposed for constructing analytical formulations between performance parameters of the oscillator and parameters of the circuit. The model consists of an equivalent circuit of fundamental wave and that of second harmonic wave. Each of the circuits comprises circuit models of main cavity, transmission waveguide, and transmission cavity. Absorbing material placed between the transmission waveguide and the transmission cavity can suppress additional resonances originated from transmission cavity. The behavior of the second harmonic oscillator can be effectively described by the circuit model. Furthermore, based on this model, mechanical tuning characteristics have been studied at first, and then analytical formulas for quality factor and efficiency depending on circuit parameters have been derived. The circuit parameters can be conveniently extracted by electromagnetic field simulation. Hence the formulas exhibit both compact form and enough accuracy. Thereafter, general rules of performance parameters varying with circuit parameters have been deduced for the harmonic oscillators. Then some design considerations have been derived according to the corresponding analysis. The equivalent circuit model is useful for designing and adjusting millimeter wave second harmonic stabilizing oscillator with a transmission cavity.     

Analysis and application of NRD-guide excitors

In this paper, the NRD-guide excitors using a length of coaxial line which consists of a rectangle external conductor and a thin round inner conductor are presented, and the characteristics of the input impedances of LSE ₁₁ ^o mode and LSM ₁₁ ^o mode at the insertion points are analysed and calculated by using the method of the rector Green's function respectively. The computation results show that the mode excitors are able to improve the operating modes matching between NRD-guide and the quasi-TEM transmission line. The structures are suitable for fabrication LSE ₁₁ ^o mode oscillator and LSM ₁₁ ^o mode oscillator to compose millimeter wave NRD-guide integrated front ends. An example of an LSE ₁₁ ^o mode oscillator in Ka-band is given. Its oscillation frequency and output power could be easily adjusted by changing the parameters of the excitor.     

Millimeter wave monolithic high electron mobility (HEM) varactor diode-grid frequency tripler arrays

High Electron Mobility (HEM) varactor structures have been studied for millimeter-wave monolithic diode-grid frequency tripler array applications. The improved HEM varactor diode structures provide a highly nonlinear C-V characteristic (i.e., a steep slope of the C-V curve and a large capacitance ratio) which produces high harmonic generation efficiency and reduce the power requirement for efficiently pumping each device. The effects of the light illumination on the C-V characteristics of the Barrier-Intrinsic-N⁺ (BIN) varactor diode have also been studied and the results will be discussed in this paper. In the development of a monolithic diode-grid frequency multiplier array, the low-loss quasioptical configuration is used for the construction of the multiplier circuit. The study of the effects of the light illumination on the C-V characteristics of varactor diode is important in understanding the potential applications of the quasi-optical varactor diode-grid frequency multiplier array circuit.     

Integrated image-line steerable active antennas

This paper reports a varactor-tuned Gunn oscillator constructed in an image-line circuit and integrated with a grating antenna. The combined circuit exhibited a 2 percent electronic tuning range corresponding to a frequency steerable range of 4 degrees. The active antenna demonstrated a clean spectrum with low side lobes. The beam angles agree very well with the prediction. The circuit can be produced at low cost and should have many applications at millimeter-wave frequencies     

Development of Broad Band VCO at Ka-Band

A broad band VCO has been developed at Ka-band for FMCW Radar applications. To achieve a wide range of frequency variation, VCO has been designed in series configuration. Design steps have been presented. VCO exhibits a tuning range of 600 MHz with the power output of 50 mw, when the controlled varactor voltage varies from 7.5 volts to 15 volts. Frequency drift with temperature has been contained within 30 MHz using a proportionally controlled DC heater module over the temperature range of 0°C to +55°C. Phase Noise of the oscillator measured at the mid and extreme frequencies is about -70 dBc/kHz at 10 kHz away from the carrier. The experimental circuit and measured performance is also presented.     

Closed-Form Expression for the Exact Period of a Nonlinear Oscillator Typified by a Mass Attached to a Stretched Wire

The exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential, is obtained. Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire. The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.     

Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.     

基于参数展开的同伦分析法在强非线性随机动力系统中的应用

将基于参数展开的同伦分析法(PE-HAM)进行了推广，使之适用于谐和激励与随机噪声联合作用下的强非线性随机动力系统. 通过构造合适的同伦映射，将对强非线性随机动力系统响应的求解转化为对一组线性随机微分方程的求解. 进一步研究了受到谐和与Gauss白噪声激励的强非线性Duffing振子，由PE-HAM得到了该系统的解过程和稳态概率密度的解析表达式. 数值模拟的结果说明了PE-HAM方法的精确性. 关键词： PE-HAM方法 强非线性随机动力系统 稳态概率密度 解过程 随机激励     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

Theoretical and experimental research on nonradiative dielectric waveguide directional couplers

A practical design of nonradiative dielectric (NRD) waveguide directional couplers is presented. The dispersion characteristics of coupled NRD-guides and the scattering coefficients of NRD-guide directional couplers are studied as well as the transitions from metal rectangular wave-guide to NRD-guide couplers. Experiments of 3-dB and 10-dB NRD-guide couplers in ka-band show that the couplers are of good performance and promise in millimeter-wave integrated circuits.     

Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities

This paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. The Galerkin method is employed to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived, and the incremental harmonic balance (IHB) method is employed to obtain the steady-state response of the curved beam. The cases investigated include softening stiffness, hardening stiffness and modal energy transfer. The stability of the periodic solutions for given parameters is determined by the multi-variable Floquet theory using Hsu's method. Particular attention is paid to the anti-symmetric response with and without excitation, as the excitation frequency is close to the first and third natural frequencies of the system. The results obtained with the IHB method compare very well with those obtained via numerical integration.     

Exact and Approximate Values of the Period for a "Truly Nonlinear" Oscillator: $\ddot{x} + x + x^{1/3} = 0$

We investigate the mathematical properties of a "truly nonlinear" oscillator differential equation. In particular, using phase-space methods, it is shown that all solutions are periodic and the fixed-point is a nonlinear center. We calculate both exact and approximate analytical expressions for the period, where the exact solution is given in terms of elliptic functions and the method of harmonic balance is used to calculate the approximate formula.     

对称非线性弹簧振子的周期特性

通过计算机编程(Quick Basic)描绘对称非线性弹簧振子振动的特性曲线,使难懂的物理过程变得直观、形象。对称非线性弹簧振子的振动是一种周期性振动,但不是严格的简谐振动,其振动周期随非线性系数、振幅的变化而偏离简谐振动的周期。