Graphical Representation of the Power Transfer Efficiency of Lumped-Element Circuits Based on Hyperbolic Geometry

Abstract

By adding a passive element to a power source, an inevitable power attenuation caused by the dissipative loss occurs. It is important for some applications to reduce this undesirable attenuation. A representative example of one of these kinds of applications is impedance matching, for which we transform the source's impedance to a different preferable one with passive elements, and the attenuation is to be as low as possible. To give a visible way to design better circuits in such situations, we propose a graphical representation to understand the attenuation from the viewpoint of hyperbolic geometry. We reveal that the attenuation in logarithmic scale (typically, dB or Np) is proportionate to the hyperbolic length of the path representing the movement of the reflection coefficient, and the constant of proportionality is determined by the unloaded Q-factor of the connected elements. Exploiting the result, we can find preferable topologies by using the Smith chart with an intuition. As well as this graphical representation, we also reveal the lower bound of the attenuation in terms of the hyperbolic distance between the two reflection coefficients which we want to match. A new usage of the Smith chart to estimate the loss and its lower bound is given in this study.Index Terms-Smith chart, power efficiency, impedance matching, hyperbolic geometry.