TX to Ae

Ch.3 Contents

Part 2


3. Inductors and Transformers: Part 1.
Revised July 2008
Contents:
3-0. Introduction.
3-1. The current-sheet solenoid.
3-2. Equivalent current-sheet length.
3-3. Effective current-sheet diameter (LF).
3-4. Effective diameter (HF).
3.5. Internal inductance.
3.5a. LF-HF transition frequency.
3-5b. Internal inductance factor.
3-6. Field non-uniformity. (Nagaoka's coeff.)
3-6a. Asymptotically correct approximations.		 3-6b. Wheeler's formulae.
3-6c. Using inductance formulae.
3-6d. Bad formulae.
3-7. Rosa's round-wire corrections.
3-7a. Self-inductance correction.
3-7b. Mutual inductance correction.
3-8a. Apparent inductance.
3-8b. Correcting for self-capacitance.

Part 2 >>> .

3-0. Introduction:
When modelling and using inductive devices, it is important to be aware that the concept of lumped inductance is only strictly applicable at low frequencies. The construction of an inductor involves cramming a large amount of wire into a small volume, and at radio frequencies, this means that the wavelength is likely to be comparable to the length of the wire. In such circumstances, it cannot be said that any given point within the device is in instantaneous communication with every other part of the device; in which case, the lumped component theory cannot provide an accurate description. This does not necessarily preclude the use of simple approaches to circuit design; but it does mean that lumped element analysis should be applied with caution.
     What particularly undermines the validity of the lumped approach is the propensity for inductors to exhibit dispersive behaviour. The term 'dispersion' comes from the field of optics, where a 'dispersion region' is a range of frequencies over which the refractive index of a medium changes; this being the the reason why a prism disperses white light into its component colours. The refractive index is the geometric mean of the relative permeability and permittivity (i.e., n=Ö[mr er] ); and so, in an electrical context, where a dispersion region is a frequency range over which permeability or permittivity is changing, the meaning is exactly the same. Coils with magnetic cores are inevitably dispersive, due to the complicated behaviour of ferromagnetic crystals. What is less well recognised however, is that simple coils of wire are dispersive also.
     The term 'refractive index' is not much used in electrical engineering; but many will be familiar with 'velocity factor', which is its reciprocal. This begs the question; "what has velocity got to do with inductance?" to which the answer is; "rather a lot". The traditional understanding of coils depends on the idea that they are effectively electromagnets, and that they have reactance because energy is stored in the surrounding magnetic field. This picture is mostly wrong, even though it suffices at low frequencies. If we may take the liberty of using the word 'light' to mean electromagnetic radiation of any frequency; what a coil really does is to modify the refractive index of space in its vicinity in such a way as to bend light and force it to follow the electrical conductor. All electrical circuits do that of course, but in inductors, the path is deliberately made long. Hence a coil is a waveguide or transmission line, which stores energy by trapping and detaining radiation which would otherwise have made a much shorter journey.
     The static magnetic conception of inductance works at low frequencies because the length of the wire used to make the coil is much shorter than the wavelength of the radiation. This means that a wave entering the coil at one terminal will emerge from the other terminal with almost exactly the same phase. Thus an instantaneous view of the magnetic field surrounding the coil will be almost identical to the field produced by a direct current; in which case, the energy stored (LI²/2) will be the same as in the DC case and the inductance can be calculated accordingly. From an electrical point of view therefore, a coil operating at low frequencies looks like a lumped inductance in series with the DC resistance of the wire.
     The first dispersion-related impedance variation (assuming that there are no ferromagnetic materials or lossy dielectrics to complicate matters) occurs at the onset of the skin effect; i.e., when the current ceases to be distributed uniformly throughout the wire cross-section and starts to concentrate at the surface. The frequency at which this change occurs depends on the diameter, resistivity and permeability of the wire, but it is usually somewhere between the audio and low short-wave radio regions. We can go part of the way towards understanding what happens by separating the total inductance into external and internal parts: where external inductance is that due to energy stored in the magnetic field which permeates the surrounding medium; and internal inductance is that associated with the field within the body of the wire itself. Inductance in electrical circuits is associated with current, and where there is no current there is no inductance. Hence, as the current within the bulk of the conductor diminishes with increasing frequency, so too does the internal inductance. There is a little more to it than that however, because the redistribution of current is also affected by the magnetic fields produced by adjacent turns. This leads to a substantial second-order effect, known as the proximity effect; which gives rise to a reduction in the effective area enclosed by each turn of wire, and hence a reduction in the external inductance.
     Thus the onset of the skin effect gives rise to a distinct transition from low-frequency to high-frequency behaviour; after which both the inductance and the resistance become frequency dependent. This does not necessarily preclude the use of the lumped component model however; because most of the decline in inductance occurs in the first two decades of frequency above the onset. Once out of the dispersion region, the inductance (now, strictly; the equivalent lumped inductance) settles down for a few octaves, and becomes reasonably (but never quite) constant.
     In the high-frequency region, it is no longer possible to treat the coil as though its reactance is purely inductive; the reason being that a wave emerging from the coil is now significantly delayed, and therefore has a phase which differs from its phase on entry. One observable outcome is that the impedance at the coil terminals looks the same as that of an inductance (with series loss resistance) in parallel with a capacitance. This capacitance is known as the 'self-capacitance' (or sometimes, misleadingly, as the 'distributed capacitance') of the coil. Presuming that the measured impedance has been corrected for strays, and that the coil is wound in a single layer (i.e., there are no overlapping turns), then the self capacitance has no identifiable electrostatic origin. It is hypothetical, evoked in order to repair the lumped component model, and should be accorded no existence beyond that. It remains reasonably constant over several octaves however, it can be predicted, and it is therefore useful for the purpose of circuit analysis.
     Unfortunately, the electrical literature abounds with articles which claim that the self capacitance of a coil is due to the capacitance between adjacent turns. This hypothesis is easily refuted, because it makes the wholly incorrect prediction; that coils which have closely-spaced turns will have much greater self-capacitance than those which do not. The static component of self capacitance is small in single-layer coils, because a wave travelling along the wire does so with its electric vector nearly perpendicular to the coil axis, i.e., the electric field component parallel to the axis is almost negligible in comparison to the radial component. Nevertheless, the static capacitance idea appears to be so intellectually compelling, that there are at least two examples, in the peer-reviewed literature, where researchers have been motivated to fabricate experimental evidence in order to support it.
     The inclusion of self-capacitance into the lumped-component model gives rise to the prediction that a coil will still exhibit parallel resonance in the absence of an external circuit. This is indeed correct; except that, unless the coil is extremely long and thin, the actual self-resonance frequency (SRF) is considerably greater than predicted. This failure of the lumped component theory is mainly due to the onset of another dispersion-related effect; this time in which the apparent inductance declines (presuming that we adopt the view that the self-capacitance is constant) in such a manner that the SRF is pulled to the frequency at which the wire in the coil is very nearly one half-wavelength long.
     This time, there is no reprieve for the lumped element theory. The SRF occurs at the electrical half-wavelength point because that is the frequency at which a wave, trapped in the coil by reflection from the impedance discontinuities which occur at the terminals, arrives back at its starting point in phase with itself. The pulling effect can be understood by considering the overall field pattern as the superposition (combination) of two waves, one travelling along the coil axis and the other following the helix. At low frequencies, the axial wave dominates and the helical wave is forced to keep up. This causes the phase velocity (i.e., the apparent velocity) of the helical wave to be several times the speed of light. As the frequency increases, the helical velocity falls steadily as propagation along the helix becomes increasingly important, but the change is smooth and corresponds to an impedance characteristic consistent with the lumped component model. As the SRF is approached however, the scattering cross-section of the coil suddenly increases and the axial wave is overwhealmed. Hence the impedance characteristic deviates as the coil 'locks-on' to the half-wave resonance.
     From now on up, only a fully electromagnetic model can describe the coil's behaviour. Above the SRF, radiation follows the wire at approximately the speed of light for the surrounding medium. What then occurs is a sequence of alternating parallel and series resonances, at frequencies where the electrical length of the wire corresponds to a half-integer multiple of wavelengths. From the lowest parallel resonance (the SRF) to the first series resonance, the reactance is capacitive. It then switches back to being inductive until the next parallel resonance; and so on, almost ad infinitum, except that the length of a single turn will eventually become comparable to the wavelength and further complexities will arise. It follows, that coils have interesting properties at frequencies around and above the fundamental SRF, but lumped component theory is of no help in understanding such phenomena.
     That coils are best regarded as transmission lines has long been known; but the art of characterising them as such is hampered by the difficulty in solving Maxwell's equations for practical coils of arbitrary geometry. The problem is not completely intractable however; and can be usefully adressed by treating the coil as a surface waveguide constrained to conduct only in the helical direction. This model is known as the Ollendorf sheath-helix. An overview of this subject is given by the Corum Brothers (of Tesla coil fame) [1], and additional information is given by Ramo et al. [4] (sect. 9.8) and elsewhere [18], [19], [TA3.3]. The sheath-helix model points to a unification of the static magnetic and the transmission-line approaches, it accounts for the phase velocity profile around the SRF, and it also explains a useful but widely unrecognised phenomenon; which is that the resonant voltage magnification of a coil with minimal external capacitance is much greater than the lumped component theory predicts.
     The downside of the sheath helix approach is that it involves simplifying assumptions and lacks certain important corrections. Also, it has to be said, that traditional modelling methods, when properly applied, are astonishingly accurate at frequencies well below the SRF. Consequently, in the discussion to follow, we will adopt the view that an essentially magnetic approach to coil modelling (albeit without the misconceptions) is adequate in the majority of situations, and that transmission-line concepts are best used to extend rather than replace what is well established.

3-1. The current-sheet solenoid:
In the design of high Q inductors for radio-frequency applications, the physical configuration most commonly adopted is the single-layer solenoid. The word 'solen' is an old-fashioned term meaning 'drainage channel', which eventually came to acquire the additional meaning 'drain-pipe'. The word 'cylinder' comes from the same root. Hence a solenoid is a pipe-like coil, usually wound with the aid of an actual pipe known as the coil-former. Winding the wire in a single layer produces an inductor with minimal parasitic capacitance, and hence gives the highest possible self-resonant frequency (SRF). Striving to obtain a high SRF and low losses is the key to producing coils which have radio-frequency properties bearing some useful resemblance to pure inductance.
     The basis for the calculation of the properties of practical coils is the inductance of a theoretical solenoid constructed using infinitely thin conducting tape wound, in a single layer, with zero spacing (but no electrical connection) between turns. Such a coil is known as a current-sheet inductor [2], [3]. A very long current-sheet inductor (at a low frequencies) has the property that the the magnetic field along its length is practically uniform, in which case its inductance is given by a very simple expression:
Ls = m A N² /     Henrys

1.1
Where the constant of proportionality m (in Henrys/metre) is the magnetic permeability of the environment outside the conductor (m=m0mr) and can be replaced with the permeability of free-space, m0 ("mu nought") in the absence of ferromagnetic material. A is the cross-sectional area of the cylinder, N is the number of turns, and is the cylinder length.

     Recall that the inductance of a coil can be expressed as an inductance factor AL, defined by the relationship:
L = AL N²
For the long current-sheet therefore:
AL = m A /  Henrys/turn²
Since turns are dimensionless and may be omitted from the units, this is exactly analogous to the expression for the capacitance of a capacitor (section 2-11):
C = e A / h    Farads
Note that permeability, like permittivity, is strictly complex; but for the sake of simplicity we can consider it to be real when not taking magnetic losses into account. Hence we should use the symbol m (in bold) when including losses in the permeability factor, and the symbol m when not. Notice also that the factor A/ has units of [length²/length]=[length], and since AL is an inductance, it is this which dictates that the units of m are Henrys/metre.
     Equation (1.1) tells us that inductance is proportional to the cross-sectional area of a coil (strictly, the area enclosed by the current loop). The optimal cross-sectional shape for a coil is that which gives the maximum amount of inductance using the minimum length of wire (maximum ratio of reactance / resistance), i.e., a former of circular cross-section is best. For a cylindrical coil, where A=pr², r being the coil radius, the long-current-sheet formula can be written:
Ls = m p r² N² /      Henrys

1.2
We can also write this expression using the coil diameter D instead of the radius; noting that, since D=2r, the appropriate substitution is r²=D²/4, i.e.:
Ls = m p D² N² / 4      Henrys

1.2a


Although the long current-sheet is an excellent starting-point for the calculation of inductance from physical dimensions, the equations given above require modification if we are to obtain expressions accurate for practical coils. This entails the inclusion of various correction terms and factors as will be explained in the discussion to follow. At least six types of correction are required in principle, although some of these can be neglected under certain circumstances. The main corrections are listed below with the parameters which will be introduced in order to apply them:

'Frequency independent':
· kL - field non-uniformity correction for short coils.
· ks - self-inductance correction for round wire.
· km - mutual inductance correction for round wire.
Frequency dependent:
· D or r - effective loop diameter (or radius).
· Li - internal inductance of the wire.
· CL - self-capacitance (i.e., phase-delay modelled as a negative parallel reactance).

Note that the 'frequency independent' corrections are only so in the sense that the errors inherent in failing to include frequency dependence are reasonably small.

3-2. Equivalent current-sheet length:
In the extensive literature on the subject of inductance calculation, one recurrent omission is that of an unambiguous definition for the coil length. The length required is that of the equivalent current-sheet from which the inductance will be calculated; but the problem is that a current-sheet inductor, being a hypothetical device, can be defined without considering the method of connection. The correct definition is given by Grover (ref. [3] p149), but requires interpretation.
     The equivalent current-sheet length is obtained by considering each turn of the coil to lie at the centre of a corresponding turn of the current sheet. This means that if the length of the coil is measured on the side where the connecting wires are brought out (assuming a whole-number of turns) then the distance required is that from centre to centre of the emerging wires, i.e., it is the length of the coil measured from the outside of the winding less the diameter of the wire. This length is equal to Np, where N is the number of turns, and p is the winding pitch-distance.

     Rosa and Grover [Formulas and Tables for the Calculation of Mutual and Self Induction, 1911] appear to give a definition which is different to that given above [see p119], but an ambiguity arises because the electrical termination is not considered. The instruction given is effectively; that the length can be obtained by measuring to the outside of the winding, then subtracting the wire diameter and adding the pitch. This length is stated to be equal to Np as above, but it is only so if the measurement is made on the side of the coil opposite to the side where the connecting wires are brought out.
     Note incidentally, that all of the expressions for solenoid inductance so far given (and to be given) contain a factor 1/. This factor goes to infinity as the length of the coil goes to zero, whereas the field non-uniformity correction (kL) to be introduced shortly goes to zero at this point. Hence the inductance of a zero length coil tends to 0/0 and is undefined. This condition does not happen in practice, because the length of the equivalent current sheet can never be less than the diameter of the wire. The ambiguity arises because winding pitch (and hence solenoid length) is not defined unless a coil has more than one turn. The inductance of a single turn coil is best obtained using a loop inductance formula (see section 2-6).

3-3. Effective current-sheet diameter (LF):
When a coil is wound with a thin flat conductor (broadside to the coil former), its radius (r = D/2) is well defined. When a coil is wound with round (i.e., cylindrical) wire, the equivalent current sheet radius will obviously be obtained by measuring from the solenoid axis to some point which lies within the body of the wire, but it is by no means obvious where that point should be. Referring to the diagram: If the radius of the wire (excluding any insulation) is rw, and the average radius of the helix (measured from the solenoid axis to the wire axis) is ra; then there is a radial conduction zone which extends from r = ra-rw to r = ra+rw. The effective current sheet radius must lie within that range.

     It is traditional to assume that the effective radius is the same as the average radius ra (at least at low frequencies), and this is the basis for most inductance calculations. It has however been pointed out by Fraga et al. [15], that the conduction path on the outside of the coil (at r = ra+rw) is longer than the path on the inside (at r = ra-rw). This means that the current-density in the wire will be biased towards the inside of the coil; and the equivalent current sheet diameter will be consequently less than ra. To that observation, we can also add, that the act of winding the wire around a cylindrical former causes the metal on the outside of the coil to become stretched relative to the metal on the inside. When metal wire is stretched (particularly in the case of soft copper), it does not so much shrink in diameter as increase in resistivity; i.e., the microcrystals within the material tend to rearrange and become less densely packed (until the plastic limit is reached). Hence the solenoid develops a radial resistivity gradient, the bulk resistivity being greatest at r = ra+rw and something close to the native value at r = ra-rw. The effect, once again, is to bias the current distribution towards the inside, with consequent reduction in the effective radius.
     From considerations relating to the boundary conditions of Maxwell's equations, the effective current sheet radius is defined as: 'that distance measured perpendicular to the solenoid axis at which the current flowing on the outside is equal to the current flowing on the inside'. A derivation on that basis, including the effects of path length and strain, is given in the theory appendix [TA3.1] and results in the following formula:

r0 = ra [1 - (rw/ra)²]

Equivalent current-sheet radius at low frequencies.
ra/rw >4 ,  2pra >> p

(3.1)

This, of course, can also be stated in terms of coil and wire diameters:

D0 = Da [1 - (d/Da)²]

Equivalent current-sheet diameter at low frequencies.
Da/d >4 ,  pDa >> p

(3.1a)

Notice that when the average coil diameter (Da) is much greater than the wire diameter (d), then D0 » Da. High Q coils however, tend to be wound with relatively thick wire; in which case, inductance calculations which use Da instead of D0 will exhibit a systematic error. Such error, although usually small, is exacerbated by the fact that inductance is proportional to D².

3-4. Effective current-sheet diameter (HF):
In an isolated wire, at low frequencies, the current is distributed uniformly throughout the material. At high frequencies however, due to the inability of a good conductor to support an electric field within its bulk, the current is confined to a thin layer close to the surface. This is the well-known skin effect.
     In coiled-wire inductors, the skin effect is perturbed (i.e., modified); not only by the conductivity gradient discussed in the previous section, but by an interaction with the external magnetic field known as the proximity effect. Presuming that the number of turns is reasonably large; the currents in adjacent turns are very nearly in phase, even at frequencies approaching the SRF. Under such conditions, there is a repulsion between adjacent current streams and a further interaction with the overall magnetic field. The result is that the current, in turns close to the middle of the coil at least, tends to crowd towards the coil axis [3, Ch. 24] [Medhurst 1947]. This means, of course, that there will be a further reduction in the effective current sheet radius at high frequencies.
     It is important to be aware that dispersive phenomena have both real and imaginary parts. In the case of the proximity effect; the real part is that which causes the AC resistance of the wire to be greater than that predicted from the skin effect alone. The imaginary part is that which reduces the internal inductance of the wire (next section) and reduces the effective current sheet radius. It follows that the skin and proximity effects are not strictly separable. When the transition from low frequency to high frequency behaviour occurs (usually somewhere in the high audio to low radio frequency range); it is the proximity of other conductors, and the phases of the currents in them, which dictates how the current is distributed over the surface of the wire once it can no longer penetrate significantly into the body.
     From its name, it should be obvious that the proximity effect is greatest in coils with closely-spaced turns. That part of it associated with a reduction in effective current sheet radius is also greatest when the wire diameter is significant in comparison to the coil radius. In high Q coils; which require the use of relatively thick wire to keep the AC resistance down, and have plenty of turns to maximise the inductance obtained in a given volume; variation between the actual and the effective diameter may cause a difference of several percent between the low-frequency and the high-frequency inductance.
     Due to the complexity of the underlying physics, the effective coil radius at high frequencies is difficult to predict from first principles. Fraga et al. [15] approximate the situation by treating the coil as a modified current sheet with finite conductor thickness and resistivity. This approach has considerable merit, but is not completely realistic. It has also been suggested that, for modelling purposes, the wire can be considered to shrink towards the inner radius (ra-rw) as the frequency increases; but this is unconvincing. For those who are interested in this problem, it is important to understand that current still flows all over the surface of the wire when the proximity effect is present. This is physically necessary, because the wire remains an imperfect conductor, with the bulk shielded from external fields by conduction at the surface. Thus it is a matter of current redistribution, rather than of parts of the wire ceasing to conduct. Precise determination of the effective radius therefore involves finding an expression which defines the current density at any point in the wire cross section, and then noting that the integral of the current density from the inner radius (ra-rw) to the effective radius is the same as the integral from the effective radius to the outer radius (ra+rw).
     Since the effective solenoid diameter at high-frequencies is difficult to determine, and since the difference between the average diameter (Da) and the low frequency effective diameter (D0) is not generally appreciated; inductance calculations are usually based on the average diameter. We can do a little better than that however; there being no great difficulty in determining limits within which the actual inductance must lie. We start by noting that the current-sheet diameter D should be replaced by a mathematical function which depends on the winding-pitch to wire-diameter ratio (p/d), and on the wire diameter to solenoid diameter ratio (Da/d), and varies between the low frequency value (D0) and some high frequency limiting value, which we will call D¥. We cannot easily determine D¥ by mathematical analysis; but we can at least say that, for turns in the middle of the winding, it can never be smaller than the inner diameter (i.e., Da-d). Furthermore, for the two turns at the ends of the coil, the current stream will be repelled from the next turn in, and so the effective diameter will remain as D0. Hence we can define an absolute minimum effective diameter as the average of N-2 turns with a diameter of Da-d and 2 turns with a diameter of D0, i.e.;

Dmin = [ (N-2)(Da-d) + 2D0 ] / N
This expression will always underestimate D¥, and it will continue to do so if we use the approximation D0=Da, i.e.;
Dmin = [ (N-2)(Da-d) + 2Da ] / N
which simplifies to:
Dmin = Da-d +2d/N

(4.1)

It is a straightforward matter (with the aid of a computer) to perform two inductance calculations; one with D set to D0, and one with D set to Dmin. From this we will obtain two inductances, L0 and Lmin (say); the former being accurate at low frequencies and providing an upper uncertainty boundary for the high frequency inductance (L¥), and the latter (presuming that the model is otherwise correct) giving the lower uncertainty boundary for L¥.
     It is, of course, tempting to try to define a semi-empirical formula for D¥. For that, it is useful to know that D¥ is fairly close to Dmin when the p/d ratio is close to 1, and almost the same as D0 when p/d>10. It follows that the accuracy of the inductance prediction will always be improved by taking the weighted average of D0 and Dmin in such a way that Dmin dominates when p/d=1 and progressively loses its influence as p/d increases. Such a formula can be obtained by direct deduction, i.e;

D¥ =

D0 + Dmin a / [ (p/d)-1]
1 + a / [ (p/d)-1]

a = 2
d >> di
(see next section)

(4.2)

Note that in practical inductors, (p/d)-1 is always > 0 because coils with closely spaced turns must be wound with insulated wire. Hence using the reciprocal of (p/d)-1 in the weighting coefficient will not cause divide-by-zero errors. The constant 'a' is determined empirically, and setting it to 2 allows the HF inductance of typical radio coils to be predicted to within a few parts per 1000 when the radius of the wire is greater than 3 times the skin depth.
     Note that the value for 'a' given above is not final. A frequency dependent, method for taking the weighted average of D0 and D¥ is described in section 3-5b, and requires a different value.

3-5. Internal inductance:
The 'external inductance' of a coil is the inductance due to the storage of energy in the magnetic field which permeates the surrounding medium. The 'internal inductance' is due to the magnetic energy stored within the body of the conductor itself. Internal inductance diminishes with frequency because it depends on the current distribution within the wire; i.e., it is the imaginary counterpart of the skin effect.
     The conducting strip in the theoretical current-sheet is infinitely thin and therefore has no internal inductance. Wire, on the other hand, does have internal inductance, as is discussed in section 2-6 and (for example) ref [4]. The internal contribution to overall inductance is generally small, and is therefore usually neglected in approximate calculations; but it can amount to several % of the total under certain circumstances. The following points may help when considering its importance:
 Internal inductance is proportional to the wire-length, and therefore to the number of turns N; whereas external inductance, being enhanced by winding the wire into a helix, is proportional to N². Hence, internal inductance is most likely to be significant in coils which have a low number of turns.
External inductance is enhanced by the use of a magnetic core, whereas internal inductance is unaffected. Hence internal inductance is not usually significant if the coil has a high-permeability core.
Internal inductance diminishes with frequency more rapidly in thick wire than it does in thin wire; i.e., thick wire coils have the skin effect dispersion at lower frequencies than thin wire coils. For coils made from wire of less than 1mm diameter, internal inductance may still be significant at the low end of the short-wave region (see next section).

From the discussion of section 2-6: the internal inductance of a round wire at DC is given by:
Li(dc) = w m(i) / 8p     Henrys
where w is the length, and m(i) is the permeability of the wire material. For copper and other non-ferromagnetic conductors, m(i) can be taken to be the same as m0, i.e., 4px10H/m, which means that the low-frequency internal inductance of any non-magnetic round wire is 50nH/m.

The internal inductance of a wire at high frequencies is given by:
Li(hf) = w (m(i) /2p) (di / d)    Henrys
where d is the diameter of the wire, and di is the skin depth given by:
di = Ö[ r / ( p f m(i) )] 
r being the resistivity of the wire. Hence, at high frequencies, internal inductance becomes proportional to the reciprocal of the square root of the frequency.

The general problem of calculating internal inductance is discussed in [A2.1]. A suitable formula for solenoid modelling is the ACA4ML approximation, which is accurate to within ±0.12%. Since the internal inductance contribution to the total inductance is typically <1%, the error in the ACA4ML approximation contributes less than 1 part in 10 to the overall error.

Li = m(i)
2p

di
d
 [ 1 - exp{ -[d/(4di)] } ] (1 - y') [ H / m ]

±0.12%

ACA4ML
(5.1)

y' = 0.02758 / [1 + 2( z - z )² ]³   ,   z = d/(4.763di)   ,   di = Ö[r/(pfm(i))]

where di is the skin depth, r is the resistivity of the wire, and f is the frequency.

Note that the formula gives the internal inductance per unit length. The actual internal inductance is:
Li = w Li
where w is the length of the winding wire and is given by the expression:
w = Ö[ (2prN)² + ² ]
(see section 3-10 for derivation).

The total inductance of a current loop is the sum of the internal and external inductances. For coils however, there will be an additional term (analogous to mutual inductance) due to the external fields of adjacent turns passing through the wire; i.e., a there will be a perturbation due to the proximity effect (as mentioned in the previous section). The proximity of other current-carrying conductors has no effect at low frequencies, but reduces the internal inductance at high frequencies. Fortunately, internal inductance makes a relatively small contribution to the overall inductance, and so the error in using an isolated wire model for internal inductance (i.e., ignoring the perturbation caused by the proximity effect) is usually small.

3-5a. LF-HF Transition frequency:
When deciding whether to use a low or a high-frequency inductance formula, it is necessary to be able to locate the intervening dispersion region. A simple rule for doing so can be obtained by examining the graph below, which shows the relationship between internal inductance and the ratio of wire radius to skin depth. The calculation (using equation 5.1) is for an isolated wire (see spreadsheet: Li_aca4ml.ods), but while the proximity effect will steepen the inductance decline, it will not greatly affect the frequency at which the change begins.



The graph confirms a rather obvious proposition, which is that the current distribution within the wire will be substantially uniform until the skin depth becomes less than the wire radius. Hence we can define a transition frequency (fs) at which DC inductance formulae begin to break down. Skin depth is given by:
di = Ö[ r / ( p f m(i) )] 
and, from the graph above, we need to start making high-frequency corrections when rw=di. Hence, to work out the wire diameter needed to achieve particular fs (noting that d=2rw):
d = 2 Ö[ r / ( p fs m(i) )] 

(5.2)
And to work out fs for a particular wire diameter:
fs = 4 r / [ p m(i) d²] 

(5.3)
Where m(i) = m0 = 4px10 H/m for non-ferromagnetic wire.

The relationship between the dispersion onset frequency fs and wire diameter is shown below for solid copper wire (r = 17.241nWm, m(i) = m0. Spreadsheet calc.: Li_aca4ml.ods, sheet 2):



With quick reference to the graph, and using equation (5.2) for accuracy, we find (for example) that a coil wound with 1mm diameter copper wire will continue to exhibit DC behaviour up to 17.5KHz, whereas using 0.1mm wire will push the limit up to 1.75MHz (although it will be necessary to include self-capacitance in the model to calculate the correct reactance in that case). While this type of information might be useful for the purpose of designing low-frequency reference inductors however, it is not so good for deciding the frequency above the dispersion region at which the inductance can one again be considered to be constant. A fair rule of thumb is to adopt the point where rw/di=10, which occurs two decades above fs; but if the objective is (say) to design an accurate high-frequency reference coil, it is better minimise the proximity effect by using a large p/d ratio and include internal inductance in the model (the latter approach is developed in [A3.3]).

3-5b. Internal inductance factor.
Although internal inductance is perturbed by the proximity effect, the frequency interval over which the major part of the dispersion occurs is not greatly affected. We can therefore usefully define an internal inductance factor Q (Theta) which will tell us whereabouts we are in the dispersion region. Thus:
Q = Li / Li(dc)
Li is given by equation (5.1). and
Li(dc) = (m(i) / 2p)(¼)
Hence:

Q =

4

di
d
 [ 1 - exp{ -[d/(4di)] } ] (1 - y') [ H / m ]

±0.12%

ACA4ML
(5.4)

y' = 0.02758 / [1 + 2( z - z )² ]³   ,   z = d/(4.763di)   ,   di = Ö[r/(pfm(i))]

and
Li = (m(i) / 8p) Q
Note that Q=1 when the wire radius d/2 is greater than the skin depth di, and
Q4di/d (which is small) as f¥ .

The skin effect and proximity effect dispersions are interlinked and so occur on the same frequency interval. Therefore, at least as a good first-order approximation, we can use Q to weight the change in effective diameter from D0 to D¥. Obviously, when d/2<di, then DD0, and when d/2>>di, then DD¥. Hence, to track the diameter change through the dispersion region:
D = D0 Q + D¥ (1 - Q )
i.e.;
D = Q (D0 - D¥ ) + D¥

(5.5)

where D0 is given by equation (3.1a) as:
D0 = Da [1 - (d/Da)²]
(where Da is the average coil diameter), and D¥ is given by equation (4.2) as:

D¥ =

D0 + Dmin a / [ (p/d)-1]
1 + a / [ (p/d)-1]

a = ? see text below.

(5.6)

Where:
Dmin = [ (N-2)(Da-d) + 2D0 ] / N

The empirical parameter 'a' was given in section 3-4 as 2, for HF only calculations. Now, since we are removing the requirement that d >> di when calculating the effective diameter, we need to bias D¥ to be somewhat closer to Dmin. This can be done by increasing 'a' to give a good average match to the most accurate HF inductance measurements we can obtain.

>>> value of 'a' to be determined.

3-6. Magnetic field non-uniformity (Nagaoka's coefficient):
By far the greatest correction to the long-current sheet formula is that which allows for the magnetic-field non-uniformity which appears when the length of the coil becomes comparable to its diameter (i.e., when the coil is short). This modification is analogous to the Maxwell fringing-field correction for a parallel-plate capacitor (see section 2-12), but is a gross rather than a minor effect. It can be implemented by including a dimensionless factor (a special type of relative permeability), which we will here call kL. Thus, for coils of arbitrary length/diameter ratio (/D):
Ls = m p r² N² kL /      Henrys

6.1
where the inductance Ls retains its subscript as a reminder that it is still a current-sheet inductance and should only be regarded as an approximation to the inductance of a practical coil.
     The subscript L in kL can be taken to stand for 'Lorenz'; because it was Ludwig Lorenz, in 1879, who was the first to find an exact analytical expression for the inductance of current sheet of arbitrary length [see Rosa and Grover, 1911, p117]. The factor kL however (usually given elsewhere without a subscript) is most commonly known as Nagaoka's coefficient, because it was H. Nagaoka [Inductance Coefficients of Solenoids, 1909] who introduced the current-sheet formula in this convenient form and developed a practical method for calculating the correction factor. The coefficient depends only on the length/Diameter ratio (/D) of the coil and varies between 0 and 1 as /D varies between 0 and ¥. Nagaoka's procedure for calculating kL is complicated, but the results are tabulated to 6 decimal places in his paper [Nagaoka 1909, page 31] and elsewhere [3], [Rosa & Grover 1911, page 224].
     A formula which allows straightforward calculation of Nagaoka's coefficient when coded into a computer program or entered into a spreadsheet was derived by Richard Lundin [5] (see box below). This is known as Lundin's Handbook Formula , and is in the form of two expressions, one for /D£1 (short coils), and one for /D³1 (long coils). Both expressions can be used to calculate kL for /D=1 and will give kL=0.688423 (+0, -0.000001) at this point if correctly transcribed. Lundin's formula gives the field non-uniformity correction accurate to better than 3 parts-per-million (±0.0003%), this being generally superior to the accuracy with which a coil can be made or measured, and less than the error due to dimensional variation with temperature.

6.2 Lundin's Handbook Formula
(±0.0003%)
Short coils, (D³):

kL = (2/p)(/D)

[ ln(4D/) -½ ] [1 +0.383901(/D)² +0.017108(/D) ]
[ 1+0.258952(/D)² ]
  
 

+0.093842(/D)² +0.002029(/D) -0.000801(/D)

Long coils, (³D):

kL =

[1 +0.383901(D/)² +0.017108(D/) ]
[ 1+0.258952(D/)² ]

-

4 (D/)
3p
  


When /D=1, kL=0.688423

Nagaoka's coefficient (calculated using Lundin's formula) is shown plotted below; first on a linear scale; and then on a logarithmic scale for the range 0.1 £ /D £ 10, which covers most of the coils used in normal practice. When /D®¥, kL®1.





The inductance of a cylindrical current-sheet inductor, in the absence of ferromagnetic material is:
Ls = m0 p r² N² kL /      Henrys

6.3
m0 = 4p´10 Henrys/metre
m0 p = 3.94784176´10 H/m

3-6a. Asymptotically-correct approximations for kL
Some additional formulae for kL are given below. These are asymptotically correct approximations; i.e., they converge with Nagaoka's coefficient in the limit of a very long or a very short coil, but are not exact for coils of intermediate length. They provide a useful check on the coding or transcription of Lundin's formula, since agreement within the stated limits between two different expressions is a very good test of correctness. They can also be truncated and otherwise modified to produce calculator-friendly formulae of sufficient accuracy for engineering purposes.

6.4 Rayleigh-Niven formula (for short coils). [Rosa & Grover 1911, p116].
Coincident with Nagaoka's coefficient as /D®0. +0.08% when /D=0.5.
+0.28% when /D=0.7.

kL = (2/p)(/D)[ ln(4D/) -½ +[(1/8)(/D)²][ln(4D/)+¼] ]

6.5 Coffin's Formula (for short coils). [Rosa & Grover 1911, p117].
Extended version of the Rayleigh-Niven formula.
Coincident with Nagaoka's coefficient as /D®0. -0.21% when /D=1

kL = (2/p)(/D)[ ln(4D/) -½ +[(1/8)(/D)²][ln(4D/)+¼] -[(1/64)(/D)][ln(4D/)-2/3] 
             +[(5/1024)(/D)][ln(4D/)-109/120] -[(35/16384)(/D)][ln(4D/)-431/420] ]

6.6 Webster-Havelock formula (for long coils). [Rosa & Grover 1911, p121].
Coincident with Nagaoka's coefficient as /D®¥. +0.06% when /D=1

kL = 1 -(4/3p)(D/) +(1/8)(D/)² -(1/64)(D/) +(5/1024)(D/)
                                                                   -(35/16384)(D/) +(147/131072)(D/)

3-6b. Wheeler's Formulae:
Although Nagaoka's coefficient can be calculated easily using a computer, it is always useful to have methods suitable for hand calculation. The most widely known of these is a remarkable formula due to Harold A Wheeler [6], which can be written (for coils without a magnetic core):

Ls =

m0 p r² N²
l [1 + 0.45(D/)]

Henrys
Wheeler's approximation for Nagaoka's coefficient is therefore:
kL = 1/[1 + 0.45(D/)]
and has the obvious virtue that it is asymptotically correct in the limit /D®¥. The percentage difference between this approximation and Lundin's formula is shown in the graph below:


Despite its simplicity, Wheeler's formula gives an accuracy of ±0.33% for /D ³ 0.4, a range which covers many practical situations; and being good to about 3 parts in 1000 is certainly better than the dimensional accuracy obtained when making coils by hand. Wheeler determined the approximation in 1928, and even with modern calculating aids there is very little which can be done to improve it. A small adjustment of the empirical coefficient to 0.4502 makes the maximum excursions above and below Lundin's value equal in the 0.4 £ /D £ ¥ range and improves the accuracy to ±0.32%, but that is all. We might as well include the adjustment however, and so Wheeler's formula, very slightly optimised, becomes:

Ls =

m0 p r² N²
[1 + 0.4502(D/)]

Henrys
±0.32%, ³0.4D

6.7
Wheeler's long-coil formula
with
kL = 1/[1 + 0.4502(D/)]

6.8
Note that for /D less than 0.4, the accuracy deteriorates rapidly, the formula giving an inductance which is 1.33% too low when =0.3D, 3.8% too low when =0.2D, and 10.6% too low when =0.1D. We will therefore refer to equation (6.7) as Wheeler's long-coil formula.
     In a later paper [7] (1982) Wheeler gave a long-coil expression which is very similar to his original formula:
kL = 1/[1 + (4/3p)(D/)]
This is related, by inversion and a change of sign, to the first term of the Webster-Havelock formula (6.6). The quantity 4/3p however evaluates as 0.4244, and use of this in place of the original empirically-determined coefficient gives a grossly inferior fit to the actual curve of kL. Wheeler's intention however, was not to use this new expression directly, but to increase its accuracy by combining it with other formulae.
     Wheeler [7] went on to obtain various continuous approximate expressions for solenoid inductance by combining weighted averages of the long-coil formula and the first term of the Rayleigh-Niven (short-coil) formula (6.4). The most accurate of the resulting expressions is:
Ls = m0 N² r [ ln(1 + pr/) + 1/[2.3004 + 1.6/r + 0.4409(/r)²] ]
Where
2.3004 = 1/[ ln(8/p) - ½ ]
0.4409 = 6/(3p² - 16)
and the coefficient 1.6 is empirical. The formula is asymptotically correct for very short and very long coils, and using the empirical coefficient given is accurate to 0.1%. On comparing the formula with Lundin's formula and calculating the differences however, it was noted that all of the differences were positive. The empirical coefficient was therefore adjusted, and it was found that a value of 1.622 made the greatest errors above and below Nagaoka's coefficient about equal and made the formula accurate to ±0.05%. With this optimisation, and a rearrangement in favour of /D as the coil-shape parameter, Wheeler's continuous formula can be given as an expression for kL as in the box below. The percentage deviation from Lundin's formula is shown in the graph.

6.9 Wheeler's Continuous Formula (optimised):
Coincident with Nagaoka's coefficient as /D®0 and as /D®¥.
Maximum error ±0.047%

kL = (2/p)(/D)[ln[1 + (p/2)(D/)] + 1/[2.3004 + 3.244/D + 1.7636(/D)²] ]

After evaluating constants, this becomes:
kL = 0.63662(/D)[ln(1+1.5708D/) + 1/[2.3004 + 3.244/D + 1.7636(/D)²] ]
(no improvement is possible by increasing on the number of decimal places given).


3-6c. Using inductance formulae:
Note that many of the formulae given above for kL are in the form:
kL = (2/p)(/D) [expression]
if we call the expression in square brackets k', then we have:
kL = (2/p)(/D) k'
notice also, that D=2r, and so:
kL = (1/p)(/r) k'
Inserting this into the general current sheet formula (6.1) we have:
Ls = m p r² N² (1/ )(1/p)(/r) k'
i.e.,
Ls = m r N² k'
The point being that the factor (2/p)(/D) does not always have to be evaluated explicitly when calculating Ls.

3-6d. Defective formulae:
Wheeler's long-coil formula is adequate for many engineering calculations when /D ³ 0.4; and his continuous formula is both highly accurate and manageable using a scientific calculator. That does not stop people from looking for alternatives to it however, sometimes with unfortunate consequences. One such alternative was given by Meyer [8]; on the basis that Wheeler's long-coil formula did not agree with his measurements (the real problem being that he used the diameter of the coil former instead of the mean diameter of the coil):
D ³ 2 kL = 0.9694(D/)

2 ³ D kL = 0.9617exp{-0.2913(D/)}

where exp{x}=e
Using Lundin's handbook formula as datum, a comparison between this formula and Wheeler's long-coil formula is shown in the graph below:



Meyer's long-coil formula (for >0.5D) is grossly inferior to Wheeler's long-coil formula (which it claims to supplant) and has no merit whatsoever. The short coil formula (for <0.5D) however offers an accuracy of ±5% in the /D range from 0.02 to 0.3, and so might appear to have some utility as a crude approximation were it not for the fact that a far better simple formula already exists. The curve marked 'Rayleigh-Niven truncated' is obtained by using only the first two terms of the Rayleigh-Niven formula (6.4). This was reproduced in Meyer's article as part of an erroneously transcribed version of Coffin's formula, i.e.:
kL = (2/p)(/D)[ ln(4D/) -½ ]
and so he should have been aware of it. If this formula is used for /D up to 0.3, and Wheeler's long-coil formula us used for /D greater than 0.3, the maximum error is about 1.5%. This is still poor of course, and Wheeler's continuous formula (6.9) is generally to be preferred.
     What is particularly problematic about Meyer's approximation is that it has been used in at least one computer program distributed to Radio Amateurs. Its accuracy in the most important region from /D=0.3 to 3 is lamentable; and its promotion as an alleged improvement over Wheeler's formula makes it necessary to inspect the source code or otherwise verify the accuracy of inductance calculation programs obtained via the Amateur Radio community. It cannot be stressed too strongly, that when coding a program, there is no excuse for using anything less accurate than Lundin's Handbook Formula.

3-7. Rosa's round-wire corrections:
Apart from the inclusion of internal inductance; all of the formulae given so far have been aimed at determining the inductance of a coil from the inductance of the equivalent current sheet. Practical coils however, no matter how good the choice of effective diameter, do not conform exactly to the current-sheet model. This means that additional corrections are required, and these were given for the low-frequency case by E B Rosa in 1906 [see Rosa & Grover 1911, p122]. Rosa's corrections result in a general expression for solenoid inductance which takes the form:

L = Ls - mrN( ks + km )     Henrys

7.1

Ls is the current-sheet inductance as given by equation (6.1), ks is a correction for the difference between the self-inductance of a round wire loop and that of a single-turn current sheet, and km is a correction for the difference between the mutual inductance of pairs of round wire loops and the mutual inductance of pairs of current sheet loops. In reference [3], ks and km are called G and H, and in refs [2] and [Rosa & Grover 1911, p122] they are called A and B; but, since those letters have other meanings in a modern electronics context, the notation has been altered here.

3-7a. Self-inductance correction.
Rosa's correction term for self-inductance (ks) is derived from the difference in inductance between a single-turn loop of round wire and a single-turn current sheet.
     The inductance of a single-turn current sheet solenoid can be obtained using the Rayleigh-Niven formula (6.4). In this special case, only the first two terms are required, and we can identify the length of the cylinder as being equivalent to the pitch (p) of the inductor to which the correction will be applied. Hence, substituting (6.4) into (6.1), setting N=1, D=2r and substituting p for we get:
L1s = m(e) p (r²/p) (2/p)[p/(2r)][ ln(8r/p) -½]
i.e.;
L1s = m(e) r [ ln(8r/p) -½]
Where m(e) is the external permeability (i.e., the permeability of the medium local to the coil).
     The external inductance of a single turn of round wire was given in section 2-6: Modified to conform to the present notation it becomes:
L1w(e) = m(e) r { ln[8r/(d/2)] - 2} 
Rosa however, also included the DC internal inductance in his expression. This was given in section 3-4 as:
Li(dc) = w m(i) / 8p
In this case, the length of the wire is 2pr (the circumference of the loop), and so:
L1i(dc) = m(i) r / 4
For the case where the wire is non magnetic, and the coil is not in proximity to any magnetic materials:
m(e) = m(i) = m0
Hence, the total inductance of a loop of non-magnetic round wire, at low frequencies, in the absence of a magnetic core is:
L1w(dc) = m0 r { ln[8r/(d/2)] - 2 + ¼}

The correction for self inductance given by Rosa is obtained by subtracting the inductance of a single-turn current sheet from the inductance of a single turn wire loop and multiplying by the number of turns (N); i.e.;
N (L1w(dc) - L1s) = m0 r N { ln[8r/(d/2)] - 2 + ¼ - ln(8r/p) + ½}
Observe here that subtracting logarithms is the same as performing a division of the numbers within. Hence:
N (L1w(dc) - L1s) = m0 r N [ ln(2p/d) - (3/2) + ¼ ]
The ¼ being the internal inductance term. Note however, that Rosa defined his correction in equation (7.1) as negative. Thus:
ks(dc) = -(L1w(dc) - L1s) / (m0 r)
Hence, Rosa's self-inductance correction term, ks(dc) (also known as G or A), depends only on the wire-pitch/wire-diameter ratio and is given by:
ks(dc) = (5/4) - ln(2p/d)

7.2
which is the same as:
ks(dc) = ln(1.74517148 d/p)

7.2a
p/d is always >1, since the pitch can never be smaller than the wire diameter and close-spaced coils must be wound using insulated wire. When p/d = 1.745, ks = 0, and so coils with a gap between turns of about 3/4 of the wire diameter require no self-inductance correction at low frequencies.

Rosa's self-inductance correction, given as above in various textbooks, was, of course, never intended to be used for high-frequency calculations. Having preserved the identity of the internal inductance component in the derivation above however, we are in a position to modify the formula to improve its versatility. The most obvious way in which to do that is to make a complete separation between the calculation of the internal and external inductances. By so doing, it becomes possible to have different external and internal permeabilities, allowing the model to deal with magnetic cores (or even magnetic wires); and the full frequency dependence of internal inductance can be included if required.
     The self inductance correction, for external inductance only, is obtained by adding ¼ to equation (7.2):
ks(e) = (3/2) - ln(2p/d)

7.3
Using this expression, the inductance of a round-wire solenoid becomes:
L = Ls - m(e) r N( ks(e) + km ) + Li     Henrys

7.4
Li can be calculated by one of the methods described in section 3-5 and [A2.1]. The calculation requires the total length of the wire in the solenoid, which is given in section 3-10.
     The separate calculation of internal and external inductance is slightly more accurate than Rosa's method, even at low frequencies. By comparing (7.2) and (7.3) we can see that the internal inductance term included by Rosa is:
Li(dc) = m(i) r N / 4
this expression being obtained by estimating the length of the wire in the coil as N times the circumference, i.e.;
w = 2prN
This estimate is only accurate when the number of turns, N, is large. When N is small, as it sometimes is in radio coils, it is better to use the exact expression for the wire length, which is shown in section 3-10 to be:
w = Ö[ (2prN)² + ² ]
where is the solenoid length.
     Note that, even when the internal inductance component is removed from the correction term, there is still a mechanism by which ks(e) can vary with frequency. This is the repulsion between current streams in adjacent conductors (another manifestation of the proximity effect) which will cause the effective pitch / diameter ratio to increase. Noting the sign conventions used in equations (7.3) and (7.4), such repulsion will give rise to a small increase in the self inductance of a turn, but this will be offset by the error incurred by neglecting the proximity-induced reduction in the internal inductance (section 3-5).

3-7b. Mutual-inductance correction.
Rosa's mutual-inductance correction term, km (also known as H or B), is calculated by the method of Geometric Mean Distance (ref [3] p14-16 and Ch. 3) and depends only on the number of turns N. Grover has given it in the form of a table (ref [3] p150), which is reproduced below:

Table 7#1. Rosa's Correction for the Mutual Inductance of Round Wires

N

km

N

km

N

km

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

0.0000
0.1137
0.1663
0.1973
0.2180
0.2329
0.2443
0.2532
0.2604
0.2664
0.2715
0.2758
0.2795
0.2828
0.2857
0.2883
0.2906
0.2927
0.2946
0.2964
0.2980
0.2994
0.3008
0.3020
0.3032
0.3043
0.3053
0.3062
0.3071
0.3079

31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
55
60
65
70
75
80
85
90
95
100

0.3087
0.3095
0.3102
0.3109
0.3115
0.3121
0.3127
0.3132
0.3137
0.3142
0.3147
0.3152
0.3156
0.3160
0.3164
0.3168
0.3172
0.3175
0.3179
0.3182
0.3197
0.3210
0.3221
0.3230
0.3238
0.3246
0.3253
0.3259
0.3264
0.3269

110
120
130
140
150
160
170
180
190
200
220
240
260
280
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
¥

0.3278
0.3285
0.3291
0.3296
0.3301
0.3305
0.3309
0.3312
0.3315
0.3318
0.3323
0.3327
0.3330
0.3333
0.3336
0.3341
0.3346
0.3349
0.3351
0.3354
0.3356
0.3357
0.3358
0.3360
0.3361
0.3362
0.3362
0.3363
0.3364
0.3379

Values of km are also given in [Rosa & Grover 1911, p199], but differ from Grover's table by as much as 0.0011. The table above, being the most recent (1946) is presumed to be authoritative.
     The table allows the correction for most coils to be read directly; but for the coding of computer programs and spreadsheets it is useful to have some kind of formula. Unfortunately, the generating function for the table involves a complicated summation and is prone to the accumulation of computer-arithmetic rounding errors. It was found possible however, to determine a relatively simple fitting function which reproduces the values in Grover's table with a maximum difference of 0.000062. The function is given in the box below, and the procedure by which it was obtained is described in appendix [TA3.2].

7.5 Mutual-inductance correction formula:
Reproduces the values in Grover's table (ref [3] p150) with a maximum difference of 0.000062. (s =0.00003)

km = 0.337883

1 - 

0.9754
N-0.0246

+ loge(N)

-0.16725
N

+

0.0033
N²

 

Note that, since the table has rounding errors, whereas the fitting function is continuous and of low order (i.e., does not involve N raised to a high power and so cannot change rapidly enough to follow noise fluctuations in the data); the function has the effect of smoothing-out the noise, and the values it produces are consequently a little more accurate than those in the table.