The Adiabatic Equation Explained
The adiabatic equation checks that a protective conductor is big enough to survive the heat of a fault current before the protective device cuts it off.
A fault dumps a large current into the circuit protective conductor for a fraction of a second before the fuse or breaker clears it. That current heats the conductor, and if the CPC is too thin it can reach a temperature that damages its insulation or the conductor itself. The adiabatic equation is how you prove it won’t.
This guide sets out the equation S = √(I²t)/k, explains what each term means, gives the k values you look up for common conductor and insulation combinations, shows where in a design you actually need it, and works through a numerical example so the method is clear.
Key takeaways
- The adiabatic equation, S = √(I²t)/k, gives the minimum protective conductor cross-section for a given fault.
- It confirms the CPC survives the fault energy (I²t) let through before the device disconnects.
- k is a material constant from BS 7671 tables — it depends on the conductor and its insulation.
- Copper with 70°C thermoplastic insulation has k ≈ 115; 90°C thermosetting ≈ 143.
- You use it mainly to justify a reduced-size CPC — a CPC sized by the table method rarely needs checking.
What the equation checks
During an earth fault the protective conductor carries the full fault current for the time the device takes to operate. Because this is so brief, effectively no heat escapes to the surroundings — the process is adiabatic — so all the energy goes into raising the conductor’s temperature. The adiabatic equation calculates the smallest conductor that can absorb that energy without exceeding its safe limiting temperature.
In other words, it links three things: how much fault current flows (I), how long it flows for (t), and how well the chosen material and size cope (S and k). Get the CPC too small and the equation fails, warning you the conductor could be damaged before the fault is cleared.
The equation and the k factor
The formula from BS 7671 is S = √(I²t) ÷ k, where S is the minimum cross-sectional area in mm², I is the fault current in amps, t is the disconnection time in seconds, and k is a factor for the conductor material and insulation. The √(I²t) part is the energy let through by the device — the same I²t value you read from the manufacturer’s let-through characteristics.
The k value comes from the tables in Chapter 54. It captures the conductor’s material and the maximum temperature its insulation can tolerate, so a conductor with higher-temperature insulation gets a higher k and can therefore be smaller for the same fault.
| Conductor / insulation | k value | Typical application |
|---|---|---|
| Copper, 70°C thermoplastic (PVC) | 115 | CPC within a PVC-insulated cable |
| Copper, 90°C thermosetting (XLPE) | 143 | CPC within a thermosetting cable |
| Copper, bare / not in a cable | 159–228 | Bare protective conductor, value depends on surroundings |
| Aluminium, 70°C thermoplastic | 76 | Aluminium CPC in a PVC cable |
| Steel conduit / trunking (as CPC) | ≈ 47 | Metallic containment used as the protective conductor |
Where you use it
You do not need the adiabatic check on every circuit. BS 7671 gives two ways to size a CPC: the simple table method, which pairs each line conductor size with a minimum CPC size, and the calculation method using this equation. If you follow the table, the CPC is already proven adequate and no calculation is required.
The equation earns its keep when you want a CPC smaller than the table would give — for example the reduced CPC in a twin-and-earth cable, or a separate protective conductor you want to keep economical. It is also the check you reach for when a cable has an unusually high fault current or a slow-clearing device, where the standard pairing might not hold.
Size it once, size it right
TradePlanr’s adiabatic calculator takes the fault current, disconnection time and k value and returns the minimum CPC size, and the cable sizing calculator carries the result through so the CPC you specify matches the line conductor you’ve chosen.
Worked example
Take a circuit where the earth fault current is 1,500 A and the protective device clears it in 0.1 s. The CPC is copper inside a 70°C thermoplastic cable, so k = 115. First find √(I²t): I²t = 1500² × 0.1 = 225,000, and √225,000 ≈ 474.
Now divide by k: S = 474 ÷ 115 ≈ 4.1 mm². So the CPC must be at least 4.1 mm², meaning you would select the next standard size up — a 6 mm² CPC — to comply. If the cable had thermosetting insulation (k = 143), the same fault would need only 474 ÷ 143 ≈ 3.3 mm², showing how the higher k lets a smaller conductor pass.
Round up, never down
The equation gives a minimum. Always specify the next standard conductor size at or above the calculated value — a CPC exactly on the calculated figure leaves no margin for measurement or manufacturing tolerance.
Frequently asked questions
What does adiabatic mean in this context?
It means no heat is lost to the surroundings during the fault. Because the fault clears in a fraction of a second, there is no time for heat to dissipate, so all the fault energy raises the conductor’s temperature. That assumption is what makes the simple S = √(I²t)/k relationship valid.
Do I always have to do the adiabatic calculation?
No. If you size the CPC using the standard table method in BS 7671, it is already adequate and no calculation is needed. You only use the adiabatic equation when you want to justify a smaller CPC than the table gives, or in unusual high-fault or slow-clearing situations.
Where does the k value come from?
From the tables in Chapter 54 of BS 7671. Each combines the conductor material and its insulation’s maximum temperature — for example copper with 70°C thermoplastic insulation is about 115, and copper with 90°C thermosetting insulation is about 143.
From guidance to action
Related guides
Cable Sizing to BS 7671
The full cable sizing method — design current, correction factors, tabulated ratings and volt drop.
What Is Prospective Fault Current (PFC)?
PSCC vs PEFC, how to measure PFC, and why breaking capacity has to exceed it.
Main Protective Bonding Explained
What main protective bonding does, conductor sizes, and where bonding is required to BS 7671.
