We have examined residual magnetic field vectors observed in Saturn's magnetosphere during the first 2 years of the Cassini mission and have fit them to a simple axisymmetric model of the ring current in the middle magnetosphere. We then examine the variations of the ring current parameters with size of the magnetosphere. In addition, we obtain secondary parameters, including the value of the axial field at the center of the ring (equivalently Saturn's Dst) Bz0, the total current IT flowing in the modeled ring current region, and the ratio of the ring current magnetic moment relative to the magnetic moment of Saturn's dipole field, kRC. Results show that the derived parameters increase significantly with system size, due principally to the increasing radius of the outer edge of the ring. We consider the implications of the response of the magnetic moment of the ring current to changing magnetospheric size, by theoretical consideration of the magnetic moment of individual particles in the ring current. The strong positive correlation of the ring current magnetic moment with system size suggests a system in which the ring current is dominated by inertia currents, rather than by thermal effects as in the case of the Earth, with magnetosphere-ionosphere coupling maintaining the angular velocity of the plasma. The variations of Saturn's ring current parameters with system size found in this study are shown to be closely compatible with the size variations in response to the solar wind dynamic pressure recently determined from Cassini data.