International Review of Mechanical Engineering (I.RE.M.E,), Vol. xx, n. x
Determination of Stationary Temperature Distribution
in Shielded Cables of Finite Length
Florian Loos1 , Karl Dvorsky2 , Hans-Dieter Ließ3
Abstract – Thermal energy management is one of the most important aspects for the interior automobile
design and often represents a bottleneck concerning the integration of electrical components in modern cars.
Thus, the size of current bars and cables has to be dimensioned correctly. On the one hand, these connecting
structures must not be too small for thermal reasons. Inadequate materials and component dimensions result
in hotspot generation and overheating which could entail fatal damages. On the other hand, oversizing has to
be avoided for reasons of space and weight reduction.
In this paper, we present a new approach to quickly and accurately compute the heat distribution in electric
shielded cables of finite length. The derivation of a nonlinear system of ordinary differential equations allows
computing the cable temperatures for the stationary case by a fixed point method. The heat generated inside
the cable by the Joule effect is taken into account as well as the thermal energy emitted via the surface by
convection and radiation and to adjacent components by conduction.
To compare the simulation results to realistic settings, an experimental study was performed. Apart from
the good accordance of computations and measurements, we show further advantages concerning calculation
times and industrial practicability.
Keywords: Shielded Electric Cables, Joule Heating, Heat Transfer Enhancement
I.
Introduction
The great number of electrical devices in modern cars
increases the importance of connecting structures like cables, busbars and current bars [1]. Heat generated by the
Joule effect has to be reduced by larger cable diameters
on the one hand, whereas on the other hand material consumption shall be minimized to save costs and weight.
Not only but especially in electric cars, an adequate dimensioning of the current connecting structures is indispensable.
In the past and up till today, most of this dimensioning process has been done experimentally which is very
expensive. Consequently, the manufacturers start more
and more to compute the generated heat a priori and to
produce the cable layout according to their calculations.
Often, there is a lack of means, knowledge and time to develop accurate models and efficient solution methods for
the complex physical processes. Therefore, we present
a new approach, evolved in cooperation with our industrial partners, to compute the heat generation in shielded
cables of finite length, considering different environment
temperatures and the influence of connected objects.
Shielded cables, i.e. conductors with a metallic, current carrying layer in the insulation part, find practical
use in the high frequency technology. It is the task of the
shielding to ’separate’ the cable from the environment regarding both, the radiation from exterior into the cable
and from inside to outside. Radiation like electromagnetic induction by alternating current or radio waves is
concerned.
Manuscript received December 2012
Our calculation method is kept general in order to be
applicable to very short cables of only a few centimetres of length and also to longer wires over several metres
distributed over the entire car. For shorter conductors,
the temperatures of attached devices play a main role
whereas for longer cables different environment conditions essentially influence the temperature profile. In this
article, we confine to the stationary case with a constant
direct current over a longer time period.
Before going into detail, let us state some results from
literature. The first calculations on heat generation in
electric cables were published by Neher-McGrath [2] in
1957 which form the basis for many cable application
guidelines and regulations. Details about heat transfer
used for our application are explained in [3]. In [4],
more mathematical details concerning the modelling and
simulation with appropriate boundary conditions are described. Anders [5], [6] summarizes general computation
methods and advanced techniques to calculate electric cable ratings. In [7], the modelling and calculation of temperatures in unshielded cables of infinite length are described, using a temperature dependent heat transfer coefficient. We adopt this approach to our simulation model
in order to incorporate convection and radiation at the exterior insulation boundary. Furthermore, there exists a
great number of publications concerning shielded cables,
e.g. [8], but most are focused on other aspects than the
thermal development. Since today cables and shieldings
carry higher currents in the high voltage technology, the
importance of the heat generation in shielded cables has
augmented the last few year.
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
F. Loos, K. Dvorsky, H.-D. Ließ
The paper is organized as follows: In section II, the
physical problem and the governing equations are formulated. Division in subsections, appropriate boundary
conditions and the calculation method to solve the resulting equation system are described in section III. In
section IV, we present a comparison of the measurement
and calculation results for different cable types. Further
calculations and application of the method are subject of
section V. Finally, we conclude and give an outlook to
future work in section VI.
sectional areas Ai , i = 2, . . . , 5. The stationary thermal
identification of the system requires the heat conductivities λi , i = 2, . . . , 5. Furthermore, the important electrical quantities are the resistivity ρ0 at reference temperature T0 (normally 20 ◦ C), the temperature coefficient of
the electrical resistivity αρ for the cable core and the resistivity ρs for the shielding. The rise of the electrical
resistance in the cable core for higher temperatures is approximated linearly:
ρ = ρ0 [1 + αρ (T5 − T0 )]
II.
Problem formulation
Our aim is to approximate the axial temperature distribution at characteristic points in shielded cables of finite
length. These temperatures are essentially determined by
the electrical current in the cable core I and shielding Is ,
the cable material and its dimensions. Moreover, the attached devices at the ends of the cable and the environment temperatures in different cable sections influence
the temperature profile. Fig. 1 shows the cross section
of a shielded cable in radial direction.
T1
A33
A
T
T4 4
A44
A
d3
Tl
T11
d4
T12
T13
Tr
Inner conductor
TT33
Insulation
Shielding
connector
T
A55
A
d2
The temperature of attached devices can either be
known which finally results in a Dirichlet boundary condition or is approximated by an approach presented in
sec. III.2, yielding a Robin boundary condition. For the
first case, depicted in Fig. 2(a), the fixed temperatures T l
at the left end and T r at the right end are given. The attached objects in the second case, cf. Fig. 2(b), are characterized by the heat conductivities λlad respectively λrad ,
l respectively T r and the
the asymptotic temperatures Tad
ad
l
r
contact resistances Rk and Rk .
T
T2 2
A22
A
(1)
T5
l1
l2
l3
Shielding
(a) Cable with known temperatures at the ends
d5
T11
T12
T13
l1
l2
l3
Aλ5 5
λ44
A
λ33
A
λ22
A
Contact
resistance
Attached
object
(b) Cable with approximated temperatures at the ends
Fig. 2 Axial cable cross section
Fig. 1 Radial cross section of a shielded cable
II.1.
Notations and annotations
Throughout this paper, the temperatures and material
parameters for the different cable cross section areas are
indexed by 1 for the environment, 2 for the exterior insulation, 3 for the shielding, 4 for the inner insulation and 5
for the core.
T1 denotes the environment temperature, T2 the temperature at the exterior boundary of the outer insulation
layer, T3 the shielding temperature, T4 the temperature in
the inner insulation layer and T5 the core temperature1 .
The geometrical properties of the cable cross section are
determined by the diameters di , i = 2, . . . , 5, and the cross
1 The
temperature in the core in radial direction is approximately constant because of the high heat conductivity of metals.
International Review of Mechanical Engineering, Vol. xx, n. x
A shielded cable can be placed in the entire car with
varying environment temperature for different subsections. In this paper, we restrict the model to three different sections, which is, according to our industrial partner, sufficient. An extension to more subsections is easily
possible. The subsections with lengths l1 , l2 , l3 have the
environment temperatures T11 , T12 , T13 .
II.2.
Heat power balance approach
In order to derive the appropriate equation for the heat
distribution in the shielded cable, we consider a volume
element of infinitesimal small length dx, shown in Fig. 3.
The heat balance in the volume element reads as follows:
dPk + dPs
| {z }
produced heat power
=
dPx + dPr
| {z }
(2)
conducted heat power
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
F. Loos, K. Dvorsky, H.-D. Ließ
the interval (0, d5 /2) makes no heat power contribution
in radial direction. λ = λ(s) represents the different heat
conductivities in the layers of the main, i.e.


for d5 /2 ≤ s ≤ d4 /2 ,
λ4
λ(s) = λ3
(9)
for d4 /2 ≤ s ≤ d3 /2 ,


λ2
for d3 /2 ≤ s ≤ d2 /2 .
r
z
y
x
dx
Radial symmetry of the geometry and of the temperature
profile yields
Fig. 3 Volume element of length dx
Pr = − As λ(s)
with dPk the heat power generated by current flow in the
cable core, dPs the heat power produced in the shielding,
dPx the conducted heat power in axial direction and dPr
the heat power conducted in radial direction.
The formula for dPk is
dPk =
ρ0 I 2 (1 + αρ (T5 − T0 ))
A5
dx.
(3)
To reduce the complexity of our system, we neglect the
dependence of the electrical shielding resistance on the
temperature T3 and obtain
dPs =
ρs Is2
dx.
A3
(4)
This simplification is justified by smaller current and consequently lower produced heat power in the shielding
compared to the core.
We identify Px via Fourier’s law by
5
Px = − ∑ λ j A j
j=2
dT j
.
dx
(5)
Since the main contribution of Px is given by λ5 A5 ddTx5 ,
dT
it is reasonable to approximate dxj = ddTx5 , j = 2, . . . , 4.
Thus, we obtain
Px = −Λ
⇒ dPx = −Λ
(6)
d 2 T5
.
dx
(7)
The heat power conducted in radial direction in a cable of
length lx is also given by Fourier’s law via
Pr = −
Z
λ(s)
∂T
dσ,
∂s
Pr
∂T
=−
.
∂s
2 π λ(s) s lx
(10)
Integration over s ∈ (d5 /2, d2 /2) provides
Pr
T5 − T2 =
lx
!
dj
1
ln
.
∑
d j+1
j=2 2 π λ j
4
(11)
In order to include the heat transfer from the conductor
surface to the environment, we consider the temperature
difference T2 − T1 , given by
T2 − T1 =
Pα
α lx π d2
(12)
where α is the heat transfer coefficient and Pα the heat
power emitted to the environment. Second is equal to the
heat power Pr conducted form the centre to the surface.
Hence, we get
!
4
dj
Pr
1
1
T5 − T1 =
∑ 2 π λ j ln d j+1 + α π d2 . (13)
lx j=2
Replacing lx by the infinitesimal length dx, we finally obtain
π (T5 − T1 )
dPr =
(14)
dx .
4
dj
1
1
1
α d2 + 2 ∑ λ j ln d j+1
j=2
dT5
dx
with Λ := ∑5j=2 λ j A j . Assuming T5 twice differentiable,
we get
dPx
d2 T5
= −Λ 2
dx
dx
⇒
∂T
∂s
d5 /2 ≤ s ≤ d2 /2.
(8)
As
As = 2 π s lx denotes the heat transition surface with distance s to the centre of the conductor. Since we assume
a constant temperature profile in the core (0 ≤ s ≤ d5 /2),
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
The heat transfer coefficient α is a temperature and geometry dependent quantity that summarizes the heat proportion emitted or absorbed via the surface by convection
and radiation:
α = αconv + αrad .
(15)
In [7], details about the heat transfer coefficient are explained. The radiative part is calculated according to
the Stefan-Boltzmann law, the convective part is approximated by formulas in [9]. They are based on similitude
where the temperature dependent quantities are approximated by fitting formulas consisting of polynomials of
fourth degree. The constant values of α in the subsections l1 , l2 and l3 represent an idealizing assumption that
enables an explicit resolvability of (2). They are determined by an a posteriori mean value iteration over the
surface temperatures, described in sec. III.3.
International Review of Mechanical Engineering, Vol. xx, n. x
F. Loos, K. Dvorsky, H.-D. Ließ
Replacing the infinitesimal heat powers in (2), substitution of T5 by T and division by −Λ dx provides the stationary equation for heat transfer in axial direction:
1
C=
Λ
ρ0 αρ I 2
π
−
ρw
A5
,
(17)
π T1 ρ0 (1 − αρ T0 ) I 2 ρ0s Is2
+
+
ρw
A5
A3
(i) 0
(Li ) = T
(21)
(Li )
(22)
for i = 1, 2 where (21) represents the equality of the temperatures at the interface, (22) the equality of the heat
fluxes.
The boundary conditions for the first case are
T (1) (0) = T l ,
.
(18)
T (3) (L3 ) = T r .
Π1 γ = b1
1
1
ρw =
+
α d2 2
1
∑ λ j ln(d j /d j+1 )
j=2
(19)
Calculation method
In the following, we divide the cable in three parts and
assume piecewise constant coefficients. For the subsections of length l1 , l2 , l3 , we define the axial variable core
temperatures T (1) , T (2) , T (3) .
Piecewise constant approximation of equation
defining coefficients
We refer to the coefficients B and C in (17) and (18) for
each subsection as Bi and Ci :
Bi = B(αi ) and Ci = C(T1i ),
with the unknown vector γ = (γ1 , . . . , γ6 )T ∈ R6 , matrix
Π1 ∈ R6×6 respectively right hand side b1 ∈ R6 . Defining
√
τi j := e Bi L j ,
√
σi j := Bi τi j ,
√
τ−i j := e− Bi L j ,
√
σ−i j := Bi τ−i j ,
Π1 and b1 write as follows:

1
1
0
0
0
0
 τ11 τ−11 −τ21 −τ−21 0
0 


 σ11 −σ−11 −σ21 σ−21 0
0 
 , (25)
Π1 = 
 0
0
τ22 τ−22 −τ32 −τ−32 


 0
0
σ22 −σ−22 −σ32 σ−32 
0
0
0
0
τ33 τ−33
C1 C2 C1
C3 C2
C3 T
l
r
b1 = T − ,
− , 0,
− , 0, T −
.
B1 B2 B1
B3 B2
B3
(26)

i = 1, 2, 3.
The evaluation of αi is explained in sec. III.3. A general
solution of the inhomogeneous linear differential equation in (16) provides
p
p
C1
T (1) (x) = γ1 exp( B1 x) + γ2 exp(− B1 x) + , x ∈ (0, L1 ),
B1
p
p
C
2
T (2) (x) = γ3 exp( B2 x) + γ4 exp(− B2 x) + , x ∈ (L1 , L2 ),
B2
p
p
C3
T (3) (x) = γ5 exp( B3 x) + γ6 exp(− B3 x) + , x ∈ (L2 , L3 )
B3
(20)
The contact resistances and the attached objects in the
second case are considered at the exterior boundary of
the cable, i.e. at x = 0 and x = L3 , yielding the following
power balance2 :
l
Pcab − Pad
= Pkl .
Pcab = −Λ T 0 (0),
l
Pad
Pkl
Boundary and interface conditions
To get a unique solution in (20), appropriate boundary and interface conditions have to be formulated. As
in some applications the temperatures of adjacent objects
are known, we use Dirichlet boundary conditions for the
first case (cf. Fig. 4(a)). In the second case, we prescribe
the heat flow at the tails of the main modeled by Robin
boundary conditions (cf. Fig. 4(b)).
International Review of Mechanical Engineering, Vol. xx, n. x
(27)
Pcab denotes the heat power emitted at the contact boundl the heat power absorbed by the attached material
ary, Pad
l
and Pk the heat power produced by the contact resistance:
with γ1 , . . . , γ6 to be determined.
III.2.
(24)
4
denotes the heat resistance of the cable core to the environment in radial direction.
III.
(23)
With (20), this results in the linear equation system
Therein,
III.1.
(i+1) 0
(16)
with
1
Λ
T (i) (Li ) = T (i+1) (Li ),
T
d2 T (x)
− B T (x) +C = 0
dx2
B=
The interface conditions read as
0
= −λlad Ak Tad
(0),
= Rlk I 2 .
(28)
(29)
(30)
Solving (27) for T 0 (0) provides
− T 0 (0) =
l
Rl I 2
Pad
+ k .
Λ
Λ
(31)
2 We
restrict to the boundary condition for x = 0, the one for x = L3 is
derived analogously
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
F. Loos, K. Dvorsky, H.-D. Ließ
T11
T12
T13
T11
T(1)"-B1T(1)+C1=0 T(2)"-B2T(2)+C2=0 T(3)"-B3T(3)+C3=0
x=0
x=l1=L1
T(1)(0)=T l
x=l1+l2=L2
x=l1+l2+l3=L3
T(1)(L1)=T(2)(L1) T(2)(L2)=T(3)(L2)
T(1)'(L1)=T(2)'(L1) T(2)'(L2)=T(3)'(L2)
T12
T13
T(1)"-B1T(1)+C1=0 T(2)"-B2T(2)+C2=0 T(3)"-B3T(3)+C3=0
T(3)(L3)=T r
x=0
x=l1=L1
x=l1+l2=L2
x=l1+l2+l3=L3
T(1)'(0)=f(T(1)(0)) T(1)(L1)=T(2)(L1) T(2)(L2)=T(3)(L2) T(3)'(L3)=f(T(3)(L3))
T(1)'(L1)=T(2)'(L1) T(2)'(L2)=T(3)'(L2)
(a) Cable with fixed temperatures at both ends
(b) Cable with attached objects
Fig. 4 Cross sections in axial direction with governing equations, boundary and interface conditions
x=0
Using a function f for the heat flux description at both
cable ends (cf. Fig. 4(b)) and R0 = d2 /2, the boundary
conditions look as follows:
T=T(x)
Plad
Pcab
Tlad=Tlad(x)
P lk
π λlad d2
Λ
r d
π
λ
ad 2
f (T (3) (L3 )) =
Λ
f (T (1) (0)) =
Fig. 5 Heat powers at the left exterior boundary
In order to formulate (31) explicitly, we have to determine
l dependent on the asymptotic temperature T l and the
Pad
ad
contact temperature T (0). We suppose a spheric heat expansion in the attached material. According to Fourier’s
law,
l
dTad
dR
Rl I 2
l
T (1) (0) − Tad
− k , (36)
Λ
Rr I 2
r
Tad
− T (3) (L3 ) + k . (37)
Λ
Thus, the equation system for the second case is
Π2 γ = b2
(38)
with
= −T 0 (0) implicates

l
= λlad AR
Pad
l
dTad
dR
R ∈ (R0 , ∞) .
,
(32)
Hence, the isotherms are located on hemispherical shells
Conductor
Ak T(0)
R0
R

Π(11) Π(12) 0
0
0
0
 τ11 τ−11 −τ21 −τ−21 0
0 


 σ11 −σ−11 −σ21 σ−21 0
0 
 , (39)
Π2 = 
 0
0
τ22 τ−22 −τ32 −τ−32 


 0
0
σ22 −σ−22 −σ32 σ−32 
0
0
0
0 Π(65) Π(66)
T
C2 C1
C3 C2
b2 = b(1) ,
− , 0,
− , 0, b(6)
(40)
B2 B1
B3 B2
where
AR
Attached
material
l
T ad
l
Fig. 6 Attached object characterized by λlad and Tad
with AR = 2R2 π. Separation of variables provides
dT =
l dR
−Pad
2π λlad R2
(33)
and integration over R ∈ (R0 , ∞) results in
l
Tad
− T (0) =
l
Pad
2πλlad R0
l
l
⇒ Pad
= 2πλlad R0 (Tad
− T (0)) .
(34)
√
Π(11) = π λlad d2 − Λ B1 ,
√
Π(12) = π λlad d2 + Λ B1 ,
p Π(65) = π λrad d2 + Λ B3 τ33 ,
p Π(66) = π λrad d2 − Λ B3 τ−33 ,
C1
l 2
l
l
b(1) = Rk I + π λad d2 Tad −
,
B1
C3
r
b(6) = Rrk I 2 + π λrad d2 Tad
−
.
B3
By solving the linear system (24) respectively (38), we
obtain γ1 , . . . , γ6 which enables to indicate the explicit
core temperature profile T (1) , T (2) , T (3) .
Inserting this into (31), we obtain
T 0 (0) =
2πλlad R0
Rl I 2
l
(T (0) − Tad
)− k .
Λ
Λ
(35)
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
International Review of Mechanical Engineering, Vol. xx, n. x
F. Loos, K. Dvorsky, H.-D. Ließ
III.3.
contractive in each section if
Mean value iteration
For an explicit solution of (16) in the three subsections
indexed by i = 1, 2, 3, we have to define a constant value
for the heat transfer coefficient α in each subsection. To
get adequate values α1 , α2 and α3 , we compute
T̄
(i)
1
=
li
1
=
li
ZLi
T (i) (x) dx
(41)
Li−1
γ2i−1
γ2i
Ci
√
τii − τi(i−1) − √
τ−ii − τ−i(i−1) +
Bi
Bi
Bi
with L0 = 0. This enables the determination of average
exterior insulator temperatures according to [10] by
!
4
dj
1
1
(i)
(i)
T̄2 = T̄ −
(42)
∑ λ j ln d j+1
2π j=2
!
ρ0 (1 + αρ (T̄ (i) − T0 )) I 2 ρs Is2
+
·
A5
A3
and
(i)
αi = α(T̄2 ).
(43)
(i)
Starting with T̄2,start , we solve the linear equation system (24) respectively (38), obtaining the temperature profile defined piecewise in (20). (41), (42) and (43) then
(i)
provide the iterative mean value surface temperatures T̄2,1
respectively αi . Repetition of this process leads to iter(i) atively defined sequences T̄2,k k∈N converging to final
mean temperatures. The desired temperature profile is
obtained as soon as the following stop criterion is fulfilled:
(i)
(i)
|T̄2,k − T̄2,k−1 | < ε.
(44)
To get favourable initial values for the fixed point
iteration, we calculate asymptotic surface temperatures
(i)
(i)
T̄2,start := T2,as described in [10]. Therein, methods for
the temperature determination of shielded cables without
axial temperature profile are presented.
Convergence of the iterative procedure
We define the fixed point mapping for the piecewise constant mean value temperatures T̄ (i) . Via the considerations in (3), (4) and (13), it is given by
T̄ (i) = T1i +
(i)
ρw pk (T̄ (i) ) + ps =: F (T̄ (i) )
π
(45)
where
pk =
ρ0 I 2 (1 + αρ (T̄ (i) − T0 ))
,
A5
ps =
ρs Is2
A3
(i)
(46)
and ρw is identified by (19). Hence, the mapping F is
International Review of Mechanical Engineering, Vol. xx, n. x
ρ0 αρ I 2
π
< (i) ,
A5
ρw
(47)
i.e. if Bi > 0. On the other hand, Bi > 0 is also the condition for a consistent evaluation of the temperature profiles
in (20). Thus, by the Banach fixed-point theorem, Bi > 0
yields a sufficient condition for the convergence of the
iterative procedure.
Following [11], (47) is equivalent to a subresonance
condition which ensures existence and uniqueness of solutions to a semilinear elliptic equation describing the full
stationary heat transfer problem. If (47) is not fulfilled,
the respective electrical currents yield temperatures far
from practical relevance.
IV.
Comparison of measurement and
calculation results
To test the presented calculation method concerning
practical application, laboratory measurements were performed by our industrial partner. Therein, four different
shielded cable types were loaded by constant currents for
the core and shielding over a sufficiently long time such
that stationary temperatures were reached. The four cable
types with geometrical, thermal and electrical properties
are listed in Table I. They are labelled according to their
metallic cross sectional area in the core.
TABLE I
CABLE TYPES AND THEIR PROPERTIES
Cable Type
25 mm2
35 mm2
Cable property
16 mm2
50 mm2
d2 [mm]
d3 [mm]
d4 [mm]
d5 [mm]
9.90
7.54
6.90
5.30
11.90
9.34
8.50
6.65
14.10
11.00
10.15
7.70
15.50
12.69
11.85
9.45
λ2 [W/(m · K)]
λ3 [W/(m · K)]
λ4 [W/(m · K)]
λ5 [W/(m · K)]
0.23
193
0.23
310
0.23
193
0.23
310
0.23
193
0.23
310
0.23
193
0.23
310
ρ0 [Ω · m]
ρs [Ω · m]
αρ [1/K]
2.54e-8
4.96e-8
3.93e-3
2.54e-8
4.96e-8
3.93e-3
2.54e-8
4.96e-8
3.93e-3
2.54e-8
4.96e-8
3.93e-3
Via thermocouples, temperatures of the core, the shielding and the environment were taken in the laboratories
with T1 = T11 = T12 = T13 = 25 ◦ C. An axial temperature
distribution has not been measured, only temperatures at
one position were examined. Thus, the comparison of
our method is performed with a long cable (> 10 m) to
neglect external influences by attached objects.
For comparison, we restrict to material parameters
given by our industrial partners which have standard
specifications for each cable type. All these material parameters have to be treated with caution. Consequently,
relative errors less than 5 % can hardly be expected.
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
F. Loos, K. Dvorsky, H.-D. Ließ
0
50
100
150
200
250
25 25
35 34
59 59
99 100
163 164
253 255
31 33
38 41
61 65
100 106
166 169
256 260
0
1
0
1
1
2
2.9
0.0
1.0
0.6
0.8
2
3
4
6
3
4
6.5
7.9
6.6
6.0
1.8
1.6
IS = 0 A
IS = 20 A
Tcal
200
TS cal
Tme
Temperature difference (K)
Is = 20 A
S
S
Tme
Tcal
E
e
◦
◦
[ C] [ C] [K] [%]
Temperature difference (K)
I
[A]
Is = 0 A
Tme Tcal E
e
[◦ C] [◦ C] [K] [%]
150
100
50
0
0
50
100
150
200
Electrical current (A)
200
150
100
50
0
250
TS me
0
50
100
150
200
Electrical current (A)
250
Fig. 7 Measurements, calculations and errors for 16 mm2
0
80
160
240
295
345
25 25 0
36 37 1
69 72 3
129 133 4
196 196 0
286 270 16
28 30 2 7.1
37 41 4 10.8
70 75 5 7.1
132 137 5 3.8
198 199 1 0.5
287 273 14 4.9
2.8
4.4
3.1
0
5.6
IS = 0 A
250
IS = 20 A
Tcal
200
150
100
50
0
0
TS cal
250
Tme
Temperature difference (K)
Is = 20 A
S
S
Tme
Tcal
E
e
[◦ C] [◦ C] [K] [%]
Temperature difference (K)
I
[A]
Is = 0 A
Tme Tcal E
e
[◦ C] [◦ C] [K] [%]
TS me
200
150
100
50
0
100
200
300
Electrical current (A)
0
100
200
300
Electrical current (A)
Fig. 8 Measurements, calculations and errors for 25 mm2
27 29 2 7.4
36 38 2 5.6
60 66 6 10.0
104 113 9 8.7
177 188 11 6.2
237 246 9 3.8
me
150
100
50
0
0
125
250
375
500
535
25 25 0
34 37 3 8.8
65 71 6 9.2
122 132 10 8.2
220 228 8 3.6
261 263 2 0.8
27 28 1 3.7
37 40 3 8.1
68 73 5 7.4
124 134 10 8.1
227 230 3 1.3
268 265 3 1.1
TS cal
200
T
0
100
200
300
Electrical current (A)
T
S me
150
100
50
0
400
0
100
200
300
Electrical current (A)
IS = 20 A
Tcal
200
TS cal
Tme
150
100
50
0
0
400
35 mm2
IS = 0 A
Temperature difference (K)
I
[A]
Is = 20 A
S
S
Tme
Tcal
E
e
◦
◦
[ C] [ C] [K] [%]
S
Tcal
200
Fig. 9 Measurements, calculations and errors for
Is = 0 A
Tme Tcal E
e
[◦ C] [◦ C] [K] [%]
I = 20 A
S
Temperature difference (K)
25 25 0
33 35 2 6.1
57 63 6 10.5
100 111 11 11.0
172 186 14 8.1
233 243 10 4.3
I =0A
Temperature difference (K)
0
87.5
175
262.5
350
400
Is = 20 A
S
S
Tme
Tcal
E
e
◦
◦
[ C] [ C] [K] [%]
Temperature difference (K)
I
[A]
Is = 0 A
Tme Tcal E
e
[◦ C] [◦ C] [K] [%]
100 200 300
400
Electrical current (A)
500
TS me
200
150
100
50
0
0
100 200
300
400
Electrical current (A)
500
Fig. 10 Measurements, calculations and errors for 50 mm2
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
International Review of Mechanical Engineering, Vol. xx, n. x
F. Loos, K. Dvorsky, H.-D. Ließ
16mm2
25mm2
35mm2
50mm2
Core temperature (°C)
200
150
100
50
0
0.5
1
1.5
2
2.5
3
3.5
Axial cable position (m)
Fig. 12 Comparison of different cables under same conditions
the manufacturer shall choose the cable type 35 mm2 .
Since the connected electrical components only work
at specific voltages, information about the voltage drop
on the cable are necessary. It is calculated via Ohm’s law
∆U = R · I where R denotes the temperature dependent
resistance (1).
1.4
1.2
Voltage drop (V)
Fig. 7-10 show measurement and calculation results
for the four cable types. Therein, Tme denotes the measured temperature, Tcal the computed temperature, E the
absolute difference between measurement and calculaS repretion and e the relative error. Furthermore, Tme
sents the measured core temperature with current carryS the corresponding computed teming shielding and Tcal
perature. The curves illustrate the differences of the
measured and computed temperatures to the environment
temperature.
A maximum absolute difference of 16 K and a maximum relative error of 11 % confirm good accordance of
the calculations and the measurements. As mentioned,
even better agreement cannot be expected for reasons of
absence of exact material parameter values. Furthermore,
the errors crept in by the measurements essentially influence the listed errors.
By tendency, the calculated temperatures are higher
than the measured ones except for cable 35 mm2 and
I = 345 A. For this case, measurement problems seem
likely as these values do not match with the other measured values. The tendency of higher computed temperatures is due to the lower real packing density than supposed in the cable specifications.
1
0.8
0.6
0.4
V.
Further calculation results
0.2
0
16mm2
25mm2
35mm2
50mm2
Cable type
Fig. 13 Voltage drop for the different cable types
Fig. 14, 15 show a variation of the left boundary temperature T l in a longer and a shorter cable.
150
Tl = 0.00 °C
Core temperature (°C)
To illustrate the possibilities of our method, we calculate temperature profiles with varying parameters.
First, we examine the influence of different currents (cf.
Fig. 11). We suppose temperatures of the attached objects
to be T l = 80 ◦ C and T r = 70 ◦ C and the environment
temperatures T11 = 60 ◦ C, T12 = 10 ◦ C and T13 = 90 ◦ C
with cable lengths l1 = 1 m, l2 = 1.50 m and l3 = 1 m.
The cable type is 16 mm2 and the shielding current is for
each case Is = 40 A.
Tl = 50.00 °C
Tl = 100.00 °C
Tl = 150.00 °C
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
Axial cable position (m)
Fig. 14 Variation of left object temperature T l for a long cable
Fig. 11 Axial core temperatures for varying currents
160
International Review of Mechanical Engineering, Vol. xx, n. x
140
Core temperature (°C)
If the manufacturer wants to choose the appropriate cable type for fixed currents, the temperature distribution in
different cable types can be compared (cf. Fig. 12). We
take the same parameters as in the precedent case with a
fixed core current of I = 200 A.
As expected, the temperature increases for smaller cable cross sections. Suppose the temperature in the cable must not exceed the critical value of 150 ◦ C – corresponding to the melting temperature of the insulation –
120
100
80
60
Tl = 0.00 °C
40
Tl = 50.00 °C
Tl = 100.00 °C
20
0
Tl = 150.00 °C
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Axial cable position (m)
Fig. 15 Variation of left object temperature T l for a short cable
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
F. Loos, K. Dvorsky, H.-D. Ließ
Observe that the influence of the boundary temperature
in the shorter cable is more significant than in the longer
one.
One expects that higher current densities imply higher
temperatures at the boundaries. Hence, the temperature
profiles with Dirichlet conditions in Fig. 12 are rather
non-physical at the boundaries. Therefore, the same simulations are performed with respect to Robin boundary
conditions discussed in sec. III.2.
16mm2
25mm2
35mm2
50mm2
Core temperature (°C)
200
100
50
0.5
1
1.5
2
2.5
3
3.5
Axial cable position (m)
Fig. 16 Comparison of different cables under same conditions
with Robin boundary conditions
l = 80 ◦ C reIn Fig. 16, asymptotic temperatures are Tad
r = 70 ◦ C, contact resistances Rl = 1 mΩ
spectively Tad
k
respectively Rrk = 1 mΩ and the heat conductivities of
the two attached objects λlad = 10 W/(m · K) respectively
λrad = 10 W/(m · K). The changed boundary conditions
hardly influence the inner cable temperature profile, but
yield more realistic results at both ends of the cable. The
reason for the minor influence in the middle section is the
great length of the cable.
VI.
The authors wish to thank the project partners VW AG
and Labco GmbH for financial support, for their industrial input contributing to this work and the measurement
data taken in the laboratories of Labco.
References
[1] F. Loos, H.-D. Ließ, K. Dvorsky, Simulation Methods for
Heat Transfer Processes in mechanical and electrical
Connections, Proc. 1st Intern. Elec. Drives Production
Conf. 2011 (EDPC), Nuremberg, 2011, pp. 214-220.
[2] J. H. Neher, M. H. McGrath, The Calculation of the
Temperature Rise and Load Capability of Cable Systems, AIEE Transactions, Vol. 76, p. 3, pp. 752–772,
1957.
150
0
Acknowledgments
Conclusion
We presented a new method for calculation of stationary temperature profiles in shielded cables of finite
length. The influence of possibly differing environment
temperatures and attached objects is respected. Due to
the elaborated convergence conditions, we concluded that
the method converges for the practically relevant range of
parameters. Moreover, it is extremely fast – computation
times are less than one second – and flexible. Changes in
geometry can be handled much more easily compared to
the, e.g., finite element method.
Good agreement between measurements and calculations was observed which qualifies the method for direct
industrial application. It has been implemented in a simulation tool which excels by mastering complex industrial
tasks.
For the near future, we intend measurements and comparison concerning the axial temperature distribution.
Extension of the method to time dependent problems has
partly been realized and is still in progress. Thus, non stationary current profiles can be treated which essentially
enlarges the field of application.
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
[3] F. P. Incropera, D. P. DeWitt,T. L. Bergman, A. S.
Lavine, Fundamentals of Heat and Mass Transfer (8th
edition, Wiley, 2007).
[4] J. Taler, P. Duda, Solving Direct and Inverse Heat Conduction Problems (1st edition, Springer-Verlag, 2006).
[5] G. J. Anders, Rating of Electric Power Cables: Ampacity Computations for Transmission, Distribution, and Industrial Applications (1st edition, McGraw-Hill, 1997).
[6] G. J. Anders, Rating of electric power cables in unfavorable thermal environment (1st edition, IEEE Press,
2005).
[7] A. Ilgevicius, Analytical and numerical analysis and
simulation of heat transfer in electrical conductors and
fuses, Ph.D. dissertation, Fac. Elec. Inform. Tech., Univ.
German Federal Armed Forces, Munich, 2004.
[8] B. Démoulin, L. Koné, Shielded Cables Transfer
Impedance Measurement, IEEE-EMC Newsletter, p.
38–49, 2010.
[9] H. Klan, Wärmeübergang durch freie Konvektion an
umströmten Körpern, In VDI-Wärmeatlas, 9, (Berlin:
Springer-Verlag, 2002, part Fa).
[10] M. Stahl, Rechnergestützte Wärmeleitungsberechnung
in stromdurchflossenen, abgeschirmten Einzelleitungen,
Bachelor thesis, Fac. Elec. Inform. Tech., Univ. German
Federal Armed Forces, Munich, 2010.
[11] K. Dvorsky, Analysis of a Nonlinear Boundary Value
Problem with Application to Heat Transfer in Electric
Cables, Ph.D. dissertation, Fac. Aerosp. Tech., Univ.
German Federal Armed Forces, Munich, 2012.
International Review of Mechanical Engineering, Vol. xx, n. x
F. Loos, K. Dvorsky, H.-D. Ließ
Authors’ information
Florian Loos was born in Nuremberg,
Germany, the 26th march 1982. He finished his studies at the University of
Bayreuth, Germany, and the University
of Marne-La-Vallée, France, with the
diploma in mathematics and informatics
in 2008. Since march 2009, he is a PhD
student at the University of the Bundeswehr in Munich, Germany. The working title of his PhD thesis is ’Modelling, simulation and optimization of Joule heating problems for electrical
devices and connections’.
He develops and investigates simulation methods for the calculation of heat transfer in current carrying cables, busbars and
safety fuses in order to correctly dimension electrical devices
and connections.
His research fields are, apart from thermodynamics, numerical
mathematics, shape optimization and optimal control problems.
Karl Dvorsky was born in Prague,
Czech Republic, the 13th may 1982.
He studied mathematics and physics at
the Humboldt University of Berlin, Germany, from April 2002 to April 2007. In
June 2007, he started working as a PhD
student at the University of the Bundeswehr in Munich. The title of his PhD thesis is ’On a nonlinear boundary value problem with application to heat transfer
in electric cables’ which he is about to finish.
His main research subject is the thermal analysis of conductive
and convective heat transfer via partial differential equations.
Furthermore, he supports research and teaching at the university
in the branch ’mathematical engineering’.
Hans-Dieter Ließ was born in Dresden, Germany, the 17th september 1934.
He studied electrical engineering and
physics at the Technical University of
Munich and the Technical University of
Berlin from 1954 till 1963. From 1963
till 1968, he was head of development
of an industrial company in the field of
electronic components. From 1968 to 1972, he became head
of the components section of the European space organisation
ESA-ESTEC in Noordwijk, Netherlands. After his return to
Germany, he worked as technical director of a manufacturer of
electronic devices. In 1976, he was appointed as professor for
electrical engineering at the University of the Bundeswehr in
Munich.
His research field started with medical electronics, was extended to chemical and physical sensors and is at present mainly
concentrated on the area of mathematical engineering.
International Review of Mechanical Engineering, Vol. xx, n. x
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved