IEEE C37.112 Curves Explained: Constants, the TD/7 Normalization and Coordinating Against IEC

An IEEE inverse-time curve and an IEC inverse-time curve of the same name are not the same curve. They can be very close in the middle of their range and meaningfully different at the ends, and the difference comes down to one extra constant in the equation plus a different convention for the time setting. If you coordinate across the boundary — and on a multi-vendor feeder you probably do — that difference is the thing that quietly eats your coordination margin.

This article covers what IEEE C37.112 is, the equation and its constants, the time-dial normalization and where it came from, the reset model, and how the two families behave when made to work together. It closes with worked arithmetic you can check.

Key takeaways

• The commonly implemented C37.112 form is t = (TD/7) × ( A/(M^p − 1) + B ).
• The B term is the substantive difference from the classic IEC curves: it puts a non-zero floor under the operating time at high multiples.
• Commonly published constants: Moderately Inverse (A = 0.0515, B = 0.1140, p = 0.02); Very Inverse (A = 19.61, B = 0.491, p = 2); Extremely Inverse (A = 28.2, B = 0.1217, p = 2).
• The /7 normalizes the time dial to a reference setting inherited from electromechanical practice. Implementations vary — the relay manual governs.
• C37.112 also defines an electromechanical-style reset so a digital relay can emulate the spin-back of the induction disc it replaced.
• IEC and IEEE curves of the same nominal name can cross. Verify coordination intervals across the whole fault-current range, not at one point.
• A settings sheet that says “Very Inverse, TD 5” without naming the standard is ambiguous and should be treated as incomplete.

Why North American practice diverged

The IEC families are analytic shapes that were chosen, then implemented. The IEEE families are the reverse: analytic approximations to shapes that already existed, produced by the electromechanical induction-disc relays that dominated North American protection for most of the twentieth century.

An induction-disc relay’s timing came from physics — the torque a current-driven flux exerted on an aluminium disc, opposed by a spring, with a damping magnet and a contact gap set by the time dial. Each manufacturer’s mechanical design produced its own family of curves, published as printed sheets, and coordination studies were done by laying those sheets over each other. There was no equation; there was a curve you traced.

When microprocessor relays arrived they had to reproduce those curves closely enough that existing studies stayed valid. C37.112’s stated purpose is exactly that: to review the analytic techniques used to represent relay curve shapes and propose formula representations of the typical shapes, so the curves available in microprocessor relays become standardized rather than vendor-specific. The 2018 edition, available via IEEE Xplore, superseded the 1996 edition.

That history explains every awkward feature of the equation. The B constant is there because the mechanical curves flattened at high current — the disc could only travel so fast. The time-dial normalization is there because the setting on the front of the old relay was a dial, not a multiplier. IEC, starting from a clean sheet, had neither constraint.

The equation

t = (TD / 7) × ( A / (M^p − 1) + B )

• t — operating time, in seconds.
• TD — the time dial setting. On North American relays this has traditionally spanned roughly 0.5 to 10 or 11, matching the calibrated dial range of the electromechanical devices.
• M — the current as a multiple of pickup, i.e. I / Is. Same quantity as in the IEC equation.
• A — the coefficient of the inverse term. Sets the time scale of the steep part of the curve.
• p — the exponent. Sets the steepness, exactly as α does in the IEC form.
• B — the added constant. Because it is not divided by anything that grows with M, it survives as M becomes large: t → (TD/7) × B. This is the time floor.

About the /7

The division by 7 is a normalization: A, B and p are defined such that a time dial setting of 7 reproduces the reference curve shape directly, and other dial settings scale it linearly. Seven was chosen because it sat in the working part of the traditional dial range, so the reference curve was one engineers recognised rather than an extreme.

Be careful here, and this is a genuine hedge rather than a formality: implementations differ, and the manufacturer’s manual governs. Some relays implement precisely the TD/7 form above. Some publish the same family with the normalization folded into their own constants, so their A and B differ numerically from the table below while producing the same curve. Some offer a “time dial” that is functionally a direct multiplier with no normalization at all. You cannot take a TD value off a settings sheet, drop it into the equation above, and assume the answer. Read the relay’s characteristic equation as its own documentation states it, and confirm your expected values against the manufacturer’s published curve set before building a test plan or a coordination study on them.

The standard IEEE curve constants

Characteristic	A	B	p
Moderately Inverse	0.0515	0.1140	0.02
Very Inverse	19.61	0.491	2
Extremely Inverse	28.2	0.1217	2

These are the values as commonly published in relay manuals, textbooks and test-software curve libraries. They are widely consistent, but they are not ours to warrant — for the authoritative constants, reference conditions and tolerance framework, purchase C37.112. Where a relay manual disagrees with the table above, the manual wins for that relay, because it describes what the firmware does.

Two structural observations:

• Moderately Inverse is very shallow. With p = 0.02 it shares its exponent with the IEC Standard Inverse curve, so the two families’ shallowest members are similar in shape — but MI’s B = 0.1140 still gives it a floor that SI does not have.
• Very Inverse is nearly definite-time at high current. At 20x pickup its inverse term contributes about 0.049 while B contributes 0.491 — the constant dominates by an order of magnitude.

Reset

C37.112 addresses reset explicitly, because the relays it was modelling had a characteristic reset of their own. Two models are in play.

Instantaneous (or definite-time) reset. Accumulated timing is discarded as soon as, or a fixed delay after, current drops below pickup. The natural behaviour of a digital integrator and the default in most modern applications.

Electromechanical reset. Accumulated timing decays back toward zero over a finite period, emulating the disc spinning back under spring tension — not instant, and dependent on how far the disc had travelled. The standard provides a reset characteristic so designers of microprocessor relays can match the reset behaviour of the electromechanical relays they replace. The commonly implemented form is a further expression of the same normalized shape:

t_reset = (TD / 7) × ( t_r / (1 − M²) )      for M < 1

with a per-curve reset constant t_r. The values commonly published alongside the three families are 4.85 for Moderately Inverse, 21.6 for Very Inverse and 29.1 for Extremely Inverse. Treat both the form and those values as needing confirmation against the standard and your relay’s manual — reset conventions are the least consistently documented part of the subject, and some relays offer a simple settable reset time in place of any curve-based model.

Why this matters more than it looks:

• Reclosers. A distribution recloser performs several operations in sequence, sometimes on different curves. An upstream relay with slow electromechanical reset carries residual timing between shots and can operate on a later shot even though no single shot would have driven it to operation. Whether that is designed behaviour or an unpleasant surprise depends on whether anybody modelled it.
• Fuse coordination. Fuses do not reset; their damage accumulates. Coordinating a fast-resetting relay against a fuse across a repeated-fault sequence is a different problem from coordinating them at a single fault current, and the reset model is the variable.
• Replacing an electromechanical relay in a coordinated scheme. Swap the relay, keep the settings, leave reset on the digital default, and you have changed the scheme’s behaviour under repeated faults without changing a number on the settings sheet.

IEC versus IEEE, in practice

The shapes differ in two ways. The B term gives the IEEE curves a floor at high multiples. And because the constants were fitted to different reference shapes, the mid-range slopes do not match either, even where the exponents nominally do.

The practical consequence is that two curves matched at one multiple will not stay matched. Here is the arithmetic, using IEC Extremely Inverse (k = 80, α = 2) and IEEE Extremely Inverse (A = 28.2, B = 0.1217, p = 2), with the IEEE relay on TD = 3 and the IEC relay’s TMS chosen so the two agree exactly at 6x pickup.

IEEE EI, TD = 3, M = 6:
  M^p − 1     = 36 − 1 = 35
  28.2 / 35   = 0.805714
  + B         = 0.805714 + 0.1217 = 0.927414
  × TD/7      = 0.927414 × 3/7 = 0.397 s

IEC EI, matched at M = 6:
  80 / 35     = 2.285714
  TMS         = 0.397463 / 2.285714 = 0.1739

Now evaluate both at 2x and at 20x:

Multiple	IEC EI (TMS 0.1739)	IEEE EI (TD 3)
2x	4.64 s	4.08 s
6x	0.397 s	0.397 s
20x	0.035 s	0.082 s

At M = 2:
  IEC : 80/(4−1) = 26.6667;  × 0.1739 = 4.64 s
  IEEE: 28.2/3 = 9.4;  + 0.1217 = 9.5217;  × 3/7 = 4.08 s

At M = 20:
  IEC : 80/399 = 0.20050;  × 0.1739 = 0.0349 s
  IEEE: 28.2/399 = 0.070677;  + 0.1217 = 0.192377;  × 3/7 = 0.0824 s

Read the table again. The curves cross. At 2x the IEC relay is the slower of the two; at 20x it is more than twice as fast. If you set your coordination margin at 6x — a perfectly reasonable place to check — you have a margin of exactly zero where you checked and 47 ms of unmodelled divergence at 20x, in a direction that may or may not be the one you wanted.

The trap on the settings sheet

“51P: Very Inverse, TD 5, pickup 480 A.”

That line is incomplete, and it is extremely common. Very Inverse under IEC means k = 13.5, α = 1, no floor, scaled by a TMS that usually runs 0.05–1.0. Very Inverse under IEEE means A = 19.61, B = 0.491, p = 2, a substantial floor, scaled by a dial that usually runs 0.5–11 with a normalization. A value of “5” is plausible on one convention and impossible on the other, and the two curves are not close.

Do not guess from the numeric range of the time setting. Read the curve family selection out of the relay, or out of the relay’s settings file. That one-minute check prevents a commissioning day spent proving a relay wrong against the wrong expected values.

Where both families show up in the same substation

• Imported IEDs. A relay designed for a European market and applied in North America typically offers both libraries, selectable. Whoever configured it made a choice, and that choice is not always recorded anywhere except in the relay.
• Multi-vendor feeders. A feeder with relays from three eras and two continents has coordination points spanning both conventions, and the as-built settings may not match the study.
• Retrofits. Replacing one device in a coordinated chain means matching the old device’s curve and its reset behaviour, on whichever convention the study assumed.
• Programmable curves. Many IEDs allow a user-defined curve as a table of points or an equation with settable constants — useful, and also a place where the applied characteristic exists nowhere except in the relay’s configuration.

Testing implications

Verify the family, don’t assume it. The first test on any inverse-time element should establish which curve the relay is applying. The fastest way is to test at two well-separated multiples and compare against both candidate families: at 2x and 20x the IEC and IEEE curves of the same name give clearly distinguishable times, as the table above shows. A single mid-range point cannot distinguish them.

Test enough points to see the shape. The argument for the IEC families applies with more force here, because the B floor changes behaviour specifically at the fast end, where a single low-multiple test will never look. Include a high-multiple point if your test set can deliver it into the relay’s burden.

Plot three things on one set of axes. The manufacturer’s published curve, the standard curve computed from the constants, and your measured points. Divergence between the first two says the vendor implemented something other than the textbook equation — usually a documented internal limit, occasionally a normalization difference. Divergence between the second and third tells you about the relay in front of you. Keeping them separate is how you avoid attributing a modelling error to hardware.

Test reset if the scheme recloses. A single timing shot cannot detect a reset-model mismatch; a multi-state sequence can.

Record the family. “51P timing verified” is much less useful to an auditor than “51P, IEEE Extremely Inverse, TD 3, pickup 480 A, verified at 2/6/20x against C37.112 constants, tolerance basis ±5% of expected or 20 ms whichever greater, as-found and as-left.” See PRC-005 evidence checklist.

Software and simulation

Curve verification benefits obviously from automation, because the value of the test scales with the number of points while the cost of each point is almost entirely setup. A characteristic sweep computes the expected time at each multiple from the settings, injects, measures, and issues a per-point verdict against an explicitly declared tolerance basis. The tooling landscape and what to ask a vendor are in relay testing software; the IEC side of the arithmetic is in IEC 60255-151 inverse-time curve constants.

GridAPM maintains a standards reference for ProtectionAI covering the curve families it implements and how each is computed. Two limits plainly: published v1.3 executes this workflow against its built-in simulator, and the v1.4 source candidate’s bounded OMICRON path is not a physically qualified curve-test workflow. That makes it useful for resolving the question this article is about — which family is this settings sheet describing, and what times should I expect at which multiples — before anyone connects test leads, and for training an engineer on why two curves of the same name cross. It is not a substitute for qualified injection.

Sources and further reading

• IEEE C37.112-2018 — the authoritative equations, constants and reset characteristics. The Xplore landing page carries the scope and abstract.
• IEC 60255-151:2009 — the corresponding IEC functional requirements for over/under current protection.
• Your relay’s own instruction manual, which governs where any of the above disagrees with the firmware in front of you.