HVE script_v01.pdf

7/27/2019 HVE script_v01.pdf

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Chair of Energy Distribution and High-Voltage EngineeringProf. Dr.-Ing. Harald Schwarz

High-Voltage Technique and Insulating Materials Page 1

High-Voltage Technique and Insulating Materials

- Lecture notes -

Brandenburg University of Technology CottbusChair of Energy Distribution and High-Voltage EngineeringWalther-Pauer-Str.5

D-03046 Cottbus

GERMANY


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1. Introduction................................................................................................................. 41.1. Chair and Teaching Programme......................................................................... 4

1.2. Objective and Structure of the Lectures ............................................................. 71.3. Fundamental Principles ...................................................................................... 9

1.3.1. Maxwells Equations.................................................................................... 91.3.2. Static Fields............................................................................................... 111.3.3. Stationary (Steady-State) Fields ............................................................... 111.3.4. Slowly Varying (Quasi-Stationary) Fields.................................................. 121.3.5. Rapidly Varying Fields .............................................................................. 14

2. Determination of the Electric Field Distribution........................................................ 192.1. Analytical Calculations...................................................................................... 19

2.1.1. Coaxial Cylinders and Spheres................................................................. 192.1.2. Boundary Problem for Plate Electrodes.................................................... 20

2.1.3. Influence of Space Charges...................................................................... 212.1.4. Schwaigers utilization factor..................................................................... 232.2. Graphic Determination of Field Distribution...................................................... 282.3. Measurement of the Field Distribution.............................................................. 302.4. Method of Conformal Mapping ......................................................................... 312.5. Method of Substitution Charges ....................................................................... 322.6. Differential Method............................................................................................ 342.7. Method of Finite Elements................................................................................ 35

3. Boundary surfaces and imperfections in high-voltage insulators............................. 363.1. Boundary conditions ......................................................................................... 363.2. Laminated Dielectric ......................................................................................... 373.3. Tangential Fields at Boundary Surfaces........................................................... 423.4. Imperfections (Defects)..................................................................................... 43

4. Discharge Reactions in Gases (Basic Mechanisms) ............................................... 464.1. Statistical Basics............................................................................................... 464.2. Non-Self-Maintained Gas Discharge................................................................ 514.3. Self-Maintained Gas Discharge........................................................................ 544.4. Towsend Discharge.......................................................................................... 594.5. Streamer Mechanism........................................................................................ 66

5. Discharge Reactions in Gases (technical details).................................................... 695.1. Breakdown of Mixed Gases.............................................................................. 695.2. Influence of the Electrode Roughness.............................................................. 705.3. Breakdown in Inhomogeneous Fields .............................................................. 72

5.4. Streamer and Leader discharge ....................................................................... 755.4.1. Positive Streamer Discharge..................................................................... 755.4.2. Negative Streamer Discharge................................................................... 765.4.3. Leader Discharge...................................................................................... 76

5.5. Breakdown Behaviour for Transient Voltages .................................................. 785.6. Spark Discharge and Arc Discharge ................................................................ 80

5.6.1. Spark Discharge........................................................................................ 805.6.2. Arc Discharge............................................................................................ 84

5.7. Surface Discharges .......................................................................................... 875.7.1. Breakover (Flash-Over)............................................................................. 875.7.2. Pollution Layer Breakover ......................................................................... 88

5.7.3. Surface Discharge (Sliding Discharge) ..................................................... 916. Breakdown Reactions in Solid and Fluid Insulating Materials ................................. 93

6.1. Purely Electrical Breakdown............................................................................. 93


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6.2. Global Thermal Breakdown .............................................................................. 966.3. Masked Gas Breakdown................................................................................... 986.4. Local Thermal Breakdown................................................................................ 99

6.5. Fibre-Bridge Breakdown................................................................................. 1006.6. Erosion Breakdown......................................................................................... 1046.7. Partial Discharges........................................................................................... 1067.1. Gases.............................................................................................................. 113

7.1.1. Natural Gases ......................................................................................... 1137.1.2. Liquefied Gases ...................................................................................... 1157.1.3. SF6 (Sulfurhexafluorid)............................................................................ 116

7.2. Insulating Fluids.............................................................................................. 1187.2.1. Physical and Chemical Parameters ........................................................ 1187.2.2. Insulating Oil Made from Mineral Oils ..................................................... 1287.2.3. Synthetic Insulating Fluids ...................................................................... 134

7.2.4. Other Insulating Fluids ............................................................................ 1357.3. Solid Insulating Materials................................................................................ 1367.3.1. Physical and Chemical Parameters ........................................................ 1367.3.2. Inorganic Solid Insulating Materials ........................................................ 1417.3.3. Organic Solid Insulating Materials........................................................... 154

7.4. Mischdielektrika .............................................................................................. 1697.4.1. Imprgnierte Foliendielektrika................................................................. 1697.4.2. Oil Paper Dielectrics................................................................................ 170

8. Testing Insulating Materials.................................................................................... 1718.1. Dielectric Measurement.................................................................................. 171

8.1.1. Dielectric Loss Factor and Capacitance ................................................. 1718.1.2. Insulation Resistance .............................................................................. 174

8.2. Disruptive Discharge Test............................................................................... 1758.3. Creep Tracking Resistance ............................................................................ 176

8.3.1. Comparative tracking index (CTI) ........................................................... 1768.3.2. Tracking under Difficult Conditions ......................................................... 177

8.4. Resistance to Arcing....................................................................................... 1778.5. Chemical Analysis .......................................................................................... 179

8.5.1. Water Content ......................................................................................... 1798.5.2. Gas-in-Oil Analysis.................................................................................. 180


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1. Introduction

1.1. Chair and Teaching Programme

Head ofchair:

Prof. Harald Schwarz 69-4502 [email protected]

Secretary: Marika Scholz 69-4502 [email protected]

Scientific

assistants:

Dipl.-Ing. Dirk Lehmann 69-4032 [email protected]

Dipl.-Ing. (FH) Maik Honscha 69-4029 [email protected]

Dr.-Ing. Klaus Pfeiffer 69-4035 [email protected]

Dipl.-Ing. Stefan Fenske 69-3580 [email protected]

Dr.-Ing. Gunnar Lhning 69-4030 [email protected]

Dipl.-Ing. Henryk Strmer 69-3528 [email protected]

Dipl.-Ing. Lars Roskoden 69-4044 [email protected]

Technicians: Dipl.-Ing. (FH) Alexander Feige 69-4029 [email protected]

Dipl.-Ing. (FH) Lothar Kleinod 69-4025 [email protected]

Dipl.-Ing. (FH) Holger Husler 69-4027 [email protected]

Electrician: Karl-Heinz Kleinschmidt


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Ground plan

The Chair of Energy Distribution and High-Voltage Engineering is situated at thebuilding no. 3, Walther-Pauer-Strae 5 (figure 1).

Figure 1: Ground plan of the BTU Cottbus (detail)


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Teaching Programme of the Chair of Energy Distribution and

High-Voltage Engineering

o Basics of Electrical Energy Technique (L/T 3rd semester)

o High-Voltage Engineering and InsulatingMaterials (L/T 5

thsemester)

o High-Voltage Devices and Switchgear (L/T 6th

semester)

o Planning of Energy Transmission Networks (L/T 5th/7th semester)

o Protection of Energy Transmission

Networks (L/T 6th

/8th

semester)o EMC in Plants and Systems (L/T 7

thsemester)

o High-Voltage Measuring and Testing Devices (L/T 8th

semester)

o Selected Topics from Energy Transmission andHigh-Voltage Engineering (T 8

th/9

thsemester)

o Low- and Medium-Voltage Engineering LA (L 7th/8

thsemester)

o Power Automation LA (L 7th/8

thsemester)

o EU-East Expansion and Intercultural Competence (Excursion)

Figure 2: Route of the technical excursion


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1.2. Objective and Structure of the Lectures

Objective of the Lectures

- Calculation of elektric fields- Discharge and breakdown reactions in gases, fluids and solid

materials- Insulating materials and their electrical, physical and chemical

parameters

Structure of Lectures (L) and Tutorials (T)

L01 Introduction, fundamental principlesL02 Determination of the electric field distributionL03 Boundary surfaces and imperfections in high-voltage insulatorsL04 Discharge reactions in gases (basic mechanisms)

(Statistical basics; non-self-maintained discharge; self-maintaineddischarge)

L05 Discharge reactions in gases (basic mechanisms)(Townsend discharge, Streamer discharge)

L06 Discharge reactions in gases (technical details)

(Mixed gases; electrode roughness; inhomogeneous field;streamer; leader; transient voltages)L07 Discharge reactions in gases (technical details)

(Spark discharge; arc discharge; surface discharge)L08 Breakdown reactions in solid and liquid insulating materials

(Purely electrical breakdown; global thermal breakdown; maskedgas breakdown; local thermal breakdown; fibre-bridge breakdown)

L09 Breakdown reactions in solid and fluid insulating materials(Erosion breakdown; Partial discharge)

L10 Insulating materials(Gases; insulating fluids)

L11 Insulating materials(Insulating fluids; solid insulating materials)

L12 Insulating materials(Anorganic solid insulants)

L13 Insulating materials(Organic solid insulants)

L14 Testing insulating materials


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T 01 -T 02 Repetition of fundamental principles / Field patterns

T 03 Analytical calculationsT 04 Calculation of electric fieldsT 05 Boundary surfacesT 06 Statistics / Drift velocityT 07 Gas dischargesT 08 Breakdown reactionsT 09 Laboratory experiment 1 (Introduction into high-voltage

testing devices; determination of the breakdown field strengthEd for various electrode configurations) - Group A

T 10 Laboratory experiment 1 - Group BT 11 Laboratory experiment 2 (Paschen curve, impulse voltage-

time characteristic) Group AT 12 Laboratory experiment 2 - Group BT 13 Laboratory experiment 3 (Partial discharges, Surface

discharges) - Group AT 14 Laboratory experiment 3 - Group BT 15 Repetitions


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1.3. Fundamental Principles

1.3.1. Maxwells Equations

=x A

dABt

dxE

Faradays law of induction

+=x A

dAt

DJdxH

Amperes law

Integral form of Maxwells Equations (field equations : Interconnection between electric andmagnetic field quantities by the law of induction (left side) and Amperes law (right side).

Law of induction

A changing magnetic flux ( ) dAB produces a rotational electric field E.

Amperes law

An electric current

+A

dAt

DJ produces a rotational magnetic field H.


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=

A

dAB 0

Continuity equation for themagnetic flux density

=

+

A

dAt

DJ 0

Continuity equation for conductionand displacement current density

Integral form of the continuity equations for the magnetic flux density (left side, three-dimensional view) and the conduction and displacement current density (right side, sectionalview)

Continuity of magnetic flux density

The magnetic field is source free, i.e. there are no magnetic monopoles.The magnetic field lines must be closed loops. Given any volumeelement, the magnetic flux entering the surface must be equal to themagnetic flux emerging from the surface.

Continuity of conduction current density and displacement currentdensity

A temporal changing conduction current in conducting materials con-tinues as displacement current in a non-conducting material.

Material equations

HBr

0

=

EDr

0

= EJ = Material equations for magnetic and electric field quantities


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1.3.2. Static Fields

- absolutely no temporal changes- no displacement current- no conduction current- no energy transport

Examples:

- magnetic fields of permanent magnets,- electric fields of separated charges, provided that the conductivity

of the dielectric material is 0= and there is no charge equaliza-tion.

1.3.3. Stationary (Steady-State) Fields

- In contrast to the static fields a constant conduction current density(direct current) is permitted.

- The law of induction has the form

=

0dxE

From there the loop rule (Kirchhoffs Voltage Law) of the networktheory is derived

=i

iU 0

- Amperes law has the form

== A

dAJdxH

- The continuity equation for conduction and displacement currenthas the form

=A dAJ 0


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From there Kirchhoffs Current Law (Kirchhoffs point rule) isderived

0=i

i

- The continuity equation for the magnetic flux density remains un-changed.

1.3.4. Slowly Varying (Quasi-Stationary) Fields

1.3.4.1. Inductive Fields in Conductors

In materials of high conductivity the displacement current tD / can beneglected in comparison with the conduction current (for frequencies upto the GHz range).

Inductive fields

=x A

dABt

dxE

Law of induction

=x A

dAJdxH

Amperes law

0= dABA


0 dAJA

Continuity equation for theconduction current density

Maxwells equations for slowly varying inductive fields (disregarding the displacementcurrent in conductors)

Quasi-stationary inductive fields can be found in transformer windings,conductive connectors and electrodes of high-voltage devices.


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1.3.4.2. Capacitive Fields in Insulating Materials

In high-performance insulating materials with low residual conductivitythe conduction current density is very low in comparison with the dis-placement current density. That means that the electric field is mainly asource field and the induced electric field strength can be neglected.

Capacitive fields

0 dxEx

Law of induction

+=

Ax

dA

t

DJdxH

Amperes law

0= dABA


=

+A

dAt

DJ 0

Continuity equation for theconduction current density

Maxwells equations for slowly varying capacitive fields (disregarding the magnetic

induction)

The transition from inductive to capacitive fields shall be demonstratedfor the example of an open conductor loop.

Slowly varying fields inside and outside of an open conductor loop (inductive und capacitive

fields).


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Quasi-stationary fields can be found at high-voltage devices for

D.C. voltage, A.C. voltage (50 Hz), Switching impulse voltage 250 / 2500 s and Lightning impulse voltage 1,2 / 50 s

if the physical size of the devices is in the range of several meters.

1.3.5. Rapidly Varying Fields

The travel time of an electromagnetic wave for a distance x is given by

v

x=

with v = propagation speed of the wave.

In energy distribution systems the wave propagation speed is

1===r

rrr

mitcc

v

c = velocity of light

r= relative permittivity of the insulation

i.e.

ns

m

s

m

s

kmcv 3,0300000300 ====

for air insulation and


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ns

m

s

m

s

kmcv 2,0200000200

5,1

====

for cable insulation (r= 2,3) respectively.

For lightning impulse voltages, which are rapidly changing in themicrosecond range, rapidly varying fields can be found at system sizesof around 100 meters. Should transient voltages with fluctuations in thenanosecond range occur, such fields can be found at system sizes ofseveral meters.

- Maxwells equations in the complete form have to be used.

- The coupling between electric and magnetic field becomes time-and space-dependent.

- Travelling waves occur.

In general the following equations are valid:


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rr

cv

==1

Propagation speed

=z

Wave impedance

- In energy systems travelling waves can be found mainly at longlines.

- In the equivalent circuits of the lines the electric and magneticfields are respresented by inductances and capacitances.

- The wave impedance z of a line is given:

''

''

CjG

LjRz

++

=

- Neglecting R and G results in

'

'

C

L

z = ''1

CLv =


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- Typical values of the wave impedance are

z

300

overhead linez 30 cable

- If a wave front (incoming wave) comes to the connection of twolines (cables) with different wave impedances, the ratio of currentto voltage is changed at the reflection point.

Reflection and refraction of an incoming travelling wave at a discontinuity of the line waveimpedance

- The the current-to-voltage ratios of the refracted and the passingwave are determined by the wave impedances of the two lines.

- If the incoming and the passing wave have different voltageamplitudes, a refracted wave has to be superimposed to theincoming wave so that the amplitudes at the reflection point are

equal.


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- Naturally these refracted voltage wave is accompanied by a re-fracted current wave.

It holds

Current i Voltage u

Refraction factor

21

12

zz

zbi +

= 21

22

zz

zbu +

=

Reflection factor

21

21

zz

zz

ri +

= 2112

zz

zz

ru +

=

Reflection and refraction of an incoming travelling wave at a discontinuity of the waveimpedance for three special cases: open-ended line, short-circuited line and match-

terminated line


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2. Determination of the Electric Field Distribution

2.1. Analytical Calculations

2.1.1. Coaxial Cylinders and Spheres

Cylinder

( )

i

a

rrr

UrE

ln

=

Sphere

( )

=

ai

rrr

UrE

112

Boundary problem for the termination of a coaxial cylinder by a hemi-sphere because of

Cylinder Sphere

Emax

i

ai

r

rr

U

ln

a

i

ir

rr

U

1

Optimal radius ratio71,2== e

r

r

i

a

2=i

a

r

r


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2.1.2. Boundary Problem for Plate Electrodes

Field pattern at the boundary region of a parallel-plate capacitor

Field strength at the boundary region of a plate electrode arrangement

Using the method of conformal mapping a profile can be found, whichguarantees that the field strength in the boundary region is not higherthan at the homogeneous region.


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Rogowski profile for 2/ =

+= x

s

ssy

exp

2 valid for one sparking distance only

2.1.3. Influence of Space Charges

One-dimensional electric field

Poisson equation

=

=

2x


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For f (x) (homogeneous space charge)

it follows 212

2

1cxcx ++=

Boundary conditions

( ) ( )

+===

== 22

2

1

10

0

s

x

s

x

ss

x

Usx

Ux

Field strength

xss

UE

+= 2

1

Damage risk for

withoutzulE

s

U

Emax > Ezul with


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2.1.4. Schwaigers utilization factor

Emax = f (U, s, remaining term )

Parallel-plate electrodes Emax =s

U* (1)

Coaxial cylinders Emax =

irar

ir

ir

ar

s

U

ln

Concentric spheres Emax =

a

r

ir

ir

ir

ar

s

U

1

General expression

Emax =max

averageaverage

* E

EE

s

U==

Definition of Schwaigers utilization factor (degree of homogeneity)


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With the variables

bigger radius Rsmaller radius r

r

rsp

r

Rq

+=

=

= f (q, p)

Diagrams

Air unit capacitance CLE = f (p, q)

Cylinder C = r* l * CLE

Spheres C = r* r * CLE


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Utilization factor for cylinder arrangements


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2.2. Graphic Determination of Field Distribution

Objective: fast determination of a qualitative field distribution

Prerequisite: two-dimensional electrode arrangement

Graphic determination of field lines and equipotential lines for two-dimensional fields

- Field lines and equipotential lines are perpendicular to each other.

- Electrode surfaces are equipotential lines with 0 % (ground side) or100 % (high-voltage side) respectively.

- The distance a between two equipotential lines corresponds always to

the same potential difference U.

- The distance b between two field lines (displacement flux density

lines) corresponds always to the same charge Q at the electrodes.

U

QC

= is constant for the whole field map.

b/a = const.

For b/a = 1 the field determination can be drawn using circles.


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Step 1

First the known potential distribution at the homo-geneous part of the field is drawn (1). The furtherdrawing of equipotential lines is oriented to theform of the electrodes (2).Note: It is sensible to start with only a few equipo-tential lines (e.g. with the lines for 0 %, 25 %, 50%,75% and 100%). Afterwards the drawn field distri-bution can be further refined by interpolation.

Step 2

Field lines are added perpendicular to the equipo-tential lines, observing the ratio b/a = 1. It is sensi-ble to work along one electrode (e. g. the high-volt-

age electrode). By drawing circles between fieldlines and equipotential lines it can be found, thatthe ratio b/a does not equal 1 in most cases (3).

Step 3

The correction of the first picture is made by in-creasing the distance between the 25%-line andthe lower electrode toward the outside of the elec-trode arrangement (4). The 75%-line is drawncloser to the edge of the upper electrode, while thedistance to the upper side of the electrode is in-creased considerably (5).

It should be noted that the field strength at theedge of the electrode decreases from the uppertowards the lower electrode, i.e. the distancebetween the field lines shall increase. Checking theratios of sides and angles shows the necessity offurther refinements.

Step 4

By iterative refinements of the field distributionaccording to the drawing rules the final picture isdrawn.In the current example it is sensible to draw the

circles at the homogeneous part of the field first.Afterwards the drawing can be continued at theinhomogeneous region (6).


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2.3. Measurement of the Field Distribution

Basis: Analogy of slowly varying dielectric displacement fields (A.C.voltage) and stationary electric flow field (D.C. voltage)

EJEDrrrr

==

== AdJAdDQrrrr

Potential distributions of dielectric displacement fields (caused byseparated charges) are equivalent to potential distributions ofstationary electric flow fields.

Measurement at semiconductive paper (resistance paper)

- Drawing of conductive electrode outlines;

- Applying a D.C. voltage to the electrodes;

- Measuring of equipotential lines with measuring bridge and nullindicator;

- Modelling of different values rby using multiple layers of theresistance paper;

- Usable for two-dimensional fields

Measurement in semiconductive liquids (electrolytic tank)

- Immersion of the electrode arrangement in a semiconductive liquid;

- Usable for three-dimensional fields;

- Measurements require a lot of time and money.


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2.4. Method of Conformal Mapping

- Analytical calculation of several important field configurations

- Especially important before the advance of numerical field calculation

Idea

- Transformation of a complex electrode arrangement at the x,y-planeinto a simpler arrangement at an u,v-plane,

- Calculation of the simpler electrode arrangement at the u,v-plane,

- Inverse transformation of the results into the x,y-plane

Example: Cylinder in a corner

- with ( )22 jyxzw +== mapping as parallel-plate electrodes

Conformal mapping of field lines and equipotential lines for a rectangular electrode:2

zw=

After inverse transformation

22

2

2

* yxaUE +=


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2.5. Method of Substitution Charges

Idea: Modelling of potential field by superposition of singlepoint, line and surface charges

Example: Field of two point charges

- Modelling the equipotential surfaces by conductive spheres would notchange the field distribution.

- For given electrode outlines the position of the substitution chargescan be manipulated iteratively until the boundary conditions are met.


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2.6. Differential Method

Solution of Laplace equation withdifferential formulation and Taylorseries expansion

Square formula:

=

=4

14

1

iio

Diagonal formula

=

=8

54

1

iio

- Covering of the field space with a square grid;

- Modelling of the electrodes using a square grid;

- Set-up of a system of linear equations;

- Insertion of boundary conditions (electrode potentials).


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2.7. Method of Finite Elements

- Triangle or tetrahedron as basic elements;- Iterative optimisation of field distribution for minimum field energy.

Field distribution of a disconnector of a metal-enclosed switchgear assembly with SF6-insulation: a Mesh grid of the field space; b Equipotential lines in the field space


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3. Boundary surfaces and imperfections in high-voltageinsulators

3.1. Boundary conditions

Et1 = Et2 electric field strength

JWn1 = JWn2 current density

with t

EE

t

DE

WJ

+=

+=

rr

rr

a) periodical alternating (A.C.) field and 0

1

2

2

121

==

nE

nE

DD nn

b) constant (D.C.) field

1

2

2

1

=

nE

nE


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Vectors of electric field strength at the boundary surface between twoinsulating materials

1nE

1tE1E

2E

21

2nE

2tE

21 >

)( 21 >

3.2. Laminated Dielectric

a) A.C. voltage

21

1 2

1s 2s


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from = EdsU follows

U = E1s1 + E2s2

With boundary surface conditions follows

111

21

1/2

1 +

=

sss

UE

11

1

21

1

2

+

=

s

ss

UE

Example for 41

2 =

and ss

9

11

=

Range field strength depending on position of boundary layer


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Conclusions (general validity)

- For inhomogeneous dielectric the E-field depends on the electrodearrangement and the properties of the insulating materials.

- High field strengths can be found at areas of small physical

dimensions with small r.

- The electric field between parallel-plate electrodes becomes

inhomogeneous for12.

- In inhomogeneous fields the displacement current density D is no

direct measure of E.

b) D.C. voltage

Displacement current density is discontinuous

11221

2211

21 ==

E

E

D

D

from Maxwells equation

=QAdDrr

follows the boundary surface charge

Qg = A (D2 D1) 0


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Example

Short-circuiting the electrodes for a short period of time gives residualfield strengths E1R, E2R

2211**0 sEsE

RR+=

or

1

2

2

1

s

s

E

E

R

R =

Residual field strength for short circuit (s1 = s2)


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From there it follows

112

21

22

11

2

1 ==ss

EE

DD

R

R

i.e. there exists an interfacial charge. After clearing the short circuit thischarge will generate influence charges at the electrodes.

Attention

- Devices with laminated dielectrics have to be permanently short-circuited after a D.C. voltage was applied.

- Short-circuiting for a short period of time will neutralize the electrodecharges but not the interfacial charge.

- After a short-time short-circuiting the interfacial charge will generateinfluence charges at the electrodes, which will result in dangeroushigh voltages at the device.

- There will be no interfacial charge for the special case

2

1

2

1

= only.

- For high-voltage devices usually different rated voltages are defined

for A.C. and D.C.


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3.3. Tangential Fields at Boundary Surfaces

Conclusions

- Inclination reduces the tangential field strength = surface fieldstrength.

- Field problem in gas insulation

B: Tapered insulator at inner conductor (high-voltage) Increase of high initial field strength.

C: Tapered insulator at outer conductor (ground potential) Increase of low initial field strength.


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3.4. Imperfections (Defects)

Inclusion of a defect with rs results in a mixed dielectric.

Assumption: Small defects cause no alteration of basic electrodefield Eo.

Inner effects

- Disk-type defect (Field lines perpendicular to defect area)

Esi = 0E

rs

r

Example: Gas enclosure in cast resin

1;4 ==rsr

Esi = 4 Eo

At the same time reduced dielectric strength at the gas space.

- Spherical defect

Esi =02

3E

rrs

r

+


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1;4 == rsr Esi = 1.33 EoAt the same time reduced dielectric strength at the gas space.

- Cylindrical defect (Axis Field)

Esi =0

2E

rrs

r

+


1;4 == rsr Esi = 1.6 EoAt the same time reduced dielectric strength at the gas space.

In general: Es > Eo forrs < r!


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Outer effects

Field strength in the vicinity of the defect

o

rrs

r

r

rs

saEE

2

3

+=

for sphere

Example: Metal inclusion or electrode roughness

rs for metallic sphere

osaEE 3= also valid in front of a metallic hemisphere at an

electrode.


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4. Discharge Reactions in Gases (Basic Mechanisms)

4.1. Statistical Basics

Statistical methods are used because of large variation of e.g.

Inception / Extinction Breakdown voltage Breakdown time

Examples for the statistical character of discharge reactions


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For the mathematical treatment a theoretical distribution function has tobe chosen.

Gaussian (normal) distribution Weibull distribution

Gaussian (normal) distribution with densityfunction D(x) and distribution function F(x)

Weibull distribution with density functionD(x) and distribution function F(x)

Density function

( )( )

=

22

2

exp2

1

xxD

Distribution function

( ) ( )dxx

xDxF

=

Density function

( )( )

dx

xFdxD =

Distribution function

( )

=

063

0exp1xx

xxxF


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Probability paper is used for testing the validity of the chosen distribu-tion function.

Display of a theoretical distribution function(top) as a line in a probability net (below)withdistribution tests of two measurement series

Probability net for the Weibull distribution withlogarithmic scaling of the axis


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Examples

Voltage inkV

Frequency Cumulativefrequency

929394

101

0,050

0,05

112

0,050,050,1

959697

201

0,10

0,05

445

0,20,2

0,25

9899

100

22

3

0,10,1

0,15

79

12

0,350,45

0,6101102103

121

0,050,1

0,05

131516

0,650,750,8

104105106

200

0,100

181818

0,90,90,9

107108109

100

0,0500

191919

0,950,950,95

Distribution table for measured values Comparison of the empirical distribution function(cumulative frequency polygon) with a theoretical

distribution function (Gaussian distribution)

Class in kV Frequency Relativecumulativefrequency

Absolute Relative Related to

> 91,5 - 94,5 2 0,1 0,033/kV 0,1

> 94,5 - 97,5 3 0,15 0,050 /kV 0,25

> 97,5 100,5 7 0,35 0,117 /kV 0,6

>100,5 -103,5 4 0,2 0,067 /kV 0,8>104,5 -106,5 2 0,1 0,033 /kV 0,9

>106,5 -109,5 1 0,05 0,017 /kV 0,95

Formation of classes from frequency tables Comparison of an empirical density function(related frequency) with a theoretical densityfunction (Gaussian distribution)


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Remarks

- Gaussian distribution

= expectation value = average value = standard deviation

Useful for many natural processes.

Range of values from - to +

Application in high-voltage engineering

Withstand voltage ud0 for x = - 3 = 0.13 %

Breakdown voltage ud50 for x = = 50 %

Guaranteed breakdown voltage ud100 for x = + 3 = 99.87 %

- Weibull distribution

xo = initial valuex63 = 63 % value = Weibull exponent

There is a lower limit for the range of values.

Well suited for the determination of the withstand voltage because

F (x) = 0 for x x0


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4.2. Non-Self-Maintained Gas Discharge

Generation of free charge carriers by radiation energy

Ws = h * f h = 6.62 * 1034

Ws2

Planck constant of action

Energy input into a gas molecule results in a increase of the orbitalradius of the shell electrons

If the ionization energy Wi is supplied an electron leaves the orbit

Gas 02 C02 N2 SF6 1 eV = 1.6 10-19

Ws

Wi 12.8 14.4 15.8 19.3 eV

Ionization rate due to cosmic radiation and natural radioactivity

5 20 electronionpairs per cm3 * s

Continuous recombination processes by trapping of e-

by electronega-tive gases (02, SF6)

Lifetime of free electrons 10-8 sLifetime pos./neg. ions 18 sIon density in the field-free space 500 / cm3

Directional motion of charge carriers caused by an external field

Coulomb force EqF

rr

= Assumption: initial velocity after generation vo = 0

Acceleration of charge carriers

* Positive charge carriers in Field direction* Negative charge carriers against Field direction


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Drift velocity

In vacuum the charge carriers can be accelerated without retardation. Inreal gases collisions occur, resulting in an average velocity (drift veloc-ity).

VD

V

x

EbD

vrr

= with b = mobility

Electron: b =cmV

scm

/

/500

Ion: b =cmV

scm

/

/1

Large ion:

b =cmV

scm

/

/10...10 14

(Dust partikel)

Dark current

Motion of charge carriers in gases Gas dischargeNon-self-maintained gas discharge Charge carriers generated by

external influencesNo luminous effect Dark discharge

Dark current density

Dev

eq

en

Dvqn

DvqnJ

rrrr+

+++

+

=

Assumption: singly charged

E

e

b

e

nbnbn

e

qJ +

++

+

=

r


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Specific conductivity of the gas

( )

( )

116

10

1910*6,1

=


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4.3. Self-Maintained Gas Discharge

Collision ionization

At high field strength the probability for the trapping of electrons de-creases.

rB

rB

rA

1 2

( )2BAS rrA +=

Ladungstrger

Gasmolekhl

rB

Model of effective cross section and free path

length

Density of molecules kT

p

n =

p = pressure, T = absolute temperature

k = Boltzmann constant = 1.37 * 10-23

K

Ws

Cylinder in field direction V = (rA + rB)2l

Gas molecules in thecylinder N = nV(statistical average)

Statistical spread of free path length between two collisions

Mean free path length ( )2

1

BArrnnVN +

===

ll


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Charge carrier = electron rA


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Ionization Coefficient

Number of collision ionizations per path length

/1ie

=

with

Ionization path length Eq

Wi

e

i=

=

kTEe

q

pri

W

kT

pr BB

22

exp

By joining all gas specific values

=

pE

BA

p /exp

A inPacm+

1B in

Pacm

kV

+for E/p in

Pacm

kV

*

Air 64.5 * 10-3 1.9 * 10-3 0.3 1.4 * 10-3

N2 94.5 * 10-3 2.56 * 10-3 1.1 4.5 * 10-3

SF6 113 * 10-3

2.37 * 10-3

0.8 2.2 * 10-3


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Effective Ionization Coefficient

Trapping of free electrons

Aeff =

A = attachement coefficient

for SF6

=

p

E

Pacm

kV

kVp

A 310*87,24

for air

=

p

E

Pacm

kV

kVp

A 310*8,0434,0

A decreases with increasing field strength E.

- For strongly electronegative gases effbecomes nearly linear.

- There is a minimum value E/p for ionization.


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( ) ( )op

E

p

Ek

p

effi =

for SF6 cm

kV4.88

o

;1

7.27 =

=

p

E

kVik

MinimumfieldstrengthE=88,4 cmkV

foreff>0!

for air cm

kV25

o

=

p

E


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4.4. Towsend Discharge

Electron avalanche

Change of the number of electrons

effeN

dx

edN

=

)()(and

)(ofbecausexf

effxfE

Efeff ==

=

it follows

=e

N

eoN

dxx

oeffe

dN

eN

1

= dx

x

oeffeo

Ne

N exp*


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Physical model of the Townsend discharge

- Ions drifting to the cathode

- No impact ionization by ions because free path length is to small

- Impact of ions on electrodes work function of conduction electrons(of the metal electrodes) is exceeded starting electrons for secon-dary avalanche


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Work function Wa of different cathode materials

Material Wa in eV

Barium oxide 1.0

Aluminium 3.95 1.77

Copper 4.82 3.89

Copper oxide 5.34

Silver 4.74 3.09

Gold 4.90 4.33

Iron 4.79 3.92

Nickel 5.02 3.68Molybdenum 4.15 3.22

Primary avalanche: Electrons

= dxs

effeN

eN

0

exp01

Ions =1

0exp01 dx

s

effeNN

Secondary avalanche: Electrons

= 1

0

exp02

dxs

effeN

eN

Al Cu Fe = Reaction coefficientN2 0.1 0.065 0.06

Air 0.035 0.025 0.02


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Ignition Condition

01

0

exp0

e

Ns

dxeffe

N >

i.e.

( ) =>s kdxeff n0

4...5,21

1

with the guide values for

for homogeneous field eff* s > k

The reaction coefficient takes into account the following effects:

Releasing of electrons by positive ions,

Releasing of electrons by the photo-effect,

Releasing of electrons by neutral atoms,

Ion emission of the anode,

Field emission.

Documents

HVE script_v01.pdf