# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 3 Propagation of Electromagnetic Perturbations into the Resonator
speed is equal to \(c_{0}\), regardless of the shield position. We want to emphasize that the velocity factor is frequency dependent, so the propagation is dispersive. To put things in perspective, we can consider, for example, that for the industrial frequency \(f=27.12\) MHz, the plane wave wavelength is \(11\) m, so \(k=0.567\) m\({}^{-1}\). For a coil with a diameter of \(10\) cm, this gives \(kb=2.8\times 10^{-2}\). This can be in the region where all longitudinal velocity factors collapse on a single curve or not, depending on the pitch angle value. Before concluding this section, it is worth addressing in more detail two limit cases. The first one is the case without the shield, which, in our formulation, corresponds to \(c/b\rightarrow\infty\). It is straightforward to show that in this case, the eigenvalue equation reduces to the following:  
\[B_{z1} = B_{1}I_{0}(\tau r)\]6  
This result was quoted by several authors, in the context of analyzing the self-capacitance of a helical coil [24]. The rest of the analysis proceeds as before, but it is worth mentioning that the axial velocity factor can be approximated reasonably well, at least in the transition region, by the following expression [24]:  
\[B_{z1} = B_{1}I_{0}(\tau r)\]7  
where \(d=2b\) is the coil diameter. This is illustrated in Figure 4, which shows a superposition of the exact longitudinal velocity factor obtained from the resolution of the eigenvalue equation superposed to that obtained from the empirical formula above. It can be seen that, as long as the frequency is low enough so that the longitudinal velocity factor is larger than \(1\), the approximated value is reasonably close to the exact one. \[B_{z1} = B_{1}I_{0}(\tau r)\]8 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 \(\psi=5^{\circ}\). For this particular case, an analytic formula for the characteristic impedance of the resonator seen as a transmission line was derived [24]. This is given by the following (expressed in Ohm):  
\[B_{z1} = B_{1}I_{0}(\tau r)\]9  
It is worth noting that the product of Bessel functions that appears in this formula is a decreasing function of its argument. The characteristic impedance for the case without shield is also plotted in Figure 4, as a function of the normalized frequency \(kb\), for different pitch angles. It can be seen that the characteristic impedance decreases with frequency. For high frequencies, the curves all collapse on the following value:  
\[E_{z2} = A_{2}I_{0}(\tau r)+A_{2}^{\prime}K_{0}(\tau r)\]0  
whereas at low frequencies, they fall below this curve. Typical characteristic impedance values for wavelengths much larger than the coil radius fall indicatively in the range of the k\(\Omega\) for pitch angles between \(1^{\circ}\) and \(10^{\circ}\). The second limit case is that in which the shield is directly over the coil, that is, \(c/b=1\). In this case, the eigenvalue equation simply reduces to the following:  
\[E_{z2} = A_{2}I_{0}(\tau r)+A_{2}^{\prime}K_{0}(\tau r)\]1  
so that no numerical resolution is required. The dispersion relation becomes nondispersive. \[E_{z2} = A_{2}I_{0}(\tau r)+A_{2}^{\prime}K_{0}(\tau r)\]2  
with a velocity factor \(V_{f}=\sin\psi\), and therefore a longitudinal velocity factor \(V_{f}^{L}=1\): the voltage pulses propagate along the conductor at the speed of light in vacuum. We conclude this section by remarking that, in general, both the velocity factor and the characteristic impedance are functions of the frequency of the pitch angle \(\psi\) (which is the only coil geometrical parameter entering the treatment) and of the shield proximity \(c/b\).