Proximity Effect III

HIGH-SPEED DIGITAL DESIGN - online newsletter -
Vol. 4 Issue 8

*---------------------(QUESTION)---------------------*

PROXIMITY EFFECT III

Erwin Wechsler writes:

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.

Thank you.

*----------------(DR. JOHNSON REPLIES)---------------*

Dear Erwin,

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/(pi*H))/(1 + (D/H)^^2),

where I0 is the (AC) current in the signal trace,

where D is the distance from the trace to the point under examination,

where H is the height of the trace above the plane, and

where the symbol "^^2" means "squared" (i.e., raised-to-the-second-power) in ASCII-speak.

Where the symbol "pi" means 3.1415926...

Approximation [5.1] is asymptotically exact only for a very small, skinny wire (W<<H).

What I can do for you is show that [5.1] is the one and only correct 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 someone ever figured out that this was the correct solution to what will prove below to be a complicated and messy set of field equations).

The problem proceeds 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 small, skinny trace at some height H above an infinite, solid, conducting plane.

I will first calculate (using Ampere's law) the B- field at every point in space generated by the current flowing in the small, skinny trace.

Then I will 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.

Next, I will show 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.

That's the boundary condition for a solid conducting plane. There is one and only one pattern of current flow on the plane that can generate this boundary condition. This 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 the eddy currents will act to null out any perpendicular magnetic fields.

The only stable current pattern is the one that generates no perpendicular component of the magnetic field.

(If this part isn't clear to you, then you can just skip the detailed derivation because it won't make any sense either. Otherwise, please see the detailed derivation at proximity3.htm.)

I realize that the mathematical details in the .jpg file obscure the basic fact that equation [5.1] describes really well what currents in a flat plane tend to do. There actually isn't any other intuitive justification for it, except to point out a couple of facts.

First, the distribution falls off as 1/(x-squared). If you think about it, I'm sure you will recall that the field density from a single wire falls off as 1/x, so the field density from a pair of currents (the signal current and its return path along the plane) must 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 use a wider trace (W comparable to H) you will see the flat region in the middle extend wider, with the 1/(x-squared) tails falling off to each side.

Given that equation [5.1] on page 190 is true, and given that the crosstalk will be proportional to the local magnetic field strength near the reference plane [5.1] times the height of the victim above the reference plane, you can derive equation [5.2] for crosstalk on page 192.

I hope these comments are helpful to you.

Best regards,
Dr. Howard Johnson

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