Electrodynamics of planar Archimedean spiral resonator

Abstract

We present a theoretical and experimental study of electrodynamics of a planar spiral superconducting resonator of a finite length. The resonator is made in the form of a monofilar Archimedean spiral. By making use of a general model of inhomogeneous alternating current flowing along the resonator and specific boundary conditions on the surface of the strip, we obtain analytically the frequencies fn of resonances which can be excited in such system. We also calculate corresponding inhomogeneous RF current distributions ψn(r), where r is the coordinate across a spiral. We show that the resonant frequencies and current distributions are well described by simple relationships fn = f1n, and ψn(r) ≃ sin[πn(r/Re) 2 ], where n = 1, 2..., and Re is the external radius of the spiral. Our analysis of electrodynamic properties of spiral resonators' is in good accord with direct numerical simulations and measurements made using specifically designed magnetic probe and laser scanning microscope.

Meta-metallic coils and resonators: Methods for high Q-value resonant geometries

“…Filamentary analogs of the foil SRS are of current interest for meta-material meta-atoms. 13,14 These references calculate resonance frequency but not Q-value.…”

Effects of Dielectric Substrates and Ground Planes on Resonance Frequency of Archimedean Spirals

“…In Table III, spirals SR1 and SR2 from [7] are evaluated using the method presented above. The spiral from [12] is also used to evaluate the effectiveness of this formula.…”

“…When used as HTS NMR probe coils specifically, and in some other applications, it is typical for the spiral to be mounted on a dielectric substrate [6]. In many of the cases the effect of the substrate is modeled through the use of an effective dielectric constant [7], [8]. This approach uses a substitute value to model the effect of a dielectric instead of the actual dielectric constant.…”

“…In order to be able to predict the mode frequencies of most actual devices, the effect of the substrate also needs to be taken into account. For spirals on thick substrates, an effective dielectric constant given by (1+ ε r )/2 has been found to produce accurate results [7, 8]. However, the resonances of spirals on thin substrates are not consistently predicted with accuracy this way since the modes are not uniformly affected by the substrate.…”

An Empirical Expression to Predict the Resonant Frequencies of Archimedean Spirals

“…It is straightforward to derive (5) which gives the length L of an Archimedean spiral. $$\begin{array}{cc}right& leftL=\left(\frac{r_o}{2}\right)\sqrt{\frac{r_o^2}{{\alpha }^2}+1}+\left(\frac{\alpha }{2}\right)\text{ln}\left(\frac{r_o}{\alpha }+\sqrt{\frac{r_2^o}{{\alpha }^2}+1}\right)-\\ right& left\left(\frac{r_i}{2}\right)\sqrt{\frac{r_i^2}{{\alpha }^2}+1}-\left(\frac{\alpha }{2}\right)\text{ln}\left(\frac{r_i}{\alpha }+\sqrt{\frac{r_2^i}{{\alpha }^2}+1}\right)\end{array}$$ If the spiral is supported by a dielectric substrate, the velocity v can be calculated by dividing the speed of light by the square root of an effective dielectric constant [18]. For thick substrates a good approximation for an effective dielectric constant can be obtained from (6).…”