# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 7 Experimental Results  
The theoretical results described up to now were checked against the properties of actual helical resonators, both with and without a conducting shield, in unloaded and loaded conditions. In these experiments, the input impedance was measured using a PocketVNA Vector Network Analyzer, in 1-port reflection only mode, which was properly calibrated. Furthermore, the voltage amplification factor at resonance was measured by sending RF power to the resonator and measuring the input voltage with a 10:1 oscilloscope probe and the output voltage with a 1000:1 Tektronix P6015a high voltage (HV) probe. For this latter experiment, the frequency was adjusted so as to achieve the maximum \(V_{out}/V_{in}\) ratio. The first resonator (resonator 1) was built, using a 1 mm diameter insulated wire wound on a polyurethane tube of a 10 mm inner diameter and 12 mm outer diameter. Initially, 110 turns of wire were wound, and the tap point was located after 10 turns (\(\delta=0.091\)). By measuring the resonator axial length, the average pitch was evaluated to be 1.04 mm, in good agreement with the wire thickness. Subsequently, while the tap point was kept fixed, the number of turns on the top part of the resonator was decreased from 100 to 90, then to 80, and so on up to 60 (\(\delta=0.143\)). The resonator input impedance modulus and phase were measured using the VNA as a function of frequency. The results are shown in the top row of Figure 11. The curves were fitted with expression (44). However, this expression is given as a function of the wavenumber \(\beta\), which is related to the frequency by \(k=\beta V_{f}\). Thus, in order to be able to perform a fit on the data given as a function of frequency, \(\beta H\) was replaced in expression (44) by \(\pi f/2f_{0}\), introducing the new parameter \(f_{0}\), which represents the frequency at which the electrical length of the entire resonator \(\beta H\) is equal to \(\pi/2\), i.e., the frequency where the maximum of the input impedance is located. The fit thus led to opti  
Figure 10: **Left**: plot of the input resistance normalized to the characteristic impedance of the fully shielded resonator with capacitive load, achieved in the purely resistive condition near the resonance, as a function of the relative tap point position \(\delta\), for different values of \(\omega_{0}\tau_{L}\). (**Right**): voltage amplification factor for the same conditions. The attenuation factor is \(\alpha H=0.02\). mal values of the parameters \(Z_{c}\), \(\alpha H\) and \(f_{0}\), and then \(V_{f}\) was computed as \(4 Hf_{0}/c_{0}\). In fact, \(V_{f}\) is frequency dependent, and this dependence should be included in the fit. However, this would lead to a complex procedure, due to the need to solve the eigenvalue equation at each step. It was thus decided to assume \(V_{f}\) to be almost constant in the vicinity of the resonance. It can be seen that the resulting curves fit reasonably well with the experimental ones, both for the amplitude and the phase of the input impedance. Subsequently, a grounded shield made from an aluminum tube with an internal diameter of 16 mm and external diameter of 18.3 mm (average diameter of 17.15 mm and thickness of 2.3 mm) was applied to the resonator, and the fitting procedure was repeated as shown in the bottom row of Figure 11. The fitting curves also follow reasonably well the experimental ones. It can be noted that the curves for the shielded resonator are much sharper with stronger gradients, and the phases reach more negative values at the resonance, all indications of a reduced dissipation. This can be understood by assuming that the main source of dissipation is radiation from the sides of the device, which is prevented by the conducting screen. In this respect, it is important to notice, for experimenters wishing to repeat this work, that the wire used to ensure the grounding of the shield should be positioned so as to minimize the formation of loops, which can act as antennas.