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Calculation of the Eddy Current and Hysteresis Losses in Sheathed Cables Inside
a Steel Pipe
Article in IEEE Transactions on Power Delivery · November 2010
DOI: 10.1109/TPWRD.2010.2049509 · Source: IEEE Xplore
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1
Calculation of the eddy current and hysteresis
losses in sheathed cables inside a steel pipe
Wael Moutassem, Graduate Student Member, IEEE, and George J. Anders, Fellow, IEEE
1Abstract--
This paper presents an analytical method for
approximating the eddy current and hysteresis losses in a system
of multiple sheathed cables placed in any configuration inside a
steel pipe. The method is based on the theory of images, which
this paper expands to apply to pipes having both high magnetic
permeability and high electric conductivity at the same time. The
method of images, in combination with approximating the cable
conductors and sheaths as multiple physical filaments, is used to
compute the final current distributions in the cables and pipe and
thus the associated losses. The accuracy of the proposed method
is assessed against numerical solutions obtained using the
Maxwell finite element program by Ansoft.
Index Terms--cable ampacity, cables in steel casings, eddy
current losses, hysteresis losses.
I. INTRODUCTION
T
HE maximum current that a cable can carry without
overheating, also known as the cable ampacity, is
determined by the cable design and the laying conditions. In
multi-cable installations, heat produced by one cable affects
neighboring cables. If the cables are placed inside a steel pipe,
the interactions become more complicated, and the calculation
of the ampacity becomes even more nontrivial.
Installing large steel pipes, often referred to as casings,
containing many cables is becoming a growing trend and is
especially useful in railway or river crossings. The ease and
quickness of their installation makes them a very attractive
option. Moreover, the steel pipe provides magnetic shielding,
reducing the magnetic fields emitted to the surroundings.
However, the presence of the magnetic steel pipe and
metallic cable sheaths causes eddy and circulating currents in
the pipe and cables, and that makes the ampacity calculation
very complicated. Moreover, in some cases, the eddy current
losses in the pipe and in the sheaths are very significant, and
this results in a considerable reduction in the ampacity of the
cables.
The IEEE and IEC standards [1-3] deal with installations
involving three-cables-in-a-pipe, providing equations for the
steel pipe loss factor for the triangular and cradled
configurations with the cables located at the bottom of the
pipe. However, the results reported in this paper show that, for
asymmetrical arrangements, the losses computed with the
1 Wael Moutassem is with the Dept. of Electrical and Computer Engineering
at the University of Toronto, email: wael.moutassem@utoronto.ca
George Anders is with the Dept. of Electrical and Computer Engineering at
the University of Toronto and Kinectrics, email: George.anders@attglobal.net
standard approach are grossly underestimated, even by an
order of magnitude. This issue clearly points out to the need
for more accurate calculations.
Various methods to account for the presence of a magnetic
steel pipe surrounding solid cylindrical conductors are
described in [4-8]. References [4-5] describe analytical
approaches, whereas [6-8] provide numerical solutions.
Reference [4] provides an analytical method for calculation
of the eddy current loss in a magnetic pipe containing nonsheathed insulated conductors. The method replaces the
magnetic pipe and conductors with a cylindrical magnetic
sheet flush with a cylindrical current sheet carrying a current
density equivalent to that of the inner surface of the pipe.
Eddy currents are then calculated from the Helmholtz
equation. Mekjian et al. [5] present an analytical method
based on the theory of images. The steel pipe is replaced with
solid conductors, which are the images of the conductors
inside the pipe. The resulting system of conductors and their
images is relatively easier to solve using fundamental
electromagnetic theory. References [4] and [5] ignore the
effect of cable sheaths, because they assume that the sheath
losses will be smaller than 5% of the total losses. However, in
circuits with cable sheaths bonded at multiple points, the
sheath losses can become very significant. Mekjian et al. [5]
make a conceptual mistake of assuming that the image
produced with respect to a material of high permeability and
low conductivity can accurately represent an image produced
with respect to a steel pipe, which has high permeability and
high conductivity.
References [6-8] provide numerical solutions that are based
on the Finite Element (FEM) method. The application of the
FEM technique is the most accurate approach for solving the
problem at hand. Normally, one of the available commercial
programs could be used for this purpose. However, FEM
technique requires careful preparation of the input data that
has to be manually changed each time a new cable
configuration inside the casing is studied and it requires
significant simulation times. In addition, the numerical
methods are less insightful than the analytical approaches and
are not suitable for standardization purposes.
The goal of the work presented in this paper is to develop
an analytical solution for the approximation of losses in a steel
casing with multiple cables inside the pipe and an arbitrary
cable arrangement. Such installations may have very high
sheath and pipe losses, and because these installations are used
2
more and more frequently, a practical, relatively simple, and
fairly accurate solution is required.
The analytical method developed in this paper is based on
the theory of images and on the method of filaments, both well
known to engineers and scientists.
In a system of sheathed cables inside a magnetic pipe, the
eddy currents induced in the pipe affect the distribution of
currents in the sheathed cables, and vice versa. These mutual
effects are difficult to formulate analytically, but using the
physical filaments approximation for the cables and sheaths,
the analysis becomes much simpler. Accurate approximation
of the pipe with physical filaments would require a very large
number of filaments, due to its size and the very small skin
depth of the pipe. A computationally less intensive approach
is to use the image method described in Section II to model the
effect of the steel pipe on the inside cables by replacing the
pipe with image filaments. This process allows one to analyze
the final distribution of currents in the cables, by using the
method outlined in Section III for the combination of all
physical and image filaments.
Section IV shows the computation of the eddy current and
hysteresis losses in the cables and pipe, based on the
calculated current density. To simplify the problem, an
important approximation is made here with an assumption that
the pipe has a constant, uniform “effective” magnetic
permeability value. This assumption allows the system to be
solved analytically through Maxwell’s equations with the use
of phasors. Section V presents test cases for comparing the
accuracy of the proposed analytical solution against numerical
solutions given by the commercial program Maxwell by
Ansoft. Section VI concludes the paper.
II. METHOD OF IMAGES
interior of the pipe (region I) is given by:
AI (r ,  ) 


0 I  d  1 
0 I
n
  B r  2  r  n  cos(n( -  )) - 2 ln(r )  C
n
n
n 1 

(2)
where Bn , C are constants to be determined.
Because the current in steel flows close to the surface, the
pipe material is assumed to extend radially to infinity. This
assumption simplifies the mathematical derivations and is
justified for practical magnetic steel pipes that are thicker than
3 mm [10]. Within the pipe material (region II), the general
solution for the magnetic vector potential is as follows:
AII (r ,  ) 

 A K (kr ) cos(n(   ))
n
n
(3)
n 0
where k 
2
p
j

e 4 , and  p 
2
0 r p
.  p is the skin depth, and
for the pipe it is the radial distance from the inner surface of
the pipe at which the magnitude of the eddy current drops by a
factor of
1
e
from its surface value.
 p is the electrical
conductivity of the steel pipe. An is the constant coefficient
to be calculated.
Kn is the modified Bessel function of the
second kind of order n [11].
The constants An , Bn , A0 , and C are determined by
imposing boundary conditions, which are the continuity of the
magnetic vector potential Az and of 1 Az between regions I
 r
The method of images is presented for obtaining a system
that is equivalent and easier to analyze than the original system
containing the pipe. This equivalent system replaces the pipe
with virtual current filaments that produce approximately the
same electromagnetic fields in the interior of the pipe.
A. Mathematical analysis of the original system with pipe
Maxwell’s equations can be manipulated to arrive at the
following differential equation for a system with a steel pipe:
2
1   
 1  A(r ,  )
r
A
(
r
,
θ
)

 0 r j A(r ,θ )
 2
r r  r
 2
 r
vector potential, with a standard solution [9].
For a filament carrying current I and located at cylindrical
coordinates (d ,  ) inside the pipe, the solution to (1) in the
(1)
where:
0 = magnetic permeability of free space = 4 107 H m1 ,
r = relative magnetic permeability of steel,
 = angular frequency = 2 (60) rad  s 1 ,
A(r ,θ ) = magnetic vector potential ( T m ),
and II. The resulting expressions for the constants are as
follows:
0 I  d 
2r


2  b  nr K n (kb) - kbK n (kb)
(4)
0 I d n 1 nr K n (kb)  kbK n (kb)
2 n b 2 n nr K n (kb) - kbK n (kb)
(5)
0 I
r
2 b kK 0 (kb)
(6)

0 I  r K 0 (kb)
 ln(b) 

2  kbK 0 (kb)

(7)
n
An 
Bn 
A0  -
C  -
where b (m) is the inner radius of the pipe and Kn is the first
derivate of Kn with respect to its argument kb . If multiple
1
 =electrical conductivity of steel pipe or free space ( S m ),
and r ( m ) and θ ( rad ) are the cylindrical coordinates.
Equation (1) is a Bessel equation, in cylindrical
coordinates, in terms of the z-component of the magnetic
current filaments are inside the magnetic pipe, then the
superposition is used to obtain the total magnetic vector
potential in regions I and II.
B. Mathematical analysis of equivalent system
The presence of the steel pipe affects the magnetic vector
3
distribution in its interior; a single physical current filament
inside a steel pipe displaced from its center will induce
currents in the pipe, which will, in turn, affect the magnetic
vector distribution inside. The effect of the steel pipe on its
interior, which is the region of interest for calculating the final
distributions of currents in the cables, can be reproduced by
replacing the pipe with a single imaginary filament at a
specific location outside the pipe and carrying a specific
current. This imaginary filament is called the image of the
original physical filament. The system consisting of the pipe
and the physical filament is called the original system whereas
the system consisting of the image and the physical filaments
is called the equivalent system. The equivalent system will
have the same effect in the pipe interior as the original system
if the electromagnetic field distributions of the two systems are
equal there. In a limiting case, when the pipe has infinite
relative permeability and zero electrical conductivity, the
electromagnetic fields can be reproduced exactly in the pipe
interior by the equivalent system if the image carries a current
equal in magnitude and phase to the current of the original
filament and located according to (8), [14].
d 
b2
,   
d
(8)
 d ,    are the cylindrical coordinates of the image filament
with respect to the pipe center. The position of the image can
vary from just outside the pipe up to infinity, depending on the
position of the original filament, varying from the radius of the
pipe to the center of the pipe, respectively.
On the other hand, assuming that the pipe relative
permeability is equal to one and the pipe has infinite electrical
conductivity, the electromagnetic fields can be reproduced
exactly in the pipe interior if the image carries a current equal
in magnitude to the current of the original filament but with a
180o phase difference [14]. However, in practical cases [5-8,
12, 13], the steel pipe has simultaneously finite high magnetic
permeability and high electrical conductivity. It becomes
impossible to exactly reproduce the fields by a single image
filament and, therefore, one must settle for a best
approximation. The best approximation is achieved through
an approximate matching of the magnetic vector potential of
the two systems at the pipe inner surface boundary, and thus
everywhere in the interior of the boundary. The location and
current of the image that would give rise to such best
approximation is given by (8) and (9) (the mathematical basis
and derivation is given in Appendix A).

B b
n
n
I 
n 1

0  b  1
2  d   n
n 1

n
(9)
I  is the current assigned to the image filament. The current
of the image is equal to that of the original filament multiplied
by a factor that depends on the relative permeability of the
pipe, the pipe radius, and the constant ‘k’, as given by (9).
Mekjian et al. [5] assumed the aforementioned equivalent
system for infinite r and zero conductivity also applies in the
case of high r and high conductivity, which is inaccurate as
shown in Appendix A.
Note that the radius of the image filament is assumed equal
to the radius of the original physical filament (see Section IV).
If the pipe contains multiple current filaments, the interior
can be modeled by finding the image of every filament and
then replacing the pipe with all the images. The method of
filaments is discussed next.
III. METHOD OF FILAMENTS
If a cable consists of a conductor and a metallic sheath, then
the ac current in the conductor flows close to the conductor’s
surface and induces eddy currents in the sheath. The former
phenomenon is called the skin effect. The skin effect and the
sheath eddy currents arise due to the varying magnetic flux in
the sheath and conductor that is caused by the sinusoidal
current in the conductor.
When such a sheathed cable is placed in proximity of
another one, then the currents in the conductors of each cable
influence the redistribution of the current in the conductor and
sheath of the other cable. This is known as the proximity
effect. If various cables are placed beside each other, the
interaction becomes more complex. Calculating the final
distribution of the currents in the cable conductors and sheaths
becomes a very difficult task, especially if the system is further
complicated by the presence of a magnetic steel pipe.
The filament method simplifies this task [15]. Basically, it
is based on physically approximating the area of each sheath
and conductor by a group of thin cylindrical wires or
filaments, with each filament experiencing almost no skin or
proximity effects and thus having an almost constant current
density across it. This condition of constant current density
can be satisfied by choosing a filament radius that is relatively
small (around half the skin depth). The current in each
filament is not known a priori, but the total sum of currents in
the filaments must be equal to the total current in the sheath or
conductor. The interaction between current-carrying filaments
is well approximated by a relation that depends on the
distances between them. With total currents in the conductors
and sheaths known a priori, the final current value in each
filament is calculated analytically. A detailed description of
this process follows.
In what ensues, the cable conductors and sheaths will be
called composites, whereas the cylindrical wires that
approximately comprise them will be called filaments. For a
system of “m” composites and “n” filaments, I1c , , I mc define
the composites’ currents, and I1 ,
, I n define the filament
currents. Using the same numbering convention, E1c ,
define the composite voltages, and E1 ,
, Emc
, En define the
filament voltages. Vector variables can be assigned to these
arrays, as follows:
4
 I1c 
 E1c 
 I1 
 E1 
 c
 c
I 
E 
I 
E 
I c   2  , I   2  , Ec   2  , E   2 
 
 
 
 
 
 
I c 
Ec 
 In 
 En 
 m
 m
1
(10)
The above four vectors are related to each other as follows:
I c  M  I, E = MT  Ec
(11)
Matrix M , and its transpose MT , ensure that the total
composite current is equal to the sum of its filament currents,
and that the composite voltage and filament voltages are equal.
For example, for a two-cable case, with each cable containing
two composites, namely the conductor and the sheath, M is
defined as follows:
1
0
M
0

0
0
1
1 0
0 1
0
0 

0  
m = 4
0 

1  
0
1
n/2
(12)
 1
 ln s
 11
 1
G  ln
 s21



1
ln
s12
 R1
R d   0

0
R2




This section presents the method for calculating the
classical eddy current losses in sheathed cables and steel pipe.
Also, calculation of the hysteresis losses in the pipe is shown.
density, J II ( A  m 2 ) .
J II  - j p AII
(17)
Thus, the pipe eddy-current power loss can be obtained as
[5]:
P


2 Z 3p
2


Y
n
 n 2 r2  Z 2p  nr 2 Z p 
4 p b 2 n 1




1
(18)
where:
Yn
1









IV. CALCULATING LOSSES IN A SYSTEM OF SHEATHED
CABLES INSIDE A STEEL PIPE
The following relation is used to obtain the pipe current
In this case, each cable conductor is assumed to be
n
composed of a single filament and the sheath of " -1"
2
filaments. Therefore, the first and second rows of (12)
correspond to one cable conductor and the sheath,
respectively, while the bottom two rows correspond to the
other cable.
Self and mutual impedances of the filaments and (11) are
used to relate the composite voltages, E c , and their currents,
I c , as follows, [15]:
 

 
Ec   M  R d  j 0 G  M T  I c
2 
 

The above equation relates the unknown filament currents,
I , to known constants and known composite currents, I c .
The values of the composite currents I c are known a priori, as
follows: 1) Total composite conductor currents are always
given. 2) Total composite sheath current is zero for every
cable if the cable sheaths are single-point bonded. If the
sheaths are multiple-point bonded, the sum of the currents in
all of the multiple point bonded sheaths should add up to zero.
A. Classical eddy current losses
n/2
1
1
1
 

0 
0 

T

I  R d  j
G  M M R d  j
G  MT  I c (16)
2 
2 
 


2
q

 di 
2n
  b  I 
2
i
i 1
(13)
q

Zp  k b  2
(14)
n
 di d j 
 2  I i I j cos(n(i -  j )) cos( i -  j )

j i i 1  b
q
2
,
b ,
p
where:
di = radial position of the physical filament ‘i’,
i = angular position of the physical filament ‘i’,
Ii = current flowing in the physical filament ‘i’,
 i = phase of the current flowing in the physical filament ‘i’,
(15)
sij (m) is the distance between filaments ‘i’ and ‘j’. Ri () is
the electrical resistance of filament ‘i’, which could be a
conductor filament or a sheath filament.
Using (11) and (13) and through some simple algebraic
manipulations, the following expression is obtained, relating
the composite and filament currents, [15]:
q = number of physical filaments in the interior of the pipe.
The goal for using the image and filament methods is to
obtain the final distribution of currents in the conductors and
sheaths, including the effects of the magnetic pipe. Then, (17)
and (18), which are based on current-carrying wires, are used
to obtain the current distribution and the losses in the pipe
material.
When only eddy currents exist in the sheaths, then from the
experience of the authors, it is enough to approximate the
images of the cable conductor and sheath filaments by a single
5
filament located at the image position of the center of the cable
conductor and carrying a current related to the given total
current in the cable conductor according to (9). This is
because, when only eddy currents exist, the sheath image
filaments do not carry significant currents that would alter the
losses in the system. This makes the equivalent image system
easier to analyze. However, when circulating currents are
flowing in the sheaths, this approximation cannot be made, as
discussed next.
If the cable sheaths are multiple-point bonded in a threephase circuit, the currents induced in them will flow from one
sheath to another. This circulating sheath current causes
considerably higher losses than the local eddy current losses.
The final distribution of the currents in the cable sheaths that
are multiple-point bonded is calculated using the same
filament and image methods described in the previous
sections, but with the imposed condition that the sum of all of
the sheath currents equals zero. In this case, it is important to
use the images of all the original cable conductor and sheath
filaments, because the currents in the sheath image filaments
are large enough to alter the losses in the system.
B. Hysteresis loss
The second component of pipe losses is called hysteresis
loss and arises due to the phenomenon of magnetic hysteresis
[16-26].
This section discusses how to approximate
analytically the hysteresis loss for a system of cables inside a
magnetic pipe.
The B-H curve for a material experiencing magnetic
hysteresis typically looks as shown in Figure 1. The magnetic
field intensity, H, fluctuates sinusoidally at a specific
frequency f, usually 50 or 60 Hz for power systems. So, one
B-H loop is completed every 1/f seconds. The area enclosed
by this loop is equal to the heat dissipated per unit volume of
the material due to hysteresis. Multiplying the area by
frequency, one obtains the dissipated power density.
The amplitude of the magnetic field intensity is nonuniform in
the pipe. Its peak value affects the loop shape and area, and it
is generally very complicated to obtain precisely the area of
the loop from the peak of the magnetic field intensity.
However, Steinmetz discovered in 1890 that the loop area is
related to the peak value of the magnetic flux density, B,
through the following empirical relationship, [28]:
loop area=g B
h
(19)
where ‘g’ and ‘h’ are constants, ‘h’ usually being around 1.6.
Equation (19) holds for magnetic flux density values up to 1.21.4T [29], which are not exceeded in typical pipe installations.
The values of magnetic flux densities in the steel pipe
depend on the magnetic permeability of the pipe, which, in
turn, depends on the peak value of the magnetic field intensity
and the hysteresis loop shape. The value of the magnetic
intensity is also related to the magnetic flux density. Time
stepping is required to solve such a highly nonlinear problem.
Fig. 1 B-H curve for a magnetic material [27].
To simplify the problem, an important approximation is made
here: assume that the pipe has a constant, uniform “effective”
magnetic permeability value. This assumption allows the
system to be solved analytically through Maxwell’s equations
with the use of phasors. The final analytical solutions for the
fields and power losses will be of closed form, which are faster
to calculate than by using the finite element or finite difference
methods, and with comparable accuracy. This value of
constant permeability is calculated so that the resulting eddy
current losses agree with the actual nonlinear system. The
calculation of this effective permeability is discussed in the
next section. Its validity is shown here as well.
With the approximation of constant permeability, the
magnetic flux density distribution is computed. Then, the
hysteresis loss is calculated from (19).
The magnetic flux inside the pipe material, due to a single
current filament in the interior (superposition is used for
multiple filaments), is given by:

Br 
1 AII
1
An K n (kr )n sin(n(   ))
r 
r n 0

(20)

B  
AII
 - An k K n (kr ) cos(n(   ))
r
n 0

(21)
Since k  1 implies B  Br , then:
B  B
h
B 

 A k K  (kr ) cos(n(   ))
n
(22)
h
(23)
n
n 0
Therefore, the hysteresis loop area becomes:

loop area  g
 A k K  (kr ) cos(n(   ))
n
n
h
(24)
n 0
The values of ‘g’ and ‘h’ are obtained for a specific steel
material, given in [16], using curve fitting and are equal to:
g  5.913, h  1.765
(25)
6
C. Effective magnetic permeability of a steel pipe
P

f  0 r H2 (b)  2 b p

2 
1200
Pipe eddy current loss (W/m)
Steel pipe has a non-constant magnetic permeability that
depends on the strength of the magnetic field excitation. The
permeability of the steel pipe is chosen to provide a good
approximation for the eddy current loss when compared with
the actual nonlinear pipe case. The method presented here for
calculating this effective permeability is developed in [12].
As shown in Section IV-A, the eddy current loss in the pipe
can be calculated using (18). It can also be shown that, for a
pipe with constant permeability, the eddy current loss is
calculated by the following equation [12]:
Eq. (18)
Eq. (26)
1000
800
600
400
200
X: 1383
Y: 53.83
(26)
0
0
500
1000
Relative permeability
where H2 (b) is the average squared tangential magnetic field
intensity at the pipe inner surface. The experimental B-H
curve can be used to calculate r at different values of the
magnetic field intensity H. Because, in the pipe the value of
the radial magnetic field intensity, H r , is much smaller than
1500
Fig.2 Illustration of the effective permeability calculation
A. Classical eddy current loss
also be obtained by applying (18). This plot and the previous
plot can be drawn on the same axes, and because the two
equations must give the same pipe loss for some specific value
of r , the intersection of the two graphs determines the value
This section presents test cases with constant, arbitrary
value of the steel pipe permeability. The purpose of these test
cases is to compare the eddy current loss calculation presented
in this paper against results obtained by the Ansoft Maxwell
(referred to from this point forward as Ansoft for short).
The following cable systems were examined:
1. Three sheathed cables at the center of the pipe,
2. Three sheathed cables at the bottom of the pipe,
3. Three sheathed cables at the bottom of the pipe with
circulating currents,
4. Two circuits, two sheathed cables per phase and
circulating currents.
In all cases, the parameters assumed for the steel pipe and
cables are shown in Table I.
of r , as illustrated in Fig. 2, for the system described in
TABLE I
PARAMETERS COMMON FOR ALL TESTS
H ,
it
can
be
assumed
that
H  H
(where
H  H r2  H2 ). From known values of r for different
values of H , the pipe eddy current loss is calculated for
different r . Using (26), one can then produce a plot of pipe
eddy current loss versus r .
On the other hand, a plot of the pipe loss versus r can
Appendix A and shown in Fig. A1. This r is called the
effective permeability. The effective permeability value, in
conjunction with equations (18) and (24), is used for the
calculation of pipe eddy current and hysteresis losses.
V. RESULTS
Several tests were performed to determine the accuracy of
the proposed analytical solution. The results obtained using
(9), (16), (18), and (24), were compared with the losses
computed using the “Eddy Current” solver of the commercial
finite element program Maxwell by Ansoft [30]. The tests
were conducted in two phases. First, only eddy current losses
in the pipe were considered, followed by the tests with both
eddy current and hysteresis losses. The following sections
describe the tests and the obtained results.
(Hz)
Conductor
diameter
(mm)
Sheath
thickness
(mm)
Diameter
over the
insulation
(mm)
Conductor,
sheath
conductivity
(S/m)
60
26
2
48
4.46×107
Pipe
diameter
Filament
diameter
Pipe
thickness
Pipe
conductivity
(mm)
(mm)
(mm)
(S/m)
300
1
5
7.413×106
Frequency
Pipe relative
permeability
1500
The last value in the table, representing rather large relative
permeability, is based on a magnetic curve in [12] for a
carbon-steel pipe that has a maximum permeability of about
1750 with an effective value of 1580.
The percentage difference in the total losses computed by
the proposed approach and Ansoft is calculated as follows:
 Analytical Method  Ansoft 
% Difference  100* 

Ansoft


(27)
In the following sections, “Error of Ansoft” refers to how
7
close the numerical solution comes to satisfying the
electromagnetic equations that are being solved within the
Ansoft program itself. At the end of every iteration, the
numerical solution is plugged back in to the field equations,
and the residual error is computed. If the residual error is
zero, then the numerical solution is exact. Ansoft also
computes the total field energy and the field energy
contributed to this by the residual error. If the ratio of the
residual error energy to the total field energy is less than a
preset value, then the iterations stop, and the simulation is
assumed to have converged. Otherwise, a finer mesh is
generated, especially in the areas with large residual error, and
the simulation continues. The lower the final residual error
and the finer the mesh, the more accurate the solution, but the
longer the simulation time will be. So, it is a compromise
between the speed and accuracy. Generally, in solving an
eddy current problem with the interest of calculating pipe
losses, a very small residual error is sought (less than 0.5%
from the experience of the authors). The reason is that, in
many cases, most of the residual error can be concentrated in
the pipe region due to the very small skin depth of the eddy
currents in the pipe. Therefore, a very fine mesh is needed
there to obtain an accurate value for the pipe losses. The
number of mesh triangles across the thickness of the pipe is
maintained to be at least ten for all simulations.
The test cases results are shown in Tables II and III.
for the analytical method, and a case where the pipe is set to
have the experimental nonlinear magnetic curve.
TABLE II
COMPARISON FOR 3 SHEATHED CABLES AT CENTER AND BOTTOM OF PIPE
Ansoft can solve directly for eddy current losses, but does
not have its own method of calculating hysteresis losses. So,
to test the accuracy of the proposed method of calculating the
hysteresis losses using equation (24), which is based on a
constant “effective” permeability, the empirical equation (19)
is invoked in Ansoft for both cases when the pipe has constant
permeability and nonlinear magnetic curve.
The parameters shown in Table IV were assumed for the
steel pipe and the cables.
3 sheathed cables, center of pipe
Proposed
Ansoft
Calculation
Time (s)
2
46
Error of Ansoft
(%)
N/A
3 sheathed cables, bottom of pipe
Proposed
Ansoft
Calculation
Time (s)
1.8
90
0.18
Error of Ansoft
(%)
N/A
0.14
Avg. conductor
loss / cable
14.81
(W/m)
15.14
Avg. conductor
loss / cable
(W/m)
15.49
15.45
Avg. sheath
loss /cable
(W/m)
3.20
Avg. sheath
loss /cable
(W/m)
4.53
4.62
0.71
Total pipe eddy
loss (W/m)
1.35
3.05
Total pipe eddy
0.68
loss (W/m)
Total loss
difference
-2.60%
Total loss
difference
1.45
-0.52%
B. Eddy Current and Hysteresis Losses
This section presents a test case where eddy current and
hysteresis losses of a nonlinear pipe are considered. The
losses calculated by the proposed analytical method are
compared with those obtained using Ansoft. Ansoft is used to
simulate both a case where the pipe is set to have a magnetic
permeability equal to the “effective” permeability calculated
TABLE III
COMPARISON FOR 3 AND 12 CABLES WITH CIRCULATING CURRENTS
3 sheathed cables, bottom of pipe
with circulating currents
Proposed
Ansoft
Calculation
Time (s)
4.3
40
Error of Ansoft
(%)
N/A
Avg. conductor
loss / cable
(W/m)
2 circuits, 2 cables /phase with
circulating currents
Proposed
Ansoft
Calculation
Time (s)
44
193
0.29
Error of Ansoft
(%)
N/A
0.31
14.84
14.91
Avg.
conductor loss
/ cable (W/m)
4.72
4.72
Avg. sheath
loss /cable
(W/m)
10.19
9.84
Avg. sheath
loss /cable
(W/m)
5.04
5.15
Total pipe eddy
loss (W/m)
0.60
0.69
Total pipe
eddy loss
(W/m)
0.70
0.82
Total loss
difference
0.90%
Total loss
difference
1.30%
TABLE IV
PARAMETERS COMMON FOR ALL TESTS
Frequency
(Hz)
Pipe
diameter
(mm)
Pipe
thickness
(mm)
Pipe
conductivity
(S/m)
Conductor,
sheath
conductivity
(S/m)
60
743
8.5
9×106
4.3×107
Conductor
diameter
(mm)
Conductor
filament
diameter
(mm)
Sheath
thickness
(mm)
Sheath
filament
diameter
(mm)
Cable
diameter (up
to insulation)
(mm)
43.8
10
3.75
8.76
106.1
The configuration of the cables is as shown in Fig. 3, and
the polar coordinates of their centers with respect to the pipe
center are given in Table V. The cable conductors carry a
balanced current of 1830A peak.
Because the proposed method splits the cable conductors
and sheaths into multiple physical filaments, skin and
8
proximity effects are accounted for. To illustrate these effects,
consider Figs. 4 and 5, which show conductor and sheath
filament currents obtained using the proposed method.
The horizontal and vertical axes show the Cartesian
coordinates of the filaments, with respect to the pipe center, in
meters. The current magnitudes are in Amperes.
As can been seen from Fig. 4, the amplitudes of the currents
are largest in the outer filaments. This observation is
explained by the skin effect. Also, it is apparent that the
current distribution is not circularly symmetric, and this can be
explained by the proximity effect due to the other cables and
the steel pipe.
As can be seen from Fig. 5, the value of the current is
different in different parts of the sheath. This is due to the
proximity effect of neighboring cables and the steel pipe.
Table VI shows the results for the losses. “Ansoft +curve”
refers to results obtained using Ansoft with the pipe set to have
a realistic nonlinear magnetic permeability curve provided in
[16]. This is the actual practical case, and the “Proposed
method” results are aimed to match these values. An excellent
agreement can be observed between these two approaches.
To provide a comparison with another analytical method
available in the literature, the same problem is solved using
Mekjian’s approach [5], which neglects sheath losses and pipe
hysteresis losses. Using the method of [5] the following results
are obtained. Total conductor loss = 110 W/m, total pipe loss
= 136 W/m. The corresponding total installation loss of
246W/m is 12.5% smaller than the actual loss of 282 W/m
obtained using the “Ansoft +curve”.
TABLE V
POLAR COORDINATES OF CABLES
(291.4 mm,-45.0o)
Cable 2
(280 mm, 160.0 o)
Cable 3
(290 mm, 200.0 o)
TABLE VI
EDDY CURRENT AND HYSTERESIS LOSSES FOR SYSTEM IN FIG. 3
Special Study
Proposed
Ansoft
Ansoft
+curve
1338
1338
[16]
Calculation Time (s)
2
1920
2700
Error of Ansoft (%)
N/A
0.52
0.24
Total conductor loss
(W/m)
102
103
103
Total sheath loss (W/m)
21
19
19
Pipe eddy loss (W/m)
113
116
120
Pipe hysteresis loss
(W/m)
49
49
40
Total loss (W/m)
285
288
282
Total loss difference (%)
N/A
-1.0
1.1
Pipe relative
permeability
APPENDIX A. DERIVATION OF IMAGE CURRENT VALUE
Polar coordinate
w.r.t. pipe center
Cable 1
image and filament methods as well as the concept of
equivalent permeability. The results obtained with the
proposed approach are in close agreement with the solution
obtained by application of the finite element method to solve
the same problem. The total loss difference of less than 3% is
observed for all tested cases. This closed form analytical
solution is more insightful than its numerical counterparts and
could provide a basis for standardization of the computation of
losses in a system of cables inside a steel pipe.
The expression derived for the magnetic vector potential in
the interior of the pipe is given by (2) and is repeated here:
n

 I d  1
 I
 Bn r n  0    cos(n( -  )) - 0 ln(r )  C (A.1)

2  r  n 
2
n 1 

Fig. 3. Three cables from Table V
Cable 1 conductor filaments currents (A)
Cable 1 sheath filaments currents (A)
-0.18
-0.14
125
-0.185
45
-0.16
120
-0.19
40
115
-0.195
-0.2
110
-0.205
105
-0.21
100
-0.215
95
-0.22
90
-0.225
85
-0.23
0.18
80
Cartesian y (m)
Cartesian y (m)
AI 
-0.18
35
30
-0.2
25
-0.22
20
-0.24
0.19
0.2
0.21
0.22
0.23
-0.26
15
10
0.16
0.18
0.2
0.22
0.24

This expression accounts for the magnetic vector potential
contribution due to the physical source-current filament and
the contribution due to the induced current in the pipe.
If the source current filament is located at the polar
coordinates (d ,  ) in the interior of the pipe, and if the pipe is
replaced by an image filament located at (d ,  ) outside the
pipe, the magnetic vector potential, in the region interior to the
pipe (i.e. 0  r  b ), due to the source and image filaments is
given by:
0.26
Cartesian x (m)
Cartesian x (m)
Fig. 4 Cable 1 conductor filaments
Fig. 5. Cable 1 sheath filaments
VI. CONCLUSIONS
This paper presents an analytical solution for the
calculation of losses in a steel pipe caused by an arbitrary
arrangement of cables inside. The approximation uses the

AI 

0 I  d  1
0 I
 r  n cos(n( -  )) - 2 ln( r )
2

 
n 1
n

0 I   r  1
 I

cos(n( -  )) - 0 ln( d )


2  d   n
2
n 1

n
(A.2)
where I  is the current carried by the image filament and is to
9
then the original and the equivalent systems will have the same
electromagnetic fields in the pipe interior region. However, it
is impossible to find values that could achieve this. One can
only achieve a best approximation. The reason for this is that,
to have the two equations match exactly, one would need to
 I  1  1
 I
have Bn  0  
for all ‘n’ and C   0 ln  d   .
2  d   n
2
Bn has been determined previously to be a complex number
n
that is given by a quotient of two polynomials in ‘n’. Although
I  is a complex number, there is no way to represent exactly a
quotient of two polynomials in ‘n’ by a quotient of an
n
1 1
exponential function and the function ‘n’, i.e.  
. This
 d  n
is not possible for every integer n  1, ,  . An important
assumption is made here: the values of d ,  , I  will be
chosen so that the magnetic vector potentials for the original
system, (A.1), and the equivalent “image” system, (A.2), are
approximately matched at the pipe inner surface. This
assumption is achieved by equating the magnetic vector
potentials at the point on the pipe inner surface that is closest
to the filament—i.e., at (b,  ) —and, thus, the condition
simplifies to (8) and (9).
Equations (8) and (9) include only the contribution of the
magnetic vector potential terms that fall under the summation,
because they capture the angular variation of the magnetic
vector potential.
To test the above approximation for the accuracy of the
electromagnetic fields in the interior of the pipe, consider the
system shown in Fig. A1.
1
Magnetic vector potential (T.m)
that the above expression is equal to (A.1) given for the
original system of a source filament and a pipe. If the values
of d ,  , I  can be found to make these two expressions equal,
and the proposed approximation given by (8) and (9), 3)
Analytically for the equivalent “images system” using
Mekjian’s images approximation (from Section II-B). The
magnetic vector potential at the inner surface of the pipe using
the three different methods is plotted in Figure A2.
Filament 2
(249 mm, 149.8 o)
Filament 3
(221 mm, 213.0 o)
The system consists of a steel pipe containing three
filaments carrying a balanced current of 1554 A peak. The
relevant system parameters are as in Table IV, except for a
pipe conductivity of 6.41×106 S  m1 and a pipe permeability of
700. The filaments are located as given in Table VI. The
system is solved using the following approaches:
1) Analytically for the original system using (A.1), 2)
Analytically for the equivalent “images system” using (A.2)
Analytical- actual
Analytical- images
Analytical - Mekjian
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
angle (rad)
Fig. A2.
methods
Magnetic vector potential at pipe inner radius using different
1.5
-3
x 10 Angular magnetic field intensity vs angular position at pipe inner surface
Analytical- actual
Analytical- Mekjian
Analytical- images
1
0.5
0
0
1
2
3
4
5
6
7
angle (rad)
Fig. A3. Angular magnetic field at pipe inner radius using different methods
1.2
Radial magnetic field intensity (A/m)
Fig. A1. Three filaments inside a
steel pipe
(202 mm, -29.7o)
Magnetic vector potential vs angular position at pipe inner surface
0.7
Polar coordinate
w.r.t. pipe center
Filament 1
-3
0.8
TABLE VI
POLAR COORDINATES OF FILAMENTS
IN FIG. A1.
x 10
0.9
Angular magnetic field intensity (A/m)
be determined. There is flexibility in choosing d ,  , I  , so
-3
x 10 Radial magnetic field intensity vs angular position at pipe inner surface
Analytical- actual
Analytical- Mekjian
Analytical- images
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
angle (rad)
5
6
7
Fig. A4. Radial magnetic field at pipe inner radius using different methods
10
It can be observed that the magnetic vector potential for the [18] D. C. Jiles and D. L. Atherton, “Theory of ferromagnetic hysteresis,”
Journal of Magnetism and Magnetic Materials, Vol. 61, No. 1-2, 1986,
image method has good agreement with that for the actual
pp. 48-60.
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Figures A3 and A4 show the angular and radial magnetic [20] G. Bertotti and I. D. Mayergoyz, The science of hysteresis, 3 v.,
Amsterdam ; Boston ; London : Academic, 2006.
field intensities at the inner surface of the pipe for the three
[21] A. A. Adly, I. D. Mayergoyz, R. D. Gomez, and E. R. Burke,
different methods.
“Computation of magnetic fields in hysteretic media,” IEEE Transactions
Note that the angular magnetic field in Fig. A3 for the
on Magnetics, Vol. 29, No. 6, November 1993, pp. 2380-2382.
Mekjian’s method is zero. It can be seen that the proposed [22] B. C. W. McGee and F. E. Vermeulen, “Power losses in steel pipe
delivering very large currents,” IEEE Transactions on Power Delivery,
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of the original system than given by Mekjian’s approximation.
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[1] ANSI/IEEE Standard 575 (1988), “Application of sheath-bonding methods
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VII. BIOGRAPHIES
current losses in nonlinear ferromagnetic sheaths of three-phase power
cables,” IEEE Transactions on Power Delivery, Vol. 7, No. 3, July 1992,
pp. 1060-1067.
Wael Moutassem (S’06) received a B.A.Sc.
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unbalanced currents,” IEEE Transactions on Power Delivery, Vol. 17,
(Electrical Option) in 2005, and M.A.Sc. in
No. 2, April 2002, pp. 313-317.
Electrical Engineering in 2007, both from the
[9] R. Harrington, Time-harmonic electromagnetic fields. New York ;
University of Toronto, ON, Canada.
In
Toronto, McGraw-Hill, 1961.
September 2005, he joined the Energy Systems
[10] K. Kawasaki, M. Inami and T. Ishikawa, “Theoretical considerations on
Group at the University of Toronto, where he is
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currently a Ph.D. candidate.
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[11] M. Abramowitz and I. Stegun, “Handbook of mathematical functions and
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received a M.Sc. degree in EE from the
Standards, Applied Mathematics Series-55, 1965
Technical University of Lodz in Poland in 1973,
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and a M.Sc. degree in Mathematics and Ph.D.
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Since 1975, he has been employed by Ontario
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