389IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 2, February 1983

TRANSIENT ANALYSIS OF GROUNDING SYSTEMS

A. P. MeliopoulosIEEE, Member

M. G. MoharamIEEE, Member

School of Electrical EngineeringGeorgia Institute of Technology

Atlanta, Georgia 30332

Abstract

This paper addresses the problem of computing theground potential rise of grounding systems duringtransients. Finite element analysis is employed tomodel the constituent parts of a grounding system.Short lengths of earth embedded electrodes are charac-terized as transmission lines with distributed induc-tance, capacitance and leakage resistance to earth.Leakage resistance to earth is accurately computed withthe method of moments. The other parameters of thefinite element, namely inductance and capacitance, arecomputed from the resistance utilizing Maxwell's equa-tions. This modeling enables the computation of thetransient response of substation grounding systems tofast or slow waves striking the substation. The resultis obtained in terms of a convolution of the step re-sponse of the system and the striking wave. In this waythe impedance of substation systems to 60 cycles isaccurately computed. Results demonstrate the depen-dence of the 60 cycle impedance on system parameters.The methodology allows to interface this model of asubstation ground mat with the Electromagnetic Trans-ient Analysis Program thus, allowing explicit represen-tation of earth effects in electromagmatic transientscomputations.

1. INTRODUCTION

The transient response characteristics of ground-ing systems play an important role in the protection ofelectrical installations. For example, the voltagedrop along a ground rod connecting a surge arrester andthe transformer it is protecting can obtain a valuewhich is a substantial percentage of the basic impulselevel of the transformer insulation. Depending on theconfiguration, the surge arrester experiences an over-voltage which is less than the one reaching the trans-former. Thus system protection is reduced. The intro-duction of solid state arresters and the every shrink-ing safety margins demand more accurate analysis proce-dures for substation design and protection. In thiscontext, analysis procedures predicting the transientresponse of substation grounding systems are very im-portant.

The transient response of grounding structures hasbeen studied many years ago by Rudenberg [1], Bewley[2], Sunde [3] and others. The classical experimentsperformed by Bewley [21 on counterpoises provide muchinformation about the transient characteristic of

82 SM 369-7 A paper recommended and approved by theIEEE Substations Committee of the IEEE Power Engineer-ing Society for presentation at the IEEE PES 1982Summer Meeting, San Francisco, California, July 18-23,1982. Manuscript submitted February 4, 1982; made avail-able for prinfting April 19, 1982.

grounding systems. Verma and Mukhedkar [51 showed thatdistributed resistance and inductance models of buriedground wires predict transient response of such systemsin agreement with the experiments of Bewley. However,they do not provide any models for practical substationgrounding systems. Kostaluk, Loboda and Mukhedkar [15]provide experimental data for transient ground impe-dances. Similarly, Rogers [6] reports on actual systemtransient response of a large tower footing. Bellashiet al. [8], [91, [10], have given a complete treatmentof driven rods characteristics. Gupta and Thapar [7]provide empirical formulae for the impulse impedance ofsubstation ground grids, defined as the ratio of thepeak value of the voltage developed at the feedingpoint to the peak value of the current. This defini-tion of impulse impedance leads to uncertainty becausethe peak values of voltage and current do not necessar-ily occur at the same time. The so defined impulseimpedance strongly depends on the rise time of the waveconsidered, the mesh size of the grid, soil resistivityand permittivity, the feeding point, etc. This paperpresents data which further illustrate the point.Thus, the definition of impulse impedance of reference[71 is at best ambiguous.

The work reported in this paper addresses theproblem of transient analysis of practical groundingsystems consisting of ground mats, ground rods, etc.The developed models are in good agreement with experi-mental results.

The paper is organized as follows. First, thesimple case of an earth embedded conductor is treated.This case is extended to the case of a substationground mat. These two cases clearly illustrate themethodology. Sample test cases are presented and com-pared to known experimental data. The comparison isfavorable. Finally, a methodology is outlined for theinterface of the grounding system models of this paperwith the EMTP computer program which enables the studyof the impact of grounding systems on electromagnetictransients.

2. TRANSIENT RESPONSE OF AN EARTH EMBEDDED CONDUCTOR

2.1 Problem Formulation

Development of models of grounding structuressuitable for the computation of their transient re-sponse can be demonstrated with the simple system of asingle buried conductor. Such a system is illustratedin Figure 1. A small segment of length Q of theconductor of Figure 1, is characterized with a seriesresistance Ar, a series inductance AL, conductance Agto remote earth and capacitance AC. This representa-tion is illustrated in Figure 2. These parameters aredistributed along the length Q of the segment. Thethick solid line signifies the tact.

The numerical values of the quantities Ag, AL, ACcan be directly computed from two quantities, namelythe conductance Ag and the speed of electromagneticwaves in the soil V , as follows. The speed V is

0018-9510/83/0002-0389$01.00 ( 1983 IEEE

AIR/ /t ts/ / / / I I J J -1 / /

eQs / (x,y,z)hIs.Tej EARTH) (conductivity a)

Figure 1. Single Conductor Buried in Uniform Soil.A Short Segment of Length R. is Indicated.

AC = a Ag

AL = a S0 0

(3)

(4)

E0= E/cEr permittivity for free space.It is obvious that knowledge of Ag and soil pro-

perties suffice to determine the parameters of the fi-nite element. Computation of Ag is outlined in Section2.2.

Applying Kirchoff's laws to a small section Ax,one obtains the usual equation of a distributed param-eter line:

321 tr 321+ (Ar-AC + AL-Ag)31i + Ar.Ag I

0

There are two approaches of solving this equation:

(5)

Ar AL- - ~~~~~~~~Figure 2. Representation of a Finite Element with Circuit Elements.

readily computed from the soil properties with the aidof Equation (1).

I~~~~

where

CV = s ,

rr

(1)

C0 is the speed of light in free space, andE is the relative permittivity of the soil.r

The computation of the conductances Ag has beenreported in an earlier publication [13] and it is sum-marized in Section 2.2 of this paper.

Maxwell's equations dictate that

AC C (2)Aga

where: soil permittivity

C : soil conductivity

Also, considering the segment Q as a transmission linewith distributed inductance ASL and capacitance ACyields:

Q__ Cs = 0

/ AL *AC xwhere r

Q length of the finite element under consid er-ation

Above relationships yield the inductance and capaci-tance of the finite element:

(a) By direct solution (i.e. FFT) which leads tocomputationally infeasible procedure forthis problem; and

(b) Using approximate analysis techniques.

The latter approach will be described. Consider againa finite element of the conductor. Since the elementis very short the circuit of Figure 2 can be approxi-mated with the circuit of Figure 3. The middle part canbe recognized as a lossless transmission line. Theequivalent circuit of Figure 3 is the basis for thedevelopment of the methodology. To this purposeDommel's method [14], can be directly employed to yieldthe resistive equivalent circuit of Figure 4. The pasthistory current sources of Figure 4 are defined as fol-lows:

O 0o

I e (t-T) + 2g e (t-T) +2Z ik(tT) (6b)+ ) ik(t-T)

Lossless Transmission Line

Ar/2 AL Ar/2

g I TAg/2

Figure 3. Approximate Equivalent Circuit of a Short Lengthof an Earth Embedded Conductor.

390

dS ~ dldS

..-. -1

(i) 'k"t; i 'X m(t) / i+ ikm(t) imk(t)

Bk(t) Ag/2 > Z0< ' Ik(tT) ,zo Ag/2 em(t)

Figure 4. Resistive Equivalent Circuit of a ShortLength of an Earth Embedded Conductor.

391TABLE 1. Algorithm for the Computation of the

Transient Response of Grounding Systems.

Step 1: Partition the grounding system into finiteelements.

Step 2: Compute the parameters of each finite element.

Step 3: Compute the equivalent resistive networkparameters for each element.

Step 4: Compute the admittance matrix Y. Invert ma-trix Y using sparcity techniques. Let k=O.

Derivation of above formulae is given in the appendix.

The equivalent circuit of Figure 4, which will bereferred to as the equivalent resistive network, is thebasis of the method. Consider an earth embedded con-ductor of length Q. Assume a partition of this conduc-tor into n segments. n is selected according to thedesired degree of accuracy in the computations. Eachone of the segments can be represented with the equiva-lent circuit of Figure 4, and associated equations.The resulting equivalent circuit is resistive. Thus,nodal analysis is most suitably applied to yield:

Y e(t) = i(t) + b(t-h) (7)where

ye(t)i(t)b(t-h)

is the admittance matrix of the circuitis the vector of voltages at the nodes ofthe circuit (terminals of the segments)is the vector of currents injected at thenodes of the circuitis the vector of past history.

In this particular application, the admittance matrixhas a special structure. All entries are zero exceptthe diagonal and those which are located one positionover or under the diagonal

Y. . #01 1

Y #0 for every i

Y #0Yi+l,i all others zero.

This special structure of the matrix Y admits theefficient use of sparcity techniques for the solutionof Equation (7). The vector of past history is con-structed from the current sources I (t-h). It is ex-pedient to select h=T, where - is thme wave travel timealong any one of the finite elements.

The nodal equations enable the solution of thevoltages e(t) at the various nodes of the system if thecurrent injections i(t) are known. Table 1 illustratesthe algorithm for the computation of the transient re-sponse for a period of tmax seconds with time step h.

The same analysis methodology can be applied to aset of interconnected earth embedded conductors forminga ground mat. This analysis is presented in Section 3.

Step 5: Let k=k+l. Compute the past history currentsources of the equivalent circuits at timet=(k-l)h.

Step 6: Compute the external current sources at timet=kh. Compute the vector i(kh) + b((k-l)h).

Step 7: Solve for the voltages e(kh) = Y 1(i(kh) +b((k-l)h)).

Step 8: If kh > tmax terminate. Otherwise go to Step

2.2 Computation of the-Ground Resistance

This section describes the procedure for the com-putation of the conductance Ag of a finite element of agrounding system. It is based on the rigorous solutionof Laplace's equation in the seminfinite conductingmedium of the earth. The description of the method israther sketchy. More details can be found in [13].

The computation of the ground resistance includesthe following steps. Consider an earth embedded con-ductor. Further consider an infinitesimal surface dSof the conductor emanating total current dI. The flowof current dl generates a voltage field in earth whichis governed by Laplace's equation

V2V(x,y,z) = 0 (8)The solution for the voltage at point (x,y,z) due

to current dI has the following general form:

dV(x,y,z) = dI f(x,y,z,ds) (9)where f is a function of point (x,y,z), the infini-tesimal surface dS and the soil properties.

Now consider a finite length of the earth embeddedconductor of length i . Under the assumption of uni-form current distribution on the surface of this seg-ment the voltage at (x,y,z) due to the current emanat-ing from the outside surface of the finite segment is

V1 (x,y,z) = f dV(x,y,z)whchiscopuedt b

which is computed to be l13]

Vsl(x,y,z) = Ri(x,y,z,j)I.

(10)

(11)

Consider now that the conductor is partitioned in-to n segments of lengths k1,i ...v Q , respectively.Further assume that the current is uni%ormly emanatingfrom the surface of each segment and has a total value11 I29" In- The voltage at point (x,y,z) shall becomputed from the superposition of all contributions,i.e.

392

V(x,y,z) = z V (x,y,z) = R (x,y,z,i)I. (12)i

1

Specifically, the voltage of segment k can be computedas:

Vk = Vi(kxky,kz i) =zRkiIi (13)

Writing one such equation for every segment weobtain

[VI = [R][I]where

wherE

[ 1

V in

(14)

I = i]LIni

V. is the voltage of the outside surface of1ment i;

Ii is the current emanating from the surfacsegment i; and[RI is an nxn matrix which is symmetric.

Above matrix equation can be inverted to yiel

[I = [Y[V]

[Y[ = [RI1

Matrix Y represents an admittance matrix wcorresponds to an equivalent circuit for the e[131, as follows:

(a) Entry y.. of [Y] equals the negative contance oiJa element connected between segi and j; and

(b) yYi. equals the conductance of an equivadirEAit element connected between segmeand remote earth.

The equivalent conductance -y.. between remote elemi and j is in general very sirJall and can be omitThus, for every segment an equivalent conductancremote earth is computed. This conductance provthe basis for the computation of the other parameof the f inite element as it has been shown in Sec2.1.

3. TRANSIENT ANALYSIS OF GROUND MATS

seg-

-e of

Y e(t) = i(t) + b(t-h) (16)

where the admittance matrix Y is highly sparse (maximumof three non-zero elements per row), i(t) is the exter-nally injected currents. Solution of above equationfor times t=0, h, 2h, 3h, ... yields the voltages e(t)everywhere in the substation ground system, as it isoutlined in Table 1.

In Equation (16) depending on the excitation ofthe grounding system, the known quantities will be:

(a) The externally injected current vector i(t)(for example, a lightning current wave im-pending at a certain location);

(b) Some of the entries of vector e(t) (for exam-ple, a voltage wave impeding the groundingsystem); and

(c) Combination of above.In general, every type of surge injected in the

grounding system can be accommodated with Equation(16). In Section 6, the procedure will be generalizedto the extent of interfacing this model of the groundmat with the Electromagnetic Transient Analysis Program[141.

4. 60-CYCLE IMPEDANCE OF GROUNDING SYSTEMS

The developed models are suitable for the computa-d tion of the power frequency impedance of grounding sys-

tems. The 60-cycle impedance of a grounding system (or(15) the impedance at any given frequency) can be computed

in two ways:

(a) Inject a sinusoidal current wave (peak valuehich I ) to the grounding system and compute its

marth voltage elevation. This voltage will also be

sinusoidal at steady state. The impedance ofthe grounding system is computed from the

Lduc- peak value of the voltage wave, V , and themtment phase difference, C, between voltage and cur-

rent:V

lent z (17)nti I

(b) Compute the current response, s(t), of thegrounding system to a unit step voltage (see

Lents Figure 11). The impedance of the grounding:ted. system at frequency f is thene toridesters:tion

The transient response of substation ground matscan be similarly computed with the finite element anal-ysis procedure described. To this purpose the conduc-tors of the substation ground mat, ground rods, fences,etc., are segmented into a number of finite elements.The equivalent circuit representation of the earth as-sociated with above segmentation of the substationground mat is then computed with the procedure outlinedin Section 2.2. Then using Equations (3) and (4), eachfinite element is represented with a lossless transmis-sion line, series resistance and shunt conductance asit is shown in Section 2.1. Next each finite element isrepresented with the equivalent circuit of Figure 4.Nodal analysis for the resulting equivalent circuityields

f s

Z = 1.0/| ejj27rft s' (t)dt (18)where s'(t) is the time derivative of thestep response s(t).

Computationally, the second way is more efficient be-cause the first method requires the simulation of thegrounding system response for a long time until sinu-soidal steady state is achieved. Computation of theintegral of Equation (18) is straightforward and compu-tationally efficient. A computer program has been de-veloped for the computation of the Equation (18).

5. TEST RESULTS

The methodology described in this paper has beenimplemented and a number of grounding systems have beenstudied. These are:

(1) A 60 meter long earth embedded 4/0 copperconductor (burial depth = 0.6 meters).

(2) A 6 x 6 mesh ground mat with 10 meter squaremeshes buried at 0.6 meters under the earthsurface. This system will be referred to asMAT A.

(3) A 10 x 10 mesh ground mat with 6 meter squaremeshes buried at 0.6 meters under the earthsurface. This system will be referred to asMAT B.

(4) A 10 x 10 mesh ground mat with 12 metersquare mesh buried at 0.6 meters under theearth surface. This system will be referredto as MAT C.

Mats A, B, and C are assumed to be constructed from 2/0copper conductor.

Figures 5 and 6 illustrate the response of a 60meter long 4/0 copper conductor embedded in 125 Qm soilat depth of 0.6 meters, to a step and a 1/20 us currentwave respectively: l 0 t < 0

i (t) =a 1 kA 1 t > 0

-0.4t -1.8tib (t) = 1.1152 (e - e ) AIn both cases the current is injected at one end of theconductor. The voltage at both ends of the conductorand the middle is plotted versus time.

Figure 7 and Figure 8 curve A, illustrate thetransient response of MAT A to a step current and a 1/20ps current wave respectively. Figure 7 illustrates thevoltage at the feeding point, corner and a middlepoint.

The following general observations apply:

(a) During the rise time of the current surge theconductor demonstrates an impedance equal toits characteristic iInpedance for the stepsurge and a lower value for the exponentialwave.

(b) As time progresses in Figures 5 and 7, thevoltage of the conductor approaches a steadyvalue which is verif ied to be equal to RIwhere R is the dc resistance to remote earthof the grounding system. In these cases, theresistance is computed to be 4.1777 and1.0104 ohms respectively.

(c) The earth embedded conductor behaves as ahighly lossy transmission line. For example,in Figure 5 it appears that the time totravel from point A to point C is more thantwice the time to travel from point A topoint B. This is in conformity with experi-mental results carried out by Bewley andothers. The phenomenon is due to the jointeffects of the self-inductance and ground re-sistance which leads to a lower and ever de-creasing wave velocity with length.

Figure 8 illustrates, on a common system of coor-dinates, the responses of ground mats A and B to a 1/20Pis current surge. These two ground mats cover the samearea and have approximately equal DC resistance toearth (1.0104 and .9876 ohms respectively). However,their transient response is different. Specifically,the 1/20 impulse generates a much higher overvoltage onmat A than mat B. This is due to the fact that theconductors of mat B are closer spaced than the conduc-tors of mat A. Results obtained with the models de-

393scribed in this paper indicate the following. Thelevel of overvoltages resulting from direct strokes onsubstation depend strongly on: (a) conductor spacing,(b) rise time of stroke, (c) soil resistivity and per-mittivity, and (d) others. These characteristics ofground mats are very important in the design of over-voltage protection. Closer spacing of ground mat con-ductors yields lesser overvoltages and thus reduces thepossibility of backstroke in the case of direct light-ning stroke on a substation grounded structure.

The ac impedance of the test grounding systemshave been computed and listed in Table 2. The effec-tive resistance, reactance, and inductance for 60 Hz aswell as for a number of harmonics are tabulated. Thesoil is assumed to be dry or wet with the indicatedparameters. From the table, it is obvious that thereactance of a grounding system depends strongly on itslength and the soil permittivity. For medium sizegrounding systems the reactance at 60 Hz is substan-tial. Wet soil, which is characterized with greaterpermittivity values tends to decrease the inductance ofgrounding systems. The resistance, as it is expectedis approximately proportional to the soil resistivity.These results are in agreement with results obtainedthrough sophisticated measurements techniques of powersystem ground impedances [12].

TABLE 2. Impedance of Test Systems

A. Dry Soil: p - 1000 ohm-m, Sr M 9.0

Frequency(Hz)3r-

60 meter 60Conductor 120

180240300

Mat A 60120180240300

Mat B

Mat C

60120180240300

60120180240300

B.

60120180240300

60 meterConductor

Mat A

Mat B

Mat c

Resistance(ohms)}33 .42133.42133.42133.42133.421

8.0838.0838.0838.0838.083

7.6617.6617.6637.6657.668

3.9543.9693.9944.0284.072

Wet Soil: p - 100 nm,

3.3423.3423.3423.3423.342

Reactance-(ohms)

.0349

.0698

.1047

.1396

.1745

.0356

.0712

.1068

.1424

.1780

.0571

.1143

.1714

.2284

.2854

.1366

.2717

.4037

.5308

.6512

C-

36.0r

.0348

.0697

.1045

.1394

.1743

.808 .0360

.808 .0721

.808 .1082

.808 .1443

.809 .1805

60120180240300

60120180240300

60120180240300

.766

.766

.767

.767

.768

.0438

.0875

.1311

.1745

.2176

.395 .0867

.397 .1733

.400 .2587

.402 .3421

.402 .4256

Inductance(EHenry)_.0926.0926.0926.0926.0926

.0944

.0944

.0944

.0944

.0944

.1516

.1516

.1515

.1515

.15 14

.3623

.3603

.3569

.3520

.3455

.0924

.0924

.0924

.0924

.0924

.0956

.0956

.0956

.0956

.0956

.1163

.1161

.1159

.1157

.1154

.2299

.2298

.2288

.2268

.2257

\0 A a CAtHeGomo_

8

8

918u-

8.

8-

00 2o 4.0 6.0 16.0 12.0 14.0

TIME (MICRO SECONDS)

FIGURE S. TRANSIENT RESPONSE OF A 60 METER, 410 COPPERCONDUCTOR TO A STEP CURRENT OF i KA.

0.0 20 4.0 .0 8.0 10.0 ZO 14.0 16.0TIME(MICRO SECONDS)

FIGURE6

TRANSIENT RESPONSE OF A 60 METER, 4/0 COPPERCONDUCTOR TO A 1/20 As. 1 KA CREST SURGE.

8

i'$

8

8.

8.

60 10.0 20.0 26.0 40.0 50.0 60.o 70.0TIME lMICRO SECONDS)

FIGURE 7. TRANSIENT RESPONSE OF MAT A TOA STEP CURRENT OF 1 KA.

r-

leqt-h) t G

l ~ ~ ~ AA- Y -l

G = YAAAB BB YBA

leq(t-h) = bA(t-h)- AByBB bB(t-h)

Figure 9. Equivalent Representation of a GroundingSystem Compatible with the EMTP.

0.0 2.0 4.0 .0 a'o I0. i20TIME (MICRO SECONDS)

FIGURE 6. TRANSIENT RESPONSE OF GROUND MATS A AND BTO A lJ20 s; 1 KA CREST SURGE.

ad'

.; -

zf 0:

8.

0.0 20. 40.0 6.0 60.0 100.0 120.0TIME (MICRO SECONDS)

FIGURE 10. TRANSIENT CURRENT RESPONSB TO A STEP VOLTAGE OF kI.

3948

g

8

S.

8

8

8

8

8

V

8

8

0

8

8

A B

so

395

6. INTERFACE WITH THE EMTP

The methodology described in this paper for thecomputation of the transient response of grounding sys-tems is compatible with the methodology of the Electro-magnetic Transient Analysis Program (EMTP), developedby BPA. Thus, it can be interfaced with this computerprogram. In this case, the effects of substationgrounding systems on overvoltages, surge arrester per-formance and, in general, transient performance of sub-station can be evaluated. The introduction of newtechnologies in substation design, such as gaplesssurge arresters and computers and the ever shrinkingsafety margins demand more accurate transient analysisprocedures for susbtation design and protection. Thus,the specific modeling of substation grounding systemsin the EMTP is important. This section briefly out-lines the process by which the substation ground modeldescribed in this paper can be interfaced with theEMTP.

There are at least two ways to interface thismodel with the EMTP:

(a) The direct method; and(b) With linear convolution.

While the direct method is straightforward, the methodbased on time-domain linear convolutions is numericallymore efficient. Both methods will be' described.

6.1. The Direct Method

Consider Equation (7) which is repeated here forconvenience

Y e(t) = i(t) + b(t-h)

Assume there are m connections of the grounding systemto the rest of the system. Let the vector eA(t) repre-sent the voltages at the interconnections. Equation(7) can be rearranged in the following form:

FYAA yAB1 eA(t)l bA(t-h)l iA(tTh BA BBJ eB(t)j bB(t-h) BL 0 (18)

The form of the vector i(t) should be obvious sincecurrent injections will occur only at the connections.

From the last equations, the vector e (t) can beeliminated to yield: B

(YAA YABYBBYBA)eA(t) bA(t-h)-1

AY

B b (t-h) + i (t)ABYBB B A (19)Above equation can be directly interfaced with theEMTP. If for simplicity it is assumed that the mat isconnected with only one connector to the overhead skywire, above equation is a scalar equation. In thiscase it represents the equivalent circuit of Figure 9.Note that in above representation, the equivalent cur-rent source Ieq(t-h) needs to be computed at each iter-ation.

assumed that there is only one connection of thegrounding system to the system. Generalization istrivial.

The step response of the grounding system is com-puted with the methodology of this paper. Figure 10illustrates the response s(t) to a unit step voltage inthe middle of a 10 x 10 mesh ground mat. If a timedependent voltage V(t) is applied to the mat, the elec-tric current response will be [11].

i(t) = V(t) s(0) +f v(t-T) ds(T) dT0o

(20)

For numerical calculations, above expression canbe written as:

N

where

i(t) = v(t) s(0) + I v(t - kAT) d(kA)k=0 dT

(21)

N = tNTor

i(t) =v(t) [s(0)+ ds () ]+ E v (t-kAT) ls(kA AT (22)dt k=l OLet N

I (t-h) = E v(t - kA,) ds(kAT) ATk=l d

(23)

and observe that it depends on past time values of thevoltage and the known function s(t). Now above expres-sion represents the equivalent circuit of Figure 11.Obviously, this circuit can be interfaced with theEMTP. In this approach, at every time step the quanti-ty Ic(t-h) need to be computed with a numerical convo-lution. The computation of this convolution can beperformed much more efficiently than the computation ofIe (t-h) in the previous approach. Explicit expres-si ns of the linear convolutions are developed next.

I ~~~~~~~~~~~.- _.to system

I(t-h) t ',GC

G gms(O) + dS(O ATdi

IC(t-h) z: I v(t-kat) -sdskr) ArTk=1 1

6.2. Linear-Convolutions

This method requires the knowledge of the step orimpulse response of the grounding system. The methodhas' been successfully applied to model lossy transmis-sion lines in the EMTP program [111. The method will bedemonstrated assuming knowledge of the step response ofthe grounding system. For simplicity, it will also be

Figure 11. Equivalent Circuit Representation of a GroundingSystem Compatible with EMTP (Convolution Approach).

396

Let the step response, s(t), of the system be ap-proximated with a piecewise linear function as in Fig-ure 10. The function is defined with a number, m, ofpoints: (t.,s(t.)), i=1,2,...,m . For usual ground-ing systems he step response can be accurately approx-imated with a maximum of 20 linear segments (m=20).

ds(kATr)In this case, the derivative of Equation(23) will be constant for an4_RWte val of time t. < t