A System Stability Analysis for a Time-delayed Four Meander Line Antenna Legs Birdcage for Helicon Excitation with Titanium Alloy and Copper Legs

A four meander line antenna legs birdcage for helicon excitation with titanium alloy and copper legs is used in many applications. Helicon waves are electromagnetic waves which propagate in magnetized plasma. An approach for helicon plasma source improvement consists in optimizing helicon wave excitation by means of specific antenna designs. The antenna is composed of N₁ identical meander line antenna legs equally distributed on a cylinder of radius R_c. We consider an \mathrm{N}-{1}=4 design. Due to the RF antenna leg parasitic effect we get each leg's current with delay \tau-{k} (k is a leg number index, \mathrm{k}=1,\ldots,4). For simplicity, we assume all time delays are equal, \tau-{k}=\tau for k = 1, 2, 3, 4. We present a practical guide which combines graphical information with theoretical analysis to effectively study the local stability of models involving delay dependent parameters. The stability of a given steady state is determined by the graphs of some function of \tau.