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Determining thermal stress in cables and conductors

Feature

By Philippe Aupetit and Jérôme Mullie, Product Managers, Trace Software International

Checking for thermal stresses in a cable or a conductor is part of the design of an electrical installation – this ensures the safety of equipment and people.

Here, we discuss the fundamentals of determining thermal stresses, and the most commonly-experienced thermal constraints.

A cable or conductor has many characteristics, but there are two values that are essential for the size:

  • the maximum temperature of the core in steady state, which allows to determine ampacity; and
  • the maximum temperature of the core in short‑circuit, i.e. the temperature beyond which the insulation begins to deteriorate.

For most cables, these values ​​are imposed by a standard, for example for cables insulated with EPR/XLPE the values are 90°C and 250°C respectively. Therefore, it is necessary to check that short-circuit current (Ik) during disconnection time (t) will not raise the cable’s core beyond its maximum temperature.

For disconnection time of less than five seconds (hence the maximum imposed in the installation standards), the heating is considered “adiabatic”, which means the heat produced stays at the conductor’s core and does not have time to dissipate to the other parts of the cable. The thermal stress borne by the conductors is then determined by:

Let-through energy = Ik2 × t in A2s

The thermal stress admissible by the conductor is determined with k2 × S2, where S is the cross-section of the conductor in mm2, k is a factor that takes into account the resistivity, temperature coefficient and resistance to heating of the conductor’s material, as well as the initial temperatures (maximum core temperature in steady state for a charged conductor or a PE incorporated in the cable, or ambient temperature for a separate PE) and final (maximum temperature of the short‑circuited core) of the conductor. Therefore, it should be verified that in all short-circuit cases the relationship is:

Ik2 × t < k2 × S2

For easy determination, there are tables available that list the values of k in certain cases.

Trace Software’s Elec-Calc too

Simpler ways of checking thermal stresses

For simpler measurements of thermal stresses in a cable, Trace Software developed the Elec Calc software, which determines the maximum let-through energy in conductors, and compares it to their thermal withstand value according to k²×S². This verification applies to phases, neutral and PE, and if the condition is not met, a thermal stress error is displayed.

When a thermal stress problem is encountered, a cross-section of the conductor can be oversized, thus increasing its admissible thermal stress. However, there are other possibilities to achieve this without increasing the cross-section, including:

  1. Using fuses: The melting time of a fuse is generally much shorter than the operating time of a circuit breaker for high short‑circuit currents. There is therefore a natural limitation of the let-through energy. Note that in the event of fuse protection, maximum energy can occur during the shortest short‑circuit time because the melting time may be longer. Hence, it’s necessary to check the cable’s withstand for all types of short‑circuit current.
  2. Using limiting circuit-breakers: Certain equipment pieces are designed to prevent fault currents by letting only a current of limited intensity. The limitation performance of a circuit-breaker is given by the manufacturer in the form of limitation curves:
  3. The curve representing the peak value of the limited current as a function of the rms value of the alternating component of the prospective fault current (useful for checking electrodynamic stresses); and
  4. The curve representing the value of the limited let‑through energy as a function of the rms value of the alternating component of the prospective fault current. It is this limited value that must be compared with the admissible thermal stress of the conductor.

Elec Calc’s multi-manufacturer catalogue shows limitation curves for various protection; as soon as a reference with a limitation is associated with a protection device, the software is able to recover the limited let‑through energy associated with the prospective short‑circuit current. It should be noted that most miniature circuit‑breakers have limitation capacities, making it possible to solve the thermal stress problems in small-section cables most affected by thermal stress problems.

Example

An electrical installation includes a lighting circuit with a U1000R2V-3G2.5 cable, protected by a 16A curve C circuit‑breaker. The maximum short‑circuit cable current is 5.63kA and the protection tripping time 10ms; hence, the let‑through energy is Ik² × t = 5.63² × 0.01 or 316,969A²s. The thermal withstand of the cable is k² × S² = 143² × 2.5² or 127,806A²s. Check Ik² × t > k² × S² => problem of thermal stress on the cable: in the event of short‑circuit, the cable will be damaged.

  1. Fuse case:

If the circuit‑breaker is replaced with a 16AgG fuse, the melting time is 4.10-5s. The maximum let‑through energy is therefore Ik² × t = 5.63² × 0.00004 or 1268A²s. This indicates no thermal stress on the cable. In our case, we also checked that it is the maximum short‑circuit current that gives the maximum energy.

  • Integration of a circuit‑breaker reference with limitation:

Including the manufacturer’s reference on the protection, from the energy-limiting curve, the software can read the let-through energy for a short‑circuit current of 5.63kA.

In our case the residual energy is 15,120A²s, which is lower than the conductor thermal withstand. So, there’s no longer thermal stress on the cable.

Greater accuracy

In all the equations mentioned here, we considered that the value of Ik is rms value of the AC component of a prospective fault current. For greater accuracy, the DC component of the fault current should be considered, since its influence is especially noticeable when the point of fault is close to the sources. Since it depends on the disconnection time and the value of the X/R ratio of the circuit at the fault point, it is advisable to calculate an equivalent thermal current, which replaces the current Ik in the equations.

In the case of multiple sources causing faults, each contributing to the short‑circuit current, their protections may not react at the same time to the short‑circuit current the sources generate. The accurate calculation of the let‑through energy must therefore integrate a chronological cumulation of the energies produced by each source. This is what the Elec Calc software does, in order to get as close as possible to the real cause.

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