# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 4 The Helical Resonator as a Transmission Line  
Armed with the expressions for the velocity factor and, for the shieldless case, of the characteristic impedance, we are now in the position to model our helical resonator as a transmission line, with the helical coil as one conductor and the shield as the other, and with voltage \(V_{in}\) applied to the tap point. The basic concepts of transmission line modeling are recalled, for the reader's convenience, in Appendix A. The aim of this model is to obtain expressions for the two quantities, which are to be optimized for plasma generation, that is, the input impedance, which ideally should match the output impedance of the power supply (typically 50 \(\Omega\)), and the voltage amplification factor that we want to maximize. We start from basic Equations (A1) and (A2), describing the spatial behavior of voltage and current for a perturbation at frequency \(\omega\):  
\[V(z) = V_{0}^{+}e^{-\gamma z}+V_{0}^{-}e^{\gamma z} \tag{32}\] \[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]  
where \(z\) is a coordinate running along the cylinder axis and \(\gamma=\alpha+i\beta\), with \(\alpha\) being the attenuation constant and \(\beta\) the wave number of the propagating perturbation. We consider separately the bottom and the top part of the resonator as two different transmission lines, each with its own boundary conditions. The bottom part, being short circuited (\(V=0\)), gives the conditions at the grounded end (\(z=0\)) and the tap point (\(z=H\delta\)):  
\[V_{0}^{+}+V_{0}^{-}=0 \tag{34}\] \[V_{0}^{+}e^{-\gamma h}+V_{0}^{-}e^{\gamma h}=V_{in} \tag{35}\]  
where we have introduced \(h=H\delta\). These yield the following solution:  
\[V_{b}(z) = V_{in}\frac{\sinh(\gamma z)}{\sinh(\gamma h)} \tag{36}\] \[I_{b}(z) = \frac{V_{in}}{Z_{c}}\frac{\cosh(\gamma z)}{\sinh(\gamma h)}. \tag{37}\]  
For the top part, which is open (\(I=0\)) we have the following:  
\[V_{0}^{+}e^{-\gamma h}+V_{0}^{-}e^{\gamma h}=V_{in} \tag{38}\] \[\frac{V_{0}^{+}}{Z_{c}}e^{-\gamma H}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma H}=0 \tag{39}\]  
yielding the following:  
\[V_{t}(z) = V_{in}\frac{\cosh(\gamma(H-z))}{\cosh(\gamma(H-h))} \tag{40}\] \[I_{t}(z) = \frac{V_{in}}{Z_{c}}\frac{\sinh(\gamma(H-z))}{\cosh(\gamma(H-h))}. \tag{41}\]  
The voltage at the line endpoint is \(V_{out}=V_{t}(H)\), so the following holds:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]0  
We shall call \"voltage amplification factor\" the modulus of this ratio. The total input current at the tap point is the following:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]1so that the input impedance of the resonator is given by the following:  
\[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]2  
This is generally complex, with a resistive and a reactive part. The expressions found for the input impedance and for the amplification factor are functions of the frequency, first because the wave number \(\beta\) is related to the frequency through the velocity factor (\(k=\beta V_{f}\)), and also because the velocity factor and the characteristic impedance are both frequency dependent. The dependence on the frequency of the attenuation coefficient \(\alpha\), which also exists, is neglected in the following. In order to understand the behavior of the expressions found above, we first make the assumption that \(Z_{c}\) is independent of frequency, and we plot the relevant quantities as a function of the \"electrical length\" \(\beta H\) (which, we remind, is equal to \(\beta^{L}L\)), expressed in units of \(2\pi\). This quantity is related to the frequency through the velocity factor. The results are shown in Figure 5 for three different values of the attenuation factor \(\alpha H\). The modulus of the input impedance displays a maximum and a minimum.