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DEGREE PROJECT, IN , SECOND LEVELELECTRIC POWER ENGINEERING
STOCKHOLM, SWEDEN 2015
Propagation and mitigation of VeryFast Transient Overvoltage in GasInsulated Substation
FRANÇOIS GALLIANO
KTH ROYAL INSTITUTE OF TECHNOLOGY
ELECTRICAL ENGINEERING
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KTH - SUPERGRID INSTITUTE
Master Thesis Propagation and mitigation of Very Fast Transient Overvoltage in Gas Insulated
Substation
GALLIANO François
14/01/2015
Examiner and academic responsible : Hans Edin Supervisors : Paul Vinson – Thomas Berteloot
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Acknowledgments
I would like to thank my supervisor in KTH, Hans Edin, for giving me the opportunity to do
this master thesis.
Thank you very much to all members of Supergrid Institute and especially to Paul Vinson
and Thomas Berteloot who provided me with general supervision, constant support and most
interesting discussions.
I would also like to thank all my colleagues for their availability, the quality of their advice,
and for making this degree project such a nice experience.
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Acknowledgments .................................................................................................................... 2
Abstract ..................................................................................................................................... 6
1 Introduction ...................................................................................................................... 9
1.1 Background and motivation ....................................................................................... 9
1.2 Aim and objectives ................................................................................................... 11
1.3 Thesis disposition ..................................................................................................... 12
2 Mitigation of VFTO in a coating .................................................................................. 13
2.1 Multilayer coaxial structure model .......................................................................... 13
2.1.1 Electromagnetic model (infinite geometry) ......................................................... 13
2.1.2 Comprehension of the phenomenon .................................................................... 18
2.1.3 Equivalent electric representation ........................................................................ 25
2.1.4 Theoretical optimisation ...................................................................................... 32
2.2 Mitigation of the VFT............................................................................................... 37
2.2.1 Arc resistance ....................................................................................................... 38
2.2.2 Damping system model........................................................................................ 39
2.2.3 Distributed line model – constant parameters ...................................................... 43
2.2.4 Influence of the electric parameters ..................................................................... 45
2.3 Choice of material.................................................................................................... 48
2.3.1 General considerations ......................................................................................... 48
2.3.2 Ferromagnetic materials: typical properties ......................................................... 50
3 Mitigation of VFTO using magnetic rings ................................................................... 55
3.1 Total core loss – state of the art............................................................................... 55
3.2 Classical eddy losses................................................................................................ 57
3.2.1 Eddy currents in a single lamination .................................................................... 57
3.2.2 Impedance of nanocrystalline rings ..................................................................... 62
3.2.3 Performance ......................................................................................................... 67
3.3 Instantaneous losses................................................................................................. 72
4 Conclusion ...................................................................................................................... 77
Appendix: FEM validation of the principles with COMSOL ............................................ 79
Bibliography ........................................................................................................................... 83
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Abstract
Very Fast Transients are surge overvoltages created for example when opening or closing a
disconnector switch (DS). Because of the very high frequency stress they exert on the
equipment and their magnitude (up to 3 pu), they constitute an important issue in the design
of ultra-high voltage Gas Insulated Substations.
This thesis offers a general understanding of Very Fast Transient Overvoltages (VFTO or
VFT) and explores different mitigation techniques, mainly based on dissipation of the energy
associated with it.
Two solutions are mainly discussed for attenuating the VFTO: applying a semi resistive and
magnetic coating or using magnetic rings. Both rely on the non-linearity of skin effect and
magnetic losses and are treated in similar manners. An electromagnetic model of both
systems is proposed as well as a way to model the influence of the device on the transient
behaviour of the system.
The performances of both systems still have to be determined by experimentation but they
both should significantly reduce the impact of the VFTO.
Sammanfattning
Mycket snabba transienter (eng. Very Fast Transients) är överspänningar som skapas till
exempel när en frånskiljare öppnas eller stängs (DS). På grund av de mycket högfrekventa
påkänningar som de utövar på utrustningen och deras storlek (upp till 3 pu), utgör de en
viktig fråga i utformningen av ultrahögspännings gasisolerade ställverk.
Denna avhandling ger en allmän förståelse för mycket snabba transienta överspänningar
(VFTO eller VFT) och undersöker olika metoder för dämpning, främst baserade på förlust av
energi. .
Främst diskuteras två lösningar för dämpning av VFTO: applicering av en semi resistiv och
magnetisk beläggning kring ledaren eller använda magnetiska ringar. Båda är beroende av
icke-linjäritet av strömförträngningseffekt och magnetiska förluster, båda
metodernabehandlas på liknande sätt. En elektromagnetisk modell av båda systemen föreslås
liksom ett sätt att modellera inverkan av anordningen på transienta systemets beteende.
Utförandet av båda systemen måste verifieras genom experiment, men de båda bör avsevärt
minska effekterna av VFTO.
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1 Introduction
This section provides with the motivation of studying and mitigating Very Fast Transient
Overvoltages (VFTO) in Gas Insulated Substations (GIS). It also presents a few typical
quantities associated with VFTOs.
1.1 Background and motivation
Gas Insulated Substations are highly reliable and compact and thus have spread widely in
transmission and distribution networks. The levels of voltages now reached in GIS (up to
1200 kV AC) present new issues, and in particular VFTOs. Very Fast Transients are surge
overvoltages created for example when opening or closing a disconnector switch (DS). They
can also occur during the operation of a circuit breaker or because of an earth fault, but
because of their low operating speed, disconnector switches create an important amount of
restrikes between the two electrodes and are considered as the main source of VFTO [1]; [2].
A disconnector switch is used to disconnect a set of equipment (typically a circuit breaker)
from the network. It is not used to interrupt a fault but to disconnect a part of the system in
which there is normally no current flowing. A disconnector switch has two electrodes which
are relatively slowly separated (or joined) when opening (or closing). One electrode is under
nominal 50 Hz (or 60 Hz) voltage, while the other one is at a floating potential. When the
potential difference between the two electrodes is higher than the breakdown voltage, an arc
(restrike) is created which suddenly equalizes the two electrode potentials and creates two
waves with a steep front to propagate along the GIS (see Figure 2). Of course the breakdown
voltage depends on the spacing between the two electrodes. As the two electrodes are
separated, the breakdown voltage increases and the number of restrikes per second decreases
while their magnitude increase, as illustrated in Figure 1.
Figure 1 - Electrode voltages in an opening switch [3]
0.15 pu trapped charge
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Figure 2 - Equalization of a switch elctrodes' potentials
The typical rise time of the wave in AIS (Air Insulated System) is in the order of . Because
GIS are filled with SF6, breakdown occurs at higher values and are steeper, and the rise time
of the created wave is about to , against around for lightning impulse. One
other critical aspect is the multiple reflections and refractions of these waves that exist in GIS
because of its geometry and the numerous impedance changes. The peak value can then be as
high as 3 p.u. with very high frequency components, up to [2]. Figure 3 illustrates
the typical waveform of a VFTO measured at the load electrode of the DS (V0, red) and at
the open end of the GIS (V2, green).
Figure 3 - Typical VFTO wave form. V0: voltage at the source electrode of the DS; V2: voltage at the open end
The peak value and the waveform of the VFTO depend on the nominal value of the system
and the trapped charge in the bus, as well as its geometry and length.
Such overvoltages can excess the Lightning Impulse Withstand Level of the system and
seriously damage equipment upstream or downstream of the disconnector switch, which
makes VFTOs an important topic of study for ultra-high voltage systems. As shown in Figure
US
UL
UE
Source side Load side
(open)
Voltages at prestrike/restrike
Time
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4, for 800 kV systems and above, VFTO magnitudes can be higher than that of lightning
impulse (LI). The International Council on Large Electric Systems (CIGRÉ) demands that all
equipment be designed so that VFTO do not exceed a percentage value of the typical
lightning impulse.
Figure 4 - Lightning impulse voltage levels, VFTO levels and safety levels of overvoltages fixed by CIGRE vs nominal
voltage
1.2 Aim and objectives
The creation of an arc (restrike or prestrike) is intrinsic to the functioning of a disconnector
switch. The objective is not to prevent it from happening but rather control its impact on the
equipment by mitigating it. Some methods have already been implemented to mitigate VFTO,
such as damping resistors or resonators etc [4]. The main challenge is here to study compact
and easily manageable solutions to damp VFTO without degrading the system’s performance
in its nominal regime. The specifications would then be:
The solution must mitigate only high frequency components (higher than 100 kHz)
and not affect the 50-60 Hz regime.
The solution must be compact and easily integrated in an existing Gas Insulated
Substation - GIS (the design for a damping resistor in parallel with the disconnector
switch and put in series when operating it is not very efficient on this point of view).
The solution must prevent VFTO from propagating and damaging other equipment by
significantly trapping it or dissipating it.
The solution must dissipate the energy associated with the VFTO in a time constant to
be determined. In particular if the VFTO is trapped, the energy associated with it can
be dissipated slowly. Other methods relying only on dissipation should have a very
small time constant (the resonator tends to trap the VFTO which is then slowly
Max
imu
m V
olt
age
[kV
p]
Nominal Voltage of the apparatus [kVrms]
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dissipated in the natural resistance of the bus bar, but this solution doesn’t seem to
efficiently reduce the peak of the VFTO).
1.3 Thesis disposition
Chapter 1 here above provides with the basic knowledge about VFTO, necessary to
understand the objectives and applications of the mitigation systems envisaged, which are
presented in chapters 2 and 0.
Chapter 2 presents the necessary steps for developing an optimal mitigation system based on
a coated GIS bus bar. An analytical model of wave propagation in a multilayer coaxial
structure is proposed in section 2.1, which allows analysing the effects of the coating on the
system’s properties. The conducted method for transient simulation of the system is then
presented in section 2.2. It presents how, from the field calculations conducted in section 2.1,
one can model the system by an equivalent electrical diagram that can then be implemented
in a simulation software (ATP in this thesis). Finally, a few elements of comparison of
different ferromagnetic materials are given in section 2.3 that could guide the reader in his
choice of material for developing an actual prototype.
Chapter 3 is based on the same logic as chapter 2. It presents how magnetic rings can mitigate
VFTO. Section 3.1 provides with different modelling techniques that could be used as well as
references. Section 3.2 presents the modelling method chosen for this paper, that is, a
complete calculation of the electromagnetic fields in the laminated magnetic rings. Brief
elements of evaluation of the possible performance are presented as well. Section 3.3 finally
presents extensions of the magnetic considerations that should be taken into account for a
more precise model.
Chapter 4 is a brief conclusion on the principles, performances and optimisation of the two
mitigation techniques and also mentions other points of interest that should be considered for
developing a proper prototype.
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2 Mitigation of VFTO in a coating
The very first approach envisaged to limit the amplitude of VFTO is by application of a
dissipating frequency selective device based on skin effect. By applying a proper semi-
conducting coating on an existing bus bar, it should be possible to damp the high frequency
components of the VFT without affecting the 50-60 Hz properties of the system. The
underlying physical phenomena associated with this idea being eddy-currents and
electromagnetism, it seems natural to start with Maxwell’s equations.
2.1 Multilayer coaxial structure model
In order to understand how VFT can be dissipated in a resistive layer set on an aluminium
conductor, it is essential to understand the propagation phenomena of an electromagnetic
wave inside a conductor. The main idea with such a device is to dissipate the energy
associated with the overvoltage by resistive losses. It is then of main interest to compute the
global resistance per unit length (sometimes abusively called resistivity) of the coaxial system
and see how it can be affected by the properties of an additional outer layer. This is done here
by studying the electromagnetic fields inside the conductor.
2.1.1 Electromagnetic model (infinite geometry)
The following calculations are given for a section of a cylindrical coaxial structure in which
two layers (the aluminium conductor and its coating) are considered to be electrically
conductive and in perfect contact (see Figure 5). As justified later, the influence of the
metallic envelope on the conductor can be neglected. The objective is to obtain a frequency
dependent equivalent electric representation of this structure. The phenomena associated with
wave propagation, reflections and transients will then be computed from the electric
equivalent diagram and presented in section 2.1.3.4.
Figure 5 - Cross section of cylindrical coaxial structure
One major principle under study is the skin effect, since it already presents some desirable
characteristics for VFTO mitigation, that is, an increase of resistance and decrease of internal
inductance when the frequency increases.
Conductor
Coating
Metallic
enclosure
SF6
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2.1.1.1 Cylindrical geometry
The electromagnetic fields are particularly interesting inside the two conductive layers. The
proximity effect of the metallic envelope on the conductor is supposed to be negligible here
considering the wide spacing between them. The physical properties and coefficients
referring to the different media are indexed 1 for the conducting bus bar and 2 for the coating.
Bessel functions are indexed 0 and 1 but this refers to their order, not the medium they are
considered in. The set of cylindrical coordinates is used.
Starting from Maxwell’s equations and sinusoidal states:
(2.1)
Displacement current can typically be neglected but can also be taken into account by writing:
(2.2)
The mathematical relationship:
(2.3)
Yields :
(2.4)
It is important to note here that all quantities are considered to be sinusoidal and that we are
working with their magnitude. Since only a section of the system is considered here (or an
infinite length), the electric field associated with the conductor current is supposed to be
longitudinal (along z-axis) and independent of the angle . This is justified as long as
where d is the conductor diameter, its length, and the wavelength (see [5]).
Figure 6 represents the current density at the edge interface. One can see that the contribution
of the radial currents is negligible when sufficiently far from the edge. For VFTO application,
the coating length shall be of 1 m or more, the conductor diameter of about 100 mm and the
wavelength of approximately
(2.5)
This approximation therefore seems valid (see also Appendix).
(2.6)
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Figure 6 - Schematic current density at the edge interface
Using the cylindrical set of coordinates , the equation becomes:
(2.7)
Or
(2.8)
The previous equation admits for general solution:
(2.9)
Where and are constants to be determined by the boundary conditions, are
Bessel’s functions of first and second type, of order 0 and 1. The magnetic field can be
supposed to be azimuthal and computed by:
(2.10)
And considering that is only along z,
(2.11)
Where Bessel’s functions property is used.
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These expressions of the electric and magnetic fields are true for both media (where k
depends on the properties of the medium), and the coefficients and can be found by
applying the appropriate boundary conditions. First, both materials are real and not ideal
conductors, which means that the tangential components of and are continuous at the
interfaces. Then, no current is supposed to flow in the hollow part of the conductor (filled
with SF6). Ampere’s theorem applied on the inner radius of the conductor yields:
(2.12)
With the inner radius of the inner conductor. The continuity of the tangential components
of and at the interface between aluminium and the coating (at ) yields:
(2.13)
Where the relation has been used again. Also, note that and designate
the (complex) magnetic permeability of, respectively, the conductor and the coating.
The fourth boundary condition concerns the current. All of the current flows through the
two layers. This current will be used as a link to the equivalent electric diagram later on
but can as well be normalized and set equal to 1.
(2.14)
(2.15)
Or
(2.16)
Bessel’s functions property has been used here.
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These four conditions can be written in the form of a linear system to find the coefficients
. Bessel’s functions diverge when becomes important though, which means
they are numerically unstable. One could take the computation from there and pursue it
analytically and write down Bessel’s approximation functions at low and high frequencies.
The approach in this paper is to reconsider the former calculation with an approximated
planar geometry.
2.1.1.2 Approximated planar geometry
A more numerically stable calculation is obtained by considering the curvature radius
sufficiently big compared to the other dimensions. The worst case for this approximation
under consideration will be the one with a 90 mm outer diameter aluminium bar and 7.5 mm
thickness as they represent the smallest conductors typically used in GIS where VFTO are of
concern. As shown below, this approximation presents little error.
The set of initial equations is unchanged:
(2.17)
is then introduced:
(2.18)
Where is the skin depth
(2.19)
So that the equation can be written:
(2.20)
(Keeping in mind that , when displacement currents are neglected)
With a planar geometry, this equation admits a simple exponential solution:
(2.21)
The term is associated to the depth in the conductor and can be replaced by in both
media (only changing the constant and ). In each layer then:
(2.22)
The coefficients can be computed by the same boundary conditions as before: no current
is flowing in the hollow part of the conductor, and are continuous at the interface, and
the total current flows through both layers.
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(2.23)
(2.24)
(2.25)
(2.26)
The four conditions expressed above can be written in the form of a linear system to get
and thus and fields in both layers.
2.1.2 Comprehension of the phenomenon
2.1.2.1 Current density
As verification, the planar geometry model (also referred to as the exponential solution model)
is compared to the more complete cylindrical geometry solution for some frequencies and
sets of parameters and for the coating. The approximation is expected to be more correct
at higher frequencies (smaller skin depth) and bigger dimensions for the bar. The worst case
to be considered for this specific application is then for low frequencies and with an outer
diameter for the aluminium bar of 90 mm and thickness 7.5 mm, which corresponds to a
small bar for 245 kV systems. With higher voltage systems, the bars have a bigger cross
section and the error is therefore smaller.
The following curves were obtained with “Mathcad Software”. They show a good coherence
of the two solutions, with a single hollow conductor as well as with a two-layer system, even
at low frequencies and with a small conductor. The cylindrical geometry solution can also be
referred to as “Bessel” solution, and the approximated planar solution by “Exponential”
solution according to the typical functions the solutions are made of. Aluminium has a
conductivity of S/m. In figures 7, 8, and 9, the current density is normalised by its
value at the inner radius of the conductor ( ) and at 1 Hz.
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Figure 7 – Current density in a hollow aluminium bus bar vs radius for both cylindrical and planar geometries
Figure 7 reveals several typical characteristics that were expected. First, the skin effect is
increased at higher frequencies, that is, the current tends to concentrate at the periphery of the
conductor when the frequency increases, because of its own induced field variations and eddy
currents. Then, one can check the good adequacy of the two solutions, even at very low
frequencies. The integral of the relative error for current density is in the order of 0.1% (local
worst case is 4%). The current density is normalised by its value at low frequencies.
To check the validity of the boundary conditions, Figure 8 shows the current density obtained
with an aluminium coating. Of course, no discontinuity is observable which tends to confirm
the validity of the calculation for a multilayer conductor. One can also notice that the planar
geometry solution (“exponential”) is more correct for a thicker conductor at the same
frequencies. To be more precise, its accuracy depends on the ratio of the thickness and skin
depth. Also, one can notice that at the same frequencies, the skin effect is more pronounced
in a thicker conductor since eddy currents are less restricted.
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Figure 8 - Current density with a 2 cm thick aluminium bar (treated as made of 2 layers of the same material)
The objective is of course to use a material with different properties from aluminium, as
represented in Figure 9. The difference of permeability, conductivity and thickness will
control the current density in both layers. These are the parameters to be optimized for an
optimal VFT mitigation system. They shall be carefully chosen so as to dissipate as much
energy as possible at high frequencies and the least possible at 50Hz. There is a compromise
to make on the resistivity of the coating for example since it controls the losses in the coating
at high frequencies but also the “penetration” of the current in the layer. The skin depth is of
particular interest as will be shown later on. In Figure 9 one can note the discontinuity of
current density while the electric field was said to be continuous. This is due to the change of
material’s conductivity.
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Figure 9 - Current density with a 2 mm thick coating ( )
2.1.2.2 Influence of the parameters
The following graphs and considerations should allow the reader to better appreciate the
parameter’s influence on the current distributions and frequency dependence of the dissipated
energy. Knowing the current distribution in the two layers, the dissipative losses can be
calculated. In particular the losses in the coating are compared to the global losses and the
ratio is represented below (which is a function of frequency, coating permeability,
conductivity and thickness, in that order).
(2.27)
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Figure 10 - Proportion of power loss in the coating vs frequency for different values of and constant ;
thickness = 2 mm
Figure 11 - Proportion of power loss in the coating vs frequency for different values of and constant ;
thickness = 2 mm
One can already see from Figure 10 and Figure 11 how the ratio depends more on than .
increasing
from1 to 10000
increasing from
10 to S/m
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Figure 12 - Proportion of power loss in the coating vs frequency for constant ( ); t = 2 mm
These are plotted for a 2 mm thick coating without frequency dependence of the magnetic
permeability. It appears that when the permeability increases, all other factors being equal,
the current tends to concentrate on the outer periphery of the coating at lower frequencies.
The same tendency is also observable for constant and increasing conductivity.
At constant thickness, the proportion of energy dissipated in the coating appears to be mainly
controlled by the product , as shown by Figure 12 in which two sets of 3 curves with
constant product are plotted. The higher the product, the easier it is for the current to
penetrate the coating.
Qualitatively, the main feature requested by a VFTO mitigation system seems to be respected.
Indeed, with the appropriate parameters, no additional resistive losses are induced by the
coating at low frequencies while they are completely dominated by it at high frequencies. It
appears then that the product should be kept in a certain limit not to induce additional
losses at 50 Hz but high enough so the current flows in the coating at high frequencies. Also,
resistive losses at high frequencies are controlled by the inverse of conductivity so should
be low in order to maximize losses at high frequency. is typically limited by the choice of
material and sensitive to frequency and saturation effects.
2.1.2.3 Magnetic considerations
In order to account for the magnetic losses induced by microscopic phenomena in the
magnetic material (domain wall displacement, hysteresis losses…), a complex permeability is
classically introduced. The real and imaginary part can be chosen in series or in parallel. Here
the series expression is used, so the magnetic losses shall be represented by a series resistance
[6] :
increasing
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(2.28)
This allows characterizing the material over a wide frequency range. At low frequency, the
permeability is real, and phasors and are parallel to each other. At high frequencies, the
permeability is complex and and phasors are not parallel anymore. The complex
permeability may be used to represent all types of magnetic core losses. The core loss
processes are conveniently modelled as the imaginary component of the complex
permeability. Both real and imaginary parts are function of frequency.
Landau and Lifshitz described the microscopic phenomena of magnetization and predicted
the existence of ferromagnetic resonance: when the frequency of the AC magnetic field
coincides with the precession frequency of electrons, the energy is transferred from the AC
magnetic field to the precessing electrons. Precession is the change of electrons orbit axis so
that their magnetic moment is aligned with the applied magnetic field. Precessing electrons
then dissipate energy in internal friction and heat. Also, a ferromagnetic material is divided in
magnetic domains, separated by boundaries, called domain walls or Bloch walls. The
movements of Bloch walls are discontinuous and the magnetization changes in many small
discontinuous jumps. The jumps are very fast and induce eddy losses referred to as excess
losses. These phenomena justify the lagging of the magnetic field density and additional
losses and they can be represented by a complex relativity as said earlier.
An approximation of the series complex relative permeability is used here:
(2.29)
Where is the low-frequency real series permeability and is the 3-dB frequency of .
Those two parameters can widely vary from one type of ferromagnetic material to another,
with values up to for and from a few kHz to several MHz. This expression
accurately represents ferrites but may not be suitable for all materials. For nanocrystalline
materials for example, the frequency dependence is less important, even though the cut-off
frequency is lower. Experimental data given by the suppliers should be used.
Figure 13 - Typical ferrite permeability dependence:
Frequency (Hz)
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A similar expression can be used for permittivity but is of limited interest here.
2.1.3 Equivalent electric representation
2.1.3.1 Parameters computation
A convenient way to represent and simulate the effect of the proposed device is to use an
equivalent electric diagram. The coaxial structure naturally leads to a distributed line
representation as in Figure 14.
Figure 14 - Equivalent circuit for an incremental length
The electric parameters can be calculated by identification of the different types of energy.
The series resistance (per length) is of main interest for the coating. It is found by
identification of the resistive losses in the electromagnetic model and the electric circuit.
(2.30)
It is important to keep in mind that all quantities are considered sinusoidal and referred to by
their amplitude. Also, since only a section was considered, the electric parameters shall be
computed per unit length.
The other types of losses than eddy losses are magnetic losses and dielectric losses. The latter
is represented by the conductance but the magnetic losses will be added up to the series
resistance (see Poynting’s theorem [7] and section 3.2.2 for expression of the magnetic
losses). Magnetic losses are considerable when the imaginary part of permeability reaches its
peak value.
The resistivity of the outer envelope due to eddy losses can be added as well, even though it
will be negligible at higher frequencies. It represents the losses induced in the envelope by
the return current in a GIS. The proximity effect of it on the main conductor was neglected,
as confirmed by finite elements modelling (see
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Appendix: FEM validation of the principles). A good approximation at high frequencies is to
introduce a skin resistance.
(2.31)
With the inner radius of the envelope, and
(2.32)
The global resistivity is then computed by the following expression:
(2.33)
The first term represents the eddy losses in the aluminium conductor and its coating and the
second term represents the magnetic losses in the coating. As justified later, the series
resistance is the main attenuation factor and will be given a particular attention.
The inductance can be computed by identification of the time-average magnetic energy:
(2.34)
Where, again, quantities are supposed sinusoidal and represented by their amplitude.
One can distinguish two terms for the inductance. The internal impedance corresponds to the
magnetic energy stored inside the conducting layers. It is function of frequency. The external
impedance corresponds to the magnetic energy stored in the gas between the conductor and
the outer metallic envelope and is only function of the geometry. The external inductance is
the predominant one in typical systems and is simply computed from the expression of the
field in the gas:
(2.35)
Which then gives
(2.36)
is the real part of the gas, in practice equal to the permeability of vacuum .
The internal impedance adds in series to the external impedance and is a decreasing function
of frequency. It is also directly function of the permeability of the material.
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(2.37)
(2.38)
Finally, the contact between the conductor and the coating is supposed ideal, which means no
admittance is introduced in between. This might need further studies but the main influence
of an imperfect contact would be on the dielectric characteristics which mainly induce
parasitic losses and deterioration of the material. For the application under consideration,
these aspects are neglected for now. That being said, the admittance of the conductor and
coating would be much bigger than that of the insulating gas, which means the global
admittance of the system can be computed by that of the SF6. It is therefore really close from
the one computed without any coating. The SF6 gas is considered to have very good
dielectric properties, which means the conductance G is neglected.
(2.39)
The capacitance per length is determined by using geometrical considerations for a coaxial
arrangement:
(2.40)
Note: several other methods can be used to arrive to an equivalent electric representation, cf
[8] , [9] or [10]
2.1.3.2 Validation: basic formulae
Since the resistance is the main aspect, its accuracy was verified by comparison to well-
known simple formulae and finite element modelling.
First the computed resistivity of the bar with the previous model is compared to well-known
skin effect expressions for a single round conductor. Two simple expressions for it are the
introduction of skin resistance (which is only true at high frequencies, [6]) and Levasseur
formula, which is a mathematical approximation of Kelvin’s complete formula [11]. These
can be expressed by
(2.41)
With
(2.42)
Levasseur approximation formula is:
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(2.43)
Where S is the surface of a section of the conductor, p its outer perimeter, and the skin
depth
(2.44)
Figure 15 shows the resistance of a single aluminium hollow bar computed with the
previously presented models (cylindrical and planar geometries) and with the skin resistance
and Levasseur formulae. One can see the very good cohesion of the cylindrical and planar
geometry solutions. Also, the divergence of Bessel’s functions lead to absurd values and
impossibility to calculate at higher frequencies.
Figure 15 - Resistance of a single hollow cylindrical conductor with different models
2.1.3.3 Validation: FEM with COMSOL
Finally, the equivalent global resistivity for a multilayer coaxial structure is computed by
numerical simulation (with a 2D COMSOL Multiphysics model). The mesh quality is of
crucial importance at high frequencies, because if the skin depth is smaller than the typical
meshing dimension, the obtained results are wrong, as can be observed on Figure 16. The
angular division should be fine as well as the radial meshing (‘better triangle” essentially
refers to a finer triangular basis).
Frequency (Hz)
29
Frequency (Hz)
Figure 16 – Resistance per unit length of a hollow aluminum busbar vs frequency: analytical and FEM
By setting correctly COMSOL’s parameters, one can find a perfect adequacy between the
theoretical expected results (Rexp on the graph) and simulation results, even with a
multilayer system. Note that the outer metallic envelope is added in the simulation but does
not change the current repartition in the bar and coating system. The resistivity is computed
in both cases by identification of the resistive losses in the two conducting layers.
Note: the mesh consists of a triangular base and additional boundary layers at the interfaces.
Even with a significant amount of boundary layers, if the basic triangular mesh is too coarse
the results will not be correct due to the poor angular division of the circular geometry.
30
Frequency (Hz)
Figure 17 – Resistance per length of an aluminum bar with a 1 mm thick semi-conductive coating ( ) as a
functoin of frequency
Similar adequacy was found for the magnetic energy and computation of the inductance.
2.1.3.4 Wave propagation and attenuation
Figure 18 - Electric circuit for an incremental length
The equations governing the distribution of transverse voltages and currents along the line are
the telegrapher’s equations. The same circuit is considered for the previously described
structure. The parameters are expected to be frequency dependent though.
(2.45)
31
For cosine harmonic time dependence and infinitesimal length it becomes:
(2.46)
This can be written as:
(2.47)
Where
(2.48)
The electrical quantities are then:
(2.49)
By injecting these expressions in the first differential equations, there comes:
(2.50)
Which gives:
(2.51)
The characteristic impedance is defined from here by:
(2.52)
So one can write the following relations:
(2.53)
The attenuation of the wave in the conductor is studied here. Only the wave propagating in
the z-positive direction is considered. It is attenuated by the real part of according to the
following expression:
(2.54)
32
Since conductance G is neglected, the real part of is
.
Figure 19 - Attenuation constant of a typical busbar
The attenuation is therefore an increasing function of the resistance and a decreasing
function of . This justifies the approach of maximizing the resistance. The inductance shall
be controlled too but is of secondary priority.
2.1.4 Theoretical optimisation
As explained above, the main factor of attenuation is the resistance. Its dependence on the
conductivity, permeability and thickness of the coating are under study now. The final
objective is of course to find the optimal existing material, with properties as close as
possible from the theoretical optimum. Most magnetic materials are good electric conductors
though.
An equivalent resistance per unit length is computed by identification of the resistive and
magnetic losses,
(2.55)
Some aspects have already been seen and are studied more in depth here. First, all other
factors being equal, the global resistance per unit length presents an optimum with the
coating conductivity (Figure 20). This can be seen as a compromise between current
penetration in the coating which is facilitated by a higher conductivity, and the inverse
dependence of skin resistivity with conductivity when the skin effect is predominant in the
coating.
Frequency (Hz)
33
Figure 20 - Global resistance at 1 MHz, vs coating conductivity for several values of permeability
There is no such direct optimum with permeability. Indeed, permeability only enhances the
skin effect, which increases the global resistance (the coating conductivity is always
supposed lower than aluminium). A high permeability material would induce a high
inductance at low frequencies though, the effect of which should be watched. The observed
knees on Figure 21 traduce the transition between different regimes: one where the coating
contribution is negligible because eddy currents in it are not important enough compared to
the ones in the bar; one transition region for which the coating and aluminium bar have
comparable contributions to the losses, and one region for which the dominating component
is the skin resistance of the coating. The permeability has to be controlled though in order not
to increase the resistivity at 50 Hz.
Figure 21 – Resistance (per unit length)at 1 MHz vs for different conductivities
Finally, the thickness is also of major importance. Actually, the three regimes described
above can be linked to the comparison of the skin depth and thickness. As long as the
thickness of the coating is smaller than the skin depth , eddy currents are limited by this
thickness and the losses in the coating are small. For of the same order of magnitude as ,
eddy currents appear in the coating and the global resistivity increases until eddy currents are
freely induced in the layer. Then, increasing the thickness further only decreases slowly the
resistivity according to the expression of the skin resistance of the coating which is by then
the main component:
34
(2.56)
With
(2.57)
And is the outer radius of the coating.
Figure 22 - Resistivity at 1 MHz vs coating thickness for different values of and constant
As an example, the global resistivity with a 1mm thick coating of semi-conductive material
( ) is decomposed in Figure 23.
Figure 23 - Example of resistance per unit length of a coated conductor vs frequency
Four different regions are observable on Figure 23. Region 1 corresponds to the DC
behaviour of the aluminium bar: at low frequencies, its thickness is smaller than its skin depth
1 2
3
4
Frequency (Hz)
35
so the resistivity is controlled by its geometry and is approximately equal to its DC resistance.
Then, skin effect appears in the aluminium conductor and the resistivity increases with :
(region 2). The behaviour of the system there is little or not influenced by the
coating. Then, the current is flowing through both the conductor and the coating and the
resistivity is in a “transition region” (region 3). Finally, all current flows in the coating and
the global resistivity is approximately equal to its skin resistance (region 4). In a log-log scale,
the slope appears to be the same as in region 1 but shifted by a factor depending on the
coating parameters of course. The frequencies corresponding to the different knees can
overlap and mask the existence of a region. This is actually a desirable feature to increase
more rapidly the resistivity with frequency (rapid transition from 1 to 4).
Optimisation concerns all three parameters with different but equally important contributions.
The resistivity is optimized at 1MHz because the energy of the VFT is concentrated around
and above these frequencies as will be shown later. The main constraint is not to change the
50 Hz behaviour of the system, in order not to alter the very function of a GIS. The limit
chosen for variation of the 50 Hz resistance is 5%. This can also be adjusted depending on the
length of coating. Thermal conductivity and restrictions should also be taken into
consideration for an actual development of the solution.
The optimization is computed numerically and is summed up as follows:
Maximisation of the 1MHz resistivity within the following constraints:
(2.58)
(2.59)
The limits set on the coating thickness are actually depending on the coating technology
chosen, thermal conductivity and other parameters. The global theoretical optimum is studied
as well as the optimal thickness for a few materials. Some typical material’s properties are
reviewed too. (Within one category of a material, a wide range of values of conductivity and
permeability may exist. The following values should not be taken as references but as an
illustrative example). The reference bus bar has an outer diameter of 90 mm and 7.5 mm
thickness.
36
Material Conductivity
(S/m) Relative
Permeability
r
Optimal
coating
thickness
(mm)
Resistivity at
100kHz
Ratio at
100kHz Resistivity at
Ratio at
1MHz
No coating
1,7E-04 1 5,4E-04 1
Iron 1,00E+07 5,00E+03 3,5E-03 0,004 23 0,08 150
Permalloy 5,00E+06 1,50E+05 1E-03 0,037 217 0,62 1150
Nickel 1,40E+07 6,00E+02 0,026 0,007 43 0,022 40
Pure Iron 1,00E+07 2,00E+05 1,5E-03 0,156 915 0,466 870
Nanocrystall
ine 8,70E+05 2,00E+05 1,9E-03 0,08 470 1,721 3200
Iron powder 1,00E+03 5,00E+02 1,1E-03 0.239 659 5.3 4660
Theoretical
optima
5%
constraint 1,3E-03 3,7E+04 10 0,65 3800 65 120000
Limited
thickness 0,084 1,6E+06 Limited :
0,2 mm 0,65 3800 65 120000
3%
constraint 1,4E-03 5,0E+04 6,3 0,35 2060 35 65000 Table 1 - Optimal thickness and performance for a few materials
The optimal material would appear to be very magnetic and very resistive. Unfortunately, this
is not so common. Besides, other very important features neglected so far are frequency
dependence of the permeability and saturation. They play a very important role in the
choosing of the material and supplier. In particular, the saturation induction is very
important for two reasons: first, to avoid a important dropping of the equivalent relative
permeability and second because the cut-off frequency of the permeability (defined in section
2.1.2.3), for a given material, is linked to by Snoek’s criteria: [6]. A material
with much lower permeability than the theoretical optimum will be chosen, and with a as
high as possible, keeping in mind that the electric resistivity should be high enough. The
following figures show the importance of the cut-off frequency on the global resistivity for an
iron powder material. The resistivity of a single aluminium bar is plotted for reference as well.
37
Figure 24 - Resistivity with a coating material having
Figure 25 - Resistivity with the same coating properties but , lower than 0.1 /m at 10 MHz
The magnetic material should be chosen with a permeability cut-off frequency as high as
possible, ideally equal to a few MHz for the total resistance per unit length reaches a limit
after that.
2.2 Mitigation of the VFT
From the electric circuit established above, it is possible to study the influence of the coating,
and more specifically its attenuation, on the form and magnitude of a VFTO in a GIS. This
study is conducted with ATP software (“Alternative Transients Program”), which allows
modelling the whole system and its transient behaviour. The following diagram for
simulation was used. This would have to be adapted depending on the exact geometry of the
system but already represents the main elements present in a GIS. The reader should keep in
mind that a disconnector switch is operated when the breaker is open, which is represented by
an important capacitance at the end of the circuit.
Frequency (Hz)
Frequency (Hz)
38
Figure 26 - ATP diagram for VFTO simulation
2.2.1 Arc resistance
The arc is first represented by a voltage controlled switch. A breakdown can indeed be seen
as a very fast equalization of the two electrodes potentials, if the voltage is higher than the
breakdown voltage.
This is not accurate enough though. Tests were run on an experimental device corresponding
to the previous diagram. Modeling the arc by an ideal switch induces very high frequency
parasites that do not exist in practice. Also the rise time of the surge voltage induced by a
restrike is known to be in the order of 3 to 20 ns. Numerous studies have been conducted to
accurately model an electric arc and its resistance. They are not presented in depth here but
one model of arc resistance has been selected whose parameters allow a more realistic
representation of the voltage drop, even though it is still very conservative. It is implemented
in ATP as a time variable resistance.
Before the appearance of an arc, the switch can be modelled by a very high resistance and
capacitance in series. When the arc is created, an arc resistance is introduced (time origin
defined as the moment when , 20 ns in Figure 28)
(2.60)
The parameters are chosen to fit the experiment and in a range of possible values found in the
literature. Here, .
39
Figure 27 - Initial arc resistance
The equalization of the electrodes potentials then has the expected shape: the rise time is of a
few ns and, since the immediately adjacent pieces to the disconnector are of the same
geometry, their impedances are equal, which sets the surge amplitude to half the voltage
difference (without any trapped charges).
Figure 28 - Electrodes potential equalization
2.2.2 Damping system model
The electric model of the coated system can now be implemented in the GIS structure. The
electromagnetic considerations led to a frequency dependent distributed line model.
2.2.2.1 From the frequency domain to the time domain
The electric elements in the software should be time dependent since the aim of the
simulation is to represent the transient regime. Several mathematical tools exist to assure the
transition from one domain to another such as Laplace transform or Fourier transform. The
model is not easily described by a simple transfer function since reflexions and other physical
considerations are to be represented. The following approach is then considered:
40
First, the frequency dependence of the inductance is neglected. It is in practice, quickly equal
to its “external” geometrical component ( in 2.1.3.1). Then, when the conductance is
neglected, a resistive line has been proved to be accurately modelled by one or several
sections of lossless lines combined with lumped resistances. This approximation is
particularly correct when the line is smaller than a few hundred meters which is the case here
since the typical length of the coating shall be in the order of one meter. Figure 29 represents
the chosen architecture (cf [12]).
Figure 29 - Representation of a lossy line with lossless sections and lumped resistances
Now for the frequency dependence of the resistance, the attempted solution was to compute
the instantaneous frequency decomposition of the incident voltage (or current) signal and to
pilot the value of the resistance from the curve obtained besides. When decomposing
the current into harmonics for example, one can compute the global losses by:
(2.61)
Where is the resistivity calculated at the harmonic.
The discrete Fourier Transform (or Short Time Fourier Transform) of a discrete signal is:
(2.62)
N is the number of points considered for the calculation (window size), n is the index of the
last sample available. The frequency spectrum precision is controlled by the sample
frequency ( ) of the signal and the number of points. The index in the expression refers to
the frequency component:
. The
term is the frequency precision of the
spectrum. A precision of at least is desired.
In order to efficiently calculate Short Time Fourier Transform at every time step, a recursive
expression is used:
(2.63)
For the N first points, the expression is used by considering the terms equal to zero.
This is consistent with the “zero-padding” technique on the first terms, which is known to
increase the frequency spectrum quality.
41
Note that this is the decomposition of a signal sampled on N points, with a window centred
on the time at which it is computed. The underlying approximation is that the instantaneous
frequency decomposition of a signal at time n can be approached by that at instant
.
Unfortunately, this calculation seems to need a certain amount of “initialization points” to be
valid. The instantaneous spectrum is usually used a posteriori on a recorded signal, which
doesn’t pose these problems. The results of the computed instantaneous main frequency
component are illustrated below.
Figure 30 - Voltage signal of a VFT and main its main computed frequency component
For this example, and N=2500. One can check that the obtained results are
coherent but delayed by 5 .The observable delay is incompressible though because the
same parameters directly control the frequency precision:
. This track was then
aborted because of its intrinsic delay. The instantaneous frequency is not correctly
represented. A deeper study of signal analysis might solve the problem though, by some kind
of exponential window for example.
2.2.2.2 EMTP cable model
Some frequency dependent cable models are already implemented in ATP/EMTP. The main
advantage is that they offer a turnkey solution for transient simulation while the underlying
theoretical calculations are conducted in the frequency domain. In particular, JMarti’s model
[12] comes in very handy for this application. The system’s parameters are calculated in the
frequency domain and, by integration of Bergeron’s approach and approximation of these
parameters by products of first order transfer functions, a time response is obtained. For more
details, see EMTP Theory Book, section 4.2.2.6 [12].
42
Figure 31 - ATP window for cable settings
With this model, the coating is modelled by the sheath and the metallic envelope by the
armour, which is grounded. Within the model it is not possible to implement the same outer
radius of the core and inner radius of the sheath. An insulating layer has to be present, even if
very thin but, by connecting the sheath and the core at both ends of the cable, the coated
conductor should be represented correctly. Actually this is used to model a coated cable
transient behaviour [8].
The previous work to find the optimal coating is still crucial though. Indeed, there is no
optimization module available that allows varying the parameters (such as coating
conductivity or thickness) in an ATPDraw module, the results of which are used again after
in the EMTP logic. Optimization tools do exist in ATPDraw but they can only be used to
vary direct electric parameters of ATPDraw models. For the following simulation, a 2 m long
bar was considered right downstream of the disconnector (see Figure 26). The VFT is
measured at both bar ends and is plotted below, with and without damping system.
43
Figure 32 - VFTO at both ends of the bar without coating
Figure 33 - VFTO at both ends of the bar with a magnetic semi conductive coating (parameters shown in Figure 31)
The breakdown here happens at the peak of the 50 Hz-100 kVp curve. The coating is made of
theoretical magnetic semi-conductive material with and
thickness. No saturation effect or frequency dependence of the permeability is
considered in this model.
It appears that the VFTO peak value can efficiently be damped by an appropriate coating.
The shape of the wave is also modified, and in particular at the right end of the bar (v1). The
transmitted surge voltage has a smoother shape, which is a desirable feature for protection of
the equipment. Indeed, some studies have shown that breakdown aren’t necessarily caused by
the VFTO magnitude but by the high voltage stress oscillations [1].
2.2.3 Distributed line model – constant parameters
It was possible to check a certain amount of results using ATP coaxial models. One
particularly interesting feature is the use of a simple distributed line model in which the
44
parameters are constant. By decomposing the VFTO signal into its Fourier components and
taking the frequency transmitting the highest energy, it was also possible to represent the
influence of the coating on the VFTO by a distributed line model with constant parameters
analytically calculated at that frequency (around 8 MHz).
Figure 34 - VFTO simulation circuit
First, the simulation is run without any mitigation system. Its Fourier decomposition is then
analyzed. The observation time is chosen between 1 and 2 µs and a Hanning window is used.
The results are presented in Figure 35. X0, X1, X2 are the STFT of the voltages V0,V1 and
V2.
Figure 35 - Fourier decomposition of the VFTO without any mitigation system
The 1-2 MHz components correspond to the LC resonance of the circuit. Higher frequencies
are to be associated with the reflections and refractions of the wave.
Above a few MHz, all parameters other than the resistance per unit length of the system can
be considered to be constant. The system can then be correctly represented by a distributed
line model whose parameters are computed at the main frequency, here 8 MHz.
Frequency (MHz)
45
Figure 36 - VFTO with a 1.5 mm thick coating, µ=300, on 2 1m, with JMarti’s frequency
dependent model
Figure 37 - VFTO with a 1.5 mm thick coating, µ=300, ρ = 0.001 Ω.m on 2×1m with a constant distributed line model
calculated at 8MHz
The difference between the two models (Figure 36 and Figure 37) resides in small very high
frequency oscillations that should indeed be attenuated by a higher resistance at higher
frequencies. The peak value and general shape are well represented though. Besides, ATP
cannot represent magnetic losses with such a model. This approach is then interesting to
simulate the effect of the coating when magnetic losses are an important part of the
attenuation. One should make sure that the system without any mitigation system is well
represented first, and extract its main frequency component.
2.2.4 Influence of the electric parameters
As discussed earlier, one important factor under consideration is the attenuation constant
which directly depends on . This affirmation didn’t take
into account any reflection / refraction aspects or the influence of the rest of the circuit. In
46
order to check quickly and efficiently that the resistance is indeed the main factor to
concentrate on, the damping system was simply modelled by a 1 m long PI-model. The
capacitance was fixed to (approximate value for a 245kV GIS) and the influence of
and was studied. The parameters were varied by a program which launches as many
simulations as the number of parameters combinations, and extracts the results. The program
was internally developed for this application. The peak value of the VFTO at point v1 (see
Figure 26) is represented for the different values of and .
Figure 38 - Influence of R and L on the peak of the VFTO
As expected, the higher the resistance, the better the attenuation. It was also previously stated
that the attenuation is a decreasing function of , which is globally confirmed here.
Considering the range of variations of and in Figure 38, the resistivity will still be
considered as the main factor. Indeed, the inductance should not be much higher than a few
(scale in mH on the graph).
The rest of the circuit is also influent. Indeed, one can see the system as a filter, the
capacitance being the combination of the stray capacitances of the bars and that of the open
end, and and the resistance and inductance of the lines and damping system. The system
would then more or less react as a low-pass second order filter to a step voltage. This is the
main reason why the VFTO doesn’t appear to be damped at all for high values of inductance:
the LC resonance of the circuit becomes preponderant, as shown in Figure 39.
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
0,00E+00
2,00E+04
4,00E+04
6,00E+04
8,00E+04
1,00E+05
1,20E+05
1,40E+05
1,60E+05
1,80E+05
2,00E+05
1.00 2.51
6.31 15.85
39.81 100.00
200.00
L (mH)
maxV1(V)
R(Ohm)
47
Figure 39 - VFTO for high inductance. R and L placed right after the arc
From a filter point of view, the damping factor
should be increased.
Increasing the capacitance might be risky for the dielectric strength of the system; decreasing
the inductance doesn’t appear easy, so the effort will be concentrated on the resistance.
The inductance can be a good way to protect a specific device though. It doesn’t dissipate
energy but reflects part of the incoming surge and the transmitted one is less steep. An
incoming wave facing an impedance change is indeed partly reflected and partly transferred,
according to the following coefficients:
(2.64)
(2.65)
A simple system of an inductance between two different lines ( ) is considered. The
propagation time of the inductance can be neglected. The surge voltage is created at the
switch: V0. V1 and V2 are the voltages at the left and right end of the inductance.
Figure 40 - Refraction on an inductance, circuit
(2.66)
48
(2.67)
Figure 41 - Reflection and refraction of a surge voltage on an inductance
The amplitudes of the reflected and refracted waves are controlled by their difference of
impedance (here ) and the rise time by
.
2.3 Choice of material
The confrontation of ideal expectations and actually available materials was rough. As it
happens, good magnetic materials are in general good electrical conductors too. Plus, the
frequency dependence of permeability and saturation aspects were only slightly discussed so
far but are really important in practice. Finally, not all ferromagnetic solutions are applicable
as a coating. A few typical characteristics of different ferromagnetic materials are reviewed
now. This section might not be extensive but classical aspects of the most common materials
are treated. Discussions with providers were also conducted during this thesis to find the most
appropriate solution that could be used as a coating.
2.3.1 General considerations
The most critical aspect may be the frequency dependence of permeability (see Figure 13,
Figure 25). The frequency dependence of the permeability for a material shall be given by the
supplier. Also, one can consider a saturated material by an equivalent permeability. An
approximated expression of the magnetic flux density with magnetic field is proposed (it
doesn’t represent hysteresis effects because only soft ferromagnetic materials are considered
for this application).
49
r
(2.68)
is the saturation induction, the low frequency relative permeability of the material, and
the void permeability.
Figure 42 – Example of magnetic flux density B as a function of magnetic field H
When considering an equivalent relative permeability so that , the following
expression can be used:
r
(2.69)
For high field densities, it is more important to use a material with a high saturation induction
than a high initial permeability, as illustrated in Figure 43.
50
Figure 43 - Relative permeability as a function of H for different materials
The three materials considered for Figure 43 are representative of a typical ferrite (
), an example of nanocrystalline core ( ), and a good ferromagnetic powder
( =300). One can see the importance of the saturation induction level. Besides, the cut-off
frequency is usually linked to the saturation induction level, by Snoek’s relation. For high
frequency applications, it might be more interesting to use a material with a lower initial
permeability and higher saturation induction to get a higher cut-off frequency.
2.3.2 Ferromagnetic materials: typical properties
For more information about ferromagnetic materials, one can refer to the following
publications: [6], [13], [14]. Some properties are discussed here.
2.3.2.1 Iron
Iron cores consist of alloys of iron (Fe), and small amounts of nickel (Ni), cobalt (Co) and
chrome (Cr). They have high relative permeability and saturation induction and are
good electric conductors.
2.3.2.2 Ferrosilicon
Silicon steels are made by introduction of a small percentage of silicone (Si). Depending on
the proportion of silicon added, the properties of the alloy can be different but mainly, they
have a resistivity from to , relative permeability from 2500 to 5000 and
saturation induction from 0.5 to 2 T. They have lower losses at low frequencies than iron.
51
2.3.2.3 Amorphous Alloys
They are characterized by a completely random microscopic structure that resembles glass.
They are sometimes called metallic glasses. They are made of various percentages of iron,
cobalt, nickel, and transition metals, such as boron, silicon, niobium or manganese. They can
have very high relative permeability, up to , high saturation induction (0.7 to 1.8 T)
and are relatively conductor ( from to ).
2.3.2.4 Nickel-Iron and Cobalt-Iron
Typically, nickel is used in an alloy to increase its relative permeability while cobalt
increases its saturation induction. NiFe alloys typically have high permeabillities (
), and saturation induction from 0.8 to 1.5 T. They are also good electrical conductors
( from to ). Much higher values of permeability can be reached
through heat treatment for example like for Permalloy or Mumetal. CoFe alloys have very
high saturation magnetic flux density (2.4 T) and typical permeability and
resistivity .
2.3.2.5 Ferrite
Ferrites are hard and brittle polycrystalline ceramics made of iron oxides mainly. Other
oxides are introduced such as manganese, zinc, nickel, or zinc. The most common
compositions are NiZnFe2O4 and MnZnFe2O4. They can have high relative permeability (40
to ), a very large range of high resistivity (1 to ). The typical saturation
induction though is . Their other properties make them good cores for low voltage
applications though. They are not very appropriate high voltage applications because of their
low saturation induction.
2.3.2.6 Powder coating
These are made from powdered iron alloys by grinding the material into small grains. These
particles are then coated with an inert insulating layer, typically by chemical treatment. This
corresponds to adding a distributed air gap to the material, which increases its saturation
induction and decreases its relative permeability. The grain size is from 5 to and the
insulating layer thickness 0.1 to 0.3 . Eddy currents are then restrained by the grain size
but contact points typically exist in the component, as illustrated in Figure 44. Typical values
of are from 3 to 550, ranges from 0.3 to 1.6 T. An overall resistivity is measured as
well, from to . Powder bulks are usually manufactured by very strong
compression (800 MPa) and heat treatment to assure that the particles be isolated from one
another. The final properties can be very different depending on the level of compression heat
treatment or the type of insulating matrix. Well known powders are Sendust, KoolM , or
Molybdenum permalloy powder (MPP). MPP offer high operating frequencies (1MHz).
52
Figure 44 - Particle and Bulk Eddy currents in a powder core;
2.3.2.7 Nanocrystalline material
Nanocrystalline cores are usually tape-wound, like amorphous alloy cores and correspond to
a particular treatment of metallic glasses. They are characterized by a crystalline structure
with typical crystal size of 7 to 20 nm. The strip thickness is usually around . They
can have very high permeability (5 000 to 150 000) and high saturation induction (1.2-1.5
T).Their resistivity is . They have a very soft magnetic behavior (low
hysteresis losses) and are progressively replacing amorphous alloy tape wound cores because
they can have equally good magnetic properties but lower magnetostriction and they age
better. When aged under saturation (submitted for a long time to high magnitude magnetic
field) and a relatively high temperature, amorphous alloy materials see their permeability
increase much more than for nanocrystalline cores. When aged at remanence (no externally
applied field), amorphous alloys present a more rapid decrease of permeability than
nanocrystalline. The difference of behaviour is even more visible at higher temperatures. For
these reasons, nanocrystalline cores tend to replace amorphous cores in most applications.
Note: Magnetostriction is a property that ferromagnetic materials have to change their shape
and dimensions during the magnetization process. It actually corresponds to the rotation of
magnetic domains and the domain wall displacement, which both cause a change in the
material’s dimensions. The magnetostrictive strain is usually described by its saturation value
: the dimensionless fractional change in length as magnetization increases from zero to
saturation value. This effect causes losses due to frictional heating. It is also responsible for
the typical noise transformers make.
53
Figure 45 - Relative change of permeability vs the time of thermal aging [15]
Table 2 shows the different typical frequency applications of the ferromagnetic materials
when used as cores [6].
Material Frequency range
Iron alloys 50 - 3000 Hz
FeNi alloys 50 - 20 000 Hz
FeCo alloys 1 – 100 kHz
Nanocrystalline 0.4 – 150 kHz
Amorphous alloys 0.4 – 250 kHz
MnZn ferrites 10 – 2 000 kHz
Iron powders 0.1 – 100 MHz
NiZn ferrites 0.2 – 100 MHz
Table 2- Frequency range of ferromagnetic materials
54
55
3 Mitigation of VFTO using magnetic
rings
Another VFTO mitigation method under study is the use of magnetic rings, disposed around
the main conductor. Experimental results of such a system have already been presented in [4].
It has been shown that, by disposing nanocrystalline rings around the bar, right below the
disconnector switch, it is possible to efficiently damp the VFTO, particularly at the open end.
The major results to keep in mind are that using more rings with a higher permeability
mitigates the VFTO more efficiently.
Figure 46 - Magnetic ring mitigation system
Figure 47 - VFTO measured at the open end, with 8 and 3 rings of permeability 45000 and comparison of the effect of
8 rings of permeability 45000 and 8000
In the following sections, a model of the effect of magnetic rings is proposed.
3.1 Total core loss – state of the art
Once again, the objective is to dissipate the energy associated with the VFTO by dissipative
losses. This is actually one of the differences between the typical use of ferrites for low
voltage systems and nanocrystallines for VFTO. For ferrites, the cut-off frequency of
permeability is rather high (a few MHz typically), which means permeability is mainly real
and the impedance presented by a ferrite core is mainly inductive. For nanocrystalline rings,
56
the cut-off frequency is much lower (10-100 kHz) but the slope is less important, which
means that at high frequencies, the imaginary part of permeability is still important, and is
associated to losses.
Figure 48 - Differences in the variations of μ’ and μ’’ for nanocrystalline and ferrite lead to different attenuation
mechanisms
Special attention shall therefore be given to losses in the magnetic core. Magnetism
phenomena are particularly complex and hard to predict. Some important work has been
conducted to describe the microscopic physics of magnetization [16] and can be used to
predict losses by coupling Maxwell and Landau-Lifshitz-Gilbert for example [17]. Such
models are not within the scope of this thesis.
A widely spread method is Steinmetz empirical equation for total core loss. It is based on a
physical understanding of magnetization but uses experimental data to fit certain parameters.
Indeed, the losses can be divided into three components: hysteresis losses, classical eddy
current losses, and excess losses. Excess losses are the least well understood today but can be
linked to the eddy currents induced by the motion of domain walls. The core loss per unit
volume expressed for sinusoidal induction of amplitude is then expressed by:
(3.1)
It can also be expressed by manufacturers in a more compact form:
(3.2)
Where is somewhat larger than 1 and somewhat larger than 2. In general, the coefficients
can be given by the manufacturer or found using experimental data and the method described
in [18] for example. Steinmetz empirical formula can also be adapted to describe non-
sinusoidal induction waveform [19]. It has been shown though that, for nanocrystalline cores
57
for example, eddy current loss has the biggest contribution to the total losses for frequencies
above a few kHz [20]. Magnetic losses in the core shall still be accounted for by a complex
permeability.
3.2 Classical eddy losses
Eddy currents in a laminated core are now considered. The following calculations are
presented with a real permeability for simplification. They were actually conducted using a
complex permeability but it was verified that the resistive losses due to eddy currents can
accurately be represented by a real permeability. In the following calculations, the term
should be interpreted as the module of the complex permeability (frequency dependent). For
calculations of the magnetic energy though, it is important to introduce a complex
permeability. The real part will correspond to the inductance while the imaginary are
additional losses.
3.2.1 Eddy currents in a single lamination
3.2.1.1 Fields in a lamination
Figure 49 - Cross section of a single lamination
Since the thickness of the laminations is typically very small ( ), the curvature of
the core is not considered and the calculations are made in the Cartesian coordinate system
. To get back to the cylindrical geometry, one should assimilate the axis to the φ
dy
h
lc
w
y
x
dx
y
x
Je
H
z
58
axis, to and to . Another approximation is that, at the scale of one lamination, the
external field is homogenous, that is . Indeed, is at least the radius of the
bar, , and is the thickness of the lamination, . The field at the interface
between the lamination and surrounding medium (insulating layer for nanocrystalline cores)
is continuous and equal to
where is the current flowing in the bar and the
distance between the center of the bar and the middle of the lamination, set as the origin for
the calculations. The difference of magnetic field between the two plates of the lamination is
neglected. For the total core loss though, it will be supposed that the field is not the same
from one lamination to another (section 3.2.2).
The last approximation is that, considering the shape of the lamination ( ), eddy
currents are well described by their component. Also, is along the z axis. That being said,
Maxwell’s equations can be used as in section 2.1.1.2 but this time using the magnetic field,
to get Helmholtz equation:
(3.3)
It is supposed once again that all quantities are sinusoidal and described by their amplitude.
Displacement currents are also neglected and is considered a constant of time and space
(though it can depend on frequency and current density with local saturation).
The following quantities are introduced:
(3.4)
Where is the skin depth of the core, and its permeability and conductivity. The general
solution is:
(3.5)
According to the previously stated hypothesis, and by placing the origin at the middle of the
lamination, and can be calculated:
(3.6)
So
(3.7)
Using the continuity of the magnetic field:
(3.8)
For typical inductors, where the coil is wound around the core, where is the
current flowing in the winding. In that case:
59
(3.9)
r being the distance between the centre of the bar and the middle of the lamination.
By substitution, the magnetic field in the lamination is described by the following expression:
(3.10)
Then, considering the directions of the current and magnetic field, gives:
(3.11)
For a complex permeability, one should use the expression
where
. Still, it is interesting to conduct the calculation with a real permeability since
the error introduced is small.
The magnitude of the current density is
(3.12)
Figure 50 - Magnetic field and current density in a lamination for
Using Ohm’s law , we obtain the time-average eddy-current power loss density
60
(3.13)
Figure 51 - Power loss density in a lamination for δ/w = 1; 1/2 and 1/4
It is clear that when the frequency increases, the skin depth decreases and the losses increase.
3.2.1.2 Approximated formulae and optimum
The time-average power loss dissipated in one single lamination is
(3.14)
At low frequencies or for very thin laminations, , Taylor series approximation give
(3.15)
Therefore, the time-average eddy-current loss dissipated in the lamination is
(3.16)
is the core resistivity and its volume. From the same general expression, losses
at very high frequencies, or thick plate ( ) can be approximated by
61
(3.17)
The time-average eddy-current losses in that case are approximately
(3.18)
And
(3.19)
The expression of eddy current losses in laminations at low frequency can easily be found in
the literature and is widely used for transformers. The calculation usually starts by
considering that the magnetic field is uniform in the material. One should verify that the
condition is fulfilled though.
From the general expression
(3.20)
One can derive the optimal thickness for a given conductor at a given frequency, by
calculating
(3.21)
(3.22)
For odd values of , reaches a local maximum and for even values of , reaches a local
minimum (by study of the second derivative). The largest maximum is for , that is
(3.23)
And by substitution
(3.24)
The local minimum is for and is
62
(3.25)
For the application envisaged here, the thickness should be so that the losses be maximal at
the main frequency component (~10 MHz), which means a little higher than at this
frequency. Indeed, the decrease for is much slower than the decrease for , as
shown in Figure 52. For nanocrystalline rings with an initial permeability of 100 000 and a
permeability at 10 MHz equal to 2000, .
Figure 52 - Eddy losses in a lamination as a fuction of w/
3.2.2 Impedance of nanocrystalline rings
When considering classical inductors, with a winding around the core, an efficient way to
compute its impedance is by using the linkage flux. The impedance of an inductor would then
be calculated by
(3.26)
Where
(3.27)
is the number of turns, the number of laminations, the flux through one lamination,
the cross section of one lamination. Using the expression of derived in (3.10) with
, it comes
63
(3.28)
With
(3.29)
In our case, with a core around the bus bar, there is no “turn” as in a classical inductor and the
linkage flux is not easily calculated. The resistance and inductance presented by one
lamination can be calculated by identification of resistive losses and magnetic energy as in
part 2.1.3.1. For a single lamination
(3.30)
The losses in the magnetic core can therefore be represented in the global circuit by an
additional series resistance
(3.31)
Now is actually equal to ( the radius) for the considered geometry, which gives
(3.32)
The time-average magnetic energy stored in one lamination should also be calculated to get
the inductance. is the amplitude of the magnetic field created by a current of magnitude .
(3.33)
Here, it is important to introduce a complex permeability to distinguish inductance from
magnetic losses. The following calculations are still expressed for a real permeability but a
complex one was implemented for actual quantitative calculations. Hereafter, we consider
(3.34)
And
64
(3.35)
Therefore,
(3.36)
It is rather correct to consider a real permeability for the calculation of resistance but not for
the inductance. The upper expression is only valid below the cut-off frequency of the
permeability of the material considered. The proper expressions to start from when
considering a complex permeability are
(3.37)
(3.38)
Then, magnetic losses can actually be accounted for by introducing a complex permeability,
typically given by the manufacturer. Magnetic losses are then equal to
(3.39)
They can be represented by a series resistance
(3.40)
One could also account for dielectric losses,
(3.41)
(See Poynting’s theorem, [7]: where is the power delivered
by the source, the transmitted power,
the magnetic energy,
the electric energy. The total losses (Joules, magnetic and dielectric)
are
)). Again, dielectric losses are neglected
considering the importance of eddy losses and magnetic losses. They would be represented
by a parallel conductance in the line model.
65
Since there is no net current flowing through a lamination and that the laminations are
considered ideally insulated from one another, the losses of each lamination can be added up
to get the total losses. Mathematically, we considered that the values of the field at both
interfaces of a lamination are equal
.
By laminating the core into layers indexed by insulated from one another, the
total losses are
(3.42)
(3.43)
Considering that and
, the total losses can be expressed as
(3.44)
can be considered equal to
, for starting at 1 and being the inner radius of
the core. By introducing the outer radius of the core, constitute a subdivision
of . One can then recognize a particular case of Riemann’s sum:
r
(3.45)
Where . Here,
We then choose
(3.46)
So that
66
r
(3.47)
is approximately which is considered very small, so
(3.48)
The mathematical relative error introduced here is negligible ( ). Finally, the total losses
in one magnetic ring are
(3.49)
Where is the ribbon thickness (typically ), is the length between the centers of
two adjacent laminations (nearly equal to ; the insulating layer is approximately ),
and are the outer and inner radius of the magnetic ring, h the tape width (typically a
few cm), the electrical resistivity and , with complex.
To compute the series resistance of the ring, these losses are to be identified with the losses in
the equivalent electrical representation
So the additional series resistance that one magnetic ring represents is equal to
(3.50)
(Divide by to get it per length for ATP simulations).
In the same way, the magnetic energy stored in the magnetic rings is equal to
(3.51)
Which gives
67
(3.52)
For materials with a high permeability cut-off frequency, magnetic losses can be ignored in a
certain frequency range. For nanocrystalline cores, the ratio increases more slowly
than for other materials. It represents the fact that magnetic domains rotate quickly and
magnetic flux density doesn’t lag much the magnetic field.
Magnetic and eddy current losses are compared below. One of the difficulties is to find an
appropriate expression for the complex permeability. For ferrites it can accurately be
approximated by
(3.53)
For other materials, such as nanocrystalline, the slope can be different. Indeed, at high
frequencies, the real and imaginary parts of permeability have approximately the same slope
(see Figure 53).
Figure 53 - Typical shape of permeability vs frequency for ferrites and nanocrystalline material
3.2.3 Performance
3.2.3.1 Mitigation of VFTO
Magnetic losses have already been mentioned. They are given a closer attention now. They
can actually have an important contribution to the total losses especially when considering
nanocrystalline laminations. An example of comparison is given in Figure 54 between the
68
eddy losses and magnetic losses for nanocrystalline rings and silicon steel laminations. The
same volume of magnetic material was considered but with 800 laminations of thick
laminations of nanocrystalline material or with 8 laminations of thick silicon steel.
With this example, one can already note that nanocrystalline material has a more desirable
characteristic. Indeed, the total losses at low frequencies (50 Hz) are less important and more
important at higher frequencies, which is more appropriate for VFTO mitigation since the
optimal mitigation method does not affect the 50 Hz behaviour of the system at all but
absorbs and dissipates the most possible energy above 1 MHz.
Figure 54 - Eddy losses and magnetic losses for a same volume of nanocrystalline strips or silicon steel laminations.
Nanocrystals: 800 laminations of 20 µm thick tape, with , =60000 and ;
Silicon steel: 8 laminations of 2 mm thick tape, with , =10000 and
Also, another interesting feature is the influence of the number and thickness of laminations.
It is a common rule that for transformers, laminating a given volume of core in laminations
rougly reduces eddy losses by . While this is true for the typical frequencies and
thicknesses of a transformer core, it is not the case at very high frequencies. The limit
between the two regions is at . At low frequencies, eddy currents are limited by the
lack of “space”. A thinner lamination thus has lower losses. At higher frequencies, when
, the thickness of the layer doesn’t influence the losses anymore which explains why
all four curves in Figure 55 have the same asymptote. This is easily understandable by
looking at the low and high-frequency approximations of eddy losses in one lamination:
(3.54)
(3.55)
W
Frequency (Hz)
69
Since the laminations are supposed ideally insulated, using laminations roughly multiplies
the losses by (see 3.2.2 for more detailed calculation). This explains why nanocrystalline
material may not be very efficient when used as a coating but, because its optimal thickness
for filtering the 10 MHz frequency range is very thin (a few ), a tape wound ring of
hundreds of layers may be very efficient and still manageable in terms of volume.
Figure 55 - Resistance of one lamination for different thicknesses as a function of frequency
Figure 56 - Resistance of a same volume of material, divided into laminations of different thicknesses
A typical nanocrystalline ring would have an initial permeability of 45000, a cut-off
frequency around 50 kHz. Typical dimensions are
. That gives approximately 750 laminations. The additional resistance and inductance
presented by the rings then have the following form, as a function of frequency
Frequency (Hz)
Frequency (Hz)
70
Figure 57 - Additional resistance of a nanocrystalline ring as a function of frequency
Figure 58 - Inductance of a nanocrystalline ring as a function of frequency
These are in adequacy with data available on different manufacturers websites. The values
are a little lower than what could be obtained for classical common core chokes because of
the decrease of the external magnetic field with radius and the fact that the conductor does
not make any turns around the core. The point of inflexion in the resistance curve
corresponds to the cut-off frequency of the permeability. A source of error could be the
complex permeability though. In order to represent the permeability of nanocrystalline
materials, the following expression was used
(3.56)
A better fit should be used for accurate predictions, in particular in the region of the cut-off
frequency. Interpolation of the manufacturer’s data may be a good solution.
The rings can be modelled by an additional series inductor and resistor. Once again, the main
frequency component (~10 MHz) is considered for calculation of the values of and . If
10 of the rings mentioned above were to be used, the equivalent series resistance and
inductance would be
Frequency (Hz)
Frequency (Hz)
71
The VFTO would then be efficiently damped as shown in Figure 59 and Figure 60 (red:
reference, without rings; green: with 10 rings)
Figure 59 - Influence of nanocrystalline rings on the VFTO at the source electrode of the DS [before the rings]
Figure 60 - Influence of nanocrystalline rings on the VFTO at the open end [after the rings]
The influences of the two components and are observable. The resistance dissipates the
energy associated with the VFTO which explains the general damping effect on the VFTO.
The inductance opposes the propagation of the very steep first front and has still a sufficiently
low value so that LC resonance is not of significant importance. This explains the shape of
the VFTO at the open end with rings (Figure 60).
Finally, considering the same constraints on magnetic permeability, cut-off frequency and
total volume, the performance of both solutions are of the same order of magnitude The use
of insulated layers of a magnetic material extends the possibilities in the choice of material.
More conductive materials, used in thin layers can be considered.
72
3.3 Instantaneous losses
So far, all calculations were conducted in sinusoidal regime. It is especially handy
considering that all waveforms can be approached by their harmonic decomposition but
because of the very high fields the material is subject to, it will quickly saturate, making the
magnetic field density nonsinusoidal. Under saturation, some papers approximate the
losses by the low-frequency approximation
(3.57)
Replacing by the saturation induction . This formula seems incorrect in that case, in
particular because it does not justify why higher permeability rings would mitigate more the
VFTO. Even with very high magnitude magnetic fields, the permeability of the material
should influence the time-average losses as justified below.
It is assumed that the magnetic field density is uniform in a lamination, which corresponds
to . In that case, a more simple derivation of the losses can be conducted. Since the
induction is considered uniform, the electromotive force is easy to calculate (refer to Figure
49 for schema)
(3.58)
The amplitude of the eddy current in the incremental strip of thickness and area is
(3.59)
And the amplitude of the eddy-current density is
(3.60)
The time-average eddy losses in one lamination are then
(3.61)
The above formula is always true, for a periodic waveform of period . Since we considered
uniform in the lamination,
(3.62)
With the dimensions expressed in Figure 49 the time-average losses in one lamination are
73
(3.63)
Finally,
(3.64)
This gives the same expression as the low-frequency approximation given above for
sinusoidal induction, in which case
(3.65)
Because of saturation effects, the induction is not sinusoidal in time, even if it is uniform (in
space). The following expression of induction as a function of the magnetic field is used:
r
(3.66)
Figure 61 - Ideally soft magnetic cycle
A sinusoidal magnetic field does not always result in a sinusoidal magnetic flux as shown in
Figure 62. For sufficiently low permeability, the material is not saturated, and a sinusoidal
magnetic field excitation gives a sinusoidal magnetic flux density. For very high permeability,
the material is nearly instantly saturated which gives a more square shape to . Since the
losses are proportional to
, one can already see how considering that is
sinusoidal of magnitude will lead to significant errors for highly saturated materials.
74
Figure 62 - Magnetic flux density as a function of time for sinusoidal magnetic field
As an example, the magnetic field magnitude is set as (corresponds to
Amps flowing in a bar of 90 mm diameter). The saturation induction is chosen
equal to 1.2 T (nanocrystalline) and different values of permeabilities are considered. The
instantaneous losses are proportional to
.
Figure 63 - dB/dt for sinusoidal induction and actual saturated cycle
The higher the permeability, the narrower and higher the peaks of dB/dt. Therefore, the time-
average losses do depend on permeability. The term
is plotted as a function
of permeability in Figure 64 and compared to the one obtained with a sinusoidal form
. The magnitude of the magnetic field is chosen equal to 30000 A/m and
the saturation induction to 1.2 T.
t(s)
B(T
)
t(s)
T/s
75
Figure 64 - Time-average losses as a function of permeability. Sinusoidal induction taken as reference
The above calculation still supposes that the magnetic flux density is uniform in the material
which is only correct for (which is not true for nanocrystalline rings at several MHz)
and does not consider any delay between and . This delay can be important at high
frequency, and is represented by the imaginary part of the permeability. The phase delay is
r
(3.67)
For nanocrystalline rings it can be as high as at the considered frequencies. For
simulation of transients, it may be interesting to get an accurate formula of the instantaneous
losses or once again calculate them analytically at the main frequency of the VFTO but the
equivalent resistance would still depend on the current magnitude.
(W)
76
77
4 Conclusion
The issues of VFTO were presented and very general specifications of a good mitigation
system were expressed. Then, two solutions were studied that would potentially meet the
requirements: coating the bus bar with a semi-conducting, magnetic material and setting
magnetic rings around the conductor.
Detailed electromagnetic models were proposed for both systems, from which it was possible
to deduce the main features of the optimal material for VFTO attenuation. Then an equivalent
electrical representation was proposed for transient modelling.
Both solutions rely on the same principle: the dissipation of energy in eddy losses and
magnetic losses and addition of a series inductance for protection of the material downstream
of the mitigation system. The second solution allows more flexibility in the choice of material
though, mainly because several layers can be wound around the conductor to multiply its
effect. It is also easier to find magnetic materials that are good conductors and whose optimal
thickness is therefore very thin. The limits on such a system are the saturation of the magnetic
material and the dissipation of heat, which would both have to be studied more in depth
(maybe by Finite Element Modelling, where a formula of permeability could be introduced
taking into account the effects of frequency and saturation).
78
79
Appendix: FEM validation of the
principles with COMSOL
Figure 65 - Current density in the conductor and the coating at low frequency
At low frequencies, the skin effect has little influence on the distribution of the current
density. The coating is chosen more resistive than the aluminium so almost no current flows
through it.
80
Figure 66 - Current density in the conductor and in the coating in the "transition region"
At intermediate frequencies, the current flows both in the conductor and in the coating. Also,
it is verified that the geometrical zone where the radial currents are important is small (a few
mm). Neglecting edge effects was therefore a valid approximation.
81
Figure 67 - Current density in the conductor and in the coating at high frequency
At very high frequency, the current is localized on the periphery of the system, that is, in the
coating. Radial currents are localized at the very edge of the coating and re-penetrate in the
conductor.
82
Figure 68 - Current density in the conductor and a magnetic layer, separated by an insulating material
When the magnetic material is separated from the conductor by an insulating layer, there is
no path for eddy currents to flow back in the conductor. Therefore they complete their loop in
the lamination. At high frequency, they are concentrated at the periphery of the material. The
distribution of the current density in the conductor is not modified because no net current
flows in the magnetic layer. The magnetic material is influenced by the proximity effect of
the conductor but there is no reciprocal.
83
Bibliography
[1] Working Group CIGRE 15.03, “GIS Insulation Properties in case of VFT and DC
Stress,” CIGRE, 1996.
[2] Z. Liu, Ultra-high Voltage AC/DC Grids, Academic Press, 2014.
[3] M. Mohan Rao, H. S. Jain, S. Rengarajan, K. R. S. Sheriff and S. C. Gupta,
“Measurement of very fast transient overvoltages (VFTO) in a gis module,” London,
1999.
[4] S. Burow, W. Köhler, S. Tenbohlen and U. Straumann, “Damping of VFTO by RF
Resonator and Nanocrystalline Materials,” 2013.
[5] A. Gromov, Impedance of Soft Magnetic Multilayers: Application to GHz Thin Film
Inductors, Stockholm, 2001.
[6] M. K. Kazimierczuk, High-Frequency Magnetic Components, Dayton, Ohio, USA:
Wiley, 2014.
[7] D. M. Pozar, Microwave Engineering, JohnWiley & Sons, Inc., 4th Edition, 2012.
[8] Ametani, Miyamoto and Nagaoka, “Semiconducting Layer Impedance and its Effect on
Cable Wave-Propagation and Transient Characteristics,” IEEE TRANSACTIONS ON
POWER DELIVERY, vol. 19, no. 4, 2004.
[9] C. E. Baum and S. J. Tyo, “Transient Skin Effect in Cables,” Phillips Laboratory, 1996.
[10] S. Nordebo, B. Nilsson, T. Biro, G. Cinar, M. Gustafsson, S. Gustafsson, A. Karlsson
and M. Sjöberg, "Electromagnetic dispersion modeling and measurements for HVDC
powercables," Lund University, Lund Sweden, 2011.
[11] A. Levasseur, “Nouvelles formules valables à toutes les fréquences pour le calcul rapide
de l'effet Kelvin,” 1929.
[12] H. W. Dommel, EMTP Theory Book, 1996.
[13] S. Burgerhartstraat, Handbook of Magnetic Materials, vol. 10, Amsterdam: Elsevier
Science B.V, 1997.
[14] R. C. O'Handley, Modern Magnetic Materials, Wiley, 2000.
[15] M. E. McHenry, M. A. Willard and V. E. Laughlin, Amorphous and nanocrystalline
84
materials forapplication as soft magnets, 1999.
[16] G. Bertotti, Hysteresis in Magnetism, 1998.
[17] A. Magni, F. Fiorillo, E. Ferrara, A. Caprila, O. Bottauscio and C. Beatrice, “Domain
wall processes, rotations, and high-frequency losses in thin laminations,” Torino, 2013.
[18] Y. Chen and P. Pillay, “An Improved Formula for Lamination Core Loss Calculations in
Machines Operating with High Frequency and High Flux Density Excitation,” Industry
Applications Conference, no. IEEE, 2002.
[19] J. Li, T. Abdallah and C. R. Sullivan, “Improved Calculation of Core Loss With
Nonsinusoidal Waveforms,” IEEE Industry Applications Society Annual Meeting, 2001.
[20] T. Trupp and P. Zoltan, “Effect of ribbon thickness on power losses and high frequency
behavior of nanocrystalline FINEMET-type cores,” 2013.
[21] M. K. Kazimierczuk, G. Sancineto, G. Grandi, U. Reggiani and A. Massarini, “High-
Frequency Small-Signal Model of Ferrite Core Inductors,” IEEE Transactions on
Magnetics, vol. 35, no. 5, 1999.
[22] M. Gerlin, "Pertes supplémentaires dans les conducteurs pour forte intensité par effet de
peau et de proximité," 2002.
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