Analytic model of a magnetically insulated transmission line with collisional flow electrons

We have developed a relativistic-fluid model of the flow-electron plasma in a steady-state one-dimensional magnetically insulated transmission line (MITL). The model assumes that the electrons are collisional and, as a result, drift toward the anode. The model predicts that in the limit of fully developed collisional flow, the relation between the voltage ${V}_{a}$, anode current ${I}_{a}$, cathode current ${I}_{k}$, and geometric impedance ${Z}_{0}$ of a 1D planar MITL can be expressed as ${V}_{a}={I}_{a}{Z}_{0}h(\ensuremath{\chi})$, where $h(\ensuremath{\chi})\ensuremath{\equiv}[(\ensuremath{\chi}+1)/4(\ensuremath{\chi}\ensuremath{-}1){]}^{1/2}\ensuremath{-}\mathrm{ln}\ensuremath{\lfloor}\ensuremath{\chi}+({\ensuremath{\chi}}^{2}\ensuremath{-}1{)}^{1/2}\ensuremath{\rfloor}/2\ensuremath{\chi}(\ensuremath{\chi}\ensuremath{-}1)$ and $\ensuremath{\chi}\ensuremath{\equiv}{I}_{a}/{I}_{k}$. The relation is valid when ${V}_{a}\ensuremath{\gtrsim}1\text{ }\mathrm{MV}$. In the minimally insulated limit, the anode current ${I}_{a,\mathrm{min}}=1.78{V}_{a}/{Z}_{0}$, the electron-flow current ${I}_{f,\mathrm{min}}=1.25{V}_{a}/{Z}_{0}$, and the flow impedance ${Z}_{f,\mathrm{min}}=0.588{Z}_{0}$. {The electron-flow current ${I}_{f}\ensuremath{\equiv}{I}_{a}\ensuremath{-}{I}_{k}$. Following Mendel and Rosenthal [Phys. Plasmas 2, 1332 (1995)], we define the flow impedance ${Z}_{f}$ as ${V}_{a}/({I}_{a}^{2}\ensuremath{-}{I}_{k}^{2}{)}^{1/2}$.} In the well-insulated limit (i.e., when ${I}_{a}\ensuremath{\gg}{I}_{a,\mathrm{min}}$), the electron-flow current ${I}_{f}=9{V}_{a}^{2}/8{I}_{a}{Z}_{0}^{2}$ and the flow impedance ${Z}_{f}=2{Z}_{0}/3$. Similar results are obtained for a 1D collisional MITL with coaxial cylindrical electrodes, when the inner conductor is at a negative potential with respect to the outer, and ${Z}_{0}\ensuremath{\lesssim}40\text{ }\ensuremath{\Omega}$. We compare the predictions of the collisional model to those of several MITL models that assume the flow electrons are collisionless. We find that at given values of ${V}_{a}$ and ${Z}_{0}$, collisions can significantly increase both ${I}_{a,\mathrm{min}}$ and ${I}_{f,\mathrm{min}}$ above the values predicted by the collisionless models, and decrease ${Z}_{f,\mathrm{min}}$. When ${I}_{a}\ensuremath{\gg}{I}_{a,\mathrm{min}}$, we find that, at given values of ${V}_{a}$, ${Z}_{0}$, and ${I}_{a}$, collisions can significantly increase ${I}_{f}$ and decrease ${Z}_{f}$. Since the steady-state collisional model is valid only when the drift of electrons toward the anode has had sufficient time to establish fully developed collisional flow, and collisionless models assume there is no net electron drift toward the anode, we expect these two types of models to provide theoretical bounds on ${I}_{a}$, ${I}_{f}$, and ${Z}_{f}$.