By solving the wave equation for the radial electric field with constant poloidal electric field around the resonance layer of the fast Alfvén wave, various complex characteristics of mode conversion physics can be elucidated and analyzed for ion cyclotron heating of tokamaks. The validity of the Budden and tunnelling model [Ngan, Y. C. and Swanson, D. G., Phys. Fluids 20, 1920 (1977)] for the conversion studies is explored, and the conversion coefficient for the ion-ion hybrid resonance in the presence of cyclotron damping is found in closed form. The analytical results are compared with the numerical solution of the full wave equations expanded to second order in ion Larmor radius. It is found that the standard tunnelling solutions can be erroneous, not only in the case of strong damping, but also when the linearization of the plasma parameters around the resonance, peculiar to the tunnelling model, becomes inaccurate. The effects of the damping and cavity resonances on the conversion are separated in the derived analytical estimates, and the limits of the local theory of conversion are determined.