cables theory design

7/16/2019 Cables Theory Design

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P A R T 1Theory, D esign and Principles

Common to all Cable Types



C H A P T E R 2

Basic Electrical Theory

Applicable to Cable Design

I n a ll e n g i n e e r i n g u n d e r t a k i n g s , e c o n o m i c a l , t e c h n i c a l a n d p r a c t i c a l a s p e c t s a r e t a k e n

i n t o c o n s i d e r a t i o n t o e s t a b l i s h t h e o p t i m u m s o l u t i o n o r d e s i g n . F o r t h e t r a n s m i s s i o n ,

d i s t r i b u t i o n a n d u t i l is a t i o n o f e l e c tr i c al p o w e r , t h e c h o i c e n o r m a l l y l ie s b e t w e e n t h e u s e

o f o v e r h e a d l in es a n d u n d e r g r o u n d c a b le s .F o r e c o n o m i c re a s o n s , o v e r h e a d l in es a r e u s ed e x t e n s iv e l y f o r t h e t r a n s m i s s i o n a n d

d i s t ri b u t io n o f e le c t ri c it y i n ru r a l a r e a s w h e r e e n v i r o n m e n t a l o r p r a c t i c a l c o n s i d e r a t i o n s

d o n o t d i c t a t e o th e r w i s e . H o w e v e r , i n u r b a n a r e a s it is m o r e u s u a l t o i n s t a ll i n s u l a t e d

c a b le s w h i c h , i n t h e m a i n , a r e b u r i e d u n d e r g r o u n d . T h e u t i l i s a t i o n o f e l e c t r ic i t y i n

f a c t o ri e s , d o m e s t i c p re m i s e s a n d o t h e r l o c a t i o n s is a l s o m a i n l y b y c a b l e s a s t h e y p r e s e n t

t h e m o s t p r a c t ic a l m e a n s o f c o n v e y i n g e l ec t ri c al p o w e r t o e q u i p m e n t , t o o l s a n d

a p p l ia n c e s o f a ll ty p e s. C a b l e d e s ig n s v a r y e n o r m o u s l y t o m e e t t h e d i v e rs e r e q u i r e m e n t s

b u t t h e r e a r e c e r t a i n c o m p o n e n t s w h i c h a r e c o m m o n t o a l l .A l l t y p e s o f e l e ct r ic c a b l e c o n s i st e s s e n ti a l ly o f a l o w r e s i s t a n c e c o n d u c t o r t o c a r r y t h e

c u r r e n t e x c e p t i n s p e c ia l c a se s , s u c h a s h e a t i n g c a b l es , a n d i n s u l a t i o n t o i s o l a t e t h e

c o n d u c t o r s f r o m e a c h o t h e r a n d f r o m t h e i r s u r r o u n d i n g s . I n s e v e r a l t y p e s , s u c h a s

s in g le - co r e w i r i n g c a bl e s, t h e t w o c o m p o n e n t s f o r m t h e f in i s he d c a b l e , b u t g e n e r a l ly a s

t h e v o l t a g e i n c r e a s e s t h e c o n s t r u c t i o n b e c o m e s m u c h m o r e c o m p l e x .

O t h e r m a i n c o m p o n e n t s m a y i n c l u d e s c re e n i n g t o o b t a i n a r a d i a l e l e c t ro s t a t ic fi eld , a

m e t a l s h e a t h t o k e e p o u t m o i s t u r e o r t o c o n t a i n a p r e s s u r i s i n g m e d i u m , a r m o u r i n g f o r

m e c h a n i c a l p r o t e c t i o n , c o r r o s i o n p r o t e c t i o n f o r th e m e t a ll ic c o m p o n e n t s a n d a v a r i e t y

o f a d d it i o n s e x te n d i n g , f o r e x a m p l e , t o i n t e r n a l a n d e x t e r n a l p i p e s t o r e m o v e t h e h e a t

g e n e r a t e d i n t h e c a b l e .

T h i s c h a p t e r c o n t a i n s s o m e o f t h e e le c t r ic a l t h e o r y a p p l i c a b l e t o a l l c a b l e t y p e s .

F u r t h e r d e t ai ls o f in d i v id u a l c a b le d e si g ns a n d c o m p o n e n t s a r e g i v e n in l a t e r c h a p t e r s.

V O L T A G E D E S I G N A T I O N

I n t h e e a r l y d a y s o f el e c tr i c p o w e r u t i l i s a t io n , d i r e c t c u r r e n t w a s w i d e l y u s e d , b u t l i tt le

n o w r e m a i n s e x c e p t f o r sp e ci al a p p l i c a t io n s a n d f o r a f e w i n t e r c o n n e c t i o n s i n



10 Electric Cables Handbook

transmission networks. Alternating current has many advantages and 3-phase

alternating current is used almost exclusively throughout the world.

So that suitable insulation and cable construction can be specified for the required 3-

phase a.c. service performance, the design voltages for cables are expressed in the form

Uo/U (formerly Eo/E). Uo is the power frequency voltage between conductor and earth

and U is the power frequency voltage between phase conductors for which the cable isdesigned, U0 and U both being r.m.s, values.

Power cables in British Standards are thus designated 600/1000 V, 1900/3300 V, 3800/

6600 V, 6350/11000 V, 8700] 15000 V, 12700/22000 V and 19000/33000 V. For transmis-

sion voltages above this it is normal to quote only the value of U and thus the higher

standard voltages in the UK are 66, 132, 275 and 400 kV. The maximum voltage can be

10% greater than the above values for voltages up to and including 275 kV and 5%

greater for 400 kV.

Although the local distribution voltage in the UK is 240/415V, the cables aredesigned for 600/1000 V, largely because during manufacture and installation this grade

of cable requires an insulation designed on mechanical rather than electrical

parameters.

Standardisation o f system voltages has not been achieved worldwide although there

is some move towards this. IEC has published voltage designations which are

approaching universal acceptance.

D.C. system voltages, by which is meant d.c. voltages with not more than 3% ripple,

are designated by the positive and negative value of the voltage above and below earth

potential. The symbol /-10 is used for the rated d.c. voltage between conductor and the

earthed core screen.

Many references will be found to cables described as low voltage (LV), medium voltage

(MV), high voltage (HV) and even EHV or UHV. Apart from low voltage, which is

defined internationally, these terms do not have generally accepted precise meanings and

can be misleading. In some countries MV has in the past applied to 600/1000 V cables

(these now fall clearly within the LV designation), whereas others have taken

it to mean 6/10kV or 8.7/15 kV and misunderstanding could arise. For precision it is

best to use the actual voltage rating of the cable.

CONDUCTOR RESISTANCE

D . C . r e s i s t a n c e

Factors affecting d.c. conductor resistance in terms of material resistivity and purity are

discussed in chapter 3 and those relating to conductor design in chapter 4. The latter areassociated with the fact that the prime path of the current is a helical one following the

individual wires in the conductor. Hence if an attempt is made to calculate the

resistance of a length of stranded conductor a factor must be applied to cater for the

linear length of wire in the conductor to allow for extra length caused by the stranding

effect. In a multicore cable an additional factor must be applied to allow for the

additional length due to the lay of the cores.

The d.c. resistance is also dependent on temperature as given by

Rt --- R20[l + az0(t - 20)] (2.1)



Basic Elec t r ica l Th eor y Ap pl icable to Cable De s ign 11

w h e r e Rt : c o n d u c t o r r e s i s t a n c e a t t ° C ( ~ )

R 2 0 = c o n d u c t o r r e s i s t a n c e a t 2 0 ° C ( D )

a 20 = t e m p e r a t u r e c o e ff ic i en t o f r e s is t a n c e o f th e c o n d u c t o r m a t e r i a l a t 2 0 ° C

t = c o n d u c t o r t e m p e r a t u r e ( °C )

A.C. res i s tance

I f a c o n d u c t o r is c a r r y i n g h ig h a l t e r n a t i n g c u r r e n t s , t h e d i s t r i b u t i o n o f c u r r e n t i s n o t

e v e n l y d i s p o se d t h r o u g h o u t t h e c r o s s - se c t i o n o f t h e c o n d u c t o r . T h i s is d u e t o t w o

i n d e p e n d e n t e ff e ct s k n o w n a s th e ' s k i n e f f e c t' a n d t h e ' p r o x i m i t y e ff e c t '.

I f t h e c o n d u c t o r i s c o n s id e r e d t o b e c o m p o s e d o f a la rg e n u m b e r o f c o n c e n t r ic

c i r c u la r e le m e n t s, t h o s e a t t h e c e n t r e o f th e c o n d u c t o r w i ll b e e n v e l o p e d b y a g r e a t e r

m a g n e t i c fl ux t h a n t h o s e o n t h e o u t si d e . C o n s e q u e n t l y t h e s e l f - in d u c e d b a c k e . m . f , w i ll

b e g r e a t e r t o w a r d s t h e c e n t r e o f t h e c o n d u c t o r , t h u s c a u s i n g t h e c u r r e n t d e n s i t y t o b ele ss a t t h e c e n tr e t h a n a t t h e c o n d u c t o r s u r f a c e . T h i s e x t ra c o n c e n t r a t i o n a t t h e s u r fa c e

i s t h e s k i n e f f e c t a n d i t r e s u l t s in a n i n c r e a s e i n t h e e f f e c t i v e r e s i s t a n c e o f t h e c o n d u c t o r .

T h e m a g n i t u d e o f t h e s k i n e f fe c t is i n f lu e n c e d b y t h e f r e q u e n c y , t h e s i ze o f th e

c o n d u c t o r , t h e a m o u n t o f c u r r e n t fl ow i n g a n d t h e d i a m e t e r o f th e c o n d u c t o r .

T h e p r o x i m i t y e ff e c t a l s o i n c r e a se s t h e e f f ec t iv e r e s i s ta n c e a n d is a s s o c i a t e d w i t h t h e

m a g n e t i c f ie ld s o f tw o c o n d u c t o r s w h i c h a r e c l o s e t o g e t h e r . I f e a c h c a r r i e s a c u r r e n t i n

t h e s a m e d i r e c ti o n , th e h a l v es o f t h e c o n d u c t o r s i n cl os e p r o x i m i t y a r e c u t b y m o r e

m a g n e t i c fl ux t h a n t h e r e m o t e h a l v es . C o n s e q u e n t l y , t h e c u r r e n t d i s t r i b u t i o n is n o t e v e nt h r o u g h o u t t h e c r o s s- s e ct io n , a g r e a t e r p r o p o r t i o n b e i n g c a r r i e d b y t h e r e m o t e h a l v es . I f

t h e c u r r e n t s a r e i n o p p o s i t e d i r e c t i o n s t h e h a l v e s i n c l o s e r p r o x i m i t y c a r r y t h e g r e a t e r

d e n s i t y o f c u r r e n t . I n b o t h c a se s t h e o v e r a l l e f fe c t r e s u lt s i n a n i n c r e a s e i n t h e e f f ec t iv e

r e s is t an c e o f t h e c o n d u c t o r . T h e p r o x i m i t y e f f e ct d e c r e as e s w i t h i n c r e a se in s p a c i n g

b e t w e e n c a b l e s.

M a t h e m a t i c a l t r e a t m e n t o f th e s e ef fe c ts is c o m p l i c a t e d b e c a u s e o f th e l a r g e n u m b e r

o f p o ss ib l e v a r i a ti o n s b u t A r n o l d I h a s p r o d u c e d a c o m p r e h e n s i v e r e p o r t ( se e al so

c h a p t e r 8 ) .

S k i n a n d p r o x i m i t y e ff ec ts m a y b e i g n o r e d w i th s m a l l c o n d u c t o r s c a r r y i n g l o w

c u r r e n t s . T h e y b e c o m e i n c r e a s i n g l y s i g n i f i c a n t w i t h l a r g e r c o n d u c t o r s a n d i t i s o f t e n

d e s i ra b l e fo r te c h n ic a l a n d e c o n o m i c r e a s o n s t o d e s ig n t h e c o n d u c t o r s t o m i n im i s e

t h e m . T h e M i l li k e n c o n d u c t o r , w h i c h r e d u c e s s k i n a n d p r o x i m i t y e f fe c ts , is d e s c r i b e d i n

c h a p t e r 4 .

A . C . r es is ta n c es a r e i m p o r t a n t f o r c a l c u l a ti o n o f c u r r e n t c a r r y i n g c a p a c i t y . V a l u e s f o r

s t a n d a r d d e s ig n s o f d i s t r i b u t i o n a n d t r a n s m i s s i o n c a b l es a r e i n c l u d e d i n t h e t a b l es i n

a p p e n d i c e s 1 2 - 1 6 .

I N D U C T A N C E

T h e i n d u c t a n c e L p e r c o r e o f a 3 - c o r e c a b le o r o f t h r e e s in g l e - c o r e c a b l e s c o m p r i s e s t w o

p a r t s, t h e s e lf - in d u c t a nc e o f th e c o n d u c t o r a n d t h e m u t u a l i n d u c t a n c e w i t h o t h e r c o r e s.

I t i s g iv e n b y

2 SL = K + 0 .2 l og e ~ - ( m H / k m ) ( 2 .2 )




w h e r e K = a c o n s ta n t r e la t in g t o th e c o n d u c t o r f o r m a t i o n ( t a b le 2 .1 )

S = a x ia l s p a c i n g b e t w e e n c o n d u c t o r s w i t h i n t h e c a b l e (r a m ) , o r

= a x i al s p a c i n g b e t w e e n c o n d u c t o r s o f a t r e fo i l g r o u p o f s i n g le - c o r e c a b l e s

( m m ) , o r

= 1 .2 6 x p h a s e s p a c i n g f o r a f i a t f o r m a t i o n o f t h r e e s i n g l e - c o r e c a b l e s ( m m )

d = c o n d u c t o r d i a m e t e r o r f o r s h a p e d d e si gn s t h e d i a m e t e r o f an e q u i v a l e n tc i rc u l ar c o n d u c t o r ( m m )

F o r 2 - c o r e, 3 - c or e a n d 4 - c o r e c a b le s , t h e i n d u c t a n c e o b t a i n e d f r o m t h e f o r m u l a

s h o u l d b e m u l ti p l ie d b y 1 .0 2 i f t h e c o n d u c t o r s a r e c i r c u l a r o r s e c t o r - sh a p e d , a n d b y 0 . 9 7

f o r 3 - c o r e o v a l c o n d u c t o r s .

Tab l e 2 .1 Typ ica l va lues fo r cons tan t K fo r d i f fe ren t

s t randed condu c to rs (a t 50 Hz )

Nu m b e r o f w ire s i n c o n d u c to r K

3 0.0778

7 0.0642

19 0.0554

37 0.0528

61 an d ov er 0.0514

1 (solid ) 0.0500Ho l low-core cond uc to r , 12m m duc t 0 .0383

R E A C T A N C E

T h e i n d u c t i v e r e a c t a n c e p e r p h a s e o f a c a b le , o r o f t h e s in g l e -c o r e c a b le s c o m p r i s i n g t h e

c i r c u i t , m a y b e o b t a i n e d f r o m t h e f o r m u l a

X = 2~rfL x 10 -3 ( f~ / .km) (2 .3 )

w h e r e f = f r e q u e n c y ( H z )

L = i n d u c t a n c e ( m H / k m )

I M P E D A N C E

T h e p h a s e i m p e d a n c e Z o f a c a b l e, o r o f t h e s i n g l e - c o r e c a b l e s c o m p r i s i n g t h e c i r c u i t , is

g i v e n b y

Z = (R 2 + X 2) 1/2 ( f t / k m ) ( 2 . 4 )

w h e r e R = a .c . r e s i s ta n c e a t o p e r a t i n g t e m p e r a t u r e (9 t / k m )X = r e a c t a n c e ( f U k m )



B a s i c E l e c t r ic a l T h e o r y A p p l i c a b l e t o C a b l e D e s i g n 1 3

I N S U L A T I O N R E S I S T A N C E

I n s u l a t i o n r e s i st a n c e is t h e re s i s ta n c e t o t h e p a s s a g e o f d i r e c t c u r r e n t t h r o u g h t h e

d i e l e ct r ic b e t w e e n t w o e l e c t ro d e s . I n t h e c a s e o f a n e l e c tr i c c a b l e i t is t h e v a l u e o f t h e

r e si st a nc e b e tw e e n t h e c o n d u c t o r a n d t h e e a r t h e d c o r e s c re e n , m e t a l l ic s h e a t h , a r m o u r

o r a d j a c e n t c o n d u c t o r s .C o n s i d e r a u n i t l e n g th o f si n gl e -c o r e c a b l e w i t h c o n d u c t o r r a d i u s r a n d r a d i u s o v e r

i n s u l a t i o n R (f ig . 2 .1 ) . T h e s u r f a c e a r e a o f a r i n g o f i n s u l a t i o n o f r a d i a l t h i c k n e s s 6 x a t

r a d i u s x is 2 7rx t i m e s t h e u n i t l e n g t h . T h e i n s u l a t i o n r e s i s t a n c e 6 R o f th i s r i n g i s g i v e n b y

6 R - p 6 x27rx (f~) (2.5 )

w h e r e p i s t h e s p e c i f ic r e s i s t i v i t y ( f~ m) .

T h u s t h e i n s u l a t i o n r e s is t a n c e D R o f r a d i a l t h i c k n e s s R - r f o r 1 m c a b l e l e n g t h isg i v e n b y

w h i c h e v o l v e s t o

P d xl (12) (2.6 )

D R = ~ J r x

p RDR = ~ log e --r (f~)

o r f o r c a b l e l e n g t h 1

p Rloge --r (f ' /) (2.7 )

C o r r e c t i o n f o r t e m p e r a t u r e m a y b e m a d e a c c o r d i n g t o

Pt ":- P20 e x p ( - a t )

w h e re p20 = sp e c i f i c r e s i s t i v i t y a t 2 0 ° C

c~ = t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t iv i t y p e r d e g r e e C e l s i u s a t 2 0 ° C

t = t e m p e r a t u r e ( ° C )

S p e c if ic r e s is t iv i t y is al s o d e p e n d e n t o n e l e c tr i c s t re s s a n d h e n c e t h e a b o v e d e r i v a t i o n

is a s i m p l i fi c a ti o n . H o w e v e r , t h e e ff e c t is m u c h l es s t h a n t h a t o f t e m p e r a t u r e a n d i n

m o s t c a s e s i t c a n b e n e g le c t ed . I t c a n b e o f i m p o r t a n c e i n t h e d e s i g n o f h i g h v o l t a g e d . c .

Fig. 2.1 Calcu la t ion of insula t ion resistance of an insula ted co nd uc tor



14 Elec t r i c Cab le s Handbook

p a p e r c a b le s a n d t hi s is d e a l t w i t h in c h a p t e r 3 8. In m a k i n g m e a s u r e m e n t s o f i n s u l a ti o n

r e s i s ta n c e i t is n e c e s s a r y to m a i n t a i n t h e d . c . t e s t v o l t a g e f o r s u f f ic i en t t i m e t o e n s u r e

t h a t a n y t r a n s i e n t c u r r e n t s a s s o c i a t e d w i t h c h a r g i n g o f t h e c a b l e a r e o f n e g l i g ib l e v a l u e .

I n a . c . c a b l e s t h e r e w i l l b e a d d i t i o n a l c u r r e n t s c h a r g i n g t h e i n s u l a t i o n d u e t o t h e

c a p a c i t a n c e o f th e i n s u la t i o n a n d f o r h i g h v o l t a g e d. c. p a p e r c a b l e s a n e x t r a f a c t o r

is i n t r o d u c e d t o t a k e a c c o u n t o f t h e v a r i a t i o n o f r e si s ti v it y w i t h a p p l i e d v o l t a g e( s e e c h a p t e r 3 8 ) .

C A P A C I T A N C E

S i n g l e - c o r e c a b l e s o r c i r cu l a r s c r e e n e d c o r e s

M e d i u m v o l t a g e a n d h i g h v o l t a g e c a b l e m a y b e s i n g l e - c o r e c a b l e w i t h a n e a r t h e d

m e t a l li c la y e r a r o u n d t h e c o re , o r 3 - c o re c a b l e w i t h a n e a r t h e d m e t a l li c s c re e n a r o u n d

e a c h c o r e . I n b o t h c a s e s, t h e e l e c t r o s t a t ic f ie ld is c o n t a i n e d w i t h i n t h e e a r t h e d s c r e e n

a n d i s s u b s t a n t i a l l y r a d i a l .

C o n s i d e r a s i n gl e -c o re c a b le w i t h a s m o o t h c o n d u c t o r o f r a d i u s r ( m ) a n d a n i n t e r n a l

s c re e n ra d i u s o f R ( m ) (fig . 2 .2 ). A s s u m e t h a t t h e c o n d u c t o r c a r r ie s a c h a r g e o f q

( c o u l o m b / m ) , t h e n t h e e l ec tr ic f lu x e m a n a t e s f r o m t h e c o n d u c t o r r a d i a ll y , g iv i n g a fl u x

d e n s i t y a t r a d iu s x ( m ) f r o m t h e c e n t r e o f t h e c o n d u c t o r o f

Dx = q ( c o u l o m b / m 2)27rx

T h e e l e c t r i c f i e l d i n t e n s i t y a t r a d i u s x i s

Dx q

Ex - Eocr - 27rxE0cr (2.8)

w h e r e co = p e r m i t t i v i t y o f f r e e s p a c e , 1 0 -9 / 36 7 r

cr = r e l a ti v e p e r m i t t i v i t y o f t h e i n s u l a t i o n

T h e w o r k d o n e i n m o v i n g a u n i t p o s i t iv e c h a r g e t h e d i s t a n c e d x i n a n e l e c tr ic fi el d o fi n t e n s i t y E i s g i v e n b y

dW = -Exdx

i . e . t h e c h a n g e i n p o t e n t i a l a l o n g d x i s

dV = -Ex dx

Fig. 2 .2 Cross-sect ion of cable core for der ivat ion o f capa ci tance



B a s i c E l e c t r i c a l T h e o r y A p p l i c a b l e t o C a b l e D e s i g n 1 5

T h e r e f o r e th e w o r k d o n e i n m o v i n g a u n i t c h a rg e f r o m t h e c o n d u c t o r s u r fa c e t o t h e

o u t e r s u r f ac e o f t h e i n s u l a t io n is g o v e r n e d b y

V = - E x d xR

_ q f r d x

J7reoer R x

- q(2 .9 )

27r~0~r

T h e c a p a c i t a n c e o f t he c a b l e p e r m e t r e l e n g t h i s g i v en b y

qC =--

V

27re0~r

- - l o g o ( R / r ) ( F / m )

27rer x 10 -9

- - 3 6 7 r l o g ~ ( R / r ) ( F / m )

o r

~r

C - 1 8 l o g o ( D / d ) ( g F / k m ) ( 2 . 1 0 )

w h e r e D = d i a m e t e r o v e r t h e i n s u la t io n ( m )

d = d i a m e te r o v e r t he c o n d u c t o r ( m )

e r = r e l a t i v e p e r m i t t i v i t y

T h e r e l a t i v e p e r m i t t i v i t y is a c h a r a c t e r i s t i c o f t h e i n s u l a t i o n m a t e r i a l a n d is d e p e n d e n t

o n t e m p e r a t u r e a n d f r e q u e n c y . F o r p o w e r f r e q u e n c i e s a n d n o r m a l o p e r a t i n g

t e m p e r a t u r e s t h e e f fe c t is s m a l l a n d c a n b e i g n o r e d f o r m o s t e n g i n e e r i n g c a l c u l a t i o n s .T h e c a p a c i t a n c e c a l c u l a t e d u s i n g e q u a t i o n ( 2 . 1 0 ) i s r e f e r r e d t o a s e i t h e r t h e n o m i n a l

c a p a c i t a n c e p e r c o r e o r t h e n o m i n a l s t a r c a p a c i t a n c e . T h i s e q u a t i o n i s a l s o u s e d t o

c a l cu l a te t h e c a p a c i t a n c e o f lo w v o l t a g e s i n g l e- c o re c a bl e s w i t h o u t a n y m e t a l l i c o u t e r

l a y e r . T h i s i s j u s ti f ie d o n t h e b a s is t h a t i t is i m p o s s i b l e t o k n o w w h a t t h e a d d i t i o n a l

c a p a c i t a n c e is i n a p a r t i c u l a r i n s t a l l a t i o n b e t w e e n t h e o u t e r s u r f a c e o f t h e i n s u l a t e d c o r e

a n d a n y s u r r o u n d i n g e a r t h r e f e r e n c e . I n a n y c a se , su c h a n a d d i t i o n a l c a p a c i t a n c e i n

s e r i e s w i t h C w o u l d r e d u c e t h e t o t a l s t a r c a p a c i t a n c e .

Three-core be l ted type cables

T h e e q u a t i o n f o r ca l c u la t in g t h e c a p a c i t a n c e o f a b e lt e d t y p e c a b l e is n o t r e a d i l y

f o r m u l a t e d (s ee l a te r f o r f ie ld t h e o r y f o r p a p e r i n s u l a t e d c a b le s ), b u t a n a p p r o x i m a t i o n

o f th e ca p a c i ta n c e b e tw e e n o n e c o n d u c t o r a n d t h e o t h e r c o n d u c t o r s c o n n e c t e d t o t h e

m e t a l li c sh e a t h , s c re e n o r a r m o u r c a n b e o b t a i n e d f r o m e q u a t i o n ( 2.1 0 ) i f D is t a k e n a s

D = d i a m e t e r o f o n e c o n d u c t o r p l u s t h e t h ic k n e s s o f in s u l a t i o n b e t w e e n c o n d u c t o r s

p l us t h e t h ic k n e ss o f i n s u l a t i o n b e t w e e n a n y c o n d u c t o r a n d t h e m e t a l s h e a t h ,s c r e e n o r a r m o u r




T h e v a r i o u s o t h e r c a p a c i t a n c es o f a b e l te d t y p e c a b l e m a y b e o b t a i n e d , t o a c l o se

a p p r o x i m a t i o n , b y c a lc u l a ti n g C b y e q u a t i o n ( 2.1 0 ) a n d u s i n g t h e f o l l o w i n g f a c to r s :

C l = 0 .8 3 × C = c a p a c i t a n c e o f o n e c o n d u c t o r t o a ll o t h e r c o n d u c t o r s a n d o u t e r

m e t a l l i c l a y e r

C2 = 0 .5 0 x C = c a p a c i t a n c e o f o n e c o n d u c t o r t o o n e o t h e r c o n d u c t o r , w i t hr e m a i n i n g c o n d u c t o r s a n d o u t e r m e t a ll ic l a y e r f l o a ti n g

C 3 = 1 .5 0 x C = c a p a c i t a n c e o f a ll c o n d u c t o r s ( b u n c h e d ) t o o u t e r m e t a l li c l a y e r

D I E L E C T R I C P O W E R F A C T O R ( D I E L E C T R I C L O S S A N G L E )

I t is o f g r e a t i m p o r t a n c e t h a t t h e p o w e r f a c t o r o f t h e d i e le c t ri c o f c a b le s f o r v o l t a g e s o f

3 3 k V a n d a b o v e i s o f a v e r y l o w v a l u e . T h e p o w e r f a c t o r o f a d i e l e c tr i c is th e r a t i o

l o ss i n d i e l e c t r i c ( w a t t )

v o l t s x a m p s

R e f e r r i n g t o f i g . 2 . 3 , w h e n a v o l t a g e i s a p p l i e d t o a c a b l e w i t h a ' p e r f e c t ' d i e l e c t r i c , a

c h a r g i n g c u r r e n t I c f lo w s w h i c h is i n le a d i n g q u a d r a t u r e w i t h t h e v o l t a g e . I n s u c h a

' p e r fe c t ' d i el ec t ri c t h e re w o u l d b e n o c o m p o n e n t o f t h is c u r r e n t i n p h a s e w i t h U .

H o w e v e r , p e r f e c t i o n i n d i e le c tr ic s h a s n o t b e e n a c h i e v e d a n d t h e r e is a s m a l l c u r r e n t I R

w h i c h i s i n p h a s e w i t h U . ( S e e l a t e r s e c t i o n o n d i e l e c t r i c l o s s e s . ) T h i s c u r r e n t c a u s e slosses IRU i n th e d i e l e c t ri c w h i c h g e n e r a t e h e a t . T h e l o s se s in t h e d i e l e c t r i c a r e

p r o p o r t i o n a l t o t h e co s in e o f t h e a n gl e b e t w e e n t h e r e s u l t a n t c u r r e n t I t a n d a p p l i e d

v o l t a g e U .

N o w

a n d

I t = ( /R 2 + IC2) 1/2

IRU = / t U c o s 4 ~

I R= ( 2 . 1 1 )

cosq~ (I R 2 q- IC2 )1/2

As q~ is c lo se t o 9 0 ° , c o s q~ e q u a te s a p p ro x im a t e l y t o t a n (9 0 - ~b), i .e . e q u a t e s

( a p p r o x i m a t e l y ) t o t a n 6, a n d t h e d i e le c t r ic p o w e r f a c t o r o f a c a b l e i s f r e q u e n t l y r e f e r r e d

t o a s t a n 6, w h e r e 6 is k n o w n a s t h e d i e l e c t r ic lo s s a n g l e ( D L A ) .

c o s # - ta n

S = 9 0 - - ~ "

Fig. 2 .3 V ector d iagram to represen t d ie lec tric loss angle

IR



Basic Electrical Theory Applicable to Cable Design 17

The dielectric loss in watts per kilometre per phase is given by

D -- 27rfC U02 tan 6 10-6 (watt/km per phase) (2.12)

It will be seen from this equation that, for a specified design of cable in which values

off , C and U0 are fixed, the dielectric loss angle must be kept to an absolute minimum

to achieve low dielectric losses.

In addition to establishing the dielectric loss angle of the cable to determine the

dielectric losses for cable rating purposes, valuable information can be obtained by

testing the power factor of the cable in discrete voltage steps within the voltage range

0.5U0-2U0. Such a test gives information on the ionisation which takes place in the

insulation because ionisation increases with increase in applied voltage (see later under

'Mechanism of insulation breakdown in paper insulated cables').

In the case of paper insulated cables the DLA is a funct ion of the densi ty of the paper

and the contamination in the fluid and paper. The fluid can readily be cleaned and thecontaminants arise mostly from the paper as ionisable salts. To obtain the lowest DLA

in transmission cables at high temperature, deionised water is used in paper

manufacture.

ELECTRICAL STRESS DISTRIBUTION AND CALCULATION

The flux distribution in a.c. belted cable insulation is complex and is showndiagrammatically in fig. 2.4. The stress is a maximum at the conductor surface and

varies throughout the insulation, decreasing with distance from the conductor surface,

but not in a clearly defined manner because of the differing permittivities of the

components and the distribution of the flux at various times during the voltage phase

rotation. The screened cable used for alternating voltages has a clearly defined stress

pattern, while that of cables used on d.c. transmission has a changing pat tern depending

on temperature due to cable loading.

C o r e i n s u l a t i o n

F i l l e r

F l u x t a n g e n t i a lB e l t i n s u la t io n c o m p o n e n t in

t h i s r e g i o n

Fig. 2.4 Paper insulated belted cable with top conductor at peak potential




A . C . s tr e s s d i st r ib u t i o n in s in g l e - c o r e a n d s c r e e n e d m u l t ic o r e c a b l e s

T h e s t r a n d in g e f fe c t o f u n s c r e e n e d c o n d u c t o r s g iv es a s l i g ht u n e v e n n e s s o f s t re s s

d i s t r ib u t i o n a r o u n d t h e p e r i p h e r y o f t h e c o n d u c t o r b e c a u s e o f th e s m a l l r a d iu s o f t h e

i n d i v i d u a l w i r e s. N e g l e c t i n g t h i s e ff e c t, a s c r e e n e d c o r e c o n s i s t s o f a c y l i n d r i c a l

c a p a c i t o r w i t h t h e c o n d u c t o r a s t h e i n n e r e l e c t r o d e a n d t h e c o r e s c r e e n a s t h e o u t e r

e l e ct r od e . A s s u m i n g a u n i f o r m p e r m i t t i v i ty , t h e r a d i al s t r es s d i s t r i b u t i o n c u r v e o f a

c i r c u l a r c o r e i s d e r i v e d a s b e l o w . I t w i l l b e s e e n t o b e m a x i m u m a t t h e c o n d u c t o r

s u r f a c e , r e d u c i n g i n a h y p e r b o l i c c u r v e ( fi g. 2.5 ) . A s t h e p e r m i t t i v i t y o f t h e d i e l e c tr i c is

s u b s t a n ti a ll y c o n s t a n t t h r o u g h o u t t h e o p e r a t i n g t e m p e r a t u r e r a n g e o f t h e ca b l e, th e

s t r e s s d i s t r i b u t i o n r e m a i n s c o n s t a n t a t a l l o p e r a t i n g c o n d i t i o n s .

U s i n g t h e n o t a t i o n g i v e n p r e v i o u s l y , i . e .

r = r a d iu s o f c o n d u c t o r ( m )

R = i n t e r n a l r a d i u s o f s h e a t h ( m )co = p e r m i t t i v i t y o f f r e e s p a c e

c r = r e l a t i v e p e r m i t t i v i t y o f t h e d i e l e c t r i c

V = p o t e n t i a l o f c o n d u c t o r r e l a ti v e to t h e s h e a t h ( V )

q = c h a r g e ( c o u l o m b / m o f a x ia l l e n gt h )

f r o m e q u a t i o n ( 2 . 9 )

T h e r e f o r e

V = q l o g e ( R ) ( V )

27rCOCr

q V

2 7 r c o c r l o g e ( R / r )

S t re s s m a y b e c a l c u l a t e d f r o m e q u a t i o n ( 2. 8) w h i c h s h o w e d t h e e l e c t r ic f ie l d i n t e n s i t y

E x a t r a d i u s x t o b e

E x - q27rxcocr

S t r e s s a t c o n d u c t o r

\

)

Fig. 2.5 El ect ros tatic stress in single-core cab le

. S t r e s s a t s h e a t h

~ P o t e n t i a l d i f fe r e n ce = V



B a s i c E l e c t r i c a l T h e o r y A p p l i c a b l e t o C a b le D e s ig n 1 9

By substitution, the stress at radius x is

VE x =

x l o g e ( R / r )

It will be noted that the range is

Vmaximum -

r l o g e ( R / r )

Vminimum -

R l o g e ( R / r )

Several features arising from this are as follows.

(MV/m) (2.13)

at conductor surface

at sheath inner surface

Stress at conductor sur face

In the above derivation of the stress distribution it was assumed that the cond uc to r

surface was smooth. However, the effects of the radius of the wires in a stranded,

uncompacted conductor may increase the stress substantially. In high and medium

voltage cables, therefore, it is usual to apply a semiconducting screen to a stranded

conductor to obtain a smooth surface. This screen would be a carbon paper tape on

paper cables or an extruded semiconducting polymer on polymeric cables. When using

the equation (2.13), the value of r would be the radius over this conductor screen.

Conductor diameter

The equation (2.13) derived for the stress within the cable dielectric showed the stress to

be a maximum at the conductor surface. The equation may be developed further to

obtain the ratio of the diameter ove r the cond uc tor to that over the insulation a nd

hence the conductor diameter which gives the minimum stress at the conductor surface

for a specified voltage and diameter over the insulation:

VE r - r l o g e ( R / r )

For minimum stress at the conductor for constant values of V and R

d E r= 0

dr

i.e.

Therefore

d I v ]r l o g ~ R / r ) ~- 0

~ l ogeR- - r loge =0

- V(loge R - loge r - 1) = 0

(r log~R - r loge r) 2

l o g ~ R - l o g e r - I = 0

l o g ~ ( R / r ) = 1



2 0 Electric Ca bles Handbook

rmin R /e

C o n d u c t o r r a d iu s ( r)

Fig. 2 .6 Va ria t ion of s tress w ith co nd uc tor radius

a n d

R / r = e (2 .14 )

T h e s tr es s a t t he c o n d u c t o r is a t a m i n i m u m w h e n t h e r a t io R /r e q u a l s e . B y s u b s t i t u t i n g

t h i s in t h e s t r e s s e q u a t i o n i t w il l b e s e e n t h a t t h e v a l u e o f th e s t r e s s is V/r. T h i s i s

i l lus t ra ted in f ig . 2 .6 .

T h e c o n d u c t o r c r o s s - s e c t i o n a l a r e a i s d e t e r m i n e d b y t h e c u r r e n t w h i c h h a s t o b e

c a r r i e d . W h e n d e s i g n i n g c a b l e s , p a r t i c u l a r l y f o r t h e h i g h e r t r a n s m i s s i o n v o l t a g e s , i t i s

p o s s ib l e t h a t t h e r a d i u s o f t h e c o n d u c t o r s iz e n e e d e d t o g i v e t h e c u r r e n t c a r r y i n g

c a p a c i t y c a ll ed f o r w i ll b e s m a l le r t h a n t h a t t o g i v e o p t i m u m r a t i o o f d i a m e t e r o v e r

c o n d u c t o r t o d i a m e t e r o v e r i n s u la t i o n . T h i s n e c e s si ta t es t h e a p p l i c a t i o n o f t h i c k e r

i n s u la t i o n to m a i n t a i n a n e q u a l m a x i m u m s tr es s. F r e q u e n t l y t h e sm a l l e r c o n d u c t o r s

w i t h i n a s p ec if ic v o l t a g e r a n g e h a v e a g r e a t e r i n s u l a t i o n t h i c k n e s s t h a n t h e l a r g e r s iz es .

A n a l t e rn a t i v e d e si g n p r o c e d u r e is to i n c r ea s e th e c o n d u c t o r d i a m e t e r t o a t t a i n t h e

o p t i m u m r a t io . F o r e x a m p l e , in t h e c a se o f f lu i d -f il le d c a b l e s , th e d i a m e t e r o f t h e c e n t r a l

f l u i d d u c t m a y b e i n c r e a s e d .

Grading o f insulationI n h i g h v o l ta g e i m p r e g n a t e d p a p e r i n s u l a t ed c a b l e s it is a n a d v a n t a g e t o h a v e m a t e r i a l

o f h ig h e l e ct r ic s t r e n g t h n e a r t h e c o n d u c t o r t o r e d u c e t h e t o t a l i n s u l a t i o n t h i c k n e s s .

T h i s i s n o r m a l l y a c h i e v e d b y h a v i n g t h i n n e r h i g h q u a l i t y , h i g h e r d e n s i t y p a p e r s i n

t hi s re g io n . T h e u s e o f g r a d e d i n s u l a t io n a ls o g iv e s i m p r o v e d b e n d i n g p r o p e r t i e s

( c h a p t e r 1 9 ) .

D . C . s t r e s s d i s t r i b u t i o n

T h e s t r e s s d i s t r i b u t i o n w i t h i n a d . c . c a b l e i s d e t e r m i n e d , inter alia, b y i n s u l a t i o n

r e s i s t i v i t y , w h i c h , a s i n d i c a t e d p r e v i o u s l y , i s i n f l u e n c e d b y t h e t e m p e r a t u r e o f t h e

i n s u l a t i o n a n d a l s o t h e s t r e s s . T h e s t r e s s p a t t e r n w i t h i n a d . c . c a b l e t h u s a l t e r s w i t ht h e l o a d . T h i s i s d i s c u s s e d i n c h a p t e r 3 8 .



Basic Electr ical Theory Applicable to Cable Design 21

FIELD CONTROL IN PAPER INSULATED CABLES

In the early days of electrical power transmission the belted type cable was used

extensively but, with the increase in system voltage to 33 kV in the early 1920s, the

shortcomings of the belted construction became apparent and the screened type

(sometimes called H or Hochstadter type) cable was introduced and has been usedsuccessfully for this voltage to the present day. As transmission voltages increased,

'pressure-assisted' type cables using the screened construction were designed and are

employed at voltages of 33 kV and above.

B e l t e d t y p e c a b l e s

The construction of the 3-core belted type cable is shown in fig. 2.7 (left). The insulation

between conductors is twice the radial thickness of the insulation around eachconductor and the insulation between any conductor and the earthed sheath is the

radial thickness of the insulation around a conductor plus the thickness of the belt. As

the phase to phase voltage is ,/3 times the phase to earth voltage, the total thickness of

insulation between conductors and that between conductors and sheath are designed in

approximately the same ratio.

For voltages up to 22kV the belted construction has been used as the electrical

stresses are acceptably low. For cables used at higher voltages it is necessary to raise the

operating stresses for economical and practical reasons and this brings to light defects

in the belted cable design which are described in the later section, 'Mechanism of

insulation breakdown in paper insulated cables'. Nowadays screened cables are

generally used at 15 kV and above.

S c r e e n e d o r H t y p e c a b l e

To eliminate the weaknesses of the belted type cable a design of cable in which each

core is individually surrounded by an earthed metallic layer was introduced and is

shown in fig. 2.7 (right). This design of cable, first patented by Hochstadter in 1914,

013

[ Shea th

Fig. 2.7 3-core belted type cable (left) and screened cable (right)

don

L e a d s h e a th



2 2 Electric Ca bles Handbook

e n s u r es t h a t t h e s t re s s is s u b s t a n t ia l l y r a d i a l a n d h e n c e n o r m a l t o t h e p a p e r s u r f a c e a n d

is a ls o c o n t a i n e d w i t h i n t h e m a c h i n e l a p p e d i n s u l a t i o n , w h i c h i s e l e c t r ic a l l y s t r o n g . T h e

p o s s ib i li ty o f s e p a r a t i o n o f t h e c o r es d u e t o t h e r m a l e x c u r s i o n s a n d m e c h a n i c a l

h a n d l i n g , w h i le n o t e l i m i n a t e d , d o e s n o t p r e s e n t a h a z a r d a s a ll e l e c t r ic a l f l u x is

C o n t a i n e d w i t h i n t h e e a r t h e d s c r e e n s .

F I E L D C O N T R O L I N P O L Y M E R I C I N S U L A T E D C A B L E S

D u r i n g t he d e v e l o p m e n t o f m e d i u m v o l ta g e X L P E c a b le s , i t w a s s o o n r e a l i se d t h a t

b e l t e d d e s i g n s a b o v e 1 . 9 / 3 . 3 k V r a t i n g e x h i b i t e d s i m i l a r w e a k n e s s e s t o t h o s e o f

b e l t e d p a p e r c a b l e s a b o v e 1 2 .7 / 22 k V r a t i n g . T h a t i s , t h e e l e c t r i c a l s t r e ss i n t h e f i ll e rs

a r o u n d t h e c o r e s p r o d u c e d p a r t ia l d i s c h a r g e a c t iv i t y w h i c h i m p i n g e d o n t h e s u r f a c e

o f th e X L P E i n s u l a ti o n , s t e a d il y e r o d i n g t h e d i e le c tr ic a n d r e s u l ti n g i n p r e m a t u r e

c a b l e f a i l u r e .

F o r m o s t n o r m a l a p p l i c a t io n s , t h e r e fo r e , p o l y m e r i c c a b le s r a te d a t 3 . 8 / 6 . 6 k V a n d

a b o v e h a v e s c r e e n e d c o re s . F u r t h e r m o r e , t h e se c ab l es n e a r l y a l w a y s h a v e c i r c u l a r c o r e s

s o t h e c a b l e s e c t i o n i s s i m i l a r t o t h a t i n F i g . 2 . 7 ( r i g h t ) .

T h e m a i n f u n c t i o n a l e l e m e n t o f t h e c o r e s c re e n is a s e m i c o n d u c t i n g l a y e r i n i n ti m a t e

c o n t a c t w i t h t h e i n s u l a t i o n . T h i s c a n c o m p r i s e a s e m i c o n d u c t i n g v a r n i s h i n

c o m b i n a t i o n w i th a se m i c o n d u c t i n g ta p e b u t , n o w a d a y s , is i n v a r i a b l y a n e x t r u d e d

s e m i c o n d u c t i n g la y e r. T h e c o m b i n a t i o n o f c o n d u c t o r s c r ee n , i n s u l a t i o n a n d c o r e s c re e n

a r e p r o d u c e d i n o n e p r o d u c t i o n p r o c es s t o e n s u re a c o m p o s i t e e x t r u si o n w i t h o u t v o i d sa t t h e i n s u l a t i o n s u r f a c e s .

A s a r e s u l t, t h e e l e c t r ic f ie l d i n th e s e c a b l e s i s k e p t w i t h i n a h o m o g e n e o u s d i e l e c t r i c

a n d h a s a ra d i a l p a t t e r n . I n o r d e r t o p r o v i d e a c o n t i n u o u s e a r t h r e f e r e n c e f o r t h e

s e m i c o n d u c t i n g c o r e s cr e e n an d t o s u p p l y t h e c h a r g i n g c u r r e n t s f o r al l p a r t s o f t h e c a b le

l e n g t h , a l a y e r o f m e t a l l i c t a p e o r w i r e s is a p p l i e d o v e r t h e c o r e s c r e e n .

S O U R C E S O F E L E C T R I C A L L O S S E S

A n e le c tr ic p o w e r c a b l e co n s is ts o f t h r e e b a si c c o m p o n e n t s , n a m e l y t h e c o n d u c t o r ( s ) ,

t h e d ie l e c tr i c a n d t h e o u t e r m e t a ll i c la y e r (s ) . W h e n t h e c a b l e i s e n e r g i s e d a n d c a r r y i n g

l o a d, h e a t, w h i c h m u s t b e d is s ip a t e d t o t h e s u r r o u n d i n g m e d i u m , i s g e n e r a t e d b y e a c h

o f th e se c o m p o n e n t s . T h e e ff ec ts o n c u r r e n t c a r r y i n g c a p a c i t y a r e d i s c u s s ed i n c h a p t e r 8.

C o n d u c t o r lo s s e s

T h e c o n d u c t o r l o s s e s a r e o h m i c l o s s e s a n d a r e e q u a l t o

nl2Ro ( w a t t )

w h e re I = c u r r e n t c a r ri ed b y t he c o n d u c t o r ( A )

Ro = o h m i c a .c . re s i s ta n c e o f t h e c o n d u c t o r a t 0 ° C (f~ )

n = n u m b e r o f c o re s

W h e n h i g h a .c . c u r r e n t s a r e tr a n s m i t t e d , t h e d is t r i b u t i o n o f c u r r e n t i s n o t e v e n l y

d i s p o se d t h r o u g h o u t t h e c r o s s -s e c t io n o f t he c o n d u c t o r . T h i s is d u e t o t h e s k i n e f fe c ta n d t h e p r o x i m i t y e ff e c t a s d is c u s s e d e a r l i e r in r e l a t i o n t o a .c . r e s i s ta n c e .




D i e l e c t r i c l o s s e s

The dielectric losses of an a.c. cable are propor tional to the capacitance, the frequency,

the phase voltage and the power factor. They are given by

D : nwCU o2ta n6 10 -6 (watt/km) (see equation (2.12))

where w = 27r multiplied by frequency

C = capacitance to neutral (laF/km)

U0 = phase to neutral voltage (V)

tan 6 = dielectric power factor

The loss component of the power factor (i.e. the current in phase with the applied

voltage) is made up of

(a) leakage current flowing through the dielectric which is independent of frequencyand consequently occurs with both a.c. and d.c. voltage applications;

(b) dielectric hysteresis, by far the largest effect, which is caused by the interact ion of

the alternating field with the molecules of the constituents o f the insulation and is

only present with a.c. voltage application;

(c) ionisation, i.e. partial discharge in the dielectric.

The power factor of the cable insulation is dependent on frequency, temperature and

applied voltage. It is of a very low order and consequently for cables of up to 50 kV

operating voltage the dielectric losses are small in comparison with conductor losses.However, for cables for operation above this level the losses rise rapidly with voltage

and this must be taken into consideration when calculating the current carrying

capacity of the cables.

S h e a t h l o s se s

When single-core cables carry alternating current the magnetic field associated with the

current induces e.m.f.s in the sheath of the cable and also in the sheaths of surroundingcables. Two types of sheath losses are possible resulting from such e.m.f.s: sheath eddy

current toss and sheath circuit loss.

Sheath eddy current loss

Eddy currents are induced by the current or currents in the conductors of the cables in

close proximity to the sheath. Consider a three-phase circuit with cables R, Y and B.

The flux of cables Y and B cuts the sheath o f cable R. More lines of flux cut the sections

of the sheath of R closer to Y and B than that section remote from Y and B. Thus a

resultant e.m.f, is induced which causes current (eddy current) to flow along one side of

the sheath and return along the other.

The integral of such currents over the sheath cross-section is zero. These eddy

currents are independent of the type of sheath bonding which is applied and decrease

with the distance between the cables.

The sheath eddy current losses are given by

Se12 3I~s (2 ~) 2 ]× 10-s (watt/km per phase) (2.15)




where Se = sheath eddy current losses

I= current (A)

= 27r multiplied by frequency

dm= mean diameter of the sheath (m)

S= distance between cable centres (m)

Rs = sheath resistance (f~/km)

For single-core lead sheathed cables these losses are normally small compared with

conductor losses, but are considerably higher with aluminium sheathed cables when

they are in close proximity.

Sheath circuit loss

When the sheath of a single-core cable is bonded to earth or to other sheaths at more

than one point, a current flows in the sheath due to the e.m.f, induced by the a.c.

conductor current by 'transformer' action. This is because the sheath and return path,to which each end of the sheath is bonded, form a closed loop which is cut by the flux

associated with the current in the conductor. The magnitude of the flux which cuts the

sheath is dependent on the size of the loop which, in turn, is dependent on the spacing

between the cables or between the sheath and the mean return path of the current

through the earth or other medium.

The voltage induced in the sheath is given by

Es = IXm

where I = conductor current (A)

Xm = 27rfM x 10-3(f//km)

The mutual inductance M between conductor and sheath is given by

M - - 0 . 2 loge (2~SmS) (mH/km)

The impedance of the sheath Zs (f~/km) is given by

Z s -~ - ( R s 2 + .,] (m 2 ) /2

where Rs is the sheath resistance (f~/km). Therefore the sheath current Is is equal to

Esl s -

(Rs 2 + Xm2) 1/2

IXm

= ( R s 2 + S . m 2 ) l/ 2 (A)

The sheath current losses per phase are given by

ls2Rs- I2Xrn2RsRs 2 + Xm 2 (watt/kin) (2.16)

Therefore total sheath losses, i.e. sheath circuit losses plus sheath eddy current losses,

are given by

12 - f X ' m 2 3ur2 x 10 -8]

/¢ s ~ R s 2 - ~ ~ .m 2 + I ~ s 2 ( 2 ~ ) 2 } ( 2 .1 7 )




The heat generated by losses in the conductor, the dielectric, the sheath and armour

has to pass to the surrounding medium, which may be the ground, air, water or some

other material. As the current carrying capacity of an electric cable is normally dictated

by the maximum temperature of the conductor, the components of the cable, in

addition to meeting the electrical requirements, must also have as low a thermal

resistivity as possible to ensure that the heat can be dissipated efficiently. The subjects ofheat dissipation and operating temperatures of cable components are discussed in

chapter 8.

M E C H A N I S M O F I N S U L A T I O N B R E A K D O W N I N P A P E R

INSULATED CABLES

In addition to the obvious and by far the most usual reasons for failure, such asmechanical damage to the insulation or ingress of moisture, there are three basic

reasons for failure:

(a) breakdown due to ionisation;

(b) thermal breakdown;

(c) breakdown under transient voltage conditions.

B r e a k d o w n d u e t o i o n i sa t io n

The 'perfect' belted solid type cable would be so manufactured that the impregnant

would completely fill the interstices between the wires of the conductor, the fibres of the

paper, the gaps between papers and the filler material. In short, the whole volume

contained within the lead or aluminium sheath would be completely void free. During

installation, however, the 'perfect' belted cable undergoes mechanical manipulation,

and movement of the cores relative to each other takes place. Also, when the cable is

loaded electrically the conductors, insulation, free impregnant and lead sheath expand,

but the lead sheath does not expand with temperature to the same extent as the interiorcomponents of the cable. The sheath is thus extended physically by the pressure exerted

by the inner components. On cooling, the lead sheath does not return to its original

dimensions and consequently the interior of the sheath is no longer completely

occupied, there being voids formed within the cable.

Installation, repeated load cycling and migration of compound of inclined routes

can therefore all cause voids within the cable. Such voids are particularly hazardous

when they occur within the highly stressed zones of the insulation. As transmission

voltages increased, the insulation of the belted cables was increased to meet the higherstresses involved. On the introduction of 33kV belted cables in the early 1920s,

however, it was found that merely increasing the insulation thickness did not give

satisfactory performance as there was a high failure rate. On studying the problem,

certain weaknesses were found in the belted construction of cables for this voltage.

Because of the design of the cable, part of the flux due to the 3-phase voltage, at a

certain instant of time during the voltage phase rotation, passes radially through

electrically strong core insulation; at other parts of the insulation there is a radial and a

tangential component of the flux, and at still other parts the flux passes through thesound core insulation and into the inferior insulation (fig. 2.4).



2 6 Electric Cables Handbook

T h e t a n g e n t i a l s t r e n g t h , i.e . t h e s t r e n g t h a l o n g t h e s u r f a c e o f th e p a p e r s , o f l a p p e d

d i e l e c tr i c i s o n l y a b o u t o n e - f i f t e e n t h o f t h e r a d i a l s t r e n g t h a n d a l s o t h e f il le r i n s u l a t i o n

is m u c h w e a k e r t h a n t h e n o r m a l c o r e i n s u l a ti o n ; c o n s e q u e n t l y t h e se w e a k n e s s e s w e r e

h i g h l i g h t e d b y t h e h i g h e r s tr e ss e s i n v o l v e d i n 3 3 k V c a b l es .

A m o r e s e r i o us w e a k n e s s in t h e b e lt e d c o n s t r u c t i o n o c c u r s , h o w e v e r , w h e n t h e c o re s

w h i c h w e r e o r i g in a l ly i n c l os e c o n t a c t w i th e a c h o t h e r m o v e o r a r e f o r c e d a p a r t , e i t h e rb y m e c h a n i c a l m a n i p u l a t i o n o r b y c ab l e lo a d i n g . W h e n t hi s h a p p e n s , t h e fl u x p a s se s

t h r o u g h s o u n d c o r e i n s u l at i o n , t h r o u g h t h e s p a c e b e t w e e n t h e tw o c o r e s a n d t h e n

t h r o u g h t h e i n s u la t i o n o f t h e s e c o n d c o r e . T h e s p a c e b e t w e e n t h e c o r e s w h i c h is li k el y t o

b e d e v o i d o f c o m p o u n d i s t h e r e f o r e h i g h l y s tr e ss e d , a n d a n y g a s w h i c h m a y b e p r e s e n t

w i l l i o n i s e . T h e a d j a c e n t c o r e i n s u l a t i o n i s t h e r e b y w e a k e n e d b y i o n i c b o m b a r d m e n t

a n d f a il u r e c a n o c c u r .

T h e s c r e e n e d c a b le w a s i n t r o d u c e d t o e l im i n a t e t h e w e a k n e s s o f th e b e l t e d t y p e c a b l e

c a u s e d b y t h e w e a k f i ll er i n s u l a t i o n , t h e s p a c e s b e t w e e n c o r e s a n d t h e t a n g e n t i a l f l u x.T h e s c r e en r o u n d t h e in d i v i d u al c o r e s c o n fi n e s t h e s tr es s t o s o u n d c o r e i n s u l a t i o n a n d

a l s o e n s u r e s t h a t t h e f l u x i s s u b s t a n t i a l l y r a d i a l .

W i t h t h e i n t r o d u c t i o n o f t h e s c re e n e d o r H t y p e c a b le , m a n y o f t h e w e a k n e s s e s o f t h e

b e l te d c o n s t r u c t i o n w e r e e l im i n a t e d a n d 3 3 k V s o li d H t y p e c ab l es h a v e g i v e n e x t r e m e l y

g o o d s e rv ic e p e r f o r m a n c e f o r m a n y y e a r s. O n i n c r e a s in g t h e tr a n s m i s s i o n v o l t a g e a b o v e

3 3 k V , h o w e v e r , t he o n e w e a k n e s s r e m a i n i n g , t h o u g h n o t m a n i f e s t l y h a r m f u l a t 3 3 k V ,

h a d t o b e e l i m i n a t e d . T h i s fi n a l w e a k n e s s w a s t h e g a s e o u s i o n i s a t i o n i n v o i d s f o r m e d

w i t h in t h e i n s u l a ti o n b y c o m p o u n d m i g r a t i o n r e s u lt in g f r o m c a b l e l o a d i n g a n d s te e p

in c l i n e s .

T h e b r e a k d o w n o f p a p e r i n s u la t io n d u e t o i o n i sa t io n o c c u r s t h r o u g h t h e f o r m a t i o n

o f c a r b o n a c e o u s ' f r o n d s ' o n t h e i n s u l a t i o n p a p e r s ( fig . 2 .8 ). T h i s is g e n e r a l l y k n o w n a s

Fig. 2 .8 Treeing, i.e. car bo n t racking, on pa per insula t ing tapes



Basic Electrical Theory Applicable to Cable Design 2 7

' t r e e i n g ' . T h e c a r b o n a c e o u s p a t h s s t a r t a t a n a l m o s t i m p e r c e p t i b l e c a r b o n c o r e ,

g e n er a ll y a t th e c o n d u c t o r s u rf a ce , a n d g r a d u a l l y s p r e a d o u t w a r d s t h r o u g h t h e

i n s u l a t io n , i n c re a s in g i n w i d t h a n d c o m p l e x i t y a s p r o g r e s s i o n t a k e s p l ac e .

F i g u r e 2 .9 d ep i c ts t h e d e v e l o p m e n t o f t re e p a t h s . T h e s t e p s w h i c h l e a d to b r e a k d o w n

c o m p r i s e t h e f o l lo w i n g .

( a) I o n i s a t i o n t a k e s p la c e w i t h i n a g a s e o u s v o i d i n th e g a p b e t w e e n t h e p a p e r n e x t t o

t h e c o n d u c t o r a n d t h e c o n d u c t o r .

(b ) O w i n g to t h e i o ni c b o m b a r d m e n t o f th e s e c o n d p a p e r , t h e i m p r e g n a n t i s p a r t l y

p u s h e d o u t a n d p a r t l y c o n d e n s e d i n t o c ab l e w a x w i t h t he f o r m a t i o n o f m o r e g a s.

E v e n t u a l l y , i f t h e i o n ic b o m b a r d m e n t is s u ff ic i en t ly s ev e re , a c a r b o n a c e o u s p a t h

w i ll p e n e t r a t e b e t w e e n t h e f ib r e s o f t h e s e c o n d p a p e r .

(c) T h e r e is n o w a c o n d u c t i n g p a t h e x t e n d i n g f r o m t h e c o n d u c t o r , t h r o u g h t h e b u t t

g a p o f t h e fi rs t p a p e r a n d r e a c h i n g t h e s u r f a c e o f t h e t h i r d p a p e r n e a r e s t t o t h ec o n d u c t o r . N e g l e ct in g t he v o l t a g e d r o p a l o n g t h i s p a t h , t h e p o t e n t i a l o f t h e p a t h

f r o n t is s u b s t a n t i a l l y t h e s a m e a s t h a t o f t h e c o n d u c t o r ; t h u s t h e r e i s a p o i n t o n t h e

c o n d u c t o r s i d e o f t h e th i r d p a p e r w h i c h i s a t c o n d u c t o r p o t e n t i a l , i .e . I/1 in f ig . 2 .9 .

T h e r e i s a p o t e n t i a l g r a d i e n t t h r o u g h o u t t h e d i e l e ct ri c a n d t h u s t h e t h i r d p a p e r is a t

a p o t e n t i a l l o w e r t h a n t h a t o f th e c o n d u c t o r , i .e . I/2 i n f ig . 2 .9 . T h e r e f o r e t h e r e

e x i s ts a t a n g e n t i a l s tr e s s V l - 1/2 a c r o s s t h e s u r f a c e o f t h e t h i r d p a p e r .

( d) I o n i s a t i o n d u e t o th i s t a n g e n t i a l s tr es s sw e e p s a w a y s o m e o f th e c o m p o u n d a n d

c o n d e n se s s o m e o f th e r e m a i n d e r , f o r m i n g w a x .

•d•• tI I I : l l I : t l ' ' l L , ~

t J ! J J l

J l l l l l ~ J l l l l I ~ " - . .I l i J I I I I B ~ " S c r e e n

; t i ] 1 1 1 1 1 1I ' i f f ~I t ~ . I I I I f

I I J I T ~ e a d s ~ e a t .I f " J I I I I

i ! l I 1 ! I

' l ' 1 ' i ' l ' 1 ' l

X " ' ~ . . . . . . - - T L - - V 2 d u e t o v o l t a g e d i s t r i b u t i o nJ I ~ i n i n s u l a t i o n

S t r e s s d i s t r i b u t i o n I I

Fig. 2 .9 M echanism of breakdow n o f paper insula t ion by t ree ing




(e) Eventually carbonisation of the compound takes place. 'Fronds' o f carbon spread

out until the gap in the third paper is reached and the path proceeds, through the

gap where there is no fibrous barrier, to the fourth paper.

(f) The treeing mechanism thus progresses, increasing in severity as the distance from

the conductor increases owing to the greater tangential stress which exists.

(g) The carbon fronds at each paper continue in length along the surface of the paperuntil the voltage drop due to the current within the main track and the frond lowers

the tangential stress to a value below ionising level.

Several methods of controlling ionisation may be used. One method is to prevent the

formation o f voids throughout the life of the cable by impregnating the cable with a low

viscosity fluid and maintaining a positive fluid pressure within the cable over the

complete operating temperature range of the cable. This is attained by fitting

pressurised reservoir tanks at strategic positions throughout the cable route. At periodswhen the cable is loaded, the components within the aluminium or reinforced lead

sheath expand and the displaced fluid passes along the cable via the ducts into the

pressure tanks which accept the fluid and retain it until the cable cools. On cooling, the

pressure tanks force the fluid back into the cable, thus retaining void-free insulation.

Such a cable is called a self-contained fluid-filled cable.

Other methods of controlling ionisation are (a) to fill the voids with pressurised gas

and (b) to apply sufficient pressure externally to the cable to close up the voids. The high

pressure gas-filled cable controls ionisation by having the conductors and gap spaces

between the pre-impregnated paper tapes filled with an inert gas under high pressure.

Ionisation does not occur in the gaps because of the high pressure involved. During cable

heating, the gas expands more in the gaps close to the conductor than in those at the

outside of the insulation owing to the temperature differential, and the gas moves

towards the aluminium or reinforced lead sheath. On cooling, the gas is forced by the

differential pressure back from the outside o f the insulation towards the conductor. The

pressure within the cable is kept at all times at a value sufficient to prevent ionisation.

Ionisation within voids is prevented in the pipe type compression cable by the

application of external pressure to the cable. In this case, the mass-impregnated single-core cables are shaped so that the sheath can be deformed under pressure. The 3-core or

three single-core cables are all contained within a steel pipe and the intervening space is

filled with a gas under high pressure. When the cable expands on load, the gas absorbs

the expansion and on cooling the gas pressurises the cable and prevents void formation.

The pipe type high pressure fluid-filled cable is constructed with three unsheathed

cores within a steel pipe. The intervening space within the pipe is filled with a low

viscosity fluid under pressure which prevents the formation of voids.

T h e r ma l b r e a k d o w n i n p a p e r i n su l a t i o n

The dielectric loss angle for fluid/paper insulation has a minimum value in the region of

50-60°C. Thus, at around the operating temperature, a rise in temperature increases the

dielectric loss, so giving a larger heat generation. The rise in temperature also increases

the temperature gradient to the surroundings and raises the rate of heat dissipation. If

the rate o f rise of heat generation is greater than the rate of rise of heat dissipation, the

cable temperature will continue to increase, with the result that the dielectric willoverheat and fail electrically.




Figure 2.10 illustrates the dielectric loss angle versus temperature characteristics of

two types of impregnated paper. Paper A is typical of paper in use some years ago, while

paper B represents paper of the type now used for very high voltage cables. Paper B

exhibits comparatively little variation of DLA with temperature over the important

temperature range, whereas the DLA of paper A increases considerably above 60°C.

Figure 2.11 indicates the effect of using these papers on the thermal stability of400 kV 2000 mm 2 self-contained fluid-filled cable buried at standard depth in ground of

normal thermal resistivity. The straight line shows the relationship between the cable

sheath temperature and the amount of heat which can be dissipated through the

ground. The other curves show the relationship between the total losses generated with

the cable and the sheath temperature. These losses comprise two main components: the

I2 R loss in the conductor and the dielectric loss. The I2 R loss is approximately linearly

dependent on the cable temperature, because of the increase in conductor resistance

with temperature. The dielectric loss is proportional to the DLA in accordance with

0.006-

0.005

c 0.004.

_o 0.003.o

0.002

0.001

Paper A ~ / /

Paper B

2 ' 0 4 b go go l & o 1 ~ 0

Temperature °C)

Fig. 2.10 Dielectric loss angle versus temperature characteristics of two types of impregnatedpaper for high voltage cables

_E

c-O

E

T

/ /60- Paper A ~ ,,~ / /

~ / /Total losses / / /~"

50" generated / / , t , , ", . R + d , e , o c t , c

1 a p e r a/

/30-/ " - . ~ Cable sheath temp.

/ w i th max . heat

20-10. / d~ssipalion

, , , , , T

20 40 60 80 100 120Sheath emperature °C)

Fig. 2.11 Thermal stability of a buried 2000 mm 2, 400 kV cable with a current of 1400 A




equation (2.12). If the current is taken to be 1400 A, applied at ambient temperature,

the cable with the insulation comprising paper B would at first rise rapidly in

temperature because the total losses generated would greatly exceed the amount of heat

which could be dissipated from the sheath. The general cable temperature would

continue to rise until the heat generation and losses were in equilibrium, i.e. at the

intersection of the two lines. The temperature of the cable would then remain steady (ata temperature of 80°C on the graph).

If the cable had insulation corresponding to paper A, the initial temperature rise

would follow a similar pattern. However, when the temperature exceeded 60°C, the rate

of rise of the dielectric losses would exceed the increase in the ability of the sheath to

dissipate the losses and the cable temperature would rise. The losses generated would

continue to exceed the losses dissipated, with the result that the progressive temperature

rise would continue until thermal breakdown of the cable occurred. In this particular

example the current loading condition is somewhat exaggerated but the principleillustrates the importance of selecting paper having suitable characteristics. It has also

been assumed that the sheath bonding and spacing between cables is such that sheath

losses can be ignored.

B r e a k d o w n u n d e r t r a n si e n t c o n d i t i o n s

Cables are designed to withstand transient conditions appropriate to their operating

voltage, and this is an important aspect in the case of pressure assisted transmissioncables (chapter 29). However, should the cable insulation be subjected to transient

voltages such as lightning or switching surges, which are higher than the impulse

voltage for which the cable is designed, a failure may occur. Such a breakdown takes

the form of a puncture of the insulation and is usually very localised in nature.

B R E A K D O W N O F P L A S T I C I N S U L A T I O N

This subject is covered in chapters 3 and 24.

ELECTROMAGNETIC FIELDS

Insulated distribution and transmission cables have an advantage over overhead lines;

the external electrostatic field is zero because of the shielding effect of the conducting

insulation screen within the cable. The magnetic field external to a three-coredistribution cable carrying balanced load currents rapidly reduces to zero because the

vector sum of the spatial and time resolved components of the field is zero. A useful

degree of ferromagnetic shielding is achieved for three-core cables by the application of

steel wire armour which helps to contain the flux. The shielding effect can be significantly

increased by eliminating the air gaps with steel tape armour (suitable for small diameter

cables) or by the installation of the cable within a steel pipe (as employed with high

pressure fluid-filled and high pressure gas-filled cables).

The magnetic field external to single-core cables laid in flat formation does not sumto zero close to the cables because of the geometric asymmetry. The distribution of flux




d e n s i t y c a n b e c a l c u l a te d a n a l y t i c a l ly b y t h e a p p l i c a t i o n o f t h e B i o t - S a v a r t l a w 2 t o e a c h

i n d i v id u a l c u r r e n t c a rr y i n g c o n d u c t o r a n d m e t a l li c s h e a t h a n d b y m a k i n g a v e c t o r i a ll y

a n d t e m p o r a l l y r e s o lv e d s u m m a t i o n ( e q u a t i o n s ( 2 .1 8 ) a n d ( 2.1 9 )) . T h e s i m p l e a n a l y t i c a l

m e t h o d c a n b e u s e d in th o s e a p p l i c a t io n s w h i c h h a v e o n e v a lu e o f p e r m e a b i l i t y a n d

w h i c h d o n o t u s e e d d y c u r r e n t s h ie ld i ng . F o r m o r e c o m p l e x a p p l i c a t i o n , s u c h a s t h o s e

e m p l o y i n g f e r ro u s m a t e ri a ls , s p e c ia li se d c o m p u t e r p r o g r a m s a r e r e q u i r e d w h i c h u s u a l l ye m p l o y a f i ni te el e m e n t a l g o r i t h m w i t h t h e a b i l it y t o m o d e l a n o n - l i n e a r B-H h y s t e r e s i s

c u r ve . I t is u s u a l i n c a l c u la t i o n s t o u s e t h e p e a k v a l u e o f c u r r e n t . T h e w a v e f o r m o f t h e

r e s u l t a n t fl ux d e n s it y is c o m p l e x , c o m p r i s i n g b o t h s i n u s o id a l a n d b ia s c o m p o n e n t s a n d

w i t h a p o l a r i se d v e c t o r r o t a t i n g a b o u t a n a x is a n d p u l s a ti n g i n m a g n i t u d e . I n

c o n s e q u e n c e i t is u s u a l t o q u o t e e i t h e r t h e r .m . s , v a l u e o r t h e m e a n v a l u e o f f l u x d e n s i t y ,

t h e p r e f e r r e d u n i t b e i n g ~ tT (1 l aT = 1 0 m G ) .

J ~ = / ~ 1 " q - / ~ 2 q - ' ' " -~ - J ~ n (2 . 1 8 )

Br = #O#rll s in(w t + ~bl)2 r r r l (2 .19)

w h e r e /~r = r e s u l t a n t f l u x d e n s i t y a t r d u e to Ii, 12,..., In, f ( t , x , y ) (T )

/~ l = fl u x d e n s i ty a t r d u e to I i , f ( t , x , y ) (T )

I I = c o n d u c t o r s h e a t h c u r r e n t ( A pe ak )

r l = d is t a n c e f r o m c e n t r e - l i n e o f c o n d u c t o r 1 , f ( x , y ) ( m )

x , y = c o - o r d i n a t e s o f h o r i z o n t a l a n d v e rt i ca l d i s ta n c e ( m )

# 0 = m a g n e t i c p e r m e a b i l i t y o f f r ee s p a c e ( T / A )

# r = r el a t iv e m a g n e t i c p e r m e a b i l i t y

w = a n g u l a r f re q u e n c y o f s u p p l y ( r a d ia n s / s )

~bl = p h a s e d i s p l a c e m e n t o f 11 ( r a d i a n s )

F o r c o m p a r a t i v e p u r p o s e s i t h a s b e c o m e p r a c t i c e t o c a l c u l a t e t h e m a g n e t i c f l u x

d e n s i t y a b o v e a b u r i e d c a b l e c i r c u it a t a h e i g h t o f 1 m a b o v e g r o u n d l ev e l. A t t h i s h e i g h t

t h e r a t i o o f d i s t a n c e t o c a b l e c e n t r e - l i n e s p a c i n g i s c o m p a r a t i v e l y l a rg e s u c h t h a t t h e

m a x i m u m m a g n i t u d e o f t h e f lu x d e n s i ty is l o w a n d , c o m p a r e d w i th a n o v e r h e a d l in e ,

I - -

¢ /}t -

3 5 - -

3 0 - -

2 5 - -

2 0 - -

1 5 - -

1 0 - -

5 - -

0

(a )

I I I I I- 4 - 2 o 2 4

Ho r izon ta l d i s tance (m)

Fig . 2 .12 H or izon ta l f lux dens i ty d is t ribu t ion I m above g ro und leve l fo r 400kV und ergr oun ddo ub le cable circuits carry ing 1000 A . (T he legend is given in table 2.2.)



3 2

Table 2 .2 Effect

(see also fig . 2.12)

Electric Cables Handbook

of ins ta l la t ion conf igura t ion on magne t ic f lux dens i ty and cab le ra t ing

Conf igura t ion

(Do uble c i rcu i t)

Cab le Ci rcu i t Dep th

spacing spacing (m m )

(mm) ( ram)

Flux densi ty a t 1000A (~tT)

Max . F ig .

Ra t in g p e r

c ircui t

(A)Fla t - XB aF la t - XB b

Fla t - XB

Fla t - XB

Trefo i l - XB

Fla t - SB

300 1500 900

300 1500 900

200 800 900

300 1500 1800

300 1500 900

300 1500 900

30 2.12a 2432

23 2.12b 2432

13 2.12c 1934

11 2.12 d 1886

11 2.12e 2248

2 2.12 f 1370

a Referen ce caseb W ith steel p lates

XB Cro s s b o n d e d sh e a th

SB So l id ly bonded shea th

r a p i d l y r e d u c e s i n m a g n i t u d e o n b o t h s id e s o f t h e c a b l e c ir c u i t , i .e . w i t h i n t h e w i d t h o f a

r o a d w a y ( f i g . 2 . 1 2 ) . S h o u l d i t b e r e q u i r e d , s i g n i f i c a n t f u r t h e r r e d u c t i o n i n f l u x d e n s i t y

c a n b e a c h i e v e d , h o w e v e r t h i s i s i n v a r y i n g d e g r e e s d e t r i m e n t a l t o t h e c a b l e t h e r m a l

r a t i n g a n d t o t h e c o s t o f t h e c i rc u i t. 3 E x a m p l e s o f f lu x d e n s i t y d i s t r i b u t i o n s a r e s h o w n i n

f i g . 2 . 1 2 f o r t h e 4 0 0 k V d o u b l e c i r c u i t c o n f i g u r a t i o n g i v e n i n t a b l e 2 . 2 . T h e s i m p l e s t

m e t h o d s a r e t o l a y t h e c a b l e s c l o s e r t o g e t h e r a n d a t g r e a t e r d e p t h , t h e m o s t e f f e c t i v e

c o m p r o m i s e b e i n g t o l ay t h e ca b le s i n a n o p e n t r ef o il f o r m a t i o n .

A d e g r e e o f c a n c e l l a t i o n o f t h e m a g n e t i c f ie ld is o b t a i n e d b y s o l i d l y b o n d i n g t h e

m e t a l l ic s h e a t h s o f s in g le c o r e c a b le s t h e r e b y p e r m i t t i n g t h e i n d u c e d v o l t a g e s t o d r i v e a

c u r r e n t w h i ch , in id e al t h e o r e ti c a l c i r c u m s t a n c e s , w o u l d b e o f e q u a l m a g n i t u d e a n d i n

a n t i p h a s e t o t h e c o n d u c t o r c u r r e n t . I n p r a c t i c e t h e f in i te r e s i s ta n c e s o f t h e m e t a l l i c

s h e a t h a n d e a r t h r e t u r n w i re s, if p r e s e n t, r e d u c e t h e m a g n i t u d e a n d a l t e r t h e p h a s e o f

t h e s h e a t h c u r r e n t t h e r e b y a c h ie v i n g o n l y p a r t i a l m a g n e t i c s c r e e n i n g a n d w i t h t h ed i s a d v a n t a g e o f g e n e r a t i n g s h e a t h h e a t lo s s o f c o m p a r a t i v e l y h ig h m a g n i t u d e . F e r r o u s

s c r e e n i n g c a n b e e m p l o y e d b y p o s i t i o n i n g s t e e l p l a t e s a b o v e t h e c a b l e s , a p r a c t i c e a l s o

u s e d t o p r o t e c t c a b l e s f r o m v e r t i c a l g r o u n d l o a d i n g a t r o a d c r o s s i n g s . T h e s c r e e n i n g

e f fe c t c a n b e i n c r e a s e d b y p o s i t i o n i n g i n v e r t e d , U - s h a p e d , s t ee l t r o u g h s o v e r t h e c a b l e s

b u t t o t h e d e t r i m e n t o f t r a p p i n g a n u n w a n t e d l a y e r o f a i r a b o v e t h e c ab l es . E d d y

c u r r e n t s c r ee n i n g c a n b e e m p l o y e d b y p o s i t i o n i n g n o n - f e r r o u s p l a t e s o f h i g h e l e c tr ic a l

c o n d u c t i v i t y o v e r t h e c a b le s , h o w e v e r t h is is a n i n e ff i ci e n t s o l u t i o n o f h i g h c o s t .

R E F E R E N C E S

( 1) A r n o l d , A . H . M . ( 1 9 4 6 ) The A.C. Resistance of Non-magnetic Conductors.

N a t i o n a l P h y s i c a l L a b o r a t o r y .

( 2 ) O h a n i a n , H . C . ( 1 9 8 9 ) Physics. W . W . N o r t o n , N e w Y o r k .

(3 ) E n d e r s b y , T . M . , G a l l o w a y , S . J ., G r e g o r y , B . a n d M o h a n , N . C . ( 1 99 3 )

' E n v i r o n m e n t a l c o m p a t i b i li t y o f s u p e r t e n s i o n c a b l e s ' .

cables theory design

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