各位电缆同仁好：

虽然我们可以通过实验仪器测试出导体的直流电阻，然后在用下面的公式，很轻易地从理论上计算出导体的交流电阻，但谁可以用自己设计的实验来验证这个公式的正确性呢？

Ra.c=Rd.c(1+Ys+Yp)

              ——其中Ra.c是导体的交流电阻，Rd.c是导体的直流电阻，Ys是集肤效应因子，Yp是邻近效应因子。

现在就有一帮外国的电缆专家做了这样的事情，并把他们做的方法和结果写了出来。

他们的工作组成员是：

          Eric DORISON   (FRANCE);

          Laurant MOREAU    (FRANCE);

         Udo FROMM   (GERMANY);

         Vince BARRY  (UNITED KINGDOM);

         Christian REMY  (ITALY);

         Jose Manuel MENDEZ  (SPAIN);

        Asakiyo UEDA  (JAPAN);

        Graeme BARNEWELL  (AUSTRILIA);

       Mike SHUVALOV (RUSSIA)

他们发表的资料名叫《LARGE CROSS-SECTIONS AND COMPOSITE SCREENS DESIGN》，里面有详细的实验设计方法及计算公式，已经他们的实验结果。

CONTENTS
 
 
GENERAL
1
  Background................................................4
2  Terms of reference...............................................4
3  Brochure Content......................................4
 
 
PART 1 - LARGE CROSS SECTIONS CONDUCTORS.
4
  Introduction................................................................5
5  General considerations..................................................................6
5.1  Skin effect and proximity effect...................................................6
5.2  Skin effect in solid conductors......................................................6
5.3  Proximity effect between solid conductors..................................................7
5.4  Skin effect and proximity effect in stranded conductors...........................7
5.5  Losses in a wire influenced by a magnetic field..............................................8
5.5.1  General expression of the electromagnetic field...............................8
5.5.2  Fields and current density within the cylinder.............................9
5.5.3  Losses calculation............................10
5.5.4  Influencing current.........................................11
5.5.5  Multiple influences : using superposition principle.................................12
5.6  Segmental conductors......................................................13
5.7  Coefficients for cores using aluminium wires......................................14
5.7.1  General.......................................................................14
5.7.2  Solid conductor....................................................14
5.7.3  Segmental conductor.....................................................16
5.7.4  Segmental conductor with peripheral strands.....................................18
5.8  I.E.C. present statements.......................................19
6  Calculation Methods for segmental cores...............................21
6.1  Several possible ways......................................21
6.2  No exact solution............................................................23
6.3  A pragmatic approach..................................23
7  Derivation of practical formula for skin effect....................................24
7.1  Background.....................................................24
7.2  Skin effect in the segments.....................................................24
7.3  Proximity effect between segments.......................................25
7.4  Sugiyama!ˉs approach........................................25
7.4.1  Introduction.....................................................25
7.4.2  Basic assumptions...........................................26
7.4.3  Principles...............................................................26
7.4.4  Proximity effect : eddy currents............................................................27
7.4.5  Proximity effects : circulating currents................................29
7.4.6  Final expression...................................................30
7.5  Comparing calculation methods...........................................31
7.5.1  General..............................................31
7.5.2  Skin effect within segments and losses due to eddy currents  in the wires...................................31
7.5.3  Proximity effects between segments............................................34
7.5.4  Proximity effects between segments, using superposition.........................34
7.5.5  Looking for validation........................................36
7.6  Practical formula....................................37
7.7  Comparison with classical formula....................................38
8  Derivation of practical formula for proximity effects..............................41
8.1  Background..................................................41
8.2  Calculations...........................................41
8.2.1  General..............................................41
8.2.2  Losses generated within the strands.............................................42
8.2.3  Losses due to circulating current between the strands.................43
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2

  Conclusion.....................................................44
8.3
9  Measurement techniques.....................................45
9.1  General.........................................................45
9.2  Electric Methods.....................................45
9.2.1  Introduction............................................45
9.2.2  Measurement of conductor ac resistance using the sheath as the return conductor......................46
9.2.3  Measurement of ac resistance using an ac bridge with a current transformer...............................46
9.2.4  Methods that could be used to give greater accuracy in measurement.........................................48
9.3  The Thermal Measurement Method...........................................50
10  Recommendations...............................................52
 
PART 2 - SHORT-CIRCUIT PERFORMANCES OF COMPOSITE SCREENS
11
  Introduction....................................53
12  Calculation method.......................................53
13  Investigating test data..........................................................53
14  Recommendations.............................................................54
 
 
PART 3 - EDDY CURRENT LOSSES IN COMPOSITE SCREENS.
  Introduction............................................................55
15
16  losses in cable screen wires....................................55
17  Circulating currents in a bundle of wires due to a magnetic field.................................56
17.1  General..........................56
17.2  Wires are assumed to be laid straight.........................................57
17.3  Wires are wound.....................................58
18  Calculation of losses in cable screen wires............................59
18.1  Eddy current losses due to the magnetic fields acting on every wire.............................59
18.2  Losses due to longitudinal currents.....................61
18.2.1  General.......................................61
18.2.2  Calculations using circuit theory.........................................61
18.3  Losses in the foil and screening effect of the foil..........................................64
18.4  Calculation results..........................................65
18.5  Stranding effect...................................................................66
18.6  Inter-wire medium and counter-helix........................................................66
19  Recommendations.....................................................66
 
REFERENCES
 
APPENDIX 1 : FROM MAXWELL!ˉS EQUATINS TO EDDY CURRENTS IN A WIRE.
1
  Maxwell!ˉs quations................................................70
2  Fields expressions.........................................72
3  Boundary conditions.................................................72
4  fields and current density within the cylinder.....................................74
 
APPENDIX 2 : SUGIYAMA!ˉS CALCULATION OFSKIN EFFECT IN SEGMENTAL CORES
1
  General.......................................................75
2  Equivalent conductance............................................75
3  Skin effect in the strands of a segment alone......................................76
4  Final expression for skin effecct in segmental cores...............................77
4.1  Introduction................................................77
4.2  Dealing with Hc..............................................77
4.3  Dealing with Hs................................................78
4.4  Consolidation........................................................79
4.5  Simplifications........................................................79
4.6  Working formulae..................................80
 
APPENDIX 3 : LOSSES IN A BUNDLE OF SCREEN WIRES
1.
  Taking into account inter-wire conductivity......................................81
2.  Losses induced by the counter helix.............................................84

5  GENERAL CONSIDERATIONS.
 
5.1  Skin effect and proximity effect.
 
Skin effect is the name given to the tendency for current to flow predominantly in the periphery of a conductor
due to the internal magnetic field in the conductor.
 
Proximity effect is the tendency for current to flow along one side of a conductor due to interaction of the
magnetic fields of the current in the conductor considered and the currents in adjacent conductors.
 
The a.c. resistance of a solid circular conductor is given by :
 
RRyy'()1
sp
 (1)
where :
   R!ˉ is the d.c. resistance,   y
 is the skin effect factor and y  is the proximity effect factor
s p
 
Skin effect in solid round circular conductors and proximity effects between solid round circular conductors
were deeply investigated, specially by A.H.M. Arnold [2 - 3 - 4 - 5], and formulae were worked out [6] for ys
and yp, through tedious calculations to approximate the Bessel!ˉs functions involved in the solution of Maxwell!ˉ
equations.
 
5.2  Skin effect in solid conductors.
 
The skin effect factor ys for an isolated solid circular conductor is given by :
 
'.'. xberxbeixbeixberx
1.xy  (2)
 
s2'2'2 xbeixber
where:
fx .2 
 
R'.
 
 
R!ˉ being the d.c. resistance.
|ì being the magnetic permeability and f the frequency.
 
ber(x), bei(x) and their derivatives ber!ˉ(x) and bei!ˉ(x) are tabulated Bessefunctions (e.g. see reference [7]).
 
The derivation of this formula is given in !ì 5..  
 
 
The following approximate formulae [6] have been obtained for ys :
 
4
x
)2(8.20 a
xyx
s
4
8.0192 x
)2(20563.00177.0136.08.38.2 bxxxyx
 
s
)2(733.0354.08.3 cxxyx
s
 
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The maximum error is less than 0.6 %, and is negligible.
The above formulae (2a) and (2c), for small and large values of x respectively are taken from Arnold!ˉs work,
while formula (2b) has been derived to fit the values of y at intermediate values of x, and at the same time not to
allow an appreciable discontinuity in the calculated values of y at x=2.8 and x=3.8.
 
5.3  Proximity effect between solid conductors.
 
Derivation of  formulae for proximity effects, even in the simple case of solid conductors, is anything but
straightforward.
 
Arnold worked out formulae for various cases, using complicated tabulated functions.
For instance, for 2 solid conductors carrying single-phase current :
 
2
d
c
.
xG
S
xy
p
2
d
c
.1
xA
S
 (3)
44
012.0042.0
xx
8.2:
xif
xGxAwith
4
20
0235.01
x
4
64
x
11
 
 
For 3 circular conductors carrying three-phase current, Neher and McGrath [8] have obtained the following
formula suitable for power frequency application :
 
22
4
x
dd
18.1
 (4)
 
cc
.312.0..
xy
p
44
SS
8.0192
x
x
27.0
4
8.0192
x
 
where d
 is the core diameter and S the axial spacing between conductors.
c
 
 
5.4  Skin effect and proximity effect in stranded conductors.
 
The basic contribution in this field once again comes from Arnold.
 
Skin effect.
 
The skin effect in a single-core cable is substantially the same as the skin effect in a solid round cylindrical
conductor having the same d.c. resistance per unit length.
 
 
Proximity effect.
 
The proximity effect may be calculated using formulae developed for solid round conductors provided that the
 , which is the ratio of the resistance of the path along the
resistance of the conductor is divided by a factor k
p
strands to the resistance of the path across the strands.
 
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This factor depends on many parameters such as the surface condition of the strands, the lay of the strands, the
impregnation of the core and the tightness of the insulation on the core.

7.5  Comparing calculation methods
7.5.1  General
 
This paragraph presents a comparison of  the losses derived using Sugiyama!ˉs formula and other calculation
methods.
 
Particularly, circuit theory was applied to core structures with a ? true ? geometry; the current distribution in the
core is illustrated in following figures, for various configurations.
 
The purpose of these calculations was to check the important assumption in Sugiyama!ˉs approach, that sector
shaped segments may be considered as circular-shaped.
Of course, the actual distribution of the wires in a segment is not as fine as it may be in a simulated one, and the
cross-section of the wires is no more circular. So that the calculation results using circuit theory have also to be
carefully considered.
 
A 2000 mm2 Copper core was considered, with a current flowing in the core of 1000 A.
 
7.5.2  Skin effect within segments and losses due to eddy currents  in the wires.
 
Losses due to the current flow and skin effect within every segments  (W/m)

Segment  number                     SUGIYAMA     (IEC for circular shape)                          Circuit theory    

            4                                                          9.2                                                                              9.3    
            5                                                          9.0                                                                              9.1
            6                                                          8.8                                                                              8.9
            7                                                          8.8                                                                              8.9
 
Losses due to the skin effect within every segments  (mW/m)

Segment    number                       SUGIYAMA     (IEC for circular shape)

            4                                                         570
            5                                                         370
            6                                                         260
            7                                                         190
 

Losses inside every wires due to magnetic fields  (mW/m)

Segment            SUGIYAMA                  SUGIYAMA          From magnetic fields due to every wires  (*) 
 number               Cos  = 0.95                        Cos  = 1                                          Cos  = 1 
 
         4                         128                                   116                                                   120
         5                         105                                     95                                                     95
         6                           85                                     77                                                      78
         7                           71                                     64                                                      65
 
(*) Losses in a wire (radius a) located in a magnetic field H are given by :  

The magnetic field acting on a wire is the resultant of the magnetic fields due to the whole of other wires.
Elementary magnetic fields are derived from Ampere!ˉs teorem, assuming every wire carries the same current.

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