  Consider a meromorphic connection on P^1 over a p-adic field. In many cases,
such as those arising from Picard-Fuchs equations or Gauss-Manin connections,
this connection admits a Frobenius structure defined over a suitable rigid
analytic subspace. We give an effective convergence bound for this Frobenius
structure by studying the effect of changing the Frobenius lift. We also give
some examples indicating that our bound is essentially optimal.
