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 Z₀ of a 1D planar MITL can be expressed as V_a=I_aZ₀h(χ), where h(χ)≡[(χ+1)/4(χ-1)]^1/2-ln⁡⌊χ+(χ²-1)^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/Z₀, the electron-flow current I_f,min⁡=1.25V_a/Z₀, and the flow impedance Z_f,min⁡=0.588Z₀. {The electron-flow current I_f≡I_a-I_k. Following Mendel and Rosenthal [Phys. Plasmas 2, 1332 (1995)], we define the flow impedance Z_f as V_a/(I_a²-I_k²)^1/2.} In the well-insulated limit (i.e., when I_a≫I_a,min⁡), the electron-flow current I_f=9V_a²/8I_aZ₀² and the flow impedance Z_f=2Z₀/3. 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 Z₀≲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 Z₀, 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, Z₀, 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.