Maxime Bocher, born in 1867 in Boston, professor at Harvard University and president of the American Mathematical Society (1908-1910), was the one that finalized the theory of inversive geometry. In fact, all these planes extend to infinity, where one of the aforementioned key loads is located. It can be concluded that all the usual 2D planar representations are unable to keep these four key impedances (open circuit, short circuit, matched load and infinite mismatch) in a bounded region. Analogously, the open circuit is placed in the center of the normalized admittance or y-plane, where another important load, the short circuit, is thrown to infinity. 5 This graphical representation gives an important role to the infinite mismatch, which is placed in its center, but moves the matched load (ρ= 0) to infinity. To be able to represent the infinite mismatch, practitioners can use a planar Smith chart plotted in the 1/ρ-plane called negative Smith chart. Although it denotes a particular active load, this input impedance is important in some practical applications such as oscillator design. Since in practical applications, impedances can be found close to the open circuit, it is rather uncommon to use the z-plane to perform a visual representation of loads.įor the reflection coefficient plane (ρ-plane), the infinite region corresponds to the load with the same magnitude and opposite sign to the characteristic impedance of the line (that is the infinite mismatch or z = –1 in normalized impedance terms). The infinite region of the normalized impedance or z-plane is due to the open circuit, because the magnitude of the input impedance grows to infinity as the load tends to an open circuit. Next, the advantages of using the 3D Smith chart to represent both active and passive loads are illustrated with two examples: the stability circles of an amplifier and the impedance of a microwave oscillator. In this article, how the 2D and 3D Smith charts deal with the infinite regions are first described. These approaches turned into complicated transforming equations, making the visual and intuitive interpretation of microwave problems very difficult. The preceding theories fail to merge the active and passive worlds in a simple and rigorous manner, since they propose an empirical solution to map an infinite region into a finite surface. The 3D Smith chart differs from previous attempts 4 to generalize the planar 2D Smith chart in a fundamental way: the way in which infinity is treated. Meantime, the Greenwich meridian is the locus of pure resistive circuits (see Figure 1, where the constant resistance r and reactance x circles are drawn in blue and red, respectively). The East hemisphere is the place of inductive circuits, whereas the West hemisphere hosts the capacitive circuits. As a result, the classical 2D Smith chart including the passive loads 2 is mapped stereographically into the North hemisphere, while the circuits with negative resistance (that are outside the classical planar Smith chart) are mapped into the South one. The reflection coefficient plane is mapped stereographically through the South Pole on the surface of a unit sphere. The mathematical theory of the 3D Smith chart 1 unifies active and passive microwave circuit design on the surface of a Riemann sphere. 1 Representation of the 2D (a) and 3D (b) Smith charts.
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