# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation

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## Abstract

**:**

## 1. Introduction

## 2. Geometry

## 3. Propagation of Electromagnetic Perturbations into the Resonator

## 4. The Helical Resonator as a Transmission Line

## 5. The Fully Shielded Helical Resonator

## 6. Effect of a Capacitive Load

## 7. Experimental Results

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RF | Radio Frequency |

HV | High Voltage |

## Appendix A. Transmission Line Theory

## References

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**Figure 1.**Schematics of the helical resonator. The terminals labeled IN and OUT are the ones where the voltages ${V}_{in}$ and ${V}_{out}$ are evaluated. The external rectangle represents the screen.

**Figure 2.**

**Left**: function $g(\tau b,c/b)$ entering the eigenvalue equation, plotted as versus $\tau b$ for different values of $c/b$.

**Right**: solutions of the eigenvalue equation, plotted versus $kbcot\psi $ for different values of $c/b$.

**Figure 3.**Longitudinal velocity factor ${V}_{f}^{L}$ plotted as a function of the normalized frequency $kb$, for different values of the shield proximity $c/b$; the left panel refers to a pitch angle $\psi ={1}^{\circ}$, the right panel to $\psi ={5}^{\circ}$.

**Figure 4.**

**Left**: exact longitudinal velocity factor ${V}_{f}^{L}$ for the case without shield plotted as a function of the normalized frequency $kb$ (solid lines) and approximation obtained from Equation (27) (dashed lines), for three different values of the pitch angle $\psi $.

**Right**: characteristic impedance for the case without shield plotted as a function of the normalized frequency $kb$, for three different values of the pitch angle $\psi $.

**Figure 5.**Behavior of the helical resonator modeled as a transmission line plotted as a function of the normalized wave number $\beta H$ expressed in units of $2\pi $, for three different values of the attenuation factor $\alpha H$. The curves are computed for $\delta =0.1$. Top left: modulus of the input impedance $|{Z}_{in}|$ normalized to the characteristic impedance ${Z}_{c}$. Top right: voltage amplification factor $|{V}_{out}/{V}_{in}|$. Bottom left: phase shift in degrees between input voltage and input current. Bottom right: phase shift in degrees between the input voltage and the output voltage.

**Figure 6.**Behavior of the helical resonator modeled as a transmission line plotted as a function of the normalized wave number $\beta H$ expressed in units of $2\pi $, for three different values of the relative tap position $\delta $. The curves are computed for $\alpha H=0.02$. Top left: modulus of the input impedance $|{\mathrm{Z}}_{in}|$ normalized to the characteristic impedance ${\mathrm{Z}}_{c}$. Top right: voltage amplification factor $|{V}_{out}/{V}_{in}|$. Bottom left: phase shift in degrees between input voltage and input current. Bottom right: phase shift in degrees between the input voltage and the output voltage.

**Figure 7.**Behavior of the helical resonator at the peak resistance and at the resonance, plotted as a function of the normalized attenuation factor $\alpha H$, for three different values of the relative tap position $\delta $. Top left: input resistance at peak, normalized to the characteristic impedance ${\mathrm{Z}}_{c}$. Top right: input resistance at resonance, normalized to the characteristic impedance ${\mathrm{Z}}_{c}$. Bottom left: ratio of the previous two quantities. Bottom right: voltage amplification factor $|{V}_{out}/{V}_{in}|$.

**Figure 8.**Behavior of the helical resonator at the point of purely resistive input impedance near resonance (continuous line) and at resonance (dashed line), plotted as a function of the tap position $\delta $, for three different values of the normalized attenuation factor $\alpha H$.

**Left**: input resistance at resonance, normalized to the characteristic impedance ${\mathrm{Z}}_{c}$.

**Right**: voltage amplification factor $|{V}_{out}/{V}_{in}|$.

**Figure 9.**Behavior of the fully shielded helical resonator with capacitive load plotted as a function of frequency f, normalized to the frequency ${f}_{0}$ of maximum impedance modulus, for different values of ${\omega}_{0}{\tau}_{L}$. The curves are computed for $\alpha H=0.02$ and $\delta =0.1$. Top left: modulus of the input impedance $|{\mathrm{Z}}_{in}|$ normalized to the characteristic impedance ${\mathrm{Z}}_{c}$. Top right: voltage amplification factor $|{V}_{out}/{V}_{in}|$. Bottom left: phase shift in degrees between input voltage and input current. Bottom right: phase shift in degrees between the input voltage and the output voltage.

**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 $\alpha H=0.02$.

**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.

**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.

**Figure 13.**

**Top**: plot of the modulus (

**left**) and phase (

**right**), normalized to $\pi $, of the input impedance of resonator 2, both in standard configuration and with the HV probe attached to the open end. 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.

**Figure 14.**

**Top**: plot of the modulus (

**left**) and phase (

**right**), normalized to $\pi $, of the input impedance of resonator 3, both in standard configuration and with the HV probe attached to the open end. 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.

**Table 1.**Parameters resulting from the fit of the input impedance curves for resonators 2 and 3. The load characteristic time ${\tau}_{L}$ and the associated load capacitance C are also reported.

Shield | HV Probe | ${\mathbf{Z}}_{\mathit{c}}\left(\mathbf{\Omega}\right)$ | $\mathit{\alpha}\mathit{H}$ | ${\mathit{V}}_{\mathit{f}}$ | ${\mathit{\tau}}_{\mathit{L}}$ (ns) | C (pF) | |
---|---|---|---|---|---|---|---|

Resonator 2 | No | No | 1003 | 0.119 | 0.025 | - | - |

Resonator 2 | No | Yes | 1003 | 0.047 | 0.025 | 5.58 | 5.56 |

Resonator 2 | Yes | No | 272.4 | 0.011 | 0.016 | - | - |

Resonator 2 | Yes | Yes | 272.4 | 0.015 | 0.016 | 2.04 | 7.48 |

Resonator 3 | No | No | 1064 | 0.173 | 0.055 | - | - |

Resonator 3 | No | Yes | 1064 | 0.026 | 0.055 | 4.54 | 4.26 |

Resonator 3 | Yes | No | 501.7 | 0.004 | 0.036 | - | - |

Resonator 3 | Yes | Yes | 501.7 | 0.013 | 0.036 | 2.29 | 4.57 |

**Table 2.**Comparison of the experimental amplification factor at resonance with the expected value $1/\alpha H$ predicted from the fit outcome, for resonators 2 and 3, with and without shield.

Shield | $1/\mathit{\alpha}\mathit{H}$ | ${\mathit{V}}_{\mathit{out}}/{\mathit{V}}_{\mathit{in}}$ | |
---|---|---|---|

Resonator 2 | No | 21.4 | 41 |

Resonator 2 | Yes | 61.8 | 80 |

Resonator 3 | No | 38.0 | 30 |

Resonator 3 | Yes | 77.6 | 100 |

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**MDPI and ACS Style**

Martines, E.; Cavazzana, R.; Cordaro, L.; Zuin, M.
The Helical Resonator: A Scheme for Radio Frequency Plasma Generation. *Appl. Sci.* **2021**, *11*, 7444.
https://doi.org/10.3390/app11167444

**AMA Style**

Martines E, Cavazzana R, Cordaro L, Zuin M.
The Helical Resonator: A Scheme for Radio Frequency Plasma Generation. *Applied Sciences*. 2021; 11(16):7444.
https://doi.org/10.3390/app11167444

**Chicago/Turabian Style**

Martines, Emilio, Roberto Cavazzana, Luigi Cordaro, and Matteo Zuin.
2021. "The Helical Resonator: A Scheme for Radio Frequency Plasma Generation" *Applied Sciences* 11, no. 16: 7444.
https://doi.org/10.3390/app11167444