Dielectric Phenomena in High Voltage Engineering (1915) — part 2 of 12

g is maximum at the surface of the inner cylinder or where x — r

Qma» ""

r log. R/r

kv./cm.

(6a)

14

DIELECTRIC PHENOMENA

Parallel Wires. — Parallel wires, one of the most common prac- tical cases, will be considered in detail, in order to illustrate the general method of calculating the dielectric circuit and to show that the expressions for the permittance, fiux density, gradient, etc., are quite simple and can be written with the aid of ordinary geometry and calculus.

Equipotentidl Surfaces, Lines of Force, and Flux Density, — All of the equipotential surfaces which arise in this case are cylin- drical. Therefore, only their intersections with a normal plane need be considered, and the problem may be dealt with as affecting only the plane.

Fee. 7. — ^Lines of force between parallel wires, by superposition of inde- pendent radial fields.

The following principles will be used:

(1) The resultant field in the space between two conductors is the superposition of the two independent fields. The re- sultant field due to any number of fields may be found by combining in pairs.

Fluxes may be added directly.

(2) The potential at any point is the algebraic sum of the potentials due to the independent fields through that point. In the same way, the potential difference between two points

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 15

Fig. 8(a). — ^Lines of force and equipotential surfaces between parallel wires.

Fig. 8(b). — ^Lines of force, equipotential surfaces, and equigradient surfaces

between parallel wires.

16 DIELECTRIC PHENOMENA

is the algebraic sum of the potential differences due to the independent fields.

(3) The density or gradient at a point is the vector sum of the densities or gradients due to the independent fields.

When the conductors are infinitely small, the dielectric field may be considered as that resulting from the superposition of the two uniform radiaP fields from the conductors to an infinite cylinder.

The resultant equipotential surfaces are then cylinders whose right sections are eccentric circles which enclose the wires, and whose centers all lie in the line connecting them; and the lines of force are arcs of circles intersecting in the conductors. The independent radial fields about each, conductor are shown in Fig. 7, and the field resulting by superposition is shown in Fig. 8.

Fig. 9.

The equipotential surfaces will first be considered. It has been shown above that the permittance C between two equipotential cylinders of radii R and r is

C = i :prT farads per cm. (5)

log* R/r

This is also the permittance between any two points on these surfaces; therefore, the voltage between points distant ri and r^ cm. from the conductor center is

Considering now the field resulting from superposition: In

• By uniform radial field is meant one in which equal central angles always include equal fluxes.

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 17

Fig. 9, the potential diflference between Hi and P, due to the radial flux ^ from A\j is

_ ^ loge Xi/ a

^^* ^ 2irkK

Similarly, the potential diflference between H2 and P, due to the radial flux — ^ from A'a, is^

_ — ^ loge X2/ b

The total potential difference between H% and P due to the two radial fields, and hence due to the resultant field, is the algebraic sum of the potential difference produced by each field separately:

But if Hi, and P are points on the same equipotential surface the potential difference between them is zero, or

Cp = -

Therefore

2nrkK\

ivfea

J

a

"*^'6/

log<

Xi

t

a

= loge

Xi

b

Xi

Xi

%

a

b

Xi Xi

a " 6 ""

const.

if Hi is fixed — that is, for any one equipotential surface.

The equipotential surfaces are cylinders whose sections are circles which surround the infinitely small conductors. These circles have centers on line AiAi, but are not concentric with the conductors. This may be shown as follows:

Assume Cartesian axes through A '3, the X axis containing A'l. Let the coordinates of P be x, y,

xi = VjA'iA'i - xy + y^ = V(a + b-xy + y^ Xi = Vx^ + y^

x, - 6 "" \

• b - xy + y'

Xi b \ x^ + y^

^ The signs must always be properly placed. It is convenient to give ^ the sign of the point displacement under consideration. The distances from the displacement points to the points between which potential is sought should always be put in the same order in the log, as, Xi/a, Xi/b, etc. See problems, Case 11 and Case 12, Chap. X. 2

18

DIELECTRIC PHENOMENA

ax + ay = fc«a« + b* + bx + 2ab* - 2abx - 2bx + by (o» - b*)x* + 26*(o + b)x + (o» - b*)y* = b*{a + b)'

b* . . bHa + b)* . b*

, . 2fe«

" '^a-b^'^Xa-b)^

• y.^^J^±Sl +

, . _^^» . » &( a + ft) , (a; + :: i.) + y = — _ r^ +

o* - 6» ' (o - 6)» 6«

a — 6'

a — b

(o - 6)» \ ^ a - 6/ ^y - {a -by ~ (o - 6)'

(^ + J^) +^ =(J^r6)l (7)

This is the equation of a circle whose center has the coordinates

and whose radius is r- The circle is thus found for

o — 6

o — 6

any given a and h.

The equipotential circle through any point P (xp, j;p) is found as follows :

a = 6 =

a

o + 6

S' =

xi

a + 6 Substituting for a and b in (7) :

&' =

a:i + xj

X2

ail + iCa

S'

&'

a: +

(

\Xi + x»/

*r

r.-s'*

X,'

(Xl + Xt)* (Ji + x»)*

/S'Va;i - xj^

S'*

/ Xi - Xii \ »

Vxi + x»/

r + xi» - x,V ■•" ^ ~ (xi* - x,»)» xi« = (S' - xp)» + yi.» xj* = x/>* + yp*

{ {&' - Xf)* + yp » } (xp+« yf«)y « (S'« - 2xp)»

/ . xp « + yp» y _ {(g'-xp)« + yp«}(xp« + yp*)

TAe re^iiZtori^ ttn6« of force are arcs of circles with centers on line n and passing through the points A'l and A, This is shown as follows: Consider Fig. 10. The flux included in PA'i A't per

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 19

centimeter length of cylinder due to A'l is -^, That in PA'%

A'\ due to A '2 is -^.

The total flux between P and A\A'% is the sum of these,

The restriction that lines of force cannot cross implies that the flux between any two is constant; hence if P move along a flux line,

^p = jr- (ai + ai) = const.

from which

ai + a2 = const. Hence: a = T -- (ai + a2) = const.

i-Af

Note. — The equation of the line of force may also be found by writing the expression for Fia. 10.

the flux densities at points and imposing the condition that the component normal to the line of force is zero.

This condition defines a circular arc passing through A'l and A'i. Choosing as before the point A^ as the origin of Cartesian coordinates, the equation of these circles is:

(f -)' + (» -m)' = m«+(|) (8)

where m is the ordinate of the center of any particular circle. The equation of the line of force through (x, y) is found as follows: Call the center of circle (lines of force)

A2O = PO = radius of circle

r«= (PO)2 = (|-a:)' + (2/-m) .-.(f -:r)' + (2/-m)« = m»+ ( Through any point (xpyp).

w

2/

(8)

20 DIELECTRIC PHENOMENA

Then (f - ^^)" + ^y^ - ^)' = (f ) + ^*

22/p Substituting this value of m in (8):

The slope of the equipotential surface at {xp, yp) is found from (7a).

Evaluate y in terms of x, differentiate, and put x = Xp.

Note. — Take z always +. m— when below x axis.

dy_ ^ _ S'xp - xp^ + yp^ dXsM yp{S' - 2xp)

The slope of the line of force at (xp, yp) is found in the same way from (8a)

dy_ ^ yp(S' - 2xp)

dxif * S'xp — xp^ + yp^

It will at once be noticed that

dy dx_

dxn dyif

which shows that the line of force at any point is perpendicular to the equipotential surface at the same point.

The flux density J D, at any point in the resultant field is the vector sum of the flux densities due to i4'i and A't separately. At P (Fig. 10) the flux density due to A' i is

2kXi and due to A\ is

directed as indicated.

Note. — Subscript es refers to equipotential surface. Subscript If refers to line of force.

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 21

The triangles whose sides are Xi, X2, S' and D%, Di, D may be shown to be similar, having one angle (a) equal, and the including sides proportional.

D^i = DiXi = ^

Then

D

sv

Xi Xt ^X\ ZwXiXt

(9)

The preceding, covering infinitely small wires, is not directly applicable to the ordinary case of large parallel wires. Green's theorem, however, states that if any equipotential surface be kept at its original potential, the fiux within it may be removed without any change in the external field. In Fig. 11 the circles

Fig. 11.

represent equipotential cylinders, surrounding fiux centers A\ and A'%. These cylinders may be maintained at their original potential. The interior may be filled with a conductor. This gives parallel conductors of radius r and spacing between centers 8. The external field has not been changed, and the preceding discussion still applies. Ai and A% must be located from A'l and A' 2, since r and S are the quantities given in any actual case. This is easily done:

a ^ S — r " z b = r -- z

ft« r* - 2rz + z^ r« - 2rz + z^

z =

a — 6

zS = r^ + z^

«* - S« + r« =

S—r—z—r+z

& -2r

g 8 _

22 DIELECTRIC PHENOMENA

Since obviously z cannot be greater than 8/2^ the negative sign is taken for the radical.

a = jS — r — _ 2S - 2r - S + VS« - 4r»

^ /S - 2r + Vg» - 4r»

2 6 = r —

^ 2r - iS H- Vgg- 4r»

2

a ^ S -2r+ Vg^ - 4r» 6 2r - S + ViS2 - 4r«

(10)

^ g - 2r H- V^S?' - 4r» 2r - /S - Vg^ - 4H 2r - S + -yJS^ - 4r2 2r - S - VS^'^^lr*

^ { - 2iS« - 4rg + 2(/S - 2r) Vg^ - 4r»

4r« - 4rS + 4r*

_ (2r - S) (/S + VS« - 4r») _ S

2r(2r - S)

"2r + Vy ~

Permittance or Capcunty. — In Fig. 11 let n be a neutral plane. Represent by en the potential between circle H2 and n (or Hi and n) and by C» the corresponding permittance to neutral per centimeter length of wires. Due to A'l.

^lOge "^^

S72

Due to A' 2.

• ^log*

'S72

Cn2 =

2irifcii:

^ log.^^

27rkK ^ Zi

^ _ ^ _ 2TkK _ 2TkK

v/n — ~~ — ~~ — ~

log. |i log. I

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 23

Substituting from (10)

2'KkK

C« =

1 « (11)

log.^

2KkK

5.66iklO-" , ,

farads per cm.

'-[I+V(l)'-']

If hyperbolic tables are available, a more convenient form is

° cosh-. I (11«)

2r 5.55A;10-"

cosh-' I

farads per cm.

If S/r be large, then^(~j — 1 is nearly equal to h"' ^^^

approximately

Cn = T TTT farads per cm. (116)

log. S/r ^ ^ '

The result corresponds exactly to the form (page 22} which

would have resulted had the wires been considered very small at

the start, or

^ ^ 27rkK

" log, a/b

where o = S, 6 = r

Gradient and Flux Density. — The flux density at any point on the line joining the centers of the conductors and distant x from the inner surface of one of them is

D = ^'^ X2 = 6 + X ^Qj

2irXiX2 Xi = a — X

(o + b)4'

25r(a6 + (o - b)x - X*) „ |4rS - 8r« , /2S - 4r\ ^ ^1

24 DIELECTRIC PHENOMENA

2ir{(r + x) {S - 2r) - x^\ Since ^ = Cn^n

2irfc/e'e»

D =

{2T(r + x)(<S - 2r)-a;»}

'°^[l+V(|)-i]

kKeWS* - 4r»

{(r + X) (S - 2r) - **} log. [^ + V(|)*-l]

0.885fee«/.S»- 4r »10-" coulombs

<S . //.<j\» 1 per cm.*

{ (r+x) (S - 2r) - x*} log.[|; + -y/d.) * _ i] The gradient along the line of centers is kK

S = r^ (12)

e.Vg« - 4r* kv. per

S . //,<?\a 1 cm.

{(r + X) (S - 2r) - x»} log.[|: + -y/(|)' _ i] The gradient is greatest at the conductor surface (x = 0} and is

Qmax —

<S - 2r)log.[| + V(|)'-l] <'2«)

kv. per cm.

^ - ') '-[I + Vl^- ']

W||^ .o*.[| + >/l^

kv. per cm.

S/2r + Where hyperbolic tables are available a more convenient form is:

Qmax = — , ^ ** kv. per cm. (126)

1-1

cosh"' :r-

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 25

If S/r is large Qn^az = ^ logjg/y. (12c)

As before, this equation would have resulted bad tbe conductors been considered small in the first place. The method of drawing the lines of force, etc., is illustrated in Chapter X.

Spheres. — In studying insulation it is sometimes convenient to use spherical electrodes. The potential between concerUric spheres of radii R and r may be found as for concentric cylinders on page 13. The equipotential surfaces are spheres. Between two surfaces at distance x, dz centimeters apart the permittance is :

dC = ^kK--^kK ax ax

i = 1 f*? ^ 1 ri _ J. "1

C 47rkKJx^ ArkKlr r\

Rt

C = 4irkK

^rR = 7^ =

R-r

C 4^kK Rr

The potential difference due to ^ between two points distant ri and rs from the center of the sphere is

= ^ ^a ~ ^1

The equipotential surfaces between two point electrodes or very small equal spheres may be found as follows, using Fig. 11 : The difference of potential between Ht and P due to A'\ is

^^' 4tirkK\ xia J due to Ai IS cp, = —

4^kK xj)

If H2 and P are on the same equipotential surface

Xi — a X2 — 6

XiG xjb

X1X2 ob

Xj — Xi 6 — a = constant

Xi X2

is the equation for the equipotential surface

=

= constant

26 DIELECTRIC PHENOMENA

The fraction of the total flux toward P through the cone with apex at A\ and half angle a due to A' is^

^1 = I (1 - cos ai)

Through the cone with apex on A' 2 and half angle a^ due to A'%, it is

. ^2 = ^ (1 - cos a^)

^p = ^1 + ^2

If P follows a line of force yf/p must be constant because lines of force cannot cross.

.'. cos a\ + cos at = constant

is the eqva,ti(m of the line force.

The equations for the gradients, etc., of two large spheres of equal radii are given below: Spheres of Equal Size in Air (Non-grounded) :*

g = Y / kv. per cm. (13o)

where

g = gradient at surface of sphere in line joining centers. e = volts between spheres.

X = distance between nearest surfaces in centimeters. / = a function of X/R where R is the radius of either sphere.

Spheres of Equal Size in Air (One Sphere Grounded) :

g = ^/i ^^' P®^ ^™-

9 ^ 'y h kv. per cm. (136^

where the letters have the meaning noted above, /i being a differ- ent function of X/R. For the case of one sphere grounded, the shanks, connecting leads, ground, etc., have a much greater effect than when both are non-grounded. For this reason the theo- retical values of /i do not check closely the experimental results

^ The area of a spherical surface is 4r/2'. The area of a spherical sector with half angle a is 2iri2'(l — cos a).

Russel, Phil. Mag., Vol. XI, 1906.

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 27

and must not be used. Elxperimental values for the grounded case are given as /o in the accompanying table. / and/i may be calculated by the following simple formula:*

/ =

X/R + 1 + V (X/R + 1)« + 8

A = 1/2

X/R + v(wT-4! Sf;'„':«^,"^

V7 t v^

Calculated

Measured'

X/R

/ Non-KTOunded

h

Grounded

(Not for use)

/o Grounded

0.1

1.034

1.060

1.03

0.5

1.177

1.282

.1.18

1.0

1.366

1.617

1.41

2.0

1.781

2.339

1.97

3.0

2.225

3.252

2.59

4.0

2.686

4.201

3.21

6.0

3.640

6.143

10.0

5.600

10.091

15.0

8.080

15.086

20.0

10.680

20.081

The gradient at any point in the line joining the centers of the spheres and distance y from the mid-point of this line is

gy

-ii

2x« (x«(/ + 1) + VCf - 1)) L§

[x^{f + 1) - 42/2(/ - 1)]« / ^ ^^^^

If two equal spheres are never separated a greater distance than twice their radii , corona can never form, but the first evidence of overstress is spark-over. If the separation is greater than 2R, corona forms and it is then necessary to further increase the -voltage to cause spark-over. The condition for corona or spark- over will now be given. A wire in a cylinder will be taken, as the calculations are simpler and best illustrate the condition.

Condition for Spark-over and for Local Breakdown, or Corona. — ^Por a wire in a cylinder the maximum gradient, and thus where breakdown will first occur, is at the wire surface,

^ =

e

(6a)

r log* R/r

1 Dean, Physical Review, De6., 1912, April, 1913.

Dean, G. E. Review, March, 1913. ' Distance of ipnounded sphere to ground in these tests is 4 to 5 diametera

28

DIELECTRIC PHENOMENA

When e is of such value that g at the wire surface just exceeds the breakdown strength of air, the air at that point becomes con- ducting, or corona forms, thus in effect, increasing the size of the conductor. If this increase lowers the gradient, the breakdown will be local, and we say corona is on the wire. If the ratio R/r is such that the increase in size of the conductor by the conduct- ing air increases the gradient, the broken down area will continue to enlarge, or spark-over will occur. The condition for corona or

spark-over may be found thus:

e

r log* R/r

For a constant value of e and R find the value of r to make g a minimum

1/, = X = ?l^-«^ = r (log. B - log. r)

dx 1

^ = - (log. R - log. r - 1)

<^.

ox

-♦—I — I 'I I I I I I

.1 J .9 .4 .5 .6 .7 .8 .9 1.0 'V

R

= for extreme of z

• (log. R - log. r - 1) =

loge R/r = 1

R/r = € therefore

1/g is maximum when R/r = € or

Fig. 12. — Concentric cylin- ders. — Variation of gradient

g is minimum when R/r = e

aer8.-variation oi graaient ^^ ^^^^^ words the stress on the air at surface of inner cylinder as decreases with increasing r until R/r

iSdrcoi^ttl!'"'^"**" "'''- = *• When R/r is equal to or less

than 6 an increase in r increases g. Thus if R/r ^ « and g is brought up to the rupturing point, g progressively increases and spark- over must occur. If R/r > € corona forms and the voltage must be still further increased be- fore spark-over occurs.

This is illustrated graphically in Fig. 12. Note that this is plotted between r/R and g. Thus the minimum occurs when r/R = l/€. It is interesting to note here that with a given R a cable has maximum strength when r is made such that

R/r = €

This is not the practical ratio however, as will appear later.

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 29 COLLECTED FORMULiE FOR THE COMMON ELECTRODES

Concentric Cylinders. :

^ ., ^ 5.55fel0-" , , Capacity C = — j p-; — taraas per cm.

Gradient gs = — i bt kv. per cm.

^ X loge R/r

Max. gradient g = — | pv- kv. per cm.

Corona does not form when

R/r <2.71S

X = distance from center of cylinder in cm. R = cm. radius of outer cylinder, r = cm. radius of inner cylinder.

Parallel Wires. : Capacity or permittance to neutral

6.55fcl0-^» 5.55*10-". , Cn = zt: 1==^=^=^ = ^ — farads per cm.

COSh"^ ;r-

2r

'-[|W^)'-^]

Gradient (at x cm. distance from wire surface on line through centers)

enVS^ - 4r2 kv. per

g^ =s —

[ir + x) (S-2r) -X*} log. [| + ^(|)*- 1 ] '""• Max. gradient (at conductor surface)^

'■W-

Q = — 7. r^ .-^--^ 1 or

(i-)>-[iwi)'-]

(I - >) ""'■-'I

kv. per cm.

Corona does not form when

S/r < 6.86

• See same equations in different form, page 24. See formula 12e, page 25.

30 DIELECTRIC PHENOMENA

S = spacing between conductors centers in cm. R » conductor radius in cm. en = kv. to neutral.

Equal Spheres in Air. — Gradient (non-grounded). (At a centimeters distance from sphere surface on line joining centers.)

E

' 2X^[X\f + i)+4(^-a)V-l)]

kv. per cm.

[Xa+l)-4(^- a) (/-!)]» Max. gradient (non-grounded)

g — yS kv. per cm. Max. gradient (one grounded) 9 ^ "yh kv. per cm.

where / = ^ + 1 + y^{X/R -- \y + 8

4 and /o = see table, page 27.

Corona does not form when X/R < 2.04

R » radius of sphere in cm. X — cm. spacing between nearest surfaces. e « kv. bet. spheres. Capacity or permittance

C = ^^./ ,. 10-" farads. 36 (/ - 1)

Combination of Dielectrics of Different Permittivities. — When several dielectrics of different permittivities are cgmbinedi as is usually the case in practice, it becomes important to so proportion and shape the electrodes and insulations that one dielectric does not overstress another. This is of especial importance in insu- lators where dielectrically weak air of low permittivity is neces- sarily in combination with dielectrically strong insulations of high permittivity.

Dielectric Flux Refraction. — When dielectric flux lines pass from a dielectric of permittivity fci to another of permittivity of ki the lines are bent or refracted. This does not occur, of course,

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 31

when the lines strike the surfaoes vertically, as in a concentric cable, and between parallel plates already considered. The angle of refraction bears a definite relation to the ratio of the two permittivities and can be shown as follows:

Let AB, Fig. 13, be the common surface of the two insulations. The flux ^1 makes an angle 0i with the normal NN to the sur- face. The total flux through the equipotential surface ab is the same as the total flux through the surface cd. The voltage be- tween a and c must be the same as the voltage between b and d, because potential at a = poten- tial at b and potential at c » po- tential at d.

Therefore

^ = Di oft = Dicd In the uniform field

(14)

Fig. 13. — Dielectric flux refraction.

e e ^' = W ^' = ac

where e is

volts,

a to c = volts 6 to d

Therefore

gM = flf2ac

Combining (14) and (15)

Di ab Di cd

Qi bd Qi ac

But

01

Therefore

tan 01 tan 02

Therefore

tan 01 Ki tan 02 K%

(15)

Dielectric in Series. — Take the simple case of two parallel planes with two different dielectrics between them and neglect the flux concentration at the edges (Fig. 14, flux concentration not

32 DIELECTRIC PHENOMENA

shown). As the liaes are normal to the electrodes there is no refraction. The same flux passes from plate to plat&

c,

k,KA lAi = C,e, =. -^ — ei

. „ tJCA

*' ^"^ X. "

kiKA kJCA

i:. — *'■ = ~^ — *^ ari Xt

Xth,

e = ei + ei . x,k.

_ kJCA

Xt

'I

i

Fig. 14. — Dielectrics of difFerent pennittivitiea in » The voltages are therefore divided thus

The gradients are

('+f:)

(-a-D

e

x*(l+

xik,\ (16)

The voltages and gradients may be found in the same way for any number of insulations in series. The expression for the gradi-

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 33

ent at any point x in a combination of n insulations in series is:

'"''■t + g + - ••+!; + • ■■+!) "'^'

Where the distance between the electrodes is greater than the radius of the edges the increase in stress at the edges becomes appreciable.

Concentric Cylinders.-^For concentric cylinders the expression may be found in the same way. The flux lines in this case also are normal at every boundary surface and are hence not refracted. Consider a wire of radius Vi surrounded by n insula- tions whose inside radii are respec- tively ri, rj, . . . rn, and whose per- mittivities are ki, k^, . , . kn.

At the distance x from the center of the. wire, which falls in the dielectric of inside radius r^, outside radius rx+i, and permittivity fc„ the expression for the gradient Qs is found as follows:

1/C = 1/Ci + l/Ct + . . . + 1/Cx +. _ 1 / log. Tj/ri log. r»/r2 , "" 2irK\ ki "^ ki '•' • •

Fig. 15.

. + 1/Cn

I loge(rx+ l)/r^

. +

log. R/ u

kn

)

C-2irX;

(

log,r2/ri . log«r8/r2

fci

b

'+.

, log. rx+,/ rx . , log. R/rn

)

i>.=

A Ce

A

2rK e

2«; / \og.ri/ri . log,r»/ra log. r,.n/ r, log, R/r

\ iGi Ki Kx Kn

■)

9* =

kji.

(17)

xkr / logeft/ri ■ log, ra/ra log, r,+,/ r, . , log,ff/r,

\ *, "^ Jfc, -I-----1- j^^ -1-...-^

g/r. \

n /

DIBLECTRIC PHENOMENA

Fig. 16, — The refraction of lines of force pasBing through a porcelain inaulatot (permittivity asaumed 4).

Flo. 17. — Dielectrics in multiple.

Fig. 19.— Dielectrics i

Fia. !8. — Rod and ring with two di- electrics. Boundary of dielectrics along line of force. (Not drawn to

Bcale.)

Dielectrics in Multiple. — Where dielectrics are combined in multiple, the diviaion between the dielectrics being parallel to the linos of force (Fig. 17), the stress on either is the same as it

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 36

would be were the other not present. Fig. 18 shows a rod in- sulated from a ring by a dielectric so shaped as to make use of this fact. Where the division line is not parallel to the lines of force, some lines must pass through both dielectrics (Fig. 19). For these lines the insulators are in series, and the corresponding precautions are necessary, just as in Fig. 14.

100 K.Y.

^laolated

OK.y,

Fio. 20. — Field not changed by a thin insulated metal plate on an equi-

potential surface.

Slux Control. — In certain electrical apparatus it is very often possible to prevent or reduce dielectric flux concentration by superposing fields upon existing fields. When it is necessary to superpose two fields, as often happens in the course of design, it is important to see that it is properly done. For instance, as a simple case, suppose the two plates. A and B in Fig. 20, are at

B -

(a)

100 K.Y. B~

(b)

60 K.Y.

OK.Y. A-

-100 K.Y.

SO K.Y.

OK.Y.

Fio. 21. — (a) Field not changed if the potential of the plate is the same as that of the surface upon which it rests. (6) Field changed by plate at potential different from tne surface.

potentials of and 100 kv. respectively. A thin insulated metal plate, C, may be placed anywhere between A and B without changing the field as long as it follows an equipotential surface — that is, parallel to A and B, Unless it follows an equipotential surface flux concentration results. If C is insulated and brought to a potential of 50 kv., the field will be disturbed unless C follows the 50 kv. equipotential surface, or, in other words, is midway

36

DIELECTRIC PHENOMENA

between the two plates — otherwise the stress in part of the in- sulation will be greatly increased. See Fig. 21 (a) and (6).

Fig. 21(a) shows the position for no change.

Fig. 21(&) shows very great increase of stress on part of the insulation.

A coil of wires in which the turns are at different potentials may be placed in the field with least disturbance if the^potential of each coil corresponds to the potential of the equipotential surface upon which it rests, as shown approximately in Fig. 22(a).

(a)

5T

o-

6

i

■100 K.V. B- .80 " -80 ••

MO ••

(6)

-100 K.V.

0K.V.

Fio. 22. — Conductors placed in a uniform field.

When insulation is used around small conductors, points, etc., the stress may be very great. This stress may be reduced by superposing a uniform field. For instance, take two small paral- lel wires with voltage e between them, the stress is

Q =

2r log*

S

e

s'

If a uniform field is superposed, as in Fig. 22(b) of gradient ^i the stress on the wires becomes

gt = 2gi

These principles must be used in generator and transformer design, etc., and will be applied in a later chapter.

Imperfect Electric Elasticity. — The electric displacement has been shown to follow Hooke's law by analogy, that is.

4^ = Ce

10»\

or

strain = constant X stress.

The dielectric has so far been assumed to be perfectly elastic. For a perfectly electrically elastic material C or fc must be con-

THE DIELECTRIC FIELD AND DIELECTRIC CIRCUIT 37

stant and independent of the time that the stress 6 or g is applied. This appears to be the ease for perfectly homogeneous dielectrics as air, various gases, pure oil, etc., but is not so for non- homogeneous dielectrics. As an example of imperfectly elastic dielectrics — ^take a cable; when potentisJ is applied between core and case the displacement immediately reaches very nearly its full value, but gradually increases through an appreciable time slightly above its initial vsJue. It thus appears that energy is slowly absorbed and this phenomenon has, therefore, been termed absorpiian. When the above condenser is disconnected from the supply, and then short circuited, the potential difference becomes zero. If the short circuit is removed a very small potential difference gradually reappears as residv>al. If such a condenser is displaced (charged), and the supply is removed, the displace- ment gradually disappears by conduction or leakage. The residual is analogous to residual stretch in an imperfectly elastic metal wire. For instance, if a steel rod is stretched and the stress is removed, it immediately assumes very nearly its initial length, but there is always very small residual stretch which very gradu- ally disappears.

It has been shown theoretically that the phenomenon of ab- sorption should exist for non-homogeneous dielectrics, but not for homogeneous dielectrics.

In non-homogeneous dielectrics the effect of this residual is to cause the flux to lag behind the voltage if the voltage change is rapid as in the case of high frequency. This is analogous to damping. If the change in voltage is slow, however, the effect would not result. A loop may thus be plotted (when the change of voltage is rapid) between voltage and displacement, similar to the hysteresis loop. If the frequency is very low or the dielectric is homogeneous the loop does not result.

This loop means loss, but it is not analogous to hysteresis loss in iron which is independent of time.

CHAPTER III VISUAL CORONA

Summary

Appearance. — If ' potential is applied between the smooth conductors of a transmission line or between concentric cylin- ders and gradually increased, a voltage is finally reached at which a hissing noise, is heard, and if it is dark, a pale violet light can be seen to surround the conductors. This voltage is called the critical visual corona point. If a wattmeter is inserted in the line a loss is noticed. The loss increases very rapidly as the vol- tage is raised above this point. The glow or breakdown starts first near the conductor surface, as the dielectric flux density or gradient is greatest there. As the broken down air near the surface is conducting, the size of the conductor is, in effect, increased by conducting corona. This increases for the given voltage until the flux density or gradient is below the rupturing gradient, when it cannot spread any more. If the conductors are very close together, a spark strikes between them immediately and corona cannot form. If the conductors are far apart corona forms first, and then, if the voltage is sufficiently increased, a spark strikes across.

Whenever corona is present there is always the characteristic odor of ozone. Air consists of a mechanical mixture of oxygen (O2) and nitrogen (N). When air is overstressed electrically the oxygen molecule is split up into 0, when it becomes chemically very active. The atoms again combine by the law of probability

into = 0, (O2), the normal state, and /X , (O3) or ozone.

0—0

Oxygen in the nascent state (O) also combines with metal, organic

matter, etc., if such are present. Ozone is also not stable and is,

hence, chemically active; it splits up as O2 and O when the latter

combines readily with metals, and organic matter. If the electrical

stress is very high the oxygen enters into chemical combination

with the nitrogen, forming oxides. The energy loss by corona is

thus in a number of forms, as heat, chemical action, light, noise,

convection, etc.

A.C, and D.C. Gorona. — When alternating voltage higher than

the critical voltage is applied between two parallel polished wires,

38

1 polished pnrallel wires.

Fio. 24. — Corona on parallel wires. Iron. First polished and then oper- ated at 120 kv. for two hours to develop spots. Diameter, 0.168 cm. Spacing, 12.7 cm. StroboseopiL- photo., 80 kv,-60 v.

Fig. 23.— D. C. corona on araootU wires by ttataon. (See Fig. 79.)

VISUAL CORONA 39

the glow is quite even, as shown in Fig. 23. After operation for a short time reddish beads or tufts form along the wire, while around the surface of the wire there is a bluish-white glow. If the conductors are examined through a stroboscope, so that one wire is sJwayB seen when at the positive half of the wave, it is noticed that the reddish tufts or beads are formed when the con- ductor is negative and the smoother bluish-white glow when the conductor is positive. (See Fig. 24.) A.c. corona viewed through the stroboscope has the same appearance as d.c. . corona. (See Fig. 25, d.c. corona.) The d.c. corona on the -|- wire has exactly the same appearance as the a.c. corona on the + half of the wave; the same holds for the -^ wire.

Measure of Stress. — The gradient in kilovolts per centimeter is a measure of the stress on the dielectric. For parallel wires the gradient at the wire surface, and hence where the stress is a maxi- mum and where the ''dielectric elastic limit" is first exceeded, is

de e , s^ t

dx ^ r log. S/r

That is, if the voltage between the surface of the conductors and a point in space at infinitesimal distance dx cm. away is de^ then this gradient in the limit at the surface is

de _ _ e dx" ^ " r log. S/r

If e is e», the observed voltage to neutral at which visual corona starts, Qv is a measure of the stress at breakdown. For a wire in the center of a cylinder

^ ev

^' r loge R/r

Influence of Wire Spacing and Diameter on Apparent Strength of Air. — If the visual corona voltages are measured for a given conductor at various spacings, it is found that Qv; or the apparent strength of air, is a constant independent of the conductor spacing. It would also naturally be expected that Qv would be a constant for all conductors independent of their diameter, or in other words, air would break down under the same constant unit stress inde- pendent of the size of the conductor; just, for instance, as differ- ent sized rods of the same material would be expected to break down at the same unit stress of kilograms per square centimeter. It has long been known,' however, that air is apparcn/Zy stronger at

• Ryftn, The Conductivity of the Atmosphere at High Voltage, A.I.E.E., Feb., 1904.

40 DIELECTRIC PHENOMENA

the surface of small conductors than large ones. (The measured apparent strength curve is given in Fig. 28.) Of course this does not mean the voltage required to start corona is greater for small wires than for large ones (it is lower for the small conductors at a given spacing), but that the term g^,, or unit stress in the expression

ev = QvT log, 8/r

is greater for air around small conductors than large ones.

This apparently greater strength of air around small conductors was long attributed to a film of condensed air at the conductor surface, because such a film would have a greater relative effect for the smaller conductors. We have given another reason for this which experiments at low air densities seem to confirm. During our first investigations we found that the relation between the apparent strength of air and the radius of the conductor could be expressed by the simple formula

where Qo is a constant and is about 30 kv. per centimeter.^ This means that the stress at the conductor surface at breakdown is not the same for all diameters, as already stated, but sJways constant at a distance 0.301/r cm. from the surface. (See Fig. 29.) This follows:

g

/ 0.301 \

r log* S/r

therefore, go = ;: — ; — TTI^ — TTl 77T

' (r + 0.301 Vr) log. S/r

The gradient x cm. from center of the conductor is approxi- mately

g =

X log. S/r X = (r + 0.301/r)

therefore, g. is the gradient O.SOlv^ cm. from surface. There- fore, by substitution also

= 3o/

0.301\ 1 + — T^F '^g* 'S/r max. kilovolts to neutral

or for a sine wave

= 21.2(

1+ ' /- ]r log. S/r kilovolts effective to neutral (19)

• F. W. Peek, Law of Corona, A.I.E.E., June, 1911.

VISUAL CORONA 41

The visual corona voltage for any diameter of wire at any spacing may thus be calculated.

The explanation seems to be this: Air has a constant strength Qo for a given density, but a finite amount of energy of some form is necessary to cause rupture or to start corona. It is obvious that this definite, finite energy is necessary, as evinced by appear- ance of heat, that higher transient voltages are necessary, etc. This will be more fully discussed later. Hence the stress at the conductor surface must exceed the elastic limit go, or be increased to Qv in order to supply the necessary rupturing energy between the conductor surface and finite radial distance in space away (O.SOlVr cm.) where the stress is Qo, and breakdown occurs.

Application of the Electron Theory. — The electron theory may also be applied in agreement with the above. Briefly:

When low potential is applied between two conductors any free ions in the field are set in motion. As the potential and, therefore, the field intensity or gradient is increased the velocities of the ions increase. At a gradient of ^o = 30 kv./cm. (5 » 1) the velocity of the ions becomes sufficiently great over the mean free path to form other ions by collision with molecules. This is the property of the negative ion or electron. This gradient is constant and is called the dielectric strength of air. When ionic saturation is reached at any point the air becomes con- ducting, and glows, or there is corona or spark.

Applying this to a wire in the center of a cylinder: When a gradient Qv is reached at the wire surface any free ions are acceler- ated and produce other ions by collision with atoms or molecules, which are in turn accelerated. The ionic density is thus gradually increased by successive collisions until at 0.301 Vr cm. from the wire surface, whqj^ go ^ 30, ionic saturation is reached, or corona starts. The distance 0.301 Vr cm. is of course many times greater than the mean free path of the ion and many collisions must take place in this distance. Thus, for the wire, corona cannot form when a gradient of go is reached at the surface, as at any distance from the surface the gradient is less than go. The gradient at the sur- face must, therefore, be increased to g« so that the gradient a finite distance away from the surface, 0.301/r cm., is go. This is the same as saying that energy is necessary to start corona, and

this energy is the 2o of the energy of the moving ions necessary

to produce ionic saturation.

42 DIELECTRIC PHENOMENA

Very Small Spacings or Films. — If conductors are placed closer together than this necessary free accelerating or energy storage distance, 0.301/r cm., the rupturing force or gradient must be increased. This will be better illustrated by later experiments, in which at small spacings air has been made to withstand gradi- ents as high as 200 kv. per centimeter.

Air Density. — Thus far the discussion has been limited to air at a constant density, or, in other words, constant pressure and temperature. The above energy explanation can be still further checked by considering air at different densities. The value of Qo given is for air density at sea level (25 deg. C, 76 cm. barom- eter). This has been taken as the standard, or for the density factor 5 = 1. Air at other densities, due to change in tempera- ture and barometric pressure, is expressed as a fraction of this. The relative density for any temperature or pressure is

3.92b

5 =

273 + <

For instance, if the temperature is kept at 25 deg. C. and pressure is reduced to 38 cm.

jj 3.92 X 38 _ r> R/x

• • 273 + 25" - -^-^

or 1/2 atmosphere. As the air density, or b is decreased, the air is less able to resist the electric stress at the increased molecular spacing. Theoretically the strength of the air go in bulk between parallel planes should decrease directly with 5.

go. = 305

flfp, however, the apparent strength of air in a non-uniform field, if the energy theory is true, cannot decrease directly with 5: the energy storage distance should be 0.301 V^^ (5), or the com- plete expression should take the form

J. . 0.301 \