# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## Appendix A Transmission Line Theory  
We recall here the basic concepts of transmission line theory. A transmission line is characterized by four distributed line parameters: the resistance per unit length \(R\), the inductance per unit length \(L\), the capacitance per unit length \(C\) and the conductance per unit length \(G\) of the dielectric separating the two conductors of the line.  
The general solution at an angular frequency \(\omega\) for the voltage and current along a line, composed of the superposition of a forward-propagating wave and a backward-propagating one is the following:  
\[V(x) = V_{0}^{+}e^{-\gamma x}+V_{0}^{-}e^{\gamma x} \tag{1}\] \[I(x) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma x}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma x} \tag{2}\]  
where \(x\) is a coordinate running along the line, \(\gamma=\alpha+i\beta\) with \(\alpha\) being the attenuation constant and \(\beta\) the wave number of the propagating perturbations, and \(Z_{c}\) is the characteristic impedance of the line. These quantities are defined by the distributed line parameters, in the limit \(R\ll\omega L\) and \(G\ll\omega C\), by the following:  
\[\alpha=\frac{1}{2}\left(\frac{R}{Z_{c}}+GZ_{c}\right)\qquad\beta=\frac{\omega }{v_{p}} \tag{3}\]  
where the characteristic impedance and the phase velocity are given by the following:  
\[Z_{c}=\sqrt{\frac{L}{C}}\qquad v_{p}=\frac{1}{\sqrt{LC}}. \tag{4}\]It is worth noting that, if \(G\) is negligible,  
\[\alpha=\frac{R}{2Z_{c}}. \tag{10}\]  
The actual values of the distributed parameters are difficult to predict from the specifications of the line for non-trivial geometries, so it seems better to consider \(v_{p}\) (or the velocity factor \(V_{f}=v_{p}/c_{0}\)), \(Z_{c}\) and \(\alpha\) as unknown parameters and derive them either by studying the propagation properties of the line or by the fitting of the experimental data. The attenuation coefficient \(\alpha\), however, remains still undetermined, and cannot be trivially computed from Equation (10) because the resistance per unit length can be substantially different from the DC value and must be evaluated experimentally.  
The reflection coefficient of the line is a position-dependent quantity defined as follows:  
\[\Gamma(x)=\frac{V_{0}^{-}e^{\gamma x}}{V_{0}^{+}e^{-\gamma x}}=\frac{V_{0}^{-} }{V_{0}^{+}}e^{2\gamma x}=\Gamma(0)e^{2\gamma x} \tag{11}\]  
The wave impedance is another position-dependent quantity defined as follows:  
\[Z(x)=\frac{V(x)}{I(x)}=Z_{c}\frac{V_{0}^{+}e^{-\gamma x}+V_{0}^{-}e^{\gamma x} }{V_{0}^{+}e^{-\gamma x}-V_{0}^{-}e^{\gamma x}}=Z_{c}\frac{1+V_{0}^{-}/V_{0}^{ +}e^{2\gamma x}}{1-V_{0}^{-}/V_{0}^{+}e^{2\gamma x}}. \tag{12}\]  
Recalling the definition of the reflection coefficient, we have the following:  
\[Z(x)=Z_{c}\frac{1+\Gamma(0)e^{2\gamma x}}{1-\Gamma(0)e^{2\gamma x}}=Z_{c}\frac {1+\Gamma(x)}{1-\Gamma(x)}. \tag{13}\]  
The reflection coefficient can be written in terms of impedance as follows:  
\[\Gamma(x)=\frac{Z(x)-Z_{c}}{Z(x)+Z_{c}}. \tag{14}\]  
Voltage and current can be written in terms of the reflection coefficient as follows:  
\[V(x) = V_{0}^{+}e^{-\gamma x}(1+\Gamma(x)) \tag{15}\] \[I(x) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma x}(1-\Gamma(x)). \tag{16}\]