General Aspects

Electromagnetic (EM) modeling has been of interest to the authors of this book for a large portion of their careers. Giulio Antonini has been involved with partial element equivalent circuit (PEEC) for over 15 years at the Università degli Studi dell'Aquila, Italy, where he is now a professor. Both Albert Ruehli and Lijun Jiang worked as Research Staff members at the IBM Research Laboratory in Yorktown Heights, New York, on electrical interconnect and package modeling and electromagnetic compatibility (EMC) issues. Lijun Jiang is now a professor at the University of Hong Kong, Hong Kong, and Albert Ruehli is now an adjunct professor at the University of Science and Technology, Rolla, Missouri. We all continue to work today on different aspects of the PEEC method.

We welcome the opportunity to share the product of our experience with our readers. Fortunately, electromagnetic modeling (EMM) is a field of increasing importance. Electronic systems have been and will continue to increase in complexity over the years leading to an ever increasing set of new problems in the EM and circuit modeling areas. The number of electronic systems and applications expands every day. This leads to an ever-increasing need for electrical modeling of such systems.

EMM has been a key area of interest to the authors for quite a while. About 40 years ago, the general field of EMM was very specialized and more theoretical. The number of tools in this area and consequent applications were much more limited. Research is driven by the desire to discover new ways and potential applications as well as the need for solutions of real life problems.

Waveguides that mostly were interesting mechanically complex structures were physically large due to the lower frequencies involved. Some of the main topics of interest were antennas and waveguides as well as transmission lines. EM textbooks usually demanded an already high level of education in the theory and they were sometimes removed from realistic problems.

Transmission lines were the most accessible devices from both a theoretical and a practical point of view. Very few tools were available for practical computations especially before computers were widely available. Computers were mostly used for specialized applications. Problems were solved with a combination of theoreticalanalysis and measurements as well as insight that was a result of years of experience.

In contrast, today electromagnetic solver tools are available for the solution of a multitude of problems. Hence, the theoretical and intuitively ascertained solutions have been replaced with numerical method-based results. However, this does not eliminate the need for a thorough understanding of the EM fundamentals and the methods used in EM tools. The advanced capabilities available in the tools require a deeper understanding of the formulations on which the tools are based. We are well aware that the interaction of tools and theory leads to advances.

Textbooks such as Ramo and Whinnery [1] have evolved over many years. Meanwhile, many new excellent introductory textbooks have been written that treat different special subjects such as EMC [2]. Our book is oriented toward a diverse group of students at the senior to graduate level as well as professionals working in this general area. In our text, we clearly want to emphasize the utility of the concepts for real-life applications, and we tried to include as many relevant references as possible.

Fundamentals of EMM Solution Methods

We have to distinguish between two fundamentally different types of circuit models for electromagnetics. Some of them are based on a differential equation (DE) formulation of Maxwell's equations, while others are based on integral equation (IE) form.

The DE forms are commensurate with the system of equations that results from the formulation of a problem in terms of DEs. This results in circuit models that have neighbor-to-neighbor coupling only. The most well-known form is the finite difference time domain (FDTD) method, which is a direct numerical solution of Maxwell's equations. The advantages of DE methods is that very sparse systems of equation result. At the same time, these systems are larger than the ones obtained from IE-based methods.

On the other hand, the IE-based methods will result in systems that have element-to-element couplings. Hence, this results in smaller, denser systems of equations. The finite element (FE) method is a somewhat hybrid technique since it involves local integrations while the overall coupling is local as in the DE methods. This also results in a large and sparse system of equations. Among the formulations used today, there are two circuit-oriented ones: (a) the DE-based transmission line modeling (TLM) method; and (b) the PEEC method. In this text, we mainly consider the IE-based PEEC method.

The PEEC method has evolved over the years from its start in the early 1970s [3–5]. Interestingly, this is about the same time when the other circuit-oriented EM approach – the TLM method – was first published [6]. Some early circuit-oriented work for DE solutions of Maxwell's equations was done by Kron in the 1940s [7]. However, the solution of the large resultant systems was impossible to solve without a computer. Hence, the work was of little practical importance. Recently, matrix stamps for FDTD models have been presented [8].

Around the same time, numerical DE methods made important progress. The FDTD method was conceived in 1966 [9]. Also, the finite integration technique (FIT) technique was published in 1977 [10]. All these methods have made substantial progress since the early work was published.

More About the PEEC Method

The PEEC method evolved in a time span of more than 40 years. From the start, the approach has been tailored for EMM of electronic packages or Electronic Interconnect and Packaging called signal integrity (SI). Power integrity (PI) and noise integrity (NI) as well as EMC problems. In the beginning, only high-performance computer system modeling needed accurate models for the electrical performance of the interconnects and power distribution in the package and chips. In main frame computes the speed of the circuits was much faster than that of conventional computer circuits such as the early personal computers.

Quasistatic solutions were adequate then even for the highest performance systems. Problems such as the transient voltage drops due to large switching currents were discovered very early. This prompted and extended the work on partial inductance calculations for problems of an ever-increasing size. In the 1990s, the modeling of higher performance chips and packages became an issue with the race for higher clock rates in computer chips. This led to the need for full-wave solutions. As a consequence, stability and passivity issues became important. Today, aspects such as skin-effect loss and dielectric loss models are required for realistic models.

Numerous problems can be solved besides package and interconnect and microwave problems. Approximate physics-based PEEC equivalent circuit models can be constructed, which are very helpful for a multitude of purposes. Further, PEEC is one of the methods used in some of the EMM tools. Fortunately, PEEC models can easily be augmented with a multitude of additional circuit models. This leads to other real advantages. Further, techniques have been found to improve the efficiency of these methods. As we show, PEEC is ideally suited for small simple models. Also, the wealth of circuit solution techniques that are available today can be employed. One example of this is the use of the modified nodal analysis (MNA) approach, which helps PEEC for low-frequency and a dc solution that other techniques may not provide.

Teaching Aspects

We hope that this text can be used as an effective tool to introduce EM to new students. We think that a key advantage of the PEEC method is its suitability for an introductory course in EM.

The teaching of the PEEC method can be approached from several different points of view. It may be used as a way to introduce EMM, since most engineering students are more familiar with circuit theory rather than EM theory. This is also the case since circuit courses are taught at a lower level than EM courses. Alternatively, one may want to start with the introduction of the quasistatic PEEC models in a first EM course.

We prefer to use concepts that can be understood in lieu of the introduction of more advanced topics and mathematical notation. As a second course, general PEEC methods could be covered. This could be done, perhaps, in conjunction with introduction of concepts such as interconnect modeling and other chip and package design concepts.

Albert E. Ruehli,

Windham, New Hampshire,

USA

Giulio Antonini,

L'Aquila,

Italy

Lijun Jiang,

Hong Kong

January, 2017

References

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2. 2. C. R. Paul. Introduction to Electromagnetic Compatibility. John Wiley and Sons, Inc., New York, 1992.
3. 3. A. E. Ruehli. Inductance calculations in a complex integrated circuit environment. IBM Journal of Research and Development, 16(5):470–481, September 1972.
4. 4. A. E. Ruehli and P. A. Brennan. Efficient capacitance calculations for three-dimensional multiconductor systems. IEEE Transactions on Microwave Theory and Techniques, 21(2):76–82, February 1973.
5. 5. A. E. Ruehli. Equivalent circuit models for three dimensional multiconductor systems. IEEE Transactions on Microwave Theory and Techniques, MTT-22(3):216–221, March 1974.
6. 6. P. B. Johns and R. L. Beurle. Numerical solution of 2-dimensional scattering problems using a transmission-line matrix. Proceedings of the IEEE, 59(9):1203–1208, September 1971.
7. 7. G. Kron. Equivalent circuit for the field equations of Maxwell. Proceedings of the IRE, 32(5):289–299, May 1944.
8. 8. A. Ramachandran, A. Ramachandran, and A. C. Cangellaris. SPICE-compatible stamps for semi-discrete approximations of Maxwell's equations. In International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Volume 21, pp. 265–277, October 2008.
9. 9. K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Transactions on Antennas and Propagation, 14(5):302–307, May 1966.
10. 10. T. Weiland. Eine Methode zur Loesung der Maxwellschen Gleichungen fuer sechskompoentige Felder auf diskreter Basis. Archiv der Elektrischen Ubertragung, 31:116–120, 1977.