The coupling impedance of a toroidal beam tube with circular cross section

In this paper, the longitudinal coupling impedance of a toroidal beam tube with circular cross section is derived in the frequency domain using the toroidal coordinate system. Exact, although coupled, differential equations for the azimuthal field components are obtained. An approximate solution, valid in the limit of small curvature, is then derived. Assuming extreme relativistic energies and a beam tube with perfectly conducting walls, one finds a closed-form expression for the purely reactive coupling impedance which at low mode numbers is dependent on the tube geometry but which at very large mode numbers approaches the free space impedance