Evaluation of the impact of proximity effect in the calculation of the internal impedance of
cylindrical conductors
Alberto Pagnetti1, Alain Xemard1, Francoise Paladian2 , Carlo Alberto Nucci3
1
EDF R&D, 1, Av.du G.de Gaulle, 92141, Clamart, France email : Alberto.pagnetti@edf.fr –
Alain.xemard@edf.fr
2
Clermont University, Blaise Pascal University, LASMEA, Clermont-Ferrand, France – email :
Francoise.PALADIAN@lasmea.univ-bpclermont.fr
3
Dip.di Ingegneria Elettrica - Facoltà di Ingegneria - Università degli Studi di Bologna, Viale Risorgimento 2,
40136, Bologna, Italia email: Carloalberto.nucci@unibo.it
Abstract
In this paper we discuss how the inclusion of proximity effect influences the evaluation of the internal impedances
of cylindrical conductors.
After recalling the classical analytical formulation used for calculating internal impedance, we use a Finite Element
Method (FEM) code, in order to take into account the proximity effect. We show that in a low voltage cable, at
frequency higher than the industrial ones, and particularly above the kilohertz range, the impact of proximity effect
is rather relevant and cannot be neglected when dealing with EMC issues related to cable networks.
1. Introduction
It is well known that the description of the propagation of electromagnetic waves in transmission lines supporting
the Transverse Electromagnetic (TEM) mode is usually made in terms of voltages and currents using the circuit
parameters of the line. For uniform transmission lines it is only necessary to know the secondary circuit parameters;
i.e.: the characteristic impedance Zc, and the propagation constant Γ, these being derived from the primary per unit
length line parameters, namely G’, shunt conductance, L’, series inductance, and C’, shunt capacitance.
When using the quasi-TEM approximation, voltages/currents description can still be used, but two additional
primary parameters are added, namely a series resistance RINT’ and an inductance LINT’, to account for certain losses
in the conductors and for skin-effect.
In the area of telecommunications the effect of the shunt conductance on the propagation phenomena can generally
be neglected as well as those of the internal resistance and inductance. As a result quasi-TEM structures can be
described only in terms of L’ and C’. This is for instance the case with coaxial cables, twisted pairs and micro-strip
lines.
In the area of energy transmission, at the industrial frequency the internal resistance and inductance cannot be
neglected and this affects the secondary parameters expression, which means attenuation and distortion in the
propagation of voltages/currents waves, especially when cables are dealt with.
We will treat the case of a low-voltage cable. This is interesting for the possibility these cables have to transmit
signals; for instance a type of Power line communication (PLC) will be used in future smart meters to be installed in
France; to allow for the exchange of information with the power company. Such low voltage cables can also be used
in other appliances, where transient high frequency phenomena require an accurate description of the cable’s
propagation function, also at high frequency.
2. Eddy currents phenomena and relevant formulations
Classical formulas used to compute the internal resistance and internal inductance of the conductors assume the
conductor to be “isolated”; Schelkunoff [1] has given them for circular and tubular conductors.
The magnetic field generated by the current flowing in the conductor induces on the conductor itself eddy currents,
which modify the current distribution displacing it toward the periphery; this “skin effect” phenomenon increases
the value of the resistance in proportion to the root of the frequency (as compared to the DC resistance) . This same
phenomenon can also be generated by current/s flowing in nearby conductor/s and cannot be neglected when the
distance/s of the nearby conductor/s is/are of the same order as the conductors radiuses as they are in energy
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transport and distribution cables, and the analysis extend to the higher frequencies components of transient
phenomena, in which case it is denoted as “proximity effect”.
The modification due to this proximity effect on the internal resistance and inductance is not easily described by
means of analytical formula and hence, in order to show the importance of taking it into account, and evaluate Zint
with an increased accuracy, we have used a FEM code in the frequency domain, to evaluate the EM field present in
core conductors (which also depends upon proximity effect) and from it we have deduced the internal impedance
value.
Here we report the classical Schelkunoff formula [1] of the internal impedance for a solid cylindrical conductor:
⎛ jωμσ
⎞
(1)
Z = ⎜⎜
I r
jωμσ I r
jωμσ
2πσ r ⎟⎟ 0 int
int
1 int
int ⎠
⎝
Here ω=2πf , σ is the conductivity of the conductor, rint its radius, µ its magnetic permeability, I0 and I1 are Bessel
modified functions of the first kind respectively of 0th and 1st order.
The FEM code we used is a commercial one, Comsol [2]; the formulation it solves is in terms of magnetic vector
potential.
The quantities evaluated by Comsol in each node of the mesh are the vector potential, the current density and the
magnetic field. In order to calculate the internal impedance Zint=Rint+jωLint the following formulas are used:
1
2
Rint =
(2)
∫ J dS
σ | I |2 cond
( (
Lint =
1
| I1 |2
I=
) (
∫
))
2
μ H dS
(3)
cond
∫
J dS
(4)
cond
Where I is the current, J is the current density, H is the magnetic field.
3. Description of the low voltage cable and simulation parameters
We now analyse the parameters of a low voltage three phase cable with a simple configuration.
Fig. 1 – Description of the geometrical structure of the
low voltage cable
Fig. 2 – The mesh used in Comsol (here for 50 kHz)
The cable used in the simulation is depicted in Fig. 1 and its geometrical parameters are: rcond=1 mm , riso=1.975
mm, rsheath=5.8 mm. The conductors material is copper, whose conductivity is σ=5.814e7 S/m and magnetic
permeability µ0.
For the FEM simulation, we used a triangular free mesh, with the only constraint due to the skin penetration depth δ.
δ = 1/ π f μσ
(5)
The triangles used inside the conductors (see Fig. 2 ) are constrained to be smaller than half the skin penetration
depth, in order to better represent the current density; this means that the number of nodes (and the complexity of the
simulation) increases with frequency. Normally the mesh is also limited by the wavelength, but in our case at 100
kHz the wavelength is much greater than any of the cable’s dimensions.
4. Results
We can see in Table 1 that the internal resistance of a conductor is
modified by the proximity effect, but this effect is only relevant for
frequencies above the kHz.
It must be noted (and can be seen also in the current density plot in Fig.
4) that at 50 Hz the current distribution is uniform, because the
conductor radius is much smaller than the skin depth. In the same way as
skin effect, the proximity effect depends strongly on the frequency,
providing roughly square-root proportionality between resistance and
frequency.
In this case the distance between conductors is 4 times their radius; with
larger spacing between conductors the proximity effect becomes
unnoticeable. The spacing between conductors is due to the insulation
layer(s) present between them. In the case of power cables the insulation
thickness is dependent upon the voltage to which the insulation must
withstand. This is the reason why the proximity effect is more
pronounced for low voltage cables.
Frequency
Relative difference
(Schelkunoff - FEM) [%]
50 Hz
0.01
1 kHz
0.24
5 kHz
4.86
10 kHz
11.65
50 kHz
18.86
100 kHz
20.04
1 MHz
22.06
Table 1 – Relative difference between
the internal resistance of one
conductor obtained with analytical
formulation of Schelkunoff and FEM
results
-8
Internal resistance
8
P.u.l inductance [H/m]
P.u.l. resistance [ Ω /m]
0.06
0.05
0.04
0.03
0.02
Schelkunoff
FEM
0.01
0
0
2
4
6
Frequency [Hz]
8
10
x 10
Internal inductance
6
Schelkunoff
FEM
4
2
0
0
5
2
4
6
Frequency [Hz]
8
10
5
x 10
x 10
Fig. 3 – Graphs of internal resistance and inductance of one conductor comparing analytical formulation and
FEM results
The cable we described has a small section, and is therefore meant for home circuits, or small industrial appliances.
Low voltage cables with bigger sections exist and are used for distribution networks; these normally have sector
shaped cores (in order to reduce the overall cable radius and mitigate skin effect), but their mutual distance is
smaller than for the cable we have considered, and therefore the proximity effect should be more relevant.
5. Conclusions
The necessity of using internal parameters of conductors, resistance and inductance, in the TL description has
motivated this contribution.
The comparison performed between the results from traditional analytical formulas for the calculation of the internal
impedance and those obtained using a FEM simulation shows the importance of including the proximity effect, in
the description of cables not having a coaxial structure.
Such an effect is however only relevant for frequency in the kHz range and above.
The comparison we showed here is less straightforward for sector shaped cables, as their parameters cannot be
calculated with traditional analytical formulas (that can only be applied to circular conductors).
We have shown, however, that using a FEM code the resistance/inductance calculation can be implemented, and this
holds also for conductor with unconventional shapes.
Fig. 4 – Magnetic field (left side) and current density (right side) at 50 Hz (first line) and 50 KHz (second line)
This preliminary work represents a base for further studies. As a matter of fact, all low voltage cables are of interest
for transmitting telecommunication signals; for instance the smart meter which will be used in France communicate
using low voltage cables in the 10-100 KHz range.
Furthermore, existing Electromagnetic Transient Programs (like EMTP [3]) do not currently allow a correct
representation of cables with non coaxial geometries (as the low voltage cable we have studied here, but also for
high voltage submarine cables).
For these reasons we are developing a semi-analytical method that permits the calculation of Zint with an adequate
accuracy, starting from the knowledge of the geometrical and electrical parameters of the conductors,
6. References
[1] The electromagnetic theory of coaxial transmission lines and cylindrical shields. Schelkunoff, S.A. 1934, Bell
System Technical Journal, pp. 532-579.
[2] COMSOL AB 2005 COMSOL Multiphysics User’s Guide (Stockholm: COMSOL AB)
[3] H.W.Dommel. EMTP Theory Book. s.l. Microtran Power System Analysis Corporation, 1992.