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Inductance calculation

 An introduction to the art of Solenoid inductance calculation By David W Knight Solenoid inductance calculation (v 0.20, unfinished, 4th Feb. 2016) Open document spreadsheet files (calculations and macro routines referred to in the text): Li_transition.ods (transition from LF to HF behaviour) L_formulae.ods (accuracy and optimisation of various approximations for Nagaoka's coeff.) Loop_funcs.ods (methods for loop inductance, functions for Rosa's self-inductance coeff.) Rosa_km.ods (functions for Rosa's original mutual inductance coeff. New reference function including the effect of pitch. Empirical function for large N asymptotic value). Rosa_kmp.ods (Intermediate data reduction for mutual inductance coeff including the effect of pitch) Rosa_kmpr.ods (Final data reduction for mutual inductance coeff. In preparation) Rosa_v_summ.ods (comparison of methods, in preparation) Lcalcs.ods (solenoid inductance calculations) Abstract: (not yet written)

Formulae, calculation methods, utilities, correspondence

 Numerical methods for inductance calculation, by Bob Weaver: electronbunker.ca/CalcMethods.html . Formulae for Rosa's mutual-inductance correction for the round-wire solenoid. DWK Practical considerations in the calculation of Kelvin Bessel functions and complete elliptic integrals K and E. by Bob Weaver. Oct. 2009 (PDF)  Inductance Examples (Open Document spreadsheet). Wheeler's Formulas - correspondence with John Crabtree KC0G. Calculating the LF effective radius of a single-layer solenoid. DWK Practical continuous functions for the internal impedance of solid cylindrical conductors. DWK. Accurate formulae for AC resistance and internal inductance of wires. Grover's 'Inductance Calculations', Supplementary information and errata. GMDs of equal parralel rectangles. Re-calculation of Grover's tables 1, 2, & 3. Rodger Rosenbaum 2009. A note on an incorrect inductance formula.  Study of the Method of Geometric Mean Distances Used in Inductance Calculations. H L Seneff, MSc. Thesis, U. Missouri. 1947. (3.8 MB).  Geometric Mean Distance, Its derivation and application in inductance calculations, Robert Weaver Mar. 2016.

Online references - Maxwell, Nagaoka, NBS
(Documents in the public domain) On the Geometrical Mean Distance of two figures in a plane. J C Maxwell, Trans Roy. Soc. Edinburgh, Vol 26, 1872. [Maxwell 1872 GMD] Calculation of the self-inductance of 1-layer coils. E B Rosa, BBS Vol 2, No.2, p161-187 [BS Sci. 31]. This is the original paper giving Rosa's method for solenoid inductance, including derivation of the external GMD for current-sheet segments. The Self and Mutual Inductances of Linear Conductors. E B Rosa. BBS Vol 4, No 2, 1908 [BS Sci. 80]. The Inductance Coefficients of Solenoids. H Nagaoka. J. Coll. Sci. Tokyo, Vol 27, 6, 1909 (1.7 MB). Note that all of Nagaoka's papers can be obtained from the University of Tokyo repository . Formulas and Tables for the Calculation of Mutual and Self Induction. E B Rosa & F W Grover, revised 1916 3rd edition, 1948. [BS Sci. 169]. (15 MiB) - corrected version of BBS Vol. 8, No. 1 (1911 & 1912) (same page numbering). Additions to the formulas for the calculation of mutual and self inductance. F W Grover. BBS, Vol 14, No 4, July 12, 1919 [BS Sci. 320] (1 MB). Formula for the inductance of a helix made with wire of any section. C. Snow. [BS Sci. 537], 1926 (5.74 MB). Methods for the Derivation and Expansion of Formulas for the Mutual Inductance of Coaxial Circles and the Inductance of Single-Layer Solenoids. F W Grover, NBS J. Research, Vol 1, 1928. [BS RP16]. p487-511 Comparison of Formulas for the Calculation of Inductance of Coils and Spirals Wound with Wire of Large Cross-Section. F W Grover, JBS Vol 3. 1929. [BS RP90]. A simplified Precision Formula for the Inductance of a Helix with Corrections for the Lead-in Wires. C Snow, JBS Vol 9. 1932, p419-426. (0.5 MB). Formulas for Computing Capacitance and Inductance. C Snow. NBS Circular 544, 1954. [BS Circ. 544]. (1.1 MB).

 The following easy-to-use routine for calculating solenoid inductance by the Rosa-Nagaoka method has been added to the macro library in the spreadsheet Lcalcs.ods . The three functions called are also in the macro library and are discussed in the solenoid incuctance calculation article. Note that the calculation is for external partial inductance only. Internal inductance and lead inductance should be added to the result

 Function LxRosa(byval h as double, D as double, N as double, dw as double, p as double) 'External inductance in henrys of a round-wire solenoid, using the Nagaoka-Rosa method. 'D W Knight. v 1.00, 2016-01-11 'Calls functions W82W(), KMGO() and Rosaks90() 'h = coil length / m 'D = coil diameter / m 'N = number of turns 'dw = wire diameter / m 'p = winding pitch / m Dim kNagaoka as double, Lsheet as double, Rosacorr as double kNagaoka = W82W(D/h) Lsheet = 1e-7*pi()*pi()*D*D*N*N*kNagaoka/h Rosacorr = KMGO(N) + Rosaks90( p/dw , p/D , 0) LxRosa =Lsheet -2E-7*pi()*D*N*Rosacorr end function

 Material not yet transferred. The text below is in the process of being updated and extended. The information will at some point be transferred to the main solenoid inductance article above.

B1. Solenoid inductance calculation vs. measurement:
Using the techniques outlined in the previous sections, it is possible to calculate the low-frequency external inductance of solenoids to an accuracy of a few parts in 1000 using a micrometer or a good set of engineer's callipers and a personal computer. If a scientific calculator is used, the computation can be simplified by using one of the Wheeler unrestricted formulae with only a small loss of accuracy. In either case, the outcome is usually better than can be obtained using general-purpose measuring equipment.
All solenoid inductance calculations are dependent on the effective current-sheet diameter (D) or radius (r=D/2). In section 3 it was noted that at low frequencies D=D0, where D0≈Da and Da is the coil diameter measured from wire-centre to wire-centre, whereas at very high frequencies D=D, where D is less than D0 but always greater than the diameter at the inside of the conducting cylinder (i.e., D0-d). Consequently, the measured radio-frequency inductance of a coil (substantially below the SRF at least) will always lie between the low frequency inductance L0, which is the value calculated using D=D0; and the minimum inductance L, which is the value which obtains when D=D (presuming that we can determine D).

The inductance of a solenoid without a magnetic core, presuming that the wire diameter is << the solenoid diameter, is given within a few parts per thousand by:

 L = ( μ0 π D² N² kL / 4 ) - [ μ0 D N (ks(e) + km )/2 ] + Li B1.1

where D=2r, kL is Nagaoka's coefficient , ks(e) Rosa's round-wire self inductance correction, for external inductance only, and km is Rosa's round-wire mutual inductance correction.. The internal inductance Li , generally < 1% of the total inductance, can be calculated to within 0.016% using the Li-PACAML approximation, see Zint.pdf. (also function Lintern() described on the Zint index page)

From the previous discussion, the inductance of a solenoid, neglecting the proximity effect, is:

 L0 = ( μ0 πD0² N² kL(0) /4 ) - [ μ0 D0 N( ks(e) + km )/2 ] + Li B1.2

where kL(0) is Nagaoka's coefficient calculated using /D0.
The worst-case minimum inductance, due to reduction in the effective diameter is:

 L∞ = ( μ0 πD∞² N² kL(∞) /4 ) - [ μ0 D∞ N( ks(e) + km )/2 ] + Li B1.3

where kL(∞) is Nagaoka's coefficient calculated using /D.

Example B1a
R G Medhurst, on page 42 of his 1947 paper "H.F Resistance and Self-Capacitance of Single-Layer Solenoids", gives accurate dimensions of a test-coil (#31) used in a study of Q and self-capacitance. The coil is wound on a grooved former using 40 turns of 20SWG bare wire (d=0.9144 mm), with a diameter of 51.9 mm measured at the outside of the winding. The length of the coil is given as 70.1 mm, but the winding pitch is stated to be:
p=(70.1-d)/(N-1)
from which we may deduce that the measurement was made to the outside of the wire on the side away from the terminations, (as shown in the diagram on the right). Hence the stated length is not the same as the equivalent current sheet length, which is defined as:
 μ0 = 0.4π μH/m μ0 π = 3.94784176 μH/m  = Np
Thus it appears that Medhurst uses a definition of coil length that differs from that intended by Rosa and Grover (as discussed in section 2) but is an understandable misinterpretation of the explanation given [Sci169, p119]. Ambiguities of this type can introduce errors of several % in inductance calculations, and there is no use in claiming accuracy in parts per 1000 unless care is taken to avoid them. In this case the equivalent current-sheet length is: = 40p = 40(70.1-0.9144)/39 = 70.9596 mm.
Using this value, with Nagaoka's coefficient kL calculated using Lundin's formula , we have:

 D0 = 51.9-d = 50.9856 mm /D0 = 1.391758 kL(0) = 0.755974 D∞ > 51.9-2d = 50.0712 mm /D∞ < 1.417174 kL(∞) < 0.759386

The round-wire correction ks(e) is:
ks(e) = (3/2) - ln(2p/d)

 p = (70.1-d)/(N-1) = 1.7740 mm p/d = 1.9401 ks(e) = 0.1441

and km =0.3142.

Medhurst measured the inductance of this coil by averaging readings taken over the range 780 to 860 kHz. Taking the average frequency as 820 kHz, the internal inductance is 101.8 nH.
The results of the calculation are shown below, where the 'greater than' symbol (>) is used for the L value because the coil inside diameter (D0 - d) was used instead of the slightly larger but undefined D.

L0 = (μ0 π D0² N² kL(0) /4 ) -[μ0 D0 N (ks(e) + km )/2] + Li
= 43.7331 - 0.5873 + 0.1018 μH
 L0 = 43.248 μH

L = (μ0 π D² N² kL(∞) /4 )-[μ0 D N(ks(e) +km )/2] + Li
> 42.3694 - 0.5768 + 0.1018 μH
 L∞ > 41.894 μH

Hence we expect the measured inductance to lie in the range:
L = 43.248, +0, -1.354 μH

 Medhurst gives nine inductance measurements for this coil, taken at 10 kHz intervals over the range 780 to 860 kHz and corrected for self capacitance. These measurements are scattered about the mean value, whereas if the inductance were changing significantly with frequency in this region we would expect them to diminish progressively as the frequency increased. Hence we can infer that deviations from the mean are principally due to experimental error, and we can use this information to estimate the standard deviation of the mean value. The measurements are tabulated below:
 Measurement No. k = Frequency / kHz Measured Lk / μH Deviation Lk - Lmean Deviation² (Lk - Lmean)² 1 780 42.32 -0.0544 0.00296420 2 790 42.29 -0.0844 0.00713086 3 800 42.32 -0.0544 0.00296420 4 810 42.36 -0.0144 0.00020864 5 820 42.41 0.0356 0.00126420 6 830 42.46 0.0856 0.00731975 7 840 42.39 0.0156 0.00024198 8 850 42.40 0.0256 0.00065309 9 860 42.42 0.0456 0.00207531 Lmean = 42.3744 Σ = 0 Σ = 0.02482222

The mean (average) of the measurements is given by:

 Lmean = 1 n n Σ k=1 Lk

Where 'Σ' (capital Sigma) means "the sum of" (over the range specified below and above), Lk is the 'kth' measurement of L, and in this case the number of measurements n=9. The square of the standard deviation (known as the variance) can be estimated from:

 σ² = 1 (n-1) n Σ k=1 (Lk - Lmean)²

where we divide the sum of the squares of the deviations by n-1 rather than n because the use of the sample mean rather than the population mean (i.e.the unknown 'true' value of L) removes a degree-of-freedom from the data [see Data Analysis]. Thus, for Medhurst's measurements:
σ = √(0.02482222 / 8) = 0.0557
Hence:
 L = 42.374 ±0.056 μH

Comparing this with our calculations (43.248 > L > 41.894 [μH] ), we can observe that at 820 ±40 kHz, the inductance of the coil is lower than that predicted from physical dimensions, but within the range allowed by variation of effective diameter. It is tempting to think that the high-frequency drop in inductance has been quantified by this comparison (and possibly it has), but such is not necessarily the case. The standard deviation calculated from the scatter in the data is a measure of precision not accuracy, and unknown systematic errors, or simple uncertainties in the coil dimensions, prevent us from drawing firm conclusions.

 Example B1b Once the principle has been established, accurate inductance calculation does not have to be a laborious process.We might note from the earlier discussion however; that the available literature on the subject contains mistakes, with the consequence that existing computer programs and models cannot be trusted without verification of the numbers that they produce. It is a relatively straightforward matter however, to set up a spreadsheet to do the calculation, and once the entered formulae have been verified, new rows can be added and old ones deleted at will.      A suitable calculation template is provided by the Open-Document Spreadsheet file: L_calcs.ods (sheet 2), which can be downloaded and amended as required. The spreadsheet as provided is filled with example inductance calculations for seven coils described in the academic literature. Medhurst's coil #31 (M31) from the previous example is there , along with another coil (M32) described in the same paper (these are the only test coils for which Medhurst gave full details). There are also: a coil described in reference  (GKMR1), and four coils described in reference  (MKG1-4). Interest in the last five coils lies in the fact that the inductance measurements are accompanied by calculations using a method different to the one described here. The results are summarised in the table below:

 Coil M31 M32 GKMR1 MKG1 MKG2 MKG3 MKG4 /mm 70.9596 48.26 163.2 69.92 125.02 150.5 132.8 D0 /mm 50.9856 51.0 326 79.5 79.5 79.5 81.2 D∞ /mm >50.0712 >50.0856 >316 >78.1 >78.1 >78.1 >79.8 d /mm 0.9144 0.9144 10 1.4 1.4 1.4 1.4 p /mm 1.7740 1.27 10.2 1.84 2.66 3.01 1.66 p/d 1.9401 1.3889 1.02 1.3143 1.9 2.15 1.1857 N 40 38 16 38 47 50 80 Freq / kHz 820 720 10 1 1 1 1 Li /μH 0.102 0.103 0.216 0.475 0.587 0.624 1.020 L0 /μH 43.248 51.084 83.205 83.871 85.247 83.612 243.367 L∞ /μH >41.892 >49.550 >79.247 >81.424 >82.595 >80.969 >235.907 Lmeas /μH 42.37(6) 49.89 82.3 84.1 85.6 84.7 244.6 Other calcs L0 /μH - - 82.4 (?)  83.9  85.2  83.6  243.4 

 The main point to note is that for coils MKG1-4, low-frequency (L0) inductances calculated using the modified National Bureau of Standards (NBS) method (i.e., the method used here) are in exact agreement with the method used in ref. , and that for a measurement frequency of 1 kHz, the internal inductance is the same as the DC value. This provides a vindication, not of the NBS method which is not in doubt, but of the formula verification procedures applied here. The calculation of ref.  however, is not in agreement with the NBS method, despite the statement that it was carried out using the same method as in ref. . Although very close to the measured value, the calculation of ref.  does not correspond to L0 or L0-Li, and appears for some reason to be in error. What we find here is that the measured inductance is sandwiched between our L0 and L∞> values as in the case of coils M31 and M32, and that the very low p/d ratio of 1.02 (and probable use of copper tubing rather than wire) has given rise to an early onset of the high-frequency regime. The results for coils MKG1-4 show nothing more onerous than a measurement standard deviation of around 1%, and that the HF regime is not evident for these coils at a frequency of 1 kHz.      If there is anything to be inferred from these comparisons between measurement and calculation, it is that the skin effect and the proximity (effective diameter) effect are linked. All of the coils measured at 1 kHz have an internal inductance component that is the same as the DC value, and show no sign of the diameter effect even though some have low p/d ratios. All of the coils measured at frequencies above 1 kHz have an internal inductance component substantially less than the DC value, and the inductances lie between the L0 and L∞ values. Logically, this is to be expected, because the proximity effect modifies the current distribution within the wire and therefore changes the internal inductance. The proximity effect however, is not simply a perturbation of the internal inductance, because the total variation it causes is greater than the internal inductance. >>>>>> Calculation methods have been refined since 2008. More accurate formulae for effective current sheet diameter have been derived. New spreadsheet: Lcalcs.ods (sheet 1, still a work in progress). >>>>>>

 B2. Additional sources of deviation: Points to note when comparing inductance measurements with calculations: The presence of conducting material in the vicinity of the coil disposed in such a way as to form a shorted-turn coupled to the coil, and especially of any metal screening-can or box, will cause the effective inductance to be less than that predicted for an isolated coil. The presence of any non-conductive ferromagnetic material in the vicinity of the coil will cause the inductance to be greater than that predicted by formulae which assume that μ=μ0. Solenoid inductance is paradoxical: the reason being that inductance is only completely defined when the terminals of the inductor are coincident in space, whereas the terminals of a solenoid are by definition separated by a distance . All measured solenoid inductances therefore contain a contribution from the connecting wires. This added inductance, usually a few 10s of nH, can be estimated using the methods outlined in Comps & materials 2, (section 6) although the rough guide "add about 20 nH per inch" (8 nH/cm) will probably suffice in many instances. This implies about 12.5 cm of connecting wire before a change of 1 will occur in the first decimal place of an inductance expressed in microHenrys (and is about right for 1 mm diameter wire). The lumped inductance model breaks down when a coil is operated close to, at, or above its self-resonance frequency.

References
  "Stray Capacitances of Single-Layer Solenoid Air-Core Inductors", G. Grandi, M K Kazimierczuk, A Massarini, U Reggiani. IEEE Transactions on Industry Applications, Vol 35, No. 5, Sept/Oct 1999, p1162-1168. Available from: Classic Tesla at time of writing. www.classictesla.com/download/ia99.pdf .  "Lumped Parameter Models for Single- and Multiple-Layer Inductors", A Massarini, M K Kazimierczuk, G Grandi. Proc PESC '96, June 1996, p295-301. Available from: Classic Tesla at time of writing. www.classictesla.com/download/pesc96.pdf .

© D W Knight 2008, 2016, 2019.
David Knight asserts the right to be recognised as the author of this work.
Last updated: 2019 May 14th. TX to Ae Magnetics - Z and Q