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1.2 Fundamental equations for rf test 3 determined. These tests are done inside the cryostat where the cavity is held vertically. Ideally, these tests are done at or near critical coupling. In addition to improving the systematic errors, setting the fundamental power coupler at or near critical coupling reduces the rf power requirement to a value close to that required for cavity wall losses. The critical variable for calculating the rf parameters of a superconducting cavity is the shunt impedance, which relates the stored energy to the effective accelerating gradient. It, along with cavity geometry, is the parameter necessary for calculating peak electric field, and peak magnetic field for any given mode. In our case it is determined using the electromagnetic simulation tool called Superfish and all important parameters determined for 1.5 and 1.3 GHz cavities are collected in tables 1.1 and 1.2 . Symbol Variable name Units r/q Geometric shunt impedance Ω/m G Geometry factor Ω E Electric field V/m L Electrical lenght m ω0 cavity frequency s−1 U Stored energy name J Rs Surface resistance Ω Tc Critical temperature K Pemit Emitted power W R Shunt impedance Ω T Operational temperature K Rres Residual surface resistance Ω Q0 Intrinsic quality factor Qcpl Fundamental Power coupler coupling factor Qpk Field probe coupling factor RC Coupling impedance Ω/m Pdiss Dissipated power W τ Decay time s r Shunt impedance per unit length Ω/m Table 1.1: Common variables when discussing rf cavities [4]. When a cavity mode oscillates with a resonant frequency ω0 , a stored energy U and rf losses on the cavity walls, Pd , the quality factor can be defined as: Q0 = ω0 U Pd (1.10) Q0 is 2π times the ratio of the stored energy and the energy consumed in one period. In the frequency domain the Q0 can also be expressed as
