HEURISTIC TREATMENT OF THE NONRESONANT COUPLING IMPEDANCE OF A TOROIDAL BEAM TUBE

Analytical expressions for the longitudinal coupling impedance of a to roidal beam tube with rectangular cross section and loss-less walls ha ve been derived by complementary methods, emphasizing the non-resonant region. A simple analytical expression for the longitudinal coupling impedance below the resonance region has been obtained by a perturbati on method in which the current-induced electromagnetic fields are expa nded into a power series in curvature and substituted directly into Ma xwell's equations. Rigorous solutions valid at all mode numbers and fr equencies have been found by the methods of field matching as well as eigenfunction expansion. From the exact solutions, simple analytical e xpressions for the non-resonant region have been derived in the limit of small curvature using asymptotic expansions of cross-product Bessel functions dedicated to the present problem. The resulting expression for the coupling impedance consists of the space charge impedance of t he straight beam pipe plus a correction term due to the curvature. It has been verified numerically that this result is in full agreement wi th the exact solution below the first resonance.