Ramping Profile for Exact Beam Loading Energy Compensation in the NLC Detuned Structure

This note determines the profile of the power input into an accelerator section during one fill time prior to beam injection required to reduce the transient beam energy spread to zero, for any group velocity profile along the section, and for any variation of shunt resistance and Q with group velocity. It confirms that the field (the square root of the power) has to be linear for a true constant gradient (CG) section, but will deviate slightly from linearity for detuned and constant impedance (CZ) sections. In this Revision, a somewhat different approximation is used to model the NLC detuned structure. Also, in the original note the & and shunt impedance were assumed to vary linearly along the length of the structure, whereas in the present note these parameters are obtained from a polynomial fit. However, the final result for required input power, beam loading gradient, etc, differ only slightly from the values obtained in the original note. Theory