The ac resistance of conductors

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In a previous article I looked at the dc resistance of conductors and in this article we turn our attention to ac resistance. If you have not read the previous article, now may be a good time to do so. The ac resistance of a conductor is always larger than the dc resistance. The primary reasons for this are 'skin effect' and 'proximity effect', both of which are discussed in more detail below.

In a previous article I looked at the dc resistance of conductors and in this article we turn our attention to ac resistance.  If you have not read the previous article, now may be a good time to do so.

The ac resistance of a conductor is always larger than the dc resistance.  The primary reasons for this are 'skin effect' and 'proximity effect', both of which are discussed in more detail below.

There are various methods to take into account these effects, but I'll concentrate on those given in IEC 60287 (Electrical cables - calculation of the current rating). In addition to being widely accepted, the method is also part of a standard which gives it some legitimacy.

The standard take skin and proximity effects into account with the following formulae:

Where:

R = the ac resistance of the conductor
R' = the dc resistance of the conductor
ys = a skin effect factor
yp = a proximity effect factor

While the above formulae is pretty straight forward, working out the skin and proximity effect factors is a little more involved, but still not too difficult.

Skin Effect

As the frequency of current increases, the flow of electricity tends to become more concentrated around the outside of a conductor. At very high frequencies, often hollow conductors are used primarily for this reason. At power frequencies (typically 50 or 60 Hz), while less pronounced the change in resistance due to skin effect is still noticeable.

The skin effect factor ys is given by:

where:

f    = supply frequency in Hertz
ks  = skin effect coefficient from the table below

Proximity Effect

Proximity effect is associated with the magnetic fields of conductors which are close together. The distribution of the magnetic field is not even, but depends on the physical arrangement of the conductors.  With the flux cutting the conductors not being even, this forces the current distribution throughout the conduit to be uneven and alters the resistance.

The formulae for the proximity effect factor differs dependant on wether we are talking about two or three cores.

$γ p = X p 4 192+0.8 X p 4 ( d c S ) 2 ×2.9$

- two core cables or two single core cables

$γ p = X p 4 192+0.8 X p 4 ( d c S ) 2 [ 0312 ( d c S ) 2 + 1.18 X p 4 192+0.8 X p 4 +0.27 ]$

- for three core cables or three single core cables

Where (for both cases):

$X p 8πf R ′ 10 −7 k p$

dc   = diameter of the conductor (mm)
s    = distance between conductor axis (mm)
kp  = proximity effect coefficient from the table below

Note:
1. for three single core with uneven spacing s = √(s1 x s2)
2. for shaped conductors, yp is two thirds the value calculated above, with
dc = dx = diameter of equivalent circular conductor of same cross sectional area (mm)
s = (dx + t), where t is the thickness of insulation between conductors (mm)

Coefficients ks and kp

ks  kp
Copper Round stranded or solid 1 1
Round segmental 0.435 0.37
Sector-shaped 1 1
Aluminium Round stranded or solid 1 1
Round 4 segment 0.28 0.37
Round 5 segment 0.19 0.37
Round 6 segment 0.12 0.37

Summary

Should you need to calculate the resistance of a conductor, I hope this article (and the accompanying dc article) can help.  In summary the procedure is fairly straight forward:

1. Look up the dc resistance at 20 0C from the IEC 60287 table, see first article
2. Adjust for operating temperature as show in the first article

<- The dc resistance of conductors

More interesting Notes:

Steven has over twenty five years experience working on some of the largest construction projects. He has a deep technical understanding of electrical engineering and is keen to share this knowledge. About the author

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1. Chris says:
7/6/2012 11:44 PM

I can't help but notice that the cause of the proximity effect is very similar to cause of inductive reactance for a coil. When you're working out calculations for a coil, especially one for high frequencies, is the proximity effect factor independent of the inductive reactance, or is it a part of the reactance?

• Steven says:
7/8/2012 8:46 AM

If I remember the reasoning correctly – if the two conductors are close, the magnetic field from one will induce eddy currents in the other. These eddy currents squash the normal current into a smaller area, thereby by increasing the resistance. The magnetic interaction producing the eddy currents is due to mutual inductance and could be the reason the equations look familiar to you

• Chris says:
7/8/2012 8:16 PM

Ok, thanks. Now I understand it better.

2. John says:
8/25/2012 3:05 AM

Is there any specific instructions about the equations of the skin effect and proxity effect, I have seen these equations many times ,but i don't konw its source.

• Steven says:
8/26/2012 11:49 AM

The equations I have quoted are from IEC 60287 - Electrical cables - calculation of the current rating.

You can also find a few books and papers on the subject if you search around. If I remember correctly (have lost my copy), a book by S.Y.King and N.Halfer, titled 'Underground Power Cables' has some discussion on this.

3. Notes says:
2/1/2013 10:30 AM

IEC 60287 "Calculation of the continuous current rating of cables (100% load factor)" is the International Standard which defines the procedures and equations to be used in determining the current carry capacity of cable. The standard is applicable... ...

4. mani says:
5/23/2013 4:56 PM

4mm wire and 0.5 meter length and both are close to 2mm
after some frequency the Rac became canstant why ?

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