## Proximity Effect II

I am reading your excellent book and would like to ask a couple of questions related to the Return Current issue presented in volume 3, #11 of your newsletter Return Current in Plane.

1. Do you have any references dealing with the proximity effect in more detail (other than Terman) and more specifically with the current density distribution in a ground plane under a high frequency signal trace including the derivation of formula [5.2]? I am interested in both treatments mentioned in your answer: numerical derivation as well as the use of partial differential equations.

2. Your statements concerning the current always following the path of least impedance sounds as you mentioned it fundamental. Is there a mathematical proof for the general case including distributed circuits? References again would much appreciated.

Thank you in advance for your reply.

Thanks for your interest in High-Speed Digital Design.

To answer both questions, I will have to point you in the direction of the literature on electromagnetic field simulation. Both effects are consequences of Maxwell's equations. Both can be observed by simulating various field patterns. There is no simple, correct explanation for either effect except merely to present, in a behavioral sense, what the currents tend to do. Good references for E&M field behavior include:

- Alain Bossavit (1998).
*Computational Electromagnetism*. Academic Press, ISBN 0-12-118710-1. - Constantine Balanis (1989).
*Advanced Engineering Electromagnetics*. John Wiley, ISBN 0-471-62194-3.

WARNING: both are highly mathematical.

Here I'll outline one method of simulating the fields for the proximity-effect problem, in the hope that you may glean from this method some insights into how currents flow in solid conductors.

My method is to first model the surface of each conductor involved in the problem. Since the skin effect causes high-frequency currents to flow only near the surface of each conductor, a surface-only model should be adequate for most transmission-line problems. Furthermore, if we assume that the traces proceed in the z direction, and that the current distribution around the circumference of each conductor is constant with linear position z, we need only model the x-y cross-sectional view of the surface of each conductor. So, I begin by breaking up the cross-section of each conductor (traces and planes) into a succession of little elements, and I assume the current density over each element is constant.

Now I can represent the problem as one of finding the current for each little element that satisfies a number of constraints, namely:

- The sum of all currents in the signal conductor equals I.
- The sum of all currents in the return conductor(s) equals -I.
- On the surface of the signal conductor, the total magnetic flux penetrating each segment is zero. This is true if the voltage potential everywhere along the surface of the conductor remains constant, which is pretty much does for a good conductor.
- On the surface of the return conductor(s), the total magnetic flux penetrating each segment is zero.

Another way to state constraints (3) and (4) is to say that the magnetic field is parallel to the conducting surfaces at all points (or that the component of the magnetic field perpendicular to the surfaces must be zero).

In terms of a solution algorithm, if you have N conductor segments, it would be convenient to work with exactly N constraints. We know that (1) and (2) together comprise precisely two constraints, so for constraints (3 and 4), one would normally pick a total of N-2 points at which to evaluate these constraints. Now you have N variables, and N constraints, and there exists one unique solution.

One simple solution procedure calculates the perpendicular magnetic field at each constraint point as a linear function of the currents on all the other segments, constructs a big matrix representing all the constraints, and then inverts it to find a final solution. This is neither the most efficient nor the most accurate solution method, but it's the easiest to visualize. If you know how to write an expression for the magnetic field surrounding one element (a long, straight wire) you can program this in MathCad.

Another way to find the solution is to guess some basic distribution of currents, and then calculate the magnetic fields. If this current pattern were to exist in nature, then wherever the perpendicular magnetic field between two adjacent current elements is non-zero, it would cause an increase in the current in one element and a decrease in the other. Make appropriate adjustments. Then re- compute the field patterns, and adjust again. Iterate until you arrive at the final solution. This is how nature solves the problem.

For the case of an infinitely small, thin trace near a solid reference plane the correct distribution of currents on the solid plane is given by:

^{2}))[1]

**Where:**

- J(D) is the current density at any spot on the plane,
- I0 = the signal current, in amps,
- p = 3.1415926... ,
- H = height of aggressor and victim traces about the nearest reference plane, and
- D = horizontal position relative to centerline of the aggressor trace.

This relation is the basis for my simple crosstalk estimates (and especially the conclusion that, given D much bigger than H, the crosstalk falls off with 1/D squared).

Modeling a realistic-sized trace above a solid plane you will find that the current density is slightly greater on the reference-plane side of the trace than on the back side. This sometimes called the proximity effect. The same thing happens for two skinny wires placed in close proximity: the current tends to concentrate on the two facing surfaces. The proximity effect is a simple manifestation of the general rule that, given a choice, high-speed current tends to concentrate near its return path.

I have presented just an outline of the simulation
procedure. If you want **all** the details, start your research here:

- Bruce Archambeault (1998).
*EMI/EMC Computational Modeling Handbook*. Kluwer Academic Publishers, ISBN 0-412-12541-2.

Best Regards,

Dr. Howard Johnson