View source for Calculation of minimum levels of short-circuit current

{{Menu_Sizing_and_protection_of_conductors}}__TOC__
{{Highlightbox|
If a protective device in a circuit is intended only to protect against short-circuit faults, it is essential that it will operate with certainty at the lowest possible level of short-circuit current that can occur on the circuit}}

In general, on LV circuits, a single protective device protects against all levels of current, from the overload threshold through the maximum rated short-circuit current breaking capability of the device. The protection device should be able to operate in a maximum time to ensure people and circuit safety, for all short-circuit current or fault current that may occur. To check that behavior, calculation of minimal short-circuit current or fault current is mandatory. 

In addition, in certain cases overload protective devices and separate short-circuit protective devices are used.

== Examples of such arrangements  ==

{{FigureRef|G43}} to {{FigureRef|G45}} show some common arrangements where overload and short-circuit protections are achieved by separate devices.

{{FigImage|DB422320_EN|svg|G43|Circuit protected by aM fuses}}

{{FigImage|DB422321_EN|svg|G44|Circuit protected by circuit breaker without thermal overload relay}}

{{FigImage|DB422322_EN|svg|G45|Circuit breaker D provides protection against short-circuit faults as far as and including the load}}

As shown in {{FigureRef|G43}} and {{FigureRef|G44}}, the most common circuits using separate devices control and protect motors.

{{FigureRef|G45}} constitutes a derogation to the basic protection rules, and is generally used on circuits of busways (busbar trunking systems), lighting rails, etc.

=== Variable speed drive ===
{{FigureRef|G46}} shows the functions provided by the variable speed drive, and if necessary some additional functions provided by devices such as circuit-breaker, thermal relay, RCD.

{{tb-start|id=Tab1208|num=G46|title=Protection to be provided for variable speeed drive applications|cols=3}}
{| class="wikitable"
|-
! Protection to be provided 
! Protection generally provided by the variable speed drive 
! Additional protection if not provided by the variable speed drive
|-
| Cable overload 
| Yes  
| CB / Thermal relay
|-
| Motor overload 
| Yes 
| CB / Thermal relay
|-
| Downstream short-circuit 
| Yes 
| 
|-
| Variable speed drive overload 
| Yes 
| 
|-
| Overvoltage 
| Yes 
| 
|-
| Undervoltage 
| Yes 
| 
|-
| Loss of phase 
| Yes 
| 
|-
| Upstream short-circuit 
| 
| Circuit-breaker 
(short-circuit tripping)
|-
| Internal fault 
| 
| Circuit-breaker 
(short-circuit and overload tripping)
|-
| Downstream earth fault (indirect contact) 
| (self protection) 
| RCD ≥ 300 mA or CB in TN earthing system
|-
| Direct contact fault 
| 
| RCD ≤ 30 mA
|}

== Conditions to be fulfilled  ==

{{Highlightbox|
The protective device must fulfill: 
*instantaneous trip setting Im < Isc<sub>min</sub> for a circuit-breaker 
*fusion current Ia &lt; Isc<sub>min</sub> for a fuse}}

The protective device must therefore satisfy the two following conditions: 
*Its breaking capacity must be greater than Isc, the 3-phase short-circuit current at its point of installation 
*Elimination of the minimum short-circuit current possible in the circuit, in a time tc compatible with the thermal constraints of the circuit conductors:

: <math>tc \le \frac{k^2S^2}{Isc_{min}\, ^2}</math> (valid for tc < 5 seconds) 

where S is the cross section area of the cable, k is a factor depending of the cable conductor material, the insulation material and initial temperature.

'''Example:''' for copper XLPE, initial temperature 90 °C, k = 143 (see IEC60364-4-43 §434.3.2 table 43A and {{FigureRef|G52}}).

Comparison of the tripping or fusing performance curve of protective devices, with the limit curves of thermal constraint for a conductor shows that this condition is satisfied if: 
*Isc (min) > Im (instantaneous or short time delay circuit-breaker trip setting current level), (see {{FigRef|G47}} ) 
*Isc (min) > Ia for protection by fuses. The value of the current Ia corresponds to the crossing point of the fuse curve and the cable thermal withstand curve (see {{FigRef|G48}} and {{FigRef|G49}})

{{FigImage|DB422323|svg|G47|Protection by circuit breaker}}

{{FigImage|DB422324|svg|G48|Protection by aM-type fuses}}

{{FigImage|DB422325|svg|G49|Protection by gG-type fuses}}

== Practical method of calculating Lmax ==

{{Highlightbox|
In practice this means that the length of circuit downstream of the protective device must not exceed a calculated maximum length: <math>L_{max}=\frac{0.8\ U\ S_{ph} }{2 \rho I_{m} }</math>}}

The limiting effect of the impedance of long circuit conductors on the value of short-circuit currents must be checked and the length of a circuit must be restricted accordingly.

For protection of people (fault protection or indirect contacts), the methods to calculate the maximum circuit length are presented in chapter F, for [[TN system - Earth-fault current calculation|TN system]] and [[IT system - Implementation of protections#Maximum circuit length|IT system (second fault)]].

Two other cases are considered below, for phase-to-phase and phase-to-neutral short-circuits.

=== 1 - Calculation of L<sub>max</sub> for a 3-phase 3-wire circuit===

The minimum short-circuit current will occur when two phase wires are short-circuited at the remote end of the circuit (see {{FigRef|G50}}).

{{FigImage|DB422326_EN|svg|G50|Definition of L for a 3-phase 3-wire circuit}}

Using the “conventional method”, the voltage at the point of protection P is assumed to be 80% of the nominal voltage during a short-circuit fault, so that 0.8 U = Isc Zd, where:

{{def
|Zd| impedance of the fault loop 
|Isc| short-circuit current (ph/ph) 
|U| phase-to-phase nominal voltage }}

For cables ≤ 120 mm<sup>2</sup>, reactance may be neglected, so that <math>Zd=\rho \frac{2L}{Sph}</math>{{fn|1}}

where:

{{def
|ρ| resistivity of conductor material at the average temperature during a short-circuit, in Ω⋅mm² / m. In practice, for circuits protected by a circuit breaker, the value to be used is ρ1, the resistivity of conductors at the maximum permissible steady-state operating temperature (see [[3-phase short-circuit current (Isc) at any point within a LV installation#Tab1204|{{FigRef|G38}}]]),
|Sph| c.s.a. of a phase conductor in mm<sup>2</sup>
|L| length in metres}}

The condition for the cable protection is Im ≤ Isc with Im = trip current which guarantees instantaneous operation of the circuit breaker..

This leads to&nbsp;<math>Im \le \frac{0.8 U}{Zd}</math>&nbsp; which gives <math>L \le \frac{0.8\ U\ S_{ph}}{2 \rho I_{m}}</math>

For conductors of similar nature, U and ρ are constants (U = 400 V for phase-to phase fault, ρ = 0.0237 Ω.mm²/m{{fn|2}} for copper conductors), so the upper formula can be written as:

<math>L_{max}=\frac {k\ S_{ph}}{I_{m}}</math> 

with Lmax = maximum circuit length in metres

For industrial circuit breakers (IEC 60947-2), the value of Im is given with ±20% tolerance, so Lmax should be calculated for Im+20% (worst case).

k factor values are provided in the following table, for copper cables, taking into account these 20%, and as a function of cross-section for Sph > 120 mm²{{fn|1}}

{{tb-start|id=Tab1209|num=|title=|cols=3}}
{| class="wikitable"
|-
! Cross-section (mm<sup>2</sup>)
! ≤ 120
! 150
! 185
! 240
! 300
|-
| k (for 400 V)
| 5800
| 5040
| 4830
| 4640
| 4460
|}

=== 2 - Calculation of L<sub>max</sub> for a 3-phase 4-wire 230/400 V circuit===

The minimum Isc will occur when the short-circuit is between a phase conductor and the neutral at the end of the circuit.

A calculation similar to that of example 1 above is required, but for a single-phase fault (230V).

* If Sn (neutral cross-section) = Sph

Lmax = k Sph / Im with k calculated for 230V, as shown in the table below

{{tb-start|id=Tab1210|num=|title=|cols=3}}
{| class="wikitable"
|-
! Cross-section (mm<sup>2</sup>)
! ≤ 120
! 150
! 185
! 240
! 300
|-
| k (for 230 V)
| 3333
| 2898
| 2777
| 2668
| 2565
|}

* If Sn (neutral cross-section) < Sph, then (for cable cross-section ≤ 120mm<sup>2</sup>)

<math>L_{max}= 6666 \frac{Sph}{Im}\frac{1}{1+m}</math>

<math>m=\frac{Sph}{Sn}</math>

== Tabulated values for Lmax  ==

Based on the practical calculation method detailed in previous paragraph, pre-calculated tables can be prepared.

In practice, the tables {{FigRef|F25}} to {{FigRef|F28}} already used in chapter [[Protection against electric shocks and electrical fires]] for earth-fault calculation can also be used here, but applying the correction factors in {{FigRef|G51}} below, to obtain Lmax value related to phase-to-phase or phase-to-neutral short-circuits.

'''Note''': for aluminium conductors, the lengths obtained must be multiplied again by 0.62.

{{tb-start|id=Tab1215|num=G51|title=Correction factor to apply to lengths obtained from {{FigRef|F25}} to {{FigRef|F28}}, to obtain Lmax considering phase-to-phase or phase-to-neutral short-circuits|cols=5}}
{| class="wikitable"
|-
! colspan="2" | Circuit detail 
! 
|-
| 3-phase 3-wire 400 V circuit or 1-phase 2-wire 400 V circuit (no neutral) 
| 
| 1.73
|-
| 1-phase 2-wire (phase and neutral) 230 V circuit 
| 
| 1
|-
| rowspan="2" | 3-phase 4-wire 230/400 V circuit or 2-phase 3-wire 230/400 V circuit (i.e with neutral) 
| Sph / S neutral = 1 
| 1
|-
| Sph / S neutral = 2 
| 0.67
|}

== Examples  ==

=== Example 1 ===
In a 3-phase 3-wire 400 V installation the short-circuit protection of a 22 kW (50 A) motor is provided by a magnetic circuit breaker type GV4L, the instantaneous short-circuit current trip is set at 700 A (accuracy of ±20 %), i.e. in the worst case would require 700 x 1.2 = 840 A to trip.

The cable c.s.a. = 10 mm² and the conductor material is copper.

In {{FigRef|F25}} the column Im = 700 A crosses the row c.s.a. = 10 mm² at the value for Lmax of 48 m. {{FigRef|G51}} gives a factor 1.73 to apply to this value for 3-phase 3-wire circuit (no neutral). The circuit breaker protects the cable against short-circuit faults, therefore, provided that its length does not exceed 48 x 1.73 = 83 metres.

=== Example 2 ===
In a 3L+N 400 V circuit, the protection is provided by a 220 A circuit breaker type NSX250N with micrologic 2 trip unit having instantaneous short-circuit protection set at 3000A (±20 %), i.e. a worst case of 3600 A to be certain of tripping.

The cable c.s.a. = 120 mm² and the conductor material is aluminium.

In {{FigRef|F25}} the column Im = 3200 A (first value > 3000A, as the table already includes the +20% on Im in its calculation) crosses the row c.s.a. = 120 mm² at the value for Lmax of 125 m. Being a 3-phase 4-wire 400 V circuit (with neutral), the correction factor from {{FigRef|G51}} to apply is 1. In complement, as the conductor is aluminium, a factor of 0.62 has to be applied.

The circuit breaker will therefore protect the cable against short-circuit current, provided that its length does not exceed 125 x 0.62 = 77 metres.

{{footnotes}}
<references>
{{fn-detail|1|For c.s.a. > 120 mm<sup>2</sup>, the resistance calculated for the conductors must be increased to account for the non-uniform current density in the conductor (due to "skin" and "proximity" effects). Suitable values are as follows:
* 150 mm<sup>2</sup>: R + 15 %
* 185 mm<sup>2</sup>: R + 20 %
* 240 mm<sup>2</sup>: R + 25 %
* 300 mm<sup>2</sup>: R + 30 %}}
{{fn-detail|2|Resistivity for copper EPR/XLPE cables when passing short-circuit current, eg for the max temperature they can withstand {{=}} 90°C (cf [[3-phase short-circuit current (Isc) at any point within a LV installation#Tab1204|{{FigureRef|G38}}]]).}}
</references>