The tensorial approach remains one of the less explored circuit theories to solve the modern printed circuit board (PCB) problem. Chapter 2 of the proposed book will open the door allowing us to the non-specialist and even non-expert of circuit design to learn and to practice with their own easy way and rapidly the tensorial analysis of networks (TAN) approach. The fundamental approach to use the tensor algebra will be introduced. The elementary and basic knowledges necessary to elaborate the TAN concept will be defined. An easy concept of physical, classical, graph topological and tensorial object representation of the primitive elements to treat the PCB signal integrity (SI), power integrity (PI) and electromagnetic compatibility (EMC) problems will be described. Several basic illustrative examples of key understanding lumped RC, RL, LC and RLC network-based circuits will be treated in the chapter. The methodologies integrating the topological graphs of the problem will be explained. The different basic steps to practice the TAN approach are indicated. For the concrete comprehension, the different practical cases for solving PCB SI, PI and EMC problems are introduced.

Despite the numerous works done about the transmission line modelling of the printed circuit board (PCB) electrical interconnects, huge efforts are still open to the signal integrity, power integrity and electromagnetic compatibility engineers to predict the PCB trace effects during the design phase. The extraction method of the interconnect network as a multiport system will be defined based on a topological algebra. The existing theory and the simulation tools are either not flexible enough or do not allow one to understand the electrical behaviours of the PCB interconnects. The system and circuit theory about the impedance (Z), admittance (Y), transfer (T) and scattering (S) matrices will be provided in the present chapter. The study will be applied to different topologies of interconnect structures as multiport graph topologies. Pedagogical approaches enabling one to practice easily the tensorial analysis of networks (TAN) concept in function of the interconnect structures will be treated in this chapter. The theory representing the interconnect as tensorial objects in branch and mesh spaces will be provided. The problem resolution via the calculations of mesh currents into the Z/Y/T/S-matrices will be developed. Illustrative application examples of microstrip, coplanar and multilayer PCB interconnect will be presented. The possibility of using the TAN model for predicting the interconnect behaviour in broadband frequency from DC to several GHz will be demonstrated at the end of the chapter.

The time domain (TD) modelling of the lumped components-based low and medium speed printed circuit board (PCB) is examined in the present chapter. The TD translation of the physical-classical electric circuit-tensorial analysis of networks (TAN) graph dictionary will be established. Then, the workflow of the methodology describing the routine algorithm of the TD Kron's model will be introduced. By considering pulse input signals, the computation methods of the TAN method resolution based on the time-different iterative discrete solver are established. The feasibility of the TAN TD model will be verified with examples of PCB systems excited by the previously cited test signals. Discussion will be made about the accuracy, advantages and limits of the TAN TD computation methods.

This chapter is focused on modelling of radiated electromagnetic compatibility (EMC) coupling onto the multilayer printed circuit board (PCB). Kron's method integrates the electromagnetic (EM) emission, Taylor's and field-to-interconnect coupling models. The equivalent graph of the field-to-interconnect coupling is established. The modelling methodology consists in defining the primitive subnetwork elements. These primitive elements are represented by vias, interconnect lines and pads. Kron's graph equivalent to the EMC problem is elaborated. Finally, the coupling voltages are calculated via the tensorial equation translated from the graph. The radiated EMC Kron's model is validated with a four-layer PCB from 0.4 to 1.4 GHz by two scenarios of EM radiation. As proof of concept, a prototype of four-layer PCB was designed, fabricated, tested and simulated in full wave with a commercial three-dimensional EM tool. For the first case, the multilayer PCB was illuminated by plane wave emission propagating in different directions. The numerical computation from Kron's formalism was compared with simulation and measurement. The other case is the field-to-interconnect coupling between a microstrip I-line PCB, as an EM field emitter, and the multilayer PCB, as a receiver, in 1-m distance. For both cases, the simulated and calculated voltage couplings onto the multilayer PCB are in good agreement.

The complementary aspect of the electromagnetic compatibility (EMC) emission is the susceptibility issue. The present chapter will focus on the EMC conducted susceptibility (CS) of hybrid printed circuit board (PCB) system with specifications described in the previous chapter. The tensorial description of the PCB susceptible components as mathematical sensitive functions integrating subdomain aspect will be originally introduced. The subdomain functions act as sigmoidal mathematical functions depending on the specification of the EMC perturbations. After the analytical description of the susceptibility functions, the workflow of the tensorial analysis of networks (TAN) modelling methodology indicating the routine algorithm of the EMC analysis will be provided. Then, random risk analyses table, including the objective functions indicating the EMC severity and the damage severity, will be addressed. To validate the EMC CSTAN model, illustration example of PCB system with the prediction of EMC severity quantification with classes A, B and C will be discussed. Then, the accuracy, advantages and limits of the EMC CS TAN model will be presented at the end of the chapter.

This chapter introduces coaxial cable S-matrix modelling with tensorial analysis of networks (TAN). The main objective of the modelling is to determine the shielding effectiveness (SE) of shielded cable coupled with a loop probe. TheTAN methodology from the equivalent graph elaboration to the Kron's branch and mesh space analyses and ended by Z-matrix is elaborated. The SE is formulated innovatively from the S-matrix. A proof-of-concept constituted by a centimetre-length braid shielded cable illuminated by a proximate millimetre-radius circular was designed and simulated with commercial full-wave simulator. The wide band computed and simulated results with parametric analysis in function of the loop probe position are in good agreement.

Using near-field (NF) scan data to predict the far-field (FF) behaviour of radiating electronic systems represents a novel method to accompany the whole RF design process. This approach involves so-called Huygens' box as an efficient radiation model inside an electromagnetic (EM) simulation tool and then transforms the scanned NF measured data into the FF. For this, the basic idea of the Huygens'box principle and the NF-to-FF transformation are briefly presented. The NF is measured on the Huygens' box around a device under test using anNF scanner, recording the magnitude and phase of the site-related magnetic and electric components. A comparison between a fullwave simulation and the measurement results shows a good similarity in both the NF and the simulated and transformed FF.Thus, this method is applicable to predict the FF behaviour of any electronic system by measuring the NF. With this knowledge, the RF design can be improved due to allowing a significant reduction of EM compatibility failure at the end of the development flow. In addition, the very efficient FF radiation model can be used for detailed investigations in various environments and the impact of such an equivalent radiation source on other electronic systems can be assessed.

The contents of these chapters illustrate the investigation on the unfamiliar tensorial analysis of networks approach proposed in this book. This final chapter summarizes the main potential contributions of all chapters (Chapters 2-15) of this book. In the fields of the electromagnetic compatibility (EMC), signal integrity and power integrity engineering, the book provides new ways to analyse the printed circuit board (PCB) problems. One of the main modellings is developed in the first 11 chapters (Chapters 2-13). Chapter 14 presents a technique of EMC conducted susceptibility analysis of digital components used regularly in the PCBs. Then, investigations on the EM NF radiations from the planar PCBs based on the scanning technique are developed in Chapters 15 and 16. Then, the concluding technical chapter proposes an overview of PCB numerical modelling.