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We derive integral expressions of the equivalent impedance of ring-type coils for induction heating systems. These expressions are obtained by integrating the analytical solution of the field generated by a circular filamentary current placed between two linear and homogeneous semi-infinite media. Constant current density is assumed for the coils, since they are wound with an appropriate Litz wire. The model also gives the magnetic field in the cross section of the windings, which is necessary to calculate the proximity losses in the cables. The solution is given in terms of Bessel function integrals, showing an improvement of the convergence with respect to the filamentary model solution. We constructed several prototypes having different turns and layers and tested them with different substrates. Our measurements confirm the validity of the analytical models.