# Maxwell's influence on signal integrity: after almost 150 years, Maxwell's equations still have a strong influence on modern SI tools and methodologies.

IN THE LATE 1800s, James Clerk Maxwell formalized the wave theory of electromagnetic radiation and arranged a set of formulae (called Maxwell's equations) to describe such wave motion (1). 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 spectrum (1) includes radio waves (< 3E9 Hz), microwaves (3E9 to 3E11 Hz), terahertz waves (3E11 to 3E12 Hz), infrared (3E11 to 4E14 Hz), optical/visible light (4E14 to 7.5E14 Hz), ultraviolet (7.5E14 to 3E16 Hz), X-rays (3E16 to 3E19 Hz) and gamma rays (>3E19 Hz).

Maxwell's formulae are available in several styles including time and frequency domain versions (2). 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/[m.sup.2]), D=Electric Flux Density (C/[m.sup.2]), [rho]v=Electric Charge Density (C/[m.sup.3]), B=Magnetic Flux Density (W/[m.sup.2]) and [omega] = Angular Frequency (Radians/sec). Maxwell's equations encompass a wide range of applications and can describe electrical properties (2) 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 law (3). 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 applications (4). 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 capacitance (2), impedance (4), and propagation delay in terms of its physical parameters. A coaxial cable includes four parts (5) as indicated by FIGURE 2.

[FIGURE 2 OMITTED]

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 applied (3) 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 manufacturing (6) 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 simulators (2) 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/ DC magnetic fields and eddy-current solvers applicable for analyses and the design of numerous electronic components.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Let's apply the eddy-current solver of Maxwell SV to compute the field distribution inside a microstrip trace at a frequency of 400 MHz. The geometry as depicted by FIGURE 4 includes a trace having the following: width W = 5 mils, thickness T = 2 mils, height H = 4 mils. The substrate and conductor materials are FR-4 and copper, respectively.

The simulation outcome is presented by FIGURE 5. Current density is described by a color map (with red being the highest and blue being lowest). The results indicate that current intensity is largest close to the conductor surface, due to skin effect.

For skin effect analyses using Maxwell SV, one pitfall to avoid is not to assign a "perfect conductor" as the material property for the trace and the ground plane. A perfectly conducting structure (i.e., infinite conductivity) allows only surface currents with skin depth approaching zero. Subsequently, to produce accurate eddy current results, the trace and ground plane need to be assigned finite conductivity (i.e., for copper conductivity o = 5.6E7 Siemens/m).

Skin depth ([delta]) is defined as the depth below the surface of the conductor at which the current density decays to 1/e (approximately 0.37) of the current density at the surface.

It can be calculated (2) via:

[delta] = 1/sqrt ([TEXT NOT REPRODUCIBLE IN ASCII]) EQUATION 1

Where, [delta] is in meter, [sigma] is conductor's conductivity (in Siemens/m), f is frequency (in Hz), [mu]o is free space permeability (1.257E-6 H/m) and pr is conductor's relative permeability.

For copper (with [sigma] = 5.6E7 Siemens/m, [mu]r = 1), a simpler equation can be obtained:

[delta] = 66/sqrt(f) EQUATION 2

where [delta] is in microns and f in MHz. At f = 400 MHz, for copper Equation 2 yields skin depth [delta] = 3.3 micron (or 0.129 mil).

[FIGURE 5 OMITTED]

The skin effect has important/practical applications in several engineering branches such as design of radio frequency and microwave circuits. The current distribution in most PCB interconnects will be skin-depth limited (2), and resistance will be frequency-dependent for frequency components exceeding 10 MHz. Skin effect can be measured (7) by applying Time Domain Transmission (TDT) techniques.

The skin effect losses may be compensated (8) by plating the outer conductor surfaces with gold (for microwave/RF PCBs) to enhance conductivity, or by increasing the width of traces (applicable for logic PCBs) to enlarge the surface area.

ACKNOWLEDGEMENT

I would like to thank the members of Broadcom SI Council, especially Sam Karikalan (chairman) and Amit Agrawal (co-chairman).

REFERENCES

(1.) Andrew Zimmerman Jones, "Electromagnetic Spectrum of Light," About.com: Physics.

(2.) Eric Bogatin, Signal Integrity--Simplified, Prentice Hall, 2004, pp. 23-26, pp. 109-192, pp. 340-342.

(3.) Peter Signell, "Gauss's Law For Spherical Symmetry" Physnet, MISN-0-132, Michigan State University.

(4.) "Introduction to Common Printed Circuit Transmission Lines:' Dallas Semiconductor/ Maxim Application Note 2093, June 2003.

(5.) Howard Johnson and Martin Graham, High-Speed Signal Propagation: Advanced Black Magic, Prentice Hall, 2003, pp. 513-514.

(6.) Timothy Hochberg and Henri Merkelo, "Propagation Characteristics of Coaxial Cable with a Helically Wound Ground Shield," 1996 IEEE, pp. 888-892.

(7.) Roy G. Leventhal and Lynne Green, Semiconductor Modeling For Simulating Signal, Power, and Electromagnetic Integrity, Springer, 2006, pp. 34 and 126.

(8.) Lee W. Ritchey, Right the First Time: A Practical Handbook on High-Speed PCB and System Design, Vol. 1, Speeding Edge, 2003, p. 19.

DR. ABE (ABBAS) RIAZI is a senior staff electronic design scientist with SewerWorks (a Broadcom Company) in Santa Clara, CA; ariazi@serverworks.com.
```FIGURE 1. Maxwell's equations.

[DELTA] x E = -j[omega]B Faraday's law

[DELTA] x H = J + j[omega]D Ampere's law

[DELTA] x D = [[rho].sub.lambda] Gauss' law

[DELTA] x B = 0 Gauss' law (magnetic)
```
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