Figure 3
Reflection coefficient orientation changes and the sign of the oriented curvature for different circuits. (a) Smith chart representation of the reflection coefficient for a 1-port negative capacitance (purely non-Foster circuit) and a positive inductance (purely Foster). For a capacitor with purely negative capacitance and an inductor with positive inductance their reflection coefficients \({{\Gamma }}_{1{zm}}(j{\omega })\) overlap on the Smith chart on a wide frequency range. Their opposite orientation is given by the different sign of their oriented curvature \({{k}}_{{{\Gamma }}_{1{zm}}}({\omega })\). Their same path is given by the same absolute value of their oriented curvature. (b) On the newly implemented frequency dependent 3D Smith chart one can see the clockwise motion with increasing frequency for the inductor with positive inductance and the counter-clockwise motion for the negative valued capacitor, the motion is on the contour of the equatorial plane (lossless circuits). (c) Mixed motion for a fabricated circuit containing non-Foster (lossy elements). (d) Mixed clockwise and counter-clockwise motion of the reflection coefficient of a passive lossy network described by the positive real function zm(s) with the 1 port reflection coefficient (for s = jω) \({{\Gamma }}_{1{zm}}(j{\omega })\). The reflection coefficient has a clockwise orientation from −2 < ω < −0.28 and for 0.28 < ω < 2, while counter-clockwise orientation for −0.28 < ω < 0.28. The sign changes of its 1-port reflection coefficient curvature \({{k}}_{{{\Gamma }}_{1{zm}}}({\omega })\)(5) generates the changes of orientation of its path on the Smith chart. It is interesting to notice that mixed motion can exist on limited bandwidth also for lossy circuits with only Foster elements and thus that the counter-clockwise motion is by no means a prove of an existence of a non-Foster element in the network. A more detailed description on oriented curvature and 1-port and two port networks is given in Supplementary Section 1.
