Printed Circuit Design & Fab - December 2007 - (Page 22)

Maxwell’s Influence on Signal Integrity After almost 150 years, Maxwell’s equations still have a strong influence on modern SI tools and methodologies. INTHE LATE 1800s, James Clerk Maxwell formalized the wave theory of electromagnetic radiation and arranged a set of formulae (called Maxwell’s equations) DR. ABE to describe such wave (ABBAS) RIAZI motion1. Maxwell’s equations form the foundation of classical work in circuits and electromagnetics. They portray the interrelationship between electric fields, magnetic fields, electric charge and electric current. Subsequently, these formulae illustrate the nature of interaction between electromagnetic fields, conductors and dielectric materials. The electromagnetic spectrum1 includes radio waves ( 3E19 Hz). Maxwell’s formulae are available in several styles including time and frequency domain versions2. One form is presented in FIGURE 1. In Figure 1, E=Electric Field Intensity (V/m), H=Magnetic Field Intensity (A/m), J=Current Density (A/m2), D=Electric Flux Density (C/m2), ρv=Electric Charge Density (C/m3), B=Magnetic Flux Density (W/m2) and ω=Angular Frequency (Radians/sec). Maxwell’s equations encompass a wide range of Δ x E = -jωB x H = J + jωD • D = ργ •B=0 Faraday’s law Ampere’s law Gauss’ law Gauss’ law (magnetic) applications and can describe electrical properties2 of interconnects. Let us briefly consider the significance of one of these four great laws of electricity and magnetism, namely Gauss’s law. It is formulated in terms of continuous charge distributions and can be utilized to derive Coulomb’s law3. Gauss’s law is well suited for ascertaining the electric fields at various points in space when the charges producing the fields are symmetrically distributed. Such situations frequently happen in the design of electronic components. An example is offered by coaxial cable, which is widely used in engineering laboratories and also possesses some interesting PCB applications4. For instance: (i) the pad for a PCB via and its surrounding ground plane cutout can form a very short section of a coaxial line (allowing use of coaxial line formulae to achieve impedance matching); (ii). Semi-rigid coax lines with very small outer diameters are available that can provide a fully shielded path for the signals on PCB in low-volume production, prototyping and reworks. The coaxial cable forms a uniform two-conductor (controlled impedance) transmission line and exact mathematical expressions exist for its capacitance2, impedance4, and propagation delay in terms of its physical parameters. A coaxial cable includes four parts5 as indicated by FIGURE 2. An enlarged view of an inner conductor and the surrounding insulation/ dielectric layer is illustrated by FIGURE 3. Figure 3 demonstrates that the charges within the dielectric region possess cylindrical symmetry. Gauss’s law can then be applied3 to ascertain the electric field inside or outside the cable. The field computation results would then reveal how the cable affects the signal traveling along its length, and how to ascertain the optimum cable parameters. Sometimes the coaxial cables depart from an ideal symmetrical geometry due to manufacturing6 limitations. Simulation may be then required for determining various characteristics (such as propagation delay) of the coax cable. For complex geometries, Maxwell’s equations usually demand simulation. Three types of electrical simulators2 include: (i) behavioral simulators, (ii) circuit simulators, and (iii) electromagnetic (EM) simulators. The EM simulators are well suited for solving Maxwell’s equations. Speed 2000 (Sigrity), Sonnet Lite (Sonnet Software Inc.) and Maxwell (Ansoft) are among electromagnetic software programs. A free version of Maxwell 2D software (called Maxwell SV) is available from the Ansoft Web site. The program contains advanced 2D electric fields, AC/ 4 3 2 1 FIGURE 1. Maxwell’s equations. 22 PRINTED CIRCUIT DESIGN & FAB Δ Δ Δ FIGURE 2. Cross section of a an RG174 coax cable: 1) center conductor; 2) insulation; 3) shields; 4) cover. FIGURE 3. Coax cross section displaying symmetrical charge distribution. DECEMBER 2007

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Printed Circuit Design & Fab - December 2007

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