Nonlinear circuit analysis using the method of harmonic balance—a review of the art. II. Advanced concepts

The harmonic balance method is a technique for the numerical solution of nonlinear analog circuits operating in a periodic, or quasi-periodic, steady-state regime. The method can be used to efficiently derive the continuous-wave response of numerous nonlinear microwave components including amplifiers, mixers, and oscillators. Its efficiency derives from imposing a predetermined steady-state form for the circuit response onto the nonlinear equations representing the network, and solving for the set of unknown coefficients in the response equation. Its attractiveness for nonlinear microwave applications results from its speed and ability to simply represent the dispersive, distributed elements that are common at high frequencies. The last decade has seen the development and application of harmonic balance techniques to model analog circuits, particularly microwave circuits. The first part of this article reviewed the fundamental achievements made during this time. In this part, the extension of the method to quasi-periodic regimes, optimization analysis, oscillator analysis, studies of various convergence strategies, and practical applications are discussed. A critical assessment of the various types of harmonic balance techniques is given. Examples of designs which have been modeled using the harmonic balance technique and built both in hybrid and MMIC form are presented.     

Computer-aided analysis of nonlinear microwave circuits using frequency-domain nonlinear analysis techniques: The state of the art

Frequency-domain nonlinear analysis techniques for the simulation of active microwave circuits solve the linear and nonlinear network equations entirely in the frequency-domain. by so doing, they avoid the aliasing problems inherent in piecewise harmonic balance approaches. Consequently, frequency-domain techniques have extremely wide dynamic range and easily accommodate high order multitone excitation. However, this is at the expense of requiring more restrictive nonlinear device models. There are a large number of frequency-domain nonlinear analysis techniques but all are based on functional expansions which enable the frequency components of the output spectrum to be calculated directly from the input spectrum. These techniques have been used to analyze many nonlinear circuits and are the only candidates for the hierarchical simulation of nonlinear microwave circuits. This paper first uses frequency-domain concepts to discuss nonlinear distortion phenomena, then, a review of the frequency-domain nonlinear analysis literature is made with the aim of presenting the major advances in these techniques.     

Complex modes based numerical analysis of viscoelastic sandwich plates vibrations

In this paper, a numerical method for linear and nonlinear vibrations analysis of viscoelastic sandwich beams and plates is developed with finite element based solution. This method couples the harmonic balance technique to complex mode Galerkin’s procedure. This results in a scalar nonlinear complex amplitude–frequency relationship involving numerical computation of three coefficients. A general formulation taking into account the frequency dependence of the viscoelastic behaviour allowing to intoduce any viscoelastic law is given. Complex eigenmodes are numerically computed in a general procedure and used as Galerkin’s basis. The free and steady-state vibrations analyses of viscoelastic sandwich beams and plates are investigated for constant and frequency dependent viscoelastic laws and for various boundary conditions. The equivalent frequencies and loss factors as well as forced harmonic response and phase curves are performed. The obtained results show the efficiency of the present approach to large amplitudes vibrations of viscoelastic sandwich structures with nonlinear frequency dependence.     

Application of fault modeling to continuous built-in test (c-BIT) for microwave and MMIC circuits

A simulation-before-test method has been applied to the linear and nonlinear microwave built-in test (BIT) problem. Fault simulation via a harmonic balance simulator is used to analyze and optimize BIT hardware placement based on figures of merit such as fault coverage, ambiguity size, and part isolation. Fault models for a prototype microwave mini-system were developed along with models for the BIT hardware itself. The search method developed to automate the placement of BIT hardware is an essential design-for-test tool. The techniques developed were applied to an example system and results are given.     

Steady-state analysis of nonlinear forced and autonomous microwave circuits using the compression approach

This article discusses techniques for the analysis of nonlinear microwave circuits that contain lumped nonlinear elements. Emphasis is put on specific harmonic balance-based technique which allows us to retain the field theoretical accuracy in cases where lumped nonlinear elements are embedded closely within complex metallization structures. © 1995 John Wiley & Sons, Inc.     

New behavioral‐level simulation technique for RF/microwave applications. Part I: Basic concepts

The quadrature modeling technique is nowadays widely used for the nonlinear simulation of RF/microwave communication circuits and systems at the behavioral (system) level. It allows one to simulate the circuit/system performance under real‐world conditions and signals (using several thousand sample frequencies) and to predict such parameters as adjacent channel power ratio, spectral regrowth, and error vector magnitude in a computationally efficient way. But it is a narrowband technique and, consequently, cannot predict harmonics of the carrier frequency and even‐order nonlinear products, to account for the circuit/system frequency response and the bias decoupling network effect. Here, we propose a new behavioral‐level simulation technique (instantaneous quadrature technique) that overcomes these drawbacks, and demonstrate its validity through measurements and harmonic balance simulation. The transformation of envelope transfer characteristics into instantaneous ones is also discussed in detail. © 2000 John Wiley & Sons, Inc. Int J RF and Microwave CAE 10: 221–237, 2000.     

Harmonic Balance Nonlinear Identification of a Capacitive Dual-Backplate MEMS Microphone

This paper describes the application of a nonlinear identification method to extract model parameters from the steady-state response of a capacitive dual-backplate microelectromechanical systems microphone. The microphone is modeled as a single-degree-of-freedom second-order system with both electrostatic and mechanical nonlinearities. A harmonic balance approach is applied to the nonlinear governing equation to obtain a set of algebraic equations that relate the unknown system parameters to the steady-state response of the microphone. Numerical simulations of the governing equation are also performed, using theoretical system parameters, to validate the accuracy of the harmonic balance solution for a weakly nonlinear microphone system with low damping. Finally, the microphone is experimentally characterized by extracting the system parameters from the response amplitude and phase relationships of the experimental data.     

Optimization technology for nonlinear microwave circuits integrating electromagnetic simulations

We review relevant concepts, formulations, and algorithms for microwave circuit optimization. Emphasis is given to recent advances in the state of the art: Automated electromagnetic (EM) design, Space Mapping, Huber optimization, an integrated CAD environment, and parallel computation. We address integration of previously disjoint simulation technologies for automated EM optimization of linear and nonlinear microwave circuits. We incorporate EM analyses of passive microstrip structures and SPICE models of active devices into harmonic balance optimization of nonlinear circuits, even for yield-driven design. Designs of a Class B frequency doubler, a broadband small-signal amplifier, and an attenuator illustrate the integrated approach. © 1997 John Wiley & Sons, Inc. Int J Microwave Millimeter-Wave CAE 7: 6–28, 1997.     

消隔组合X型减振器设计与动力学分析

本文将一种X型结构与线性弹簧阻尼减振器相结合构成一种兼具消振和隔振性能的新型X型减振器.基于拉格朗日方法建立单自由度线性振子耦合消隔组合X型减振器系统的动力学方程,应用谐波平衡法得到系统稳态响应的近似解析解,并通过Runge Kutta法得到系统的数值解验证解析解的正确性.讨论了不同的基础激励下新型X型减振器对系统的响应幅值以及位移传递率的减振效果.此外,分析了不同参数对系统的幅频响应曲线以及位移传递率的影响.研究结果表明,新型X型减振器不仅可以将线性刚度转化为非线性刚度,还可以为系统提供负刚度,拥有优秀的消振和超低频隔振性能.     

Coupled nonlinear/electromagnetic CAD of injection‐locked self‐oscillating microstrip antennas

This article presents a systematic approach to the simulation and design of self‐oscillating integrated antennas, based on the combination of state‐of‐the‐art nonlinear and electromagnetic (EM) CAD techniques. The linear subnetwork, including the oscillator circuit and the antenna, is treated as a whole and its admittance matrix is computed at all frequencies of interest (including harmonics) by EM analysis. The oscillating subsystem is then analysed by harmonic balance (HB) for autonomous circuits. The design problem is turned into the solution of a nonlinear system, with a significant reduction in the overall number of EM analyses with respect to a conventional optimization. A simple linear model of the radiated far field in terms of the exciting voltages is generated by inexpensive post‐processing of the data generated by EM analysis. This allows the far‐field properties to be directly specified during the design step. The conditions for stable injection locking are then determined by nonlinear methods based on the numerical implementation of bifurcation theory. Finally, when the antenna is injection‐locked by a digitally modulated RF/microwave carrier, the system response in terms of radiated far‐field is efficiently computed by envelope‐oriented HB analysis. © 2003 Wiley Periodicals, Inc. Int J RF and Microwave CAE 13, 398–414, 2003.     

Nonlinear identification of systems with parametric excitation

In this paper, the incremental harmonic balance nonlinear identification (IHBNID) is presented for modelling and parametric identification of nonlinear systems. The effects of harmonic balance nonlinear identification (HBNID) and IHBNID are also studied and compared by using numerical simulation. The effectiveness of the IHBNID is verified through the Mathieu-Duffing equation as an example. With the aid of the new method, the derivation procedure of the incremental harmonic balance method is simplified. The...     

Steady state analysis of free or forced oscillators by harmonic balance and stability investigation of periodic and quasi-periodic regimes

This article presents a practical and effective method to analyze the steady state regimes, periodic or quasi-periodic, and the stability of free and forced oscillators, based on the use of voltage and current probes. The efficiency of this method comes from the fact that it converts the analysis of an autonomous circuit into the analysis of a series of closely related nonautonomous circuits, which in turn relies on the power of the harmonic balance equation for these kinds of systems. The effectiveness of the method is shown by full analysis of a practical example.     

Computation of the 1‐dB compression point in radio frequency amplifier circuits using moments analysis

Nonlinear distortion analysis is one of the most computationally challenging aspects in the simulation of radio frequency circuits. Recently, an efficient moments‐based approach to determine the third‐order intercept point from the general harmonic balance equations, without the need to perform a harmonic balance simulation, was presented. In this article, the efficient computation of another important performance figure of merit, that of the 1‐dB compression point, using moments analysis is presented. In addition, the computations of the second‐ and fifth‐order intercept points are also presented. As a result, moments analysis becomes a comprehensive approach for quantifying nonlinear distortion using several key distortion figures of merit with a significant speed‐up over traditional harmonic balance methods. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:10–20, 2015.     

Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method

Although the Van der Pol oscillator, which was originally proposed as a model of vacuum tube circuits, has been widely used in electronics, biology and acoustics, its characteristics in fractional order formulations are not clearly explained even now. This paper is interested in gaining insights of approximate expressions of the periodic solutions in a fractional order Van der Pol oscillator. The presence of fractional derivatives requires the use of suitable criteria, which usually makes the analytical work much hard. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of fractional order systems. In this paper, based on the residue harmonic balance method, a detailed analysis on approximations to the periodic oscillations of the fractional order Van der Pol equation is investigated. The relations that express the frequency and amplitude of the generated oscillations as functions of the orders and parameters are shown. Moreover, some examples are provided for comparing approximations with numerical solutions of the periodic oscillations. Numerical results reveal that the residue harmonic balance method is very effective for obtaining approximate solutions of fractional oscillations.     

轮-带驱动系统稳态周期响应谐波平衡分析

研究含有单向离合器、两滑轮及附件的轮-带驱动系统稳定稳态周期响应.通过单向离合器连接从动轮与附属系统,并计入传送带的横向振动的影响,导出了由偏微分-积分方程与分段常微分方程组成的连续-离散型非线性耦合方程组.利用Galerkin方法将连续非线性方程组截断为一组非线性常微分方程组,再运用谐波平衡法得到轮-带驱动系统耦合非线性振动的稳态响应.通过比较有无单向离合器装置的系统稳定稳态幅频响应曲线,研究了单向离合器对驱动系统以及轮-带系统非线性动态特性的影响.并首次研究了高频激励下轮-带系统的稳态响应.最后,运用Runge-Kutta方法对比验证了基于谐波平衡法得到的稳态响应.     

A cad method applied to the development of microwave oscillators

In this article, a method is proposed that allows the prediction of characteristic parameters of integrated microwave oscillators, i.e., operating frequency and distribution of the output power due to harmonic frequencies. This method combines rigorous field theoretical analysis of the linear environment with the “harmonic balance” approach. Relations between these areas are shown and discussed by means of numerical and experimental results. © 1994 John Wiley & Sons, Inc.     

Simplified symbolic transfer function factorization using combined artificial bee colony and simulated annealing

Symbolic circuit analysis inherits the exponential growth of transfer function complexity with the circuit size. Therefore, symbolic simplification is an NP-hard problem. Although many simplification techniques have been presented, the simplified transfer functions are not written in a factorized form, and consequently, it is difficult to assess the contribution of poles and zeros on the circuit behavior. In this paper, a swarm intelligence based methodology is presented for the simplified factorized symbolic analysis of analog circuits. In this method, an extension of the root splitting technique is utilized to rewrite the expanded transfer function of the circuit into a factorized form comprising DC-gain, poles, and zeros. Then, the derived factorized transfer function is simplified using a hybrid Global and Local search algorithm based on Artificial Bee Colony and Simulated Annealing (named GLABCSA). The objective function is defined to minimize the complexity of the symbolic factorized transfer function while minimizing the DC-gain error and pole/zero displacements. The presented approach has been successfully developed in MATLAB. The program can derive the simplified factorized symbolic transfer function automatically from the input text netlist of the circuit. Symbolic and numerical results over two analog amplifiers are given to illustrate the efficiency of the presented methodology.     

Recent advances in microwave active filter design. Part 1: Low‐frequency techniques and noise optimization

This two‐part article presents a representative sample of recent advances in microwave active filter design. In part one, we discuss design techniques based on both analog and digital low‐frequency methods that have been adapted to microwaves. From circuits with analog origins, we present simulated results, with some experimental verification, for a frequency‐dependent negative resistance (FDNR) and active‐inductance MMIC bandpass filters. From circuits with digital origins, we present experimental results for recursive filters, including MMIC bandpass and bandstop structures as well as higher‐order cascaded sections. Part one concludes with a discussion of the noise‐wave formalism and experimental results for active‐recursive and tunable ring‐resonator filters with minimum noise figure. © 2002 Wiley Periodicals, Inc. Int J RF and Microwave CAE 12: 159–176, 2002.     

Physical/electromagnetic analysis of multifinger MOSFETs with SB‐SP combined methods

The frequency‐domain spectral balance technique has been demonstrated to be a viable alternative to the mixed‐domain Harmonic Balance technique. It has already been applied to the space‐domain polynomial expansion of the physical quantities inside the semiconductor, for the solution of steady‐state nonlinear differential equations, for the physical analysis of high‐frequency semiconductor devices. In this article it is coupled to a commercial electromagnetic solver, for the combined physical and electromagnetic analysis of multifinger MOSFET devices in linear and nonlinear regime. This method allows a fast and effective CAD analysis both in DC and RF periodic regime for very high frequencies. A quasi‐2D hydrodynamic formulation is used for a 0.35 μm gate length with a total 30 μm periphery, three finger MOSFET device; results are compared with those of a standard physical time‐domain, a harmonic balance and a spectral balance analysis for comparison of numerical efficiency. Moreover, comparison of S‐parameters with a commercial CAD tool with a compact model for circuit analysis is also given. © 2009 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.     

A robust composite nonlinear control scheme for servomotor speed regulation

A parameterised design of robust composite nonlinear controller is proposed for typical second-order servo systems subject to unknown constant disturbance and control input saturation. The control law consists of a linear feedback part for achieving fast response, a nonlinear feedback part for suppressing the overshoot, and a disturbance-compensation mechanism for erasing the steady-state error. An extended state observer is adopted to estimate the unknown disturbance. The closed-loop stability is analysed theoretically. The control scheme is applied to the speed regulation of permanent magnet synchronous motor, and numerical simulations are carried out. The results confirm that the proposed control scheme can achieve fast, smooth, and accurate speed regulation, and has a certain degree of robustness with respect to the amplitude of disturbances and the perturbations of system parameters.