# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 4 The Helical Resonator as a Transmission Line
The maximum is found for an electrical length as follows:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]3  
This corresponds to the condition of the entire coil length equal to \(\lambda/4\) (either considering the axial direction, with wavelength \(2\pi/\beta\) and length \(H\), or the longitudinal direction, with wavelength \(2\pi/\beta^{L}\) and length \(L\)). It is worth noting that this point corresponds to a  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]4 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. rather large impedance, of the order of the characteristic impedance, and thus in the range of the k\(\Omega\). The voltage amplification factor displays a maximum, corresponding to the minimum of the impedance modulus. By analyzing expression (42), it can be seen that the maxima of the amplification factor occur at the following resonant wave numbers:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]5  
which correspond to increasing resonant frequencies. These are \(\lambda/4\) resonances computed by taking into account only the length of the top part of the resonator. The peak displayed in the graph, and the only one which will be considered in the following, is \(\beta_{r}H(1-\delta)=\pi/2\). The extension of the results to higher order resonances is straightforward. Other authors have studied resonances of a higher order in the context of plasma production [25]. It is clear how the distance between the two critical points, that is, the point of maximum impedance and the resonance, increases as \(\delta\) is increased. The amplification factor at the resonance is the following:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]6  
Since in practical cases, \(\alpha H\ll 1\), this can be approximated by the following:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]7  
This expression elucidates the importance of achieving low dissipation in order to obtain a large voltage amplification. Looking at the phases, it can be seen that the input voltage and current are in quadrature at low frequencies, then go in phase at the impedance maximum; then the phase is inverted, has another zero at the amplification factor maximum and returns to \(\pi/2\) at high frequencies. Thus, the two critical points, that is, the maximum and minimum of the input impedance modulus, both approximately correspond to a real impedance. It is worth also noting that the extension of the phase reversal region between the two depends on the attenuation factor, so that this can be used as an indirect way of measuring it. Indeed, if the dissipation is too high, the phase will never become negative, and no purely resistive input impedance is possible. Finally, the input and output voltages are in phase at low frequencies, become in quadrature at the amplification factor maximum, as expected from Equation (48), and then go in phase opposition. As is shown in the next section, the distance between the two critical points depends on the parameter \(\delta\). In the following, we are concerned mainly with the point of maximum output voltage, which is related to a resonance of the system, and which represents the optimal condition for plasma generation.