### Heating of Cables

Under working conditions, the temperature of the cables increases due to the following factors,
1. The heat produced within the cables.
2. The dissipation of the heat upto the periphery of the cables.
3. The heat dissipation to the surrounding medium.
4. The current carried by the cables.
5. The various load conditions like continuous, distributed, intermittent etc.
Out of all these factors, the heat produced within the cables is most important from the point of view of heating of the cables. The heat is produced within the underground cables due to following losses,
a. Copper loss which is also called I2R loss or core loss.
b. Dielectric loss.
c. Sheath loss.
1.1 Copper Loss in Cables
The copper loss is determined by the expression I2R. The resistance of the conductor changes as the temperature changes. The resistance increases as the temperature increases. Hence to find copper loss it is necessary to obtain the resistance value correctly. It is determined by considering the following factors,
1. The resistance at any temperature is given by,
Where    R1 = Resistance at t1
α1 = Resistance temperature coefficient of material at t1
Δt = Temperature rise
So knowing R1 i.e. resistance at ambient temperature and assuming temperature rise about of 50oC, the resistance is determined.
2. The length of outermost strand is more than the central strand. To allow for stranding, the resistance value calculated as per as the central strand is multiplied by 1.02.
3. The effective area of cross section is smaller than actual section hence the resistance value is further increased by multiplying it by 1.02.
Thus finally cross losses are determined as I2R where R is effective resistance considering all the factors discussed above.
1.2 Dielectric Loss
There exists a capacitance between a conductor and the sheath, with a dielectric medium in between the two. This is represented as C. The leakage resistance is denoted as R. The equivalent circuit of the cable is a parallel combination of R and C. So there are two currents, one perpendicular to voltage V which is leading capacitive current Ic while other is in phase which voltage V which is resistive current Id representing dielectric loss. This is shown in the Fig. 1(a) and (b). Fig. 1

The dielectric loss is loss due to leakage resistance given by,
W = V2/R
Now        tanδ = Id/Ic = (V/R) / (V/Xc)
...                  V/R = (V/Xc) tan δ = VωCtanδ

Where      δ = Dielectric loss angle in radius
Generally δ is very small and hence tan δδ. For low voltage cable dielectric loss can be neglected as it is small but for high voltage cables it must be considered.
The angle Φd is the power factor angle of dielectric.
cosΦd = cos (90-δ) = sinδ
It depends on the temperature and voltage stress to which dielectric is subjected.
1.3 Sheath Loss
In a.c. transmission, alternating currents flowing through the cable produce pulsating magnetic field. This electromagnetic pulsating field links with the lead sheath and induces current in it. The value of this current depends on the frequency of pulsating field, sheath resistance, arrangement of cables and sheath conditions whether it is bounded or unbounded. These sheath currents produce the sheath losses.
There are two types of currents in the sheath,
1. Sheath eddy currents having both inword and outword directions and flow totally in the sheath of same cable.
2. Sheath circuit currents which flow from sheath of one cable to the sheath of other cable.
The unbounded cable means having one end or no ends, electrically shorted hence sheath circuit currents are absent in unbounded cable. The bounded cables means the two different cables have a sheath electrically connected at both the ends hence both the types of currents are present in them.
The approximate formula to calculate sheath losses due to sheath eddy current is suggested by Arnold as,

Where          I = Current per conductor in Amp
rm = Mean radius of sheath
Rs = Sheath resistance in
d = Spacing between conductors
These losses are practically very small and hence generally neglected.
Key Point : Thus core loss, dielectric loss and sheath loss together constitute to the heating of the cables.

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