There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

Introduction to Electromagnetic Compatibility

Four classes of models of the lightning return stroke are reviewed. These four classes are: (1) the gas dynamic models; (2) the electromagnetic models; (3) the distributed-circuit models; and (4) the "engineering" models. Validation of the reviewed models is discussed. For the gas dynamic models, validation is based on observations of the optical power and spectral output from natural lightning. The electromagnetic, distributed-circuit, and "engineering" models are most conveniently validated using measured electric and magnetic fields from natural and triggered lightning. Based on the entirety of the validation results and on mathematical simplicity, we rank the "engineering" models in the following descending order: MTLL, DU, MTLE, BG, and TL. When only the initial peak values of the channel-base current and remote electric or magnetic field are concerned, the TL model is preferred. Additionally discussed are several issues in lightning return-stroke modeling that either have been ignored to keep the modeling straightforward or have not been recognized, such as the treatment of the upper, in-cloud portion of the lightning channel, the boundary conditions at the ground, including the presence of a vertically extended strike object, the return-stroke speed at early times, the initial bi-directional extension of the return stroke channel, and the relation between leader and return stroke models. Various aspects of the calculation of lightning electric and magnetic fields in which return stroke models are used to specify the source are considered, including equations for fields and channel-base current, as well as a discussion of channel tortuosity and branches.

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Review and evaluation of lightning return stroke models including some aspects of their application

Modern packaging design requires extensive signal integrity simulations in order to assess the electrical performance of the system. The feasibility of such simulations is granted only when accurate and efficient models are available for all system parts and components having a significant influence on the signals. Unfortunately, model derivation is still a challenging task, despite the extensive research that has been devoted to this topic. In fact, it is a common experience that modeling or simulation tasks sometimes fail, often without a clear understanding of the main reason. This paper presents the fundamental properties of causality, stability, and passivity that electrical interconnect models must satisfy in order to be physically consistent. All basic definitions are reviewed in time domain, Laplace domain, and frequency domain, and all significant interrelations between these properties are outlined. This background material is used to interpret several common situations where either model derivation or model use in a computer-aided design environment fails dramatically. We show that the root cause for these difficulties can always be traced back to the lack of stability, causality, or passivity in the data providing the structure characterization and/or in the model itself.

/pdf/stability-causality-and-passivity-in-electrical-interconnect-2tf464oivo.pdf

Stability, Causality, and Passivity in Electrical Interconnect Models

This paper reviews the fundamentals and latest progress of modeling, analysis, and design technologies for signal integrity and electromagnetic compatibility on PCB and package in the past decades. Most results in this field are based on the very rich and highly educational literature produced by Prof. C. Paul in his long scientific career. The inclusion of parameters variability effects is also considered, and it is demonstrated how statistical simulations can become affordable by means of recently-introduced stochastic methods. Finally, the necessity of practical training of designers is mentioned, and an experience relying on realistic PCB demonstrators is illustrated.

Overview of Signal Integrity and EMC Design Technologies on PCB: Fundamentals and Latest Progress

This paper is devoted to transient analysis of lossy transmission lines characterized by frequency-dependent parameters. A public dataset of parameters for three line examples (a module, a board, and a cable) is used, and a new example of on-chip interconnect is introduced. This dataset provides a well established and realistic benchmark for accuracy and timing analysis of interconnect analysis tools. Particular attention is devoted to the intrinsic consistency and causality of these parameters. Several implementations based on generalizations of the well-known method-of-characteristics are presented. The key feature of such techniques is the extraction of the line modal delays. Therefore, the method is highly optimized for long interconnects characterized by significant propagation delay. Nonetheless, the method is also successfully applied here to a short high/loss on-chip line, for which other approaches based on lumped matrix rational approximations can also be used with high efficiency. This paper shows that the efficiency of delay extraction techniques is strongly dependent on the particular circuit implementation and several practical issues including generation of rational approximations and time step control are discussed in detail.

/pdf/transient-analysis-of-lossy-transmission-lines-an-efficient-4yfmc4yyvq.pdf

Transient analysis of lossy transmission lines: an efficient approach based on the method of Characteristics

This paper provides an effective solution for the simulation of cables and interconnects with the inclusion of the effects of parameter uncertainties. The problem formulation is based on the telegraphers equations with stochastic coefficients, whose solution requires an expansion of the unknown parameters in terms of orthogonal polynomials of random variables. The proposed method offers accuracy and improved efficiency in computing the parameter variability effects on system responses with respect to the conventional Monte Carlo approach. The approach is validated against results available in the literature, and applied to the stochastic analysis of a commercial multiconductor flat cable.

/pdf/stochastic-analysis-of-multiconductor-cables-and-4ke8dugivg.pdf

Stochastic Analysis of Multiconductor Cables and Interconnects

This paper addresses the development of computational models of digital integrated circuit input and output buffers via the identification of nonlinear parametric models. The obtained models run in standard circuit simulation environments, offer improved accuracy and good numerical efficiency, and do not disclose information on the structure of the modeled devices. The paper reviews the basics of the parametric identification approach and illustrates its most recent extensions to handle temperature and supply voltage variations as well as power supply ports and tristate devices.

Flavio Canavero

Papers

Stability, Causality, and Passivity in Electrical Interconnect Models

Overview of Signal Integrity and EMC Design Technologies on PCB: Fundamentals and Latest Progress

Transient analysis of lossy transmission lines: an efficient approach based on the method of Characteristics

Stochastic Analysis of Multiconductor Cables and Interconnects

M/spl pi/log, macromodeling via parametric identification of logic gates