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

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