# TN system - Earth-fault current calculation

## Methods of determining levels of fault current

In TN-earthed systems, a fault to earth will, in principle, provide sufficient current to operate an overcurrent device.

The source and supply mains impedances are much lower than those of the installation circuits, so that any restriction in the magnitude of earth-fault currents will be mainly caused by the installation conductors (long flexible leads to appliances greatly increase the “fault-loop” impedance, with a corresponding reduction of the fault current).

The most recent IEC recommendations for fault protection on TN earthing systems only relates maximum allowable tripping times to the nominal system voltage (see Figure F13).

The reasoning behind these recommendations is that, for TN systems, the current which must flow in order to raise the potential of an exposed conductive part to 50 V or more is so high that one of two possibilities will occur:

• Either the fault path will blow itself clear, practically instantaneously, or
• The conductor will weld itself into a solid fault and provide adequate current to operate overcurrent devices.

To ensure correct operation of overcurrent devices in the latter case, a reasonably accurate assessment of earth-fault current levels must be determined at the design stage of a project.

A rigorous analysis requires the use of phase-sequence-component techniques applied to every circuit in turn. The principle is straightforward, but the amount of computation is not considered justifiable, especially since the zero-phase sequence impedances are extremely difficult to determine with any reasonable degree of accuracy in an average LV installation.

Other simpler methods of adequate accuracy are preferred.

Three practical methods are:

• The “method of impedances”, based on the summation of all the impedances (positive-phase-sequence only) around the fault loop, for each circuit
• The “method of composition”, which is an estimation of short-circuit current at the remote end of a loop, when the short-circuit current level at the near end of the loop is known
• The “conventional method” of calculating the minimum levels of earth-fault currents, together with the use of tables of values for obtaining rapid results.

These methods are only reliable for the case in which the cables that make up the earth-fault-current loop are in close proximity (to each other) and not separated by ferromagnetic materials.

Note: Schneider Electric’s Ecodial software is based on the “method of impedances”.

## Method of impedances

This method summates the positive-sequence impedances of each item (cable, PE conductor, transformer, etc.) included in the earth-fault loop circuit from which the earth-fault current is calculated, using the formula:

$\displaystyle{ I=\frac{U_0}{\sqrt{\left ( \sum R \right )^2 + \left ( \sum X \right )^2 } } }$

where
U0 = nominal system phase-to-neutral voltage.
(ΣR)2 = (the sum of all resistances in the loop)2 at the design stage of a project.
(ΣX)2 = (the sum of all inductive reactances in the loop)2

The application of the method is not always easy, because it supposes a knowledge of all parameter values and characteristics of the elements in the loop.

In many cases, a national guide can supply typical values for estimation purposes.

## Method of composition

This method permits the determination of the short-circuit current at the end of a loop from the known value of short-circuit at the sending end, by means of the approximate formula:

$\displaystyle{ I_{sc}=\frac {I.U_0}{U_0 + Z_s.I} }$

where

Isc = upstream short-circuit current
I = end-of-loop short-circuit current
U0 = nominal system phase voltage
Zs = impedance of loop

Note: In this method the individual impedances are added arithmetically[1] as opposed to the previous “method of impedances” procedure.

## Conventional method

This method is generally considered to be sufficiently accurate to fix the upper limit of cable lengths.

Principle

The short-circuit current calculation is based on the assumption that the voltage at the origin of the circuit concerned (i.e. at the point at which the circuit protective device is located) remains at 80% or more of the nominal phase to neutral voltage.

The 80% value is used, together with the circuit loop impedance, to compute the short-circuit current.

This coefficient takes all voltage drops upstream of the point considered. In LV cables, when all conductors of a 3-phase 4-wire circuit are in close proximity (which is the normal case), the inductive reactance internal to and between conductors is negligibly small compared to the cable resistance.

This approximation is considered to be valid for cable sizes up to 120 mm².

Above that size, the resistance value R is increased as follows:

Core size (mm2) Value of resistance
S = 150 mm2 R+15%
S = 185 mm2 R+20%
S = 240 mm2 R+25%

The maximum length of a circuit in a TN-earthed installation is given by the formula:

$\displaystyle{ Lmax=\frac{0.8\ U_0\ S_{ph}}{\rho \left ( 1+m \right )I_a} }$

where:

Lmax = maximum length in metres
U0 = phase volts = 230 V for a 230/400 V system
ρ = resistivity at normal working temperature in ohm-mm2/meter (= 23.7 10-3 for copper; = 37.6 10-3 for aluminium)
Ia = trip current setting for the instantaneous operation of a circuit-breaker,

or:

Ia = the current which assures operation of the protective fuse concerned, in the specified time.
Sph = cross-sectional area of the phase conductors of the circuit concerned in mm²
SPE = cross-sectional area of the protective conductor concerned in mm².

$\displaystyle{ m=\frac{S_{ph}}{S_{PE}} }$

(see Fig. F23)

Fig. F23 – ﻿Calculation of L max. for a TN-earthed system, using the conventional method

## Tables

The following tables, applicable to TN systems, have been established according to the “conventional method” described above.

The tables give maximum circuit lengths, beyond which the ohmic resistance of the conductors will limit the magnitude of the short-circuit current to a level below that required to trip the circuit breaker (or to blow the fuse) protecting the circuit, with sufficient rapidity to ensure fault protection.

Note: for industrial circuit breakers (IEC 60947-2), a 20% tolerance is taken concerning the magnetic trip current, i.e. the real trip level Ia may be 20% higher (or lower) than the magnetic trip setting Im of the circuit breaker. Table Fig. F25 includes this 20% tolerance and calculates the maximum length of the circuit for the worst-case, which is for Ia = Im x 1.2. For domestic circuit breakers (IEC 60898), the tripping value is stated without tolerance (for example, Ia = Im = 10 In for C curve), so Tables Fig. F26 to Fig. F28 are calculated with short-circuit value exactly equal to Im, without tolerance.

### Correction factor m

Fig. F24 indicates the correction factor to apply to the values given in Fig. F25 to Fig. F28, according to the ratio Sph/SPE, the type of circuit, and the conductor materials.

The tables take into account:

• The type of protection: circuit breakers or fuses
• Operating-current settings
• Cross-sectional area of phase conductors and protective conductors
• Type of earthing system (see Fig. F16)
• Type of circuit breaker (i.e. B, C or D)[2]

The tables may be used for 230/400 V systems.

Similar tables for protection by Schneider Electric Compact and Acti 9 circuit breakers are included in the relevant catalogues.

Fig. F24 – Correction factor to apply to the lengths given in tables Fig. F25 to Fig. F28 for TN systems
Circuit Conductor material m = Sph/SPE (or PEN)
m = 1 m = 2 m = 3 m = 4
3P + N or P + N Copper 1 0.67 0.50 0.40
Aluminium 0.62 0.42 0.31 0.25
Fig. F25a – Maximum circuit lengths (in metres) for different sizes of copper conductor and instantaneous-tripping-current settings for industrial circuit breakers (IEC 60947-2) in 230/400 V TN system with m = 1
Nominal cross- sectional area of conductors Instantaneous or short-time-delayed CB tripping current setting Im (amperes)
mm2 50 63 80 100 125 160 200 250 320 400 500 560 630 700 800 875
1.5 100 79 63 50 40 31 25 20 16 13 10 9 8 7 6 6
2.5 167 133 104 83 67 52 42 33 26 21 17 15 13 12 10 10
4 267 212 167 133 107 83 67 53 42 33 27 24 21 19 17 15
6 400 317 250 200 160 125 100 80 63 50 40 36 32 29 25 23
10 417 333 267 208 167 133 104 83 67 60 53 48 42 38
16 427 333 267 213 167 133 107 95 85 76 67 61
25 417 333 260 208 167 149 132 119 104 95
35 467 365 292 233 208 185 167 146 133
50 495 396 317 283 251 226 198 181
70 417 370 333 292 267
95 452 396 362
120 457
• Note: this table is calculated according to IEC60947-2, thus includes a 20% tolerance on the actual tripping current compared to the circuit breaker tripping setting (see upper note)
Fig. F25b – Maximum circuit lengths (in metres) for different sizes of copper conductor and instantaneous-tripping-current settings for industrial circuit breakers (IEC 60947-2) in 230/400 V TN system with m = 1
Nominal cross- sectional area of conductors Instantaneous or short-time-delayed CB tripping current setting Im (amperes)
mm2 1000 1120 1250 1600 2000 2500 3200 4000 5000 6300 8000 10000 12500
1.5 5 4 4
2.5 8 7 7 5 4
4 13 12 11 8 7 5 4
6 20 18 16 13 10 8 6 5 4
10 33 30 27 21 17 13 10 8 7 5 4
16 53 48 43 33 27 21 17 13 11 8 7 5 4
25 83 74 67 52 42 33 26 21 17 13 10 8 7
35 117 104 93 73 58 47 36 29 23 19 15 12 9
50 158 141 127 99 79 63 49 40 32 25 20 16 13
70 233 208 187 146 117 93 73 58 47 37 29 23 19
95 317 283 263 198 158 127 99 79 63 50 40 32 25
120 400 357 320 250 200 160 125 100 80 63 50 40 32
150 435 388 348 272 217 174 136 109 87 69 54 43 35
185 459 411 321 257 206 161 128 103 82 64 51 41
240 400 320 256 200 160 128 102 80 64 51
• Note: this table is calculated according to IEC60947-2, thus includes a 20% tolerance on the actual tripping current compared to the circuit breaker tripping setting (see upper note)
Fig. F26 – Maximum circuit lengths (in metres) for different sizes of copper conductor and rated currents for type B[2] domestic circuit breakers (IEC 60898) in a 230/400 V single-phase or three-phase TN system with m = 1
Sph Rated current (A)
mm2 1 2 3 4 6 10 16 20 25 32 40 50 63 80 100 125
1.5 1200 600 400 300 200 120 75 60 48 37 30 24 19 15 12 10
2.5 1000 666 500 333 200 125 100 80 62 50 40 32 25 20 16
4 1066 800 533 320 200 160 128 100 80 64 51 40 32 26
6 1200 800 480 300 240 192 150 120 96 76 60 48 38
10 800 500 400 320 250 200 160 127 100 80 64
16 800 640 512 400 320 256 203 160 128 102
25 800 625 500 400 317 250 200 160
35 875 700 560 444 350 280 224
50 760 603 475 380 304
Fig. F27 – Maximum circuit lengths (in metres) for different sizes of copper conductor and rated currents for type C[2] domestic circuit breakers (IEC 60898) in a 230/400 V single-phase or three-phase TN system with m = 1
Sph Rated current (A)
mm2 1 2 3 4 6 10 16 20 25 32 40 50 63 80 100 125
1.5 600 300 200 150 100 60 37 30 24 18 15 12 9 7 6 5
2.5 500 333 250 167 100 62 50 40 31 25 20 16 12 10 8
4 533 400 267 160 100 80 64 50 40 32 25 20 16 13
6 600 400 240 150 120 96 75 60 48 38 30 24 19
10 677 400 250 200 160 125 100 80 63 50 40 32
16 640 400 320 256 200 160 128 101 80 64 51
25 625 500 400 312 250 200 159 125 100 80
35 875 700 560 437 350 280 222 175 140 112
50 760 594 475 380 301 237 190 152
Fig. F28 – Maximum circuit lengths (in metres) for different sizes of copper conductor and rated currents for type D[2] domestic circuit breakers (IEC 60898) in a 230/400 V single-phase or three-phase TN system with m = 1
Sph Rated current (A)
mm2 1 2 3 4 6 10 16 20 25 32 40 50 63 80 100 125
1.5 429 214 143 107 71 43 27 21 17 13 11 9 7 5 4 3
2.5 714 357 238 179 119 71 45 36 29 22 18 14 11 9 7 6
4 571 381 286 190 114 71 57 46 36 29 23 18 14 11 9
6 857 571 429 286 171 107 86 69 54 43 34 27 21 17 14
10 952 714 476 286 179 143 114 89 71 57 45 36 29 23
16 762 457 286 229 183 143 114 91 73 57 46 37
25 714 446 357 286 223 179 143 113 89 71 57
35 625 500 400 313 250 200 159 125 100 80
50 679 543 424 339 271 215 170 136 109

### Example

A 3-phase 4-wire (230/400 V) installation is TN-C earthed. A circuit is protected by a type B circuit breaker rated at 63 A, and consists of an aluminium core cable with 50 mm² phase conductors and a neutral conductor (PEN) of 25 mm².

What is the maximum length of circuit, below which fault protection is assured by the instantaneous magnetic tripping relay of the circuit breaker?

Figure F26 gives, for 50 mm2 and a 63 A type B circuit-breaker, 603 metres, to which must be applied a factor of 0.42 (Figure F24 for $\displaystyle{ m=\frac{Sph}{SPE}=2 }$).

The maximum length of circuit is therefore:

603 x 0.42 = 253 meters.

### Particular case where one (or more) exposed conductive part(s) is (are) earthed to a separate earth electrode

Fault Protection must be provided by a RCD at the origin of any circuit supplying an appliance or group of appliances, the exposed conductive parts of which are connected to an independent earth electrode.

The sensitivity of the RCD must be adapted to the earth electrode resistance (RA2 in Figure F16). See specifications applicable to TT system.

## Notes

1. ^ This results in a calculated current value which is less than what would actually flow. If the overcurrent settings are based on this calculated value, then operation of the relay, or fuse, is assured.
2. ^ 1 2 3 4 For the definition of type B, C, D circuit breakers, refer to Fundamental characteristics of a circuit-breaker