The same dual-slope 87T characteristic correctly restrains through normal load with CT mismatch, then trips decisively for a genuine internal fault.
| Minimum pickup (Id0) | 0.3 p.u. |
| First slope | 25%, up to a restraint current of 2.0 p.u. |
| Second slope | 60%, above 2.0 p.u. restraint |
| Case A — normal load | I1 = 1.05 p.u. in, I2 = 0.98 p.u. out (typical CT ratio-error mismatch) |
| Case B — internal fault | I1 = 6.0 p.u. in, I2 = 0.5 p.u. out (winding short collapses the outflow) |
Currents are already referred to a common base (CT secondary, ratio and vector-group compensated) as the relay itself would see them.
Case A's restraint (1.015 p.u.) is below the 2.0 p.u. knee, so only the first slope applies. Case B's restraint (3.25 p.u.) is above the knee, so the threshold is the value at the knee plus the second, steeper slope beyond it.
| Case | Id | Threshold | Verdict |
|---|---|---|---|
| A — normal load, CT mismatch | 0.07 p.u. | 0.554 p.u. | Restrained — no trip (correct) |
| B — internal fault | 5.5 p.u. | 1.55 p.u. | Trips (correct) |
| Check | Requirement | Actual | Status |
|---|---|---|---|
| Case A — stays restrained through normal load | Id < threshold | 0.07 p.u. < 0.554 p.u. | ✓ PASS |
| Case B — trips for a genuine internal fault | Id ≥ threshold | 5.5 p.u. ≥ 1.55 p.u. | ✓ PASS |
Key insight: A flat, fixed-pickup differential element would have to be set high enough to ride through worst-case CT mismatch at maximum through-load — which would make it dangerously insensitive to genuine internal faults. The percentage-bias characteristic solves this by letting the pickup threshold rise with restraint current, staying tight (sensitive) at low load and only relaxing (for security against CT saturation) at high through-fault currents.
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Open Transformer Differential Protection (87T) calculator →Real CTs have small ratio errors that scale with the current flowing through them, and those errors get worse (relative CT saturation) during high through-fault current. If the pickup stayed fixed at a low value, a large external fault could saturate the CTs enough to look like a differential current and cause a false trip. The second, steeper slope specifically defends against that scenario.
First slopes of 20-30% and second slopes of 50-80% above a knee around 2-8 p.u. restraint current are common starting points, but the right values depend on the specific CTs, their accuracy class and knee-point voltage, and the transformer's tap-changer range — always confirm against manufacturer guidance and a proper CT saturation study for a real installation.