# Inductance of Shielded Helical Resonators

I found an equation in a thesis which calculates the inductance for a helical coil inside of a cylindrical shield:

$$L = (\pi N d_{coil})^{2}\frac{1}{l_{coil} +0.45 d_{coil}}\left(1 -\frac{d_{coil}^{3}}{D_{shield}^{3}}\right)\left(1- \frac{l_{coil}^{2}}{2L_{shield}^{2}}\right)$$

It is surprisingly accurate, I'm wondering if anyone has seen this before and if they have, is there a source or a derivation where this equation is explained?

• Handbook of Filter Synthesis by Anatol I. Zverev covers helical resonators. His mathematical analysis is extensive, and I would not be surprised if its genesis will be found there. Jan 25, 2017 at 23:58
• Nice, I will check it out and get back to you - I'm blown away by how accurate this formula is accurate to 0.01 $\mu$H Jan 26, 2017 at 0:00
• @glen_geek Thanks for the suggestion, I managed to get a copy but unfortunately the section on helical resonators is very limited. Do you have any other ideas where I might find such a formula? Feb 2, 2017 at 15:24

This formula may be considered as having two parts: Wheeler's formula for the inductance of a solenoid, and a correction term to account for the shield.

Wheeler's formula [1] states:

$$L \approx \frac{a^2n^2}{9a+10b} \text{ }\text{ }\mu\text{H}$$

Where $a$ is the coil radius in inches and $b$ is the coil length in inches. It is valid for long coils, in particular when $b>0.8a$. Wheeler's formula may be interpreted as increasing the effective solenoid length, as compared to its physical length, by 90% of the radius [2].

The second part of your formula is a term to account for the effects of the shield. It appears to be an empirical correction suggested by Hayman [3], who states, "Whilst these factors make no pretense to theoretical form or to the correct division of the effect as between the side and ends of the screen, they do enable a coil to be designed to have a given inductance when screened."

[1] H. A. Wheeler, "Simple Inductance Formulas for Radio Coils," in Proceedings of the Institute of Radio Engineers, vol. 16, no. 10, pp. 1398-1400, Oct. 1928.

[2] T. H. Lee. "Planar Microwave Engineering: A Practical Guide to Theory, Measurement, and Circuits, Volume 1," pp. 141-142. Cambridge University Press, Aug 30, 2004.

[3] W. G. Hayman. "Inductance of Solenoids in Cylindrical Screen Boxes," in Wireless Engineer, vol. 11, no. 127, pp. 190, Apr. 1934. http://www.americanradiohistory.com/Archive-Experimental%20Wireless/30s/Wireless-Engineer-1934-04.pdf

• Very nice answer, I will do some reading in the articles you list. Thanks! Mar 26, 2017 at 22:04