IEC 60255-151 Inverse-Time Curve Constants: Equations, Values and How to Test Them

Two numbers define an IEC inverse-time overcurrent curve: a coefficient and an exponent. Everything else in the settings sheet — the pickup, the time multiplier, any definite-time adder — scales or shifts a shape that those two constants have already fixed. This article gives the equation, the constants for the four classic IEC curve families, what the standard has to say about verifying them, and two worked calculations you can check by hand.

It is written for the engineer who is about to either implement a curve in a test plan or argue with a relay about why its measured time does not match the spreadsheet.

Key takeaways

• The general IEC form is t = TMS × k / ((I/Is)^α − 1). The curve family lives entirely in k and α.
• The four classic families are Standard Inverse (k = 0.14, α = 0.02), Very Inverse (k = 13.5, α = 1), Extremely Inverse (k = 80, α = 2) and Long-Time Inverse (k = 120, α = 1).
• IEC 60255-151 specifies both the characteristic equations and the accuracy requirements against which an implementation is verified. The curve and its tolerance come from the same place.
• Test at several multiples of setting, not one. Curves that agree at 5x can differ meaningfully at 2x and at 20x.
• Very low multiples give poor repeatability by construction, because the denominator approaches zero.
• Reset characteristic matters as much as operate time wherever reclosing or repeated faults are in play.
• Always cross-check against the relay manufacturer’s published characteristic. Vendors ship IEC curves, IEEE curves and proprietary curves side by side, and a settings sheet that names only “Very Inverse” is ambiguous.

What IEC 60255-151 actually covers

IEC 60255-151:2009 sets out functional requirements for over/under current protection: the specification of the protection function, the measurement characteristic, and the time-delay characteristic, together with the influencing factors that affect accuracy under steady-state conditions and behaviour under dynamic conditions. It took over the ground previously held, for inverse-time overcurrent relays, by IEC 60255-3. The wider series sits on top of IEC 60255-1, which carries the common requirements shared across the functional parts. Megger’s overview of the IEC 60255-1xx functional standards is a readable orientation to how the parts fit together.

The point worth internalising is that the document does two jobs at once. It tells you what the curve is, and it tells you how closely a relay must follow it and under what conditions that is checked. Those two halves are why “the relay is 40 ms slow” is not by itself a finding: 40 ms may be comfortably inside the accuracy class at that multiple, or well outside it, depending on the operating time you are 40 ms away from.

The authoritative wording, the accuracy classes, and the tabulated values are the standard’s own. Buy it. What follows is the engineering content as it is commonly implemented and taught, written here in our own words — not a reproduction of the standard’s tables.

The equation, symbol by symbol

t = TMS × k / ( (I / Is)^α − 1 )

• t — operating time of the element, in seconds.
• TMS — time multiplier setting. A dimensionless scalar that slides the whole curve up and down the time axis without changing its shape. Some vendors and some regional practice call this the time dial or TD; on IEC curves it is almost always TMS and typically ranges from around 0.025 or 0.05 up to 1.0 or a few units.
• I — the current the relay is measuring, in the same units as Is.
• Is — the set current, or pickup, of the element. The current at which the element theoretically starts to time.
• I / Is — the current multiple, usually written M. This is the argument of the curve. A curve is a function of the ratio, not of amperes, which is why the same curve serves any pickup.
• k — the coefficient. Sets the overall time scale of the family.
• α — the exponent. Sets the steepness: how sharply operating time falls as current rises.

Some curve families in wider use — including the IEEE families and several vendor-proprietary sets — add a constant term inside the expression:

t = TMS × ( k / ((I/Is)^α − 1) + c )

That added c acts as a floor: as M becomes large the first term collapses toward zero and the operating time tends toward TMS × c rather than toward zero. The four classic IEC families do not use it (c = 0), which is one of the substantive differences between IEC and IEEE curve behaviour at high multiples. That comparison is developed in IEEE C37.112 curves explained.

The standard IEC curve constants

Characteristic	k	α	c
Standard Inverse (SI)	0.14	0.02	0
Very Inverse (VI)	13.5	1	0
Extremely Inverse (EI)	80	2	0
Long-Time Inverse (LTI)	120	1	0

These are the IEC-defined values as they are universally published and implemented. Two cautions, both of which bite in practice:

Check the manufacturer’s characteristic, not just the family name. A modern IED will typically offer the IEC set, the IEEE set, one or more proprietary curves, and often a user-programmable curve, all selectable from the same menu. Some vendors implement the IEC families with additional internal limits — a minimum operate time, a maximum time cap, a curve that stops following the equation above some multiple. Some allow the added c term to be set on curves that are otherwise IEC-shaped. The relay manual’s published characteristic governs what the relay will actually do.

Long-Time Inverse is the least consistently handled. It appears in fewer relays than SI/VI/EI, and where it does appear the constant set is worth confirming explicitly against the manual rather than assumed. If a settings sheet calls for LTI, verify the constants the relay is applying before you build the test plan around them.

Reset characteristics

Operate time is half the story. Reset behaviour — what the timing element does when current falls back below pickup before it has operated — is the other half, and it is where implementations differ most quietly. Two models are in common use.

Definite-time reset. Accumulated timing is discarded after a fixed delay once current drops below pickup (often settable, sometimes zero, i.e. instantaneous). Digital relays default to this because it is simple and predictable.

Inverse reset. Accumulated timing decays back toward zero over a period depending on current, emulating the inertia of an induction-disc relay whose disc spins back under spring tension. It exists so a microprocessor relay can coordinate against, or replace, an electromechanical relay without changing the coordination study.

Why it matters:

• Reclosing. On a reclose onto a persistent fault, an element with slow inverse reset carries residual time from the first fault into the second and operates faster the second time; one with instantaneous reset starts from scratch. Materially different coordination outcomes, invisible in a single-shot timing test.
• Intermittent and repeated faults. A flashing fault that repeatedly crosses pickup ratchets an inverse-reset element toward operation while leaving a definite-time-reset element permanently short of it. Whether that is desirable depends on what you are protecting.
• Coordination against upstream electromechanical relays. If the upstream device resets slowly and the downstream one instantly, the margin you calculated for the steady-state case does not survive a reclose sequence.

Testing this requires a multi-state sequence — pick up, hold below operate, drop out for a controlled interval, re-apply — not a single ramp or timing shot. Any competent relay testing software will do it; the reason it goes untested is that nobody put it in the test plan.

What the standard expects you to verify

The verification philosophy in IEC 60255-151 is that operating time is confirmed at defined multiples of the set current, and the measured value is judged against a stated accuracy — expressed for time as an accuracy class, typically a percentage of expected operating time with an absolute floor in milliseconds so that very short times are not held to an impossible relative tolerance. The standard also identifies the influencing quantities (frequency, harmonics, auxiliary supply, temperature) whose variation must be considered when accuracy is claimed. For the exact classes, reference conditions and tabulated tolerances, consult the standard — those are its normative content and we are not going to paraphrase numbers into it.

The practical consequence is the part that gets missed. Because the families differ mainly in α, two implementations can be nearly indistinguishable at one multiple and clearly different at another. Testing a single point at 2x verifies one arithmetic result. It does not verify that the relay is applying the curve you think it is, that the exponent is right, or that no internal minimum-time limit is truncating the fast end. Several points across the working range do verify that, and they cost almost nothing extra once the test is automated.

Choosing multiples in practice

• Span the range that matters. Typical practice covers roughly 1.5x to 20x for a phase overcurrent element, chosen so the points bracket the fault currents the element will see and the coordination points against adjacent devices. The upper limit is usually set by what the test set can deliver into the relay’s burden, not by what you would like to test.
• Avoid the region just above pickup. At M = 1.05 a Standard Inverse denominator is about 0.00098, so the expected time is enormous and hypersensitive: a one percent error in injected current, or a pickup calibrated one percent off, shifts the expectation by tens of percent. You are measuring the test, not the relay. Test pickup with a ramp and reset-ratio check, and test timing where the curve is well-conditioned.
• Respect the relay’s own algorithm. A digital relay measures RMS or fundamental-frequency current over a window, filters it, and updates its integrator at a finite rate. That chain has a settling time which appears as a small additive component at high multiples where the theoretical time is short. A relay with a 20 ms minimum operate time will not deliver an 8 ms theoretical result, and it is not broken.
• Know where the definite-time adder sits. Many elements allow a fixed delay added to the inverse characteristic; some allow a definite-time minimum that overrides the inverse portion below it. Which you have changes the expected value at every point, and mixing them up is a common cause of a plan that “fails” a healthy relay.
• Check what is being measured. If the test source is not clean, or the relay measures true RMS while your expected value assumes fundamental only, you have introduced an error before the first shot.

Worked example 1: Very Inverse

Element set to Is = 1.0 A secondary, Very Inverse curve, TMS = 0.2. Inject 5.0 A, so M = 5.

k = 13.5,  α = 1
M^α − 1   = 5^1 − 1 = 4
k / 4     = 13.5 / 4 = 3.375 s
t         = TMS × 3.375 = 0.2 × 3.375 = 0.675 s

Expected operating time: 0.675 s. Because α = 1 for this family, the arithmetic is exact and easy to check — one reason VI is a good first curve to sanity-check a new test template against.

Worked example 2: Standard Inverse at two multiples

Same pickup, Standard Inverse curve, TMS = 0.5. Compute the expected time at M = 2 and at M = 10.

k = 0.14,  α = 0.02

At M = 2:
  2^0.02      = 1.013960
  denominator = 0.013960
  0.14 / 0.013960 = 10.0290
  t = 0.5 × 10.0290 = 5.014 s

At M = 10:
  10^0.02     = 1.047129
  denominator = 0.047129
  0.14 / 0.047129 = 2.9705
  t = 0.5 × 2.9705 = 1.485 s

Expected times: 5.014 s at 2x and 1.485 s at 10x. A five-fold increase in current produces only a 3.4-fold reduction in time — that is what α = 0.02 means, and it is why Standard Inverse is such a shallow curve compared with the others. Run the same two multiples through Extremely Inverse (k = 80, α = 2) and the time ratio between them is 99/3 = 33 to 1. Same equation, same TMS mechanics, completely different coordination behaviour.

Both examples are worth re-deriving on your own calculator before you trust a template. Fractional exponents are exactly where transcription errors hide, because a wrong α still produces a plausible-looking number.

Testing the curve rather than the point

Once you accept that a curve needs several points, the test stops being a stopwatch exercise and becomes a sweep: the software computes the expected time at each chosen multiple from the settings, injects each point, records the measured time, and issues a per-point verdict against the tolerance basis you configured — percentage of expected time, absolute milliseconds, or the larger of the two. Plotted on log-log axes, the computed characteristic is a smooth line with the measured points sitting on it inside visible tolerance bands, so systematic error looks different from random scatter. A wrong exponent tilts the whole set. A wrong TMS offsets them uniformly. A minimum-operate-time limit flattens the fast end. You can see which you have.

That is also where re-typing settings stops being acceptable. If the pickup, curve family, TMS and adder were entered by hand from a settings sheet, the expected values are only as good as the transcription, and a mistyped TMS produces a clean, confident, wrong verdict on every point at once. Settings-driven plans — curve parameters taken from the imported relay settings rather than a technician’s keyboard — remove that failure mode; see XRIO templates explained. CT behaviour under high-multiple injection is its own subject, covered in CT saturation and relay testing.

GridAPM’s ProtectionAI runs characteristic sweeps of exactly this kind — computed curve, injected points, per-point verdicts against a declared tolerance basis, log-log overlay — against its built-in simulator. Published v1.3 is simulator-only; the v1.4 source candidate adds bounded OMICRON I/O and software ramps but not physical-CMC-qualified characteristic testing. That makes it useful for building and validating a test plan, training on curve behaviour, and checking a settings interpretation before anyone connects to a relay, rather than claiming a qualified hardware injection.

Where to go from here

The IEC families are only half the world. If your fleet contains imported IEDs, or you coordinate against North American practice, the IEEE families behave differently at the fast end in a way that matters — see IEEE C37.112 curves explained. If you are choosing tooling rather than settings, relay testing software covers what the software layer does, what it couples you to, and what to ask a vendor.

For the authoritative equations, accuracy classes and verification conditions, purchase IEC 60255-151. Nothing here substitutes for it.