Then a stereographic projection form the point (0, 0, -1) from the two sheet hyperboloid onto the unit disc (used in the Poincare unit disc model of hyperbolic geometry), is then used to compact the entire surface of the superior part of the two sheet hyperboloid onto the unit disc.Ī sequence of vertical pulling/stereographical projection, as shown in Figure 1, is applied (inspired by Beltrami-Klein model of non Euclidean geometry while using Poincare hyperboloid model). The infinite generalized 2D Smith chart is regarded in the 3D Euclidean plane as lying on a plane and is mapped onto the superior part of two sheet hyperboloid using an immersion (vertical pull) onto the infinite surface of the hyperboloid.
#Smith chart reflection coefficient software
In order to overcome the visual problems generated by this issue, two solutions were previously successfully proposed: the 3D Smith chart which implies the usage of sphere and the negative Smith chart, however both having their inconveniences the 3D Smith chart, because it cannot be completely exploited without the means of a software tool while the negative Smith chart because it cannot be used at the same time as the regular Smith chart the scales used for one are meaningless for the other. Circuits exhibiting negative resistance have the magnitude of the voltage reflection coefficient bigger than unity and thus are mapped into the exterior of the classical Smith chart. This is done by the means of single equation given by the Poincare disc model of hyperbolic geometry. The creators of the 3D Smith chart have now extended the tool by using a hyperbolic Smith chart suitable for representing both circuits with negative and positive resistance with the magnitude of the voltage reflection coefficient ρof any possible value (0<ρ<∞) in a compact manner in the unit disc. Figure 2 Input impedance (in black) of a microwave oscillator based on an Infineon bipolar transistor on the traditional Smith chart, the input impedance at the oscillation frequency is thrown towards infinity.
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