Capacitor Fuse I2t: Melting, Clearing, Outrush and Parallel Energy, With Worked Numbers

What I2t measures

I2t, ampere-squared seconds, is the integral of current squared over time. One manufacturer's capacitor fuse selection guide explains that fuse melting is an energy function and that I2t is proportional to energy: the energy in the element is I2t x R, with element cold resistance typically 10 to 50 milliohms. For example, with a 20 milliohm element, an I2t of 1,000 A2s puts about 1,000 x 0.020 = 20 J into the element.

The same guide says I2t is used only for fuse operations that occur faster than about 0.1 or 0.01 seconds. Slower than that, the time-current curve is the right tool; faster than that, the comparison is made in A2s rather than on the curve.

An IEC-oriented fuse guide splits the total into two parts: the pre-arcing (melting) Joule integral, up to the moment an arc forms, and the arcing Joule integral, while the fuse interrupts. Their sum is the total operating, or clearing, I2t. Manufacturers publish the minimum pre-arcing integral and the total operating integral, and the same guide gives time-current tolerance as plus or minus 20 percent, or plus or minus 10 percent with tight manufacturing control. A fuse maker's application note adds that the I2t passed by a given fuse decreases as the application voltage decreases, so let-through figures are tied to a test voltage.

The two checks, side by side

Every capacitor fuse sits between two limits. An IEEE fuse working group tutorial lists both on its capacitor checklist: inrush levels, using fuse size selection to prevent damage to the fuse, and let-through energy versus case rupture. A capacitor bank maker puts the same balance this way: the fuse must be small enough to prevent case rupture, yet large enough to prevent spurious or false operation from inrush and outrush.

Worked example 1: parallel energy per kvar

When one unit in a parallel group fails, the healthy units discharge into it. One manufacturer's selection guide calculates that energy by assuming the parallel capacitance is charged to 1.1 times the crest of rated voltage, giving 3.19 joules per kvar, or 3.80 at 20 percent overvoltage. The figure can be rebuilt from first principles. For a single-phase unit, C = kvar x 1000 / (2 pi f V squared), with V the rated rms voltage in volts. The crest at 1.1 per unit is 1.1 x 1.414 x V. Energy is one half C times crest squared, so E = 0.5 x C x 2 x 1.21 x V squared = 1.21 x kvar x 1000 / (2 pi f).

At 60 Hz, 1.21 x 1000 / 377 = 3.21 J per kvar; at 1.2 per unit, 1.44 x 1000 / 377 = 3.82 J. The guide's 3.19 and 3.80 differ only by rounding in its base constant. At 50 Hz the same arithmetic gives 1.21 x 1000 / 314.2 = 3.85 J per kvar, about 20 percent more than at 60 Hz for the same kvar.

Applying it: the guide's own example has 10 units of 200 kvar in a group, so 9 healthy units discharge into the failed one: 3.19 x 9 x 200 = 5,742 J. A group of 12 units of 400 kvar puts 11 x 400 = 4,400 kvar behind a failed unit: 4,400 x 3.19 = 14,036 J, under the 15 kJ limit a published protection paper gives for all-film dielectric but over its 10 kJ limit for paper or paper-film units.

Note the units. Parallel energy is in joules and is compared with the fuse's parallel energy rating and the group limits. I2t is in A2s and is compared with melting and rupture figures. The guide explains how the energy is shared: an expulsion fuse conducts to a natural current zero, so most of the energy is absorbed in the failed capacitor; a current-limiting fuse that develops back voltage absorbs a large portion of it; a current-chopping design leaves trapped voltage on the other units.

Worked example 2: n fuses in parallel

Large capacitor units sometimes need two fuses in parallel, and published fusing tables mark some large-unit ratings that way. An IEC-oriented fuse guide gives the rule: the I2t of n identical fuse-links in parallel is n squared times the I2t of one. The reason is that the current divides n ways, so each element carries 1 / n of the total, and the total current needed to bring each element to the same I2t is n times higher, which squares to n squared.

Take an illustrative single fuse with a minimum melting I2t of 4,000 A2s and a total clearing I2t of 20,000 A2s. These are round numbers for the arithmetic, not DENCO ratings. Two in parallel melt at 2 squared x 4,000 = 16,000 A2s and clear at 4 x 20,000 = 80,000 A2s. Three in parallel melt at 9 x 4,000 = 36,000 A2s and clear at 9 x 20,000 = 180,000 A2s.

Both sides of the check move. The pair is four times harder to melt with outrush, which is often why it was chosen, but the failed unit also sees four times the let-through. Run the case rupture check with the paralleled clearing figure, not the single-fuse figure, and ask the fuse maker to confirm that the published n squared figure applies to its paralleled assembly.

Worked example 3: tank rupture limits, as one maker's curves

Tank rupture data cannot be calculated by the engineer; a software vendor's application guide notes that case rupture curves are commonly available from the capacitor manufacturer, and that faults cleared to the left of the curve indicate minimal likelihood of rupture. The figures below are from one capacitor maker's published tank rupture curves and a second maker's brochure. They describe those makers' units only.

Dividing the stated minimum I2t by the square of the current gives the time that I2t allows at the top of each range. For standard and heavy duty units, 2,000,000 / 10,000 squared = 0.020 s, which is 1.2 cycles at 60 Hz. At 5,000 A the same 2,000,000 A2s allows 2,000,000 / 25,000,000 = 0.08 s. The same curves note a minimum clearing time of 0.8 cycle (13.3 ms) for safe coordination with expulsion fuses, and say that for times shorter than 0.1 second, asymmetrical rms amperes should be used.

Asymmetry matters because a fault's first loops can be much larger than its symmetrical rms value suggests. The IEEE C37.48.1 tutorial notes that the crest of a symmetrical fault is 1.414 times the rms symmetrical value, and the crest of a fully asymmetrical fault may be double that, 2.8 times, depending on X/R. A software vendor's example rupture curve for a 300 kvar, 13,800 V unit shows the short-time end near constant I2t: 4,000 A at 0.08 s is 1.28 x 10^6 A2s, and 10,000 A at 0.0133 s is 1.33 x 10^6 A2s, lower than the 2,000,000 A2s figure above. Curves differ by maker and unit, so use the curve for the actual unit.

Outrush and inrush checks, described in words

The other side of the problem is keeping healthy units in. A published shunt capacitor protection paper puts the outrush requirement plainly: the fuse on the healthy elements must not blow when discharging into an adjacent faulted capacitor. When a unit fails, every healthy unit in the group discharges through its own fuse toward the failed unit. Each healthy fuse carries its own unit's share of that oscillating discharge, while the failed unit's fuse carries the sum. The healthy fuses must survive their share with margin.

The magnitudes vary widely. A switchgear maker's application guide describes single-bank energization inrush as generally less than 5 kA and 1 kHz, and back-to-back inrush as on the order of tens of kiloamperes and tens of kilohertz. PJM transmission studies found 17.8 to 30.8 kA at 5.6 to 9.5 kHz with two 230 kV banks back-to-back. A pole-bank fusing guide adds discharge into external faults, mainly on grounded wye systems, and lightning to the list of transients a fuse must withstand.

The check in words: take the peak, frequency and duration of each transient from the switching study; turn each into an I2t (the study tool or the fuse maker can do this); compare it with the fuse's published minimum melting I2t, adjusted for paralleled fuses if any; and leave room for repetition. One IEC-oriented fuse guide states the energization version of this rule as requiring the fuse's pre-arcing Joule integral to be greater than the energy generated during bank energization, expressed through the peak phase voltage, phase capacitance and source inductance. That formula is not reproduced here; use the method in your study.

Small units are the hard case. One manufacturer's fusing table explains that for 50 kvar capacitors it is difficult to choose reasonably sized fuses that withstand the I2t outrush, because I2t withstand goes down exponentially with fuse rating rather than linearly, and limits its 50 kvar recommendations to units up to 9,960 V. Inrush reactors are the usual remedy at the bank level: a university design standard requires them to keep high-frequency inrush below the I2t rating of the fuses.

Fatigue, production tests and hidden damage

Margin matters because fuses wear. One manufacturer's guide says each time the temperature of a fuse element is raised near its melting point, the fuse deteriorates slightly, and that a curve can be plotted of the expected number of operations against the percent of one-shot melting I2t. For applications with many such exposures it supplies elements with a higher I2t to melt. No source loaded for this guide gives the curve values or a numeric margin, so ask the fuse maker for the figure that applies.

One current-limiting fuse maker's technical paper describes a production I2t test: each fuse is pulsed twice with a pulse calibrated to contain 95 percent of the published minimum melting I2t, and a good, undamaged fuse withstands an indefinite number of such pulses. The same paper shows why visual and resistance checks are not enough: a nick extending halfway from the element edge to one of its holes reduces minimum melting I2t by 44 percent while changing resistance by only 0.75 percent. It lists miscoordination, high current surges and clearing multiple phase faults as causes of in-service element damage.

Common mistakes

These errors recur in I2t checks on capacitor fuses.

• Comparing melting I2t with the rupture limit. The failed unit sees the total clearing I2t, which is always larger.
• Using symmetrical current on the rupture curve below 0.1 s, where one maker's curves call for asymmetrical rms amperes.
• Forgetting n squared. Paralleled fuses raise both the melting and the let-through I2t by n squared.
• Mixing joules and A2s. Parallel energy in joules goes against energy ratings and group limits; I2t goes against melting and rupture data.
• Using 60 Hz energy per kvar on a 50 Hz system. The 50 Hz figure is about 20 percent higher.
• Assuming a fuse that passed a resistance check is undamaged. A 44 percent loss of melting I2t can hide behind a 0.75 percent resistance change.

Frequently asked questions