Extended Capabilities of the 3-D Smith Chart With Group Delay and Resonator Quality Factor

Abstract

Abstract-This paper extends the capabilities of the 3-D Smith chart for representing positive and negative differential-phase group delay and the associated loaded resonator quality factor, displayed simultaneously with scattering (S)-parameters. Here, mathematical concepts, inspired from elementary differential geometry and topology, are used to implement 3-D projections. It is shown that a condition for a circuit to exploit negative differential-phase group delay is that its S-parameter winding number should be ≥ 0 (relative to its origin). Finally, exemplar network responses that exhibit both positive and negative differential-phase group delay and loaded resonator quality factor are shown with the 3-D Smith chart. The convenience of being able to simultaneously display a wider range of parameters on one visualization platform, with the 3-D Smith chart, may help to speed-up the design and analysis of microwave circuits by the user.Index Terms-Computer-aided design (CAD), differentialphase group delay, negative group delay (NGD), non-Foster, quality factor, Smith chart.

3D Smith Chart Constant Quality Factor Semi-Circles Contours for Positive and Negative Resistance Circuits

“…By using the stereographic projection (9), we can get images of the coaxal circles on the 3D Smith chart located on the unit space sphere (surface) centered in the origin (0,0,0) [19][20][21][22].…”

“…centre-radius-Q dependency. We go on to prove that these circle arcs represent simple semicircles on the North hemisphere for all passive circuits, when analysed on the 3D Smith chart computer aided design (CAD) tool [19][20][21]. After displaying the constant Q-arcs for negative resistance circuits, we prove that these represent semi-circles in the South hemisphere.…”

A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart

“…Described in References [7][8][9][10], the 3D Smith chart has its roots in the group theory of inversive geometry [11] where the point at infinity seen as a pole on a sphere plays a key role. However, the 3D Smith chart has some inconveniencies, for example it cannot be easy used without a software tool, while the idea of using a 3D embedded surface may be still discouraging for the electrical engineering community which is used to the 2D Smith chart [12].…”

“…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][8][9][10], the 3D Smith chart has its roots in the group theory of inversive geometry [11] where the point at infinity seen as a pole on a sphere plays a key role.…”

Ka-Band Lightweight High-Efficiency Wideband 3D Printed Reflector Antenna

“…In [15][16][17], several 3D printed antennas are designed and fabricated operating at millimetre wave to terahertz frequency bands, using two different printing technologies: binder jetting/sintering on 316 L stainless steel and the selective laser melting (SLM) on Cu-15Sn. In [18,19], different microwave devices and guiding structure are designed using 3D printing technology. In [20], an offset-fed dual band 3D printed circularly polarized stepped reflector has been designed and measured.…”