TN system - Earth-fault current calculation

Three methods of calculation are commonly used:

• The method of impedances, based on the trigonometric addition of the system resistances and inductive reactances
• The method of composition
• The conventional method, based on an assumed voltage drop and the use of prepared tables

Methods of determining levels of short-circuit current

In TN-earthed systems, a short-circuit to earth will, in principle, always 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 short-circuit current).

The most recent IEC recommendations for indirect-contact protection on TN earthing systems only relates maximum allowable tripping times to the nominal system voltage.(see Figure F12) 

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 short-circuit 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 ferro-magnetic materials.

Method of impedances

For calculations, modern practice is to use software agreed by National Authorities, and based on the method of impedances, such as Ecodial. National Authorities generally also publish Guides, which include typical values, conductor lengths, etc.

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 short-circuit earth-fault current is calculated, using the formula:
[math]\displaystyle{ I=\frac{Uo}{\sqrt{\left ( \sum R \right )^2 + \left ( \sum X \right )^2 } } }[/math] 
where
(ΣR) ² = (the sum of all resistances in the loop)² at the design stage of a project.
and (ΣX) ² = (the sum of all inductive reactances in the loop)²
and Uo = nominal system phase-to-neutral voltage.
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:
[math]\displaystyle{ Isc=I\frac{Uo}{U+Zs\ Isc} }[/math]
where
Isc = upstream short-circuit current
I = end-of-loop short-circuit current
Uo = 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.

(1) This results in a calculated current value which is less than that it would actually flow. If the overcurrent settings are based on this calculated value, then operation of the relay, or fuse, is assured.

Conventional method

The maximum length of any circuit of a TN-earthed installation is: [math]\displaystyle{ \definecolor{bggrey}{RGB}{234,234,234}\pagecolor{bggrey}\frac{0.8\ Uo\ Sph}{\rho \left ( 1+m \right )Ia} }[/math]

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

Principle The principle bases the short-circuit current calculation 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 account of 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:
[math]\displaystyle{ Lmax=\frac{0.8\ Uo\ Sph}{\rho \left ( 1+m \right )Ia} }[/math] 
where:
Lmax = maximum length in metres
Uo = phase volts = 230 V for a 230/400 V system
ρ = resistivity at normal working temperature in ohm-mm²/metre
(= 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.
[math]\displaystyle{ m=\frac{Sph}{SPE} }[/math]
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.
(see Fig. F39)

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Fig. F39: Calculation of L max. for a TN-earthed system, using the conventional method

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Tables

The following tables give the length of circuit which must not be exceeded, in order that persons be protected against indirect contact hazards by protective devices

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 safety against indirect contact.
Correction factor m
Figure F40 indicates the correction factor to apply to the values given in Figures F41 to F44, 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 system earthing (see Fig.F45 )
• Type of circuit-breaker (i.e. B, C or D)⁽¹⁾

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

Equivalent tables for protection by Compact and Acti 9 circuit-breakers (Schneider Electric) are included in the relevant catalogues.

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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. F40: Correction factor to apply to the lengths given in tables F41 to F44 for TN systems

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Circuits protected by general purpose circuit-breakers (Fig. F41)

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Nominal cross- sectional area of conductors	Instantaneous or short-time-delayed tripping current Im (amperes)
mm2	50	63	80	100	125	160	200	250	320	400	500	560	630	700	800	875	1000	1120	1250	1600	2000	2500	3200	4000	5000	6300	8000	10000	12500
1.5	100	79	63	50	40	31	25	20	16	13	10	9	8	7	6	6	5	4	4
2.5	167	133	104	83	67	52	42	33	26	21	17	15	13	12	10	10	8	7	7	5	4
4	267	212	167	133	107	83	67	53	42	33	27	24	21	19	17	15	13	12	11	8	7	5	4
6	400	317	250	200	160	125	100	80	63	50	40	36	32	29	25	23	20	18	16	13	10	8	6	5	4
10			417	333	267	208	167	133	104	83	67	60	53	48	42	38	33	30	27	21	17	13	10	8	7	5	4
16					427	333	267	213	167	133	107	95	85	76	67	61	53	48	43	33	27	21	17	13	11	8	7	5	4
25							417	333	260	208	167	149	132	119	104	95	83	74	67	52	42	33	26	21	17	13	10	8	7
35								467	365	292	233	208	185	167	146	133	117	104	93	73	58	47	36	29	23	19	15	12	9
50									495	396	317	283	251	226	198	181	158	141	127	99	79	63	49	40	32	25	20	16	13
70												417	370	333	292	267	233	208	187	146	117	93	73	58	47	37	29	23	19
95														452	396	362	317	283	263	198	158	127	99	79	63	50	40	32	25
120																457	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

Fig. F41:  Maximum circuit lengths (in metres) for different sizes of copper conductor and instantaneous-tripping-current settings for general-purpose circuit-breakers in 230/400 V TN system with m = 1

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Circuits protected by Compact or Acti 9 circuit-breakers for industrial or domestic use (Fig. F42 to Fig. F44)

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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. F42:  Maximum circuit lengths (in meters) for different sizes of copper conductor and rated currents for type B ⁽¹⁾ circuit-breakers in a 230/400 V single-phase or three-phase TN system with m = 1

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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. F43:  Maximum circuit lengths (in metres) for different sizes of copper conductor and rated currents for type C ⁽¹⁾ circuit-breakers in a 230/400 V single-phase or three-phase TN system with m = 1

(1) For the definition of type B and C circuit-breakers refer to chapter H

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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	80	46	36	29	23	18	14	11	9
6		857	571	429	286	171	107	120	69	54	43	34	27	21	17	14
10			952	714	476	284	179	200	114	89	71	57	45	36	29	23
16					762	457	286	320	183	143	114	91	73	57	46	37
25						714	446	500	286	223	179	143	113	89	71	57
35							625	700	400	313	250	200	159	125	80	100
50								848	543	424	339	271	215	170	136	109

Fig. F44: Maximum circuit lengths (in metres) for different sizes of copper conductor and rated currents for type D ⁽¹⁾ circuit-breakers in a 230/400 V single-phase or three-phase TN system with m = 1

(1) For the definition of type D circuit-breaker refer to chapter H

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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 cored cable with 50 mm² phase conductors and a neutral conductor (PEN) of 25 mm².
What is the maximum length of circuit, below which protection of persons against indirect-contact hazards is assured by the instantaneous magnetic tripping relay of the circuit-breaker?
Figure F42 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 F40 for [math]\displaystyle{ m=\frac{Sph}{SPE}=2 }[/math]).
The maximum length of circuit is therefore:
603 x 0.42 = 253 metres.

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

Protection must be provided against indirect contact 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 (R_A2 in Figure F45). See specifications applicable to TT system.

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Fig. F45: Separate earth electrode

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