The Impedance of Multi-layer Vacuum Chambers

Many components of the LHC vacuum chamber have multi-layered walls : the copper coated cold beam screen, the titanium coated ceramic chamber of the dump kickers, the ceramic chamber of the injection kickers coated with copper stripes, only to name a few. Theories and computer programs are available for some time already to evaluate the impedance of these elements. Nevertheless, the algorithm developed in this paper is more convenient in its application and has been used extensively in the design phase of multi-layer LHC vacuum chamber elements. It is based on classical transmission line theory. Closed expressions are derived for simple layer configurations, while beam pipes involving many layers demand a chain calculation. The algorithm has been tested with a number of published examples and was verified with experimental data as well.