# IEC 60287-1-2:2023

(Main)## Electric cables - Calculation of the current rating - Part 1-2: Current rating equations (100 % load factor) and calculations of losses - Sheath eddy current loss factors for two circuits in flat formation

## Electric cables - Calculation of the current rating - Part 1-2: Current rating equations (100 % load factor) and calculations of losses - Sheath eddy current loss factors for two circuits in flat formation

IEC 60287-1-2:2023 provides a method for calculating the eddy current losses in the metallic sheaths of single-core cables arranged as a three-phase double circuit in flat formation. The sheaths are bonded at one point or are cross-bonded so that there are no significant sheath circulating currents. Where metallic sheaths are bonded at both ends there are significant circulating currents which result in a lower current-carrying capacity. A method of calculating circulating current losses for double circuits is provided in IEC 60287-1-3.

The method descibed in this document provides coefficients which are applied as corrections to the loss factors for the sheaths of one isolated three-phase circuit. These corrections are negligible for cables where the parameter m is less than approximately 0,1 (m = ω/(Rs · 107)), which corresponds to a sheath longitudinal resistance higher than 314 µΩ/m at 50 Hz.

Consequently, the method is used for most sizes of aluminium-sheathed cables, but is not required for lead-sheathed cables unless they are unusually large.

The coefficients are provided in tabular form and have been computed from fundamental formulae for sheath losses, the evaluation of which calls for expertise in computer programming which will possibly not be readily available in general commercial situations. The development of simplified formulae for some of the tabulated coefficients is under consideration.

Losses for cables in a single circuit is covered in IEC 60287-1-1.

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IEC 60287-1-2

®

Edition 2.0 2023-05

INTERNATIONAL

STANDARD

Electric cables – Calculation of the current rating –

Part 1-2: Current rating equations (100 % load factor) and calculations of losses –

Sheath eddy current loss factors for two circuits in flat formation

IEC 60287-1-2:2023-05(en)

---------------------- Page: 1 ----------------------

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---------------------- Page: 2 ----------------------

IEC 60287-1-2

®

Edition 2.0 2023-05

INTERNATIONAL

STANDARD

Electric cables – Calculation of the current rating –

Part 1-2: Current rating equations (100 % load factor) and calculations of losses –

Sheath eddy current loss factors for two circuits in flat formation

INTERNATIONAL

ELECTROTECHNICAL

COMMISSION

ICS 29.060.20 ISBN 978-2-8322-6972-5

Warning! Make sure that you obtained this publication from an authorized distributor.

® Registered trademark of the International Electrotechnical Commission

---------------------- Page: 3 ----------------------

– 2 – IEC 60287-1-2:2023 © IEC 2023

CONTENTS

FOREWORD . 3

1 Scope . 5

2 Normative references . 5

3 Terms, definitions and symbols. 5

3.1 Terms and definitions . 5

3.2 Symbols . 6

4 Description of method . 7

4.1 General . 7

4.2 Outline of method . 7

4.3 Criteria for use of formulae and coefficients . 8

5 Formulae for sheath loss factors for high-resistance sheaths in a single circuit, λ . 8

0s

6 Calculation of the coefficients C , C and C . 9

H N J

6.1 Allocation of coefficients to each cable, time sequence and phase

identification . 9

6.2 Calculation of coefficients C (1, 2 and 3), Table 1 . 10

H

6.3 Calculation of coefficients C (1, 2, 3, 4, 5 and 6), Table 2 . 11

N

6.4 Calculation of coefficients C (1, 2, 3, 4, 5 and 6), Table 3 to Table 11 . 11

J

6.5 Calculation of coefficients G and g . 13

s s

7 Notes on transposition of cables . 14

8 Worked examples of calculation of eddy current losses . 14

8.1 Overview . 14

8.2 Example 1 . 14

8.3 Example 2 . 16

Bibliography . 30

Figure 1 – Cable configuration . 8

Table 1 – C coefficients . 19

H

Table 2 – C coefficients . 20

N

Table 3 – C coefficients . 21

J

Table 4 – C coefficients . 22

J

Table 5 – C coefficients . 23

J

Table 6 – C coefficients . 24

J

Table 7 – C coefficients . 25

J

Table 8 – C coefficients . 26

J

Table 9 – C coefficients . 27

J

Table 10 – C coefficients . 28

J

Table 11 – C coefficients . 29

J

---------------------- Page: 4 ----------------------

IEC 60287-1-2:2023 © IEC 2023 – 3 –

INTERNATIONAL ELECTROTECHNICAL COMMISSION

____________

ELECTRIC CABLES –

CALCULATION OF THE CURRENT RATING –

Part 1-2: Current rating equations

(100 % load factor) and calculation of losses –

Sheath eddy current loss factors for two circuits in flat formation

FOREWORD

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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is

indispensable for the correct application of this publication.

9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of patent

rights. IEC shall not be held responsible for identifying any or all such patent rights.

IEC 60287-1-2 has been prepared by IEC technical committee 20: Electric cables. It is an

International Standard.

This second edition cancels and replaces the first edition published in 1993. This edition

constitutes a technical revision.

This edition includes the following significant technical changes with respect to the previous

edition:

a) the symbols have been harmonized and aligned with the symbols used in the IEC 60287

and IEC 60853 series.

---------------------- Page: 5 ----------------------

– 4 – IEC 60287-1-2:2023 © IEC 2023

The text of this International Standard is based on the following documents:

Draft Report on voting

20/2097/FDIS 20/2104/RVD

Full information on the voting for its approval can be found in the report on voting indicated in

the above table.

The language used for the development of this International Standard is English.

This document was drafted in accordance with ISO/IEC Directives, Part 2, and developed in

accordance with ISO/IEC Directives, Part 1 and ISO/IEC Directives, IEC Supplement, available

at www.iec.ch/members_experts/refdocs. The main document types developed by IEC are

described in greater detail at www.iec.ch/publications.

A list of all parts in the IEC 60287 series, published under the general title Electric cables –

Calculation of the current rating, can be found on the IEC website.

The committee has decided that the contents of this document will remain unchanged until the

stability date indicated on the IEC website under webstore.iec.ch in the data related to the

specific document. At this date, the document will be

• reconfirmed,

• withdrawn,

• replaced by a revised edition, or

• amended.

---------------------- Page: 6 ----------------------

IEC 60287-1-2:2023 © IEC 2023 – 5 –

ELECTRIC CABLES –

CALCULATION OF THE CURRENT RATING –

Part 1-2: Current rating equations

(100 % load factor) and calculation of losses –

Sheath eddy current loss factors for two circuits in flat formation

1 Scope

This part of IEC 60287 provides a method for calculating the eddy current losses in the metallic

sheaths of single-core cables arranged as a three-phase double circuit in flat formation. The

sheaths are bonded at one point or are cross-bonded so that there are no significant sheath

circulating currents. Where metallic sheaths are bonded at both ends there are significant

circulating currents which result in a lower current-carrying capacity. A method of calculating

circulating current losses for double circuits is provided in IEC 60287-1-3.

The method descibed in this document provides coefficients which are applied as corrections

to the loss factors for the sheaths of one isolated three-phase circuit. These corrections are

7

negligible for cables where the parameter m is less than approximately 0,1 (m = ω/(R · 10 )),

s

which corresponds to a sheath longitudinal resistance higher than 314 µΩ/m at 50 Hz.

Consequently, the method is used for most sizes of aluminium-sheathed cables, but is not

required for lead-sheathed cables unless they are unusually large.

The coefficients are provided in tabular form and have been computed from fundamental

formulae for sheath losses, the evaluation of which calls for expertise in computer programming

which will possibly not be readily available in general commercial situations. The development

of simplified formulae for some of the tabulated coefficients is under consideration.

Losses for cables in a single circuit is covered in IEC 60287-1-1.

2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies.

For undated references, the latest edition of the referenced document (including any

amendments) applies.

IEC 60287-1-1:2023, Electric cables – Calculation of the current rating – Part 1-1: Current rating

equations (100 % load factor) and calculation of losses – General

3 Terms, definitions and symbols

3.1 Terms and definitions

No terms and definitions are listed in this document.

ISO and IEC maintain terminology databases for use in standardization at the following

addresses:

• IEC Electropedia: available at http://www.electropedia.org/

• ISO Online browsing platform: available at http://www.iso.org/obp

---------------------- Page: 7 ----------------------

– 6 – IEC 60287-1-2:2023 © IEC 2023

3.2 Symbols

C , C , C , C coefficients used to interpolate for C and C

A0 B0 C0 D0 H J

C , C , C coefficients which correct for sheath resistance, the

H1 H2 H3

values obtained relate to cables 1, 2 or 3 in a single

circuit

C , C , C , C , C , coefficients which introduce the mutual influences

N1 N2 N3 N4 N5

between circuits and are therefore dependent on the

C

N6

relative phase sequences of cables 1 to 3 and 4 to 6

C , C , C , C , C , coefficients which depend on the cable positions 1 to 3

J1 J2 J3 J4 J5

and 4 to 6 in each circuit

C

J6

C C coefficients used for calculation of coefficients C

M0, Z0 H

C , C , C , C coefficients used to interpolate for C

S T U V J

C coefficient used for calculation of coefficients C

Y J

C coefficient used for calculation of coefficients G and g

ß1 s s

D external diameter of the metal sheath (mm)

s

D the diameter of the imaginary cylinder which just (mm)

it

touches the inside surface of the troughs of a

corrugated sheath

the diameter of the imaginary coaxial cylinder which just

D (mm)

oc

touches the crests of a corrugated sheath

coefficient which accounts for losses due to eddy

G

s

currents across the thickness of the sheath due to the

current in the conductor

R alternating current resistance of the conductor at its (Ω/m)

c

maximum operating temperature

R resistance of the sheath (Ω/m)

s

c distance between centres of cables in adjoining circuits (mm)

1

d mean diameter of sheath or screen (mm)

f system frequency (Hz)

g coefficient which accounts for losses due to eddy

s

currents across the thickness of the sheath, due to

currents in adjacent cables

m parameter used in calculation of loss factor

s distance between centres of cables in the same circuit (mm)

t thickness of the sheath (mm)

s

y equal to s/c

1 1

z equal to d/2 · s

λ sheath loss factor for a high-resistance sheath in a

0s

single circuit

λ ′′ sheath loss factor for a low-resistance sheath in a single

1s

circuit

λ ′′ sheath loss factor for a low-resistance sheath in a

1d

double circuit

ρ electrical resistivity of sheath material at operating (Ω · m)

s

temperature

ω angular frequency of system (2πf) (l/s)

---------------------- Page: 8 ----------------------

IEC 60287-1-2:2023 © IEC 2023 – 7 –

4 Description of method

4.1 General

The method proceeds in a way similar to that used for single circuits in IEC 60287-1-1. There,

formulae for loss factors applicable to sheaths having a longitudinal resistance such that m is

less than 0,1 (R = 314 µΩ/m at 50 Hz) are given, together with empirical formulae to calculate

s

the correction coefficient for lower resistance sheaths.

However, for double circuits, accurate empirical formulae covering the complete range of

coefficients would contain so many terms that their use would show little or no advantage over

the use of precise, tabulated coefficients with interpolation, as necessary. This latter course

has the advantage that the accuracy of the loss factors can be closely equal to that of the

original calculations and is better than 1 %.

The development of empirical formulae for a limited range of coefficients is under consideration.

In order to explain the method, it is described here in a way appropriate to manual evaluation

of the arithmetic. However, because of the appreciable effort required to provide loss factors

for six cables, it is to be expected that calculations will usually be effected by means of a

computer. Under these circumstances, the decision to use interpolation (as necessary) between

tabulated values is fully justified.

However, in many cases, values of the relevant parameters will be such that interpolation is

unnecessary or can be accomplished with sufficient accuracy by inspection.

Corrections to cover the effect of eddy currents circulating within the thickness of a sheath are

derived with the use of the same formulae as those used in IEC 60287-1-1.

4.2 Outline of method

The loss factor for the sheath of a given cable in a double-circuit flat formation (see Figure 1)

is evaluated as follows:

R

s

λ′′ λC⋅ 1 to 3⋅C 1 to 6⋅C 1 to 6⋅+g G

( ) ( ) ( )

1d 0s H N J s s

R

c

The tasks performed by coefficients C and C are not directly related to any physical function

N J

but have been selected to simplify the tabulation. The nomenclature is arbitrary.

Values of C , C and C are obtained from Table 1 to Table 11 and are chosen according

H N J

the following parameters together with the position of the cable and the phase sequence

to

of the currents in the conductors.

ω

−7

m ×10

R

s

ω= 2πf

d

z=

2s

=

=

---------------------- Page: 9 ----------------------

– 8 – IEC 60287-1-2:2023 © IEC 2023

s

y =

1

c

1

where

c is the distance between the centres of cables in adjoining circuits (see Figure 1) (mm).

1

Figure 1 – Cable configuration

NOTE The factors for a single circuit having low-resistance sheaths can be obtained by using the coefficients

C (1, 2 and 3) only, as follows:

H

R

s

′′

λ λC⋅ (1 to 3)⋅+g G

1s 0s H s s

R

c

4.3 Criteria for use of formulae and coefficients

For sheaths for which the value of m is less than 0,1, which includes most lead-sheathed cables,

it can be assumed that the coefficients C , C , C and g are unity and G is zero. In such

H N J s s

circumstances, λ can be used for twin circuits without correction.

0s

When the value of m is equal to 0,1 or greater, which is generally the case for all but the smaller

aluminium-sheathed cables, values for C , C , C and g shall be calculated. The coefficient

H N J s

G is important only when the value of m is 1,0 or higher.

s

5 Formulae for sheath loss factors for high-resistance sheaths in a single

circuit, λ

0s

The sheath loss factor λ is given by

0s

2

2

md

λC=

0s C0

2

2s

1+ m

( )

For three single-core cables in flat formation, the coefficient C is given by:

C0

Cable Coefficient C

C0

Centre cable 6

Outer cables 1,5

=

---------------------- Page: 10 ----------------------

IEC 60287-1-2:2023 © IEC 2023 – 9 –

6 Calculation of the coefficients C , C and C

H N J

6.1 Allocation of coefficients to each cable, time sequence and phase identification

, C and C are dependent on the

It is important to note the way in which the coefficients C

H N J

time sequence of the currents and the physical position of the conductors.

The cables shall be numbered according to Figure 1.

The coefficients C , C and C of Table 1 are allocated on a basis of time sequence

H1 H2 H3

associated with the positions of the cables, so that the following single-circuit arrangements

have the same time sequence:

Cable number 1 2 3

Sequence R S T

Or S T R

Or T R S

With coefficients C C C

H1 H2 H3

In the above example, cable 1 is always the outer conductor on a leading phase and takes

coefficient C . Cable 3 is the outer conductor on a lagging phase and takes coefficient C .

H1 H3

It will be seen that, for these cases, the phase identification implied by the letters R, S and

T is not important, it is only the time sequence which is of significance.

In double circuits, if either circuit has a reversed sequence, the values of C shall be allocated

H

to the cables in the reverse order. The allocation of coefficient C is dependent on the time

H

sequence within each circuit.

In a double-circuit configuration, the phase identification implied by the symbols is significant

to the extent that the phase identification in relation to the cable position in one circuit shall

be either the same as, in the forward sequence, or a mirror image of, in the reverse

sequence, that in the other.

Two sets of coefficients C to C are given in Table 2 corresponding to the forward and

N1 N6

reverse sequences. If the cable positions are labelled sequentially and the phase

identification rules are adhered to, the coefficients are allocated on the same basis as

coefficient C . Note that the values for cables 4, 5 and 6 in the reversed sequence are a

H

reflection of the values for cables 1, 2 and 3.

The number of input parameters involved for the coefficients C to C makes it desirable to

J1 J6

use several tables. Table 3 to Table 8 are for each cable for the forward sequence installation.

For the reverse sequence, Table 9 to Table 11 are provided and the coefficients for cables 1

to 3 are also used for cables 6 to 4, in that order. The allocation is on the same lines as those

for coefficient C .

N

The following tables give examples of four common cases, where letters R, S, T are used for

convenience and are equivalent to other well-known sets of symbols to denote time sequence

and phase identification, such as L1, L2, L3; a, b, c; R, Y, B.

---------------------- Page: 11 ----------------------

– 10 – IEC 60287-1-2:2023 © IEC 2023

a) Forward sequence

Cable number 1 2 3 4 5 6

Sequence R S T R S T

Allocation C C C C C C C Table 1

H H1 H2 H3 H1 H2 H3

Allocation C C C C C C C Table 2, forward

N N1 N2 N3 N4 N5 N6

Allocation C C C C C C C Table 3 to Table 8, forward

J J1 J2 J3 J4 J5 J6

b) Forward sequence

Cable number 1 2 3 4 5 6

Sequence T S R T S R

Allocation C C C C C C C Table 1

H H3 H2 H1 H3 H2 H1

Allocation C C C C C C C Table 2, forward

N N6 N5 N4 N3 N2 N1

Allocation C C C C C C C Table 3 to Table 8, forward

J J6 J5 J4 J3 J2 J1

c) Reverse sequence

Cable number 1 2 3 4 5 6

Sequence R S T T S R

Allocation C C C C C C C Table 1

H H1 H2 H3 H3 H2 H1

Allocation C C C C C C C Table 2, forward

N N1 N2 N2 N4 N5 N6

Allocation C C C C C C C Table 9 to Table 11, forward

J J1 J2 J3 J4 J5 J6

d) Reverse sequence

Cable number 1 2 3 4 5 6

Sequence T S R R S T

Allocation C C C C C C C Table 1

H H1 H2 H3 H3 H2 H1

Allocation C C C C C C C Table 2, forward

N N6 N5 N4 N3 N2 N1

Allocation C C C C C C C Table 9 to Table 11, forward

J J6 J5 J4 J3 J2 J1

6.2 Calculation of coefficients C (1, 2 and 3), Table 1

H

Each coefficient C is obtained from Table 1 using the parameters m and z as well as the

H

position of each cable (see 6.1).

When values of m and z involve interpolation between values in Table 1, the following procedure

should be used where interpolation by inspection is not desired.

From the relevant part of Table 1, values for C (a, b, c, d) are obtained as shown in the

H

following diagram:

z z z

0 1

m C C

0 Ha Hc

m C

H

m C C

1 Hb Hd

where m , m , z and z are tabulated values smaller and larger than the values of m and z.

0 1 0 1

---------------------- Page: 12 ----------------------

IEC 60287-1-2:2023 © IEC 2023 – 11 –

Tabulate:

m .

0

. C = (m − m ) .

m

1 M0 1 0

z .

0

z . C = (z − z ) .

1 Z0 1 0

C .

Ha

C .

Hb

C .

Hc

C .

Hd

Then:

C = C = .

A Ha

C = (C – C )/C = .

B Hb Ha M0

C = (C – C )/C = .

C Hc Ha Z0

C = (C + C – C – C )/C · C = .

D Hd Ha Hc Hb M0 Z0

Add together:

C = .

A

+C · (m – m ) = .

B 0

+C · (z – z ) = .

C 0

+C · (m – m ) · (z – z ) = .

D 0 0

Coefficient C = total = .

H

This process shall be repeated for each of the three cables in a circuit to obtain C , C and

H1 H2

C .

H3

6.3 Calculation of coefficients C (1, 2, 3, 4, 5 and 6), Table 2

N

Values of coefficient C are obtained from Table 2, using parameter y for each cable. The

N

table has values for the forward and reverse sequences. Note that in the latter case the

coefficients for cables 4, 5 and 6 form a mirror image of those for cables 1 to 3.

Where interpolation is required, a linear one-dimensional interpolation is adequate.

6.4 Calculation of coefficients C (1, 2, 3, 4, 5 and 6), Table 3 to Table 11

J

Values of coefficient C for each cable are obtained from Table 3 to Table 11, according to

J

the sequence of the currents and the parameters m, z and y .

1

Table 3 to Table 8 apply to the six cables when the currents in the conductors follow the

forward sequence. But, in a reversed sequence, Table 9 to Table 11 apply to cables 1 to 3

and also to cables 6 to 4, in that order.

---------------------- Page: 13 ----------------------

– 12 – IEC 60287-1-2:2023 © IEC 2023

Interpolation between all three input parameters can be necessary and the following scheme

for a three-dimensional interpolation can be used.

The tables for each cable are arranged in groups, one for each value of the parameter y . Two

1

groups can be chosen, one for a value of y smaller, and one for a value larger, than the input

1

value. For each group, values of C (a to d) and C (e to f) are required (in a similar way to

J J

the interpolation for C ), as shown in the following diagrams.

H

z z z z

z z

0 1 0 1

m C C m C C

0 Ja Jc 0 Je Jg

m * m *

m C C m C C

1 Jb Jd 1 Jf Jh

Group for y Group for y

0 1

Interpolation between the values marked * gives t

**...**

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