S S symmetry
Article A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart
María Jose Pérez-Peñalver 1 , Esther Sanabria-Codesal 1 , Florica Moldoveanu 2, Alin Moldoveanu 2,* , Victor Asavei 2 , Andrei A. Muller 3 and Adrian Ionescu 3 1 Universitat Politècnica de València, Matemática Aplicada, 46022 Valencia, Spain; [email protected] (M.J.P.-P.); [email protected] (E.S.-C.) 2 Department of Computer Science and Engineering, Faculty of Automatic Control and Computers, University POLITEHNICA of Bucharest, 060042 Bucharest, Romania; fl[email protected] (F.M.); [email protected] (V.A.) 3 Ecole Polytechnique Federale de Lausanne, Nanoelectronic Devices Laboratory, 1015 Lausanne, Switzerland; andrei.muller@epfl.ch (A.A.M.); adrian.ionescu@epfl.ch (A.I.) * Correspondence: [email protected]
Received: 3 September 2018; Accepted: 28 September 2018; Published: 2 October 2018
Abstract: 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.
Keywords: hyperbolic geometry; Möbius transformation; Smith chart
1. Introduction The 2D Smith chart, one of the most helpful and recognized microwave engineering tools, was proposed by Philip H. Smith in 1939 [1] and further developed by the same author throughout the years [2] before becoming an icon of high frequency engineering [3,4]. Starting the last decades of the 20 century it became used widely in major computer aided design CAD/testing tools as add on but also as a key part in stand-alone tools [5,6]. The Smith chart is bounded within the unit circle of the reflection coefficient’s plane and provides by its practical size an excellent visual approach of the microwave problems [1,6]. But it also has some disadvantages, such as that loads with voltage reflection coefficient magnitude exceeding unity cannot be represented in a practical compact size. These loads appear in active circuits exhibiting negative resistance or in circuits with complex characteristic impedance and can be represented in 2D by constructing an additional reciprocal negative Smith chart [2]. However, the Smith chart and the negative Smith chart cannot be used in a unitary way in the same time since the scales used are different. The 3D Smith chart representation [7] using the Riemann sphere and the circle conserving property of Mobius transformations on it lead to the creation of the first 3D Smith chart tool which combines the design of active and passive components on the unit sphere. The reason for seeking a 3D embedded generalization was motivated by the aspiration to have a single compact chart proper for dealing with negative resistance impedances without distorting the circular shapes of the Smith chart, thus keeping its properties. Described in References [7–10], the 3D Smith chart has
Symmetry 2018, 10, 458; doi:10.3390/sym10100458 www.mdpi.com/journal/symmetry Symmetry 2018, 10, 458 2 of 13
Symmetry 2018, 10, x 2 of 13 its roots in the group theory of inversive geometry [11] where the point at infinity seen as a pole on a Described in References [7–10], the 3D Smith chart has its roots in the group theory of inversive spheregeometry plays [11] a key where role. the point at infinity seen as a pole on a sphere plays a key role. However,However, the the 3D 3D Smith Smith chart chart has has some some inconveniencies, for for example example it it cannot cannot be be easy easy used used withoutwithout a a software software tool, tool, while while the the ideaidea ofof usingusing a 3D embedded embedded surface surface may may be be still still discouraging discouraging for for thethe electrical electrical engineering engineering community community whichwhich is used to to the the 2D 2D Smith Smith chart chart [12] [12.]. In In References References [13 [13,14],14 ] wewe proposed proposed a rather a rather arts basedarts based hyperbolic hyperbolic geometry geometry model model dealing dealing with all with the reflection all the reflection coefficients withincoefficients the unit within disc. the unit disc. InIn this this article, article, we we present present thethe geometricalgeometrical treatment of of infinity infinity on on the the 2D 2D Smith Smith chart, chart, 3D 3D Smith Smith chartchart and and further further develop develop the the equations equations and properties of of the the intuitive intuitive hyperbolic hyperbolic Smith Smith chart chart drawingdrawing [ 13[13,14,14].]. ThisThis is done by by using using the the Kleinian Kleinian view view of ofgeometry, geometry, that that is, infinity is, infinity treatment treatment of ofMöbius Möbius transformations transformations [11 [,15]11, 15for] for the the 2D and 2D and 3D Smith 3D Smith chart chart and basic and elements basic elements of pure of Non pure- Non-EuclideanEuclidean geometry geometry [16–18] [16 –for18 ]the for hyperbolic the hyperbolic Smith Smith chart. chart.
2.2. Planar Planar Smith Smith Chart Chart and and MöbiusMöbius TransformationsTransformations
TheThe relation relation between between thethe normalizednormalized impedance 푧z = 푍Z//푍Z0 0==푟 r++푗푥jx andand the the voltage voltage reflection reflection coefficientcoefficientρ is휌 givenis given by by [1 ,[12,,72,,197,1]:9]: z − 1 = 푧 − 1 6= − ρ 휌 = , z , 푧 ≠ −11 (1) (1) z 푧++11 The expression (1) maps the impedances with positive normalized resistance r (belonging to The expression (1) maps the impedances with positive normalized resistance 푟 (belonging to thethe right right half half plane plane (RHP)) (RHP)) into into thethe limitedlimited areaarea of the unit circle circle and and the the impedances impedances with with negative negative resistanceresistance are are projected projected outside outside the the unitunit circle,circle, asas wewe observe in in Figure Figure 11::
(a) (b) (c)
FigureFigure 1. 1.(a ()a Normalized) Normalized grid grid of the of impedancethe impedance plane; plane; (b) Smith (b) chartSmith (voltage chart (voltage reflection reflection coefficients plane);coefficients (c) extended plane) Smith; (c) chart extended including Smith negative chart resistance includin impedancesg negative voltage resistance reflection impedances coefficient representationsvoltage reflection too, infinitecoefficient values representations are thrown in alltoo, directions infinite values of the 2D are plane. thrown in all directions of the 2D plane. The transformation (1) regarded throughout the references [1–6] and [12,19] as “conformal transformation”,The transformation “bilinear (1) map” regarded or “Möbius througho transformation”ut the references is from [1–6] a and geometrical [12,19] as perspective “conformal a verytransformation”, simple direct inversive“bilinear map” transformation or “Möbius [8 transformation”,9,11,15]. Direct is inversive from a geometrical transformations perspective are defined a very by (2)simple and together direct inversive with the transformation transformations [8,9 defined,11,15]. Direct by (3) inversive (indirect transformations inversive) they are form defined the group by (2) of inversiveand together transformations with the transformations [8,15]. defined by (3) (indirect inversive) they form the group of inversive transformations [8,15]. az + b T(z) = (2) cz푎푧++d푏 푇(푧) = (2) 푐푧az + 푑b TI(z) = (3) + c푎z푧̅ +d푏 푇퐼(푧) = (3) Both (2) and (3) are conformal transformations푐푧̅ + and푑 surely (1) is a particular case of (2). The geometry of these transformations can be educationally explored by placing instead of z a photo: 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:
SymmetrySymmetry2018 2018,,10 10,, 458 x 33of of 1313
Applying transformations of type (2) to Figure 2a we can see in both Figure 2b,c that the transformationApplying transformations (2) maps always of lines type like (2) shapes to Figure into2a circle we can arcs see [8] in (contour both Figure of the2b,c photo) that the(their transformationposition and length (2) maps being always dependent lines on like the shapes placement into of circle (a) in arcs the [complex8] (contour plane). of the This photo) is an innate (their positionproperty and of lengththe Möbius being transformations dependent on the but placement depending of (a) on in the the photo complex position plane). in This the is normalized an innate propertyplane, points of the can Möbius be sent transformations towards infinity. but In depending Figure 2b on we the can photo notice position that points in the are normalized already thrown plane, pointsoutside can of bethe sent unite towards circle since infinity. the Inposition Figure 2ofb the we photo can notice in Figure that points 2a was are not already completely thrown in the outside right ofhalf the plane unite of circle the z since plane the RHP position. Möbius of the transformations photo in Figure as2 aall was inversive not completely transformations in the right do not half always plane ofmap the circlesz plane into RHP. circles Möbius and transformations lines into circles as on all inversivea 2D sheet transformations since points can do be not thrown always not map only circles far intofrom circles the unit and circle lines but into also circles at infinity on a 2D [7, sheet15]. since points can be thrown not only far from the unit circle but also at infinity [7,15].
(a) (b) (c)
FigureFigure 2.2. ((aa)) aa photo of Andrei Andrei in in the the impedance impedance plane; plane; (b ()b mapping) mapping of of (1) (1) applied applied to tothe the photo; photo; (c) (mappingc) mapping of of(2) (2) considering considering 푎 a==1,1,푏b==−−11,, 푐c==1,1, 푑d==5.5.
IfIf wewe extendedextended thethe MöbiusMöbius transformationtransformation (1)(1) such such that that 푧 − 1 z − 1 M(푀z)(=푧) ρ==휌 = ; M ; (푀−(1−)1=) =∞∞; M; 푀(∞(∞) )==1,1, (4)(4) z +푧 1+ 1 then we have a map between the z-plane and the ρ-plane. A ρ-plane point in terms of the load then we have a map between the z-plane and the ρ-plane. A ρ-plane point in terms of the load normalizednormalized resistanceresistancer rand and reactancereactancex, x,is is given given by: by: 푟2 + 푥2 − 1 2푥 푀(푧 = 푟 + 푗푥) =r2 + x2 − 1 + 푗 2x , (5) M(z = r + jx) = (푟 + 1)2 + 푥+2 j (푟 + 1)2 + 푥,2 (5) (r + 1)2 + x2 (r + 1)2 + x2 then the z-plane circumference |푧| = 1 is sent to the ρ-plane imaginary axis. Its inverse, 푀−1(휌) = 푧 thenis the z-plane circumference |z| = 1 is sent to the ρ-plane imaginary axis. Its inverse, M−1(ρ) = z is 2 2 −1 1 + 휌 1 − 휌2푟 − 휌2푥 2휌푥 −푀1 (휌 = 휌푟 + 푗휌푥) =1 + ρ = 1 − ρr − ρx + 푗 2ρx ; 휌 ≠ 1, (6) M (ρ = ρr + jρx) = 1 − 휌= (1 − 휌 )2 + 휌 2+ j (1 − 휌 )2 + 휌 2; ρ 6= 1, (6) 푟2 2푥 푟2 2푥 1 − ρ (1 − ρr) + ρx (1 − ρr) + ρx then 푀−1: ℂ ⧵ {1} → ℂ ⧵ {−1}; 푀−1(1) = ∞; 푀−1(∞) = −1 is the inverse in the extended complex − − − thenplaneM map.1 : C\{1} → C\{−1}; M 1(1) = ∞; M 1(∞) = −1 is the inverse in the extended complex planeThe map. typical flat 2D Smith chart only considers the |휌| ≤ 1 region. Infinity on a 2D Smith chart embeddedThe typical on a flat planar 2D Smith surface chart means only Figure considers 1 which the | ρ makes| ≤ 1 region. it difficult Infinity to use on awhen 2D Smith loads chart with embeddednegative resistance on a planar occur. surface Facing means this Figure problem,1 which Maxime makes Bocher, it difficult a student to use of when Felix loadsKlein withand president negative resistanceof the American occur. Mathematical Facing this problem, Society, proposed Maxime Bocher,in Reference a student [11] the of general Felix Klein solution and for president all Möbius of thetransformations American Mathematical and indirect Society, inversive proposed transformations, in Reference so that [11 ]these the general should solutionalways map for allcircles Möbius into transformationscircles or lines into and circles. indirect inversive transformations, so that these should always map circles into circles or lines into circles.
Symmetry 2018, 10, x 4 of 13
3D Smith Chart and Infinity Symmetry 2018, 10, x 4 of 13 To construct the 3D Smith chart [7–10] we place the extended Smith chart inside the equatorial plane of the Riemann sphere (Figure 3) and using the south pole we map it on the unit sphere (Figure 3SymmetryD Smith2018 Chart, 10 ,and 458 Infinity 4 of 13 4): To construct the 3D Smith chart [7–10] we place the extended Smith chart inside the equatorial plane of the Riemann sphere (Figure 3) and using the south pole we map it on the unit sphere (Figure 3D Smith Chart and Infinity 4): To construct the 3D Smith chart [7–10] we place the extended Smith chart inside the equatorial plane of the Riemann sphere (Figure3) and using the south pole we map it on the unit sphere (Figure4):
Figure 3. Extended Smith chart (Figure 1c) placed on the equatorial plane of the Riemann sphere, Figure 3. Extended Smith chart (Figure1c) placed on the equatorial plane of the Riemann sphere, before its mapping on the sphere via the south pole. before its mapping on the sphere via the south pole. 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 theFigure South pole,4. 3D circle Smith shape chart ofobtained the Smith with chart the is3D preserved. Smith chart Tool [20], infinity of the voltage reflection is the South pole, circle shape of the Smith chart is preserved. The 3D Smith chart [20] obtained in Figure4 has a variety of convenient attributes analogous to a real EarthFigureThe globe,4. 3D 3D Smith Smith as can chartchart be summarized obtained[20] obtained with inthe in Table 3DFigure Smith1. 4 haschart a Toolvariety [20], of infinity convenient of the voltage attributes reflection analogous to a isreal the Earth South globe pole, ,circle as can shape be summarizedof the Smith chart in Table is preserved. 1.
The 3D Smith chart [20] obtained in Figure 4 has a variety of convenient attributes analogous to a real Earth globe, as can be summarized in Table 1.
Symmetry 2018, 10, x 5 of 13
Table 1. Comparative capabilities of Smith and 3D Smith charts. Symmetry 2018, 10, 458 5 of 13 Comparative Capabilities Smith Chart 3D Smith Chart
Positive resistanceTable 1. ComparativeInterior capabilities of unity of Smith circle and 3D Smith charts.North hemisphere Negative resistance (|휌| > 1) NO (towards infinity) South hemisphere Comparative Capabilities Smith Chart 3D Smith Chart Perfect match Origin North pole Positive resistance Interior of unity circle North hemisphere |휌| = ∞ NO South pole Negative resistance (|ρ| > 1) NO (towards infinity) South hemisphere PerfectInductive match Above Origin the abscissa NorthEast pole Capaciti|ρ| = ∞ve BelowNO the abscissa SouthWest pole Inductive Above the abscissa East r,x,g,b constant Circles, circle arcs, 1 line Circles Capacitive Below the abscissa West r,x,g,bPurely constant resistive Circles, circleAbscissa arcs, 1 lineGreenwich Circles meridian PowerPurely levels/group resistive delays AbscissaNO 3D space(Exterior Greenwich meridian > 0, Interior < 0) Power levels/group delays NO 3D space(Exterior > 0, Interior < 0) In Figure 5 we can see the idea of infinity more humanly. A girl and a boy start from the centre of theIn chart Figure moving5 we can on seethe ordinate the idea ofof infinitythe reflection more humanly.plane (푟 = A0 girl). When and athey boy reach start fromthe contour the centre (|휌| of= the1), their chart journey moving ends on the and ordinate they can of never the reflection meet again plane (and (r = just0). imagine When they meeting reach again the contour at infinity). (|ρ| = On1), theirthe 3D journey Smith endschart, and once they they can reach never the meet equator again they (and can just continue imagine meetingwalking againon the at same infinity). great On circle the 3Dand Smith still meet chart, at onceinfinity they (south reach pole) the equator which is they compact can continue and reachable. walking on the same great circle and still meet at infinity (south pole) which is compact and reachable.
(a) (b)
Figure 5. Intuitive representation of infinityinfinity on the 2D Smith chart (a) and 3D3D SmithSmith chart;chart; ((b)) SouthSouth pole–imagepole–image of infiniteinfinite voltagevoltage reflectionreflection coefficientcoefficient (magnitude)(magnitude) [[20].20].
We considerconsider the stereographic projection,projection, byby using the south pole S푆 == (0,0,0 0,, −11)),, on the complex plane π휋S핊::핊S⧵\{푆S}}→→ℂC, where, where 핊 Srepresentrepresent the the surface surface of of the the unit unit sphere: sphere: 푥 푦 x 핊 y 핊 (휋핊(푥핊, 푦핊, 푧)핊=) = S ++ 푗 S (7) πS xS, yS, zS 푧핊 + 1 j 푧핊 + 1 (7) zS + 1 zS + 1 This transformation is a bijection and by applying the inverse to the ρ-plane, then the equations This transformation is a bijection and by applying the inverse to the ρ-plane, then the equations of the 3D Smith chart are presented by: of the 3D Smith chart are presented by: 2 2휌푟 2휌푥 1 − |휌| 휌핊(휌 = 휌 + 푗휌 ) = ( , , !) (8) 푟 푥 |2휌ρ|2 + 1 |휌2ρ|2 + 11|−휌||2ρ+|2 1 S( = + ) = r x ρ ρ ρr jρx 2 , 2 , 2 (8) |ρ| + 1 |ρ| + 1 |ρ| + 1 핊 In the same way, by applying first the Möbius transformation (1) and then 휌 (8), we calculate the corresponding point in the 3D Smith chart of each point 푧 = 푟 + 푗푥. This is just: In the same way, by applying first the Möbius transformation (1) and then ρS (8), we calculate the 2 corresponding point in the 3D Smith chart of each point|푧| −z1= r +2푥jx. This2푟 is just: (휌핊 ∘ 푀)(푧 = 푟 + 푗푥) = ( , , ) (9) |푧|2 + 1 |푧|2 + 1 |푧|2 + 1 ! |z|2 − 1 2x 2r S ◦ ( = + ) = ρ M z r jx 2 , 2 , 2 (9) |z| + 1 |z| + 1 |z| + 1 Symmetry 2018, 10, 458 6 of 13
2 2 2 Conversely, a given point on the unit sphere (xS, yS, zS) ∈ S\{S} (i.e., xS + yS + zS = 1) is mapped to in the ρ-plane (i.e., the sphere equatorial plane) by:
xS + jyS ρ(xS, yS, zS) = πS(xS, yS, zS) = (10) zS + 1
Then,Symmetry the 2018 reflection, 10, x coefficient magnitude is connected to the coordinate zS of the sphere6 of 13 and the phase of the reflection coefficient is equal to the angle of the point (x , y ) with respect to the sphere S S 2 2 2 Conversely, a given point on the unit sphere (푥핊, 푦핊, 푧핊) ∈ 핊 ⧵ {푆} (i. e., 푥핊 + 푦핊 + 푧핊 = 1) is xS axis: mapped to in the ρ-plane (i.e., the spheres equatorial plane) by: 1 − zS 푥 + 푗y푦S |ρ| = , ϕρ = arctan핊 핊 (11) 휌(푥핊, 푦핊, 푧핊)1=+휋z핊(푥핊, 푦핊, 푧핊) = x (10) S 푧핊 + 1S
Then, the reflection coefficient magnitude is connected to the coordinate 푧핊 of the−1 sphere and Similarly, (xS, yS, zS) ∈ S\{S} is mapped into in the z-plane, considering M (6), by: the phase of the reflection coefficient is equal to the angle of the point (푥핊, 푦핊) with respect to the sphere 푥핊 axis: z + jy (M−1 ◦ ρ)(x , y , z ) = S S (12) S S S 1 − x 1 − 푧핊 푦핊 S |휌| = √ , 휑 = 푎푟푐푡푎푛 ( ) (11) 1 + 푧 휌 푥 3. A Hyperbolic Smith Chart 핊 핊 Similarly, (푥 , 푦 , 푧 ) ∈ 핊 ⧵ {푆} is mapped into in the z-plane, considering 푀−1(6), by: Here we present핊 the핊 mathematical핊 theory that lies behind the intuitive models that we have −1 푧핊 + 푗푦핊 presented in References [13,14]. (푀 ∘ 휌)(푥핊, 푦핊, 푧핊) = (12) 1 − 푥핊 The extended complex ρ-plane (Figure1c) includes also the loads outside the unit circumference |ρ| = 1 but3. A thrownHyperbolic also Smith towards Chart infinity and the compact representation requires a sphere) (Figure4), which necessitates a software tool for its practical use. Here we present the mathematical theory that lies behind the intuitive models that we have Inpresented this hyperbolic in References approach [13,14]. while maintaining a flat surface, we use models associated to the hyperbolic geometry,The extended the complex Weierstrass ρ-plane ( ModelFigure 1 andc) includes its stereographic also the loads outside projection the unit onto circumference the Poincar é unit disk in the|휌| = plane1 but [thrown15]. also towards infinity and the compact representation requires a sphere) (Figure We4), consider which necessitates extended a complex software ρtool-plane for its and practical map ituse. onto the superior part of two sheet hyperboloid using the usualIn this parametrization hyperbolic approach of this while surface. maintaining Then, a byflat usingsurface, the we stereographic use models associated projection to the from the hyperbolic geometry, the Weierstrass Model and its stereographic projection onto the Poincaré unit upper sheet hyperboloid onto the unit disc = {ρ ∈ : |ρ| > 1} (used in the Poincaré unit disc disk in the plane [15]. D C model of hyperbolicWe consider geometry), extended we complex construct ρ-plane the and Hyperbolic map it onto Smith the chart superior (Figures part of6 and two7 ). sheet jϕ A generichyperboloidρ-plane using point the usualρ = ρ parametrizationr + jρx = |ρ| e ofρ thisis mapped surface. on Then, by using the stereographic projection from the upper sheet hyperboloid onto the unit disc 픻 = {휌 ∈ ℂ ∶ |휌| > 1} (used in the n o Poincaré unit disc+ model of hyperbolic geometry),3 we2 construct2 the2 Hyperbolic Smith chart (Figures H = (xH, yH, zH) ∈ R : xH + yH + zH = −1, zH > 0 , (13) 6 and 7). A generic ρ-plane point 휌 = 휌 + 푗휌 = |휌| 푒푗휑휌 is mapped on by the usual parametrization of this surface푟 푥 t : → + : ℍ+ = {(푥 , 푦 , 푧 ) ∈ ℝ3 ∶ 푥C2 + 푦H2 + 푧 2 = −1, 푧 > 0}, ℍ ℍ ℍ ℍ ℍ ℍ ℍ (13) q 푡: ℂ → ℍ+ 2 by the usual parametrizationt( ofρ this= ρ surfacer + jρx ) = ρr,: ρx, 1 + |ρ| , (14) 2 푡(휌 = 휌푟 + 푗휌푥) = (휌푟, 휌푥, √1 + |휌| ) , (14)
++ FigureFigure 6. Relation 6. Relation between betweenH ℍ and 픻D through stereographic stereographic projection projection 푠. s.
Symmetry 2018, 10, 458 7 of 13 Symmetry 2018, 10, x 7 of 13
++ Next,Next, by by applying applying the stereographic projectionprojection s푠::ℍH →→픻D, ,fromfrom the the point point ((00,,0 0,, −−1)1,) ,onon the the complex plane to a generic point in the space (푥 , 푦 , 푧 ) ∈ ℍ++, we obtain that the point is mapped complex plane to a generic point in the space (xHℍ, yHℍ, zHℍ) ∈ H , we obtain that the point is mapped intointo the the unit unit disc disc 픻D centeredcentered onon thethe origin,origin, asas inin thethe FigureFigure6 :6: 푥ℍ 푦ℍ 푠(푥ℍ, 푦ℍ, 푧ℍ) = xH + 푗 yH (15) s(xH, yH, zH) = 푧ℍ + 1+ j 푧ℍ + 1 (15) zH + 1 zH + 1 Therefore, the hyperbolic Smith chart equation in terms of the voltage reflection coefficients is givenTherefore, by: the hyperbolic Smith chart equation in terms of the voltage reflection coefficients is given by: 휌 휌 ℎ 2 푟 푥 휌 (휌 = 휌푟 + 푗휌푥) = (푠 ∘ 푡)(휌) = 푠 (휌푟, 휌푥, √1 + |휌| ) = + 푗 q 2 2 (16) h 2 1 +ρ√r 1 + |휌| 1 +ρ√x1 + |휌| ρ (ρ = ρr + jρx) = (s ◦ t)(ρ) = s ρr, ρx, 1 + |ρ| = q + j q (16) By arithmetical manipulations we can further get the connection1 + 1 +to |theρ|2 resistance1 + 1r +and|ρ |reactance2 x: By arithmetical manipulations we can further get the connection to the resistance r and reactance x: (휌ℎ ∘ 푀)(푧 = 푟 + 푗푥) = 2 h 2 |z| −1 2x ρ ◦ M (z = r |+푧|jx−) =1 q + j 2푥 q (17)(17) (r+1)2+x2+ 2(1+|z+|2)(푗 (r+1)2+x2) (r+1)2+x2+ 2(1+|z|2)((r+1)2+x2) (푟 + 1)2 + 푥2 + √2(1 + |푧|2)((푟 + 1)2 + 푥2) (푟 + 1)2 + 푥2 + √2(1 + |푧|2)((푟 + 1)2 + 푥2)
InIn Figure Figure 77,, we observeobserve the differentdifferent stepssteps inin thethe constructionconstruction ofof thethe HyperbolicHyperbolic GeneralizedGeneralized SmithSmith chart chart:: First, First, the the generalized generalized Smith Smith chart chart is is sent sent by by tt ((14)14) on on the the upper upper sheet sheet of of the the hyperboloid hyperboloid andand then then the the stereographic stereographic projection projection s푠(15)(15) sendsend itit inin thethe PoincarPoincaréé unitunit discdisc model.model.
(a) (b)
FigFigureure 7. 7. StepsSteps in in construction construction of of the the Hyperbolic Hyperbolic Generalized Generalized Smith Smith chart chart [13,14 [13,]: (14]:a) Generalized (a) Generalized Smith Smithchart orthogonally chart orthogonally projected projected on the upperon the part upper of the part two-sheet of the two hyperboloid-sheet hyperboloid (it spreads (it to spreads infinity butto infinityon the upper but on sheet the of upper the hyperboloid); sheet of the (b hyperboloid)) stereographic; (b mapping) stereographic from the mapping upper hyperboloid from the upper on the hyperboloid2D plane (Figure on the6)—the 2D plane infinity (Figure becomes 6)—the the infinity contour becomes of the unit the circle contour (from of point the unitS = circle(0, 0, (from−1), —bluepoint 푆colour).= (0,0, −1),—blue colour).
CConversely,onversely, a ageneric generic point point 푑 =d (푥=h, 푦(hx)h ,∈yh픻), in∈ theD, unit in thedisc unit centred disc on centred the origin on, thethen origin,, 푑 = 푗휑푑 jϕ −1 + 푥then,h + 푗푦hd ==|x푑h| +푒 jyh, |푑=| | −1 2푥h 2푦h 휌(푑) = 휌ℎ (푑) = (푡−1 ∘ 푠−1)(푑) = + 푗 (19) 1 − |푑|2 1 − |푑|2 To find the expression in the z-plane, we need to consider (19) and the extended transformation 푀−1(6). Then: Symmetry 2018, 10, 458 8 of 13 Next, this point in the upper hyperboloid is mapped into C by applying the orthogonal projection −1 + t : H → C. Therefore, a given generic point d = (xh, yh) in the Hyperbolic Smith chart is mapped to the ρ-plane by: 2x 2y ρ(d) = ρh−1(d) = t−1 ◦ s−1 (d) = h + j h (19) 1 − |d|2 1 − |d|2 Symmetry 2018, 10, x 8 of 13 To find the expression in the z-plane, we need to consider (19) and the extended transformation −1 M (6). Then: −1 ℎ−1 푧(푑) = (푀 ∘ 휌 )(푑 = 푥h + 푗푦h) 2 4 2 1 − 6|푑| + |푑| 42푦h(14 − |푑| ) 2 (20) = −1 h−1 + 푗 1−6|d| +|d| 4y h(1−|d| ) z(d) = (M(1 + |◦푑ρ|2)(()1(d+=|푑x|2h)+−jy4푥h))= (1 +2 |푑|2)(2(1 + |푑+|2)j− 4푥 )2 2 (20) h (1+|d| )((1+|d| )−4xh) (1+|dh| )((1+|d| )−4xh) 4. Properties 4. Properties The Hyperbolic Smith chart has useful properties. In this section, we summarize them (Figure8). The Hyperbolic Smith chart has useful properties. In this section, we summarize them (Figure The entire Smith chart is mapped into the interior of the 0.414 radius circle of the hyperbolic chart. 8). The entire Smith chart is mapped into the interior of the 0.414 radius circle of the hyperbolic chart. Figure 8.8. Hyperbolic Hyperbolic Smith Smith chart. chart. Circuits Circuits with with the magnitudethe magnitude of the of voltage the voltage reflection reflection coefficient coefficient below below1 are mapped 1 are mapped into the into interior the of interior the 0.414 of the hyperbolic 0.414 hyperbolic circle in ρ h circle(the circuits in ρh (the with circuits blue normalized with blue normalizedresistance r and resistance with red r normalized and with redreactance normalizedx) while reactance the circuits x) with while negative the circuits normalized with resistance negative normalized(green) are mapped resistance in between(green) are the 0.414mapped radius in between circle and the the 0.414 unit circle radius (in circle this the and constant the unit normalized circle (in thisreactance the constant circle are normalized coloured inreac orange).tance circle are coloured in orange). The circumference √ ofof radiusradius oneone inin thetheρ ρ-plane,-plane,| ρ||휌=| =1,1is, is mapped mapped by by (16) (16 to) to the the circumference circumference of 1 1 ofradiu radius√ == √22−−1 1'≃0.4140.414 in in the theph ρ-plane,h-plane |, p|h휌|ℎ| == 0.4140.414 and 푎푛푑 |p |h휌|| = 00 inin | p|휌hℎ|| == 0.0. 1+ 12+√2 The Hyperbolic Smith chart contains the circuits with inductive reactance above the horizontal h (real line) line) of of the theρ -planeρh-plane and and the circuitsthe circuits with capacitivewith capacitive reactance reactance below the below real line the of real the line hyperbolic of the reflection plane, like in the Smith chart. Further, as seen from (16), the constant reflection coefficients hyperbolic reflection plane, like in the Smith chart. Further, as seen from (16), the cons tant reflection h ℎ coefficientscircles 0 ≤ circles|ρ| ≤ 10 of≤ | the휌| ≤ 2D1 Smithof the 2D chart Smith are chart projected are projected onto circles onto with circles0 ≤with ρ 0 ≤ |0.414휌 | ≤ in0.414 the inhyperbolic the hyperbolic reflection reflection coefficients coefficients plane. plane The. The1 < 1|<ρ| |휌<| <∞∞constant constant circles circles (which (which areare exteriorexterior to thethe 2D Smith chart or in the South he hemispheremisphere on the 3D Smith chart) and which are important in active circuit design are contained in limited region of the hyperbolic reflection reflection coefficients coefficients plane 0.414 < |휌hℎ | < 1 with 0.414 < ρ < 1,, asas shownshown in FigureFigure9 9,, andand theirtheir circularcircular formsforms isis unaltered.unaltered. TheThe imageimage ofof thethe infinite magnitude constant reflection coefficient circle |휌| = ∞ unimaginable to represent even on a generalized 2D Smith chart becomes the contour of the new chart |휌ℎ| = 1. Symmetry 2018, 10, 458 9 of 13 infinite magnitude constant reflection coefficient circle |ρ| = ∞ unimaginable to represent even on a h generalized 2D Smith chart becomes the contour of the new chart ρ = 1. Symmetry 2018, 10, x 9 of 13 Symmetry 2018, 10, x 9 of 13 Figure 9. 9. HyperbolicHyperbolic Smith Smith chart. chart. Constant Constant 0 0<<|휌||ρ<| <∞ ∞contourscontours are are repre representedsented by circles by circles on the on Hyperbolicthe Hyperbolic Smith Smith chart and chart entirely and entirely mapped mapped into the hyperbolic into the hyperbolic reflection coefficients reflection coefficientsplane 0 < |휌 planeℎ| < 1; |휌 |h= 0 ⟺ |휌ℎ| = 0; |h휌 | = 1 ⟺ |휌ℎ| = 0 .414h ; |휌| = ∞ ⟺ |휌ℎ| = 1 (in red are the |휌| > 1 0 < ρ < 1; |ρ| = 0 ⇔ ρ = 0 ; |ρ| = 1 ⇔ ρ = 0.414 ; |ρ| = ∞ ⇔ |ρ| = 1 (in red are the |ρ| > 1 circles while while in in black black the the |ρ휌| << 11circles). circles). Thus, Thus,infinity infinity ofof thethe magnitudemagnitude ofof thethe reflectionreflection coefficientcoefficient can be represented on on the the contour contour of of the the hyperbolic hyperbolic Smith Smith chart chart (as (asin hyperbolic in hyperbolic geometry) geometry) [15]. [15] If we consider lines of constant resistance equal to k in the z-plane, r = k, we obtain in the ρh-planeh If we consider lines of constant resistance equal to 푘 in the z-plane , 푟 = 푘, we obtain in the ρ - planethe part the inside part inside of theFigure of unit 9. the Hyperbolic circle unit ofcircle Smith the chart. of quartic Constantthe quartic curve: 0 < |휌| (a) (b) Figure 10. The (a) quartic curves are orthogonal in the (b) image of the points of the hyperbola 푟2 − Figure 10. The2 (a) quartic curves are orthogonal in the (b) image of the points of the hyperbola 푥 = 1 in the z-plane. r2 − x2 = 1 in the z-plane. In the hyperbolic(a) Smith chart, by using (19) we see that the reflection coefficient( is:b) Figure 10. The (a) quartic curves are orthogonal in the (b) image of the points of the hyperbola 푟2 − 푥2 = 1 in the z-plane. In the hyperbolic Smith chart, by using (19) we see that the reflection coefficient is: Symmetry 2018, 10, 458 10 of 13 Symmetry 2018, 10, x 10 of 13 In the hyperbolic Smith chart, by using (19) we see that the reflection coefficient is: 2|푑| 푦ℎ |휌| = , 휑 = 푎푟푐푡푎푛 ( ) (23) 1 − |푑2||2d| 휌 y푥 |ρ| = , ϕ = arctan hℎ (23) 2 ρ x We observe that the phase of the reflection1 − |d| coefficient is theh same in the ρ-plane and the ρh-plane and the magnitudes are clearly related. We observe that the phase of the reflection coefficient is the same in the ρ-plane and the ρh-plane By using (20), we obtain that the normalize impedance is: and the magnitudes are clearly related. 2 By using (20), we obtain that the normalize2 impedance4 2 is: 2 √(1 − 6|푑| + |푑| ) + (4푦h(1 − |푑| )) |푧| = q 2 2 (1 + |푑| )2((1 +4 2|푑| ) − 4푥 2) 2 (1−6|d| +|d| ) +(4yh(1−|d|h )) (24) |z| = 2 2 (1+푟|d| )((11−+|6d| 푑)−|24x+h)|푑|4 (24) 1−6|d|2+|d|4 cot (휑푧) = 푄 = = r 2 cot(ϕz) = Q푥 = x4=푦 (1 − |푑| 2) h 4yh(1−|d| ) 5.5. Application Example Example and and Discussion Discussion InIn oscillatoroscillator theory,theory, thethe activeactive circuitcircuit mustmust exhibitexhibit anan infiniteinfinite valuevalue ofof thethe reflectionreflection coefficientcoefficient atat thethe desired desired frequency. frequency. Surely, Surely, the usethe of use a 2D of Smith a 2D chart Smith based chart on Euclideanbased on geometry Euclidean is unattainablegeometry is tounattainable visualize the to phenomenonvisualize the phenomenon (as seen in Figure (as seen 11). in Figure 11). FigureFigure 11.11.Input Input impedance impedance (in (inblack) black) of a microwave of a microwave oscillator oscillator based on based an Infineon on an bipolar Infineon transistor bipolar sendtransistor towards send infinity towards in theinfinity Euclidean in the geometryEuclidean ofgeometry the 2D Smith of the chart. 2D Smith chart. InIn thethe hyperbolichyperbolic reflectionreflection plane, plane, governed governed by by Non-Euclidean Non-Euclidean geometry geometry [16 [16]] this this phenomenon phenomenon is easyis easy to to be be seen seen in Figurein Figure 12 a,12 whilea, while in Figurein Figure 12 b12 onb on the the 3D 3D Smith Smith chart. chart. Symmetry 2018, 10, 458 11 of 13 Symmetry 2018, 10, x 11 of 13 (a) (b) FigureFigure 12. 12.(a ()a Input) Input impedance impedance (in (in black) black) of of a a microwave microwave oscillator oscillator based based on on an an Infineon Infineon bipolar bipolar ℎ transistor is approaching infinity (unit circle) |휌| = ∞ ⟺h |휌 | = 1 on the hyperbolic Smith chart. transistor is approaching infinity (unit circle) |ρ| = ∞ ⇔ ρ = 1 on the hyperbolic Smith chart. Thus, itsThus, behaviour its behaviour (capacitive/ (capacitive/ inductive inductive approach approach of infinity) of infinity) can can be alwaysbe always easily easily seen seen on on this this later later hyperbolichyperbolic Smith Smith chart; chart (; b()b) 3D 3D Smith Smith chart chart representation representation of of the the input input impedance impedance (through (through a a few few discretediscrete frequency frequency points). points). TheThe hyperbolic hyperbolic Smith Smith chart chart (whose (whose main main properties properties are are summarized summarized in in Table Table2) 2) maps maps the the infinite infinite mismatchmismatch into into the the contour contour of of the the unit unit circle, circle thus, thus if if one one would would use use a a new new vocabulary vocabulary when when passing passing fromfrom one one geometry geometry to to another another [18 [1] one8] one can can translate translate infinite infinite reflection reflection coefficient coefficient (Smith (Smith chart) chart) into theinto properthe proper representation representation (Table (Table3). 3). TableTable 2. 2.Comparative Comparative capabilities capabilities of of the the Hyperbolic Hyperbolic Smith Smith chart. chart. ComparativeComparative Capabilities Capabilities HyperbolicHyperbolic Smith Smith Chart Chart PositivePositive resistance resistance InsideInside the the 0.414 0.414 radius radius circle circle NegativeNegative resistance resistance BetweenBetween the the 0.414 0.414 radius radius circle circle and and unit unit circle circle Perfect match Origin Perfect match Origin |ρ| = ∞ Unit circle | | 휌 =Inductive∞ AboveUnitxh axes circle Capacitive Bellow x axes Inductive Aboveh 푥h axes r, x, g, b constant Quartic curves and 0.414 radius circle circumference Capacitive Bellow 푥h axes Purely resistive Oxh axes r, x, g, b constant Quartic curves and 0.414 radius circle circumference Purely resistive Table 3. Vocabulary. O푥h axes | | = ChartTable Geometry 3. Vocabulary. ρ ∞ Smith Chart 2D Euclidean Not usable Extended Chart 2 Smith chart 2DGeometry Euclidean unending|흆| = ∞ Smith3D Chart Smith chart Inversive,2D Euclidean spherical South poleNot usable Hyperbolic Smith chart hyperbolic Contour of the unit circle Extended 2 Smith chart 2D Euclidean unending 3D Smith chart Inversive, spherical South pole 6. Conclusions Hyperbolic Smith chart hyperbolic Contour of the unit circle We have presented the geometric properties of the Smith chart, 3D Smith chart and Hyperbolic Smith6. Conclusions chart by means of inversive geometry and hyperbolic geometry. We have presented the geometric properties of the Smith chart, 3D Smith chart and Hyperbolic Smith chart by means of inversive geometry and hyperbolic geometry. Symmetry 2018, 10, 458 12 of 13 The infinite magnitude of the reflection coefficient is mapped by these charts (drawn when applied to the grid of the normalized impedance plane) into different locations: infinity on the 2D Smith chart, South pole on the 3D Smith chart and contour of the unit disc for the Hyperbolic Smith chart. Further we presented for the first time the equations of the Hyperbolic Smith chart, diagram which was previously just intuitively drawn. The infinity treatment exceeds the classical electrical engineering books and is based on applied pure mathematics (and not engineering manipulations of equations). In the future, the topology of the 3D Smith chart in particular can be expanded to deal with more complex phenomenon such as phase change materials [21,22] where their physical properties vary a lot as frequency or temperature changes. The compact (spherical) representation of its reflection coefficients on its surface leaves the space surrounding it free to be used for visualizing other parameters by employing 3D Euclidean geometry or space filling concepts where the notion of infinity may be treated differently. Author Contributions: A.A.M. proposed the format of the article and introduced the concepts of 3D Smith chart and hyperbolic Smith chart. M.J.P-P and E.S-C developed the further new mathematical properties (equations) (13)–(24) of the hyperbolic Smith chart starting from A.A.M initial concept. A.M., V.A. and F.M. developed the 3D Smith chart Java implementation and revised and arranged the paper, A.I. supervised the work and proposed the further development of the 3D chart to deal with further multi-parameter problems by means of other geometries. Funding: This research was partially funded by DGCYT grant number MTM2015-64013-P. 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