The ac resistance of conductors 

By on

Power Cable
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:

cffcdbb291a5c84522e4f4202e86f371

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:

3de47c07bc69acf6c724a5efb6350fc3

where:

e95f68b1053c74300dbd04ca54afe04b

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
  3. Adjust for skin and proximity effects as shown in this article

If anyone has any comments or anything to add, please do so below.

<- The dc resistance of conductors



Steven McFadyen's avatar Steven McFadyen

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

myElectrical Engineering

comments powered by Disqus

  1. Chris's avatar 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's avatar 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's avatar Chris says:
      7/8/2012 8:16 PM

      Ok, thanks. Now I understand it better.

  2. John's avatar 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's avatar 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's avatar Notes says:
    2/1/2013 10:30 AM

    Trackback from Notes

    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's avatar 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 ?


Comments are closed for this post:
  • have a question or need help, please use our Questions Section
  • spotted an error or have additional info that you think should be in this post, feel free to Contact Us



Difference Between Live and Dead Tank Circuit Breakers

A quick post in connection with an email question: Live Tank - the circuit breaker the switching unit is located in an insulator bushing which is live...

Operational Amplifier

The fundamental component of any analogue computer is the operational amplifier, or op amp. An operational amplifier (often called an op-amp,) is a high...

Robotics - Home Innovations

We have a sister note to this (Robots - Interesting Video), in which I have posted some videos of interesting robots developed by commercial corporations...

How to Check a Circuit is Dead

If you want to check a circuit is dead (not live), you should always use the three point method. First check a known live circuit, then check the dead...

Our internet address and Vanity URLs

Visitors who like to type web address rather then click menus may be interested in how our URL structure works.

What is Aircraft Ground Power

Ever wondered what kind of power an aircraft uses when parked at the airport stand. Normally the aircraft generates it own power, but when parked with...

Gas Insulated or Air Insulated Switchgear

Various arguments exist around SF6 Gas Insulated (GIS) and Air Insulated (AIS) medium voltage switchgear. Recently we had to change a GIS design to AI...

UPS - Uninterruptible Power Supply

A UPS is an uninterruptible power supply.  It is a device which maintains a continuous supply of electrical power, even in the event of failure of the...

Motor Insulation

Insulation on a motor prevents interconnection of windings and the winding to earth.  When looking at motors, it is important to understand how the insulation...

Dielectric loss in cables

Dielectrics (insulating materials for example) when subjected to a varying electric field, will have some energy loss.   The varying electric field causes...

Have some knowledge to share

If you have some expert knowledge or experience, why not consider sharing this with our community.  

By writing an electrical note, you will be educating our users and at the same time promoting your expertise within the engineering community.

To get started and understand our policy, you can read our How to Write an Electrical Note