# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 4 The Helical Resonator as a Transmission Line
The voltage amplification becomes, of course, smaller as the attenuation factor increases, so minimizing dissipation is an important part of the design of an effective resonator. A similar analysis can be performed by keeping the dissipation constant, and varying the tap point relative position \(\delta\). This is shown in \[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]8 It can be observed that the maximum modulus of impedance increases as \(\delta\) is increased, while the amplification factor peak moves to the right, as expected, and grows in magnitude. The same shift occurs to the phase shift between the input and output voltage. The region of negative \(V_{in}-I_{in}\) phase becomes deeper and wider. This last result shows that, for the given resonator parameters, there is a minimum \(\delta\) that allows a purely resistive input impedance. Since in the context of plasma production, the interest is focused on the resonance, which gives the maximum voltage amplification, we now plot the resistance at the peak, the resistance at the resonance, their ratio and the voltage amplification factor at the resonance, as a function of the dissipation factor \(\alpha H\) for three \(\delta\) values. This is shown in \[I(z) = \frac{V_{0}^{+}}{Z_{c}}e^{-\gamma z}-\frac{V_{0}^{-}}{Z_{c}}e^{ \gamma z} \tag{33}\]9 It can be seen that the peak resistance is of the same order of magnitude of the characteristic impedance, that is, of the order of the k\(\Omega\), and decreases as the dissipation grows, while it increases when the tap point is moved to the right. On the contrary, the resistance at the resonance, which is the relevant parameter for the operation of the resonator as a voltage amplifier, increases both with dissipation and with tap position. This shows that changing the tap position is a way to achieve an impedance matching of the device with the power supply. Finally, the ratio of the output voltage to the input one is shown, as could be expected from energy considerations and from the previous figures, to decrease with dissipation, with a modest dependence on the tap position in the explored range. In the figure, we have also plotted the ratio of the peak resistance to the resistance at resonance, in order to illustrate the fact that this easily measurable parameter could be used as a way to estimate the attenuation factor \(\alpha H\). This, in turn, allows us to estimate the output without actually measuring it, a useful possibility given the fact that a direct measurement with a high voltage would be perturbative due to the addition of a capacitive load (as described below). \[V_{0}^{+}+V_{0}^{-}=0 \tag{34}\]0 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 \(|Z_{in}|\) 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. We should now remark, though, that the input impedance at resonance is given by the following:  
\[V_{0}^{+}+V_{0}^{-}=0 \tag{34}\]1  
This expression clarifies that the input impedance at resonance is not fully resistive. Indeed, the point of zero \(V_{in}-I_{in}\) phase is located at a slightly lower frequency. As a consequence, the calculation of the input impedance and of the amplification factor for the point of purely resistive input impedance has to be done numerically by identifying the wave number for which the imaginary part of the impedance is zero, and then evaluating its real part at this same wave number.