Accuracy considerations in microstrip surface impedance measurements

Abstract
An approach is proposied for the design, measurement and data extraction of superconducting microstrip resonators used in determination of surface resistance and penetration depth. Major sources of error are analyzed and procedures to minimize them are given. I. INTRODUCTION Mcasurement of microstrip ring resonators has been proven to be a useful technique for characterizing surface impedancc of superconducting thin films (I) (or equivalently, their surface resistance R,y and penetration depth LL ). How- ever, there are still parameters that can be optimized in their design and features of their structure (like low-loss coupling) that can be further exploited to facilitate their measurement. In this paper we analyze the tradeoffs between accuracy and dynamic range; modify Aitken's algorithm (2) for the par- ticular case of cryogenic 1 -port resonator measurements; and present a simple method to predict the uncertainty in R, due to tolerances in the loss tangent of the dielectric substrate. 11. BASICS The basic structure of a microstrip ring resonator (Fig. 1) consists of an access line, a coupling gap and the mi- crostrip ring itself. From its symmetry, it can be argued that the current in the ring, at the point diametrically opposed to the coupling gap, must be zero. Cutting the ring at that point would not alter its electric properties and therefore, its im- pedance seen from the coupling gap has to be equal to that of two parallel, open-ended stubs whose length would be half the perimeter of the ring (Fig. 1). From this equivalent cir- cuit, it Is apparent that thcre are parallel resonances when the perimeter of the ring is a multiple of the wavelength. The lumped equivalent circuit of the ring can thus be assumed to be like that shown in Fig. 2, in which the values of LrinR,Cand Rrinx are related to the parameters of the ring

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