# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 7 Experimental Results
The resulting values of the fit parameters, both without and with shield, are shown in Figure 12. It can be seen that the characteristic impedance without shield grows with \(\delta\) from 1.15 k\(\Omega\) to 1.3 k\(\Omega\), whereas with the shield, it has a constant value of around 580 \(\Omega\). The attenuation factor \(\alpha H\) turns out to have a peaked shape, with values ranging between 0.05 and 0.11. This shape is somehow surprising: indeed, as \(\delta\) is increased by reducing the number of turns on the top end, and therefore the resonator length \(H\), one would expect a decreasing behavior of \(\alpha H\) with \(\delta\), assuming \(\alpha\) to be a constant independent of the resonator  
Figure 11: (**Top**): plot of the modulus (**left**) and phase (**right**), normalized to \(\pi\), of the input impedance of resonator 1. The different curves correspond to different number of turns above the tap point. The dashed curves are the fits obtained as described in the text. **Bottom**: same as above, but with the addition of a conducting shield around the resonator. size. It is, however, possible that other issues enter the dissipation process, including possible different cable positions, and therefore different radiation patterns between one measurement and the following. This is also in agreement with the observation that in the shielded case, the obtained value of \(\alpha H\) is constant, around 0.0085. If one assumes that the shield suppresses radiation, then this residual value could be attributed to other causes, and the case-to-case variation would disappear. The frequency \(f_{0}\) was found to be linearly increasing with \(\delta\), both for the unshielded and shielded case. This was expected, in view of a constant velocity factor. Indeed, the velocity factor turns out to be essentially constant, with a value of 0.052 for the unshielded case and 0.024 for the shielded one. An important issue to be addressed is how much the characteristic impedance (for the unshielded case) and the velocity factors match the predictions of the propagation model described in Section 3. According to expression (28), one would expect in the unshielded case a characteristic impedance of 2.25-2.45 k\(\Omega\) at the frequency where the resonance occurs in the measurements. This is more or less double than what was found from the fits. Concerning the velocity factor, from expression (25) one would expect values of 0.07 for the unshielded case, and 0.036 for the shielded one. These are both 1.5 times larger than those found from the fits. The source of these discrepancies is not clear. Subsequently, two other resonators were built (resonators 2 and 3) with a fixed number of turns (10 and 100 for the bottom and top parts, respectively), wound on quartz tubes. The tube of resonator 2 had a diameter of 12 mm, whereas the tube of resonator 3 had a diameter of 7 mm. The length \(H\) was respectively 110 mm and 103 mm. The shield was obtained from the same aluminum tube, which was used to shield resonator 1. Their input impedance was once again measured, using the Pocket VNA instrument. Since in this experiment, the aim was to measure the high voltage generated at the resonator open end,  
Figure 12: Parameters resulting from the fits of the input impedance data for resonator 1, plotted as a function of the relative distance \(\delta\) of the tap point from the bottom end. Top left: characteristic impedance. Top right: attenuation parameter. Bottom left: frequency at which the total electrical length of the resonator is \(\pi/2\). Bottom right: velocity factor, obtained from the previous quantity. In all graphs, results without and with conducting shield are reported. using a Tektronix P6015A high voltage probe, the input impedance measurements were performed both with and without the probe attached to the resonator. Indeed, the probe is expected to add a capacitive load, along the lines described in Section 6. The nominal HV probe input capacitance is, according to the technical specifications, around 3 pF.