596 IEEE Transactions on Power Delivery, vol. 4, No. 1, January 1989

A WODEL E%R SWITCEING IMPULSE LEADER INCEPTION AND BREAKDOWN OF LONG MR-GAPS

Faiouk A.M. Rizk Fellow, IEEE

I n s t i t u t de r eche rche d'Hydro-QuSbec (IREQ) 1800 montge S te - Ju l i e , Varennes, CjuEbec, Canada, JOL 2PO

Keywords: Streamer, Leader, Switching Impulse, Breakdorn, A i r I n s u l a t i o n .

Abstract - The paper in t roduces a new mathemati- cal model f o r cont inuous l e a d e r i n c e p t i o n and breakdown of long air gaps under p o s i t i v e swi t ch ing impulses wi th c r i t i c a l time-to-crest..

The model d e a l s with rod-, sphere- and conductor- p l ane gaps. I t .p rov ides novel a n a l y t i c a l expres s ions f o r cont inuous l eade r i n c e p t i o n v o l t a g e , he igh t of t h e f i n a l jump and breakdown vo l t age as well as a n a l y t i c a l t o o l s t o determine t h e c r i t i ca l e l e c t r o d e r a d i u s f o r any gap spacing.

The theo ry is e x t e n s i v e l y compared wi th previous expe r imen ta l r e s u l t s and is t e s t e d a g a i n s t s e v e r a l For- merly developed e m p i r i c a l formulae, r e l e v a n t t o s e v e r a l d i s c h a r g e parameters , f o r d iFFeren t e l e c t r o d e forms and over a wide range of gap spacings.

1N"RODLJCTION

Much work has a l r e a d y gone i n t o t h e i n v e s t i g a t i o n o f spark-over c h a r a c t e r i s t i c s OF l a r g e air-gaps under swi t ch ing impulses, p a r t i c u l a r l y rod-plane gaps under p o s i t i v e p o l a r i t y v o l t a g e s with c r i t i c a l time-to-crest [ I ] , [ Z I P 131.

The phys ica l p i c t u r e t h a t emerges from such inves- t i g a t i o n s mainly comprises t h e start of avalanche a c t i v i t i e s i n t h e v i c i n i t y of t he p o s i t i v e rod a t a c e r t a i n th re sho ld t i e l d , which may l ead t o t h e forma- t i o n of prebreakdown s t r eamers , normally termed First corona. Th i s may be fol lowed, as t h e v o l t a g e con t inues t o rise, by one o r more dark pe r iods du r ing which lit- t l e a c t i v i t y t akes place. Under c e r t a i n conditions t h a t have l a r g e l y d e f i e d a n a l y t i c a l Formulation a l e a d e r d i scha rge might form at t h e stem of t h e streamers. Leader growth may Follow which can a l s o be accompanied by one or more da rk per iods. Under c e r t a i n c i rcumstances the l eade r may con t inuous ly propagate i n t o t h e gap, preceded by l eade r corona. Immediately a f t e r l eade r corona reaches t h e plane e l e c t r o d e , break- down O F t h e gap t akes place ( f i n a l jump) with t h e usua l v o l t a g e c o l l a p s e and flow of cons ide rab le c u r r e n t .

P r a c t i c a l l y eve ry s t a g e of t h e d i scha rge de t e lop - ment b r i e f l y mentioned above has been e x t e n s i v e l y s t u d i e d most notably by members of Les Rena rd i s r e s

An important i n g r e d i e n t i n those s t u d i e s was mathematical modelling of d i f f e r e n t phases O F t he dis- charge. These included g l o b a l models [ 6 ] , [7], a d e t a i l e d streamer propagat ion model [ 8 ] , a dynamic model f o r l eade r propagat ion [9] and a streainer-plus- l eade r model [IO]. These models c o n t r i b u t e d s i g n i f i - c a n t l y t o our p re sen t day understanding O F t h e d i s -

G ~ ~ U P [41, [51.

cha rge mechanism. Fundamental t o the s u b j e c t of t h e p re sen t paper is

a model Formulated by Carrara and ThLone [ l l ] from which we would p a r t i c u l a r l y unde r l ine t h e fol lowing a s p e c t 9:

cont inuous l e a d e r i n c e p t i o n and propagat ion, a s s o c i a t e d wi th c r i t i ca l t ime-to-crest was recognized as t h e most important phase of t h e breakdown mechanism: f o r sphere- terminated rod-plane gaps and For c y l i n d r i c a l conductor-plane gaps, below a cer- t a i n c r i t i ca l e l e c t r o d e r a d i u s , breakdown v o l t - age remains p r a c t i c a l l y cons t an t independent of t he r ad ius . . For e l e c t r o d e s wi th r a d i i of c u r v a t u r e g r e a t e r than t h e c r i t i ca l r a d i u s , €irst corona and con- t i nuous l eade r i ncep t ion p r a c t i c a l l y co inc ide .

For t h e de t e rmina t ion of t h e l e a d e r i n c e p t i o n v o l t a g e of rod-plane and conductor-plane gaps Car ra ra and Thione made use of an empi r i ca l Formula f o r t he 50% sparkover v o l t a g e developed a t EdF [12] as well as ano the r empi r i ca l formula f o r t h e he igh t of t h e Einal jump due t o Lemke [13] .

An a t tempt t o phys i ca l ly e x p l a i n the concept OF c r i t i ca l r a d i u s i n terms of a c r i t i c a l corona i n t e n s i t y was made i n [SI us ing t h e streamer development model of Ga 11 imber t i .

The above i n t r o d u c t i o n l eads n a t u r a l l y t o the l e g i t i m a t e ques t ion : i f so much has a l r e a d y been accom- p l i shed what is l e f t t o j u s t i f y another paper on the s u b j e c t ?

To the a u t h o r ' s knowledge:

t h e r e e x i s t s no formula f o r cont inuous l eade r i ncep t ion vo l t age , r e l a t e d t o gap l eng th and e l e c t r o d e geometry; s e v e r a l u s e f u l empi r i ca l formulae a r e a v a i l a b l e For c a l c u l a t i o n of sparkover vo l t age of a rod- plane gap, however each i s n a t u r a l l y v a l i d w i t h i n a c e r t a i n range OF gap spacing and bea r s l i t t l e o r no phys ica l c o r r e l a t i o n t o our p re sen t day knowledge O F t he d i scha rge mechanism; c r i t i c a l r a d i u s can only be determined from experimental r e s u l t s o r empi r i ca l formulae; . t h e r e appears t o be b a s i c c o n t r a d i c t i o n between a widely used empirical formula For the he igh t of t he F ina l jump [13] and t h e la tes t sparkover d a t a of very l a r g e gaps [14] .

The p resen t paper comprises an attempt to address t h e above a s p e c t s of switching impulse d i scha rge phe- nomena in l a r g e a i r gaps.

Fundamental i n€orms t ion on swi t ch ing impulse l e a d e r c h a r a c t e r i s t i c s were provided mainly through

88 WM 096-0 A paper recommended and approved c o n t r i b u t i o n s O F Les Rendrdisres Group 11, 4, 51. by t h e I C E E Transmission and D i s t r i b u t i o n Committee Under a c r i t i c a l p o s i t i v e switching impulse, with c r e s t of t h e IESE Power Engineer ing Soc ie ty for p r e s e n t a t - vo l t age in the v i c i n i t y of 1JS0, the a s p e c t s most rele- i o n a t t h e IREE/PES 1988 Winter Meeting, Yew York, van t t o the p re sen t work inc lude : New York, January 31 - February 5, 1988. Manuscript submit ted August 26, 1987; made a v a i l a b l e f o r p r i n t i n g l e a d e r propagat ion i s a s s o c i a t e d with charge Uovember 3, 1987. i n j e c t i o n at a r a t e qL = 40-50 $/m of a x i a l

l eade r Length for rod-plane gaps. The l i n e a r

0885-8977/89/0100-05%$01 .WO 1989 IEEE

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597

charge d e n s i t y is somewhat h ighe r fo r s p h e r i c a l e l e c t r o d e s and somewhat lower f o r c y l i n d r i c a l conductors ; under c r i t i ca l c o n d i t i o n s , t h e in s t an taneous app l i ed impulse v o l t a g e i n c r e a s e s so as t o main- t a i n a s e n s i b l y cons t an t l e a d e r t i p p o t e n t i a l as t h e l eade r con t inues t o p e n e t r a t e t h e gap; t h e axial l eade r v e l o c i t y is normally in t h e range 1-2 c d p s with a f r e q u e n t l y quoted va lue of 1.5 cm/ps; w i th t h e excep t ion of probable l a c k of l o c a l thermodynamic equ i l ib r ium, t h e l e a d e r bea r s some re senb lance t o an e lectr ic arc, wi th a v o l t a g e g r a d i e n t t h a t depends on t h e l i f e t i m e , and which v a r i e s from a n i n i t i a l va lue E t of approximately 400 kV/m f o r a newly c r e a t e d l e a d e r segment t o an u l t i m a t e va lue E, of approximately 50 kV/m. An estimate of t he time cons tan t involved is 50 us [ 6 ] .

Our main i n t e r e s t in t h i s paper La t he con t inuous l e a d e r which, by d e f i n i t i o n is one t h a t succeeds t o p e n e t r a t e t h e gap u n t i l t h e f i n a l jump and complete breakdown. Contrary t o most previous i n v e s t i g a t i o n s which d e a l t w i th d i scha rge phenomena i n the v i c i n i t y o f t h e p o s i t i v e high v o l t a g e e l e c t r o d e , we s h a l l t h e r e f o r e c o n c e n t r a t e on t he l eade r c o n d i t i o n s as it approaches t h e f i n a l jump.

A s t h e l eade r propagates i n t o t h e gap, streamer d i s c h a r g e s ahead of t h e l eade r ( l e a d e r corona) w i l l i n g e n e r a l develop under the i n f l u e n c e of l eade r space cha rge being c o n t i n u a l l y i n j e c t e d i n t o t h e gap at a rate of pa C/m, t h e geometr ic Eteld and streamers own space charge. A s w i l l be shown i n t h e fol lowing, as t h e l e a d e r approaches t h e f i n a l jump, t h e f i e l d com- ponent due t o l eade r space charge becomes most promi- nent and p l ays a d e c i s i v e r o l e i n determining t h 2 h e i g h t of the f i n a l jump. We s h a l l t h e r e f o r e ana lyse t h e e lectr ic f i e l d due t o l eade r space charge.

E l e c t r i c F i e l d due to Leader Space Charge

For t h e purpose of f i e l d c a l c u l a t i o n ahead o€ t h e l e a d e r t i p due t o l e a d e r space charge, t he l eade r is s imulated by a s p a c i a l c y l i n d r i c a l d i s t r i b u t i o n of p o s i t i v e charge of l o n g i t u d i n a l d e n s i t y q a s i t u a t e d a t an e f f e c t i v e r a d i u s re around t h e gap a x i s , having a l eng th aZ along the gap, Fig. 1. To determine the a x i a l electric f i e l d E(y) a t a point a t a d i s t a n c e y from t h e l e a d e r t i p , t h e e f f e c t O E induced charge on t h e plane is taken t n t o account through image cha rges of oppos t t e p o l a r i t y . The gap spacing is d. The expres s ion of E(y) fo l lows from e l e c t r o s t a t i c s as:

whtch a€ ter performing t h e i n t e g r a t t o n becomes:

Of p a r t i c u l a r t n t e r e s t I s t he f i e l d Ep, due t o t h e l e a d e r space charge, a t the plane, f o r which y = d-a,:

( 3 )

which, f o r r i << (d - t z )2 s i m p l i f i e s to:

~ W E , [d - a, d ]

1 1 E P = % - - - (4)

Another q u a n t t t y t h a t inight be of i n t e r e s t i s t h e f i e l d a t t h e l e a d e r t i p due t o l eade r space charge:

which f o r re much smaller than t h e o t h e r d t s t a n c e s involved y i e l d s t h e f i r s t approximation:

Fig. 1: Schematic Represen ta t ion of Leader Space Charge.

I t is noted t h a t expres s ion (4 ) provides a s i m p l e means t o e v a l u a t e the e lectr ic f i e l d a t the plane due to t h e l eade r space charge as func t ion of t he R X i Q l l e a d e r l e n g t h f o r any gap l eng th d.

Before t e s t i n g expres s ion (4) a g a i n s t a c t u a l f i e l d probe measurements a t the plane, it might be menttoned t h a t e a r l i e r e f f o r t s 111 using charge s tmula t ion tech- niques € a i l e d t o reproduce the measured f i e l d va lues due t o l i n e a r l eade r charge alone. The m l n reason 9s w e know now is t h a t ear l ier i n j e c t e d charge va lues u t i l i z e d were i n c o r r e c t l y low. Cate r d t g i t a l computa- t i o n s [ I+ ] were inore s u c c e s s f u l a l though the a n a l y s i s was not pursued any f u r t h e r .

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598

I

I I I

loo0 t W)

Fig . 2: Simultaneous Records of t h e Charge (b) I n j e c t e d i n t o a 10 m Gap (500/10 000 ps impulse) and the Electric F i e l d a t t h e Cone T ip (a) and Earthed P lane (c) from Ref. [4 ] .

200

150

E 5

2 w- 100

I I - Field due to leader charge

Feory) X Experiment ( Ref. 4 ) I

-/ n I I I I I " 0 1 2 3 4 5 6 7

&, m Fig. 3: F i e l d at t h e P lane due t o Leader L inea r Space

Charge as Funct ion of Axial Leader Length. 10 m Rod-Plane Gap. Critical Switching Impulse. Tcr = 500 us.

The f i r s t p re l imina ry test of expres s ion (4 ) is based on t he osc i l l og rams of Fig. 2 , produced by Les RenardiSres Group f o r i n j e c t e d charge, f i e l d a t t h e cone e l e c t r o d e and f i e l d a t t h e plane, f o r a 10 m gap exposed t o a 500/10000 p s impulse. The i n s t a n t s of s tar t and end of l e a d e r propagat ion, are i d e n t i f i e d on t h e f i g u r e , corresponding t o an approximate d u r a t i o n of

390 ps. With an average a x i a l v e l o c i t y of 1.5 c d p s t h i s t r a v e l time corresponds t o an a x i a l l e a d e r l e n g t h o f 5.85 m. The osc i l l og rams y i e l d qg o f approxi- mately 45 pC/m, which when s u b s t i t u t e d i n t o expres s ion (4) produces a f i e l d a t t h e plane amounting t o 1.14 kV/cm. The measured va lue i n Fig. 2 is 1.32 kV/cm. The f i g u r e a l s o shows t h a t be fo re t h e start of l e a d e r propagat ion, at t h e plane t h e f i e l d was q u i t e small and was s e v e r e l y l i m i t e d a t t h e rod, i nd i - c a t i n g an e f f e c t i v e d i scha rge s h i e l d i n g e f f e c t on t h e geomet r i ca l f i e l d f o r the rod-plane conf igu ra t ion . With t h e s e encouraging r e s u l t s we proceed t o t h e more e x t e n s i v e v e r i f i c a t i o n of F igu re 3. With qQ of 50 pC/m and p a r t i c u l a r l y f o r l onge r l e a d e r l eng ths , as t h e i n s t a n t of f i n a l jump is approached, t h e r e s u l t s show t h a t t h e f i e l d due t o t h e l eade r space charge, as c a l c u l a t e d above, s a t i s f a c t o r i l y accounts f o r t he most important part of t h e f i e l d at t h e plane.

Cont inuous Leader I n c e p t i o n Voltage of Rod-Plane Gap

The l e n g t h of a streamer d i scha rge may be de te r - mined from t h e p o s i t i o n at which t h e app l i ed f i e l d drops below a c e r t a i n unknown cr i t ical va lue which is l i k e l y t o depend on t h e f i e l d c o n f i g u r a t i o n [ l ] , whi le streamer breakdown occur s when the a p p l i e d v o l t a g e amounts t o t h e product of streamer l e n g t h and mean streamer g r a d i e n t E,, which f o r low breakdown proba- b i l i t i e s is approximately 400 kV/m [ll].

As t h e cont inuous l e a d e r r eaches t h e p o s i t i o n o f f i n a l jump, two c o n d i t i o n s must t h e r e f o r e be s imulta- neously f u l f i l l e d : t h e l e n g t h of t h e streamers c o n s t i - t u t i n g l e a d e r corona must be s u f f i c i e n t t o r each t h e plane e l e c t r o d e and t h e l e a d e r t i p p o t e n t i a l must be adequate t o cause streamer breakdown t o b r idge t h e remaining gap.

A t t h e p o s i t i o n of f i n a l jump t h e r e f o r e , we f i r s t make t h e fol lowing s u b s t i t u t i o n i n (4) above:

d - Q, = hf (7 1

Then t h e l e a d e r t i p p o t e n t i a l which, under cr i t i - cal c o n d i t i o n s , is equa l t o t h e con t inuous l e a d e r i n c e p t i o n v o l t a g e Ulc w i l l be g iven by:

U l C = Eshf

F i n a l l y equa t ing t h e f i e l d at t h e plane due t o t h e l e a d e r space charge t o t h e cr i t ical f i e l d which de te r - mines t h e streamer ex ten t :

Ep = Ecr (9 1

In t roduc ing (8) and (9) i n t o (4) we g e t t h e f o l - lowing expres s ion f o r t h e cont inuous l e a d e r i n c e p t i o n v o l t age :

Recognizing t h a t t h e q u a n t i t y q ~ / 2 n o ~ has t h e dimension of p o t e n t i a l , we s u b s t i t u t e

i n t o (10) t o ge t :

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599

Our a n a l y t i c a l expres s ion (16) f o r Ulc b e a r s s t r i k i n g resemblance t o the well-known EdF empi r i ca l formula f o r U 5 0 [12]. A fundamental d i f f e r e n c e however is t h a t while t h e EdF formula p r e d i c t s s a t u r a t i o n of U50, formula (16) p r e d i c t s s a t u r a t i o n of t h e cont inuous l e a d e r i n c e p t i o n v o l t a g e while , as we s h a l l see la ter , U50 does not s a t u r a t e .

A f u r t h e r check of (17) is o f f e r e d through t h e work of Hu tz l e r 1151 who developed an empi r i ca l formula f o r t h e streamer l eng th as ahead of t h e l eade r t i p f o r any a x i a l l eade r l eng th a,:

4

5

7

10

13.5

and

U l C

800 0.500

880 0.454

1020 0.392

1108 0.361

1161 0.345

1

In terms of t h e cont inuous l eade r i ncep t ion vo l t - age U l c , (13) can be w r i t t e n as:

Equat ion (15) p r e d i c t s t h a t t h e cont inuous l e a d e r i n c e p t i o n v o l t a g e U l c of a rod-plane, under Cr i t i ca l swi t ch ing impulse cond i t ions , is r e l a t e d t o the gap spacing d i n such a way as t o render t h e L.H.S. term cops tan t . Equat ion (15) o f f e r s a s imple way t o test t h e v a l i d i t y of t h e p re sen t approach. The v e r i f i c a t i o n is c a r r i e d out i n Table 1, i n which t h e va lues of U l c of a rod-plane gap f o r spac ings in t h e range of 4 t o 13.5 m are taken from Carrara-Thione [11] . The las t column shows t h a t indeed t h e L.H.S. of equa t ion (15) is s a t i s f a c t o r i l y cons t an t with a mean va lue of 0.257 and a scatter around the mean of f 5%, which is probably t h e range of e r r o r i n t h e experimental i npu t d a t a .

Table 1 Verification of Equation (15)

0.250

0.255

0.249

0.261

0.271

There fo re t h e c o n s t a n t Ecr/Vo, f o r rod-plane gaps, is taken as 0.257. A f u r t h e r experimental check is t o use t h i s va lue ob ta ined f o r Ecr/Vo t oge the r w i th expres s ion ( 1 4 ) t o c a l c u l a t e t h e he igh t of t he f i n a l jump f o r a 2 m gap. The so -ca l cu la t ed v a l u e amounts t o 1.32 m, while t h e measured va lue quoted by Carrara-Thione [ l l ] is 1.3 m. Although t h e a b s o l u t e va lue of E,, is not necessary f o r ou r f u r t h e r anal- y s i s , i t may be i n t e r e s t i n g t o g e t an estimate of t h a t q u a n t i t y . For qa between 40 and 50 $/m t h e charac- terist ic p o t e n t i a l Vo w i l l be i n t h e range 719-899 kV which y i e l d s E,, i n t h e range 1.8 - 2.3 kV/cm. Sub- s t i t u t i n g f o r Ecr/Vo by 0.257 i n (12) and (14) and f o r E, by 400 kV/m i n (12) we g e t t he fol lowing expres s ions f o r t he cont inuous l eade r i n c e p t i o n vo l t age and he igh t of t he f i n a l jump of a rod-plane gap under Cr i t i ca l swi t ch ing impulse:

1556 3.89 1 + - d

U l C = ~

3.89 hf = - 3.89

1+ - d (17)

4.25

1 + - as = 3.5 a,

This formula can be used t o c a l c u l a t e t he he igh t of t h e f i n a l jump s i n c e at t h a t p o s i t i o n as = hf and a, = d-hf. Comparison of hf as func t ion of a, between t h e p re sen t paper and the work of Hu tz l e r is shown in Fig. 4. The agreement is S a t i s f a c t o r y par- t i c u l a r l y s i n c e f o r lower l e a d e r l e n g t h s , Hu tz l e r i n t r o d u c e s an a d d i t i o n a l term i n his formula t o account f o r t he r e s i d u a l e f f e c t s of t h e f i r s t corona, which w i l l raise h i s va lues of hf and b r ing the two cu rves even c l o s e r .

It is i n t e r e s t i n g t o no te t h a t both formulae pre- d i c t even tua l s a t u r a t i o n of t he he igh t of t h e f i n a l jump. On t h e o t h e r hand i n t h e empi r i ca l formula of Lemke [13 ] , t he he igh t of t h e f i n a l jump i n c r e a s e s con- t i nuous ly with the loga r i thm of t h e gap spacing. I f a p p l i e d t o ve ry l a r g e gaps t h i s w i l l be incompatible wi th t h e r e s u l t s of P i g i n i e t a l . [14 ] , which i n d i c a t e a l i n e a t i n c r e a s e of t h e breakdown v o l t a g e wi th gap spac ing , w i th a p r a c t i c a l l y c o n s t a n t v o l t a g e i n t e r c e p t .

3t-

l t 1 0 5 10 15

1,. m

Fig. 4: Height of F i n a l Jump as Funct ion of Ax ia l Leader Length. Rod-Plane Gaps. C r i t i ca l Switching Impulse.

Leader Voltage Drop

The second component of t h e minimum breakdown v o l t a g e UB, be s ide t h e cont inuous l e a d e r i ncep t ion v o l t a g e Ulc is the l eade r vo l t age drop A U ~ .

Due t o resemblance between t h e l e a d e r and t h e e lectr ic arc, t h e l e a d e r conductance p e r u n i t l eng th G may be assumed [6 ] to be governed by a dynamic equa t ion due t o Hochrainer [16] , which € O K a cons t an t c u r r e n t l e a d e r propagat ion y i e l d s

- t / e G ( t ) = G,+ (Gi - G,) e

where t is the l i f e time of t h e l e a d e r s e c t i o n i n q u e s t l o n , G i is t h e i n i t i a l and G, t h e u l t i m a t e va lues and e is a time c o n s t a n t , normally assumed i n t h e range 30-60 us [6 , 9 , 151.

S ince it is known t h a t t he l e a d e r , under c r i t i c a l c o n d i t i o n s , is propagat ing with an approximately con- s t a n t speed v, (19) can he expressed as:

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- Present work

(20) G(x) = G o I + (Gi - G,) e - x h o

where ~0 = ve, which wi th v = 1.5 cm/ps and 8 = 50 vs w i l l be ~0 = 0.75 m. Here x - v t .

S ince t h e l eade r g r a d i e n t E t is r e l a t e d t o t h e conductance per u n i t l e n g t h G and l e a d e r c u r r e n t it 3 600

X

i G

E t =

t h e l e a d e r v o l t a g e drop f o r any axial l e a d e r 1, w i l l be g iven by:

which a f t e r i n t e g r a t i o n y i e l d s :

where Er and E, are the i n i t i a l and u l t i m a t e

Fig. 5: Leader Voltage Drop as Funct ion of Ax ia l Leader Length.

E l ~ U B Ec r

U l c d V o d I n - Vo

(23) L - - + k s * - = - 2d

v a l u e s o f t h e i e a d e r g r a d i e n t , normally assumed 400 kV/m and 50 kV/m r e s p e c t i v e l y .

S u b s t i t u t i n g f o r t h e c o n s t a n t s i n (23) y i e l d s :

AUUQ = 50 &+37.5 ln[8-7exp(-l.33&)] (kV, m)(24)

For t, 2 2 m t h i s s i m p l i f i e s to:

M i " 50 .tZ + 78 (kv. m) (25)

A t t h e f i n a l jump, wi th E, = d-hf, s u b s t i t u t - i n g from (17):

d

1 +- & = - 3.89 d

where k, is a s h i e l d i n g f a c t o r t h a t remains t o be determined, and which was assumed t o be ze ro f o r a rod- plane gap, UB is t h e app l i ed v o l t a g e at t h e i n s t a n t of t h e f i n a l jump (UB - U l c + NE), and r is t h e e q u i v a l e n t conductor r ad ius .

It should be noted t h a t both Vo and E,, w i l l i n g e n e r a l be d i f f e r e n t from those of a rod-plane gap. What (29) says i n e f f e c t is t h a t t h e r e must e x i s t a l i n e a r r e l a t i o n s h i p between t h e l e a d e r space charge f i e l d Vo(Es/Ulc - l / d ) and t h e geometr ic f i e l d 2Ug/d l n ( 2 d / r ) a t t h e plane. I n o rde r t o test t h i s p r e d i c t i o n of t h e theory, we w i l l use v a l u e s of U l c and UB from the paper of Carrara-Thione (111 and perform t h e c a l c u l a t i o n s shown i n Table 2 , f o r a con- duc to r r a d i u s of 0.05 m as used i n r e f . [ l l ] .

T a b l e 2 V e r i f i c a t i o n of Equa t ion ( 2 9 )

which when s u b s t i t u t e d i n (24) , (25) y i e l d s :

and t h e approximate expres s ion

50 d Aut = -

1 +- 3.89 + 78 d

I n Fig. 5 , t h e l e a d e r v o l t a g e drop expressed by (24) is compared t o an e m p i r i c a l formula given by Thione [17].

Cont inuous Leader I n c e p t i o n of C o n d u c t o r P l a n e Gap

Here w e w i l l f o l low the same procedure used f o r t h e rod-plane gap wi th one except ion. In t h e rod-plane gap, we assumed t h a t due t o d i scha rge s h i e l d i n g of t h e po in t e l e c t r o d e , t he geometr ic f i e l d p l ays a minor r o l e i n determining t h e f i e l d at t h e plane. For a c y l i n d r i - cal conductor extending cons ide rab ly beyond the l e a d e r , it is more reasonable t o assume t h a t t h e geometr ic f i e l d a t t h e moment of t h e f i n a l jump p lays a c e r t a i n r o l e i n the de te rmina t ion of t h e e x t e n t of t h e streamers c o n s t i t u t i n g t h e l e a d e r corona.

The above arguments l ead t o t h e fol lowing rela- t i o n s h i p f o r t he conductor plane gap, which r e p l a c e s (15):

I I I I

4 1190 1110 0.110 117.2

0.125

0.134

0.141 65.3

0.145 56.0

F i r s t , i t is noted t h a t c o n t r a r y t o t h e rod-plane case, t h e q u a n t i t y Es/Uic - l / d r e l a t e d t o t h e l e a d e r space charge f i e l d a t t h e i n s t a n t of t h e f i n a l jump is not c o n s t a n t , but i n c r e a s e s s t e a d i l y wi th t h e gap l eng th . Second, l i n e a r r e g r e s s i o n between Es/Ulc - l / d and 2Ug/d In (2d / r ) according t o (29) y i e lded a corre- l a t i o n c o e f f i c i e n t oE 0.999 wi th Ecr/Vo = 0.178 m-' and ks/Vo = 5.767 x (kV)-'. With Vo = 719 kV, t h i s corresponds t o E,, = 1.28 kV/cm and ks = A f t e r some minor s i m p l i f i c a t t o n s , (29) y i e l d s the fol lowing expres s ion €or t h e cont inuous l eade r i n c e p t i o n v o l t a g e of a conductor-plane gap:

0.4.

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ES U l C = X

Ecr + 1 -5.- 2E Q - 2d V o d Vo I n -

S i n c e Ea depends on a,, f o r a c c u r a t e evalua- t i o n should be determined by i t e r a t i o n .

The q u a n t i t y in squa re b r a c k e t s i n (30) is normal- l y around 1.05, so t h a t s u b s t i t u t i n g f o r t h e d i f f e r e n t parameters , a f i r s t approximatioa of Ulc w i l l be:

and a f i r s t approximation f o r t h e he igh t of t h e f i n a l jump of conductor-plane gap:

5.9 hf = - (my kV/m)(32)

5.6 E a 1 + - - 6.5 10-3 2d d I n

APPLICATIONS

Breakdm Characteristic$ of Rod-Plane Gaps

Following [ll] t h e minimum breakdown v o l t a g e of a rod-plane gap can be determined as t h e sum oE U l c and N a y us ing (16 ) , (27) and (28) g iv ing :

1556 + 50d

1 +3.89 "B = d

(33)

and f o r Q, 1 2 m i.e. d 2 4 m:

1556 + 50d u B = 3.89 1 +-

+ 78

d (34 1

The 50% sparkover v o l t a g e can be determined i n t h e u s u a l manner as:

UB '50 = (35)

where U is the s t anda rd d e v i a t i o n , which f o r c r i t i c a l swi t ch ing impulses and rod-plahe gaps amounts t o approximately 5%. With t h i s va lue of U we g e t t h e Eol- lowing expres s ion f o r U 5 0 based on (33) and (35):

18 30+5 9 d -1.33d U50 = - 3.89 + 44 In [8 - exp(m)]

1 +r (kV, m) (36)

and corresponding t o (34):

1830 + 59d '50 = 3.89 + 92

l + a -

The r e s u l t s of coinputatton of U l c and U s 0 f o r rod-plane gaps i n the range 2-27 m are shown i n Fig. 6

toge the r with corresponding experimental po in t s . The agreement between theory and experiment f o r U s 0 is w i t h i n 2.5% except f o r t h e 27 m gap where the theo ry p r e d i c t s a va lue 6% h ighe r than t h e va lue r epor t ed in r e f . [14] .

4

3

r 2

1

0

I

- Present work

I + CESI-ENEL ( Ref. 14

I I

d, m 5 10 15 20 25 3

Fig. 6: Continuous Leader Incep t ion Voltage and 50% Sparkover Voltage of Rod-Plane G a p s under Cri t ical P o s i t i v e Switching Impulses.

Critical Sphere Radius

With the a v a i l a b i l i t y of an a n a l y t l c formula f o r t h e cont inuous l e a d e r i n c e p t i o n v o l t a g e U l c , t h e de t e rmina t ion of t h e c r i t i ca l sphere r a d t u s f o r any gap spac ing can be done by equa t ing [Ilc t o t h e corona i n c e p t i o n v o l t a g e U t .

A formula, based on the s t reamer theory, f o r t he corona incep t ion f i e l d i n terms of the mean r a d i u s of c u r v a t u r e of t he e l e c t r o d e was given i n [17]. For a sphere with r a d i u s r, m, t he formula r eads

Th i s f i e l d is r e l a t e d t o t h e incep t ion vo l t age U i by:

u i E t = k g y - (39 )

where kg is a geomet r i ca l Eactor r e l a t e d t o the sphe re suppor t etc. which i n eEEect Lnfluences the equ iva len t r a d i u s o f t he sphere. I n Les Renard l s r e s experiments [ 5 ] kg can be determined t o be i n t h e range 0.8-0.9 f o r sphe res with r a d i i i n t h e range 0.125-0.50 m and gap l eng ths i n the range 3-10 m.

For any gap l eng th the c r i t i c a l r a d i u s can then be determined from:

2300 0.224 1556

d

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602

r s m

0.005

Computation r e s u l t s are shown i n Fig. 7 , where it is shown t h a t t he v a r i a t i o n i n kg can account t o a g r e a t e x t e n t f o r t h e d i f f e r e n c e between t h e experimen- t a l l y determined c r i t i c a l r a d i i a t CESI and P r o j e c t UHV

Ulc, kv Auk, kV UB, kV U ~ O , kV

469 2145 2318 1676

40

30

fi 2

20 - Present work I X Y

CESI

x Project U W - lerlments

0 5 10 15 20 25 30 d. m

Fig. 7: Cri t ical Sphere-Radius as Funct ion of Gap Spacing. Sphere-Plane Gaps.

3

a 1.2

* 7 1.1

1.0 0

Fig. 8:

I I I

- Present work x Experiments ( Ret. 14 )

~ // Calculated

/ ’- I---- Empirical ( Ref. 14 )

5 10 d, m

15 20

P o s i t i v e Switching Impulse Sparkover Charac- ter is t ics of Conductor-Plane Gaps.

C h a r a c t e r i s t i c s of C o n d u c t o r P l a n e Gaps

The same procedure o u t l i n e d above was followed t o c a l c u l a t e the cont inuous l eade r i ncep t ion v o l t a g e U l c dnd U s 0 i n the range 3-20 m with one except ion: follow- ing [L4] t he s t anda rd d e v i a t i o n U f o r t h e conductor- plane gap was taken a s 2 . 5 % . The r e s u l t s a r e shown i n Fig. 8 and compared with experiments with r = 0.005 in i n both cases . Also i nd ica t ed is t h e gap f a c t o r com-

puted as t h e r a t i o of U s 0 f o r t h e conductor-plane and t h e rod-plane gaps f o r t he same spacing. Had the cal- c u l a t i o n s been made wi th U - 5% OP had K been de te r - mined as the r a t i o of U l c i n s t e a d of Use, K would have been i n the range 1.22-1.26, i n s t e a d of t h e va lue 1.15 r epor t ed in Fig. 8.

To i n v e s t i g a t e t h e e f f e c t of conductor r a d i u s on t h e sparkover c h a r a c t e r i s t i c s computations were c a r r i e d o u t , f o r a 12 m gap, w i th conductor r a d i i i n t h e range 0.005-0.08 m, which as we can see from Fig. 9 , is below t h e c r i t i c a l r a d i u s f o r t he gap i n ques t ion . The r e s u l t s are shown i n Table 3. It is remarkable t h a t i n c r e a s i n g t h e conductor r a d i u s by 16 f o l d , but s t i l l remaining below t h e cr i t ical r a d i u s , d i d no t i n c r e a s e t h e 50% sparkover v o l t a g e by more than 3%.

Tab le 3 V a r i a t i o n of Sparkover Characteristics d t h

Conductor-Radius for a 12 n Conductor-Plane Gap

F i n a l l y c a l c u l a t i o n s of t h e c r i t i ca l r a d i u s f o r conductof-plane gaps, w i th k, = 1, are shown i n

- Present work x Emplrlcal fromula ( Rat. 17 )

I I I

d, m 0 5 10 15

Fig . 9: Cri t ical Radius of Conductor-Plane Gaps Funct ion of Gap Spacing.

0

as

CONCLOSIONS

1. A new model €or cont inuous l e a d e r i ncep t ion and breakdown of long rod-plane gaps under c r i t i ca l p o s i t i v e swl t ch ing impulses has been developed, based on p resen t knowledge of d i scha rge physics .

2 . The model provides novel a n a l y t i c a l expres s ions € o r cont inuous l eade r i ncep t ion vo l t age , he igh t of t he f i n a l jump and breakdown v o l t a g e o€ rod-plane gaps as well as an a n a l y t i c a l t o o l t o determine the cr i t - ical r a d i u s f o r any gap spacing.

3 . The p resen t work p r e d i c t s s a t u r a t i o n of the cont inu- ous l e a d e r i n c e p t i o n vo l t age but no s a t u r a t i o n of UB o r a t l a r g e gap spacings.

4. The same b a s i c model has been s u c c e s s f u l l y extended t o conductor-plane gaps and has been a b l e , €of t h e f i r s t t i m e , t o account fo r t he almost cons t an t 50% sparkover vo l t age f o r conductor r a d i i below cr i t - i c a l .

5. The model cove r s a very wide range of gap spac ings

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and success fu l ly accounts fo r s eve ra l previously developed empi r i ca l formulae r e l evan t t o d i f f e r e n t a spec t s of t he discharge c h a r a c t e r i s t i c s . The present work exp la ins dev ia t ions between previ- ous experimental r e s u l t s r e l a t e d to c r i t i ca l r ad ius t h a t have not been previously accounted f o r quanti- t a t i v e l y . The theory has been subjected to ex tens ive experi- mental v e r i f i c a t i o n and with view of t he complex na tu re of t h e sub jec t , t h e agreement is very s a t i s f a c t o r y .

603

BOGBAPBP

6.

7.

BgPEBENCBS

[ I ] Les Renardizres Group, "Research on Long A i r Gap Discharges at Les Renardizres," E l e c t r a , No. 23, J u l y 1972, pp. 53-157.

[ 2 ] C. Menemenlis, G. Harbec, "Coeff ic ient of Varia- t i o n of t he Positive-Impulse Breakdown of Long Air-Gaps," IEEE Trans., Vol. PAS-93, 1974, pp. 916-927.

131 H.M. Schneider, F.J. Turner, "Switching-Surge Flashover C h a r a c t e r i s t i c s of Long Sphere-Plane Gaps f o r UW S t a t i o n Design," IEEE Trans.,

14) Les Renardisres Group, "Research on Long A i r Gap a t Les Renardizres - 1973 Resul ts ," Electra,

[51 Les Renardizres Group, "Pos i t i ve Discharges in Long A i r Gaps a t Les Renardizres - 1975 Resu l t s and Conclusions," E l e c t r a , No. 53, 1977, pp. 31-151.

[6] B. Jones, "Switching Surges and Air Insu la t ion , " Ph i l . Trans. R. Soc. Lond. A. 175, 1973,

[7] R.C. K l e w e , R.T. Waters, B. Jones, "Review of Models of Breakdown,'' IEEE Pub l i ca t ion 74 CH0920-0-PWR, 1974, pp. 29-40.

[ 8 ] I. Gal l imbert i , "A Computer Model f o r Streamer Propagation," J. Phys. D., Vol. 5 , 1972, pp. 2179-2189.

191 B. Hutzler , D. Hutzler-Barre, "Leader Propagat ion Model f o r Determination of Switching Surge Flash- over Voltage of Large A i r Gaps," LEEE Trans.,

[ l o ] I. Gal l imbert i , A. Osgualdo, "Space-Charge Distri- but ion and Leader Channel C h a r a c t e r i s t i c s , " I E E PrOC., Vol. 133, Pt-A., NO. 7, 1986, pp. 459-463.

[11] G. Carrara, L. Thione, "Switching Surge S t r eng th of Large A i r Gaps: A Physical Approach," IEEE Trans., Vol. PAS-95, No. 2, 1976, pp. 512-524.

[12] G. Gallet , G. Leroy, R. Lacey, I. Kromer, "General Expression f o r P o s i t i v e Switching Impulse S t r eng th Valid up t o Ex t ra Long A i r Gaps," IEEE Trans., Vol. PAS-94, No. 6, 1975, pp. 1989-1993.

[13] E. Lemke, "Elements f o r t h e Evaluat ion of t h e Sparkover Voltage of Long Air Gaps," ( i n German), E lek t r . Inform. Energietechnik Leipzig, Vol. 3 , No. 4, 1973, pp. 186-192.

[14] A. P i g i n i , G. Rizzi , R. Brambtlla, E. Garbagnati, "Switching Impulse S t r eng th of Very Large Air Gaps," ISH, Milan, 1979, paper No. 52.15.

[15] B. Huteler , D. Hutzler , "A Model of t he Breakdown i n Large A i r Gaps," EdF Bull. Etudes et Recher- ches , Sg r i e B, No. 4, 1982, pp. 11-39.

[ 161 A. Hochrainer , "Einige Bemerkungen zum Stromnulldurchgang in Wechselstromschaltern," EuM,

[17] L. Thione, "The Electric S t r eng th of A i r Gap Insu- l a t i o n , " in Surges i n High Voltage Networks, Edi ted by K. Ragal ler , 1979, pp. 165-205.

[18] H.M. Schneider, L. Za f fane l l a , Discussion of r e f .

Vol. PAS-94, NO. 2, 1975, pp- 551-559.

NO. 35, J u l y 1974, pp. 49-156.

pp. 165-180.

Vol. PAS-97, NO. 4, 1978, pp. 1087-1096.

Vol. 87, NO. 1, 1970, pp. 15-19.

[ I l l .

Farouk A.M. Rizk (Fellow, 82) was born i n Egypt on 6 J u l y 1934. H e holds a B.Sc.Eng. (19551, M.Sc. (1958) from Cairo Universi ty , a L i c e n t i a t e of technology degree (1960) from the Royal I n s t i t u t e of technology, Stockholm, Sweden and a doc to r of technology degree (1963) from Chalmers Un ive r s i ty of technol- ogy, Gothenburg, Sweden.

D r . Rizk worked as a research engineer with ASEA, Sweden, in t h e High power Laboratory, Ludvika, (1960-1963) and i n the Computer Department, Vasteras, (1963). H e worked f o r t h e Egyptian E l e c t r i c i t y Authori ty (1964-1971), becoming manager, High Voltage, in 1968. H e joined t h e I n s t i t u t de recherche d'Hydro-QuCbec (IREQ) as a sen io r research s c i e n t i s t i n 1972, becoming program manager in 1975, s c i e n t i f i c d i r e c t o r (1976), d i r e c t o r power t ransmission (1980) and vice-president (1986). Since January 1987 D r . Rizk ho lds t h e t i t l e of Fellow Research S c i e n t i s t a t IREQ.

D r . Rizk was awarded the Egyptian Nat ional P r i z e of Engineering Science f o r 1971 and decorated by t h e Order of Worthiness (Third Class) i n 1972 and the Order of Science ( F i r s t C la s s ) in 1973. Dr. Rizk is a regis- t e r e d p ro fes s iona l engineer in Quebec and has been, s i n c e 1984, i n t e r n a t i o n a l chairman of Technical Committee 28: I n s u l a t i o n Coordination, of t h e I n t e r n a t i o n a l E lec t ro t echn ica l Commission (IEC).

Discussion

Prof. M. Khalifa (University of Cairo, Egypt): The author should be congratulated for his excellent contribution. Now, the experimental observations about the breakdown of long air gaps under positive switching surges could be mosaiced by one more scientific formula rather than a number of empirical relations with a limited range each.

The author's model for the leader inception and breakdown in long gaps under atmospheric pressure seems to have valiantly passed numerous checks with experiment. The impression one gets is that the model is correctly based on the proper physical phenomena. The encouraging results given in the paper makes one wonder if only minor modifications are needed to make the model applicable also under switching impulses with fronts shorter, even much shorter than critical. The author's comments would be highly appreciated.

Thinking of the author's successful model fairly accurately predicting breakdown voltages of meter-gaps in atmospheric air, and thinking of Paschen's law, would the model apply equally successfully to gaps of the order of 10 cm at pressures of 10 atmospheres; as expected in GIs? or some adjustment would then be needed?

For the conductor-to-plane gaps, the author cleverly adopted a shielding factor and assigned to it a magnitude of 0.4. Accordingly, the 50% breakdown voltages of a 12 meter gap as calculated came insensitive to the conductor radius so long as it is less than the critical radius. One wonders what would be the suitable magnitude for the shielding factor in case of a gap of only one meter say. The author's comment would be highly appreciated.

Manuscript received January 28, 1988.

A. Pigini (CESI Via Rubattino 54 Milano, Italy): The Author is to be commended for the appreciable work, which led to the formulation of analytical expressions for leader inception voltage and for the breakdown voltage of rod plane and conductor plane geometries holding up to very large clearances.

Most of the approaches and models proposed in the literature, including that of the Author, deal with the various configurations separately: for instance in the paper rod plane and conductor plane geometry is dealt with separately, while some unifying criterion should exhist.

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604

UUCTPR

Rc.

0 L 4 8 12 H(m)

Fig. 1. Streamer length for sphere-plane and conductor plane at leader inception voltage for electrodes having critical tadius.

n

I ’ , , , , I- ’ H(m) lo Fig. 2. Computed critical radii for conductor bundles.

0 2 4 ‘ 6

An attempt to find a common criterion for electrode-plane geometries was made in [l]. In particular special attention was paid to the leader inception condition of rod plane and conductor plane, with electrodes having critical radii, and it was shown that, for any given electrode height on the ground plane, the streamer length at the leader inception voltage was very close for the two geometries (see Fig. 1). The assumption was made that the above streamer lengths (with related streamer charge) were the minimum ones necessary for streamer to leader transition thus allowing the determination, with a feed back procedure, the critical size of other geometries (e.g. conductor bundles, see Fig. 2). Still the method presents strong limitations due to the apparent dependence of the critical streamer length on the gap clearance, making difficult its application to configura- tions different from the “electrode-plane” one.

It is obvious that models which could permit to extrapolate the experimental results to complex configurations would be of even greater usefulness than models which substantially help in optimal interpolation of the experimental data for simple configurations.

Do the Author envisage any possibility of extending the approach such to allow the application to other geometries and configurations?

Reference

[ l ] R. Brambilla, A. Pigini “Discharge phenomena in large conductor bundles” 4th International Conference on Gas Discharges, IEE, Swansea 1976.

Manuscript received January 29, 1988.

w. D. Lampe (ASEA-BBC High Voltage Research, Ludvika, Sweden): This paper represents a significant contribution to a deepened understanding of the switching impulse strength of long airgaps. The propOsed model finds an impressive support in the available experimental results. The author is to be congratulated.

Not unexpectedly, the critical field strength in the leader and in the streamer before and during the final jump play an important role. The

degree of ionization a d the size of the discharge channel determine these fields strengths. The question therefore arises if these basic processes for charge creation and the thermodynamics could also be taken into account, which means that the formula can also include the intrinsic dielectric strength of the gas and its fundamental properties.

This might not only provide a sound basis for the density corrections in breakdown strength but also improve the predictions for other gases, in particular those with high electron detachments.

Another question regards the possibility to extrapolate the theory to stepped leaders and then also cover the long gaps bridged by lightning.

Manuscript received February 19, 1988.

Sune Rusck and Roland Eriksson (The Royal Institute of Technology, Stockholm, Sweden): The author is to be complimented to an excellent paper, which, starting from the model formulated by Carrara and Thione and using the present knowledge of the physics of long sparks, develops quantitative formulas for the description of the sparkover of long gaps subjected to switching impulses.

The author uses two assumptions for the calculation of the continuous leader inception voltage U1, viz.

U t c = E A (8)

Ep = E,, (9)

These two assumptions together with equation (4) give a formula for the calculation of the final jump hf

k hf=-

k 1 +-

d

where k = 3,89 In Table A, hf is calculated as a function of d using eq. (17) with k =

3,89 and compared with the values given by Carrara and Thione. As can be seen there is a systematic discrepancy between the values calculated by the formula given by Rizk and those quoted by Carrara and Thione. In the same table hf is also calculated by eq. (17) using k = 5 which gives an excellent agreement with the experimental values. This implies that the latter values give a constant critical field strength at the plane, which was one of the postulates of the author.

Table A hf a function of d

d 2 4 5 7 10 13,5

Rizk 1,32 1,97 2,18 2,50 2,80 3,02 Carrara-Thione 1,30 2,20 2,60 2,95 3,30 3,61 5/(1 + 5/d) 1,43 2,22 2,50 2,92 3,33 3,65

If the value k = 5 is accepted, eq. (8) must be changed to

Utc = U, + Eshf

A numerical investigation shows that U, = 100 kV and E, = 300 kV/m which give the formula

500 l W + -

d U,, =-

5 1 +-

d

This formula gives about the same values of Ut, as eq. (16). However, with k = 5 the length of the final jump and the critical leader length agrees better with the experimental results. The modified leader length should then be introduced into eq. (23) to consider the voltage drop Aut. By suitable choice of constants in eq. (23) it might be possible to improve the agreement between calculated 50% sparkover voltages and experimental results for large spacings as compared in Figure 6.

We would appreciate the author’s comments upon our suggestions on a modified formula for Ut, based upon one set of reported hf values. Are there further experimental results avaible by the author or others for comparisons with the proposed modified theoretical approach?

Manuscript received February 19, 1988.

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Herman M. Schneider (GE-EPRI High Voltage Transmission Research Center, Lenox, Massachusetts): The author should be commended for making considerable pfogress in the application of theoretical models to the switching impulse breakdown of large air gaps. It is particularly noteworthy that some of the difficulties in applying concepts such as critical radius have been resolved. However, in extending the present model to the conductor- plane gap, a new complication in the form of a “shielding factor” was introduced to account for the effect of the geometric field near the plane. What quantitative criterion is proposed for deciding on the need for the shielding factor? For example, would a shielding factor be required for a configuration such as conductor-tower, and, if so, what approach is suggested for determining it?

Manuscript received February 19, 1988.

M. Abdel-Salam (Assiut University, Assiut, Egypt): The author should be commended for this interesting paper in which he developed a mathematical model for calculating the continuous leader inception and breakdown voltages of long air gaps under positive switching impulses with critical time-to-crest. The model is based on some assumptions; namely: 1- Axial propagation of the leader. 2- Constant charge injection during leader propagation (q = 45 pclm). 3- Constant velocity of leader propagation (U = 1.5 c d p s ) 4- Resemblance between the leader and the electric arc with a conductance which varies expotentially with the life time of the leader (time constant 8 = 50 us). Subsequently, the voltage gradient within the leader varies from an initial value E, ( = 400 kV/m) to an ultimate value E, ( = 50 kVlm). 5- Constant voltage-gradient through the leader corona streamer (E, = 400 kV/m).

The agreement between the values predicted by the model including the height of the final jump, leader voltage-drop and 50% breakdown voltage and those measured experimentally is excellent in the light of the abovementioned assumptions and the unique values of q,, U, E&, E, and &. The model dealt with rod-, sphere- and conductor-plane gaps. Was any attempt made to check if the model is applicable for more realistic gap geometries such as tower-window gaps?

The model is restricted to critical conditions, namely switching impulses which lead to the minimum of breakdown voltage. Is there a way to extend the model to take into account the waveform of the applied voltage? Streak photographs of the luminosity during positive switching impulse

breakdown, in a wide variety of electrode geometries, show that the corona clouds (including the leader itself and the streamers at its tip) are conical in shape [ 1,2,3]. Would the author comment on the cylindrical representation of the leader space charge given in Fig. 1 of the paper. The field at the tip of the corona-cloud (&,) was found [3] to remain approximately constant as the cloud grows, eveh though the applied voltage increases. When the total length of the leader and streamer exceeds 0.667 of the gap spacing, the developing discharge becomes unstable and & increases. The author’s comments on these findings in the light of his model are welcomed.

References

[l] I. S. Stekolnikov and A. V. Shkilev, “Development of a Long Positive Spark in the Case of an Exponential Wavefront”, Soviet Physics-Doklay, Vol. 8, pp. 825-828, 1964.

[2] B. Hutzler, J. Jouaire and G. Leroy, ‘‘Discussion Contribution”, IEEE Trans. Vol. PAS-94, p. 1031, 1975.

[3] L. E. Kline, “Corona Cloud Model Predictions of Switching Surge Flashover Voltages Versus Electrode Geometry”, IEEE Trans., Vol. PAS-96, pp. 543-549, 1977.

Manuscript received February 22, 1988.

Gianguido Carrara , CESI, Mllano ( I t a l y ) , EC - The paper r e p r e s e n t s a n o t i c e a b l e s t ep forward in t h e so-called “physical approaches” t o e s t a b l i s h t h e model under d i s - cussion. These approaches a r e midway between the empir- i c a l formulae i n t e r p o l a t i n g t e s t r e s u l t s , and t h e high- l y d e s i r e d , but s t i l l not ex i s t ing , physical models of t h e whole d ischarge phenomenon. In my opinion, t h e suc- cess of a physical approach should be based on: a ) the number of CONSTANT parameters necessary t o c a l c u l a t e t h e d ischarge vol tages of a given number of i n s u l a t i o n conf igura t ions ; b ) whether these parameters a r e based

on physical a s p e c t s of the discharge; c ) t he v a l i d i t y of t h e model in regions f a r from t h e r e s u l t s on which t h e parameters were evaluated. The Author’s model is l i m i t e d t o rod-plane and conductor-plane, bu t , in t h i s f i e l d , meets t h e t h r e e mentioned requirements much bet- t e r than any s i m i l a r approach presented up t o now. Con- c e r n i n g point a ) : only eight parameters (q l , E G , 8 , v , k ), a l l CONSTANT, a r e necessary ab c:fi c&iat@’the disch8rge vol tages . A s concerns point b) a l l parameters are r e l a t e d t o physical elements of t he d i s - charge process, except parameter k i n formula (29) . AS concerns poin t c ) t h e parameters wl re determinad on re- s u l t s obtained up t o 13.5 m f o r rod-plane, and t o 12 m f o r conductor-plane, but g i v e f a i r l y a c c u r a t e estimates up t o 27 m and 17 m r e s p e c t i v e l y , which a r e the maximum d i s t a n c e s f o r which t e s t r e s u l t s a r e a x a i l a b l e . The 6% discrepancy of t he r e s u l t f o r rod-plane gap of 27 m i n Fig. 6 could be reduced adopting a s l i g h t l y lower value f o r G which seems more r e a l i s t i c . Furthermore, t he AuthoFPi model expla ins d i f f e r e n c e s (Flg. 7) which, i n the “ c r i t i c a l rad ius” model , were considered a s approx- imat ions t o be accepted. Now t o the ques t ions : can the Author g i v e a more d e t a i l e d i n d i c a t i o n on where the pa- rameter E is mentioned i n r e f . [ l ] ? Can t h e Author express hfg opinion on what follows? I ) My a n a l y s i s of t h e model. i i ) The physical reasons why k is not equal t o un i ty , t h a t i s why t h e r e is no superp8s i t lon of t h e e f f e c t s . i i l ) What could be t h e q u a l i t a t i v e changes re- quired t o extend t h e model t o phase-to-phase configura- t i o n s , energized by two vol tages of oppos i te p o l a r i t y . While expressing my congra tu la t ions f o r t h e exce l len t work, I h e a r t i l y encourage the Author t o devote h i s fu- t u r e e f f o r t s t o the extension of t he model.

Farouk A.M. Rizk: The author would l i k e t o thank the d i s c u s s e r s fo r t h e i r keen i n t e r e s t , va luable c o n t r i b u t i o n s and per t inent questions.

The au thor would l i k e t o br ing the following p o i n t s t o the a t t e n t i o n of D r . Abdel-Salam: 1. The c y l i n d r i c a l space charge column represent ing t h e leader in the present model should not be confused with the e l e c t r i c a l l y conducting leader channel. A s i n d i c a t e d by Hutzler in r e f . [9] of the paper, the r a d i u s of t he space charge column is estimated a t 0.5 m o r more while the rad ius of the e l e c t r i c a l conduction channel of t he leader is of the order of a mi l l imeter . 2. A s the leader is almost dark, s t r e a k photographs with c o n i c a l d i scharge shape r e f e r t o streamer d is - charges ahead of the leader t i p ( l e a d e r corona) and a s such are not re levant t o our r e p r e s e n t a t i o n of the leader space charge column. 3. A s mentioned in the paper, work by Les Renardieres Group [4 ] showed t h a t a l i n e a r space charge along the gap a x i s is adequate t o account for the e l e c t r i c f i e l d a t the plane. It follows tha t a s long as the r a d i u s of the space charge column is much smal le r than the length of the f i n a l jump, the exact shape of t he column is of minor importance t o the determination of the f i e l d a t the plane. 4 . A s noted by A. Fischer i n the d i s c u s s i o n of r e f . 3 of Abdel-Salam’s d iscuss ion , the corona cloud model s u f f e r s from a fundamental l i m i t a t i o n a s it assumes the corona-cloud t o be a per fec t conductor. That is why t h a t model has not been r e f e r r e d t o in the present paper.

Professors Rusck and Eriksson suggest t o modify formula (16) of the paper, by rep lac ing the cons tan t 3.89 by 5.0, i n order t o provide b e t t e r f i t t i n g t o hf d a t a in the paper of Carrara-Thione 1111. The author d isagrees with such proposal fo r the following reasons: 1- The d i s c u s s e r s a re under the f a l s e impression tha t hf va lues in r e f . [ll] were obtained experimentally. In f a c t , a s mentioned i n re ference [ll] and in the present paper, those hf va lues fo r d > 4 m were computed from Lemke’s model [13]:

E G

hf = 1 + l n d

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606

It is obviously meaningless to try to fit this formula with another formula of the form:

2- The suggestion of Rusck and Eriksson necessitates the introduction of an arbitrary formula for Ulc and the adoption of an unrealistically low value of E, = 300 kV/m. 3- The authors of ref. [ll] were careful not to use Lemke's model for direct calculation of the leader inception voltage but used it indirectly to determine the leader length, from which the leader voltage drop was deduced, with a relatively minor impact on the accuracy of Ulc. 4- The fundamental difference between our formu- la (16) and that of Lemke for hf is that the former predicts saturation of the final jump while the latter does not. The paper has already pointed to the incom- patibility between Lemke's formula for the final jump at very large gaps and experimental results [14] showing a linear increase of the breakdown voltage with gap length. Such experimental finding, on the other hand, has been fully accounted for by the model of the present paper.

Dr. Pigini presented some interesting results from an earlier attempt to find a common approach to deal with leader inception of electrode-plane geometries based on Gallimberti's model [8]. Pigini admitted however that the method presents strong limitations. It should be mentioned on the other hand, that the approach used in the present paper to deal with rod- plane, sphere-plane and conductor plane configurations remained fundamentally unchanged. The basic criteria for all configurations considered are: the extent of streamers constituting the final jump is determined by a critical value of the applied field and the leader inception voltage must be sufficient to cause streamer breakdown of the final jump. The fact that the rela- tive importance of the different components of the applied field as well as the numerical value of E,, varied between rod-plane and conductor-plane configura- tions does not by itself constitute a change in approach.

Professor Carrara formulated some objective crite- ria to evaluate models of physical phenomena and proceeded to show that, within its field of applica- tion, the present model meets these criteria much better than any similar approach presented to-date. This is particularly gratifying in view of Prof. Carrara's previous significant contribution to the subject of the paper. The following are the answers t o his specific questions: 1. Reference to a critical applied field can be found in the second paragraph of page 116 of ref. 111 which reads: "... there is a minimum value of the charge in the streamer tip below which propagation is not

possible. This minimum value (stability charge) depends upon the value of the applied field; at any instant a streamer can grow only if the charge in its tip is higher than the stability charge corresponding to the applied field in the position of its tip". 2. The shielding factor ks is not equal to unity mainly because in the presence of the leader space charge column, charges of opposite polarity are induced on the high voltage conductor which have the effect of reducing the geometric field component at the plane from its value in the absence of space charge. Further reduction from the theoretical geometrical value for an infinitely long conductor will be caused by the fact that laboratory experiments deal with conductors of finite length and with curved profiles, in order to favour concentration of breakdowns at mid span. Both these effects are included in k, and can, in princi- ple, be accounted for by numerical computations based on charge simulation.

In answer to Dr. Hermann Schneider, one needs to consider the geometric field component in addition to the leader space charge component at the final jump whenever the electrode configuration plus the leader space charge could not be adequately represented by charge simulation along the gap axis. The factor k, was introduced in the paper primarily because the objective was to obtain analytical expressions of the leader inception voltage. If numerical solutions are sought, as in the case of more complex configurations, then the geometric field in the presence of the leader space charge will be determined as part of the total applied field. ks will therefor be obtained implic- itly without the need for any special approach to determine it.

In answer to Professor Khalifa's question about ks, it should be noted that the leader inception voltage is not very sensitive to the value of k,. For large gaps, an error of 20% in the value of k, would typically produce an error of 3% in Ulc. The one meter gap in particular does not present any diffi- culty since it breaks down,basically through a streamer mechanism. Similarly with impulses much shorter than critical the leader does not have adequate time for significant penetration of the gap.

Dr. Lampe made an interesting suggestion to explore the relationship between the model parameters and fundamental physical properties of the gas. Such effort if successful, may permit the application of the model to other gases and higher presssures as also suggested by Prof. Khalifa.

Finally several discussers proposed to extend the model to more complex structures, phase-phase insula- tion, impulses other than critical and to lightning. These valuable suggestions will be taken into account in the planning of future work.

Manuscript received June 1, 1988.

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