Simple model for the rf field amplitude dependence of the trapped flux sensitivity in superconducting rf cavities

The improvement of the performance of RF superconducting 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 non-linear 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.