Impedance Scaling and Impedance Control

When a machine becomes really large, such as the Very Large Hadron Collider (VLHC), of which the circumference could reach the order of megameters, beam instability could be an essential bottleneck. This paper studies the scaling of the instability threshold vs. machine size when the coupling impedance scales in a ``normal`` way. It is shown that the beam would be intrinsically unstable for the VLHC. As a possible solution to this problem, it is proposed to introduce local impedance inserts for controlling the machine impedance. In the longitudinal plane, this could be done by using a heavily detuned rf cavity (e.g., a biconical structure), which could provide large imaginary impedance with the right sign (i.e., inductive or capacitive) while keeping the real part small. In the transverse direction, a carefully designed variation of the cross section of a beam pipe could generate negative impedance that would partially compensate the transverse impedance in one plane