TX to Ae

Ch. 2 Contents



Appendix 2.1
Practical continuous formulae for the internal impedance
of cylindrical conductors.

1. Introduction.
2. Choice of fitting functions.
3. Exact expression for Zi.
AC Resistance
4. AC resistance factor.
5. Exact expression for Rac.
6. Thick conductor approximations.
7. Truncated exponential decay.		 8. Variable-order boundary correction.
9. Secondary correction methods.
10. Modified Lorentzian correction.
Internal Inductance
11. Exact expression for internal inductance.
12. Limiting behaviour.
13. Asymptotically correct approximations.
14. Modified Lorentzian correction.

Abstract:
Methods for calculating the internal impedance of round wires are investigated. 'Exact' calculation using Kelvin Bessel functions runs into difficulties at radio frequencies due to the rounding errors inherent in standard computer floating-point arithmetic. Specialist techniques, such as the use of high-precision BCD arithmetic, can overcome this problem; but for general modelling, the use of approximations is indicated. The traditional 'thick-conductor approximation' for AC resistance is inaccurate and has an incorrect boundary condition. A generalised method for producing continuous asymptotically-correct approximations (ACAs) accurate to within a few percent is demonstrated. Suitable choice of ACA allows further correction using modified Lorentzian functions, leading to a family of aproximations. The best of these for AC resistance is accurate to within ±0.16%; and the best for internal inductance is accurate to within ±0.12%.

1. Introduction:
The internal impedance of a conductor is the phasor combination of AC resistance and internal inductance. It can be written:
Zi = Rac + j2pf Li
The AC resistance, Rac, is the same as the DC resistance at low frequencies, but increases with frequency as a consequence of the skin effect. The internal inductance, Li, which is associated with the magnetic field within the conductor, is m/(8p) per metre at low frequencies for all wires and conductors of circular cross-section (50nH/m for non-ferromagnetic materials), but diminishes with frequency as a consequence of the skin effect. Hence internal impedance, when furnished with expressions for Rac and Li which are suitable for practical calculation, accounts for the skin effect in the manner required for the purpose of RF circuit modelling.
     Rac, of course, must be quantified when estimating the efficiency of antennas, transmission lines, coils, transformers, and so on. Its significance is therefore generally appreciated, and its calculation is a matter of some importance. Li is more likely to be neglected; but its inclusion can usefully improve the accuracy of inductance calculations, especially when relatively thin wires are involved. It is therefore unfortunate that the exact calculation of Rac and Li is a matter of some difficulty; a situation which leads to the the use of approximations which can easily become inaccurate or invalid without vigilance on the part of the investigator.
     The exact calculation of the internal impedance of a cylindrical conductor or 'round wire' involves Bessel functions of the first kind, of zero order, with complex arguments (and derivatives thereof). This type of problem (as will be explained below) can be handled using the Kelvin Bessel functions, Ber, Bei, Ber' and Bei'. The Kelvin functions are computed from series expansions of alternating sign, and higher terms involve large powers of the argument divided by large factorials; small increments to the returned value being obtained by subtraction of one very large number from another. This, unfortunately, is exactly the kind of procedure which falls foul of the weaknesses in the standard computer representations for floating-point numbers. For example, in experiments with a spreadsheet program compiled to use the IEEE 64-bit floating-point engine, it was found that no further improvement could be obtained, indeed a degradation of accuracy occurred, for Kelvin function series of more than 9 terms. This had the effect of limiting the calculation of the internal impedance of copper wires up to 30MHz to a maximum wire diameter of about 0.25mm. The situation might be improved by recompiling the program to use an 80-bit floating-point representation; but arbitrary extension of the upper computation limit requires the use of high-precision BCD arithmetic, which is ultimately limitited by issues of execution speed and memory size. Hence the use of the exact expressions can hardly be said to be a straightforward matter.
     The troublesome nature of Bessel functions of large argument has led to the development of polynomial approximations [48]. These can give accuracies of the order of 0.1 parts-per-million; but involve large numbers of empirical parameters and the lead to formulae which cannot be applied continuously. They therefore constitute an over-complicated approach to many radio-related problems, particularly those for which accuracy may be sensibly traded for simplicity in view of the uncertainties of the input parameters.
     In order to push the calculation of internal impedance to high-frequencies (or large diameters) without special programming or undue mathematical complexity; it is common practice to use the so-called 'thick conductor' approximations. The basic thick conductor approximation for internal inductance is easily dispensed, since it amounts to the assumption that there isn't any. The approximation for AC resistance however, is based on the assumption that surface curvature is small and thickness large, so that the solution of Maxwell's equations for field penetration into a plane conductor of infinite thickness prevails. In truth, this approximation can give excellent results when correctly applied; but the most widely accepted formula, which hails from a time when radio components were specified to a tolerance of 20%, is still inaccurate at hundreds of MHz. That the traditional crude approximation is thought by some to be exact is a caveat for those who use computer programs of unknown provenance; but there is still an issue even when a better derivation is used. The problem is that thick conductor approximations behave incorrectly in the limits of low frequency or small wire diameter. Hence they can give wildly inaccurate results in the hands of the inexperienced or unsuspecting user.
     Even for those who use the available computation methods with confidence, the lack of continuous formulae for internal impedance is an inconvenience and encourages mistakes. Hence the work to be described below, which was prompted by a desire to model the RF dispersion behaviour (variation of inductance with frequency) of solenoid inductors wound with thin wire; the required accuracy being anything better than one part in 1000. Since the test coils had a low-frequency internal inductance component of about 1% of the total, the need was merely to obtain a computationally straightforward expression accurate to better than 10%. This target was comfortably exceeded; the outcome being a family of asymptotically correct approximations, using only standard functions, the best of which are accurate to within 0.2%. These formulae, being everywhere better than thick conductor approximations, may be useful to others; and a report on their origins is necessary as background to the solenoid work.

2. Choice of fitting functions:
The method by which the approximations were obtained will be described because it may have relevance in other situations. Particularly, the use of modified Lorentzian functions appears to have an application in correcting empirical formulae in cases where the error is always positive or always negative. Such is the case, or can be forced to be the case, for a range of problems when an asymptotically-correct approximation is chosen as the starting point.
     In the process of developing formulae for AC resistance, the first concern was to find a correction function linking the DC and the high-frequency cases. This problem was solved by means of a useful but physically unrealistic conception, which will be referred-to here as 'truncated exponential decay' (TED). The idea is that, since skin depth is derived from the area under an exponential decay curve extending from the surface to infinite depth, then for conductors of finite thickness, a modified skin depth parameter can be obtained by determining the area under a truncated decay curve. It was found that by truncating the curve at the wire axis (maximum depth equal to the radius), the new skin depth parameter was forced equal to the radius at low frequencies, and the expression for resistance converged with the chosen thick-conductor approximation at high frequencies, i.e., an asymptotically-correct approximation was obtained.
     On producing graphs of the difference between the TED and the 'exact' models, it was noted that the error was concentrated in a single peak of highly skewed appearance when plotted on either linear or logarithmic frequency scales. When plotted on a scale of the fourth-root of frequency however, the error distribution took on an approximately symmetric appearance of roughly Gaussian profile. This was compared against a true Guassian using initially guessed parameters for centre frequency and width (the height already being known). Then, by varying the independent input parameters; d (wire diam.) and r/m (resistivity/permeability), expressions for the centre frequency and the width were found. This first attempt produced a formula accurate to within 0.7%.
     It was noted that a good deal of the error in the TED-G model was due to residual asymmetry. The first issue here is that the actual error function must go to zero as frequency goes to zero, whereas the Gaussian does not. This problem can be resolved by modifying the independent variable (x). Specifically; a true Gaussian, normalised to have a maximum height of 1, has the form:
y = exp[ -w (x -)² ]
where w is the width-determining parameter and is the value of x at the peak. A modified form which is forced to zero when x=0 is:
y = exp{ -w [ (x/) - (/x) ]² }
Using this form did indeed improve the fit, but the asymmetry introduced by correcting the zero-frequency boundary condition was not sufficient to match the skew of the actual error curve. An arbitrary amount of skew can however be introduced by multiplying the width parameter by a dimensionless normalised function of x, i.e., the desired effect can be obtained by allowing the width to vary with x.
     A Gaussian correction function was not used in the final formulae however, because a Modified Lorentzian function was found to give better results. The reason is that the width of a normalised lineshape function can be altered by raising it to an arbitrary power. In the case of a Gaussian, raising to a power has exactly the same effect as changing the width parameter. In the case of a Lorentzian however, it transpires that raising to a power has more effect in the wings of the distribution than it has at the half-height. Hence, using a modified Lorentzian, it becomes possible to tailor the wings and the half-width independently.
     A true Lorentzian, normalised to have a maximum height of 1, has the form:
y = 1/[ 1 + w (x - )² ]
This curve is symmetric about , as is the Gaussian; and can be skewed to establish the boundary condition that y®0 as x®0 in much the same way:
y = 1/{ 1 + w [ (x/) - (/x) ]² }
This form, incidentally, is known to engineers as the electrical resonance curve. Now, raising the whole thing to an arbitrary power p, we obtain a function with two width parameters of differing effect:
y = 1/{ 1 + w [ (x/) - (/x) ]² }
Now we need to introduce variable skew, but before doing so, a discussion of the argument x is in order. It was mentioned earlier, that x was found to be proportional to the fourth-root of frequency. Supplementary to that however, it was found that the peak in the difference curve, for any values of the input parameters, always occurred when
d / di = p
where d is the wire diameter and di is the thick-conductor skin depth, defined as:
di = Ö[ r / (p f m) ]
Hence, d/di is proportional to the square-root of frequency, and so:
x = Ö( d / di )   and    = Öp
Hence:
x / = Ö[ d / ( di p )]
Some multiple or function of d/di is always the argument (input value) for the internal impedance problem; i.e., although there are several input parameters, internal impedance is actually a function of a single argument. Hence we have a single argument for the correction function, normalised (brought to unity when x=) by inclusion of the factor 1/Öp.
     Such a normalised function is also what is required in order to skew the curve, i.e., we need to multiply the width parameter w by x/ raised to some arbitrary power. Thus:
y = 1/{ 1 + w (x/) [ (x/) - (/x) ]² }
If that is the case however, then we can just as well multiply the skewing function into the bracket to its right:
y = 1/{ 1 + w [ (x/)(x/) - (x/)(x/) ]² }
(where b = Öa). Combining the powers we get:
y = 1/{ 1 + w [ (x/) - (x/) ]² }
Finally we note that even greater versatility can be obtained, without affecting the normalisation, by allowing the powers to which the two x/ terms are raised to take on arbitrary values. Also, we introduce a parameter h, which sets the height at the peak. Thus:
y = h /{ 1 + w [ (x/) - (x/) ]² }

ML
This function, in addition to its quasi-independent adjustments for skirt-width and half-width, allows the shapes of the curves on either side of the peak to be adjusted independently. It will be referred to in this document as a modified Lorentzian (ML) function.
     The TED-ML approach can be used to produce good approximations for AC resistance and internal inductance, but a refinement which gives even better results is possible. This involves modifying the power or order of the function used to achieve the initial asymptotically-correct approximation (ACA). Since, after modification, the function no longer represents truncated exponential decay, it will be referred to here as ACAn, where n is the order (and ACA1=TED). By varying n, it is possible to find ACAs which are accurate to within a few percent without further correction. The order also varies the shape of the error function, from bell-shaped curves of positive or negative height, to S-shaped curves inbetween. Adjusting n to obtain an all-positive or all-negative error curve and then applying ML correction, resulted in an approximation for AC resistance accurate to ±0.16%, and an approximation for internal inductance accurate to ±0.12%. A possible disadvantage of ACAn is that it can run into computer rounding-error problems when n is large (but only at frequencies in the region of a few Hz).

3. Exact expression for internal impedance.
The solution of Maxwell's equations for the internal impedance of a round wire of uniform composition is given by various authors. See for example Ramo et al. [9] or McLachlan [47]. Expressed using the Kelvin Bessel functions it is:
Zint = Rac + j2pf Li = 

Rs
(Ö2) p r

Ber(q) + jBei(q)
Ber'(q) + jBei'(q)

[ W / m ]

(3.1)
Where Z, R and L are in italics to indicate that they must be multiplied by the conductor length in order to give quantities in Ohms or Henries. Rs is the surface resistivity defined as:
Rs = Ö(p f m r)     [W]
and r = d/2 is the wire radius. The argument of the Kelvin functions, q, is defined as:
q = (Ö2) r / di = d / [ (Ö2) di ]
where di is the skin depth (the depth at which the current density has dropped to 1/e of its surface value) in the limit of infinite conductor thickness and zero surface curvature. di, as mentioned previously, is given by the expression:
di = Ö[ r / ( p f m )]
r being the volume resistivity, and m the magnetic permeability of the conductor.

The Kelvin functions, Ber and Bei (Bessel real and Bessel imaginary) are evaluated using the series expansions:
Ber(q) = 1 -

(q/2)
2!²

+

(q/2)
4!²

-

(q/2)
6!²

+

(q/2)
8!²

-

(q/2)
10!²

+ . . .
  
and
Bei(q) = (q/2)² -

(q/2)
3!²

+

(q/2)
5!²

-

(q/2)
7!²

+

(q/2)
9!²

-

(q/2)
11!²

+ . . .
  

The functions Ber' and Bei' are the first derivatives of Ber and Bei and may be obtained by differentiating the series term by term. Thus:

Ber'(q) = 1 -

4q
2 2!²

+

8q
2 4!²

-

12q
2 6!²

+

16q
2 8!²

-

20q
2 10!²

+ . . .
  
and
Bei'(q) = q/2 -

6q
2 3!²

+

10q
2 5!²

-

14q
2 7!²

+

18q
2 9!²

-

22q
2²² 11!²

+ . . .
  

The Kelvin functions converge rapidly for arguments (values of q) less than 1, and in that case may be evaluated to good accuracy using only the first three terms. For larger arguments however, convergence is slow and more terms are needed. For the calculations to follow, the functions were evaluated using 9 terms. Transcription was verified by comparing the returned values with McLachlan's tables [47] (which go up to q=10 and can be reproduced using 8-term expansions). As mentioned in the introduction; due to computer rounding errors, no improvement could be obtained by going beyond 9 terms; and at least one of the functions became seriously unstable for arguments greater than 15. Fortunately, the internal impedance calculation is convergent with the thick-conductor approximations used somewhat before q reaches 15; and so in a reversal of roles, the approximation serves to monitor the health of the 'exact' result.

AC Resistance
 4. AC resistance factor.
The resistance of a conductor at low frequencies is given by:
Rdc = r / A     [W]
where is the length and A is the cross-sectional area. For a conductor of circular cross-section, this becomes:
Rdc = r / ( p r² )     [W]
This relationship can also be expressed as resistance per unit length:
Rdc = r / ( p r² )     [ W / m ]
We can envisage the skin effect as reducing the area available for conduction as the frequency increases. Hence we can define AC resistance as:
Rac = r / Aeff
where Aeff, a function of frequency, is the effective area.
     One of the easiest ways in which to include AC resistance in electrical models is to capture it in a dimensionless AC resistance factor, or 'skin-effect factor'. Following Medhurst, the symbol X (Greek Xi) will be used here. Thus:
Rac = Rdc X
and
X = Rac / Rdc = ( r / Aeff ) / ( r / A )
Hence, by definition:
X =

Rac
Rdc

=

A
Aeff
   AC resistance factor
Skin-effect factor
By obtaining expressions for Aeff and X, the various formulae for AC resistance can be compared in a consistent manner.

5. Exact expression for AC resistance.
The exact expression for the AC resistance factor is obtained by rearranging equation (3.1) into a+jb form and identifying Rac as the real part. Thus:

Rac =

r
Aeff

=

Rs
(Ö2) p r

Ber(q) Bei'(q) - Bei(q) Ber'(q)
[Ber'(q)]² + [Bei'(q)]²

[ W / m ]

Where   Rs = Ö(p f m r)   ,   q = (Ö2)r/di   and   di = Ö[ r/( p f m )]
Substituting for Rs gives:

Rac =

r
Aeff

=

Ö(p f m r)
(Ö2) p r

Ber(q) Bei'(q) - Bei(q) Ber'(q)
[Ber'(q)]² + [Bei'(q)]²


Now, by multiplying top and bottom by rr, and collecting parameters to separate 1/di we get:

Rac =

r
p r²

r
(Ö2) di

Ber(q) Bei'(q) - Bei(q) Ber'(q)
[Ber'(q)]² + [Bei'(q)]²

 

r/(pr²)  is of course Rdc, and so:

X =

Rac
Rdc

=

r
(Ö2) di

Ber(q) Bei'(q) - Bei(q) Ber'(q)
[Ber'(q)]² + [Bei'(q)]²

exactly

(5.1)

Alternatively, we can create an expression for Aeff. Before doing that however, it will be convenient to make the following definition:

U(q) =

Ber(q) Bei'(q) - Bei(q) Ber'(q)
[Ber'(q)]² + [Bei'(q)]²

  where q = (Ö2)r/di

Which allows us to write:

X =

A
Aeff

= 

r U(q)
(Ö2) di
   where di = Ö[r/(p f m)]

Hence, noting that A=pr² and r =d/2:

Aeff =

(Ö2) p r di
U(q)

=

p d di
(Ö2) U(q)

[m²]

exactly

(5.2)

6. Thick conductor approximations, good and bad.
In a conductor of infinite thickness and zero curvature, with an electric field parallel to its surface, the current decays exponentially with depth. The skin depth is the hypothetical depth to which the current would penetrate if it flowed uniformly at shallow depths and then ceased abruptly. Its derivation is illustrated below.



I0 is the current density at the surface of the conductor, and the skin depth di is defined such that the area I0di is the same as the area under the decay curve:
I = I0 exp[ -d Ö(p f m / r )]
Where d is the distance from the surface. The area is determined by integration, and the standard integral required in this case is:
òexp(ax)dx = (1/a)exp(ax) + c
Thus:

¥
ò
0
 I0 exp[-d Ö(pfm/r)] dd = I0[-Ö{r/(pfm)}][exp(-¥) - exp(0)] = - I0 [Ö{r/(pfm)}] [0 - 1]  

Hence the area I0 Ö[r/(pfm)] is the same as I0 di; and di, the depth at which the current has fallen to 1/e=0.368 of its surface value is:
di = Ö[r / (p f m)]

The virtue of the abrupt cutoff model is that, presuming that the thickness of the conductor is much greater than di, it allows the effective cross-sectional area to be calculated easily. In the case of a cylindrical conductor, subject to the assumption that the radius is sufficiently large to allow the surface curvature to be ignored; the effective area is simply the total area minus the area of the 'non-conducting' inner region. Referring to the diagram on the right, the radius of the inner region is r-di, and so the effective area is:
Aeff = p r² - p (r-di)² = p ( r² - r² - di² + 2rdi)
Thus:

Aeff = p(2rdi - di²) = p(ddi - di²) [m²] r >> di

(6.1)
and

X =

A
Aeff

=

Rac
Rdc

=

r²
2rdi - di²
   Thick-conductor approximation
(good)

(6.2)

A further approximation is possible by observing that, when di is small relative to r, di² will be extremely small. Hence it is arguable that the di² term can be deleted, giving:

X =

A
Aeff

=

Rac
Rdc

=

r
2 di
   Thick-conductor approximation
(bad)

(6.3)
and

Aeff = 2pr di = p d di [m²] r/di ® ¥

(6.4)

Notice that 2prdi is the area of a rectangle of length equal to the circumference and width di. Thus it is possible to deduce this approximation without the intermediate step of deriving (6.1). This may explain why (6.4) is the preferred approximation in general use; even though (6.1) is vastly superior and only a little more complicated.

The two thick conductor approximations are compared with the exact expression (5.1) in the graph below; which shows Rac/Rdc vs Ö(d/di), the latter being proportional to the fourth-root of frequency. For calculation details see the Open-Document spreadsheet file: Xi_aprox.ods .



The 'good' approximation goes to infinity when d=2r=di , i.e., when Ö(d/di)=1 ; but then converges rapidly with the exact calculation as d/di increases. If the formula is used for Ö(d/di)>Ö2, i.e., r>di, the maximum error is 5.5%. As a practical example; for a 0.5mm diameter copper wire, r=di, at 700KHz.
     Convergence with the exact calculation within 0.1% occurs when d/di>18. Recall that the argument for the Kelvin functions is q=d/(diÖ2), and so the corresponding value for q is 12.7. It was mentioned earlier, that rounding errors start to cause problems when q>15 if standard double-precision arithmetic is used; but there is evidently a useful overlap between the the two calculation methods. Hence, for the best outcome in mathematical modelling, without the use of special program-compilation options, an AC resistance calculation routine can hand over to the good thick-conductor approximation at the point where the Bessel function calculation runs into difficulties.

It is perhaps unfair to refer to (6.3) as the 'bad' thick-conductor approximation, because it was used in some highly respected studies in the early part of the 20th Century. Those studies however, were carried out before engineers and scientists had access to electronic computers, and the correcting effect of the di² term in (6.2) may not have been appreciated. Should the approximation be discovered in the source code of a modern computer program, 'bad' is only one of a range of epithets which might be applied. The importance of the formula nowadays is that it can be used to recalculate the results of early investigations, provided that the raw data are not too heavily buried under the interpretation.
     The bad approximation has an error of -51% when r=di , reducing to -5% when d/di=20. For a copper wire of 0.5mm diameter, it does not converge with the good approximation until well into the UHF range, and the error is -5% at 7MHz. By contemporary standards, the assumption that the di² term in the effective area can be dropped is not valid for ordinary small wires.

7. Truncated exponential decay:
Even if only a rough approximation is required, the danger in using the good thick-conductor approximation is that it explodes in the region of d=di. Hence the first step in improving the formula is to find a modification which gives the correct boundary condition:
Rac®Rdc as d/di ®0
The approach adopted here is to multiply the thick-conductor skin depth di by a function chosen so that the product has a value r when d/di ®0. If we call this modified skin depth di', then equation (6.1) becomes:
Aeff = p( 2 r di' - di'²) = p(d di' - di'²)
and if di' ® r when d/di is small, then:
Aeff ® pr²
A solution can be found by considering the derivation of skin depth, as discussed in section 6. di is related to the area of a rectangle having a dimension in common with, and the same area as, an exponential decay curve. Hence, to a first approximation, the modified skin depth can be related to the area under a decay curve truncated at finite depth. The idea is illustrated below, where the area lost by truncation is subtracted from the area which determines di, resulting in a new skin depth di'.



It is by no means obvious that the truncation should occur at depth r (the wire axis), but proof will be given shortly. For now, observe that the area I0di' must be the same as the area under the decay curve from 0 to r. Thus:

I0 di' =

r
ò
0
 I0 exp(-d/di) dd = I0 (-di) [ exp(-r/di) - 1 ]  

and the modified skin depth is:
di' = di [1 - exp(-r/di) ]

(7.1)

As a first check on this solution, note that as r ®¥, exp(-r/di) ®0 and so di' ®di. The equation converges with the thick-conductor approximation as required. The behaviour as f®0 and di®¥ however, is a little more difficult to determine. If we take the equation at face value, it says that as di®¥, di'®0x¥. i.e., di' is undefined. This is reasonable however, because di' is a function of frequency; and zero frequency is, strictly, physically impossible. We can only discover the true boundary condition by moving away from f=0 by an infinitesimal amount, and that can be done by making use of the series expansion for e:

e = 1 + x +

x²
2!

+

x³
3!

+

x
4!

+

x
5!
  + . . . . . . .  

When |x| is small, the high-order terms vanish, leaving:
e » 1 + x 
Hence, for the present problem, as di®¥ , di' ® di[1-1+r/di]
i.e.,
di' ® r
The modified skin depth is the same as the wire radius at low frequencies. Hence equation (7.1) satisfies all of the necessary boundary conditions, and an asymptotically correct formula for the effective area is:

Aeff = p ( 2 r di' - di'²) = p (d di' - di'²)
where
di' = di [1 - exp(-r/di) ]

TED

(7.2)
For obvious reasons, this is referred to here as the 'truncated exponential decay' (TED) approximation.

It is perhaps worth noting that the function
x' = x[1-exp(-a/x)]
can be used to correct any formula which has the wrong boundary condition for x®¥. We can, for example, also use it to correct the bad thick conductor approximation:
Aeff = 2pr di
In this case, we want Aeff ®pr² when di®¥ , and since
[1-exp(-a/di)] ® a/di when di®¥ , we obtain the required boundary condition when a=r/2. Hence, an asymptotically correct formula based on the bad thick-conductor approximation is:

Aeff = 2pr di" = p d di"
where
di" = di [1 - exp{-r/(2di)} ]

TED2 (bad)

(7.3)

The two TED approximations are plotted for comparison with the exact calculation in the graph below. Also shown dotted are the thick-conductor approximations (TCAs) from which the asymptotically-correct forms were obtained (Spreadsheet calculation: Xi_aprox.ods ).



A clear indication of the goodness of an approximation is given by plotting the difference between it and the exact value, as in the graph below. The curves shown are all obtained from the formula:

Error / % = 100

Xapprox
Xexact

- 1
  




The TED2 approximation (7.3) is, as might have been expected, not a promising basis for further improvement. It has the correct boundary conditions, but only in the sense that it converges with the bad TCA at high frequencies and will eventually converge with the exact calculation at UHF. Except for the crossover point, it fails to approach the zero error line in the range of interest for practical impedance calculations; and it will therefore be extremely difficult to correct.
     The TED approximation obtained from the good TCA however, is remarkably well-behaved. Plotted on a scale proportional to the fourth-root of frequency (as above); the difference curve has the form of a nearly-symmetric lineshape function or a normal distribution, with principal parameters intriguingly related to natural constants; i.e., the peak occurs when:
d/di = p
and
Xted / Xexact = Ö2 = 1.189

8. Variable-order boundary correction.
The low-frequency boundary condition can also be satisfied by a modified version of equation (7.1), specifically:

dni = di [1 - exp{-(r/di)} ]

(8.1)

Now the quantity inside the square bracket goes to (r/di) as di®¥, but taking the nth root restores it to r/di, so that dni®r. This type of correction will be referred to as ACAn (asymptotically-correct approximation of order n). Note that (8.1) is the same as (7.1) when n=1, and so ACA1 is the same as TED.

Shown below is a comparison between the exact calculation, TED, ACA3, and the good TCA on which the other approximations are based (spreadsheet calculation Xi_acan.ods ):




The effect of raising the order (n) is to reduce the influence of the correcting function for higher values of d/di. This causes dni to converge with di much earlier than it otherwise would have done. The quantitative outcome is shown in the set of error curves below:



The graphs show that if the ACA is to be used without further correction, the best integer choice for n is 3. Increasing the order beyond 3 has no effect on the maximum positive error, but starts to increase the maximum negative error. Hence, a continuous formula for Aeff accurate to within 5.5% is:

Aeff = p ( 2 r d3i - d3i²) = p (d d3i - d3i²)
where
d3i = di ³Ö[1 - exp{-(r/di)³} ]

+5.43%
-0.25%

ACA3

(8.2)

9. Secondary correction methods.
For a formula (X say), for which a first approximation X1 exists, and for which the exact solution is also known, we can define an error function:

y(obs) =

X1
X

- 1
  

(9.1)

(where 'obs' stands for 'observed'). Other choices are possible, but this has the virtue that it vanishes when there is no error. Hence an exact solution for X can be written:

X =

X1
1 + y(obs)
    

In some trivial cases, an analytical solution for y(obs) will exist; but in general, an improved second approximation can be obtained by finding a function y which is a good approximation to y(obs). The new formula is then:

X2 =

X1
1 + y
   Divisor correction

Notice that when the error in X1 is positive, then y is positive, and the factor 1/(1+y) is less than 1 (and vice versa). Hence, the effect of the correction is to multiply X1 by a factor which adjusts it in an attempt to reproduce the correct value X. It follows that we can define an alternative correction function y' such that:

X2 = X1 ( 1 - y' ) Multiplier correction

y' will be similar to y, but not identical. There will also be a corresponding y'(obs), obtained from the expression:

X = X1 [ 1 - y'(obs) ]  

Hence:

y'(obs) = 1 -

X
X1
  

(9.2)

Which also vanishes when there is no error.

It is possible to find an expression for y or y' by least-squares fitting to an arbitrary polynomial. Were that the intention however, then there would be little merit in working from an initial approximation; because the investigation might just as well proceed by fitting the original function X to a polynomial. The point in applying sucessive corrections is to avoid brute-force methods; which are at best inelegant, and often result in expressions which cannot be extrapolated and are difficult to transcribe. A least-squares fit moreover, is not what is required; since the intention is to obtain the smallest possible maximum error, rather than to minimise the average error.
     An important consideration also, is to find a correction function of relatively low order; i.e., a function which does not contain terms of the argument raised to a high power and therefore cannot change direction suddenly. The issue here is that the exact function and the approximation can only be compared at a finite number of points while the formula is being developed, but the final formula in use must return a value within stated limits for any argument supplied.
     Hence the fitting method used is essentially graphical; a laborious business in the days before electronic computers, but a task for which spreadsheet programs are eminently suitable. The approach is to obtain various initial approximations, as has been done in the preceding sections; and select one, which is not necessarily the most accurate, but which has an error curve resembling a simple mathematical function. That function then becomes the correcting function (y or y'); and as discussed above, there are two ways in which it can be applied, and the final choice is dictated by whichever gives the best result.
     One additional operational point worth emphasising, is that the appearance of an error function changes according to the scale on which it is plotted. Linear and logarithmic scales are the obvious starting point, but squares, square roots, and other powers of the argument are the next line of attack. Once a promising shape is obtained, the operation performed on the argument in order to obtain an x value is also applied to the argument inserted into the correction function. In that way, the correction takes place in a space suited to the correction process, but is automatically applied in the space in which the final formula is to be used. In other words, if the exact formula and the first approximation have an argument g (say), but the error function has to be plotted on a scale of x = g to achieve a promising form, then the argument of the correcting finction is g, not g.

10. Modified Lorentzian correction:
10a. Correcting the TED approximation.
The error function for the TED approximation, as discussed in section 7, is reminiscent of a normal distribution or a lineshape function when plotted on a scale of x = Ö(d/di). Both the Gaussian and the Lorentzian however are symmetric about the mean, and can take on finite values for x<0; whereas the error function for an asymptotically correct approximation must be nailed to zero at x=0 (and also when x®¥), and the TED error function in particualar is slightly skewed to the 'high frequency' side (in the manner of an electrical resonance curve). Hence modified Gausian and Lorentzian functions, having the correct zero boundary condition and variable skew, are the best candidates as discussed in section 2. A modified Gausian function brought the maximum error to within ±0.7%; but will not be described here because it was superceded by a modified Lorentzian (ML) of the general form:
y = h /{ 1 + w [ (x/) - (x/) ]² }
which brought the maximum error to ±0.28%.

Note that from the graphical investigation of section 7, we already know that, at the peak of the error curve:
Xted / X = Ö2
Hence if we apply a divisor correction:
Xml = Xted / (1+y)
then we need to multiply Xted by 1/Ö2 at the peak in order to make Xml equal to X. Now recall that the maximum height of the ML function is h. Hence
1/(1+h) = 1/Ö2
and so, for divisor correction:
h = Ö2 - 1 = 0.1892
Conversely, if we appliy a multiplier correction:
X'ml = Xted ( 1 - y' )
then
(1-h') = 1/Ö2
and so:
h' = 1 - 1/Ö2 = 0.1591

The condition y=h occurs when x/=1. The peak in the error curve is found to occur when
Ö(d/di) = Öp .
Hence
x/ = Ö[d / (di p) ]
The basic modified Lorentzian, minimally skewed to establish the condition: y=0 when x=0 (i.e., the generic form of the electrical resonance curve) is:
y = h /{ 1 + w [ (x/) - (x/) ]² }
Hence, if we define
z = d / (di p)
then the first candidate for the correction function is:
y = h /{ 1 + w [ z - z ]² }
where h is already known (and depends on how the correction is applied), and w can easily be adjusted by hand to obtain the best fit (starting with w=1).

Although the problem has at least four adjustable parameters (w, p, p1, p2), finding solutions proved to be straightforward using a suitably organised spreadsheet. The technique used involved monitoring two graphs: one being a plot of the correction function superimposed on the error function, the other being a plot of the percentage difference between the new approximation and the exact solution. For a given correcting function, adjustable parameters were placed at the head of the column, rather than embedded in the formula, so that the effect of changing a parameter could be seen by changing a number in a single cell. Movable error-limit bars were also placed on the percentage difference graph, their purpose being to indicate whether an adjustment had made matters better or worse.

Shown below, superimposed on the corresponding error function, is the the best multiplier correcting function found for the TED approximation (spreadsheet calculation: Xi_ted-ml.ods ).



The goodness of the correction is shown in the error graph below. Also shown is the error in the best divisor-corrected formula obtained; but since the multiplier correction turned out to be superior to it, it will not be discussed further except to say that it might have been accepted as the preferred formula had not both correction methods been tried. Note the vertical axis. The curves do not represent chaotic behaviour, but tiny deviations from the exact result.



Notice also that, although the Bessel function calculation can tolerate arguments (q values) up to 15, there is evidence of rounding error problems at the parts per 1000 level for Ö(d/di) > 4.45, which corresponds to q>14. The inability to calcuate the difference correctly for q>15 does not affect confidence in the approximation, because convergence with the exact value is assured by boundary conditions.
     The final error bars are placed at ±0.28%. They could have been brought-in a little closer, but the exact and the approximate calculations are only compared at a finite number of points, and so the peak error excursions may be a little greater than they appear on the graph. Hence ±0.28% is safe and guaranteed, whereas ±0.275% is not.

The final parameterisation of the multiplier correction function was:

y' =

h
( 1 + w[z- z]² )
   h = 1 - 1/(Ö2) = 0.1591
w = 1/(p-1) = 0.4669
z = d/(dip)

A new approximation for X can perfectly well be constructed with the function in this form, but a practical improvement is possible. Since w=1/(p-1), a factor 1/(p-1) can be removed from the denominator and lumped with h. So, noting that:
[1-1/(Ö2)] [p-1] = 1.562754
and
25/16 = 1.5625
are the same to within 2 parts in 10000, we obtain the easily transcribable formula:

y' =

25
16 [ p-1 + (z- z)² ]
   z = d/(dip)

In the final error function, this change makes a difference only in the third decimal place of % (i.e., at the 10 parts-per-million level). Hence AC resistance can be computed using the continuous expression:

X =

Rac
Rdc

= 

r² [1 - 25/{16[ p-1 + (z- z)² ]}]
2rdi[1 - exp(-r/di)] - di²[1 - exp(-r/di)]²

±0.28%

Where: z = 2r/(dip)  and  di = Ö[r/(pfm)]

TED-ML

10b. Correcting the variable-order ACA.
As discussed in section 8, varying the order of the asymptotically-correct approximation can produce a variety of error curves. In some cases the error is always positive, and in others the error varies from negative to positive. For modified Lorentzian correction, an error function which does not cross zero to any significant extent must be chosen.
     It was shown in section 8 that the ACA3 approximation gives the minimum initial error and that there is no advantage in increasing the order beyond 3. ACA3 however crosses the zero axis and is unsuitable for ML correction. Hence, without resorting to arbitrary non-integer orders, corrections were tried for n=2 and n=2.5, and the latter was found to give the best result even though it does make a small excursion below zero. As before, both multiplier and divisor corrections were tried, and multiplier correction proved to be the best by a small margin.

The modified skin depth in the ACA2.5 approximation is defined as:
d2i5 = di [1 - exp{-(r/di)} ]
and the corresponding effective cross-sectional area of the conductor is:
Aeff = p (d d2i5 - d2i5²)
The AC resistance factor is:
Xaca2.5 = A / Aeff
and the error function appropriate for multiplier correction is:
y'(obs) = 1 - X / Xaca2.5

Shown below is the error function y'(obs) with the best determined fitting function y' superimposed (spreadsheet calculation Xi_aca_ml.ods ).



y' =

0.0528
( 1 + 0.91[z - z]² )²
  

and z = d/(3.871 di)

Note that the eror in the starting approximation is much smaller than in the TED case investigated in the previous section, and the outcome is correspondingly better:



The resulting formula for AC resistance is:

X =

Rac
Rdc

=

r² [1 - 0.0528/{[ 1 + 0.91(z - z)² ]²}]
2r d2i5 - d2i5²

±0.16%

ACA2.5-ML

Where:  z = d/(3.871 di) = 31d /(120di) ,  d2i5 = di [1-exp{-(r/di)} ]  
and  di = Ö[r/(pfm)]

Since it is often difficult to establish the major model input parameters (resistivity and wire diameter or radius) to an accuracy sufficient to calculate AC resistance to within ±0.16%, it is probable that the ACA2.5-ML approximation will prove adequate for many applications. An example calculation using the formula is shown below (spreadsheet: Xi_ac2a5ml_tst.ods ). The Bessel function calculation, shown for comparison, goes wrong at about 20MHz in this case, but the new formula works over a range starting at considerably less than 1Hz and extending beyond optical frequencies.




Internal Inductance
11. Exact expression for internal inductance:
The internal inductance of a conductor is related to the energy stored in the internal magnetic field. Since the effective cross-sectional area available for conduction diminishes as the frequency increases, then so too does the internal inductance. Hence internal reactance is related to the skin effect; the decrease in inductance combined with the increase in losses being classic dispersive behaviour. A corollory of this association is that, while losses are still increasing with frequency, inductance must still be diminishing. Hence, contrary to what is often stated; internal inductance, although very small at high frequencies, never actually vanishes.
     To a very good first-order approximation; internal inductance can be included in mathematical models by the simple expedient of adding it to the inductance obtained from external fields. Whether it is worthwile to do so depends on the required accuracy; the inductance of small-signal RF coils, for example, being affected at about the 1% level (within an order of magnitude) at low short-wave frequencies. It can be said however, that the issue of whether or not internal inductance is important can be settled without prevarication by putting it into the model as a matter of policy (especially if the model is part of a general-purpose computer program). This, of course, presumes that the calculation is straightforward.

The exact expression for the internal reactance of a round wire is obtained by rearranging equation (3.1) into a+jb form and identifying 2pfLi (internal reactance per unit length) as the imaginary part. Thus:

2pfLi =

Rs
(Ö2) p r

Ber(q) Ber'(q) + Bei(q) Bei'(q)
[Ber'(q)]² + [Bei'(q)]²

[ W / m ]

(11.1)

where q = (Ö2)r/di   and   Rs = Ö(p f m r)
To make the working less cumbersome, let us define:

W(q) =

Ber(q) Ber'(q) + Bei(q) Bei'(q)
[Ber'(q)]² + [Bei'(q)]²

 

Hence, substituting for Rs in (11.1) and dividing throughout by 2pf:

Li =

[Ö(p f m r)] W(q)
2(Ö2) p² f r
  

[ H / m ]
  

Removing a factor m/2p and recalling that di=Ö[r/(pfm)], gives:

Li =

m
2p

di
r Ö2

W(q)

=

m
2p

W(q)
q

[ H / m ]

exactly

(11.2)

Since the calculation Li uses exactly the same Kelvin functions as AC resistance (albeit in a different combination), the problem of evaluating Li for q>15 is the same as before.

12. Limiting behaviour of internal inductance:
At low frequencies, for a round wire:
Li ® [m/(2p)](1/4)
See, for example, ref.[9]. Note that, for non-magnetic materials, m = m0 =4p x 10 H/m. Hence, all non-magnetic round wires, regardless of diameter, have an inductance of 50nH/m at low frequencies. This might not seem like much, but a small 10mH inductor wound on a low-permeability iron-powder toroid will require about 1m of wire, and so the low-frequency internal inductance contribution will be about 0.5%. For a solenoid of similar inductance, it might take 2 or 3m of wire, and the LF internal inductance contribution will be 1% or more. Hence internal inductance is not important in rough inductance measurements, but it will cause model breakdown errors in more accurate work. Furthermore, for wires thinner than about 0.7mm diameter, the internal inductance will change at its greatest rate in the HF radio region.

At high frequencies:
Li ® [m/(2p)](di/d)
The limiting HF behaviour is for the internal inductance to become proportional to 1/Öf.

13. Asymptotically correct approximations:
The most basic allowance for internal inductance in HF circuit modelling can be made by using the high-frequency limiting case as the formula. Thus: