Dielectric Phenomena in High Voltage Engineering (1915) — part 1 of 12

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DIELECTRIC PHENOMENA

IN

HIGH VOLTAGE ENGINEERING

McGraw-Hill BoDkCompaipr

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DIELECTRIC PHENOMENA

IN

HIGH VOLTAGE ENGINEERING •

cf\X.

BY

F. W. PEEK, Jr. It

First Edition

McGRAW-HILL BOOK COMPANY, Inc.

239 WEST 39TH STREET, NEW YORK

6 BOUVERIE STREET, LONDON, E. C.

1915

Copyright, 1915, by the McGraw-Hill Book Company, Inc.

• • • «

. - - • ♦

• fc- k «

XBB MAPIiS FHSSS TOH

PREFACE

It is the object of the author to give in this book the properties of gaseous, liquid and solid insulations, and methods of utilizing these properties to the best advantage in the problems of high- voltage engineering. Such problems require a knowledge, not only of the laws and mechanism of breakdown of dielectrics as determined by experiment, but also a simple working knowledge of the dielectric circuit.

Methods that have proved useful in designing apparatus, transmission lines, insulators, bushings, etc., are discussed and illustrated by practical application. In addition, such subjects as the manner of making extensive engineering investigations and of reducing data, the measurement of high voltages, the effects of impulse and high-frequency voltages, methods of draw- ing dielectric fields, outline of modern theory, various dielectric phenomena, etc., are considered. In all cases where laws and discussions of dielectric phenomena are given, it has been thought best to accompany these with experimental data.

Much original work is given, as well as reference to other in- vestigations. The author's extensive research was made possible by facilities afforded by the Consulting Engineering Department of the General Electric Company, for which acknowledgment is made. Thanks are due Mr. H. K. Humphrey, and others, who have greatly assisted in the experiments and calculations.

F. W. P., Jb.

Schenectady, N. Y., April, 1916.

313777

CONTENTS

Page

Preface v

Dielectric Units xi

Table of Symbols xiii

CHAPTER I

Introduction 1

General discussion of energy transfer — Experimental plots of di- electric and magnetic fields — Analogy between magnetic and dielectric fields — Analogy with Hooke s Law.

CHAPTER II

The Dielectric Field and Dielectric Circuit 8

(Mathematical Consideration) General treatment of the dielectric field and dielectric circuit with discussion of principles used — Parallel planes — Field between; per- mittance, etc. — Concentric cylinders — Permittance or capacity; flux density and gradient — Parallel wires — Principles used in super- position of fields; determination of resultant fields; equation of equipotential surfaces, lines of force and flux density; permittance; gradient and equigradient surfaces — Concentric spheres — Spheres — Two small equal spheres, field of, and permittance; two large equal spheres, gradient, permittance — Conditions for spark-over ^^'^tuid local breakdown or corona — Collected formulae for common electrodes — Combinations of dielectrics of different permittivities — Dielectric flux refraction — Dielectric in series — Dielectric in multiple — Flux control — Imperfect electric elastivity or absorption in dielectrics; dielectric hysteresis.

CHAPTER III

Visual Corona 38

Qeneral Summary and Discussion — Appearance — Chemical action — A. C. and D. C. spacing and size of conductor — ^Laws of visual corona formation — Theorj' of corona — Electron theory — Air films at small spacings — Aii density — Measuring voltage by corona — Conductor material, cables, oil and water on the conductors, humidity. Ionization — Wave shape, current in wire. Experimental Study and Method of Reducing Experimental Data — Tests showing the effects of size and spacing of conductors — Air density — Temperature — Barometric pressure — Strength of air films — Effect of frequency — Conductor material, oil, water, dirt, humidity — Ionization — Current in wire — Stranded conductors — Split conductors.

Photographic and Stroboscopic Study — Positive and negative corona — Corona at different voltages — Thickness of corona — Oscillograms of corona current, etc.

vii

viii • CONTENTS

CHAPTER IV

Page

Spark-over 79

Definition — Condition for spark-over or corona — Spark-over be- tween parallel wires, wet and dry — Measurements of and method of calculating — Wires in a cylinder — Needle gap — Sphere gap — Effect of barometric pressure, temperature, humidity, moisture and rain on spark-over; measurement of voltage by spheres; calculation of curves; precautions in testing — Rupturing energy and dielectric spark lag — ^Law of spark-over, effect of high frequency, oscillatoiy, and impulse voltages on spark-over, and method of measuring such voltages — Insulators and bushings — Spark-over of; effect of alti- tude, etc.

CHAPTER V

Corona Loss 117

Method of making a large engineering investigation — Method of reducing data — The quadratic law — Loss on very small conductors — Effect of frequency, size of conductor and spacing; conductor material and surfaces; air density and humidity — The disruptive critical voltage — ^Loss near the disruptive critical voltage; the probability law — ^Loss during storm — ^Loss at very high frequency,

CHAPTER VI

Corona and Spark-over in Oil and Liquid Insulations 153

Liquids used for insulating — Physical characteristics of transformer oil — Spark-over with different electrodes; effect of moisture; tem- perature — Corona in oil — ^Law of spark-over and corona in oil — Spark-over of wires, plates and cylinders — Resistivity of oil — Dis- ruptive energy — Oil films — Transient voltages — Barriers — Com- parison of high frequency — 60 — and impulse arc over.

CHAPTER VII

Solid Insulation 166

Solids used for insulation — Dielectric loss — Insulation resistance and dielectric strength — Rupturing gradient — Methods of testing — ^Law of strength vs. thickness — Solid vs. laminated insulations — Effect of area of electrodes — Impulse voltages and high frequency — Cumulative effect of over-voltages of steep wave front — ^Law of strength vs. time of application — Permittivity of insulating materials — Energy loss in insulations at high and low frequency — Operating temperatures of insulation — Surface leakage — Solid insulating barriers in oil — Impregnation — Mechanical — Direct current — Complete data on permittivity, dielectric strength with time, thickness, etc.

CHAPTER VIII

The Electron Theory 192

Review of and example of practical application.

CONTENTS IX

CHAPTER IX

Page

Practical Corona Calculation for Transmission Lines 199

Corona and summary of various factors affecting it — Practical corona formulae and their application with problems to illustrate — Safe and economical voltages — Methods of increasing size of con- ductors — Conductors not symmetrically spaced — Voltage change along line — Agreement of calculated losses and measured losses on commercial transmission lines — The corona limit of high-voltage transmission, with working tables and curves.

CHAPTER X

Practical Considerations in the Design of Apparatus where Solid, Liquid and Gaseous Insulations Enter in Combination . 213 Breakdown caused by addition of strqnger insulation — Corona on generator coils — Corona in entrance bushings — Graded cable — Transformer bushing, oil-filled bushings, condenser type bushing — Dielectric field control by metal guard rings, shields, etc. — High frequency — Dielectric fields — Methods of plotting, lines of force, equipotential surfaces, equigradient surfaces — Dielectric fields in three dimensions, experimental determination of dielectric fields — Effect of ground on the dielectric field between wires — Three- phase dielectric fields with flat and triangular spacing of con- ductors — Occluded air in insulations — Examples of calculations of spark-over between wet wires, of sphere curves, of breakdown of insulation for transient voltages, of strength of porcelain, of energy loss in insulation, etc.

CHAPTER XI

Complete Data Appendix 238

Measured data on corona loss.

Index 257

DIELECTRIC UNITS

Electromotive force, volts Gradient

Permittance or capacitance or capacity.

Permyitivity or specific capacity

relative k (k

absolute (air) K =

e 9 *

C =

volts.

• volts/cm.

^^ = 8.84 *- 10-" farads.

X X

Elastance

Elastivity

Flux, displacement

Flux density, Intensity

Stored energy

Energy density

Permittance or capacity current

Permittance or capacity current

Permittance in series

Elastance in series Permittance in multiple

Elastance in multiple

5 = 7;

D = F

Wo =

= 1 for air)

10» -^^ = 8.84 X 10-»< farad cm. cube.

1 C

Cc = -s coulombs (or lines of force).

kKg flux or displacement per cm.' (unit not used in text).

-s- joules.

~- joules per cm. cube. ^=C^^amps.

ic = 2x/Ce amps, for sine wave.

JL

C

S

c 1

s

Ci Cj C|

1 + 1+1 <Si 5i Si

V = velocity of light x= spacing cm. A

3 X 10" cm. per sec. area in sq. cm.

NoTS. — For non-uniform fields «, x, eto. are measured over very small distances and be-

de eome de, dz, eto. Then the gradient at any point is ff"^^* ^^'

XI

TABLE OF SYMBOLS

The following is a list of the principal symbols used. The use given first is the most general one. The meaning is always given in the text for each individual case.

A area in square cm., constant. A'lA's flux foci or flux centers. a distance, constant.

b barometric pressure in cm., constant, distance. C permittance or capacity. Crifs permittance between points fi ft. Cn permittance to neutral. c constant, distance. D dielectric flux density. d distance, constant. e voltage.

e» voltage to neutral. erirt voltage between points ri rt. Cp voltage to point p. 6« visual critical corona voltage. Co disruptive critical corona voltage. Bd disruptive critical corona voltage for small wires. e* spark-over voltage. / frequency. ftfiifo coefficients used in reducing average gradient to maxi- mum — see page 28. F constant (sometimes used for dielectric field intensity), g, G gradient.

g gradient volts per cm. or kilovolts per cm. g gradient volts per mm. for solid insulations. gv visual critical gradient. go disruptive critical gradient. gd disruptive critical gradient for small wires. gm^ maximum gradient — see note below. g, spark gradient. ga gradient at point a. h constant, height. i current amperes. K dielectric constant for air

10» K ^ - - = 8.84 X 10"" farads per cm. cube.

k relative permittivity (A; = 1 for air). L inductance. I length, thickness

• • •

XIll

J I

xiv TABLE OF SYMBOLS

M constant.

m ordinate of center of line of force, m mass. m«, mo irregularity factor of conductor surface. N neutral plane, n number. O center point. P point, p power loss.

q constant.

r radius of wires or cables. R radius of spheres, of outer cylinder.

r resistance. 5, 9 spacing between conductor centers. S* distance between flux foci. S elastance — see page 11.

i temperature, thickness. T time.

i; velocity of light in cm./sec. = 3 X 10".

V velocity.

wi magnetic stored energy. tOe dielectric stored energy. w weight. X^ X cm. spacing between conductor surfaces, thickness, co- ordinate of a point. Xi, x% distance.

y coordinate of point.

z distance from the center of a conductor or an equipo- tential circle to flux foci.

a angle, constant.

/3 constant.

3 relative air density. AS difference of two sums.

c base of natural log. '9 dielectric displacement or dielectric flux.

4> magnetic flux.

^ angle, function.

B angle.

«r elastivity.

2 sum. SS sum of two sums.

w resistance, mm. millimeter, cm. centimeter. 8 approximately equal to.

Note that voltages in measured data are often given to neutral; in such cases the single phase line to line voltages are twice (2), and the three phase (symmetrical) y/z times, these values.

TABLE OF SYMBOLS XV

Pennittances or capacities are also frequently given to neutral because it is a great convenience in making calculations.

The subscript max. is often used to distinguish between the maximum and root mean square or effective. This is done because insulation breakdown generally depends upon the maximum point of the wave. Such voltages may be reduced to effective sine wave by dividing by y/2. Sometimes when the maximum gradient is referred to it means the gradient at the point in the field where the stress is a maximum. These references are made clear in the text for each individual case.

Tests were made on single-phase lines unless otherwise noted.

Views or theories advanced by the author are always accompanied by sufficient experimental data so that the reader may form conclusions independently.

DIELECTRIC PHENOMENA

CHAPTER I INTRODUCTION

It is our work as engineers to devise means of transmitting energy electrically, from one point to another point, and of con- trolling, distributing, and utilizing this energy as useful work. Conductors and insulating materials are necessary. Trans- mission problems are principally problems of high voltage and therefore of dielectrics. In order that energy may flow along a conductor, energy must be stored in the space surrounding,-tbe ^conductor. This energy is stored in two forms, electromagnetic and electrostatic. The electromagnetic energy is evinced by the action of the resulting stresses, for instance, the repulsion be- tween two parallel wires carrying current, the attraction of a suspended piece of iron when brought near the wires, or better yet, if the wires are brought up through a plane of insulating material, and this plane is dusted with iron filings, and gently tapped, the filings will tend to form in eccentric circles about the conductors. These circles picture the magnetic lines of force or magnetic field in both magnitude and direction. This field only exists when current is flowing in the conductors. If now potential is applied between the conductors, but with the far ends open circuited, energy is stored electrostatically. The resulting forces in the dielectric are evinced by an attraction between the conductors; a suspended piece of dielectric in the neighborhood is attracted. If the conductors are brought through an insulating plane as before, and this is dusted with a powdered dielectric, as mica dust, the dust will tend to form in arcs of circles beginning on one conductor and ending on the other conductor. See Fig. 1(a) and (6). The dielectric field is thus made as tangible as the magnetic field. Fig 1(c) is an experimental plot of the magnetic and dielectric fields. Fig. 1 (d) is the mathematical plot. Fig. 1(c) represents the magnetic and

1

» « •

2 ' -. . -• BlELECTUlC PHENOMENA

dielectric fields in the space surrounding two conductors which are carrying energy. The power is a function of the product of these two fields and the angle between them. In comparing Figs. 1(c) and (d) only the general direction and relative density of the fields at different points can be considered. The actual number of lines in Fig. 1(c) has no definite meaning. The Hiftlf>fif.nV. 1i>fiff of f ftrre in Fig. 1 (d) are drawn so that one t^ygnty-

fourth of the total flux is included between any two adjacent lines. Due to the dielectric fields, points in space surrounding the conductors have definite potentials. If points of a given poten- tial are connected together, a cylindrical surface is formed about the conductor; this surface is called an equipotential surface. Thus, in Fig. 1(d), the circles represent equipotential surfaces. As a matter of fact, the intersection of an equipotential surface by a plane at right angles to a conductor coincides with a magnetic line of force. The circles in Fig. 1(d), then, are the plot of the equipotential surfaces and also of the magnetic lines of force. The equipotential s urfaces are drawn so that one-twentieth of the_volta^ is between any two surfaces. For example: If 10,000 volts are placed between the two conductors, one con- ductor is at -f-5000 volts, the other at —6000 volts. The circle ( 00 radius) midway between is at 0. The potentials in space on the different equipotential surfaces, starting at the positive sur- faces, are +5000, +4600, +4000, +3500, +3000, +2600, +2000, +1600, +1000, +500,0, -600, -1000, -1500, -2000, -2500, -3000, -3500, -4000, -4600, -6000. A very thin insulated metal cylinder may be placed around an equipotential surface without disturbing the field. If this conducting sheet is con- nected to a source of potential equal to the potential of the surface which it surrounds, the field is still undisturbed. The original conductor may now be removed without disturbing the outer field.

The dielectric lines of force and the equipotential surfaces are at right angles at the points of intersection. The dielectric lines always leave the conductor surfaces at right angles. The equipotential circles have their centers on the line passing through the conductor centers, the dielectric force circles have their centers on the neutral line.

Energy does not flow unless these two fields exist together — for instance, if the dielectric field exists alone it is aptly spoken of as "static."

5*

'4

INTRODUCTION 3

The energy stored in the dielectric field is

2

where e is the voltage and C a constant of the circuit called the permittance (capacity) and the energy stored in the magnetic

2

field is where i is the current and L is a constant of the circuit called the inductance.

The energy stored in the dielectric circuit is thus greater for high voltage, and in the magnetic circuit for high currents.

When energy was first transmitted, low voltages and high cur- rents were used. The magnetic circuit and magnetic field in this way became known to engineers, but as little trouble was had with insulation, the dielectric field was therefore not generally considered. If insulation broke down, its thickness was in- creased without regard to the dielectric circuit.

A magnetic circuit is not built in which the magnetic lines are overcrowded in one place and undercrowded* in another place — in other words, badly out of balance. Since voltages have become high it is of great importance to properly proportion the dielectric circuit* Although an unbalanced magnetic field may mean energy loss, an unbalanced or too highly saturated dielec- tric field will mean broken down insulation.

The dielectric and magnetic fields may be treated in a very similar way.^ For instance, to establish a magnetic field a mag- neto-motive force is necessary; to establish a dielectric field an electro-motive force is necessary. If in a magnetic circuit the same flux passes through varying cross sections, the magneto- motive force will not divide up equally between equal lengths of the circuit. Where the lines are crowded together the magneto- motive force per unit length of magnetic circuit will be larger than where the lines are not crowded together. The magneto-motive force per unit length of magnetic circuit is called magnetizing force. Likewise for the dielectric circuit where the dielectric flux density is high a greater part of the electro-motive force per unit length of circuit is required than at parts where the flux density is low. Electro-motive force or voltage per unit length

»See Karapetoflf, "The Magnetic Circuit/' and "The Electric Circuit." Steinmetz, " Electric Dischargee, Waves and Impulses."

4 DIELECTRIC PHENOMENA

of dielectric circuit is called electrifying force, or voltage gradient. If iron or material of high permeability is placed in a magnetic circuit the flux is increased for a given magneto-motive force. If there is an air gap in the circuit the magnetizing force is much greater in the air than in the iron. If a material of high specific capacity or permittivity, as glass, is placed in the dielectric circuit, the dielectric flux is increased. If there is a gap of low permit- tivity, as air, in the circuit, the gradient is much greater in the air than in the glass. The electric circuit is also analogous, as will appear later.

A given insulation breaks down at any point when the dielec- tric flux density at that point exceeds a given value. . It is thus important to have uniform density. The flux, ^, depends upon the voltage, the permittivity, or specific capacity of the insulation, and the spacing and shape of the terminal. That is.

The flux density D, at any point, is proportional to the gradient flf, or volts per centimeter at that point, and to the permittivity of the dielectric. Thus,

also D = -J

As the density is proportional to the gradient, insulations wUl, therefore, also rupture when the gradient exceeds a given value; hence if the gradient is measured at the point of rupture it is a measure of the strength of the insulation. The strength of in* sulationis generally expressed in terms of the gradient rather than flux density.

By analogy with Hooke's Law the gradient may be thought of as a force or stress, and the flux density as a resulting electrical strain or displacement. Permittivity, k, then, is a measure of the electrical elasticity of the material. Energy is stored in the dielectric with increasing force or voltage and given back with decreasing voltage. Rupture occurs when the unit force or gradient exceeds the elastic limit. Of course, this must not be thought of as a mechanical displacement. In fact, the actual mechanism of displacement is not known.

When two insulators of different permittivities are placed in series with the same flux passing through them, the one with the

INTRODUCTION 6

lower permittivity or less electrical elasticity must take up most of the voltage, that is, the "elastic" one may be thought of as "stretching" electrically and putting the stress on the electrically stiff one. The dielectric circuit is also analogous to the electric circuit when the flux is thought of in place of the current — and the permittance as conductance. The reciprocal of the permit- tance is sometimes called the elastance {S) and corresponds to ''resistance" to the dielectric flux. It is convenient when permittances are connected in series, as the total elastance is the sum of the elastances. When permittances are connected in multiple, the total permittance is the direct sum of the permit- tances. Take two metal plates in air and apply potential between them until the flux density is almost sufficient to cause rupture. Now place a thick sheet of glass between the plates; the permittance and therefore the total flux is increased. This increases the stress on the air, which breaks down or glows. The glass does not break down. »Thus by the addition of insu- lation the air has actually been broken down. This takes place daily in practice in bushing, etc., and the glow is called static.

It is especially important in designing leads and insulators im- mersed in air to avoid overstress on the air.

It can be seen that a statement of volts and thickness does not determine the stress on the insulation. The stress on insulation does not depend altogether upon the voltage, but also upon the shape of the electrodes; as, for instance, for needle points the flux density at the point must be very great at fairly low voltages, while for large spheres a very high voltage is required to pro- duce high flux density. For this reason 200 kv. will strike 55 cm, between needle points, while it will strike only about 17 cm. between 12.5-cm. spheres. From the above it can be seen that it is much more important to design the dielectric circuit for proper flux distribution than the magnetic circuit. Local overflux density in the magnetic circuit may cause losses, but local overflux density in the dielectric circuit may cause rupture of the insulation.

Consider now the two conductors of a transmission line with voltage between them. The total dielectric flux begins on one conductor and ends on the other conductor (see Fig. 1(d)). The flux is dense at the conductor surface and less so at a distance from the conductor. Hence the voltage gradient is greatest at the surface, where the dielectric cross-section is a minimum,

6

DIELECTRIC PHENOMENA

Xi^l'^l Om

and therefore the ''flux resistance" or elastance is greatest and breakdown must first occur there. For the particular case shown in Fig. 2 one-third of the voltage is taken up by the space 12 cm. from each conductor, although the total space is 100 cm. The gradient is greatest at the wire surface. That is, if across a small distance, Xi, the voltage is measured near the wire surface and then again across the same space Xs some distance from the wire, it is found that the voltage is much higher across the small

space Xi near the wire surface than across the one farther out. In ac- tually measuring the gradient, or rather cal- culating it, X is taken very small or dx. The voltage across dx is de. The purely mathemat- ical expression for the gradient at the surface of parallel wires is:

de e

+60i

dx

2r log. J

Fig. 2. — Voltage in space between two par- allel wires.

If the conductors are close together a spark jumps across when the voltage is high enough to produce overflux density at the conductor surface; or corona and spark-over are simultaneous. If far apart, corona forms around the conductor surface and sparkover takes place at some higher voltage.

As voltages or electro-motive forces become higher the proper shaping and spacing of the conductors to prevent dielectric flux concentration becomes of more importance. The dielectric field must now be considered in the design of apparatus as the magnetic field has been considered. Certain phenomena always exist which go unnoticed because of their feeble effect, but which when conditions are changed, usually in a way to cause a greater energy density, become the controlling features. This is so with the dielectric circuit. The problem first made itself apparent to

INTRODUCTION 7

engineers in the transmission line, which will be taken to illus- trate this. When voltages were below about 60,000 the conduc- tors used had sufficient radius or circumference so that the surface flux density or gradient was not sufficient to cause breakdown. As voltages became higher the sizes of conductors remained about the same, and therefore the flux density or gradient became greater. The air broke down and caused the so-called corona and resulting loss.

As high voltage engineering problems will be, to a great extent, problems of the dielectric circuit, this will be discussed in the next chapter and calculations made for a few common forms of elec- trodes. The determination of the dielectric flux density, etc., is purely a mathematical problem. Exact calculations are diffi- cult and often impossible except for simple forms. Exact calcu- lations are not necessary in practical design work, but the general principles must be kept in mind.

CHAPTER II THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT

If two conductors placed in a dielectric, as, for instance, the two parallel wires of a transmission line in air, are connected together at one end by an electric motor, or resistance^and poten- tial is applied across the other end from an a.c. generator or other source of power supply (see Fig. 3), energy transfers take place. The motor at the far end turns and part of the energy is thus used as useful work — part appears as heat in the motor.

T T

Fig. 3. — Transmiasion line carrying energy.

As a function of the current in the transmission circuit and a constant called the resistance, energy is absorbed. This energy appears as heat in the conductors; it is proportional to the product of the square of the current and the resistance, and is commonly known as the /V loss. Hence, as it is not returned to the circuit or transferred into useful work, but is dissipated as heat, it is analogous to a friction loss. During the transmission, energy is stored in the space surrounding the conductors in the electric field in two different forms — magnetic and dielectric.

Energy is stored in the magnetic field, where it is proportional

8

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT

to the square of the current and to a constant of the circuit called the inductance:

PL

Magnetic energy is stored with increasing current and delivered back to the circuit with decreasing current. The magnetic energy becomes noticeable or large when the currents are large, or in low voltage circuits.

Due to the dielectric field, the energy is

We =

eHJ

e 0-

«'c

!AAA/

This energy is stored with increasing voltage and delivered back with decreasing voltage. A dielectric may thus by analogy be thought of as an electrically elastic material, which is dis- placed by an electric pressure, i.e., voltage. Energy is hence stored in the dielectric with in- creasing voltage or electric pressure, is maximum at the maximum point of the voltage wave and is delivered back to the circuit with decreasing voltage. When the pressure becomes too great the electric ** elastic limit" is exceeded, or the dielectric becomes dis- torted beyond this "elastic limit," and rupture occurs. The dielectric energy becomes of great importance at high voltage, and henpe in the study

of insulations, and it only will be considered here. The electric displacement may be pictured in magnitude and direction by lines of force. The dielectric lines of force for two parallel con- ductors are shown in Fig. 3, the eccentric circles (dotted) are the magnetic lines of force. The magnetic circles are also equipo- tential boundary lines for the dielectric field. The dielectric energy is sometimes said to be due to a charge on the con- ductor. This is often confusing, as the energy is stored not

lAAAA

O"* ]8D<> tn"* MO"* 120<

lima

Fig. 4. — Variation of dielectric and magnetic stored energy with voltage and current.

10

DIELECTRIC PHENOMENA

on the conductor but in the surrounding space, and may be thought of as due to an electric displacement. The nature of this displacement is not known. It can be seen that in order that a transfer of energy may take place, energy must be stored in the space surrounding the conductors in two forms — magnetic and dielectric. Energy thus flows only in a space in which there is a magnetic and a dielectric field. This energy is proportional to the product of the magnetic and dielectric field intensity and the sine of the included angle. If one field exists alone there can be no energy flow. The change of stored energy from magnetic to dielectric and back is shown in Fig. 4. Dielectric Field between Parallel Planes. — In the dielectric field the flux or total displacement is

}{/ = Ce coulombs (or lines of force)

(1)

where e is the applied e.m.f . or voltage, and C is a constant of the circuit, depending upon its dimensions, and is called the capacity.

FiQ. 5. — Dielectric field between parallel planes.

or better, permittance. If C is measured in farads, and e in volts, ^ is expressed in coulombs. Fig. 5 shows the simplest fornf of dielectric circuit. Neglecting the extra displacement at the edges, it is seen that the dielectric lines of force are everywhere parallel and the field is uniform. The dielectric circuit constant, or the permittance, is directly proportional to the area of the cross- section perpendicular to the lines of force, inversely propor- tional to the spacing along the lines of force, and directly propor- tional to the dielectric constant or the permittivity.

For large parallel planes without flux concentration at the edges

A , /10»\

-M

• — jj = y fc iC farads

(2)

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 11

where A is the area in square centimeters,

X is the distance between plates in centimeters,

k is the specific inductive capacity, or better, permittivity,

and V is the velocity of light, 3 X 10^® cm., per second.

The term in brackets is due to units. The fiux density then, or displacement per unit area, as the fiux is uniform, is:

Z> = ^ = -J- = —r—i coulombs per cm.* (3)

To establish this flux or displacement through the distance X an electromotive force e is required. The force per unit length of dielectric circuit or electrifying force is constant in the uniform

field and is then y» The gradient then is

Q = Y volts/cm. The density may thus be written:

which is analogous to Hooke's law in Mechanics,

strain =" k times stress.

The larger k is, the greater the displacement is for a given force g.

Thus k is the coefficient indicating electrical elasticity of the material, or its "conductivity" to the flux. The reciprocal of permittivity is analogous to resistivity and has been termed elastivity (<r). The reciprocal of permittance has been termed elastance {S). The dielectric circuit then becomes analagous to the electric circuit

y. volts tL — -

*flux resistance' S

It is often convenient to consider the dielectric circuit in this way, and to use <r and S, as the total elastance of a number in series is the direct sum. The total permittance of a number in multiple is the direct sum. See two methods, Case 1, Chapter X.

12 DIELECTRIC PHENOMENA

In studying insulations it is important to be able to express their relative strengths. This is naturally generally done in terms of the force or voltage gradient necessary to cause rupture. It may also be done in terms of the flux density at rupture; that is, in coulombs per square centimeter. A given insulation breaks down at any point when the flux density exceeds a certain defi- nite value at that point, or when the gradient exceeds a given definite value. For Fig. 5 where the field is uniform the gradient is

/XT 1 n ^ . C'e p. , coulombs

(7 = 6/X kv. per cm. Z) = ^-=^ = Xkg. ^^ ^^,

K = :Pi = 8.84 X lO"'^

Hence, if the voltage is increased until rupture occurs, and found to be e, the voltage gradient or the flux density at rupture is known; it would seem that this would be a good form of test piece with which to study insulation. This is not usually the case because of the extra displacement at the edges which is dif- ficult to calculate. This may, however, be made very small at small spacings by proper rounding of the edges. Equations for the voltage gradient, permittance, etc., will now be given for a few of the common electrodes. » In general, calculations are made in the same way, except that the field is usually uniform over only very small distances. The total capacity is found by taking the capacities over distances so small that conditions are still unifornL and integrating.

Concentric Cylinders. — Concentric cylinders make a conven- ient arrangement for studying dielectric strength, especially that of air and oil. On account of the symmetrical arrangement, the dielectric circuit is readily calculated. For testing, the extra displacement at the ends is eliminated by belling (see Fig. 6).

Permittance or Capacity. — In this case the lines of force are radial. The equipotential surfaces are concentric cylinders. The total flux per centimeter length of cylinder is

^ = Ce

The permittance may be thought of as made up of a number of permittances in series between r and R, each permittance being between two equipotential surfaces dx centimeters apart.

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 13

For a number of permittances in series

1_ C

V.

Lines of /"^ Force

Fig. 6. — Concentric cylinders.

The permittance of the condenser of thickness dx, over which the field is uniform (Fig. 6), per centimeter length of cylinder is

dz 4irv^ dx ^TTV^ 2v^ dx

C =

2v^

felO^ — 5.55 fclO^^' farads per centimeter 2v^ log, R/r ~ logo R/r length of cylinder.

(5)

Gradient and Flux Density. — ^The flux density is greatest at the conductor surface and, hence, the gradient must be greatest there. The flux density at any point x measured from the center is

ek

Z) = ^ = ^ = _

A A A2v* log, R/r

10»

A » 2ncX

..D =

ek

4irxv* log. R/r

10» = 0,884

ek in-i* coulombs X log, R/r per cm.*

4irt;»

.'.g ^ D —j— 10^* = — 5 ST" kv. per cm.

k X log, R/r *^

(6)