Mathematical modelling of propagating inducing and induced power due to actual pulse shape of lightning surge

Journal of Electrostatics, 15 (1984) 43--51

43

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

MATHEMATICAL MODELLING OF PROPAGATING INDUCING AND INDUCED POWER DUE TO ACTUAL PULSE SHAPE OF LIGHTNING SURGE

KHALIL DENNO

Department of Electrical Engineering, N.J. Institute of Technology, Newark, N.J. (U.S.A.) (Received December 6, 1982; accepted in revised form September 6, 1983)

Summary This paper presents a further stage of analytical work concerning the mathematical modelling of lightning phenomenon. Previous results identified solutions for all components of the inducing electromagnetic fields as well as the voltage induced on a transmission line generated by a step pulse of the conductive and convective current densities in the return stroke. In this paper, solutions are given for electromagnetic inducing and induced effects, as well as the induced voltage, due to the actual pulse shape of the current density represented by a sharp linear rise in the pulse front and a lower linear decline in the pulse tail. Also, closed form solutions are obtained for the propagating real power of both the inducing and induced effects. Phenomenological results due to electromagnetic induction presented here are considered to be the closest yet obtained to those calculated as due to lightning effects due to the actual pulse shape of the current densities produced by the return stroke.

Introduction Previous work carried o u t by this author [4] has led to the development of a mathematical model both as a function of space and time for the inducing as well as the induced voltages at a point on a transmission line, due to a lightning surge, based on the combined effects o f conduction and convection, for the return stroke. The model was developed with the aid of the Helmholtz field radiation function and by assuming a sharp-stepped function for the conductive and convective return stroke current densities. This paper extends these results by considering a model of current densities which approximates closest the return stroke, namely a linear rise for the fxont pulse followed by a dropping tail. Solutions for the electromagnetic field components are given at any point on a transmission line, developed by the inducing surge as well as the associative function and the induced voltage. Also complete solutions are established for the inducing current and the total propagating real power delivered to the power system by the actual lightning surge. 0304-3886/84/$03.00

© 1984 Elsevier Science Publishers B.V.

44

Statement of the problem [ 1--4] Figure 1 shows the configuration of the source distribution and field effects of a lightning surge and Fig. 2 identifies the conductive and convective current densities as being sharp-stepped pulses having a constant taft. Figure 3

£

cylindrical distribution of electric

current in return stroke (conductive current Jc ) ~

-

/,in. /power

-

ii ~t~£1~rc,e,z~ ~

cylindrical distribution of t i m e varying bound charges (convective current Jv )

"

I [[.111 ~ : I111

o! ~wer

~

"//////'2///////////"

Rv - ' ~

~ -Z

C u r r e n t source distribution in the r e t u r n stroke Jc = fc(rc , 8 , z )

Ground Spoce region around power line

Jv = f v ( r v ' 8 ` z ) Fig. 1. Source distribution and field effects o f lightning surge.

/ Jo

/ /

/

J(t)

i

/

--crt t

t~ t

(

t

x<

= A l ( t = t 1)

\ \ \

Fig. 2. Return stroke with constant tail. ~

Ldx

~

b ~ - - - - d× Fig. 3. Inducing voltage on power line.

OI I ~-~dx

\

45

shows a section of the transmission line for calculating the induced electromagnetic effects: Required mathematical closed form solutions of the following items produced by actual lightning surge as shown in Fig. 4.

Inducing fields due to actual lightning surge [1--12]

/i/i/

The actual lightning surge pulse approximates very closely a sharp linear front rise followed by a linear steeply dropping tail as shown in Fig. 4. /

J=~t

t

J = A(t -t 1 )

~ . . --.

A ~A I

Fig. 4. Return stroke with dropping tail.

0
Jc = A c t , = At(t-t,)

t > t,

independent of r, 9,Z

(1)

independent of r, 0,Z

(2)

And Jv = A ~ t ,

0 < t < tl

= A~(t-t,)

t > tt

where Jc, Jv = conductive and convective current density in the return stroke, respectively. Restating the Helmholtz field radiation equation for the inducing electric and magnetic components~ V 2 E + k2 E = O ,

~2~+ k:H = 0,

(3)

the scalar form is expressed as, V2¢+k~

= 0

(4)

where k = (_~?:~yn and where ~, 9 = impeditivity and admittivity of the surrounding space, respectively. By using cylindrical coordinates in eqn. (4) and identifying p, a, r, as the space coordinates at any point in space including a power line and using

46 Fourier transforms as was done in previous work which dealt with constant lightning surge [4], the following solutions for 0~ and Ov are obtained:

¢~ =

R¢ .T 2

~ ~

1 2n~ [ A c ( t - t , ) + jA~t U_, ( t - t , ) ]

n=0

(6)

X [Jn(klp - Rcl) + H(.a) (k [p -R¢ l )], ~'RvTr ~ 1 Cv 2 ~ 2n~r [A~(t-ti) +]Avt U_i ( t - t , ) ] n=0

(7)

X [Jn(k[p - Rv]) +H(.2)(klp-RvJ)l.

Electromagnetic field components at a point relatively far from the source generated by the truncated time function for Jc(t) are:

Ep = [Ac(t-t,) + jAct U_1 ( t - t , ) J ¢ ( t ) ] ( J ) '/: (p-Rc) 'l: R:~

X [

E~

E~

"~

H.

jJc(t) 2:~

2

J~(t)~ 2 +

(p-Rc) "-3/2 + j 2 " ( n - 1 ) ! ] 2n 2

2j ),/2 (lr(p -Rc) exp (-j (p -Re))

jJc(t)z (

-

(P-Re)n- ,

(p-Re) R2

= Jc(t)Rck2n 2:~

Hp

~ jnexp(-j(p-R¢)) ~] n!2 2 .=o

(

(8)

= jn(p-Rc). 2"n!

rt=O

2j )'/: ~" j"(p-Rc)" n(p-R¢) ~ 2"n! n-~O

2j I '/2

~(p--Rc) !

exp(-j(p-Rc))

n=0

~ [ 1 ~ k - - J" (p-R~)Hnt2) (p-Re) n=O nP2

(2/u)i/2 (_j)n+l/2 g n ( p _ R c ) 2

.1/2

j"(-Rc) 2"n!

-

1 nRc

- -

Hn

(:) (p

-Re)

(p -Rc) "-1 2"-'(n-1)!

1 exp (-j(p -Re)) nR¢ p -Rc 2 1-

(9)

47 where J . ( p - R c ) at large p

1

(p-Re) n

n!

2"

'

(1o)

2j . ,/2 H. (2) (p-Re) -~ \~(p-~Rc)) j" e x p ( - j ( p - R c ) ) ,

(11)

H~ = 0

(12)

Electromagnetic field components produced by time-varying bound charges, and represented by the convective current density Jv are the same as those of Ep, Ea, Er, Ha, Hr and H expressed in eqns. (8) and (9) by replacing Ac by A~ and R c by Rv.

Inducing voltage Vi [ 1--4] Vi = - f

Ep dp

(13)

where the inducing potential Vi is due to both the time-varying bound charges and the vector potential component of the source current, as was shown in previous work [4]. l J t l / 2 1 ~ o J" [P"+I/2exp(-jnp) Vi = [Jc(t) - Jv(t)]~--~] ~ - = 2"n! jn2R n+l/2

f pn-1/2 exp (-jnp)dp 4

pn-I exp (-jnp )

- n-lj.. IFD" . - 2 exp(-jnp)dp + J 2"-13~(n-1)! p3/2] (14) R = = Jc = = •]~= =

Rc for conductive Rv for convective Av t, 0 < t < t l , A t ( t - t 1 ) U_,(t) t > t,, A~ t, O < t < t , , A~ U_,(t) [ t - tl] t > t,.

(15)

Due to time retardation by to = (p -Rc)/Co, where Co = velocity of light, ;he associative function ¢ is given by = Vi (~, p, r, t) U(t - to)

(16)

48

The induced voltage V(x, t)

~2V

1 /)2V

ax 2

C 2 ~t 2

1 c~2Vi (17)

C 2 8t 2

V~ = Vi (~, p, r, t) U(t - to),

(18)

V~ = Vi (a, p, r) [J¢(t) - Jv(t)],

(19)

Je(t) = Act - Ac U ( t - t l ) = AJ-Ac(t-t,)U(t-t,),

t>t,

0
J¢(t) = A ~ t - A ~ ( t - t , ) U ( t - t , ) ,

Ot,.

(20)

(21)

From eqns. (14) and (17) and using the boundary conditions, namely that V(0, t) = 0 at the source, V(L, t) = 2 Vm~,, where L is very far from the source, the solution for V(x,t) at a point on the power line is as follows:

V ( x , t ) = V, + V: + V3 + V 4 ,

(22)

where

2 C 2V~axVi(x! 5 0 ( t ) [ A c - A v ] V~(x,t) =

V2(x,t) =

Va(x,t) =

V4(x,t) =

,

V i ( x = L ) 5~(t) 2 C 2 Vmax Vi(x)

Vi (x =L) 6~ ( t - t,) - 2 C 2 Vmax Vi(x)

V i ( x = L ) 60' ( t - t , ) 2 C 2 Vmax V i ( x )

V i ( x = L ) 5~ ( t - t 1 )

(23)

6o ( t - t ~ ) [A~ - A c ] ,

(24)

6 ~ ( t - t l ) [Av + Ac] ( t - t , ) ,

(25)

5o ( t - t , ) U-1 (t) (Av + Ac).

(26)

Induced and inducing propagated power [1--4, 7, 8] (a) Induced electromagnetic p o w e r The instantaneous power density per square meter delivered at a point on a transmission line can be expressed as P,(t) = [V(x,t]2p

(Wm-2)m

(27)

where p = resistivity of the power line. Hence

P l ( t ) = [V, + Y2 + Va + V412p.

(28)

49 To calculate th e average p o w e r delivered t hrough i nduct i on (induced power), an averaging process is carried o u t on eqn. (27) which results in t he follo wing: P, a-crate-induced

1[ =

+

2C2V"~'~Vi(x) ~o(Ac_A~) + Vi (x= L) 2 C 2 Vm,,, Vi (x)

Vi(x=L)

6o(t-tO ( A v - A c )

-

2 C 2 Vmax V~(x) 8~ (t-t1) ( t - t O (Av + Ac) + V~ (x= L ) +

2 C 2 V,~,~ Vi (x) Vi(x=L)

2

6o(t-t,) U_,(t) (Av + Ac)l J

(29)

(b ) Inducing electromagnetic power The inducing cur r ent density vector is expressed as,

(30) Jp

=

aEp,

Ja = oEa, and d~ =

o E~ .

Going f r o m a cylindrical t o a cartesian c o o r d i n a t e system, ] i s expressed as Jx = Jp ap cos a - Ja a-'~sin a ,

= dp ap sin a +Ja aa cos ~ , J~

=

J~a~

(31)

,

where x = p cos a, y = p sin a, and z = r. The solutions for Ep, Ea and ET are shown in eqn. (22). T h e r e f o r e

P (t)--151 /o =

[~+Ja+~]/o.

(32)

50 To find P2 average, an averaging process is carried out on Ep, Ea and ET in eqn. (22): P2 average =

[Ae(t-t~)] Re [jAet U - l ( t - t , ) ] X[ (p-Rc)n

(P_Rc)n-I

R~

( P - R e ) n-3/2 +

Ren

2n

~

' 2~

n=O

12.

J2n(n-1)!]] : 2 2

+

26

+~

26

[Ac(t-tt)] Re

Ac(t-t~)] Re

~'Y

~

(p-Rc)

L 29

l

exp(-j(p-Rc))

n=O

"2~'~"

]J

(p-Re)

× exp(-j(p-Rc)) ~_l jn(p-Re)" ]j : ,

(33)

where Re = the real part of (F). Similarly, the average convective power generated by the time-varying propagating bound charges is the same as the solution expressed by eqn. (33), with the replacement of Re by Rv and Ac by Av.

Discussion and conclusions [ 1--12] Previous work carried out by this author has centered on the development of mathematical modeling for the inducing and induced field effects at any point in space generated by a lightning surge pulse which considered convective currents as well as the conductive counterpart. Solutions had been obtained for all electromagnetic field components produced by a constant level pulse for both the conductive and convective current densities in the return stroke, using the field model of the Helmholtz radiation function and Fourier transforms. Solutions were also established for the inducing electric potential as well as the induced voltage at any arbitrary location on a transmission line. Work presented in this paper has considered the surge lightning pulse of the return stroke to have a sharp linear rise for the pulse front and a linearly falling pulse taft. Such a surge model indeed very closely approximates the actual surge which is characterized as being exponential in the rise period as well as in the tail. From field experimental measurements, however, it is known that the exponential variation is almost linear, due to an infinitesimal time constant for the pulse front and a substantially longer time constant for the tail.

51 Conclusions drawn from results presented in this paper may be summarized from t h e closed mathematical solutions f or the following: (1) Electric and magnetic inducing field c o m p o n e n t s at an arbitrary position. (2) The inducing electric potential. (3) Th e in d u ced electric potential. (4) The propagating real power of the inducing fields t hrough a section o f transmission line. (5) The propagating induced real p o w e r through a section of transmission line. The mathematical models obtained in this work are based on a pulse waveshape o f the conductive and convective c ur r e nt densities which closely approximates the r et ur n stroke o f the lightning surge. These models present solutions which are compatible and unique to a space poi nt structure relatively far f r o m the stroke location. Also such solutions are indeed exact in the m o s t general sense since considerations have been taken to include t he state o f time variation at t he source field effects, velocity o f propagation of t he r e t ur n stroke as well as t h e time rate o f change of the eart h-bound charges, and the geometrical space dimensions o f clouds and power-line structures. References 1 2 3 4 5 6 7 8 9 10 11 12

G.W. Brown, IEEE Trans. Power Apparatus and Systems, PAS-97 (1978) 33--38. G.W. Brown, IEEE Trans. Power Apparatus and Systems, PAS-97 (1978) 39--47. P. Chowdhuri and T.B. Gross, Proc. IEE, 114(12) (1967) 1899-1907. K. Denno, J. Electrostatics, 13 (1982) 55--69. P.R. Garabedian, Partial Differential Equations, John Wiley, 1964. G. Gallet and G. Lerogy, IEEE Trans. Power Apparatus and Systems, (Conference Paper No. C-73-408-2), 1973. R.F. Harrington, Time-Harmonic Electromagnetic Fields, New York, Chap. 5, 1961. T. Mihalkovics, IEEE Trans. Power Apparatus and Systems, (Conference Paper No. C-73-183-1), 1973. D.P. Papadopoulos, Proceedings of Midwest Power Symposium, Vol. II, 1977. R. Rudenberg, Electrical Shock Waves in Power System, Harvard University Press, 1968. T. Suzuki and K. Miyake, IEEE Trans. Power Apparatus and Systems, (Conference Paper No. C-73-382-9), 1973. S.C. Tripathy and M. Yusuf Khan, Power Record (Summary), 1974, pp. 1572--1574.