# A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart

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^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Planar Smith Chart and Möbius Transformations

#### 3D Smith Chart and Infinity

## 3. A Hyperbolic Smith Chart

## 4. Properties

^{h}-plane, |p

^{h}| = 0.414 and |p

^{h}| = 0 in |p

^{h}| = 0.

^{h}-plane and the circuits with capacitive reactance below the real line of the hyperbolic reflection plane, like in the Smith chart. Further, as seen from (16), the constant reflection coefficients circles $0\le \left|\rho \right|\le 1$ of the 2D Smith chart are projected onto circles with $0\le \left|{\rho}^{h}\right|\le 0.414$ in the hyperbolic reflection coefficients plane. The $1<\left|\rho \right|<\infty $ constant circles (which are exterior to the 2D Smith chart or in the South hemisphere on the 3D Smith chart) and which are important in active circuit design are contained in limited region of the hyperbolic reflection coefficients plane with $0.414<\left|{\rho}^{h}\right|<1$, as shown in Figure 9, and their circular forms is unaltered. The image of the infinite magnitude constant reflection coefficient circle $\left|\rho \right|=\infty $ unimaginable to represent even on a generalized 2D Smith chart becomes the contour of the new chart $\left|{\rho}^{h}\right|=1$.

^{h}-plane the part inside of the unit circle of the quartic curve:

^{h}-plane the part inside of the unit circle of the quartic curve:

^{h}-plane, corresponding to the image of the hyperbola ${r}^{2}-{x}^{2}=1$ in the z-plane.

^{h}-plane and the magnitudes are clearly related.

## 5. Application Example and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Normalized grid of the impedance plane; (

**b**) Smith chart (voltage reflection coefficients plane); (

**c**) extended Smith chart including negative resistance impedances voltage reflection coefficient representations too, infinite values are thrown in all directions of the 2D plane.

**Figure 2.**(

**a**) a photo of Andrei in the impedance plane; (

**b**) mapping of (1) applied to the photo; (

**c**) mapping of (2) considering $a=1,b=-1,$ $c=$ 1, $d=5$.

**Figure 3.**Extended Smith chart (Figure 1c) placed on the equatorial plane of the Riemann sphere, before its mapping on the sphere via the south pole.

**Figure 4.**3D Smith chart obtained with the 3D Smith chart Tool [20], infinity of the voltage reflection is the South pole, circle shape of the Smith chart is preserved.

**Figure 5.**Intuitive representation of infinity on the 2D Smith chart (

**a**) and 3D Smith chart; (

**b**) South pole–image of infinite voltage reflection coefficient (magnitude) [20].

**Figure 6.**Relation between ${\mathbb{H}}^{+}$ and $\mathbb{D}$ through stereographic projection $s$.

**Figure 7.**Steps in construction of the Hyperbolic Generalized Smith chart [13,14]: (

**a**) Generalized Smith chart orthogonally projected on the upper part of the two-sheet hyperboloid (it spreads to infinity but on the upper sheet of the hyperboloid); (

**b**) stereographic mapping from the upper hyperboloid on the 2D plane (Figure 6)—the infinity becomes the contour of the unit circle (from point $S=\left(0,0,-1\right),$—blue colour).

**Figure 8.**Hyperbolic Smith chart. Circuits with the magnitude of the voltage reflection coefficient below 1 are mapped into the interior of the 0.414 hyperbolic circle in ρ

^{h}(the circuits with blue normalized resistance r and with red normalized reactance x) while the circuits with negative normalized resistance (green) are mapped in between the 0.414 radius circle and the unit circle (in this the constant normalized reactance circle are coloured in orange).

**Figure 9.**Hyperbolic Smith chart. Constant $0<\left|\rho \right|<\infty $ contours are represented by circles on the Hyperbolic Smith chart and entirely mapped into the hyperbolic reflection coefficients plane $0<\left|{\rho}^{h}\right|<1$; $\left|\rho \right|=0\iff \left|{\rho}^{h}\right|=0$; $\left|\rho \right|=1\iff \left|{\rho}^{h}\right|=0.414$; $\left|\rho \right|=\infty \iff \left|\rho \right|=1$ (in red are the $\left|\rho \right|>1$ circles while in black the $\left|\rho \right|<1$ circles). Thus, infinity of the magnitude of the reflection coefficient can be represented on the contour of the hyperbolic Smith chart (as in hyperbolic geometry) [15]

**Figure 10.**The (

**a**) quartic curves are orthogonal in the (

**b**) image of the points of the hyperbola r

^{2}− x

^{2}= 1 in the z-plane.

**Figure 11.**Input impedance (in black) of a microwave oscillator based on an Infineon bipolar transistor send towards infinity in the Euclidean geometry of the 2D Smith chart.

**Figure 12.**(

**a**) Input impedance (in black) of a microwave oscillator based on an Infineon bipolar transistor is approaching infinity (unit circle) $\left|\rho \right|=\infty \iff \left|{\rho}^{h}\right|=1$ on the hyperbolic Smith chart. Thus, its behaviour (capacitive/ inductive approach of infinity) can be always easily seen on this later hyperbolic Smith chart; (

**b**) 3D Smith chart representation of the input impedance (through a few discrete frequency points).

Comparative Capabilities | Smith Chart | 3D Smith Chart |
---|---|---|

Positive resistance | Interior of unity circle | North hemisphere |

Negative resistance $(\left|\rho \right|>1$) | NO (towards infinity) | South hemisphere |

Perfect match | Origin | North pole |

$\left|\rho \right|=\infty $ | NO | South pole |

Inductive | Above the abscissa | East |

Capacitive | Below the abscissa | West |

r,x,g,b constant | Circles, circle arcs, 1 line | Circles |

Purely resistive | Abscissa | Greenwich meridian |

Power levels/group delays | NO | 3D space(Exterior > 0, Interior < 0) |

Comparative Capabilities | Hyperbolic Smith Chart |
---|---|

Positive resistance | Inside the 0.414 radius circle |

Negative resistance | Between the 0.414 radius circle and unit circle |

Perfect match | Origin |

$\left|\rho \right|=\infty $ | Unit circle |

Inductive | Above ${x}_{\mathrm{h}}$ axes |

Capacitive | Bellow ${x}_{\mathrm{h}}$ axes |

$\mathrm{r},\text{}\mathrm{x},\text{}\mathrm{g},\text{}\mathrm{b}$ constant | Quartic curves and 0.414 radius circle circumference |

Purely resistive | O${x}_{\mathrm{h}}$ axes |

Chart | Geometry | $\left|\rho \right|=\infty $ |
---|---|---|

Smith Chart | 2D Euclidean | Not usable |

Extended 2 Smith chart | 2D Euclidean | unending |

3D Smith chart | Inversive, spherical | South pole |

Hyperbolic Smith chart | hyperbolic | Contour of the unit circle |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pérez-Peñalver, M.J.; Sanabria-Codesal, E.; Moldoveanu, F.; Moldoveanu, A.; Asavei, V.; A. Muller, A.; Ionescu, A.
A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart. *Symmetry* **2018**, *10*, 458.
https://doi.org/10.3390/sym10100458

**AMA Style**

Pérez-Peñalver MJ, Sanabria-Codesal E, Moldoveanu F, Moldoveanu A, Asavei V, A. Muller A, Ionescu A.
A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart. *Symmetry*. 2018; 10(10):458.
https://doi.org/10.3390/sym10100458

**Chicago/Turabian Style**

Pérez-Peñalver, María Jose, Esther Sanabria-Codesal, Florica Moldoveanu, Alin Moldoveanu, Victor Asavei, Andrei A. Muller, and Adrian Ionescu.
2018. "A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart" *Symmetry* 10, no. 10: 458.
https://doi.org/10.3390/sym10100458