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The classic coplanar waveguide (CPW) is formed by a microstrip conductor strip separated from a pair of ground planes pours, all on the same layer, affixed to a dielectric medium. In the ideal case, the thickness of the dielectric is infinite. But in practice, it is thick enough so that electromagnetic fields die out before they get out of the substrate. A variant of the coplanar waveguide is formed when a ground reference plane is provided on the opposite side of the dielectric. This is referred to a conductor-backed or grounded CPW. CPWs have been used for many years in RF and microwave design as they reduce radiation loss, at extremely high frequencies, compared to traditional microstrip. And now, as edge rates continue to rise, they are coming back into vogue. In this month’s column, I will look at how conformal field theory can be used to model the electromagnetic effects of microstrip coplanar waveguides.
Simplistically, space has three dimensions. Picturing a box, we observe the three dimensions of width, height and depth (x,y,z). But, there is an obvious fourth dimension–time. The box will only exist for a certain period of time. These three spatial dimensions plus the temporal dimension are referred to as space-time. But in the intricate world of quantum physics, there can be as many as 26 dimensions used to model the complexities of quantum fields.
In 1921, Theodor Kaluza, a mathematician, proposed that our intuition has misled us and suggested that space-time actually has five dimensions. Kaluza adapted Einstein’s General Theory of Relativity that was formulated to the familiar four dimensions, and rewrote it to apply to five. Surprisingly, these terms corresponded precisely to the description of electromagnetism that James Clerk Maxwell had published decades before. By adding the extra dimension, Kaluza had unified gravitation and electromagnetism–two of the fundamental forces of nature.
This fifth dimension is not apparent to us at the macro scale, as it is a minuscule curling spatial dimension bound by the other larger dimensions. The analogy generally used, to help wrap your head around the concept, is to consider the large dimension to be like a drinking straw. At distant scales of magnification, it appears to be just a straight line. But close up, it has a perpendicular circumference that is curling around the central line of the dimension. This is the compactified small dimension. This fifth dimension represents the varying electric and magnetic fields that radiate at right angles to the central line.
To read this entire column, which appeared in the March 2017 issue of The PCB Design Magazine, click here.