IDMT Relay Curve Calculator — IEC 60255-151 & IEEE C37.112
An IDMT relay’s operating time is a function of the current multiple and the time multiplier — not a single number. This calculator applies the IEC 60255-151 and IEEE C37.112 curves, plots the characteristic and returns the injection points a secondary test should hit.
Key takeaways
- The current multiple is M = I / Is. Below pickup (M ≤ 1) the relay should not operate, and the curve is asymptotic as M approaches 1.
- IEC curves use t = TMS × k / (Mα − 1); IEEE curves add a constant term: t = TD × (A / (Mp − 1) + B).
- The time multiplier scales the whole curve vertically. Halving TMS halves every operating time on that curve.
- Test at 2×, 5× and 10× pickup to check pickup, curve shape and the instantaneous element; the plot shows what lies between them.
- The relay’s own published curve governs acceptance. Manufacturer tolerances and definite-time elements are not captured by a single formula.
1. What this calculator does — and what it does not
It computes the steady-state operating time of an inverse-time overcurrent element for a given current multiple and time multiplier, using the standard IEC and IEEE curve families. It plots the characteristic over the useful current range and tabulates the test points a secondary injection set should apply.
It is a reference implementation, not a relay model. It does not include relay-specific tolerances, definite-time elements, reset characteristics, harmonic restraint, saturation or DC offset. It assumes a steady sinusoidal current and a relay whose curve follows the declared standard family. Where the relay’s instruction manual gives a curve or a tolerance, that document governs.
2. IDMT relay curve calculator
Curves: IEC 60255-151 and IEEE C37.112. The plot is a steady-state characteristic; relay tolerances, definite-time elements and reset behaviour are not modelled. Confirm acceptance against the relay manufacturer’s published curve.
3. How to read the result
M is the multiple of pickup, not the current in amperes
A relay set to 1 A with 5 A injected sees M = 5. The same relay with 2 A injected sees M = 2. The curve is defined against M, so the shape does not change when you change the CT ratio — the CT ratio only changes the primary current that corresponds to the same secondary test current.
The time multiplier moves the curve, it does not reshape it
TMS (IEC) and TD (IEEE) scale every operating time by the same factor. If the relay is set to TMS = 0.5 and the curve gives 10 s at TMS = 1, the expected time is 5 s. This is why a timing test at one current can be enough to verify the multiplier — but a second point is still needed to prove the curve shape.
Test points are chosen to expose different failures
2× pickup checks the timing at a moderate multiple, where the curve is steep and a pickup error shows up as a large time error. 5× and 10× check the flatter part of the curve and the transition to any instantaneous element. A relay that passes at 2× but fails at 10× may have a curve-shape or instantaneous-setting problem, not a pickup problem.
The primary equivalent is an arithmetic conversion, not a test result
Primary current = secondary current × CT ratio. It is useful for matching a test sheet to a primary fault level, but it does not prove the CT is accurate or that it will not saturate. The relay only sees the secondary current you inject.
4. Formula and constants
| Family | Curve | Expression | Constants |
|---|---|---|---|
| IEC 60255-151 | Standard inverse (SI) | t = TMS × 0.14 / (M0.02 − 1) | k = 0.14, α = 0.02 |
| IEC 60255-151 | Very inverse (VI) | t = TMS × 13.5 / (M − 1) | k = 13.5, α = 1 |
| IEC 60255-151 | Extremely inverse (EI) | t = TMS × 80 / (M2 − 1) | k = 80, α = 2 |
| IEC 60255-151 | Long-time inverse (LTI) | t = TMS × 120 / (M − 1) | k = 120, α = 1 |
| IEEE C37.112 | Moderately inverse (MI) | t = TD × (0.0515 / (M0.02 − 1) + 0.114) | A = 0.0515, B = 0.114, p = 0.02 |
| IEEE C37.112 | Very inverse (VI) | t = TD × (19.61 / (M2 − 1) + 0.491) | A = 19.61, B = 0.491, p = 2 |
| IEEE C37.112 | Extremely inverse (EI) | t = TD × (28.2 / (M2 − 1) + 0.1217) | A = 28.2, B = 0.1217, p = 2 |
| Definite time | DT | t = time setting for M > 1 | Independent of current above pickup. |
The constants above are the standard curve constants, not a specific relay’s firmware. Many manufacturers implement the same families with an additional definite-time minimum, a reset characteristic and a tolerance band. Use the relay’s published curve for acceptance and this calculator for the expected value and the test plan.
5. Worked example — IEC standard inverse
Pickup 1 A, TMS 1.0, injected current 5 A, CT 400/5. M = 5, and the standard-inverse time is 0.14 / (50.02 − 1) ≈ 4.28 s. The same curve at 10× pickup gives about 2.97 s, and at 2× gives about 10.0 s. Those three points are enough to see whether the relay follows the declared curve or has a pickup or multiplier error.
6. Frequently asked questions
What is the difference between the IEC and IEEE curves?
They use different constants and, for IEEE, an additional constant term B. The shape is similar but the numbers differ, so the curve family must match the relay’s declared characteristic. Using an IEC standard-inverse formula on a relay set to an IEEE moderately inverse curve produces a plausible-looking but wrong time.
Is TMS the same as time dial?
Conceptually yes: both scale the curve vertically. IEC uses the term time multiplier setting (TMS); IEEE uses time dial (TD). The numeric ranges and the curve constants differ, so do not carry a setting from one family to the other without converting through the curve.
Why does the calculated time go to infinity at pickup?
Because the standard curve is asymptotic at M = 1. Real relays have a definite minimum operating time, a reset time and a tolerance band, and they do not actually sit at infinity. Treat the region just above pickup as a warning that the curve is steep and the timing is sensitive to small current errors — not as a literal prediction.
How do I test the curve?
Inject at 2×, 5× and 10× pickup, record the operating time at each point, and compare with the relay’s published curve at the set TMS/TD. If the relay has an instantaneous element, test its threshold separately. Use a test set with a stable current source and a timing input connected to the trip contact.
Does the CT ratio change the operating time?
No. The relay responds to the secondary current at its terminals. The CT ratio only converts that current to a primary equivalent for the test record. CT accuracy, burden and saturation affect what the relay sees in service, but they are separate from the relay’s curve.
7. Where to go next
For the test procedure, see overcurrent relay testing. For the injection method and the classic mistakes, see secondary injection testing. The VAE-430 and VAE-660 relay test systems provide the current channels these tests need; for a configuration matched to your protection schemes, request a quote.
Related tools: relay test lead voltage drop calculator · winding resistance temperature correction calculator · transformer turns ratio calculator · all test calculators.