Measuring plasma potential with an impedance probe in low density plasma

A recent rf technique for determining plasma potential, \textit{$\phi $}$_{p}$ , using an impedance probe was shown to be independent of probe geometry, magnetic field, and orientation. However, a problem which arises in low density plasma concerns a magnitude mismatch between typical network analyzer input impedance ($Z_{0 }$= 50 $\Omega )$ and the large value of ac resistance ($R_{ac})$ which is inversely proportional to $n_{e}$. The method relies on finding a minimum in \textit{Re(Z}$_{ac}$)\footnote{\textit{Phys. Plasmas} \textbf{17}, 113503 (2010).}$^,$\footnote{\textit{NRL Memorandum Report 6750-12-9413} (2012).} which is difficult if $R_{ac}$ is much larger than $Z_{0}$. For low density space plasmas (10$^{4}$ -10$^{5}$ cm$^{-3})$ values of $R_{ac}$ range to k$\Omega $ levels. We have developed numerical simulations based on solving the Boltzmann equation in spherical geometry for a given sheath size. These simulations include a presheath and predict values for $Z_{ac}$ which are then used to estimate the error as a function of input impedance based on the error associated with a 50 $\Omega $ load.