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The improvement of the performance of superconducting rf cavities has recently motivated a considerable research effort in order to elucidate the effect of trapped magnetic flux on the surface resistance ${R}_{s}$. In this paper, we show that, by introducing a nonlinear pinning force in the Gittleman-Rosenblum equations for the rf power dissipation due to a trapped magnetic flux in a superconductor, we can empirically describe the linear dependence on the rf field amplitude ${B}_{rf0}$ of the additional surface resistance ${R}_{fl}$. We also show that the proportionality between the rf-field-dependent and -independent terms ${R}_{fl}^{1}$ and ${R}_{fl}^{0}$ and the frequency dependence of ${R}_{fl}^{1}$ follow naturally from this approach.

The first cryomodule of the new HIE-ISOLDE rare isotope accelerator has recently been commissioned with beam at CERN, with the second cryomodule ready for installation. Each cryomodule contains five superconducting low-beta quarter wave cavities, produced with the technology of sputtering a thin niobium film onto the copper substrate ($\mathrm{Nb}/\mathrm{Cu}$). This technology has several benefits compared to the bulk niobium solution, but also drawbacks among which the most relevant is the increase of surface resistance with accelerating field. Recent work has established the possible connection of this phenomenon to local defects in the $\mathrm{Nb}/\mathrm{Cu}$ interface, which may lead to increased thermal impedance and thus local thermal runaway. We have analyzed the performance of the HIE-ISOLDE cavities series production, as well as of a few prototypes', in terms of this model, and found a strong correlation between the rf properties and one of the model characteristic quantities, namely the total surface having increased interface thermal impedance.