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A self-portrait of M.C. Escher (1898-1972) in spherical mirror, dating from 1935 titled Hand with Reflecting Globe.
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A conceptual relation between Circle Limit IV, an artistic creation by M.C. Escher, and Smith Chart, geographical aid for microwave engineering created by P.H. Smith, was established. The basis of Escher's art and Smith chart can both be traced back to invariance of the cross ration of four complex numbers under Möbius transformation on the domain...
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... A simple way to position any such process in a way that makes the extremalities observed easily visible seems to be via the introduction of a hyperbolic model for characerising noise sources via their proximity to a set of geodesics. This is possible using a class of conformal maps of the upper half plane on the unit disk similar to the construction of Smith charts in RF engineering which carries the same properties with [10]. This is often given in terms of some complex impedance z 0 as ...
We report on a peculiar effect regrading the use of the prime's last digit sequence which is equivalent to a quaternary symbolic sequence. This was used as a driving sequence for the recently introduced Schramm-Loewner Evolution after trying two different possible binary encodings. We report on a clear deviation from the standard space-filling curves normally expected from such a process. We also contrast this behavior with others produced via a symbolic dynamics applied on standard noise sources as well as deterministic sequences produced by simple automata. Our findings include the well known, Morse-Thue sequence as the simplest model exhibiting such behavior apart from some strongly biased Levy walks. We conjecture on an analytical condition for reproducing the particular effect based on Carleman linearization as well as on the possibility of certain continuous Langevin diffusion process that could reproduce similar behavior for appropriate noise source. We discuss the possibility oft further experiments with supercomputers further corroborating these results.
... The Smith chart is limited within the unit circle to passive circuits with positive resistance (r) (or conductance (g)) [1]; circuits with negative r (or g), which occurs in active circuits, are not covered by the conventional Smith chart [4,6]. In 2006, IEEE Microwave Magazine presented an article [7] 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 Mobius transformations in geometry within the 2D complex plane. ...
... Inversive geometry considers the space of inversive transformations, which map all the circles into circles on a 2D sheet or on the Riemann sphere, when points are thrown to infinity (∞). For this geometry, ∞ is just a point [5,[11][12][13], unlike Euclidean geometry where ∞=unending or hyperbolic geometry where ∞=circle [7]. ...
... The equation maps the constant r and constant x grid lines of the right half plane (r > 0) of the impedance plane within the unit circle of the Smith chart into arcs and circles, leaving the left half plane (r < 0) to be mapped in its exterior, as depicted in Fig. 2, while throwing points to infinity in all directions. Unlike the classical Smith chart literature which considers (1) a bilinear transformation [4,7,16] or a conformal one [17][18], here we see it as a particular case of inversive transformation: the direct inverse: D(z) (3) (a) [11]. ...
The Smith chart was primarily developed, extended, and refined by Phillip Hagar Smith [1] in a series of works published [2]-[4] between 1939 and 1969. Smith was born in Lexington, Massachusetts, in 1905. He majored in electrical communications at Tufts University and joined the Radio Research Department of Bell Telephone Laboratories (now Bell Labs) in 1928. While there, in around 1930, Smith started work on the diagram that was to become the Smith chart. He submitted the initial version to Electronics Magazine in 1937; the magazine finally published his diagram in 1939 [2]. The MIT Radiation Laboratory started using the chart. In 1940, and in 1944 Smith published a second article that incorporated further improvements, including the use of the chart with either impedance or admittance coordinates. In 1952, Smith was elevated to IEEE Fellow for his contributions to the development of antennas and the graphical analysis of transmission-line characteristics. The first issue of Microwave Journal (1958) published a biography of Smith to acknowledge the importance of his contributions. In 1969, he wrote the book Electronic Applications of the Smith Chart in Waveguide, Circuit and Component Analysis; he retired from Bell Labs in 1970. In 1975, he received the IEEE Microwave Theory and Techniques Society?s Special Recognition in Microwave Applications award for the Smith chart, and in 1994 he was elected to the New Jersey Inventors Hall of Fame.
... The Smith chart [1][2], proposed by Philip Hagar Smith in 1939, survived the passing of years, becoming an icon of microwave engineering [3], being nowadays still used in the design and measurement stage of various radio frequency or microwave range devices [4][5][6][7], for plotting a variety of frequency dependent parameters. Constant quality factor (Q) representations on the Smith chart are a visual way to determine the quality factor of various passive microwave circuits, being mostly known in the microwave frequency range community [8][9][10][11][12][13][14][15][16][17]. ...
The article proves first that the constant quality factor (Q) contours for passive circuits, while represented on a 2D Smith chart, form circle arcs on a coaxal circle family. Furthermore, these circle arcs represent semi-circles families in the north hemisphere while represented on a 3D Smith chart. On the contrary we show that the constant Q contours for active circuits with negative resistance form complementary circle arcs on the same family of coaxal circles in the exterior of the 2D Smith chart. Also, we find out that these constant Q contours represent complementary semi-circles in the south hemisphere while represented on the 3D Smith chart for negative resistance circuits. The constant Q - computer aided design (CAD) implementation of the Q semi-circles on the 3D Smith chart is then successfully used to evaluate the quality factor variations of newly fabricated Vanadium dioxide inductors first, directly from their reflection coefficient, as the temperature is increased from room temperature to 50 degrees Celsius ({\deg}C). Thus, a direct multi-parameter frequency dependent analysis is proposed including Q, inductance and reflection coefficient for inductors. Then, quality factor direct analysis is used for two tunnel diode small signal equivalent circuits analysis, allowing for the first time the Q and input impedance direct analysis on Smith chart representation of a circuit, including negative resistance
... where j stands for the imaginary unit, and ρ denotes the standing wave ratio (SWR) [5,8]. Introducing the natural logarithm of SWR, namely D = ln ρ, the above relation can be rewritten as ...
This paper presents a basic consideration of complex impedance projection. We start by recalling the traditional Smith chart. The well-known standing wave ratio is revisited by its natural logarithm, which brings Poincaré metric to geometrical distance on the chart. We then replace the distance by its double to observe the Beltrami-Klein disk model from the aspect of RF engineering. To give fundamental locus examples, we formulate how lumped-constant elements behave on hyperbolic coordinates, and find that they draw their constant-R and -X contours in straight lines or vertical ellipses. For higher frequency applications, we also formulate the behavior of transmission lines to show what trajectories they exhibit on the disk.
... The Smith chart [1], [2], which, was invented by Philip Hagar Smith in 1939, has survived the passing of years, becoming an icon of microwave engineering [3]. It is still being used in the design and measurement stage of various radio frequency or microwave range devices [4]- [7], for plotting a variety of frequency dependent parameters. ...
The article firstly proves that the constant quality factor (Q) contours for passive circuits, while represented on a 2D Smith chart, form circle arcs on a coaxal circle family. Furthermore, these circle arcs represent semi-circles families in the north hemisphere while represented on a 3D Smith chart. On the contrary, it then shows that, the constant Q contours for active circuits with negative resistance form complementary circle arcs on the same family of coaxal circles in the exterior of the 2D Smith chart. Moreover, we reveal that these constant Q contours represent complementary semi-circles in the south hemisphere while represented on the 3D Smith chart for negative resistance circuits. The constant Q semicircles implementation in the 3D Smith chart computer aided design (CAD) tool is then successfully used to evaluate the quality factor variations of newly fabricated Vanadium dioxide inductors, directly from their reflection coefficient, as the temperature is increased from room temperature to 50 degrees Celsius (°C). Thus, a direct multi-parameter frequency dependent analysis is proposed including Q, inductance and reflection coefficient for inductors. Then, quality factor direct evaluation is used for two tunnel diode small signal equivalent circuits analysis, allowing for the first time the direct analysis of the Q and input impedance on a 3D Smith chart representation of a circuit, while including negative resistance.
... However, the 3D Smith chart has some inconveniencies, for example it cannot be easy used without a software tool, while the idea of using a 3D embedded surface may be still discouraging for the electrical engineering community which is used to the 2D Smith chart [12]. In References [13,14] we proposed a rather arts based hyperbolic geometry model dealing with all the reflection coefficients within the unit disc. ...
... The expression (1) maps the impedances with positive normalized resistance (belonging to the right half plane (RHP)) into the limited area of the unit circle and the impedances with negative resistance are projected outside the unit circle, as we observe in Figure 1: The transformation (1) regarded throughout the references [1][2][3][4][5][6] and [12,19] as "conformal transformation", "bilinear map" or "Möbius transformation" is from a geometrical perspective a very simple direct inversive transformation [8,9,11,15]. Direct inversive transformations are defined by (2) and together with the transformations defined by (3) (indirect inversive) they form the group of inversive transformations [8,15]. ...
... Both (2) and (3) are conformal transformations and surely (1) is a particular case of (2). The geometry of these transformations can be educationally explored by placing instead of z a photo: The transformation (1) regarded throughout the references [1][2][3][4][5][6] and [12,19] as "conformal transformation", "bilinear map" or "Möbius transformation" is from a geometrical perspective a very simple direct inversive transformation [8,9,11,15]. Direct inversive transformations are defined by (2) and together with the transformations defined by (3) (indirect inversive) they form the group of inversive transformations [8,15]. ...
This work describes the geometry behind the Smith chart, recent 3D Smith chart tool and previously reported conceptual Hyperbolic Smith chart. We present the geometrical properties of the transformations used in creating them by means of inversive geometry and basic non-Euclidean geometry. The beauty and simplicity of this perspective are complementary to the classical way in which the Smith chart is taught in the electrical engineering community by providing a visual insight that can lead to new developments. Further we extend our previous work where we have just drawn the conceptual hyperbolic Smith chart by providing the equations for its generation and introducing additional properties.
... Quaternions can be used for threedimensional applications such as the Riemann sphere, but they do not scale to four or higher dimensions. In addition, neither quaternions nor complex numbers are adequate tools for noneuclidean geometry, which is the natural geometry for several parts of transmission line theory [3]. ...
In this paper we present the application of a projective geometry tool known as Conformal Geometric Algebra (CGA) to transmission line theory. Explicit relationships between the Smith Chart, Riemann Sphere, and CGA are developed to illustrate the evolution of projective geometry in transmission line theory. By using CGA, fundamental network operations such as adding impedance, admittance, and changing lines impedance can be implemented with rotations, and are shown to form a group. Additionally, the transformations relating different circuit representations such as impedance, admittance, and reflection coefficient are also related by rotations. Thus, the majority of relationships in transmission line theory are linearized. Conventional transmission line formulas are replaced with an operator-based framework. Use of the framework is demonstrated by analyzing the distributed element model and solving some impedance matching problems.
... The Smith chart, also called Reflection Chart, Circle Diagram, Immitance Chart and Z plane chart in its initial stages, was first published in 1939 [1] and refined throughout the years [2] before becoming a universal tool in microwave engineering design and measurement stages. So entrenched is the Smith chart in microwave engineers' conceptualization that despite their powerful and highly sophisticated computational capability, both the modern computer-aided design software and the computer controlled microwave measurement equipment continue to present results on Smith chart overlays [3]. To have a finite and practical size, the classical 2D Smith Chart is constrained to the unit circle. ...
The paper is describing and presenting the recent advances in the 3D Smith chart representations and applications and then it proposes a new conceptual model for the extended 2D Smith chart based on hyperbolic geometry by mapping the generalized Smith chart in the unit disc using the stereographic projection from a hyperboloid (Poincare disc model).
... The Smith chart [1] was proposed in 1939 and has become through the years an icon of microwave engineering, being even used in the design of logos. Although originally intended to be a graphical aid for eliminating the drudgery of computation with complex numbers, the Smith chart still remains a highly useful tool for representation and comprehension of microwave data [2], becoming an universal aid to the design of matching circuits and the display of measurements. Although the Smith chart was developed more than seven decades ago, it is still widely used by the microwave community. ...
The paper presents the applications of the recently proposed 3D Smith chart by means of a completely updated 3D Smith chart tool issued in September 2013. It is capable to deal with both active and passive microwave circuit parameters simultaneously. The 3D Smith chart key maps all the active and passive loads on the surface of the unit ball (inverted Riemann sphere), so that circuits with negative resistance are now assembled in the South hemisphere, circuits with positive resistance (classical 2D Smith chart) are mapped into the north hemisphere, whereas inductive and capacitive circuits are placed at the East and West of Greenwich meridian, respectively. The updated 3D Smith chart tool can plot simultaneously and in an unique manner scattering parameters, power wave reflection coefficients, voltage reflection coefficients, input and output stability circles, and deal with real and also complex characteristic impedances ports.
... Although invented in the late 30's the diagram survived among the time and its use has grown steadily over the years as software design tool and as a visualization instrument on the network analyzers. Its main geometrical properties are related with the beauty of the Möbius transformations and are also linked to topics from arts -Esher's art [3] and arhitecure. In the recent years several tries were done in order to increase the the capability of the original Smith chart into 3d in order to contain a wider range of impedances as in [4,5]. ...
The paper explains for the first time the visual complex analyisis propeties of
the very recently proposed 3D Smith for active and passive circuits. The authors
develop their very actual concept with elements of complex analysis applied in the
complex plane and in 3D (Riemann sphere).The geometrical properties of the
different transfromations of the complex plane are exploited and applied in
microwave design.
