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If making an RF Coil from existing formulas winds you up, try the methods described here.

This article removes much of the mystery from the design of single-layer radio-frequency coils, by providing simplified yet acceptably accurate formulas for those who usually find calculations daunting. Pre-calculated tables are included for those who prefer to avoid calculations altogether or to check the results of calculations. Worked examples in a no-frills manner are included for guidance on using the formulas.

Resonance

HF radio and rf coils are inseparable. Without the RF coil there is no LC resonance, and without resonance there is no radio. This is true even of the satellite microwave bands where microstrip and dielectric resonators prevail, and yet reception of their direct TV broadcasts requires an immediate down-conversion in frequency to the more familiar VHF and HF ranges, ie to the domain of the RF coil/capacitor (LC) resonator.

Before discussing RF coils however, consider first the familiar formula for calculation of the resonant frequency of a series LC tuned circuit, that is, an RF coil with a capacitor connected in series with it.

Assuming good quality LC components with minimal resistive losses, this formula is equally valid for a parallel resonant LC circuit, ie a coil with a capacitor connected across it.

Eq 1

It seems harmless enough at first glance, but in amateur radio applications it is far from being user friendly, in that it needs the values for inductance L to be in henries (H), and capacitance (C) to be in farads (F). Then it would give the frequency in hertz (Hz) not the more relevant megahertz (MHz).

The frequency range of interest to the typical short-wave radio enthusiast is, say, from about 2 - 200MHz. Not far below 2MHz lies the medium-wave broadcast band which requires multi-layer RF coils, and above 200MHz the coil dimensions become small enough to be replaced by other resonance techniques such as microstrip transmission lines.

Inductance values of RF coils at the frequencies mentioned, range from about 50µH at 2MHz, down to about 0.1 µH at 200MHz - ie millionths of a henry, not henries.

And similarly for the values of capacitors associated with those small values of inductance, ranging from around 200 picofarad (pF) at 2MHz, to a few pF at 200MHz - again not farads, but million-millionths of a farad.

Arithmetical handling of such miniscule dimensions can present a problem for the non-mathematically inclined. Certainly, even some scientific pocket-calculators cannot accept decimal fraction calculations in the form 0.000005 (for 5µH) multiplied by 0.00000000005 (for 50pF) to give a value for L x C as required in the formula for resonant frequency.

The alternative methods using logarithms or powers of ten with or without a calculator, may be equally unpalatable to the radio experimenter.

So, with the ordinary pocket-calculator and its user in mind, the formula for frequency of a series or parallel LC circuit resonant in the HFNHF range, has been simplified to:

Eq 2

Where L is inductance in µH and C is capacitance in pF

Example No 1

Find the resonant frequency of an LC tuned-circuit comprising a coil of inductance L = 2µH in parallel or series with a capacitor C = 50pF.

Eq 3

Most of the basic pocket-calculators can give the square root of a number as well as add, subtract, multiply and divide. Some may even have an invert (1/x) key which is useful for working out fractions, because it allows the denominator (below the line) to be calculated first, then inverted, then multiplied by the numerator (above the line).

To demonstrate this, the previous example would be calculated by pressing key entries as follows:

Example No 2

Ex 1

Inductance by calculation

A long-established formula for the calculation of the inductance of a coil of known dimensions is:

Eq 4

where R is the mean radius in inches.

Converting to metric units and simplifying gives the G3BIK formula for inductance:

Eq 5

where D = inner diameter of coil (mm)
N = Number of turns)
S = span (length) of winding (mm). Assumes S equal to or greater than approx D/2.

This simplified formula gives results which are accurate enough for practical purposes, typically to within 2% of the earlier established formula. See Fig 1 for definition of coil dimensions.

Fig 1
Fig 1: Coil Dimensions: D = Inner Diameter mm; N = Number of Turns; S = Span of Winding mm.

It does not require information on the wire with which the coil is wound. One needs only to measure in millimetres the inside diameter D of the coil (or the outside diameter of the former on which it is wound), the distance S in millimetres from end to end of the winding, and to count the number of turns by running a soft-pointed probe over them.

The calculation is quick and easy using a basic pocket-calculator first to calculate below the line, invert that value, then multiply above the line, using key entries as follows:

Ex 2

Alternatively, if the calculator does not have the inverting 1/x key, the memory key may be used thus:

Ex 3
Ex 4

For practice and to gain confidence, the above key entry sequence should be used in the following examples:

Example No 3

Calculate the inductance of an enamelled copper-wire coil of:

inner diameter D = 12.5mm
no of turns N=41
span of winding S = 21 mm

Eq 6

Example No 4

Calculate the inductance of an enamelled copper-wire coil of:

inner diameter D = 4.8mm
no of turns N = 14
span of winding S = 7mm

Eq 7

All results have been rounded up or down to the nearest whole digit.

Number of turns by calculation

The converse of finding the inductance of a given number of turns, is to determine the number of turns required to produce a required value of inductance, from given coil diameter and wire details.

Classical published formulas for the calculation of the number of turns for an RF coil, can be horrific and daunting. Take for example the following formula prior to simplification:

Eq 8

Again a not too userfriendly formula, being in dimensions of yesteryear and involving the radius rather than the more measurable diameter.

Converting to metric, and to inner diameter instead of outer radius, and simplifying, gives the G3BIK formula for number of turns of an air-cored single layer coil:

Eq 9

where
N = number of turns, air-cored
L = inductance in µH
n = turns per mm (close-wound or spaced)
D = inner diameter in mm

It may still look somewhat complex, but is in fact really easy to use, as will now be demonstrated by worked examples and guidance on how to manipulate the everyday nonscientific type of pocket-calculator.

For this formula, one needs to know the turns per millimetre, ie the winding-pitch for the coil. Assuming close-wound coils, the turns per mm values are as given in Table 1 for the most practical range of standard wire gauge enamelled copper-wires.

Note that turns per mm does not simply equal (1/dia), because it allows for the thickness of the enamel covering. The diameter of the wire is not in fact needed, and is given for interest only.

If there is any doubt about the actual gauge of the wire available, the turns per millimetre can be easily obtained by close-winding enough turns onto the coil-former or pencil to cover 10mm, then to divide by ten to get turns per mm for a close-wound coil.

The simplified formula for number of turns N may be calculated on a pocket-calculator by the following key entries:

Ex 5

Example No 5

Inductance of L = 0.5µH to be close-wound on standard former of 4.8mm dia, with air core, using 28SWG enamelled copper-wire, close-wound.

From Table 1, number of turns per mm for 28SWG wire = 2.4 t/mm = n

For formula calculations, L = 0.5 D = 4.8 n = 2.4

Using calculator, number of turns is given by:

Ex 6
Ex 7
wire on 4.8mm dia former

Example No 6

L required = 10µH
Dia D = 12.5mm
Wire = 26SWG enamelled
From Table 1 n = 1 .98 t/mm closewound

Using calculator, number of turns is given by:

Ex 8
wire on 12.5mm former

Using the tables

The preceding examples were based on the simplified formulas for inductance and formula for number of turns as appropriate.

To avoid calculations however, Tables 2 to 6 provide a quick and easy way of determining either inductance or number of turns, over the range of values of immediate interest, and assuming close-wound coils. The results are again based upon the simplified formulas and are exact enough for practical purposes.

The table entries indicated with a # were calculated from the longer established empirical formulas by others, and are included to demonstrate the comparative accuracy of the G3BIK simplified formulas.

The calculated values given in the tables have been rounded off for convenience of tabulation.

Use of the tables is straightforward, to derive the coil dimensions for a required value of inductance; or conversely the inductance value from dimensions of an existing coil.

To determine number of turns and span in millimetres for a required value of inductance L in µH:

  1. Decide diameter of coil former and select related Table
  2. Decide wire gauge to be used and go to related column heading
  3. Move down column to intersect with required L µH
  4. Read off number of turns N, and Coil-Span S mm

To determine value of L µH for an existing coil:

  1. Measure inner diameter of coil D mm
  2. Select related Table
  3. Count number of turns N
  4. Measure coil-span S mm
  5. Divide N by S to get turns per mm (t/mm = n = N divided by S)
  6. Find that t/mm column heading
  7. Look down its N column to find the nearest number of turns
  8. Read off inductance L µH

Whilst the tables apply to close-wound single layer coils, the previous G3BIK simplified formulas for inductance and number of turns do also apply to coils which are wound with spaced turns, ie the turns per millimetre value will be less than that for close-wound.

It will be appreciated that it is almost impossible to hand-wind a coil to perfection, because of unintentional gaps in spacing between turns and the fact that the connecting-tails add to the inductance - as do the connections from a resonating capacitor.

In all of the previous considerations allowance has been made for connecting-tails of circa 10mm length, but on coils of inductance less than say 0.2µH (for use at VHF) the coil tails will significantly affect the inductance value, so they must be kept as short as possible.

Fractional values for the number of turns obtained from the tables should not be taken as sacrosanct, but should in practice be rounded up or down to the nearest whole number. This also applies to the coil span dimensions.

It should be noted also that the accuracy of calculated inductance diminishes when the coil span becomes significantly greater than, say, three times the coil diameter.

Table 1: Enamelled copper wire dimensions.
SWG3028262422201816
dia mm0.310.380.460.560.710.911.221.63
t/mm2.852.401.981.641.301.020.780.58
Table 2: Inductance of 25mm coil v number of turns v Standard Wire Gauge for close-wound single-layer air-cored coil, of enamelled copper-wire (based on G3BIK simplified formula).
Inner diameter D = 25mm - N = Number of turns - S = Span of winding mm
SWG30282624222018
t/mm2.852.401.981.641.301.020.78
L µHNSNSNSNSNSNSNS
0.5.#3.171.13.201.33.331.63.32.03.42.63.53.43.64.6
0.53.311.23.331.43.371.73.42.13.52.73.63.53.74.8
1.04.81.74.82.04.92.55.03.05.14.05.35.25.67.2
2.06.92.47.02.97.23.67.44.57.75.98.17.98.711
3.08.63.38.803.79.14.69.45.79.87.610.51011.415
4.010.13.610.44.310.75.411.16.811.79.012.61213.918
5.0#11.13.911.44.811.86.012.47.513.21014.31416.121#
5.011.54.011.84.912.26.212.77.813.51014.71416.421
6.012.84.513.15.513.66.914.38.715.31216.71618.724
7.014.04.914.46:015.07.615.79.616.91318.51821.027
8.015.15.315.66.516.38.217.11118.51420.42023.330
9.016.25.716.87.017.58.918.51120.11522.22225.533
10.0#16.75.917.37.218.29.219.41221.21623.72327.535#
10.017.26.017.97.418.89.519.81221.81724.02427.236
20.026.49.327.71229.71532.02035.82841.14049.263
30.034.31235.51539.52043.22649.13857.55670.190
40.041.71544.61948.72553.93362.14873.67290.0117
50.048.61752.42257.82964.33974.95889.688112143
50.0#47.71751.62257.12963.93974.85889.988111143
# = results of non-simplified formulas
Table 3: Inductance of 20mm coil v number of turns y Standard Wire Gauge for close-wound single-layer air-cored coil, of enamelled copper-wire (based on G3BIK simplified formula).
Inner Diameter D = 20mm - N = Number of Turns - S = Span of Winding mm
SWG30282624222018
t/mm2.852.401.981.641.301.020.78
L µHNSNSNSNSNSNSNS
0.53.761.33.81.63.872.03.942.44.13.14.24.14.45.7
1.05.51.95.62.35.72.95.83.66.14.76.46.26.98.8
2.08.02.88.23.48.54.38.85.39.37.19.99.711.014
3.010.03.510.4410.8511.2712.0913.11314.719
4.011.9412.3512.86.513.5814.61116.01618.324
5.0#13.24.613.75.714.47.315.29.316.612.818.618.221.628#
5.013.64.814.15.914.87.515.69.517.013.118.918.521.828
6.015.2515.8716.6812.71119.31521.62125.232
7.016.7617.4718.4919.61221.61724.42428.637
8.018.1618.9820.11021.51323.81827.02731.940
9.019.5720.4821.71123.41426.02029.72935.245
1020.8721.9923.41225.21528.12232.33238.549
2032.81235.11538.31942.32648.73857.75771.191
30.043.51547.22052.22658.53668.65382.691103133
40.053.71958.7 2565.73374.44588.368107105136174
50.063.52270.02979.04090.15510883132129168215
# = results from non-simplified formulas.
Table 4: Inductance of 12.5mm coil v number of turns v Standard Wire Gauge for close-wound single-layer air-cored coil, of enamelled copper-wire (based on G3BIK simplified formula).
Inner Diameter D = 12.5mm - N = Number of Turns - S = Span of Winding mm
SWG30282624222018
t/mm2.852.401.981.641.301.020.78
L µHNSNSNSNSNSNSNS
0.55.11.85.22.25.42.75.63.45.94.56.36.27.08.9
1.07.62.87.83.38.14.18.65.29.37.110.21011.615
2.011.54.012.05.012.76.413.78.315.11217.21720.326
3.014.8515.7716.8918.31120.61623.92228.837
4.017.9619.1820.71022.71425.92030.33037.148
5.020.8722.3924.41226.91731.02436.83645.458
6.023.6825.41128.01431.21936.22843.24253.769
7.026.4928.51231.51635.32241.33249.64962.079
8.029.01031.51335.01839.42446.33655.95570.290
9.031.61134.51438.51943.42751.34062.36178.4101
10.0#33.71237.01541.62147.42956.54469.16887.5112
10.034.21237.41641.92147.52956.34368.66786.7111
20.058.62165.52775.33887.25310682132129169217
30.0822993391085512777155120194191251322
40.0105371205014171166101205158257252333427
50.0128451476117388205125254195320314415532
# = results from non-simplified formulas.
Table 5: Inductance of 7.1mm coil v number of turns v Standard Wire Gauge for close-wound single-layer air-cored coil, of enamelled copper-wire (based on G3BIK simplified formula).
Inner Diameter D = 7.1mm - N - Number of Turns - S = Span of Winding mm (max 25.4mm)
SWG30282624222018
t/mm2.852.401.981.641.301.020.78
L µHNSNSNSNSNSNSNS
0.5#7.72.78.23.48.84.49.55.810.88.312.512.215.119#
0.57.92.88.43.59.04.59.75.910.98.412.512.315.119
1.012.64.413.55.614.87.516.41019.01522.622.028.036
2.020.77.322.79.525.612.929.017.734.626.6
3.028.39.931.513.135.918.241.425.250.038.5
4.035.712.540.116.746.223.353.632.7
5.0#42.915.048.620.356.628.6#
5.043.015.148.620.256.328.5
6.050.217.657.023.866.533.6
7.057.320.165.427.3
8.068.824.173.830.7
9.071.525.182.134.2
10.071.525.182.134.2
10.078.627.6
# = results from non-simplified formulas.
Table 6: Inductance of 4.8mm coil y number of turns y Standard Wire Gauge for close-wound single layer air-cored coil, of enamelled copper wire (based on G3BIK simplified formula).
Inner Diameter D = 4.8mm - N = Number of Turns - S = Span of Winding mm (max 15mm)
SWG30282624222018
t/mm2.852.401.981.641.301.020.78
L µHNSNSNSNSNSNSNS
0.512.04.213.05.414.57.316.410.019.414.923.523.029.638
1.020.47.122.59.325.913.129.918.336.327.9
2.0#36.212.741.417.3#
2.036.212.741.217.2
3.051.718.2
# = results from non-simplified formulas.

Coil cores

As previously discussed, the formulas and related Tables assume an air-core within the coil. The standard 4.8mm and 7.1mm formers do have a threaded inner-surface to allow the use of a 4mm or 6mm diameter screw-slug respectively for adjustment of the inductance value.

Generalising, an iron-dust or ferrite slug increases the inductance as the slug is screwed into the coil, whilst conversely, a non-ferrous (eg brass or aluminium) slug decreases the inductance.

There are different grades of iron-dust or ferrite slug material, some of which are more suited for use at VHF or HF than others - but that is a subject in its own right and will not be expanded upon herein. Without being definitive, suffice to say that some iron-dust slugs can almost quadruple the inductance value, whereas a non-ferrous slug would typically produce only a modest decrease in induct ance to say half of its former value.

Some minor adjustment of inductance can be achieved by opening out the spacing between turns, the inductance decreasing as the spacing increases.

It should be borne in mind that increasing the value of inductance in an LC resonant circuit lowers the resonant frequency, whilst decreasing the value of inductance increases the resonant frequency.

The previous calculations and tables apply reasonably well to non-enamelled wire, on the assumption that the close-spaced turns do not electrically touch.

Winding coils

The coil diameters in the Tables cater for the use of standard coil-formers or readily available tubular materials such as plastic water-pipe or electrical conduit.

On the larger diameter formers, two small close-spaced holes drilled at each end of the coil-span would allow the wire ends to be secured by being fed into the former via one hole and out again via the adjacent hole. Alternatively, a solder tag secured by a small diameter machine-screw and nut may be used to anchor the wire at each end of the former.

Winding a coil with constant tension on the wire can be tiring to the fingers. It is a good ploy to have available a few pieces of self-adhesive tape to hold the turns in place during rest periods, or when mechanically anchoring the ends.

An inductor is only as good as its construction. If there is any slack in the finished product, a smear of epoxy-resin glue or nail varnish should hold the turns secure.

The easiest and least damaging method of straightening out enamelled copper-wire before commencing the winding, is to pull it gently through a cloth whilst nipping the wire between finger and thumb. Care must be taken when approaching loops in the wire to avoid kinking, which mechanically weakens the wire and damages the enamel insulation.

Older wire salvaged from electrical motors etc may have enamel that needs to be removed carefully prior to soldering, but the modern polyurethane enamels are said to be self-fluxing so do not need to be removed. The heat of the soldering-iron should in effect melt away the enamel coating without detriment to the quality of soldering. It is still wise to use a flux-cored solder rather than rely upon the polyurethane enamel as the fluxing agent. NOTE: It is wise to minimise the inhalation of fumes given off from the polyurethane coated wire.

G3BIK, E Chicken.