# The Helical Resonator: A Scheme for Radio Frequency Plasma Generation
## 6 Effect of a Capacitive Load  
In many practical applications, the helical resonator will not be free-standing, but its upper end will be connected to some electrode or other structure, adding to it a load that, in a first approximation, can be considered fully capacitive. Even when this is not the case, one may wish to measure the voltage of the upper end with a high voltage oscilloscope probe, which will add a capacitive load of capacitance \(C\) (typically a few pF). We thus wish to address the problem of connecting a load with impedance  
\[Z_{L}=\frac{1}{i\omega C} \tag{54}\]  
to the transmission line model described above. Using a standard result of transmission line theory, the input impedance of a line of length \(L\) and characteristic impedance \(Z_{c}\) connected to a load \(Z_{L}\) is the following:  
\[Z_{in}=Z_{c}\frac{1+\Gamma_{L}e^{-2\gamma H}}{1-\Gamma_{L}e^{-2\gamma H}} \tag{55}\]  
where \(\Gamma_{L}\) is the reflection coefficient at the load, given by the following:  
\[\Gamma_{L}=\frac{Z_{L}-Z_{c}}{Z_{L}+Z_{c}}. \tag{56}\]  
For the case of a capacitive load, the reflection coefficient takes the following form:  
\[\Gamma_{L}=\frac{1-i\omega\tau_{L}}{1+i\omega\tau_{L}} \tag{57}\]  
where the characteristic time \(\tau_{L}\) is defined as follows:  
\[\tau_{L}=Z_{c}C. \tag{58}\]  
The helical resonator can be considered the parallel of the bottom section and of the top section, the latter being connected to the capacitive load. Since the bottom section is short-circuited by the ground connection, it has \(Z_{L}=0\) and, therefore, \(\Gamma_{L}=-1\), so its input impedance is \(Z_{in}=Z_{c}\tanh\gamma H\delta\). Combining this in parallel with the top section connected to the capacitive load, one obtains the overall input impedance as follows:  
\[Z_{in}=Z_{c}\left[\coth\gamma H\delta+\frac{1-\Gamma_{L}e^{-2\gamma H(1- \delta)}}{1+\Gamma_{L}e^{-2\gamma H(1-\delta)}}\right]^{-1}. \tag{59}\]  
In the limit \(C\to 0\), \(\Gamma_{L}\) tends to 1, and the formula for \(Z_{in}\) correctly reduces to expression (44). Using the fact that  
\[\Gamma_{L}=\frac{V_{0}^{-}e^{\gamma H}}{V_{0}^{+}e^{-\gamma H}} \tag{60}\]  
it is straightforward to derive the voltage amplification factor:  
\[\frac{V_{out}}{V_{in}}=\frac{1+\Gamma_{L}}{e^{\gamma H(1-\delta)}+\Gamma_{L}e^{- \gamma H(1-\delta)}}. \tag{61}\]  
According to [PERSON] et al., this is \"probably the most important equation in all of high voltage RF engineering\" [27]. Once again, when \(\Gamma_{L}\) tends to 1, this formula reduces to expression (42). Figure 9 shows, for a fully shielded resonator, the frequency dependence of the input impedance normalized to the characteristic impedance of the amplification factor, and of the phases between \(V_{in}\) and \(I_{in}\) and between \(V_{in}\) and \(V_{out}\), for various values of \(\omega_{0}\tau_{L}\). As before, the frequency is normalized to the frequency corresponding to the wave number \(\beta_{0}\), which satisfies \(\beta_{0}H=\pi/2\), that is, the frequency of the maximum input impedance in the no-load case. It is possible to observe that as the load capacitance is increased (and, therefore, \(\omega_{0}\tau_{L}\) becomes larger) the resonance frequency decreases, and the peak impedance and peak voltage amplification are both reduced. The phase diagrams are also shifted toward lower frequencies. In general, the downshift in frequency appears to be the most relevant effect, unless the capacity becomes too large. Figure 9: Behavior of the fully shielded helical resonator with capacitive load plotted as a function of frequency \(f\), normalized to the frequency \(f_{0}\) of maximum impedance modulus, for different values of \(\omega_{0}\tau_{L}\). The curves are computed for \(\alpha H=0.02\) and \(\delta=0.1\). 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.