An improved FDTD formulation for general linear lumped microwave circuits based on matrix theory

An improved finite-difference time-domain (FDTD) formulation using the matrix theory is presented to analyze hybrid general linear lumped and distribution microwave circuits. Traditionally, since discrete current expressions of lumped circuits should be substituted into Maxwell's equations explicitly, the different discrete formulations of Maxwell's equations should be reconstructed when different active circuits are used. It will be much more difficult when high-order linear lumped circuits are considered in a hybrid system. In our improved formulation, a high-order linear lumped circuit is expressed by a serials of first-order modified integral transforms. When their interior variables together with electric fields, port voltage, and port current/current density of active circuits are considered as a vector, a local implicit and iterative matrix expression can be built. Compared with the traditional method, since our improved FDTD formulation is implicit, it easily combines with other active circuits built by the matrix method and decreases the condition number, is more stable, flexible, and general. Based on this formulation, the numerical stability and condition number are discussed. Simulation shows that the results by our improved FDTD formulation are in good agreement with that from commercial software.     

Circuit analysis of electromagnetic radiation and field couplingeffects for networks with embedded full-wave modules

With continually increasing operating frequencies, the analysis of electromagnetic interference (EMI)-related effects is becoming an important issue for high-speed designs. An algorithm is presented for fast analysis of radiation and incident field coupling effects in high-speed circuits. The proposed technique provides an efficient means for combining the solutions from full-wave field solvers such as the finite-difference time-domain (FDTD) method with circuit level simulators such as SPICE for calculating radiated/coupled fields in arbitrarily shaped interconnect structures. The technique speeds up the whole simulation process by employing a model-reduction-based approach, and also overcomes the numerical stability problems associated with the FDTD, in the presence of nonlinear terminations. In addition, the proposed algorithm provides a direct access to existing vast device libraries of SPICE in EMI analysis     

Transient Chip-Package Cosimulation of Multiscale Structures Using the Laguerre-FDTD Scheme

Transient simulation using Laguerre polynomials is unconditionally stable and is ideally suited for modeling structures containing both small and large feature sizes. The focus of this paper is on the automation of this technique and its application to chip-package cosimulation. Laguerre finite-difference time-domain (FDTD) requires using the right number of basis coefficients to generate accurate time-domain waveforms. A method for generating the optimal number of basis functions is presented in this paper. Equivalent circuit models of the FDTD grid have been developed. In addition, a method for simulation over a long time period is also presented that enables the extraction of the frequency response both at low and high frequencies. A node numbering scheme in the circuit model of the FDTD grid that is suitable for implementation has been discussed. Results from a chip-package example that shows the scalability of this technique to solve multiscale problems have been presented.      

基于FDTD的约瑟夫森结与超导传输线协同分析方法

为实现高性能处理器,超导RSFQ（快速单磁通量子）电路被提出.该电路主要由超导约瑟夫森结和超导无源传输线组成,对其建模分析是超导RSFQ电路设计的基础.本文提出了基于FDTD（时域有限差分）的约瑟夫森结与超导传输线的协同分析方法.该方法采用FDTD数值方法求解超导传输线的电报方程.在超导传输线与约瑟夫森结交界处的非线性边界条件上,采用了Newton-Raphson迭代算法.数值结果表明,本文提出的约瑟夫森结和超导传输线的协同分析方法与WRspice仿真软件相比具有相同精度,且运算效率显著提高.     

Application of the three-dimensional finite-difference time-domainmethod to the analysis of planar microstrip circuits

A direct three-dimensional finite-difference time-domain (FDTD) method is applied to the full-wave analysis of various microstrip structures. The method is shown to be an efficient tool for modeling complicated microstrip circuit components and microstrip antennas. From the time-domain results the input impedance of a line-fed rectangular patch antenna and the frequency-dependent scattering parameters of a low-pass filter and a branch-line coupler are calculated. These circuits were fabricated and the measurements made on them are compared with the FDTD results and shown to be in good agreement     

一种计算微波电路的并行算法--FDTD-Diakoptics

ＦＤＴＤ－Ｄｉａｋｏｐｔｉｃｓ将复杂的微波电路分割为若干简单的子电路,使用有限时域差分方法（ＦＤＴＤ）独立求解每个子电路的时域特性,使用并行算法连接各子电路,最终得到整个电路的特性。本方法适用于结构复杂,规模较大的微波电路的分析设计,与整个电路使用ＦＤＴＤ进行设计研究的方法比较,本算法在保证相同数值精度的条件下可以提高计算效率五倍左右,故具有广泛的应用前景。     

FDTD analysis of high frequency electronic interconnection effects

A full-wave analysis of coupled high-frequency interconnect discontinuities is presented using the finite-difference time-domain (FDTD) method. The electromagnetic effects of two via holes on microstrip lines in close proximity to one another are examined and equivalent circuits are presented. The effects of two adjacent lines with bond wires, used, for example, to connect a die to the leadframe of an integrated circuit (IC) package are also analyzed. Frequency domain results are presented by using the discrete Fourier transform of the time-domain results. Guidelines regarding the effective use of the FDTD code including the use a priori calculated electric field distribution in the excitation plane, and the use of a weighted ϵ_r,eff to minimize reflections at the absorbing boundaries are described. The obtained FDTD results and the developed equivalent circuit models show the importance of radiation effects at frequencies beyond 20-30 GHz, the possibilities of reducing the inductive effect of bond wires by using two parallel bond wires instead of one, and the importance of including mutual inductance elements in the equivalent circuit model to account for the crosstalk between parallel vias across a ground plane     

Stability characteristics of absorbing boundary conditions inmicrowave circuit simulations

In this paper, we examine the stability properties of several absorbing boundary conditions in the finite-difference time-domain (FDTD) simulations of microwave circuits. The numerical experiments show that the stability characteristics of absorbing boundary conditions, e.g., Mur's (1981) and perfectly matched layers (PML), can depend upon the discretization of the computational domain     

On-Chip Power-Grid Simulation Using Latency Insertion Method

Ensuring the integrity of the power supply in the power distribution networks (PDNs) of a chip is essential for building reliable high-performance chips. To ensure the power integrity, accurate, and memory- and time-efficient simulation approaches for simulating the power-supply noise in the on-chip PDN are essential. In this paper, a finite-difference formulation based on the latency insertion method (LIM) has been employed for simulating the power-supply noise in the on-chip PDN. A new common-mode type equivalent circuit has been proposed. In this equivalent circuit, a capacitance to ideal ground may not be present at all the nodes. Further, the nodes can be capacitively coupled to each other. To avoid inverting a large nonbanded matrix, a small capacitance to ground is added to a node that did not have any capacitance to ground, and a small series inductance is added to any floating capacitor that did not have any series inductance. Approximate closed-form expressions to compute the values of these capacitances to ground and series inductances have been proposed. The accuracy of the LIM-enabled transient simulation and the accuracy of the proposed closed-form expressions have been demonstrated. The memory and time complexity of the simulation for each time step have been shown to be O(N_n) each, where N_n is the number of nodes in the equivalent circuit. Stability condition is derived for the first time for multidimensional inhomogeneous RLC circuit. A upper bound of the time step is derived from the stability condition. Using this bound on the time step, the runtime of the overall transient simulation has been estimated to be approximately proportional to N_n ^2-2.5 for N_n in the order of millions.     

A straightforward method for the FDTD analysis of quasi-opticalarrays of active devices

A method for the finite-difference time-domain (FDTD) analysis of quasi-optical circuit arrays of active devices is presented in this paper. To analyze active devices such as varactor diodes and MESFETs, this method creates a set of field-state central finite-difference equations, and all the field and state variables are solved simultaneously at the same FDTD time step. It will be shown in this paper that this formulation is simple and straightforward. The analyses of quasi-optical back-to-back varactor tripler and MESFET oscillator arrays have been performed to show different applications of this method. Excellent agreements have been obtained among the results from this simulation method, analytical solutions and commercial software, and experimental measurements to show the validity of this method     

Three-Dimensionally Nonorthogonal Alternating-Direction Implicit Finite-Difference Time-Domain Algorithm for the Full-Wave Analysis of Microwave Monolithic Circuit Devices

An alternating-direction implicit finite-difference time-domain (FDTD) algorithm is applied to the full wave analysis of microwave integrated circuit devices. A 3-D multidomain method is developed in nonorthogonal coordinates. Nonorthogonal grids are only used for the anomalistic regions of a complex structure, whereas the standard FDTD lattice is used for the other regions. By using the Jacobian coordinate transformation, curvilinear coordinates can be converted into conventional FDTD format expediently. The perfectly matched layer is used to truncate the boundary. Accurate griddings using the new scheme are obtained, and the complexity of the algorithm is minimal. To illustrate the theory, a sinusoidal plane wave and a Gaussian pulse that propagate through a localized nonorthogonal grid space is used, and the stability of our code is examined. A newly developed compact microstrip bandpass filter is analyzed using the proposed method. The simulated results agree very well with measurements. As compared to other nonorthogonal FDTD (NFDTD) method, the proposed algorithm is much more efficient than other NFDTD counterpart when complex structures are analyzed.      

On the modeling of conducting media with the unconditionally stable ADI-FDTD method

The Courant-Friedrich-Levy stability condition has prevented the conventional finite-difference time-domain (FDTD) method from being effectively applied to conductive materials because of the fine mesh required for the conducting regions. In this paper, the recently developed unconditionally stable alternating-direction-implicit (ADI) FDTD is employed because of its capability in handling a fine mesh with a relatively large time step. The results show that the unconditionally alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method can be used as an effective universal tool in modeling a medium regardless of its conductivity. In addition, the unsplit perfectly matched layer combined with the ADI-FDTD method is implemented in the cylindrical coordinates and is proven to be very effective even with the cylindrical structures that contain open conducting media.     

Analysis of Two Alternative ADI-FDTD Formulations for Transverse-Electric Waves in Lossy Materials

A numerical dispersion analysis of the alternating-direction implicit finite-difference time-domain method for transverse-electric waves in lossy materials is presented. Two different finite-difference approximations for the conduction terms are considered: the double-average and the synchronized schemes. The numerical dispersion relation is derived in a closed form and validated through numerical simulations. This study shows that, despite its popularity, the accuracy of the double-average scheme is sensitive to how well the relaxation-time constant of the material is resolved by the time step. Poor resolutions lead to unacceptably large numerical errors. On the other hand, for good conductors, the synchronized scheme allows stability factors as large as 100 to be used without deteriorating the accuracy significantly.      

Analysis of a double step microstrip discontinuity in the substrateusing the 3D-FDTD method

The finite-difference time-domain (FDTD) method has been applied to the analysis of a double step microstrip discontinuity having thickness changes in the longitudinal direction. The discontinuity occurs in patch antenna feeds or interconnections between microwave planar circuit modules. The simulation results are compared with those computed by HFSS to show a good agreement. An equivalent circuit for the double step discontinuity is developed from the scattering parameters computed by the FDTD method     

Analysis of power delivery network constructed by irregular-shaped power/ground plane including densely populated via-hole

The high speed and low power trend has imposed more and more importance on the design of the power distribution network (PDN) using multilayer printed circuit boards (PCBs) for modern microelectronic packages. This paper presents a fast and efficient analysis methodology in frequency domain for the design of a PDN with a power/ground plane pair, which considers the effect of irregular shape of the power/ground plane and densely populated via-holes. The presented method uses parallel-plate transmission line theory with equivalent circuit model of unit-cell grid considering three-dimensional geometric boundary conditions. Characteristics of PDNs implemented by perforated planes including a densely populated via-hole structure is quantitatively determined based on full-wave analysis using the finite-difference time-domain (FDTD) periodic structure modeling method and full-wave electromagnetic field solver. Using a circuit simulator such as popularly used SPICE and equivalent circuit models for via-hole structure and perforations, the authors have analyzed input-impedance of the power/ground plane pair. Since the presented method gives an accurate and fast solution, it is very useful for an early design of multilayer PCBs.     

Analyzing the stability of the FDTD technique by combining the vonNeumann method with the Routh-Hurwitz criterion

This paper addresses the problem of stability analysis of finite-difference time-domain (FDTD) approximations for Maxwell's equations. The combination of the von Neumann method with the Routh-Hurwitz criterion is proposed as an algebraic procedure for obtaining analytical closed-form stability expressions. This technique is applied to the problem of determining the stability conditions of an extension of the FDTD method to incorporate dispersive media previously reported in the literature. Both Debye and Lorentz dispersive media are considered. It is shown that, for the former case, the stability limit of the conventional FDTD method is preserved. However, for the latter case, a more restrictive stability limit is obtained. To overcome this drawback, a new scheme is presented, which allows the stability limit of the conventional FDTD method to be maintained     

Stability of analogue circuit optimization process and control vector structure

Analogue circuit optimization process may be controlled using a generalized design methodology. Lyapunov function is an integral function that carries information on this process. The use of Lyapunov function concept for a dynamic system allowed for comparison of various design strategies with respect to their stability and convergence. Study of Lyapunov function’s and its derivative’s behavior allowed for revealing significant correlation between this function’s properties and processor time for circuit design. Analysis of processor time dependence on control vector switching points provided a possibility to find this vector’s optimal structure. Numerical results prove bright future of such approach for finding quasi-optimal algorithm of analogue circuits design.     

Modelling of Microstrip Circuit Using a Hybrid PEEC/FDTD Approach

Currently available electromagnetic analysis methods are showing their limitations when large complex circuit structures, such as modern printed circuit boards and their environments, need to be analyzed. Two methods which may be used for the purpose are the partial element equivalent circuit (PEEC) approach and the finite-difference time-domain (FDTD) method. Each of these has their own advantages and limitations. In this paper, a hybrid method is described which overcomes many of the limitations of PEEC and FDTD while retaining their strengths. Results using the method for low-pass and bandpass microstrip filters are presented.     

A Time-Domain Approach for Extracting Broadband Macro- π Models of Differential Via Holes

A novel time-domain approach is proposed to synthesize the broadband macro-pi model of the differential vias based on time-domain reflected/transmitted waveforms either measured by time-domain reflectometry (TDR) or simulated by finite-difference time-domain (FDTD) method. The step responses of the differential via are solved in terms of rational functions by the generalized pencil-of-matrix (GPOM) method. The macro-pi model in terms of the rational functions pairs is obtained through an ABCD matrix transformation. The order of the macro-pi model can be reduced without losing the accuracy according to a residue criterion. The equivalent lumped circuits of the macro-pi model is synthesized by a lumped circuit extraction method (LCEM). The stability and passivity of the extracted models can be preserved based on this time-domain approach and the macro-pi topology. Two examples, one is an asymmentric via and the other is an differential via in four-layer PCB, are used to demonstrate the broadband accuracy of the proposed approach both in time-domain and frequency-domain