After reading your recent newsletter on the proximity effect, I reviewed the corresponding chapter in your book. I would like to ask you for a reference dealing with the deduction of formulas 5.1 at p.190 and 5.2 at p.192.
Thanks for your interest in High-Speed Digital Design.
Approximation [5.1] gives the current density in the solid plane underlying a trace. It says the current density at a point H removed from the trace has an intensity approximately proportional to
- I0 is the (AC) current in the signal trace,
- D is the distance from the trace to the point under examination,
- H is the height of the trace above the plane, and
- π equals 3.1415926... .
Approximation [5.1] is asymptotically exact only for a very small, skinny wire (W<<H).
In response to your question, I shall attempt to show that [5.1] is a valid solution to the problem of finding the distribution of returning signal current flowing on a solid plane beneath a small, skinny trace. What I cannot do is explain HOW anyone ever found this solution in the first place. That's lost to history, but doesn't really matter now that we have the solution and can use it.
A professional E&M field specialist would attack the problem like this:
Assume we are looking at an X-Y cross-section of an infinitely long, uniform trace and its associated solid plane (i.e., a microstrip configuration).
Position one small, skinny trace at some height H above an infinite, solid, conducting plane.
Calculate (using Ampere's law) the B- field at every point in space generated by the current flowing in the small, skinny trace.
Next, calculate (using an integral version of Ampere's law) the B-field generated at every point in space due to the flow of current on the solid plane, assuming the current density on the plane follows equation 5.1.
Finally, observe that the Y-component of the B- field from the trace and the Y-component of the B- field from the plane cancel perfectly at the surface of the solid plane.
The Y-component cancellation is the boundary condition for a solid conducting plane. There is one and only one pattern of current flow on the plane that can satisfy the boundary condition. The boundary condition is the genesis of the proximity effect. Were the perpendicular component of the magnetic field to be non-zero, it would generate additional eddy currents, shifting the distribution of current. The good-conductor assumption tells us that eddy currents always act to null out any perpendicular magnetic fields.
The only stable current pattern possible is the one that generates no perpendicular component of the magnetic field.
Here's a detailed derivaion showing all the calculations: proximity3.pdf. (If the problem outline above is not clear to you, then you can just skip the detailed derivation because it won't make any sense either.)
The mathematical details in the .pdf file show the [5.1] is correct, but provide no intuitive justification for it. There's isn't any justification. It is what it is. The way I look at it, [5.1] just happens to describe really well what currents in a flat plane tend to do.
To conclude, let me make a couple of minor points.
As you move away from the centerline of the trace, the distribution of current in the plane falls off as 1/(x-squared). If you think about it, you might recall that the field density from a single wire falls off as 1/x, so it makes sense that the combined field density from a pair of currents (the signal current and its return path along the plane) might fall off one degree faster, or 1/(x-squared).
Also, in the region near x=0 (underneath the trace) the current is fairly constant. I like that. By the way, if you measure the return-current distrubtion underneath a wide trace (W wide comparable to H) you will see that the flat region in the middle extends wider, but the tails on either side still fall off with 1/(x-squared).
Given that equation [5.1] on page 190 is true, and given that the crosstalk at a nearby neighbor trace varies in proportion to the local magnetic field strength at the neighbor's location on the reference plane [5.1] times the height of the neighbor above its reference plane, you can derive equation [5.2] for crosstalk on page 192.
I hope these comments are helpful to you.
Dr. Howard Johnson