The Standard Resonance Equation: MHz = 1000/(SQRT(L*C)*6.2832)
Therefore: pF or uH = 1,000,000/(39.4786 *(MHz)^2 *uH or pF)
Standard impedance equations using MHz, uH, and pF
XL = MHz * uH * 6.2832 and -XC = 1,000,000/(pF * MHz * 6.2832)
Standard (ideal) Inductance Equation: uH = N^2 * d^2/(18*d + 39*length)
Where: N = Number of turns (loops) of wire, d = form diameter PLUS wire diameter,
and length is the length of the coil
Diameters and lengths are in inches
Loop Inductance in the Metric System
LOOPS are coils with a length much shorter than the diameter.
A good rule of thumb is the diameter is greater than 6x the length.
The equation for metric loops is slightly different r^2 N^2 / 18* (r + l).
Ideal inductance of a ferrite rod core inductor
uH= Ue*K*N^2*r^2 / (25.33*lr)
Where ALL measurements are in centimeters (cm).
NOTE that this answer is for low (audio) frequencies, application to the AM-BCB or HF-SW bands will increase the apparent inductance due to capacitive effects of the coil and the antenna-ground system among several factors. It is prudent to 'design low'.
Ue is the stated Effective Permeability of the rod, generally based upon the length to diameter ratio. NOTE that this may vary between manufacturers, as there are differences in production methods, materials, and recipe that allow similar general properties, yet have unique Ue. Variance noted for a ferrite rod with a 6 to 1 length to diameter ratio can be Ue = 20 to 25. CHART on next page.
K is the correction factor when lc is less than lr. K ~= n^(1/n^0.43) WHERE n = lr/lc
This formula is accurate when n is between 2 and 4. CHART on next page illustrates 1/n: corresponding values are 0.25 to 0.50. Error occurs at each endpoint but is +/- 1%. The optimum mix of K (maximizing inductance) and the figure of merit Q apparently occurs at n = 3. At n = 3 K = 1.984
N Capital N is the Number of turns or loops of wire around the ferrite rod.
lr is the length of the rod, and lc is the length of the coil winding.
r is the radius of the ferrite rod
25.33 is a consolidation of constants Uo = 4pi*10E-7 nH/cm, 10000 (Maxwells/Gauss), and pi (from the cross-sectional area formula A=r^2*pi). The result is 0.0394784 which is then inverted (1/x) and placed in the denominator as 25.33 ~= 1/0.0394784
The expanded formula can be entered into a scientific calculator as:
r^2 * N^2 * Ue * (n^(1/n^0.43)) / (25.33 * lr)
Many thanks go out to Ben Tongue and Dan McGillis for illuminating this rather tricky subject. The one or two tweaks and/or simple errors fixed make this a good starting point for construction. I'd been racking my brain over differing results using same construction.
The Charts for K and Ue are here.
Formula for Skin Effect Depth
This fomula closely approximates the depth of penetration a signal at radio frequency "f"
has in a solid copper wire loop (coil). Note that a smaller diameter wire is favored, in spite of having a higher electrical (DC) resistance.
The figure of 1.68 E-8 is the resistivity for copper. The Reletive Permeability (Ur) is also a factor, but for air-coils using copper wire this value is considered to be unity (1.00). In this formula Ur has been set to 1, and ignored. The factor of 503292 is an approximation of 1/(SQRT[pi * 4 * pi * 10E-7]). You may recognize 4pi*10E-7 as being Uo, the permeability of free space used in the ferrite inductor formula above. The value of "f" is in Hz. The final answer is in millimeters (mm). For results using inches the constant is 19814.6
503292 * SQRT(1.68E-8 / f*Ur) = mm depth
The illustration below shows the skin-effect in an ideal fashion. In reality, the shape of such region will not be a circle, but more like a banana or oval depending upon the severity of the proximity effect. The proximity effect is simply the closeness of the coil windings. The proximity effect serves to reduce the diameter of the ideal circle of the skin-effect, moving the boundary away from the center of the wire, and compacting the current density (not good for Q). The image is very close to scale but not perfect. One pixel = 0.0001", thus the picture is 253x253 pixels representing the diameter of the #22 AWG wire.
There are three frequencies shown 200kHz, 530kHz, and 1700kHz, plus two wire diameters equivilant to #22 AWG (large Red), and #32 AWG (smaller blue). As can be plainly seen, a 1700kHz signal has a skin-depth that is about 8% of the #22 wire diameter (about 1/6th the distance to the center of the wire). Under such condition, very little current flows in the center of the wire. In contrast, the #32 wire at 1700kHz has much more current flowing in the center of the wire, because the skin-effect is 25% of the wire diameter (about 1/2 way to the center). The second image idealizes the choice of #33 AWG wire, showing the same three frequencies. Should the reader build an idealized skin-effect coil for the AM-BCB or the Beacon-band (200 to 530 kHz), the turns are spaced as 50 per inch. A very good article about the skin-effect can be found at Wikipedia.