The d.c. resistance of cables can be estimated in accordance with CENELC technical report CLC/TR 50480 "Determination of corss-sectional area of conductors and selection of protective devices", dated February 2011.
For a conductor:
R= ρ 20 S
where R = d.c. resistance of the cable Ω.m-1
ρ20 = electrical resistivity of conductor material at 20 °C, Ω.m
S = cross sectional area of conductor, m2 [or 1e-6 mm2]
An alternative to calculating the d.c. resistance is given by IEC 60228 "Conductor of insulated cables". The standard has tables of maximum allowable resistance for various copper and aluminium cables. For more details see IEC 60228 DC Resistance
Typical resistivities can be found in the Useful Tables part of the Knowledge Base.
The standard IEC 60228 'Conductors of Insulated Cables' specifies the maximum allowable resistance for conductors. Values given in the IEC 60228 standard are used within myCableEngineering.com. For situations and cables not covered by IEC 60228, then resistance values are calculated using the CENELEC formulae.
Resistance calculated above, is valid for unscreened cables. For screened (or any type of magnetic shield) cables, with the metallic screen earthed at both ends, the resistance increases as depicted in IEC 60909-9 'Short-circuit currents in three-phase a.c. systems - Part 2: Data of electrical equipment for short-circuit current calculations' table 7. See Impedance for more details.
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. Calculation of the a.c. resistance is derived from equations given in IEC 60287 "Electric cables - Calculation of the current rating".
Skin and proximity effects into account with the following formulae:
R ac =R[ 1+ γ s + γ p ]
Rac = 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.
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:
γ s = X S 4 192+0.8 X S 4 with X S = 8πf R 10 −7 k s
where: f = supply frequency, Hz
ks = skin effect coefficient from the table below
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 [ 0.312 ( 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
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)
The d.c. resistance of a conductor is dependent on temperature:
R t = R 20 [ 1+ α 20 ( t−20 ) ]
where Rt = resistance of conductor at t °C
R20 = resistance of conductor at 20 °C
t = conductor temperature, °C
α20 = temperature coefficient of resistance of material at 20 °C
Typical temperature coefficients can be found in the Useful Tables part of the Knowledge Base.
At zero current the cable conductor temperature will be the same as the ambient temperature. At the the maximum sustainted current rating the cable will be at the insulation limiting temperature (typcially 70 °C for thermoplastic insulation and 90 °C for thermosetting insulation). At a current ratings between these extreems the cable temperature wil be at a vaule between ambient and the limiting temperature.
The cable operating temperature can be found from:
t= ( I b I z ) 2 ×( T c − T a )+ T a
where Ib = cable design current, A
Iz = sustained current rating of cable, A
Ta = ambient temperature, °C
Tc = conductor [insulation] limiting temperature, °C