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Payne : The ac Resistance of Rectangular Conductors (Issue 3)

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THE AC RESISTANCE OF RECTANGULAR CONDUCTORS

A simple equation is given here for the ac resistance of rectangular conductors. This is

shown to give good agreement with independent measurements for all shapes of

rectangular conductor from very thin strip to square section.

1. INTRODUCTION

A rigorous treatment of the ac resistance of conductors with a rectangular section leads to formidable

mathematical difficulties, and many authors have tried to overcome these problems over the past 100 years

but without success. Given that there is no useful theory the curves by Haefner (ref 1) are widely published,

and these are shown below. His paper was written over 80 years ago and so it is surprising that no

analytical analysis has replaced it.

The analytical approach used here is to simplify the problem by assuming that current diffuses from the

four faces independently. This along with experimentally derived factors give a simple equation which is

shown to be surprisingly accurate for all shapes of rectangular conductor from thin strip to square cross-

section.

2. HAEFNER’S MEASUREMENTS

2.1. Published graphs

Haefner measured the ac resistance of five copper conductors having width to thickness (w/t) ratios of 1, 2,

5, 10, and 2400, and over a frequency range from 0 to 8 KHz. Each conductor was around 60 ft in length,

folded back on itself with a spacing of about 2ft which he had checked would have negligible proximity

effect. In addition to his own measurements he also used measurements published by other experimenters.

A summary of his findings is given in the graph below, extracted from Terman (ref 2) :

Figure 2.1 Haefner’s results


2

The other experimenters had used different sized conductors to his and measured them at different

frequencies, and so to combine all these measurements he used the principle of ‘similitude’. This had

previously been enunciated by Dwight who found that if measurements of resistance were plotted against

the parameter p = [8π f /(109 Rdc)]

0.5, the plot can be used to predict the resistance under different conditions

to the measurements. For instance measurements made on large conductors at low frequencies are valid for

small conductors at high frequencies and vice versa.

The parameter p can be rewritten as :

p = A0.5

/ (1.26 δ) 2.1.1

where A is the cross-sectional area in mm2

δ is the skin depth = [ρ/(πfµ)]0.5

in mm (= 66.6/f 0.5

for copper)

It can now be seen that the parameter p is the ratio of two lengths, one related to the size of the conductor

and the other the skin depth. Notice that the above equation includes the permeability, which in Haefner’s

equation is assumed to be unity.

The above curves are a combination of experimental and calculated results, with the dotted portions

representing the extrapolation of low frequency measurements to join some theoretical high frequency

results by Cockroft (ref 3). However his results are unproven because he was unable to find any published

measurements against which to verify his theory, and so these extrapolations must be treated with some

caution. The above graph is widely reproduced but not always with the dotted lines, so that it is then not

clear that much of the graph consists of extrapolations.

Note :

In some versions of the above graph the horizontal axis p is given as (f/Ro)0.5

, where f is in Hz and Ro is in

µΩ/m (for instance the Copper Association ref 4). Multiplying this factor by 1.58 gives p as defined by

Equation 2.1.1. Alternatively p has been given in terms of (f/Ro 1000m)0.5

, where f is in Hz and Ro 1000m is the

dc resistance of 1000 meter length. Dividing this equation by 20 gives p as defined by Equation 2.1.1.

2.2. Extrapolation of Haefner’s measurements

A large proportion of Haefner’s graph consists of extrapolations from measurements, and so it raises the

question as to how accurate these might be. Also further extrapolation may be necessary if his curves are to

used beyond the values given.

An insight into this extrapolation can be gained by considering the round conductor, because the resistance

of this is well understood and predictable. At very high frequencies the conducting area is a narrow band

around the circumference of the wire and of thickness equal to the skin depth δ. The conducting area is

therefore A= (π d δ). The ratio of the high frequency resistance to the low frequency resistance is thus equal

to the ratio of the conducting areas :

Rac/Rdc ≈ (πd2/4)/ (π d δ) = 0.25 d/δ 2.2.1

From Equation 2.1.1, p = d/ (1.48 δ). Thus the limiting value of the slope of Haefner’s round wire curve at

high frequencies should be by this calculation :

Limiting value of [(Rac/Rdc) / p] = 0.37 2.2.2

And indeed this is what it is. However we can anticipate that the curves for the rectangular conductors will

also have this same limiting slope. This will apply as long as the frequency is sufficiently high that ‘the

radius of curvature of the conductor surface is everywhere appreciably greater than the skin depth and does

not vary too rapidly around the periphery’ (Terman ref 2, page 34). Of course this can never be true for a

rectangular conductor with perfectly sharp corners, but a practical conductor will have rounded corners.

Indeed Haefner notes that the samples he measured had ‘… edges slightly rounded, as is the case in the

commercial production of such strip conductors’.


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Indications of this limiting slope can be seen in Figure 2.1 where the slopes of the curves for w/t = 4 and

w/t = 8 are increasing towards that of the round conductor as p increases. The curve for w/t =16 also shows

this trend. The curve for w/t = 24 does not, but beyond p=5 it is linearly extrapolated and this may be

wrong. Indeed the theory presented here shows an increasing slope at this value of p (see comparisons

Section 5), and has a maximum slope of 0.37 as it does for all w/t ratios.

[Cockroft gives limiting slopes depending on the factor p (see ref 1). For instance for the square cross-

section he gives this as 0.418p for p>3.2, whereas Haefner’s curve is 0.37 p. For w/t = 8 Cockroft gives

0.29, whereas Haefner’s curve is 0.34 at p =10, and the slope is clearly still increasing. So it would seem

that Cockroft’s limiting values do not agree with experiment].

3. KEY FACTORS

3.1. Introduction

There are two key factors which determine the ac resistance of rectangular conductors : a) the diffusion of

current into the conductor from the four surfaces, and b) current crowding whereby the current concentrates

at the corners and edges. These are outlined below.

3.2. Diffusion of current

It is well known that at high frequencies the current in a conductor tends to concentrate in a thin layer

around its surface. For instance a conductor with a circular cross-section carries current in only a thin layer

around its circumference, so that its high frequency resistance is the same as that of a hollow tube with a

thickness equal to the skin depth. At all points around this circumference the current has the same density.

Skin effect arises because current diffuses into a conductor from the outside, and so the amplitude is

greatest at the surface and decreases with depth (ref 5):

Figure 3.2.1 Current density at depth

For a very thick plane conductor the current decays exponentially with depth into the conductor. This

decaying current has the same area as that of a current uniformly distributed down to depth of δ, and zero at

greater depths and this leads to the definition of skin depth :

Skin depth δ = [ρ/(πfµ)]0.5

3.2.1

This equation also applies to circular conductors for frequencies where the skin depth is small compared

with the conductor diameter, and to rectangular conductors where the skin depth is small compared with the

thickness. But at lower frequencies where this is not true the current distribution within the conductor will

not be exponential and so the skin depth concept is not valid. As an example the current distribution across

the radius of a cylindrical wire, with a radius equal to 2 δ is shown below :


4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Va

lue

r/r0

Current Distribution in Cylindrical wire

Iz

Exp

Figure 3.2.2 Current Distribution in Cylindrical wire

Also shown is the current distribution if this was exponential, and it can be seen that this greatly

underestimates the actual value of current. The actual current is described by Bessel functions but

unfortunately this leads to serious analytical difficulties, and so the simplification used here is to assume

that current diffuses exponentially but from the four faces in parallel. This clearly increases the current over

the single exponential and fortunately gives a good approximation to the true current.

There is some confusion in the literature over whether the skin effect is due to the penetration of

electromagnetic waves into the conductor or whether it is due to diffusion of current from the surface

(diffusion is defined as the movement of charge from a high concentration to a lower concentration). The

author’s paper reference 6 shows by experiment that diffusion is the true mechanism (see also Appendix 6).

3.3. Current Crowding

In addition to the skin effect, for a rectangular conductor the current concentrates at the edges and the

corners at high frequencies. This effect is known as current crowding and is illustrated below :

Figure 3.3.1 Current crowding in a conductor with a rectangular section


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Figure 3.3.2 Current density in square section conductor

Current crowding is frequency dependent, with none at dc and rising with frequency until it reaches a

limiting value at high frequencies so that the increase in resistance due to current crowding never exceeds a

value of 2.2.

4. THEORY

4.1. Introduction

In this section semi-empirical equations are developed for the increase in resistance due to diffusion and

current crowding. The overall resistance is then expressed as a modification to the dc resistance, Rdc = [ρ ℓ /

(w t )], as :

Rac = [ρ ℓ / (w t’ )] Kc 4.1.1

The thickness t’ is the apparent conductor thickness due to diffusion and Kc is the increase in resistance due

to current crowding.

4.2. Diffusion loss

At high frequencies conduction will take place in only a thin band around the periphery of the conductor,

equivalent in depth to the skin depth δ (ignoring current crowding). This conduction will have therefore an

area AHF =2 δ (w+t), where w is the conductor width and t its thickness. Consider firstly a strip conductor

so the thickness is much smaller than the width then the conducting area becomes AHF ≈ 2 w δ. So the

effective thickness t’ changes from a value of t at low frequencies to 2δ at high frequencies.

Assuming this change with frequency has an exponential relationship gives :

t' = t (1- e

–x ) 4.2.1

where x= 2δ/t

This equation is asymptotic to t at low frequencies, and to 2δ at high frequencies (the latter because ex ≈

1+x for small values of x).


6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

t eff / t

δ/t

Effective Thickness t eff of Strip Conductor

t' / t

2δ

Figure 4.2.1 Effective thickness of Strip conductor

In a thicker conductor diffusion will take place from 4 surfaces. Taking a square conductor and assuming

that the diffusion from the four surfaces acts independently then the limiting value is equal to 2 (2 δ/t). In

general for any w/t ratio the limiting value will be given by 2 δ/t (1+t/w). Equation 4.2.1 now becomes :

t’ = t (1- e –x

) 4.2.2

x = 2 δ/t (1+t/w)

So the overall resistance including diffusion will be :

Rac = [ρ ℓ / (w t’ )] = [ρ ℓ / (w t )] / [1- e –x

] 4.2.3

4.3. Current Crowding

The factor KC gives the increase in resistance due to current crowding and Terman (ref 2) gives a graph of

KC, calculated from Cockcroft (ref 3) for high frequencies. This graph can be described by the following

equation :

KCC ≈ 1.06 + 0.22 Ln w/t + 0.28 (t/w)2 4.3.1

However the author has derived a different equation, again for high frequencies (Appendix 5) :

KCP ≈ 1 + 1.2/ e 2.1 t/w

+ 1.2/ e 2.1 w/t

4.3.2

These two equations are plotted below :


7

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 10 100 1000

Kc

w/t

Resistance Crowding Factor Kc

Cockcroft

Payne

Figure 4.3.1 Current crowding factor KC

In principle the most accurate of these can be determined by experiment. However there are significant

practical difficulties but the author’s own experiments consistently point to Equation 4.3.2 as being the

more accurate (see Appendix 5).

4.4. Variation of Current Crowding with Frequency

The current crowding changes with frequency, with little or no crowding at low frequencies increasing as

the frequency is raised until it reaches the limit of 2.2 set by Equation 4.3.2 at very high frequencies.

Generally this limit is not reached in normal practical applications.

The author has not found any information on the variation of KC with frequency, but clearly it has a value

of unity at very low frequencies. It is therefore useful to express it as KC = (1+x), where the factor x varies

with frequency. So Equation 4.3.2 is conveniently expressed as :

KC = 1 + F(f) [1.2/ e 2.1 t/w

+ 1.2/ e 2.1 w/t

] 4.4.1

The factor F(f) describes the variation with frequency, being unity at very high frequencies and zero at very

low frequencies. It is difficult to determine F(f) theoretically, but we can anticipate that it varies between

these limits exponentially, and the following empirical equation gives a good match to Haefner’s curves

and the author’s measurements :

F(f) = (1- e – 0.026 p

) 4.4.2

where p is given by Equation 2.1.2

NB it was initially thought that the exponent in the above equation would be a function of t/δ ie k t/δ,

however it was then found that k needed to be dependent upon w/t for agreement with Haefner’s curves.

Equation 4.4.1 is plotted below (including Equation 4.4.2) :


8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1 10 100 1000

Kc

Haefner's factor P

Current Crowding Factor Kc Payne

w / t =1000

w / t =50

w / t =16

w / t =4

w / t =2

w / t =1

Figure 4.4.1 Current crowding factor KCP

Current crowding starts to rise when p=1, and does not reach its maximum value until p is greater than 200.

This occurs at frequencies greatly beyond that of a practical application because a) as the frequency is

increased the cross sectional area of the conductor will likely reduce, and b) the factor p increases

relatively slowly with frequency ( as √f, Equation 2.1.1) so that the normal range of p for a practical

conductor is less than 30.

4.5. Interim Equation

So the ac resistance of a rectangular conductor is given by :

Rac / Rdc ≈ KC / (1- e –x

) 4.5.1

where KC = 1 + F(f) [1.2/ e 2.1 t/w

+ 1.2/ e 2.1 w/t

]

F(f) = (1- e – 0.026 p

) where p is given by Equation 2.1.1

x= 2(1+t/w) δ/t

Rdc = [ρ ℓ / (w t )]

4.6. Improved accuracy at small values of w/t

The above equation gives good agreement with Haefner’s curves except for small w/t ratios and small

values of p. For instance for w/t= 2 the above equation gives :


9

0

1

2

3

4

5

0 2 4 6 8 10 12 14

Ra

c/R

dc

Factor p

Rac/ Rdc w/t = 2

Calc Rac /Rdc

Haefner

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 10 20 30 40 50 60 70 80

Rac /

Rd

c

Haefner Factor p

Resistance of Aluminium w/t =1154

Measured Corrected Rac /Rdcw/t =1154

Theoretical Rac/Rdc

Figure 4.6.1 Equation 4.5.1 for w/t = 2

It is seen that the calculated value is too large when p lies between about 1.5 and 5, and the reason is that

the assumption of diffusion from the four faces underestimates the actual value of current. This can be

corrected by adding a term to the value of x which applies at small values of w/t and reduces at high values

of p. Such an equation is given below. :

x= 2(1+t/w) δ/t + 8 (δ/t)3/(w/t) 4.6.1

4.7. Improved accuracy for very wide thin strips

Equation 4.5.1 becomes increasingly inaccurate at w/t ratios above about 100. As an example, the author’s

measurements on an aluminium strip with w/t =1154 is given below along with the prediction :

Figure 4.7.1 Measurements cf Equation 4.5.1

Over the range of p measured above the correlation is very poor, but at higher values of p the correlation is

very good (see Appendix 5). It is probably significant that the poor correlation occurs only when the

conductor thickness is less than the skin depth δ (in the above figure t/δ = 0.35 at p =10), whereas when the

thickness is greater than one skin depth the correlation is much better. So we can conjecture that diffusion

into the narrow edges is inhibited when the skin depth is greater than the thickness, and that diffusion is

then principally from the two wide faces only. Equation 4.5.1 thus has to be modified for this, but it is not


10

0

1

2

3

4

0 2 4 6 8 10 12

Rac

/Rd

c

Factor p

Rac/ Rdc w/t = 1

Calc Rac /Rdc

Haefner

clear whether this modification should be to the crowding factor KC or to the diffusion factor x.

Experiments with an Excel model incorporating Equation 4.5.1 show that getting this to agree with

experiment requires a complicated modification to KC, but only a relatively simple modification to x. On

the principle of Occam’s razor it is this which is chosen here with x, as given by Equation 4.5.1, modified

by an empirical equation as follows :

x = [2 δ/t (1+t/w) + 8 (δ/t)3/(w/t)] / [(w/t)

0.33 e

-3.5 t/δ +1] 4.7.1

4.8. Final Equation

So the ac resistance of a rectangular conductor is given by :

Rac ≈ [ρ ℓ / (w t )] [ KC / (1- e –x

) ] 4.8.1

where ρ is the material resistivity in Ωm (= 1.72 10-8

for copper)

ℓ is the length of the conductor in metres

KC = 1 + F(f) [1.2/ e 2.1 t/w

+ 1.2/ e 2.1 w/t

]

F(f) = (1- e – 0.026 p

) where p is given by Equation 2.1.2

x = [2 δ/t (1+t/w) + 8 (δ/t)3/(w/t)

] / [(w/t)

0.33 e

-3.5 t/δ +1]

w is the width of the conductor in metres

t is its thickness in metres

f is the frequency in Hz

It is useful to normalise this ac resistance to the dc resistance Rdc = ρ ℓ / (wt) to give the ratio Rac / Rdc :

Rac / Rdc ≈ KC / (1- e –x

) 4.8.2

NB the factor 1.2/ e 2.1 w/t

is extremely small for w/t above 3, and indeed at w/t above about 300 Excel is

unable to calculate the value and returns an error message.

5. COMPARISON WITH HAEFNER’S CURVES

5.1. Comparison with theory

The comparison of Equation 4.8.2 with Haefner’s curves is given below :


11

0

1

2

3

4

5

0 2 4 6 8 10 12 14

Ra

c/R

dc

Factor p

Rac/ Rdc w/t = 2

Calc Rac /Rdc

Haefner

0

1

2

3

4

5

0 2 4 6 8 10 12 14

Ra

c/R

dc

Factor p

Rac/ Rdc w/t = 4

Calc Rac /Rdc

Haefner

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2 4 6 8 10 12 14

Ra

c/R

dc

Factor p

Rac/ Rdc w/t = 8

Calc Rac /Rdc

Haefner


12

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14

Ra

c/R

dc

Factor p

Rac/ Rdc w/t =2400

Calc Rac /Rdc

Haefner

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20

Ra

c/R

dc

Factor p

Rac/ Rdc for w/t = 16

Calc Rac /Rdc

Haefner

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14

Ra

c/R

dc

Factor p

Rac/ Rdc w/t =24

Calc Rac /Rdc

Haefner

In the above graph for w/t =24 the slope of Haefner’s curve is less than that of the prediction. However

Haefner’s curve is an extrapolation above p =5 and therefore may not be accurate.


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0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16 18 20 22 24

Resis

tan

ce r

ati

o

Haefner Factor p

Resistance ratio of Copper Strip w/t = 128

Measured Corrected Rac /Rdc

Theoretical Rac/Rdc

The data for the above curve for w/t=2400 comes from Haefner’s original paper.

6. COMPARISON WITH AUTHOR’S MEASUREMENTS

6.1. w/t = 128

The ac resistance of a thin copper strip was measured. It was 7 metres long with a width of 5.1 mm, and a

measured thickness of 0.04 mm. This very small thickness is difficult to measure accurately and the

uncertainty was probably ±10%.

These measurements are compared with Equation 4.8.2 below:

Figure 6.1.1 Author’s measurements on Copper strip with w/t =128

Correlation with the theoretical curve is poor. However measurements of the dc resistance of the copper

strip showed that its resistivity was 2.75 10-8

Ωm compared with that of pure copper of 1.72 10-8

Ωm. This

indicated that it contained impurities and these increased its permeability above unity as a test showed (see

Appendix 3). Nickle is the most likely impurity because it is often added to improve the corrosion

resistance, and the percentage needed to explain the raised resistivity was calculated to be 1.07%, raising

the permeability to 2.2 (Appendix 4). If the skin depth in Equation 2.1.1 is calculated with this value of

permeability the Haefner factor p changes for each of the measurement frequencies and the correlation with

the theoretical improves to :


14

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16

Rac /

Rd

c

Haefner Factor p

Resistance of Thin Flat Bronze Strip w/t = 14

Measured Corrected Rac /Rdc w/t = 14

Theoretical Rac/Rdc

Haefner w/t=16

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16 18 20 22 24

Resis

tan

ce r

ati

o

Haefner Factor p

Resistance ratio of Copper Strip w/t = 128

Measured Corrected Rac/Rdc

Theoretical Rac/Rdc

Figure 6.1.2 Comparison with µr = 2.2

The measurements are subject to considerable error because the resistance was less than 2Ω, so the

correlation is probably fortuitous. However there is definitely a different slope at the larger values of p and

this is also apparent in the comparison with Haefner’s curve for w/t=24, so this may indicate a limitation of

the theory.

6.2. w/t =14

Measurements were made of a bronze strip of width 2.46 mm and thickness of 0.18 mm, so having w/t =

14. The resistivity was determined as ρ = 12 10-8

by measuring the dc resistance of a 3 meter length.

Appendix 1 gives the experimental details. The metal was found to be very slightly magnetic (Appendix 3),

and so the skin depth was calculated on the assumption that µr =1.1.

These measurements are compared with Equation 4.8.2 below :

Figure 6.2.1 Comparison between author’s measurement and theory for w/t = 14

The resistance values were very low at less than 2Ω over most of the frequency range from 0.01 to 25 MHz,

and the measurement error is assessed as ±10%.

Self–resonance determined the maximum frequency which could be measured (see Appendix 2) and this

frequency is dependent upon the length of the conductor. So a 6 metre length was used for measurements

between 0.01 and 0.5 MHz, and a 1.5 metre length for 0.5 to 20 MHz.


15

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10 11 12

Ra

c /

Rd

c

Haefner Factor p

Resistance of Copper Strip w/t = 1

Measured Corrected Rac /Rdc w/t = 1

Theoretical Rac/Rdc

Haefner w/t=1

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 10 20 30 40 50 60

Rac /

Rd

c

Haefner Factor p

Resistance of Aluminium w/t =1150

Measured Corrected Rac /Rdcw/t =1150

Theoretical Rac/Rdc

6.3. w/t = 1

Also measured was a square bronze conductor of sides 0.69 mm, with two lengths of 6 m and 1.5 m. The

resistivity was determined as ρ = 16 10-8

by measuring the dc resistance of the 6 meter length.

These measurements are compared with Equation 4.8.2 below :

Figure 6.3.1 Comparison between author’s measurement and theory for w/t=1

Measurement error was somewhat less here because the resistance was higher at between 2 and 8 Ω. The

measurements are seen to lie very close to both the Haefner curve and the theoretical curve, and so the

marker points have been made especially large to make them visible.

6.4. w/t = 1150

Confirmation of Equation 4.8.2 for wide conductors is provided by the comparison with Haefner’s

measurements for w/t = 2400 (Section 5). Further confirmation is provided by the measurements shown

below on an aluminium conductor having w/t = 1150. The conductor was standard aluminium baking foil

cut to a length of 3.06 m and width 15 mm. The thickness was determined to be 0.013 mm by measuring its

dc resistance and assuming ρ = 2.65 10-8

. The permeability was shown to be unity by the test described in

Appendix 3.

Figure 6.4.1 Comparison between author’s measurement and theory for w/t=1150


16

7. SUMMARY AND CONCLUSIONS

It has been shown that Equation 4.8.2 gives a very good prediction of the resistance of rectangular

conductors, from thin strip to square cross-section and over a frequency range from dc up to high

frequencies.


17

Appendix 1 : VNA CALIBRATION

All resistance measurements were made with an Array Solutions UHF Vector Network Analyser.

Calibration of this analyser required an open circuit, a short circuit and known resistive load, and these are

shown below.

Figure A 1.1 Calibration loads

To ensure that the calibration resistance had minimal stray reactance a thick-film resistor was used (above),

and this had the added advantage that it could be located in the same plane as the short circuit. Its value was

47 Ω ± 1%. SMA connectors were used because they are small and therefore have a small stray

capacitance, and so any error in calibrating this out would also be small.

Appendix 2 : SELF-RESONANCE

The wire to be measured needed to be folded back on itself for connection to the test equipment. It

therefore constituted a two-wire transmission-line and this will resonate when its folded length is equal to

nλ/4, where λ is the wavelength. Thus the first resonant frequency is when

fR = 300/ (2 ℓ ) A 2.1

where ℓ is the total length of the wire (ie twice the length of the line).

As this frequency is approached the measured resistance and the inductance increase above their true value

and Welsby ( ref 11, p 37) has shown that the true value is given by :

L = LM [ 1- (f / fR )2] A2.2

R = RM [ 1- (f / fR )2]2 A2.3

LM and RM are the measured values, and fR is the self-resonant frequency. Welsby developed these

equations for the self-resonance in coils and they are less accurate for a transmission-line and become

increasingly inaccurate as the resonant frequency is approached.

In practice it is better to measure the self-resonant frequency rather than calculate it from Equation A2.1

because a very small stray capacitance in the measurement jig will reduce the SRF considerably. The

reason for this high sensitivity to stray capacitance is that the characteristic impedance of the line will be


18

very high because there has to be a large spacing between the two sides of the folded conductor to minimise

the proximity loss.

So in practice it is necessary to measure the SRF, and this has to be in the measurement jig. The SRF is

defined here as the frequency where the transfer impedance goes through a zero phase angle.

Appendix 3 : TEST FOR PERMEABILITY

To test the copper strip for permeability a short length was placed onto the surface of water contained in a

glass cup, supported by surface tension. A permanent magnet was brought close and the copper strip drifted

very slowly towards it taking around 5 minutes to traverse the diameter of the glass. This showed that the

permeability of the strip was greater than unity but the slow movement suggests that it was probably not

much greater than unity, and indeed the permeability was not so high that conductor would ‘stick’ to a

powerful permanent magnet.

Figure A3.1 Permeability test

Appendix 4 : COPPER IMPURITIES

Trace elements will increase the resistivity of copper. Reference 9 (figure 1) gives curves for the resistivity

increase for the various impurities, and these curves can be represented by the following equation:

ρt = [1.68 + C1 (ΔR1) + C2 (ΔR2) + C3 (ΔR3) ……..] 10-8

A4.1

where 1.68 is the resistivity of pure copper

C1 is the concentration of first impurity in %

ΔR1 is the slope of the curve for the first impurity

etc

From the curves ref 9 :

ΔR for P = 14.3

ΔR for Fe = 9.5

ΔR for Si = 6.3

ΔR for As = 5.45


19

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 10 100 1000

Kc

w/t

Resistance Crowding Factor Kc

Cockcroft

Payne

ΔR for Cr = 4.2

ΔR for Mn = 2.8

ΔR for Ni = 0.7

The resistance of the metal strip was measured, and gave the resistivity as 2.75 10-8

. Assuming the main

impurity was nickel the concentration of this will have been (from the above equation) 1.1%.

The permeability will be increased by the presence of the nickel according to the following equation,

assuming the relative permeability of pure copper to be unity :

µRA = 1 + (µR1 -1) C1 + (µR2 -1) C2 + .............. A4.2

where µR1 and C1 are the permeability and concentration of impurity 1

µR2 and C2 are the permeability and concentration of impurity 2

etc

Assuming the only impurity is nickel and this has a concentration of 1.1% (from the resistivity

measurements above) and a permeability of 110, the above equation gives :

µRT = 1 + (110-1) * 1.1/100 = 2.2 A4.3

Appendix 5 : CURRENT CROWDING

A5.1. Crowding Equations

Terman (ref 2) gives a graph of KC, calculated from Cockcroft (ref 3) for high frequencies. This graph can

be described by the following equation :

KCC = 1.06 + 0.22 Ln w/t + 0.28 (t/w)2 A5.1.1

However the author has derived a different equation (see Payne ref 5):

KCP = 1 + 1.2/ e 2.1 t/w

+ 1.2/ e 2.1 w/t

A5.1.2

These two equations are plotted below :

Figure A5.1.1 Current crowding factor KC


20

In this appendix an experiment is described which aims to determine which of these two equations is the

most accurate.

The above crowding values are only achieved at very high frequencies where the skin depth is small

compared with the dimensions of the conductor. At these high frequencies the error due to self-resonance is

very large (Appendix 2) and this can only be reduced by ensuring that the SRF is much higher than the

measurement frequency and this requires a very short length of conductor. This in turn means that the

resistance values to be measured are very small (milliohms) making measurement very difficult. A practical

experiment must therefore compromise and use a lower frequency than would be ideal, and at these lower

frequencies the above two equations are assumed to be (see Section 4.4) :

Payne : KCP = 1 + F(fP) [1.2/ e 2.1 t/w

+ 1.2/ e 2.1 w/t

] A5.1.3

where F(fp) = (1- e – 0.022 p

)

Cockcroft : KCC = 1+ F(fc) [0.06 + 0.22 Ln w/t + 0.28 (t/w)2] A5.1.4

where F(fc) = (1- e – 0.048 p

)

The factors F(fP) and F(fc) describe the change in the crowding with frequency. They are empirical equations

and their exponents have been optimised in each case to give the best correlation with Haefner’s

measurements, and surprisingly the above two equations give very similar results. However Haefner only

measured up to p = 10 and the equations are then dominated by F(fP) or F(fc), with F(fp) equal to only 0.2 and

F(fc) equal to only 0.38 at this value of p. In contrast the experiments here should aim to have F(fP) and F(fc)

close to unity, and this would require p to be in excess of 100 (see figure 4.4.1), but this would have

required a very high frequency (>400 MHz), and a very short conductor resulting in very low resistance.

The compromise chosen here was to measure at 108 and 175 MHz, where p is equal to 38 and 48 for the

conductor described below, and F(fP) is equal to 0.61 and 0.84, and F(fc) is equal to 0.7 and 0.9.

A5.2. Conductor

The conductor must have a w/t ratio of around 10, or alternatively greater than 1000 if there is to be a

significant difference in the two predictions (see above figure). For a ratio around 10 the conductor would

have to have a very small width if its resistance was to be high enough to measure, but accurately cutting

such a width was not possible with the tools available to the author. So it was decided to investigate a

conductor with w/t> 1000.

This required a very thin conductor and copper foil was readily available on e-bay with a purity of 99.95%

and a thickness of 0.009m, as determined by measuring the resistance of a 180 mm strip 10 mm wide. So

w/t =1100.

A5.3. Measurement Jig

At the measurement frequencies of 108 and 175 MHz the strip resistance is around 0.07 Ω and this is too

small for normal single port VNA impedance measurements (ie S11). However there is an alternative

technique and this is described in reference 10. This is a two port measurement and is illustrated below :


21

Figure A5.3.1 Two Port measurement of very low impedances

Here the two ports of the VNA are connected together and the DUT is shunted across the connecting cable,

thereby reducing the signal into the second port. For a 50 Ω system the value of the impedance ZDUT is

given by :

ZDUT = 25 S21/(1-S21) ohms A5.3.1

A special jig was made using two SMA connectors (end-launch pcb) back to back, as shown in the next two

photos :


22

Figure A5.3.2 Measurement jig

The reactance of the strip conductor was tuned-out with a capacitor. This needed to have an adjustable

value of up to 25 pf, and to have a very low series resistance (ESR) because the resistance to be measured

was only around 0.07 Ω. It also needed to be as small as possible in order to minimize stray capacitance.

The capacitor used is of unknown origin (see photo) but seems to be similar to the Johanson 9374. The

copper strip is soldered to the top terminal of this capacitor and to the ground terminal of the SMA

connectors via short lengths of copper wire (see later). The loss of this capacitor needed to be measured

(along with any other jig loss) and this is discussed later.

Also shown in the upper photo is the copper wire and two launchers back-to-back used in the calibration. It

was important that the capacitor value was not changed between the two measurements of wire loop and

strip so that the capacitor loss would be unchanged. This required that the the calibration loop needed to

provide the same resonant frequency as the strip.

A5.4. Launchers

The strip to be measured is connected to the jig by soldering a short length of copper wire (0.5 mm dia x 7

mm long) to each end. There is now an unknown resistance due the connection wire and to the current

spreading out from the contact point with the wire and into the wide strip. This is called here the ‘launcher

resistance’, and the value of this resistance was determined by measuring the resistance of two launchers

back to back, each launcher consisting of a length of copper strip and a copper wire connection, and joined

strip to strip.

A5.5. Measurement Procedure

Two measurements are taken at each frequency. The first determines the combined resistance of the

capacitor, the launchers ,and any other jig resistance. To do this the capacitor reactance must be tuned-out

and this is done using a loop of copper magnet wire (shown in photo above). This has the advantage that its

resistance can be accurately calculated and then subtracted from the overall resistance. The length of this

wire, along with the length of the launchers was chosen to be similar to that of the strip to be measured

since the self-resonant frequencies (SRF) would then be similar reducing the errors (see later for discussion

of SRF). The diameter of the wire was chosen to give a resistance similar to that of the strip to reduce any

errors associated with different signal levels.

For the second measurement the wire loop and the launchers were replaced by the strip to be measured.

In analyzing the results, three other factors need to be taken into account: Proximity resistance, Self

Resonant Frequency and Radiation resistance , and these are discussed below.

A5.6. Proximity resistance


23

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Re

sis

tan

ce

R

ati

o

s /d

Resistance Ratio Opposing Currents

R/Ro Moullin

A5.6.1. Wire Loop proximity resistance

The copper wire used for calibration has to be bent into a hairpin loop and its resistance will increase due to

proximity of its two halves, and this increase is given by the following equation (see Payne ref 5) :

R / Ro = 1/ [1- (d /s)2 ]

0.5 A5.6.1

where d is the diameter of the wire

s is the spacing, centre to centre.

This equation is for high frequencies where the skin depth is small compared with the wire diameter. The

above equation is plotted below :

Figure A5.6.1 Proximity Effect in circular conductors

For the wire loop shown in Figure A5.3.2 (there shown with launcher attached) the wire diameter was 2.02

mm (measured) and the spacing averaged 14 mm so that the proximity increase was 1.01.

A5.6.2. Copper strip proximity resistance

The copper strip is also folded back on itself and so will also suffer an increased resistance due to

proximity. Unfortunately no theory has been found for this loss nor any measurements, but it seems

reasonable to assume that the loss will be less than that of a circular conductor having a diameter equal to

the maximum dimension of the strip (ie its width w). Then for a spacing of 3 times the conductor width (ie

w/s =3) the resistance ratio is less than 1.061, and for w/s = 4 is less than 1.033 (from Equation A5.6.1,

assuming w/s = d/s).

A5.7. Self –Resonant Frequency SRF

See Appendix 2.

A5.8. Radiation Resistance

There will be a small amount of radiation from the loops, and this loss will produce an equivalent resistance

in series with loop given by :

Rr = 31200 (A/λ2)

2 A5.8.1

where A is the area of the loop in m2

λ is the wavelength in m


24

As an example the 180 mm long strip was formed into a loop with dimensions of about 70 mm x 20 mm

and at 175 MHz this has a radiation resistance of 0.007 Ω. Given that the resistance of the strip was 0.073

Ω at this frequency the radiation resistance contributed about 10 % and so cannot be ignored.

A5.9. Analysis of Measurements

As described above there are a number of factors to be taken into account in the analysis of the

measurements and these are analysed in this section.

A diagram of the measurement set-up is shown below :

Figure A5.9.1 Diagram of measurement apparatus

RO : the combined resistance of the capacitor, the back-to-back launchers and the jig

S1 : the increase in resistance due to the SRF

S2 : the increase in resistance due to the SRF when the strip is in place

RC : the measured resistance when the wire and launchers are in place

RM : the measured resistance when the strip is in place

RW : the resistance of the wire, with no increase for SRF or proximity

RS : the resistance of the strip, with no increase for SRF or proximity

Initially ignoring proximity loss and radiation resistance :

RC = S1 RO +S1 RW A5.9.1

RM = S2 RO +S2 RS A5.9.2

From A5.9.1

RO = (RC - S1 RW )/ S1

= RC / S1 - RW A5.9.3

Subs into A5.9.2

RM = (S2 / S1 ) RC - S2 RW + S2 RS A5.9.4

So :

RS = (RM - (S2 / S1 ) RC + S2 RW)/ S2

= RM / S2 - RC / S1 + RW A5.9.5

(check : if S2 = S1 and RM = RC , then RS = RW. QED)

If the proximity loss ratio for the strip and wire are PS, and PW respectively (Equation A5.6.1 ), and the

radiation resistance is RRS and RRW, then the above equation becomes :


25

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 20 40 60

Rac /

Rd

c

Haefner Factor p

Measured Resistance of Thin Copper strip w/t = 1100

Theoretical Payne Rac/Rdc

Measured R (40 mm launchers)

Measured R (20 mm launchers)

Theoretical Cockcroft

RS = [(RM / S2 - RRS) - (RC / S1 - RRW ) + RW PW]/ PS A5.9.6

where RRS and RRW are given by Equation A5.8.1

PW is given by Equation A5.6.1

PS was assumed to be 1.06 (see Section A5.6.2)

NB in applying the proximity loss to Equation A5.9.5, note that for the wire it is its actual resistance which

is required, including proximity, RW PW, whereas for the strip its resistance without the effects of proximity

is required.

A5.10. Measurements

The measured values are given below, along with the modeled values assuming either Cockcroft’s

crowding factor or Payne’s (Equations A5.1.3 or A5.1.4) :

Figure A5.10.1 Measured resistance values

The rectangular conductor is connected to the test jig via a short wire at each end. There is therefore a

transition region as the current spreads out from each wire to flow across the whole width of the strip. It is

not known how far down the strip this transition occurs but was assumed initially to be equal to one strip

width, so the two launchers back to back had a length of 20 mm (for the strip width of 10 mm). The

measured resistance of the conductor assuming this are shown above in dark blue. As a check, launchers of

twice this length were also measured and the results are given above in light blue, with no significant

difference. So it would appear that the assumption of current spreading within one strip width was

reasonable.

The measurements are seen to give a very good correlation with Payne’s crowding factor, and a poor

correlation with Cockcroft’s.

During the measurements it was necessary to unsolder the wire loop (with launchers) and to solder the

copper strip in their place. It was found that immediately after soldering the loss was significantly higher,

and that a cooling period of at least 10 minutes was needed before measurements were taken.


26

The following are the values used in Equation A5.9.6 at a frequency of 175 MHz, for the measurement at

p= 48.3 in the above curve:

RM = 0.913Ω

S2 = 2.01

RRS = 0.007Ω

RC = 0.952 Ω

S1 = 2.07

RRW = 0.0085Ω

RW = 0.081Ω

PW = 1.01

PS = 1.06 (a guess, see Section A5.6.2)

RS = 0.0688 Ω (calculated from Equation A5.9.6)

A5.11. Discussion

The measurements above give strong support to Payne’s equation for current crowding (Equation A5.1.3).

However to be highly critical, the experiment does not necessarily confirm Equation A5.1.1 since it

includes the factor F(fP) and this is empirical. Also the experiment does not entirely rule-out Equation 5.1.4,

since this also contains a similar empirical factor F(fc). However for Equation 5.1.4 to agree with experiment

AND with Haefner’s curves, this factor would need to be much more complicated than the simple

exponential. On the basis of Occam’s razor this is discounted.

Appendix 6 DIFFUSION

Diffusion of current into a conductor is a very slow process, and for instance in copper at 1MHz the phase

velocity is about 415 metres per second, or about a millionth of the velocity of EM waves in air (ref 7,

p540). A current at 1 MHz thus has only 0.5 µs to diffuse into the copper before the excitation reverses

direction and in this time the current penetrates into the copper by only 0.2 mm.

So this raises the question : if diffusion is so slow, how is energy transmitted along a conductor at a

velocity close to the speed of light? This has been answered recently by Edwards & Saha (ref 8) who show

that ‘Currents are established on the surface of conductors by the propagation of electromagnetic waves in

the insulating material between them’ (my italics). This sets-up an excitation potential which travels along

the wire at the speed of light. ‘If it were not for the displacement current setting up the surface currents in

the first instance ……. energy transmission via copper conductors would be virtually impossible because of

the long diffusion process’.


27

REFERENCES

1. HAEFNER S J : ‘Alternating-Current Resistance of Rectangular Conductors’, Proceedings of the

Institute of Radio Engineers, Volume 25, No 4, April 1937.

2. TERMAN E T : ‘Radio Engineers Handbook’ McGraw-Hill Book Co, 1943

3. COCKCROFT J D : ‘Skin Effect in Rectangular Conductors at High Frequencies’, Proceedings

of the Royal Society, Vol 122, February 1929.

4. COPPER DEVELOPMENT ASSOCIATION :

http://copperalliance.org.uk/docs/librariesprovider5/pub-22-copper-for-busbars/copper-for-

busbars-all-sections.pdf?sfvrsn=2

5. PAYNE A N : ‘Skin Effect, Proximity Effect and the Resistance of Conductors’,

http://g3rbj.co.uk/

6. PAYNE A N : ‘Skin Effect : Electromagnetic Wave or Diffusion ?’, http://g3rbj.co.uk/

7. LORRAIN P, CORSON D P, LORRAIN F : ‘Electromagnetic Fields and Waves’, 3rd

Edition, W

H Freeman & Co, New York, 1988.

8. EDWARDS J, SAHA T K : ‘Diffusion of Current into Conductors’,

https://espace.library.uq.edu.au/view/UQ:9820/aupec-03-6.pdf

9. COPPER DEVELOPMENT ASSOCIATION :

http://www.copper.org/publications/newsletters/innovations/1997/12/wiremetallurgy.html

10. Agilent Application Note : Ultra Low Impedance Measurements using 2 Port Measurements

http://cp.literature.agilent.com/litweb/pdf/5989-5935EN.pdf

11. WELSBY V G : The Theory and Design of Inductance Coils’, Macdonald, London, 1960

Issue 2, April 2017 : Replacement of Cockroft’s current crowding by Payne’s equation. Equations given for

greater accuracy at small w/t and small p, and for very large w/t. Sections 3 and 4 re-written for clarity.

Issue 3 June 2017 : Error in Equations 4.6.1, 4.7.1 and 4.8.1

© Alan Payne 2017

Alan Payne asserts the right to be recognized as the author of this work.

Enquiries to [email protected]

http://copperalliance.org.uk/docs/librariesprovider5/pub-22-copper-for-busbars/copper-for-busbars-all-sections.pdf?sfvrsn=2

http://copperalliance.org.uk/docs/librariesprovider5/pub-22-copper-for-busbars/copper-for-busbars-all-sections.pdf?sfvrsn=2

http://g3rbj.co.uk/

http://g3rbj.co.uk/

https://espace.library.uq.edu.au/view/UQ:9820/aupec-03-6.pdf

http://www.copper.org/publications/newsletters/innovations/1997/12/wiremetallurgy.html

http://cp.literature.agilent.com/litweb/pdf/5989-5935EN.pdf

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