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 Va , anode current Ia , cathode current Ik , and geometric impedance Z0 of a 1D planar MITL can be expressed as Va=IaZ0h(χ) , where h(χ)≡[(χ+1)/4(χ−1)]1/2−ln⌊χ+(χ2−1)1/2⌋/2χ(χ−1) and χ≡Ia/Ik . The relation is valid when Va≳1MV . In the minimally insulated limit, the anode current Ia,min=1.78Va/Z0 , the electron-flow current If,min=1.25Va/Z0 , and the flow impedance Zf,min=0.588Z0 . {The electron-flow current If≡Ia−Ik . Following Mendel and Rosenthal [Phys. Plasmas 2, 1332 (1995)], we define the flow impedance Zf as Va/(Ia2−Ik2)1/2 .} In the well-insulated limit (i.e., when Ia≫Ia,min ), the electron-flow current If=9Va2/8IaZ02 and the flow impedance Zf=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≲40Ω . 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 Va and Z0 , collisions can significantly increase both Ia,min and If,min above the values predicted by the collisionless models, and decrease Zf,min . When Ia≫Ia,min , we find that, at given values of Va , Z0 , and Ia , collisions can significantly increase If and decrease Zf . 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 Ia , If , and Zf .