The adiabatic equation checks one thing: whether a protective conductor has enough cross-sectional area to survive the heat of a fault until the protective device cuts the supply. It is a thermal test, nothing more. A conductor that passes it still has to satisfy automatic disconnection, mechanical protection, and the other conditions that apply to the installation.
For electricians, contractors, and apprentices, the hard part is rarely the sum. It is picking inputs you can trust. Run the adiabatic equation with the wrong fault current, the wrong operating time, or the wrong k factor, and a tidy calculation hands you an unsuitable conductor.
This guide works through the maths with three examples. The fault currents, operating times, and let-through values shown here are illustrative design inputs. They are not data for any specific device or installation, so never lift them into a real design.
Where the Adiabatic Equation Fits in UK Electrical Design
For current design work, use BS 7671:2018+A4:2026 and its supporting guidance. The IET and BSI set 15 October 2026 as the withdrawal date for the previous edition, BS 7671:2018+A2:2022+A3:2024. An existing installation does not become unsafe just because it was designed to an earlier edition.
BS 7671 is a non-statutory standard. The HSE’s position is that following it is likely to achieve compliance with the relevant parts of the Electricity at Work Regulations 1989. So a calculation on its own does not prove you have met every legal or technical duty. It is one piece of evidence, not the whole case.
Protective conductor sizing follows one of two routes:
- The Calculation Route: Regulation 543.1.3 covers the adiabatic criterion, where you work out the minimum size directly.
- The Tabulated Route: Regulation 543.1.4 lets you select a size against Table 54.7 instead, by relation to the line conductor.
Each route carries its own conditions. A reduced circuit protective conductor (CPC) in an existing cable has to be judged against the right requirements. You cannot accept or reject it just by holding it up against the line conductor.
What the Adiabatic Equation Actually Says
The thermal sizing relationship is:
S ≥ √(I²t) / k
| Symbol | Meaning | Unit |
|---|---|---|
| S | Selected protective conductor cross-sectional area | mm² |
| I | Relevant fault current, using the correct RMS value for AC work | A |
| t | Device operating time for that fault current | s |
| k | Factor set by conductor material and the applicable start and end temperatures | A√s/mm² |
The expression √(I²t)/k gives the calculated minimum area. You pick an available size at least that large, then finish the rest of the design checks. The current and the operating time together describe the fault energy. The material and temperatures set k.
One assumption sits underneath all of this. Adiabatic heating treats the fault heat as if it all stays trapped in the conductor with none escaping. That short-duration method applies to fault times no longer than five seconds. Anything longer needs a different assessment.
Read the five-second figure for what it is. It is the ceiling of this method, not a green light for every circuit to take five seconds to disconnect.
Choose the k Factor for the Real Conductor Arrangement
The k factor stands for more than the metal in the conductor. Insulation, starting temperature, and the highest temperature the conductor may reach all feed into it.
Two reference cases come up again and again:
| Insulation and Material | Initial Temperature | Final Temperature | k |
|---|---|---|---|
| 70°C thermoplastic (PVC), copper | 70°C | 160°C | 115 |
| 90°C thermosetting (XLPE/EPR), copper | 90°C | 250°C | 143 |
These values match BS 7671 Table 54.3. Check the correct protective-conductor table and its notes in your copy before you use them on a job.
Here is the catch that trips people up. Both k values above assume the conductor is already sitting at the cable’s operating temperature, so 70°C or 90°C. A separately run conductor that does not carry load current may start nearer 30°C, which changes k and changes your answer. So do not read k off an insulation label, and do not reach for a bigger value just because it shrinks the result.
The same warning covers material. Copper values do not carry across to steel armour or aluminium. Where armour acts as the protective conductor, its material, effective area, terminations, and thermal conditions all need their own look.
Get Matching Fault-Current and Disconnection-Time Data
Work out the fault conditions that actually apply to the conductor and its protective arrangement. Then find out how the specified device behaves under those conditions. These two numbers have to belong to the same scenario.
A few common mistakes turn a correct formula into an undersized cable:
- The Rated Current Trap: A device's rated current is not the fault current. A 32 A breaker rating is not an instruction to enter 32 A as I.
- The Operating-Time Trap: Use the device's real time-current characteristic with its tolerance, or proper manufacturer data. Do not drop in a required maximum disconnection time and treat it as the device's actual operating time.
- The Let-Through Trap: For fast current-limiting operation, use published let-through data. Multiplying prospective current squared by a guessed operating time can badly misstate the thermal stress.
A useful alternative form compares the stress against the withstand:
I²t ≤ k²S²
Here I²t is a thermal-stress quantity in A²s, often called let-through energy. It is not an energy in joules, and treating it as joules is a classic slip. When you use manufacturer I²t data, confirm the figure covers the total clearing duty for the conductor, not just a fuse’s pre-arcing stage.
Worked Example 1: Calculate the Minimum CPC Size
Say a design assessment has fixed these inputs for a particular fault:
- Fault Current: I = 500 A
- Operating Time: t = 0.20 s
- k Factor: 115 (copper, 70°C PVC)
Step one, calculate I²t:
I²t = 500² × 0.20 = 50,000 A²s
Step two, take the root and divide by k:
S minimum = √50,000 / 115 = 1.944 mm²
The next standard size above that result is 2.5 mm². A 1.5 mm² conductor does not clear this thermal bar.
Step three, check the proposed size. For a 2.5 mm² conductor:
k²S² = 115² × 2.5² = 82,656.25 A²s
The fault duty of 50,000 A²s sits below that withstand, so 2.5 mm² passes the thermal check for these inputs. That is not the same as saying 2.5 mm² is the final answer. If a separate copper conductor runs into a 4 mm² mechanical minimum, that larger figure still wins.
Worked Example 2: Use Protective-Device Let-Through Data
Now assume device data give a total clearing I²t of 45,000 A²s for the relevant fault duty. The arrangement again needs k = 115.
You can feed the published I²t straight into the equation:
S minimum = √45,000 / 115 = 1.845 mm²
A 2.5 mm² conductor clears the calculated thermal minimum. But check an existing 1.5 mm² conductor before you trust it:
k²S² = 115² × 1.5² = 29,756.25 A²s
The duty of 45,000 A²s is bigger than 29,756.25 A²s, so the 1.5 mm² conductor fails this thermal check. The 2.5 mm² conductor holds its 82,656.25 A²s withstand, so it passes.
That 45,000 A²s figure is made up for the example. On a real job, keep the exact device reference and the manufacturer’s data you used, and confirm the data apply to the prospective fault level, voltage, and protective arrangement. Breaking capacity alone tells you nothing about let-through performance.
Why a Thermal Pass Does Not Finish the Job
A conductor can sail through the adiabatic equation and still be wrong for the installation. Three checks sit alongside it.
Check the Mechanical Minimum Sizes
Where a copper protective conductor is neither part of a cable nor in a common enclosure with the line conductor, the minimum is generally 2.5 mm² with mechanical protection and 4 mm² without it. These figures come from Regulation 543.1.1, so confirm the exact conditions in BS 7671. Do not stamp them onto every CPC that sits inside a cable.
This matters most when the maths spits out a tiny number. Good thermal withstand does nothing to stop a physically fragile conductor from being snapped during service.
Verify Loop Impedance and Automatic Disconnection
The conductor you pick feeds into earth fault loop impedance. Change its size and you change the fault current, which changes the device’s operating time.
So after you change a conductor, go back and re-run the inputs and the disconnection assessment. A fault current worked out for a different arrangement is not safe to reuse without checking it still holds.
One more point here. An RCD’s rated residual operating current, say 30 mA, is its residual-current sensitivity. It is not the earth-fault current for a conductor thermal calculation. Where residual and overcurrent protection work together, assess the actual arrangement.
Assess the Full Range of Fault Conditions
A high fault current may clear fast, while a lower current can hang on longer. The governing thermal duty depends on the device characteristics and the circuit conditions, not on a single convenient number.
So look at the credible range of fault currents and the relevant locations. Do not assume the slowest disconnection always gives the worst I²t, and do not let one fault-current value stand in for the whole circuit. Main protective bonding, PEN conductors, and buried earthing conductors carry extra requirements of their own, and these CPC examples do not cover them.
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Record a Calculation Another Electrician Can Verify
A design record earns its keep when someone else can follow it. A good one names the conductor material, insulation, installation arrangement, selected area, and the source of the k factor. It also captures the fault-current basis and the exact device data.
Keep the thermal result apart from the automatic disconnection result. Include the mechanical minimum check and any conditions tied to manufacturer coordination data.
Before certification, confirm through inspection and testing that what got installed matches what you designed. A swapped protective device, a rerouted cable, or a different conductor arrangement can undo an earlier result without anyone noticing.
The adiabatic equation earns its trust when every input traces back to the real design. The final conductor has to satisfy the thermal calculation and the other requirements together, with the evidence kept in the installation records.
