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

Abstract

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(χ), where h(χ)≡[(χ+1)/4(χ-1)]^{1/2}-ln⁡⌊χ+(χ^{2}-1)^{1/2}⌋/2χ(χ-1) and χ≡I_{a}/I_{k}. The relation is valid when V_{a}≳1 MV. In the minimally insulated limit, the anode current I_{a,min⁡}=1.78V_{a}/Z_{0}, the electron-flow current I_{f,min⁡}=1.25V_{a}/Z_{0}, and the flow impedance Z_{f,min⁡}=0.588Z_{0}. {The electron-flow current I_{f}≡I_{a}-I_{k}. Following Mendel and Rosenthal [Phys. Plasmas 2, 1332 (1995)PHPAEN1070-664X10.1063/1.871345], we define the flow impedance Z_{f} as V_{a}/(I_{a}^{2}-I_{k}^{2})^{1/2}.} In the well-insulated limit (i.e., when I_{a}≫I_{a,min⁡}), the electron-flow current I_{f}=9V_{a}^{2}/8I_{a}Z_{0}^{2} and the flow impedance Z_{f}=2Z_{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}≲40 Ω. 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,min⁡} and I_{f,min⁡} above the values predicted by the collisionless models, and decrease Z_{f,min⁡}. When I_{a}≫I_{a,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}.