Wizards of Microwave—Follow the Copper Trace Road

By Colin Warwick

There are many practical difficulties that have to be solved in a signal integrity project, but two difficulties have their roots deep in the fundamentals of Maxwell’s equations.

One is obvious: each 0 or 1 pulse is actually an electromagnetic wave that travels at finite speed. The speed is named after as that famous electromagnetic wave: light. In vacuum, this universal speed limit is about one inch per 84.7 picoseconds, or one centimeter per 33.4 picoseconds. For striplines in FR4 board it’s about half that. Although the wave is guided by the copper trace, the wave travels in the insulating material surrounding the trace. The speed is set by the dielectric properties called permeability and permittivity:

c = (μ0μre0er)-0.5

(The wave does penetrate the copper somewhat, but it dies away exponentially from the metal surface, with a decay constant equal to the skin depth.)

But another fundamental limitation is also related to these dielectric material properties. The relative magnitudes of the electric field E and the magnetic intensity H are called the impedance, defined by Z=E/H. The impedance for free propagation is:

Z = (μ0μr/e0er)0.5

For vacuum, this is about 377 O and for FR4 it’s about half that. (Hence the retro graphic at the head of this post: the old color code for a five-band 377 O resistor was orange-purplepurple- black) For a guided wave, the impedance is generally less than that for free propagation. You can divide the free...

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