The theory of magnetic braking by a stellar wind is extended to cover a variety of assumed poloidal fields, all symmetric about the rotation axis, all dipolar angular structure, but with radial structures varying from the curl-free at one extreme to the quasi-monopolar at the other. The rates of mass-loss A and angular momentum-loss J are computed as functions of four non- dimensional parameters: 1, the ratio of gravitational to thermal energy density, , the ratio of magnetic to thermal density, `C, the ratio of the centrifugal force to gravity, all measured at the coronal base r and a parameter A that describes the structure of the poloidal field. Except for small values of and large values of IC, an equatorial dead zone is formed, separated from the wind zone. It is confirmed that J'/( 5rs2) is insensitive to changes in for the dipolar field, where s is the star S angular velocity at the coronal base, but as the field structure changes from the dipolar to the radial, it becomes rather sensitive to . For I, J/( sr52AJ) increases very rapidly as K decreases. Even in a hot corona, when K is large the centrifugal force contributes as much to the braking effect of the wind as the thermal pressure. In reality, the field structure parameter A is not independent of 1, K and but will be fixed by the trans-field component of the equation of motion. This will be discussed in a later paper.