# Re: more questions about buried PCB capacitors

Guang-Tsai Lei ([email protected])
Mon, 10 Jun 1996 21:59:16 -0500

<[email protected]>

As someone remarked, it seems reasonable to suppose that the sandwich
of two power planes behaves as a "radial transmission line". When you
inject current somewhere, a circular wavefront should move away from
the point of injection.

How does the amplitude of the radiated wave "die off" as it travels?

>From what I can tell, it's not dropping "nicely" like 1/r or log(r),
but rather as something very, very nasty.

It would be interesting to know the answer, since this radiated wave
might be able to upset the supply levels of nearby components
connected to the same power-ground sandwich.

I'm curious if anyone has better training/insight/empirical evidence
here...

--Eric

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For a semi-infinitely long radial transmission line, the behavior of the
voltage (E-field) wave can be described by a POSITIVELY traveling wave or the
Bessel function (the engineering convention: the Bessel function of the second
kind) from an axi-symmetric voltage generator. The amplitude of the voltage
wave drops like a square root of a summation of the square of two Bessel
functions and the phase change is like an ac-tangent of the ratio of one
Bessel function to the other. Similar to the uniform transmission line, when
the line is terminated (not properly), both forward and reflected waves exist,
the general solution to the radial line problem becomes a linear combination
of the Bessel functions of the second and first kind. S. Ramo described the
characteristics of the wave in detail in his book, Fields and Waves in
Communication Electronics, Chp 18, 1967 edition.

When loads are arbitrary, the boundary shape is not round, and there are some
multi-frequency distributed sources, it becomes a complecated non-homogeneous
boundary value problem. It's interesting to find out the behavior of a
resonant quasi-radial transmission line in comparison to the well developed
theory for an uniform transmission line. We've been developing software
combining an analytical technique with numerical methods to see insights into
the root causes of power/ground plane noise so that simulators can be used more
efficiently. We'll have a couple of papers to be published on this topic.

Guang-Tsai Lei
Special Purpose Processor Development Group
Mayo Foundation, Rochester, MN 55905