Abstract
This chapter covers the analytical theory of thin-wire, perfectly conducting loop antennas and nano-scaled rings. It relies on the thin-wire approximation and assumes no energy loss in the material. The chapter covers the early history of the development of the theory and points out the difficulties and how they were handled. The governing equations that need to be solved and Storer’s 1956 solution for the Fourier coefficients are shown. Difficulties with the derivation are indicated and Wu’s 1962 fix is examined. A new elliptical solution that has not appeared in the literature to date is then derived and yields a result similar to Storer’s. The difficulties Storer and Wu encountered with regard to the convergence of the Fourier series are described. Wu’s solution solves the convergence problem. Much of the problem involves the non-solvability of the governing equation when certain mathematical models of the driving generator are used. These difficulties are summarised. The Storer and Wu solutions are summarised with the elliptical solution.
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Notes
- 1.
A b is added to make \(K_n\) unitless, as it is in Storer and Wu. Wu rectified Storer’s treatment with respect to the units and we follow him.
- 2.
See Appendix A. Storer and Wu both used the Janke-Emde [17] definition where \((x\mathtt {sin}\theta -m\theta )\) is used. This gives rise to a sign difference for \(\varvec{\Omega }_n(z)\) with the standard definition.
- 3.
See Sect. 3.5 for the expansion.
- 4.
It is useful to remember that the sign difference on \(\varvec{\Omega }_{2n}(x)\) is due to a difference in its definition as used by Storer and Wu. See Sect. 3.3.1.
- 5.
This is true, except for a sign change on the second term of \(\hat{\phi }\) in Eq. (3.83) compared with his Eq. (12). This appears to be due to Kanda’s coordinate system twisted by \(180^{\circ }\) compared with Eq. (2.142). In that case, our \(\gamma \) is \(\pi \) radians out of phase with his. Kanda also has an extra phase factor of \(e^{-jn\phi _p}\), which is unnecessary here since \(\phi _p\) is assimilated into \(\alpha \).
- 6.
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McKinley, A. (2019). Thin-Wire Perfectly Conducting Loops and Rings. In: The Analytical Foundations of Loop Antennas and Nano-Scaled Rings. Signals and Communication Technology. Springer, Singapore. https://doi.org/10.1007/978-981-13-5893-7_3
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