Transient thermal analysis of multilayered structures using Green's functions

This paper presents an approach to the analysis of transient thermal states in electronic circuits using an analytical solution of the heat equation. Fully three-dimensional analytical time dependent solutions are determined for multilayered structures with the help of Green's functions (GFs). The GFs required for the solution are found using a Galerkin-based integral method. The entire solution method is illustrated in detail using a practical example, where the results of transient thermal simulations of a real hybrid power module are compared with infrared measurements.     

Integrated transient thermal and mechanical analysis of molded PBGApackages during thermal shock

During thermal shock, large thermal gradients exist within a molded plastic ball grid array (PBGA) package. The conventional assumption of uniform temperature distribution becomes invalid. In this paper, an integrated thermal-mechanical analysis was performed to evaluate the transient effect of thermal shock. For comparison, an isothermal analysis was also conducted. The computational fluid dynamics (CFD) method was used to obtain the thermal boundary conditions surrounding the package. The heat transfer coefficient obtained through CFD was compared to two analytical solutions. It was found that the analytical values were not acceptable in the time period of interest. Therefore, to obtain the actual maximum die stress, CFD solution has to be used instead of analytical solutions to derive the thermal boundary condition. This boundary condition was then applied to the package and a sequentially coupled heat transfer and thermal stress analysis was performed. The transient analysis has shown that high stresses occur in the die due to thermal shock, which can not be seen under the traditional isothermal assumption. The impact of plastic ball grid array (PBGA) package parameters on transient die stress was also studied, including mold thickness and substrate thickness. The results in this paper could be applied to either wire bond or flip-chip PBGA packages     

三维集成电路稳态解和瞬态解的建模与热分析

针对三维集成电路(3D IC)热效应问题,提供了三维集成电路热模型的稳态解析解和瞬态解析解,这些解适用于N层(N≥2)模型;热源可以非均匀分布,瞬态时热源的大小可随时间变化。求解的过程使用了分离变量法、格林函数法和本征函数正交性。通过瞬态解可以获得任意时刻模型的温度场,稳态解是与时间无关的函数,通过它可以直接计算出稳态热传导的温度场。使用所获得的解析解在MATLAB中计算得到的温度场和COMSOL仿真温度场进行比较,结果表明,稳态解计算3层和5层3D IC模型得到的结果和COMSOL仿真结果最大误差在1%左右;使用所获得的瞬态解计算3层和5层3D IC模型得到的结果和COMSOL仿真结果最大误差不超过3%。     

Three-dimensional transient thermal simulation: application to delayed short circuit protection in power ICs

Thermal effects may represent a limiting factor in the development of integrated circuits. As the power dissipated by integrated circuits becomes more relevant, the need increases for accurate modeling of the stationary and transient thermal behavior of the die-package structure. An analytical solution of the three-dimensional transient thermal diffusion problem is presented for a two-layer structure, together with a simple computer program for the calculation of the solution. The program, implemented on a minicomputer, is proven to be fast and accurate. The simulation technique is then applied to the design of a new short-circuit protection of a 6A current booster.     

Transient thermal analysis of integral-heat-sink GaAs pulsed Gunn device

A technique is outlined for the estimation of the transient thermal impedance of the commonly used microwave device structure employing an integral heat sink (i.h.s.). The structure is analysed in three separate regions, approximations being made to the analytical solution of the classical heat flow equation applied to each. Theoretical predictions are compared with experimental observations on a typical i.h.s. Gunn device, and the implications on device design considered.     

On the modeling of the transient thermal behavior of semiconductordevices

A mathematical model of the transient temperature response of integrated devices is presented which takes into account the three-dimensional (3-D) nature of heat flow and the physical structure of the device. Simple analytical relations for the transient thermal impedance and thermal time constants are derived for the first time. The impact of device geometry on the transient thermal response curve is discussed, and simple guidelines for the thermal design of solid-state devices operated in transient or pulsed regime are given     

瞬变等离子体中电磁波频率漂移特性研究

推导一维瞬变磁化等离子体的时域有限差分(FDTD)递推式,并以一维金属谐振腔作为计算模型,分析了瞬变等离子体中电磁波频率漂移特性。从理论分析出发,得到了一维瞬变等离子体对电磁波作用的解析解。通过选取相同的模型和参数,将FDTD数值解与解析解进行对比,验证了所用FDTD方法的准确性,在此基础上研究了瞬变等离子体中电磁波的频率漂移规律。     

Evaluation of analytical models for thermal analysis and design of electronic packages

The objective of this study is to evaluate the use of several analytical compact heat transfer models for thermal design, optimization, and performance evaluation in electronic packaging. A model for heat spreading in orthotropic materials is developed. The developed model is used in conjunction with the other available heat transfer models in a resistance network for calculation of heat transfer rate and junction temperatures in a multi-chip module (MCM). Refrigeration cooled MCM of an IBM server is used to illustrate the methodology. Results of the analytical model and resistance network analysis are compared with a numerical solution. Capability of the analytical model in predicting the thermal field is discussed and effectiveness of using the analytical models in thermal design and optimization of electronic packages is demonstrated.     

平面电磁波对双导线传输线终端瞬态响应的求解

随着传输线方法在电磁兼容和电磁干扰分析方面的广泛应用,人们对传输线方程的求解方法也越来越感兴趣.求解传输线方程的方法通常有时域差分方法和频域差分法.而对于平面电磁波对双导线传输线终端的响应,文章给出了其频域解析解,并且运用这个解析解和Fourier变换技术,可以很容易地给出传输线终端的瞬态响应分析.通过数值仿真实验研究三个角度参数φ、α和Ψ在电磁感应中的作用:当φ和α保持不变(或者Ψ和α保持不变),随着Ψ(或者φ)的增大,终端2的感应电流的响应时间缩短,第一阶段的感应时间延长;当φ和Ψ保持不变,随着α的增大,终端2处第一阶段的感应电流也随之减小.     

Developing a Simplified Analytical Thermal Model of Multi-chip Power Module

The study of the thermal behavior of power modules has become a necessity regarding the known rapid development in modern power electronics, and the prediction of temperature variation has generally been performed using transient thermal equivalent circuits. In this paper we have developed a simplified analytical thermal model of a power hybrid module. This analytical method is used to evaluate the thermal parameters of a device. The model takes into account the thermal mutual influence between the different module chips based on the analytical method. The thermal interaction between components is dependent on the boundary condition, the dissipated power value in the different components and the number of operating chips constituting the module. This effect is modelled as a source energy and a thermal resistance simply computed tanks to reasonably low measurement applied on the module. The derived thermal models offer an excellent trade-off between accuracy, efficiency and CPU-cost.     

Transient response of an infinite cylindrical antenna

A simple analytical formula derived from Wu's exact solution for the transient response of an infinite cylindrical antenna is found to be extremely accurate. Existing results from other numerical schemes are also compared to Wu's solution and found to be in good agreement.     

The Effect of Radial Node Spacing on Finite Difference Calculations of Temperatures in Living Tissues

Finite difference calculations of the transient radial temperature distribution in living tissue are compared with an analytical solution in which metabolism, blood flow rate, and arterial blood temperature vary with time. The effects of node spacing and integration time interval on the accuracy of the finite difference calculations are investigated.     

Bayesian surrogates for integrating numerical, analytical, andexperimental data: application to inverse heat transfer in wearablecomputers

Wearable computers are portable electronics worn on the body. The increasing thermal challenges facing these compact electronics systems have motivated new cooling strategies such as transient thermal management with thermal storage materials. The ability of building models to assess quickly the effect of different design parameters is critical for effectively incorporating innovative thermal strategies into new products. System models that enable design space exploration are built from different information sources such as numerical simulations, physical experiments, analytical solutions and heuristics. These models, called surrogates, are nonlinear, adaptive, and suitable for system responses where limited information is available and few realizations of experiments or numerical simulations are feasible. This paper applies a Bayesian surrogate framework to estimate values for unknown physical parameters of an embedded electronics system. Physical experiments and numerical simulations are performed on an embedded electronics prototype system of a wearable computer. Numerical models for the experimental prototype, which involve five and three unknown parameters, are implemented with and without thermal contact resistances. Through the use of orthogonal arrays and optimal sampling, an efficient exploration of the parameter space is performed to determine thermal conductivities, thermal contact resistances and heat transfer coefficients. Surrogate models are built that combine information obtained from numerical simulations, experimental model measurements and a thermal resistance network. The integration of several information sources reduces the number of large-scale numerical simulations needed to find reliable estimates of the system parameters. For the embedded electronics case, the use of prior information from the thermal resistance network model reduces significantly the computational effort required to investigate the solution space     

Transient analysis of radiated field from electric dipoles andmicrostrip lines

The transient response of the field radiated from electric dipole sources and microstrip lines with arbitrary time waveforms is analyzed. By using the three-dimensional Fourier transform of the spectral dyadic Green's function, relative to the integrated structure, an analytical closed-form solution is obtained and equivalent circuits for the analysis of the transient phenomena in the far-field region are derived. Numerical examples are provided to obtain noticeable insights to the behavior of planar circuits working under Gaussian pulsed excitation     

Analysis of transient and steady-state processes in the series (resonant)inverter

This paper deals with analysis of the transient and steady-state processes in series (resonant) inverters. Sets of differential and difference equations governing the transient and steady state processes are obtained and solved. The solution is found in closed analytical form. Theoretical and experimental results, both for transient and steady-state conditions, are compared and satisfactory agreement shown.     

Pulse response of linear dipole antenna

Applications of transient electromagnetics include nonnuclear electromagnetic pulse (EMP) effects on electronic systems to evaluate system vulnerability. A basic component used in the mathematical modeling of systems for EMP vulnerability is the linear dipole antenna. Wu has derived an approximate analytical solution for the pulse reflected from the feed point of a linear dipole antenna for an incident ideal step function of current. The solution does not include the part associated with the very high frequency components because of an approximation used in evaluating the integral describing the reflected pulse. The approximation is eliminated and a simple analytical solution is given that describes that portion of the step function response associated with the very high frequency components. Numerical evaluation of the solution demonstrates convergence with Wu's approximate solution.     

Analytical solution for early-time transient radiation frompulse-excited parabolic reflector antennas

A closed-form analytical solution is developed for predicting the early-time transient electromagnetic fields which are generated by a perfectly conducting parabolic reflector antenna when it is illuminated by a transient step spherical wave due to an elemental Huygen's source located at the focus. This closed-form time-domain solution, which is valid both near and far from the reflector (and anywhere in the forward region) can be used via the convolution theorem to efficiently obtain the early-time transient fields generated by the same parabolic reflector antenna when it is illuminated by a realistic finite-energy pulse which emanates as a spherical wave from the focus. The transient solution is developed here by analytically inverting, in closed form, the corresponding frequency-domain solution in terms of a radiation integral that employs an asymptotic high-frequency geometrical optics (GO)-based approximation for the fields in the aperture. Numerical results are presented for the transient fields both near and far from the reflector. The fields on boresight exhibit an impulse-like behavior similar to that of the impulse radiating antenna (IRA) introduced by Baum et al. (1989, 1993)     

Thermal analysis of layered electronic circuits with Green's functions

This paper presents a method for thermal simulation of electronic circuits using an analytical solution of the three-dimensional heat equation resulting from an appropriate circuit thermal model. The temperature fields in multilayered structures are computed analytically employing the Green's functions solution method. The entire solution methodology is illustrated in detail on the particular examples of electronic circuits containing multiple heat sources. Compared to the previous papers published by the authors, the method has been extended by including the possibility of simulating imperfect layer contacts. The simulation results are validated with infra-red measurements and results obtained using other methods. Additionally, the discussion of simulation errors caused mainly by different non-linear phenomena is included.     

A new approach to the dynamic thermal modelling of semiconductor packages

This paper describes a novel approach to the dynamic thermal characterisation of semiconductor packages. Conceived as an extension of the variable angle model for multilayer structures, the thermal capacitance of each layer is evaluated at the same time and for the same volume used to calculate the thermal resistance, which guarantees the coherence. Because this volume or lump is a function of system geometry, materials and boundary conditions, the model can be said to include 3D aspects. The thermal resistances and capacitances obtained from the model, each pair representing a layer of the structure, are used as input to an electric circuit simulator to obtain the transient response of the package.Moreover, some common assumptions usually made when interpreting the transient thermal response of semiconductor packages are revised and discussed in an attempt to clarify the scene. Particularly, the log-of-time derivative of the step response is shown to give all the information about the dynamic response when the characteristic time constants of the system are widely separated. In that particular case, otherwise very common in practice, a novel analytical method for the obtention of the characteristic time constants and associated amplitudes is presented and their results evaluated for different industrial power packages.     

Integrated modeling of microwave food processing and comparison with experimental measurements.

This paper presents an integrated electromagnetic and thermal model for the microwave processing offood packages. The model is developed by combining the edge finite element formulation of the 3-D vector electromagnetic field in the frequency domain and the node finite element solution of the thermal conduction equation. Both mutual and one-way coupling solution algorithms are discussed. Mutual coupling entails the iterative solution of the electromagnetic field and the thermal field, because the physical properties are temperature-dependent. The one-way coupling is applicable when the properties are temperature independent or this dependence is weak. Mesh sensitivity and shape regularity for the edge element based formulation for computational electromagnetics are discussed in light of available analytical solutions for a simple wave guide. The integrated model has been used to study the electromagnetic and thermal phenomena in a pilot scale microwave applicator with and without the food package immersed in water. The calculated results are compared with the experimentally measured data for the thermal fields generated by the microwave heating occurring in a whey protein gel package, and reasonably good agreement between the two is obtained.