Air-core inductor calculator: inductance of a single-layer coil

The inductance of a single-layer air-core coil from its diameter, length, turns and wire, or the turns a coil needs for a target inductance. It computes the exact current-sheet result of Lorenz's formula, expressed as Nagaoka's end correction K, and applies Rosa's correction for a winding of round wire, all from the Bureau of Standards paper that tabulated them, in millimetres or inches. Beside the answer it shows how far the long-solenoid formula would be out.

Fig 1 — 10 turns of 1 mm wire at a pitch of 1.2 mm: mean diameter D = 10 mm to the wire centres, length ℓ = n·p = 12 mm, so D/ℓ = 0.8333. Right: Nagaoka's K at that D/ℓ is 0.7269, so the current sheet has 597.8 nH against 822.5 nH for the long-solenoid formula; the round-wire correction takes it to 557.5 nH.

Solve for

Length unit

Diameter entered

Mean coil diameter D (mm)

Length given as

Pitch p, turn to turn (mm)

Number of turns n

Bare wire diameter d (mm)

Inductance of the winding, current sheet less ΔL: 557.5 nH
Current-sheet inductance, Lorenz and Nagaoka: 597.8 nH
Nagaoka's K, at diameter ÷ length = 0.8333: 0.726861
Long-solenoid formula, K = 1: 822.5 nH, 37.6 % over the current sheet
Round-wire correction ΔL, subtracted: 40.27 nH, 6.737 % of the current sheet
Bare wire ÷ pitch · Rosa's A · Rosa's B: 0.8333 · 0.3745 · 0.2664
Mean diameter, to the wire centres · length · pitch: 10 mm · 12 mm · 1.2 mm
Wire in the winding, a helix, leads not included: 314.4 mm

This coil is too short for the long-solenoid formula: it overstates the inductance by 37.6 %. Nagaoka's K is what corrects for the ends.

For the frequency this coil resonates at with a capacitor, take L = 557.5 nH to the LC resonance calculator; for its impedance at a frequency, to the reactance calculator.

How this is calculated

Standard: Rosa and Grover, Formulas and Tables for the Calculation of Mutual and Self-Inductance, Bulletin of the Bureau of Standards Scientific Paper No. 169 (revised 1916), section 6, formulas (68), (72)–(73), (75), (80) and (130), Tables VII, VIII, XX and XXI; BIPM SI Brochure, 9th edition, for μ0

$L_s = \frac{4\pi n^2}{3b^2}\left\{ d\left(4a^2 - b^2\right)E + d\,b^2 F - 8a^3 \right\}, \qquad d = \sqrt{4a^2 + b^2}, \qquad k^2 = \frac{4a^2}{4a^2 + b^2}$: Lorenz's formula (73), p. 118, as printed, in the paper's units: a the mean radius and b the length in centimetres, n the whole number of turns, L_s in centimetres. F and E are the complete elliptic integrals of the first and second kind of modulus k. "Lorenz's absolute formula (73) is of course applicable to coils of all lengths" (p. 126). The calculator evaluates F and E by the arithmetic-geometric mean.
$L = 4\pi^2 a^2 n_1^2 b K = 4\pi^2 a^2 \frac{n^2}{b} K \quad\Longrightarrow\quad L_s = \frac{\mu_0 \pi a^2 n^2 K}{\ell}$: Nagaoka's formula (75), p. 119, as printed (n₁ turns per centimetre, n the whole number), and the same in SI with a and ℓ in metres. With K = 1 it is the long-solenoid formula (68), L = 4π²a²n₁²b, of which the paper warns "There is a considerable error in this formula, due to the end effect."
$K = \frac{4}{3\pi k^2 k'}\left[\left(2k^2 - 1\right)E + k'^2 F - k^3\right], \qquad k' = \sqrt{1 - k^2} = \frac{\ell}{\sqrt{D^2 + \ell^2}}, \qquad \frac{D}{\ell} = \frac{2a}{b}$: Nagaoka's K in closed form: (73) divided by (75), with d = b/k′ and a = kd/2. The paper states Nagaoka showed his expression "is equivalent to Lorenz's absolute formula (73)"; the division is done here, and its −(4/3π)(k/k′) term is the one in Nagaoka's series (76)–(78). The tests check it against Tables XX and XXI.
$L = L_s - \Delta L, \qquad \Delta L = 4\pi a n (A + B) \quad\Longrightarrow\quad \Delta L = \mu_0\, a\, n\, (A + B)$: Rosa's correction formula (80), p. 122, for a winding of round wire, in cgs as printed and in SI. "For values of d/D less than 0.58, A is negative, and in such cases when the numerical values of A are greater than the value of B, which is always positive, the correction ΔL will be negative, and hence L will be greater than L_s."
$A = \log_e\!\left(1.7452\,\frac{d}{p}\right), \qquad B = \frac{2}{n}\sum_{m=1}^{n-1} (n - m)\,\log_e\frac{m}{R_m}$: A as printed at the head of Table VII (p. 197), d the bare wire and p the pitch (the paper's D). B is Table VIII (p. 199), whose heading prints the sum as (2/n)·Σ m·log_e(m/R_m); read literally that does not reproduce the table (0.1051 for n = 3 against 0.1663), while the weighting by the n − m pairs of turns m pitches apart does, to the last digit for n = 1 to 15. From n = 20 up the table runs above this sum by up to 0.0011; the calculator uses the sum.
$\log R_m = \frac{(m + 1)^2}{2}\log(m + 1) - m^2 \log m + \frac{(m - 1)^2}{2}\log(m - 1) - \frac{3}{2}$: Formula (130), p. 168: the geometric mean distance of two equal lines in one straight line, their centres m lengths apart, here in units of the pitch. For m ≥ 20 the calculator uses the paper's equivalent series (131), log R_m = log m − [1/12m² + 1/60m⁴ + 1/168m⁶ + …], which does not lose digits to cancellation.
$1\ \text{H} = 10^9\ \text{cm}, \qquad \mu_0 = 4\pi \times 10^{-7}\ \text{H/m}$: The paper's unit: "As in all the formulas of this paper, the dimensions are in centimeters and the value of L is in centimeters" (p. 136), with 1 henry = 10⁹ cm in its examples ("101809990 cm = 0.10180999 henry", p. 129). With lengths in metres the factor is μ0. The SI Brochure (p. 128) records that μ0 was exactly 4π × 10⁻⁷ H/m under the pre-2019 ampere and was equal to it "with a relative standard uncertainty of 2.3 × 10⁻¹⁰" when the ampere was redefined.
$\ell = n\,p, \qquad D = D_{former} + d \;\text{or}\; D_{out} - d, \qquad \text{wire} = n\sqrt{(\pi D)^2 + p^2}$: The length as the paper defines it, "n times the pitch for a uniform winding of bare or covered wire" (p. 119), and the radius "to the center of the wire". The mean diameter from a former or outside measurement, and the helix length of wire in the winding, are derived here and are not in the paper.

Assumptions

A single layer of round wire at a uniform pitch, on a former of non-magnetic, non-conducting material, with no metal or magnetic material near the coil.
Low frequency: the current is spread uniformly over the wire's cross-section. The paper's formulas "apply only to conductors carrying direct current or alternating currents of frequencies so low" that its non-uniformity is negligible (p. 172). Skin effect lowers the inductance and the capacitance between turns raises it as the frequency rises; neither is modelled, and neither is the self-resonance.
The inductance of the winding alone. The leads to the coil are not included.
A whole number of turns, as Table VIII's B is defined for.
When the diameter is entered as the former or the outside of the winding, the insulation is taken as negligible: the mean diameter is the former plus one bare-wire diameter, or the outside less one.
The defaults are an illustrative small coil, not from the paper. The paper's Examples 59 and 60, the marble cylinder, give 101.7 mH here against 0.10169003 H in the paper, the difference coming from the paper's rounded d/D and its tabulated B.

The air-core inductor formula, and what sets the inductance

An air-core coil's inductance is fixed by its geometry alone: the mean diameter, the length of the winding, the number of turns, and, to a small degree, the size of the wire against its spacing. There is no core permeability to look up and no saturation, so the inductance can be calculated from measurements of the coil alone. Rosa and Grover's paper for the Bureau of Standards gives it for a single-layer coil more exactly than such a coil can usually be wound.

The starting point is the idealised coil: a winding of thin tape whose turns touch edge to edge, so that the current flows as a uniform sheet over the whole cylinder. The paper calls this the current-sheet value, L_s: "it is the value of the self-inductance if the winding were of infinitely thin tape, so that the current would cover the entire length b." For a very long coil the field inside is uniform and the textbook formula follows, L = μ₀πa²n²/ℓ with a the radius, n the turns and ℓ the length. The paper gives it as its formula (68) and warns straight away: "There is a considerable error in this formula, due to the end effect." The field spills out at the ends of a real coil, and the shorter the coil against its diameter, the more of it is lost.

Lorenz's formula (73) is the exact current-sheet result. The paper says Lorenz "first gave an exact formula for the self-inductance of a single layer solenoid. It is, like the others, a current sheet formula, and requires correction by (80) for a winding of wire, but applies to a solenoid of any length." It needs the complete elliptic integrals of the first and second kind, which in 1916 meant tables and seven-place logarithms; here they come from the arithmetic-geometric mean, exact to the last digit a double carries. Nagaoka recast the same result as the long-solenoid formula times a factor K, "which is less than unity, to take account of the effect of the ends of the coil", and tabulated K against the ratio of diameter to length. The calculator computes K from Lorenz's formula divided by the long-solenoid one, so the two are the same number and no table interpolation is involved. Its tests check the result against every entry of Nagaoka's tables that they carry.

A coil of round wire is not a current sheet. Rosa's correction (80) subtracts a term ΔL = μ₀an(A + B) from L_s. A depends on the ratio of the bare wire's diameter d to the pitch p and B on the number of turns alone. The paper explains the sign: "A winding of insulated wire or of bare wire in a screw thread may have a greater or less self-inductance than that given by the current sheet formulas above according to the ratio of the diameter of the wire to the pitch of the winding." A close winding of thick wire has a little less inductance than the current sheet; a spaced winding of thin wire can have more. The reference note below typesets every formula and says which are printed and which are derived.

Two dimensions have to be taken the way the formulas define them. The radius is "the mean radius to the center of the wire", not the radius of the former. And the length is "the over-all length including the insulation … for a close winding of insulated wire, or n times the pitch for a uniform winding of bare or covered wire, which is, of course, the same as the length from center to center of n + 1 turns." The calculator takes ℓ = n·p throughout, and lets you enter the former's diameter or the outside of the winding and adds or subtracts one wire diameter.

Wheeler's well-known approximation is not in the paper this page cites, so this page does not use it; it computes the exact current-sheet result and the paper's correction instead, which need no approximation to carry. The paper works in centimetres, with inductance itself in centimetres, one henry being 10⁹ cm. With lengths in metres that factor is μ₀ = 4π × 10⁻⁷ H/m, which is how every formula here is written in SI.

Nagaoka's K: how much the ends of a short coil cost

K is the ratio of a coil's current-sheet inductance to what the long-solenoid formula says it should have. The table gives it for a range of diameter-to-length ratios, computed from the elliptic integrals beside the value Nagaoka's Table XXI prints, and the amount by which the long-solenoid formula overstates the inductance, 1/K − 1.

D/ℓ	K, computed	K, Table XXI	Long-solenoid formula over by
0.1	0.958807	0.958807	4.3 %
0.2	0.920093	0.920093	8.7 %
0.5	0.818136	0.818136	22.2 %
1	0.688423	0.688423	45.3 %
2	0.525510	0.525510	90.3 %
5	0.319825	0.319825	212.7 %
10	0.203324	0.203324	391.8 %

The long-solenoid formula is not a long-coil formula so much as a limit. For a coil ten times as long as it is wide the formula is still 4.3 % high. A coil as long as it is wide has K = 0.6884: the formula overstates its inductance by 45.3 %. For short, wide coils, where the formula is off by a factor of two or more, only the end correction gives a usable answer.

Air-core inductance for common coil diameters and turn counts

The inductance of single-layer coils wound at a pitch of 1 mm with 0.8 mm bare wire, the wire and covering of the paper's Example 57, for a range of mean diameters and turn counts. The coil's length is the number of turns times the 1 mm pitch. For another wire or spacing, or a diameter measured on the former, enter it in the calculator.

Mean diameter D	n = 5	n = 10	n = 20	n = 50
10 mm	242.0 nH	641.7 nH	1.536 µH	4.336 µH
20 mm	686.7 nH	1.999 µH	5.277 µH	16.37 µH
30 mm	1.216 µH	3.699 µH	10.33 µH	34.41 µH
50 mm	2.422 µH	7.703 µH	22.89 µH	83.91 µH

The range is wide: from 242.0 nH for five turns on 10 mm to 83.91 µH for fifty turns on 50 mm. Doubling the turns at a fixed pitch does not quadruple the inductance, because it also doubles the length: on 20 mm, 20 turns have 2.64 times the inductance of 10, not 4.

Run the other way, the same winding gives the turns for a target inductance: the fewest whole turns that reach it, and in brackets the inductance they actually make.

Target	D = 10 mm	D = 20 mm	D = 30 mm
1 µH	15 (1.081 µH)	7 (1.166 µH)	5 (1.216 µH)
2.2 µH	28 (2.275 µH)	11 (2.299 µH)	8 (2.607 µH)
4.7 µH	54 (4.712 µH)	19 (4.929 µH)	12 (4.892 µH)
10 µH	111 (10.09 µH)	34 (10.35 µH)	20 (10.33 µH)
22 µH	238 (22.08 µH)	65 (22.07 µH)	36 (22.80 µH)

Whole turns land where they land, so the inductance comes out above the target, by up to 22 % in this table, where one turn is a large share of a small coil. Spreading the turns a little lowers it continuously, which is why the calculator can hold the length instead of the pitch and solve for the turns that fit.

Worked example: the Bureau of Standards' marble cylinder

The paper's Examples 59 and 60 work one coil through three formulas: "a single layer coil wound on an accurately measured marble cylinder belonging to the Bureau of Standards", with a mean radius of 27.0862 cm, a winding 30.5510 cm long, 440 turns, and a bare wire of 0.0634 cm. The left column is the calculator's arithmetic; the right is what the paper prints.

D/ℓ       2 × 27.0862 / 30.5510                    = 1.77318       paper: 1.77318
K         Lorenz (73) ÷ (75), from F and E         = 0.5546962     paper: 0.554696
L_s       μ0·π·a²·n²·K / ℓ                         = 0.10181014 H  paper: 0.10181010 H
K = 1     μ0·π·a²·n² / ℓ                           = 0.18354215 H 
p         30.5510 cm / 440                         = 0.6943 mm    
d/p       0.634 mm / p                             = 0.9131        paper: 0.9135
A         ln(1.7452 × d/p)                         = 0.4660        paper: 0.4664
B         Table VIII sum, n = 440                  = 0.3348        paper: 0.3353
ΔL        μ0·a·n·(A + B)                           = 119928 cm     paper: 120067 cm
L         L_s − ΔL                                 = 0.10169021 H  paper: 0.10169003 H

The current-sheet value agrees with the paper's Lorenz calculation, 0.10181010 H, to 0.3 parts in ten million, and K with Nagaoka's tables to the sixth place. Nagaoka's own series give 0.5546959 and 0.5546956 for K, and the paper remarks that this "illustrates well the advantage of obtaining K from Tables XX and XXI rather than by calculation." The long-solenoid formula would have given 0.1835 H, 80.3 % too much, for a coil that is longer than its radius.

The correction differs a little from the paper's in two places, and both are worth seeing. The paper states d/D = 0.9135, but its own dimensions give a pitch of 0.6943 mm and a ratio of 0.9131. And its B = 0.3353 is interpolated in Table VIII, where the sum that defines B gives 0.3348 for 440 turns (the next section explains why). Together they make ΔL 119928 cm where the paper has 120067 cm, and move L by 1.4 parts in a million: 0.10169021 H against the paper's 0.10169003 H. On a coil of 101.7 mH the round-wire correction is 0.12 % of the inductance.

Worked example: a short coil of ten turns

Example 57 is the other extreme: ten turns of 0.8 mm bare wire covered to 1.0 mm, close-wound to a length of 1 cm on a radius of 25 cm, a coil fifty times as wide as it is long. The paper uses Rayleigh and Niven's formula for short coils; Lorenz's, which the calculator uses, is exact for every length and gives the same number.

D/ℓ       50 cm / 1 cm                             = 50.0         
K         Lorenz (73) ÷ (75)                       = 0.061098     
L_s       μ0·π·a²·n²·K / ℓ                         = 47985.95π cm  paper: 47985.95π cm
A, B      d/p = 0.8; n = 10                        = 0.3337, 0.2664 paper: 0.3337, 0.2664
ΔL        μ0·a·n·(A + B)                           = 600.13π cm    paper: 600.16π cm
L         L_s − ΔL                                 = 47385.81π cm  paper: 47385.79π cm
          in henries                               = 148.9 µH

The paper checks this against a completely independent route, summing the self-inductance of each turn and the mutual inductance of every pair, and gets 47,385.806π cm. The calculator's 47385.814π cm is within 0.008π cm of it. The paper's comment on the agreement is the reason the correction is in the calculator at all: "The difference of less than one in a million between the results obtained by formulas (69) and (80) combined and formula (81) is a good check on the corrections of (80), which amount in this case to more than 1 per cent of the value of the self-inductance." Here ΔL is 1.25 % of L_s; leaving it out would put the answer 1.27 % high.

Where the calculator and the printed tables differ

The calculator computes K, A and B from the paper's formulas instead of interpolating its tables, and it is tested against the tables. Four places where the print and the formulas part company are documented in the tests, and none of them is large.

Table VIII, B. The table prints its definition as B = (2/n)·Σ m·log_e(m/R_m), m from 1 to n − 1, where R_m is the geometric mean distance of two current-sheet strips m pitches apart, given by the paper's formula (130). Read literally, with m as both the weight and the spacing, that sum does not give the table: for three turns it gives 0.1051 against the printed 0.1663. Weighting each spacing m by the n − m pairs of turns that are m apart, as a sum over all pairs must, reproduces the table to its last digit for every n from 1 to 15. From 20 turns up the printed values run higher than that sum, by up to 0.0011 and unevenly:

Turns n	B, Table VIII	B, the sum
10	0.2664	0.2664
15	0.2857	0.2857
20	0.2974	0.2964
100	0.3280	0.3269
1000	0.3365	0.3364

The calculator uses the sum. B enters only through ΔL, so on the marble cylinder the table's 0.3353 against the sum's 0.3348 moves L by 7.1 parts in ten million.

Table VII, A. The table prints A = log_e(1.7452·d/D) in closed form, and the calculator evaluates it as printed. The entry for d/D = 1 is printed 0.5568, and the text repeats it; log_e 1.7452 is 0.55687, which rounds to 0.5569: one unit in the fourth decimal place of A.

Table XXI, K. The table ends with a note that "several errors in the fifth and sixth places of decimals have been corrected". One entry the tests find still two units out in the sixth place is D/ℓ = 0.15, printed 0.939141, where the elliptic integrals give 0.9391427.

Example 61's arithmetic. The long coil of Example 61 (radius 10 cm, 40 cm long, 400 turns) has two misprints. Lorenz's first term is printed −87909.94, which does not add up to the printed sum of 30843.01; −79909.94 does, and it is what the elliptic integrals give. And the correction is printed as 4πna(A + B) = 9999 cm, where the paper's own A + B = 0.1988 gives 9992.8 cm. With the sum for B the calculator has 9966.4 cm. All three are 0.077 % of the coil's 12.91 mH.

Where the current-sheet model stops being valid

Frequency. Everything above is the inductance at low frequency. The paper says so plainly at the head of its high-frequency section: "the formulas of the preceding sections apply only to conductors carrying direct current or alternating currents of frequencies so low that the error, due to the assumption that the current is uniformly distributed over the cross section of the wire, is negligible." As the frequency rises the current crowds towards the surface of the wire, and "any deviation of the distribution of the current in the wire from uniformity gives rise to a decrease in the inductance", with a larger increase in resistance. The paper treats that rigorously only for straight wires and rings, not for coils.

Capacitance between turns. The same section notes the opposite effect in coils: the eddy-current effects are "negligible at low frequencies, except in the case of heavy conductors and in coils wound with stout wire in several layers", and there the loss of inductance is masked "by the effect of the capacity between the windings of the coil, which gives rise to an increase of the inductance with the frequency." What follows is general circuit behaviour, not from the paper: that turn-to-turn capacitance resonates with the coil's own inductance at its self-resonant frequency. Approaching it, the apparent inductance rises; above it the coil behaves as a capacitor. A coil for a tuned circuit should be used well below its self-resonance, which has to be measured, for instance with an impedance analyser, since neither the paper nor this calculator predicts it.

Leads and surroundings. Also general, not from the paper: the formulas cover the winding only. The leads add their own inductance, which matters on a coil of a few turns. The coil is assumed to be in free space: a metal can, a ground plane or a nearby magnetic part changes the field the formulas integrate, and a shield close to the winding lowers the inductance.

Manufacturing. The inductance goes as the square of the diameter and nearly as the square of the turns, so the tolerance of a hand-wound coil is set by how well the mean diameter and the length are held, not by the formula. The exactness of Lorenz's result is only useful if the diameter is measured to the wire centres and the length as n times the pitch.

Multilayer air-core coils

The calculator is for single-layer coils. The paper treats multilayer coils separately, in its section 7, as circular coils of rectangular section: Weinstein's and Stefan's formulas for short coils whose winding depth and length are small against the radius, and for longer ones, "When the coil is so long that the formula of Stefan is no longer accurate, the self-inductance may be accurately calculated by a method given by Rosa," which starts from the same current-sheet value used here and corrects it for the depth of the winding. That correction's last term, for the round wire, is not a formula but a short list of cases worked by Rosa: "From the following table one can interpolate for E for any particular case not included in the table." Rather than interpolate in eleven hand-picked cases, this page leaves multilayer coils out.

Common air-core inductor mistakes

Using the former's diameter as D. The formulas want the mean diameter, to the wire centres. Twenty turns of 0.8 mm wire at a 1 mm pitch on a 10 mm former have a mean diameter of 10.8 mm and 1.770 µH; entering 10 mm as the mean diameter gives 1.536 µH, 13.2 % low. The error grows as the former gets smaller against the wire.
Using the long-solenoid formula on a short coil. It overstates a coil as long as it is wide by 45.3 %, and even the paper's long coil of Example 61, twice as long as its diameter, by 22.2 %.
Mixing the paper's centimetres with SI. The paper's formulas give L in centimetres from lengths in centimetres, and 1 H = 10⁹ cm. Putting lengths in centimetres into the SI form μ₀πa²n²K/ℓ makes a²/ℓ, and the answer, 100 times too large; forgetting the 10⁹ on the paper's form is a factor of a billion.
Confusing turns per unit length with the whole number of turns. The paper's formula (68) is written with n₁, turns per centimetre, and (75) with n, the whole number; L = 4π²a²n₁²bK and L = 4π²a²n²K/b are the same thing only because n = n₁b.
Taking d as the covered wire. Rosa's A depends on the bare conductor's diameter against the pitch. Ten turns on 20 mm at 1 mm pitch have 1.999 µH with 0.8 mm bare wire; entering the 1.0 mm covered diameter instead gives 1.971 µH. Small, but it is the correction the paper's Example 57 found to be more than one per cent.
Counting the length centre to centre of the end turns. For a close-wound coil the length is over the insulation of the first and last turns, n times the pitch; centre to centre is one pitch shorter, (n − 1)·p, and overstates the inductance by most where the turns are few. Five turns at 1 mm pitch on 10 mm, measured as 4 mm instead of 5 mm, gain 12.2 % of current-sheet inductance that is not there.