R sh /Q 0 Measurements in Klystron Cavities Robson K. B. e Silva Navy Technology Center at Sao Paulo CTMSP Av. Prof. Lineu Prestes, 2468 São Paulo, SP, Brazil rkeller@usp.br Daniel T.Lopes Nuclear and Energy Research Institute IPEN Av. Prof. Lineu Prestes, 2242 São Paulo, SP, Brazil danieltl@usp.br Cláudio C. Motta University of Sao Paulo USP Av. Prof. Lineu Prestes, 2468 São Paulo, SP, Brazil ccmotta@usp.br Abstract—This paper presents the experimental results of 0 / sh R Q measurements in klystron cavities using the perturbation technique. The theory involving this technique results in an expression which depends on the natural frequency 0 f and the perturbed frequency shift f Δ , both measured using a cylindrical reentrant cavity with offset gap built in the laboratory. In addition, it is used a cylindrical cavity (pill-box) to calculate a constant that depends on the geometry of the perturbed object, as well as analytical expressions to calculate an integral factor that relates the square of the voltage on axis and the electric energy originally stored in the small volume of the perturbed object. The values measured of 0 f , f Δ and 0 / sh R Q are, respectively, 2.86 GHz, - 5.27 MHz and 79.8 Ω. They are also compared with the results simulated by a 3D eigensolver obtaining a good agreement. Keywords – Klystron cavity 0 / sh R Q , Slater’s perturbation technique, cylindrical reentrant cavity, perturbed frequency shift. I. INTRODUCTION The cavity 0 / sh R Q is one of the most important parameters to be determined in the design of a klystron amplifier. In this problem the gain of a klystron is proportional to the cavity shunt resistance sh R , while bandwidth proportional to the cavity 0 1/ Q , so that 0 / sh R Q is a useful figure of merit in describing the effect of the each cavity on the gain-bandwidth product of overall amplifier. Therefore, it is relevant when, for example, one of the design requirements is high product gain-bandwidth. Moreover, since the 0 / sh R Q quantity is independent of cavity losses it is a very relevant figure of merit. Furthermore, it is dependent on the geometric shape of the cavity and frequency. Although there are some electromagnetic field codes available now to calculate of the cavity 0 / sh R Q , the experimental measurements provide considerable insight on the role of a cavity and its gap [1]-[3] over the amplifier. Accordingly to Slater’s perturbation theorem [4], when some parameters such as the configuration of the boundary, the material in the volume or the material of the boundary change slightly, the electromagnetic system is said to be perturbed. If the solution of an unperturbed problem is known, then the solution of the perturbed problem, which is slightly different from the unperturbed one, can be obtained by means of the principle of the perturbation. There are two possible ways to do this: the cavity wall perturbation or the conductor perturbation of the cavity. The first one means to introduce a small deformation in the wall. The second one means to introduce a small conductive perturbing object into the cavity. In this work the latter technique is used, although the deformation of the wall may also be considered as a conductive perturbing object stuck on the wall. Given a cavity at resonance, it is known that average stored magnetic and electric energies are equals. If a small perturbation is made by introducing a small conductive object into the cavity, this changes one type of energy more than the other, and resonant frequency would then shift by an amount necessary to equalize the energies again. The perturbation method assumes that the actual fields of a cavity with a small shape or material perturbation are not different from those of the unperturbed cavity. So, if the cavity frequency shift due to the small object can be measured as well its geometry shape, the Slater’s perturbation theorem is useful to determine the cavity figure of merit 0 / sh R Q . This paper is organized as follows. Section II presents the mathematical formulation of the problem. The description of the experiment setup and results are shown and discussed in Section III. Finally, the conclusion is presented in Section IV. II. MATHEMATICAL FORMULATION The physical interest problem is constituted of a real resonant cavity considering small losses where the surface currents are essentially those associated with the loss-free field solutions. This cavity is formed by a surface 0 S inclosing a volume 0 V . It is considered, as an hypothesis, the volume of the perturbing object as being V Δ and the surface enclosing the perturbing object as being S Δ . The positive direction of S Δ is the outward direction of the volume V Δ . Here, the volume of the perturbed cavity is V and the surface enclosing it is S . Consider that the positive direction of S and 0 S is the outward direction of the cavity volume V and 0 V . Besides, it is considered 0 S S S = -Δ and 0 V V V = -Δ . Let 0 ω , 0 E  and 0 H  represent the natural angular frequency, the electric and magnetic fields of the unperturbed cavity, respectively, and ω , E  and H  represent the corresponding quantities of the perturbed cavity. In both cases Maxwell’s equations must be satisfied, that is 0 0 0 0 E H j E H ω μ ω         ∇× =-                 , (1) This work was supported, in part, by FINEP (Research and Projects Financing) under contract 01.10.0430.00. 762 978-1-4577-1664-5/11/$26.00 ©2011 IEEE