Superconducting self-resonant spiral structures are of current interest for applications both in metamaterials and as probe coils for nuclear magnetic resonance spectroscopy for high-sensitivity chemical analysis. Accurate spiral models are available in the literature for behavior of a spiral below and up to self-resonance. However, knowledge of the higher modes is also important. We present the relationships between the spiral parameters and the multiple mode frequencies of single-sided spirals on dielectric substrates as modeled by method of moments simulation. In the absence of a ground plane, we find that the mode frequency has linear, although not necessarily harmonic, dependence on the mode number. The effect of a thick substrate can be approximated by an effective dielectric constant. However, when the thickness is less than 20% of the spiral trace width (router - rinner), this approximation is no longer accurate. We have developed a simple empirical formula to predict the higher modes.

中文翻译：

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介电基板和地平面对阿基米德螺线共振频率的影响

超导自谐振螺旋结构目前在超材料和作为用于高灵敏度化学分析的核磁共振波谱的探针线圈中的应用都很受关注。精确的螺旋模型可在文献中获得，用于螺旋下方和自谐振的行为。然而，更高模式的知识也很重要。我们介绍了螺旋参数与电介质基板上单面螺旋的多模频率之间的关系，如通过矩模拟方法建模的。在没有接地平面的情况下，我们发现模式频率与模式数呈线性相关，尽管不一定是谐波。厚基板的影响可以通过有效介电常数来近似。然而，当厚度小于螺旋迹线宽度（路由器 - rinner）的 20% 时，这种近似不再准确。我们已经开发了一个简单的经验公式来预测更高的模式。