Loop Gain and Impedance Considerations for the LHC RF Cavities

The LHC superconducting cavities will be operated with variable power coupling which changes the loop gain of the fast RF feedback system. Therefore we have established a mathematical formula for the absolute stability limit of such a feedback system with linear delay and cross-checked it numerically with a program that assumes only very basic physical relations. Results show that to keep the beam impedance constant we have to ramp the gain G of the amplifier in the loop. Even with this gain change the loop will remain stable.