The 3D Smith Chart: From Theory to Experimental Reality
Kirt's Cogitations™ #320

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The world was introduced publicly in 2007 to the concept of a 3-dimensional Smith chart by Chris Zelley. In article published in IEEE Microwave Magazine entitled, "A Spherical Representation of the Smith Chart," the radically new concept was illustrated on the surface of a ping pong ball using a felt-tipped pen (thumbnail at left). Inspired by the sight, Andrei A. Muller and a small team of developers in 2013 created a version of the 3D Smith Chart in software using the Java language that allows it to execute on any platform. In 2017, an expanded functionality commercial version of 3D Smith Chart was released at a very modest price. A number of articles have been published on the topic extolling the unique ability of a 3-dimensional complex impedance volume to expose properties not immediately apparent in a 2-dimensional plane. Recently, Andrei A. Muller, Victor Asavei, Alin Moldoveanu, Esther Sanabria-Codesal, Riyaz A. Khadar, Cornel Popescu, Dan Dascalu, and Adrian M. Ionescu published an article entitled, "The 3D Smith Chart: From Theory to Experimental Reality" (here is a public access version). It is to date the most comprehensive and comprehendible treatise on their creation. I repeat my prediction that at some point the 3D Smith Chart will become the de facto standard integrated into professional grade simulation packages.

As of April 2020, the 3D Smith Chart tool is free undergraduate, master or PhD university students for academic projects! For the free trial version please contact us via contact form or on 3dsmithchart@gmail.com

The 3D Smith Chart: From Theory to Experimental Reality (an excerpt from the full article)

"[...]In this article, we start with several drawings of spherical Smith charts, proposed with the aim of having a generalized chart that deals with both negative and positive resistance circuits within a compact surface. Then, we introduce our suggested 3D Smith chart, discussing its main equation from an inversive geometry perspective, for which Möbius transformations are a simple type of transformation, mapping circles into circles on a special sphere (called a Riemann sphere); doing so leads to mathematical completeness and simplicity when dealing with negative resistance and infinity.

Based on this, the article takes the reader on a journey to a compact tool where all negative and positive resistances are analyzed on a preserving-circles patterned sphere. We propose to use the 3D space surrounding the sphere for multiple-complex- scalar visualization, which is useful in equivalent-circuit extractions and multiparameter optimization or simply for multiple-variable imaging and uncovering changes in frequency orientation[...]

[...]The Smith chart is limited within the unit circle to passive circuits with positive resistance (r) [or conductance (g)]; circuits with negative r (or g), which occurs in active circuits, are not covered by the conventional Smith chart.

In 2006, IEEE Microwave Magazine presented an article on drawings by the artist M.C. Escher, including a spherical self-portrait. The article pointed out the connection between the drawings of Escher and hyperbolic geometry while also emphasizing the connection between the Smith chart and Möbius transformations in geometry within the 2D complex plane.

Motivated by the desire to have a unified chart for both active and passive microwave circuits, Zelley proposed drawing an intuitive spherical Smith chart on a ping-pong ball. The drawing[...]still lacked 'mathematical rigor,' as the author acknowledged, and no equations were given for its construction.

Inspired by the beautiful drawing[...]obtained by means of skillful but complicated arithmetical and trigonometrical manipulations-spherical Smith chart theories were proposed[...] [...]the circles appearing in 2D are mapped into different curves on the spherical chart[...]

Please go to the IEEE website for access to the full article:

The 3D Smith Chart: From Theory to Experimental Reality  (login required)

* Here is a copy that does not contain IEEE edits and is publically accessible (no login).

Posted October 13, 2020