grounding resistance 1

operating temperatures than the rest of thesystem.The authors are to be commended for a

realistic, logical approach in the evaluationof motor insulation systems by actual per-formance tests.

REFERENCES

1. See reference 5 of the paper, Figures 11 and 12.

2. MOTOR INSULATION LIFE AS MEASURED BYAcCELERATED TESTS AND DIELECTRIC FATIGUE,C. J. Herman. AIEE Transactions, vol. 72, pt. III,1953 (Paper 53-301).

C. B. Leape, J. McDonald, and G. P. Gib-son: The discussers have made some per-tinent comments on the techniques of eval-uating motor insulation. Mr. Herman statesthat the technique of reversing the motor togenerate heat and mechanical shock is notwell suited for single-phase motors. This isundoubtedly the case. In evaluating insula-tion for single-phase, fractional-horsepowermotors, the method he uses with an enlargedair gap between stator and rotor woulddefinitely appear to be more suitable.1 How-ever, the larger-size 5-horsepower 3-phasemotor was used on these evaluations be-cause it was felt that the data obtainedwould be more representative of a widerrange of motor sizes than would be the

data from a smaller motor.Mr. Mathes points out that the evaluation

of complete motors may not be feasible inthe larger sizes, and therefore it would benecessary to turn to motorette evaluationsor functional tests on components. It istrue that functional tests on components arethe only practical means, aside from fieldexperience, of evaluating large motor andgenerator insulation; however, the intentionin this method has been not only to obtainevaluations of random wound motor insula-tion systems representative of a wide rangeof motor sizes using the 5-horsepowermotor but also to evaluate components andmotorette tests so that they may be appliedmore accurately to the larger motors.Mr. Mathes is of the opinion that test

motors should be started immediately afterthe humidification cycle rather than after anair-dry cycle following humidification.Starting motors wet corresponds to field ex-perience in which motors are shut downduring naturally occurring periods of highhumidity and then started later under thesame conditions with condensation on theinsulation. The effect upon the insulationof starting motors immediately after humidi-fication when condensation is present hasbeen considered by us. However, it is ouropinion that such severe conditions are notfrequently encountered in the field and

consequently will be treated as a separaterequirement of insulation systems. Sinceonly a minority of motors in the field wouldbe expected to encounter condensation onthe insulation, we feel that this severe con-dition could be overemphasized if all testmotors were subjected to it.We plan to expose insulation systems to

operating voltage, after humidification,while they are still wet, as well as to con-tinue to air-dry the motors after humidifica-tion before the starting voltage is applied.In this way the effect of condensed moistureon aged insulation will be able to beevaluated fully.

In summary, there are two main purposesfor which this method was developed. Tosatisfy these purposes a procedure was de-vised which is intended to simulate fieldoperating conditions and to give data whichcorrelate with field experience on acceptedor standard insulation systems. With thisprocedure the field performances of newinsulation systems are to be predicted. Withthe data obtained from this procedure eval-uations shall be made of motorette and othercomponent tests.

REFERENCES

1. See reference 2 of C. J. Herman's discussion.

2. See reference 5 of the paper.

Grounding Grids ror High-VoltageStations

E. T. B. GROSSFELLOW AIEE

B. V. CHITNISASSOCIATE MEMBER AIEE

Synopsis: Grounding grids are used inhigh-voltage stations when rocky groundmakes the use of driven ground rods im-practical. The characteristics of groundinggrids are investigated in this paper, and theresults are applied to the basic design ele-ments of this type of grounding system.

THE maximum value of the allowableresistance between uninsulated parts

of electric equipment of an electric powersystem and ground will depend upon thesingle-line-to-ground fault current at theswitchyard concerned. A large stationon a system with small fault current mayhave a higher allowable ground resist-ance than a small station on a system withlarge fault current. The importance ofthe magnitude of the fault current needsemphasis. It is evident that the typeof system grounding used influences theresistance level. The material coveredin this paper will be concerned with sta-tion grounds which should in most caseshave a resistance of less than a few ohms.

L. J. STRATTONASSOCIATE MEMBER AIEE

The term "resistance to ground" needsdefinition since it is the determination ofthis quantity with which this investiga-tion is concerned. The resistance toground is the "resistance between the elec-trode system and another electrode in theground at infinite spacing."' It does notrefer to the resistance of the copper wiresused in connecting the electric equip-ment to the point where these wiresenter the ground. The resistance of thecopper wires is almost always negligible,but it is the resistance that the currentencounters in going from the wires intothe ground that is appreciable.

Nature of a Grounding Grid

Grounding is usually accomplished mostreadily by using deep-driven gTound rods.Sometimes the nature of the soil is suchthat the substratum is mostly rock, thusmaking the driving of deep rods imprac-tical. For these cases, the most econom-ical method of obtaining a low ground-

ing resistance is to bury a number of con-ductors at a depth of from 1 to 2 feetparallel to the surface of the earth. Theconductors are usually arranged in acriss-cross fashion, and the configurationis spoken of as a grounding mat or gridsystem. Such a mat not only effectivelygrounds the equipment, but has the addedadvantage of controlling the voltagegradients at the surface of the earth.As the number of buried conductors is

increased, the condition of a buried plateof conducting material is approached.The use of a solid buried horizontal plateof sufficient size to produce a low resist-ance to ground would be very expensive;it is really unnecessary in view of the factthat as the number of buried conductorsis increased (thus approaching the condi-tion of a plate), the addition of more con-ductors does less and less to reduce further

Paper 53-239, recommended by the AIEE Sub-stations Committee and approved by the AIEECommittee on Technical Operations for presenta-tion at the AIEE Summer General Meeting,Atlantic City, N. J., June 15-19, 1953. Manu-script submitted March 16, 1953; made availablefor printing May 4, 1953.

E. T. B. GROSS is with Illinois Institute of Tech-nology, Chicago, Ill., B. V. CHITNIS is with theAmerican Gas and Electric Service Corporation,New York, N. Y., and L. J. STRATTON is withArmour Research Foundation, Chicago, Ill.

The practical importance of, and the need for, athorough analysis of this problem was pointed outto one of the authors by H. H. Plumb, Head, Elec-trical Division, Bureau of Reclamation, Denver,Colo., and many discussions with Mr. Plumb havegreatly advanced this investigation.

Gross, Chitnis, Stratton-Grounding Grids for High- Voltage Stations 799AUGUST 1953

Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY JAMSHEDPUR. Downloaded on September 22, 2009 at 08:05 from IEEE Xplore. Restrictions apply.

the resistance to ground. This is so be-cause a volume of ground surroundingeach conductor is necessary to distributethe current in the ground. As the con-ductors are moved closer together, theground volumes will overlap and the vol-ume which is common to both will noteffectively distribute the current.

In designing a groundirg grid, manyfactors must be taken into account, suchas the form of the material to be used, beit strips, rods, or tubes, and the relation-ships between length, depth, separation,and radius of the buried conductors. Itis an analysis of this grid system of buriedconductors which will be presented in thebody of this investigation.

Calculation of Resistance to Ground

GENERAL EQUATION FOR RESISTANCE

The resistance of any body to the flowof current through it is given by

LR=pA (1)

For the case of earth having a resistivity p,the effective length L of the current pathand the effective cross-sectional area Athrough which the current flows are verylarge. In distributing the current intoground through the grounding grid, onlya limited area of earth is in contact withthe electrodes. As the current spreadsout into ground, the cross-sectional areaand length of current path are increasing;therefore, within a short distance fromthe electrode system, the resistance isextremely small. For this reason almostall of the ground resistance is near theelectrodes. Jensen2 states that "meas-urements show that 90 per cent of thetotal electrical resistance surrounding anelectrode is generally within a radius ofsix to ten feet from the electrode." Be-cause the cross-sectional area and lengthof the current path associated with agrounding grid are of such a complexnature, a different approach of calculatingthe resistance to ground is used.

RELATIONSHIP BETWEEN RESISTANCE ANDCAPACITANCE

In order to arrive at expressions for theresistance to ground of a grid system, ananalogy will be used. This analogy isbased on the fact that the flow of currentinto ground from the electrode system hasthe same path as the emission of electricflux from a similar configuration of con-ductors having isolated charges. It hasbeen shown by Peters' that if R is theresistance to ground of an electrode sys-tem deeply buried in ground, and if C is

the capacitance of the same (isolated)electrode system in a medium of dielec-tric constant unity, then

R=-P (2)4r C

where p is the resistivity of earth assumedconstant throughout the region. If theelectrode system is near the surface ofthe earth, the flow of current is alteredand the relationship is

Rp 1 (3)27r C

where C is now the capacitance of theelectrode system and of its image withrespect to the surface of the earth. Byequation 3, the relationship between theresistance to ground of the grounding gridand its capacitance is established, and theproblem resolves itself into that of deter-mining the capacitance of the grid and itsimage.

RESISTIVITY OF EARTH p

The factor of proportionality betweenthe resistance to ground and the recipro-cal of the capacitance involves the resis-tivity of the earth represented by p. Thevalue of p covers a wide range and isdependent on many factors. The mostimportant factors are soil material,whether clay, sand, etc.; moisture contentof the soil; and temperature.The resistivity may vary from 0500

to 50,000 ohms per centimeter cubed. Itis sometimes convenient to express theresistivity in terms of ohms-centimeter.The expression "'ohms-centimeter" repre-sents the resistance between the oppositefaces of a cube of soil 1 centimeter on aside. The literature on variations of re-sistivity contains many tables which givethe approximate range of resistivity fordifferent conditions.4 For the purposesof this investigation it will be sufficient tosay that the type of soil, as determined byits chemical composition, is variable.Many times an attempt to alter the chemi-cal composition is made by salting theearth in the vicinity of the ground elec-trodes. Even though this is sufficient tolower the ground resistivity by a greatdeal in many cases, it provides only tem-porary relief at best, because the saltingprocess must be repeated periodically,sometimes as often as every 6 months.The moisture content of the soil also

alters the resistivity. Since the moisturecontent varies with the seasons, the resis-tivity is a function of the time of the year.The moisture content of the soil also de-pends on the height of the water table;therefore, in order to reduce the seasonalvariations in resistivity due to moisture

variation, an attempt is usually made toinstall the grounding electrodes in suchmanner as to reach the permanent watertable. However, this will not be possiblehere, since this investigation is concernedwith a grid arrangement buried at a depthof 1 to 2 feet. At this depth, the varia-tion of moisture may be considerable,and the range of variation of resistivitywith moisture must be determined before-hand.Temperature also greatly affects resis-

tivityand at a small depth the temperaturevariation may be considerable. Thus,with a grid arrangement buried at a depthof only 1 or 9 feet, the specific resistivitydepends on the soil composition at thatdepth. Because the shallow depth ofburial allows variations in temperatureand moisture content, the resistivity valuemust be determined for the worst condi-tions of these factors, namely, dry coldearth. After obtaining this value, it maybe assumed that it remains constant sothat the computations lead to the highestresistance which will be experienced.The resistivity of the soil at the location

of the station must be found by tests.A number of test methods have beendeveloped which may be employed, andmany references to such literature areavailable. However it is well to stressthat these tests should be conducted overa long period of time, since resistivitvwill vary greatly with the seasons. It isworth while to extend them over a periodof at least 1 year. Such tests constitutethe first steps in installing a groundinggrid.

DETERMINATION OF CAPACITANCE

The determination of the capacitanceof the grounding grid and its image pre-sents a formidable task since the elec-trode structure of a grid is rather complex.There are two very important methodsemployed in the calculation of capaci-tance. They are:

1. Howe's average potential method.52. Maxwell's method of subareas.6,7

In the case of any electt-ode systemmade up of conducting material and carry-ing an electric charge, the charge distrib-utes itself over the body of the electrodesystem in such a manner that the poten-tial everywhere on the surface of the bodyis the same. Hence the charge densitxover the different points of the body wouldnot be the same. An average value ofpotential is taken and used in calculatingthe capacitance of the body, assumingthat the charge distribution is uniform atthe start.

This method was developed by Howe

Gross, Chitnis, Stratton-Grounding Grids for High- oTltage Stations AUGUST 1953Soo


Fig. 1 (left). Typicalgrounding grid

a7

b7

C7w

C8

b5

a5C

a4 b 4c b2 a4 4 I4

aT ra30 3 C3 Ci Ila

,-SURFACE OF EARTH

Fig. 2 (right). Sub-division of grounding

grid with one mesh

and later used by Dwight.8'9 In many

cases, the values of capacitance calculatedby this method are accurate to within a

few per cent.The method of subareas was first used

by Maxwell to calculate the capacitanceof a square. The electrode system carry-

ing electric charge is divided into sub-areas Ai (i=1, 2 . . . n), each of themcarrying a charge density qi (i= 1, 2 ... n).The subdivision is so made that each sub-area is small enough to make the followingassumptions :10

1. The charge density qi over Ai is essen-

tially constant.

2. The potential Vij produced by thecharge on Ai over the area occupied by Ajmay be chosen to calculate the potential ofAj due to charge on Ai. Similarly, the po-

tential Vii produced by the charge on Aiover the area occupied by Ai is constant.

Since the potential Vij produced on

Aj by the constant charge density qi on Aiis proportional to qi, the following may bewritten

Vij=Kijqi (4)

and the total potentialn n

Vj = E] Vij = :EKi.qj (5)

i=l i=1

If in this fashion the potential of eachsubarea may be computed, obtaining nlinear equations in n variables, qi (i= 1,2 ... n). However, since the charges are

in equilibrium over this charged conduct-ing electrode system, the potential is con-

stant everywhere on it, therefore

n

Vo = EKijqi (6)i= 1

where VO is the potential of the electrode

d - 2sTZs

rIMAGEaf b3 C3 cl b a\,

,, SURFACE OF EARTH

I d,:GROUNDING GRID. - Ib-'-o3 D3 C3

L_0

system. Hence qi (i=1, 2 n) can besolved in terms of VO. The total chargeQ is then given by

n

Q =5A iqi (7)i =1

and Q is obtained in terms of VO. Finallythe capacitance C is given by

VO Q(8)The finer the subdivision, the more ac-

curate is the result. Fortunately, inmany cases, a reasonably small value of ngives results that are quite accurate.

Effect on Capacitance of Numberof Meshes of Grid

The conductor configuration to beanalyzed is a grid buried horizontally at a

depth of 1 to 2 feet. To simplify thecalculations, it will be assumed that thegrid occupies a square area rather than a

rectangular area, Fig. 1. In this way,

symmetry will greatly reduce the amountof analytical work required to determinethe capacitance.The actual grid will cover a certain

area which will be criss-crossed by thegrid wires. The density of the mesh willdepend upon the separation between thewires. If the separation between thewires is increased, all that will be left inthe limit will be four wires which enclosethe area formerly occupied by the grid.The resistance to ground of these fourconductors, forming a square, will give

a 1 -v -c a

W - - 0

an upper limit for the resistance of thegrid.As conductors are placed within the

area, the resistance to ground will decrease.Just how much the resistance is reducedfor each conductor that is added is one

of the problems of this investigation.The limiting condition is reached whenthe spacing between the grid wires be-comes zero, so that the grounding systemhas become a plate. This will establisha lower limit on the resistance to groundwhich could be obtained for the actualgrid. These limiting conditions will bediscussed first and then the grid itselfwill be analyzed.

CAPACITANCE OF A SQUARE OF WIRE

To obtain the upper limit of the resist-ance to ground of the grid, it is necessary

to determine the expression giving thecapacitance of a large square of wirewhich encloses the area occupied by thegrid. The capacitance of a square ofwire and its image may be closely ap-

proximated by the capacitance of a circleof wire and its image, if the circum-ference of the circle is equal to the perim-eter of the square. This is so because theresistance to ground of a buried conductorconfiguration depends primarily upon thelengths of wire buried in ground whichare not surrounded by adjacent wires.", 12This is most easily seen if the resistanceto ground is calculated for a straight wire,a right angle turn of wire, a 4-point star,etc. If the over-all lengths of wire buriedin ground are the same, the resistance of

Gross, Chitnis, Stratton-Grounding Grids for High- Voltage Stations

_ l _1_

a8

C6

a6

0 I 4 0

0

4

0

9 9 4-0

* * * * * * ---

AUGUST 1 953 801


04b4 b2 02

03

4

bi 0a

Fig. 3 (left).Subdivision of

a8 grounding grida with four meshes

Os

IMAGEas co al \

/-SURFACE OF EARTH

,'-GROUNDING GRIDC

b, a, Fig. 4 (right).Subdivision ofgrounding gridwith nine meshes

d - 2s

s-Jr *

SURFACE OF EARTH

GROUNDING GRID03 C GI

- ----LW o

all configurations will be almost the same

with slightly increasing values for thelatter cases. This is due primarily to themutual coupling at the junction points inthe more complex configurations. Anequation for the capacitance of a circle ofwire and its image located a distance daway is giv-en by Dwight as

1 1 4D 4D1C Ln -In dIn ( 9)

where r is the radius of the conductor andD is the diameter of the circle. If equa-

tion 9 is to approximate the reciprocal ofthe capacitance of a square of wire whichencloses an area A, then D =4/A /rsince for this condition the length of wirein ground is the same.

By the method of subareas, the capaci-tance of a square wire and its image maybe calculated to any desired degree of

accuracy. To make the analytical ex-

pressions as simple as possible, an equa-

tion for the capacitance of a square ofwire and its image is derived in AppendixIII. Comparison of equation 9 and ofequation 13 in Appendix III shows thatthe capacitance of a circle that has a cir-cumference equal to the perimeter of a

square is slightly larger, the per-cent error

decreasing for increasing areas. This isto be expected since the conductors are

mutually coupled at the corners of the

square. The calculation of the capaci-tance by either the use of a square of wireor a circle of wire gives an upper limitfor the resistance to ground for a ground-ing grid.

CAPACITANCE OF A PLATE

If the meshes of the grid are increasedwhile the over-all dimensions of the gridare maintained the same, then in the limitthe grid becomes a plate. Thus the resist-ance of a plate with the outside dimen-sions of the grid indicates the lower limitof the resistance to ground of a grid cover-

ing a specific area. Equations for a plateare given in Appendix IV.

CAPACITANCE OF A GRIDAn attempt was made to calculate the

capacitance of a grounding grid by theaverage potential method and a numberof approximations. The grounding gridwas considered to be composed of a

number of squares, the number dependingupon the number of meshes. Thecapacitance of the squares and theirimages was approximated by the use ofequivalent circles. These circles of wirewere in turn assumed to be representedby equivalent vertical rods whose lengthand diameter were functions of the dimen-sions of the circle. The capacitance ofthese vertical rods and their images was

then calculated by the use of equationswhich apply to a group of long rods. Thismethod of replacing the ground grid byan equivalent configuration gives goodresults providing the equivalent rods are

long and thin, and such is the case forlarge grids with few meshes. As thespacing between the conductors of thegrid is decreased, the equivalent configura-tion is no longer valid. Since the calcula-tion of the capacitance of the grid by the

method of subareas gives results which are

accurate to any desired degree, themethod of subareas is preferable and was

used.The subdivision of the grounding grids

is shown in Fig. 2, 3, and 4. The ground-ing grid with one mesh and its image are

divided into 48 subareas (that is, n = 48).Because of symmetry a number of sub-areas will carry the same charge densityand only three different charge densitiesare needed. All subareas carrying thesame charge density are denoted by thesame letter (a, b, or c) but distinguishedby its accompanying numerals (a,, a2, . . .

etc.). The image of a, is denoted by a1'.Hence there will be three different chargedensities, qa, qb, and q,. The groundinggrid with four meshes is subdivided into48 subareas with only four different chargedensities, q,1, qb, qC, and qd. The ground-ing grid with nine meshes is subdividedinto 48 subareas with only four differentcharge densities, qa,, qb, q, and qd. Itshould be noted that each subarea con-

sists of the curved surface of a length Lof wire of radius r. However, because ofthe large length to radius ratio, eachsubarea is, in fact, a linear element andconsequently the charge density, insteadof being a surface charge density, is a

linear charge density and Maxwell'smethod is modified to that extent.The equation used in calculating the

potential at any point due to charge on a

linear element is given in Appendix I.A sample calculation for a grounding gridof one mesh is shown in Appendix II.

GrosS, Chitnis, Stratton-Grounding Grids for High- Voltage Stations

07

b7

Wa

b5

a5

C4

4-

d3

C3

I

07

a4

03

C2

C'

C2

a'

be

b2-

b5fb62

d =2s

s

C4

06

rIMAGEa 3 . b,I , al ,

a3 b3e*- L --*1

-- W

0

S/////X//XC/?

0 0- 0 -0

C2 di cl

05

I 4

b3

AUGUST 1953S02


Before proceeding to grids with moremeshes, a closer examination of the solu-tion given in Appendix II is in order.

1. Major contribution to Kaa' is from a,and al'.

2. Any change in the radius of the conduc-tor is reflected in the coefficients Kaa', Kbb',and Kcc' through the contribution from al,b1, and cl respectively. The other co-efficients remain unaltered. This is evidentfrom the equations given in Appendix I.

3. Any change in the depth of the grid isreflected in all the coefficients in varyingdegrees. Whereas the change in Kaa',Kbb', and K11' with any change in the depthof the grid is of major importance, thechange in the mutual coefficient is so smallthat it can be neglected. The difference be-tween the contribution of any subarea onthe main grid and the contribution from thecorresponding subarea on the image of thegrid (for example, a2 and a2') indicateswhether any change in the depth of the gridwill affect the coefficients to an appreciabledegree. When the distance between thecenter of the subarea from the point atwhich the potential is calculated is verylarge compared with the depth of the grid,this difference is quite small. For example,for the sample calculation given in AppendixII, the contributions of a, and a,' to Kaa' are5.7914 and 1.8115 respectively, and those ofa7 and a2' are 0.0725 and 0.0725 respectively.If the depth of the grid is now changedfrom 1.5 feet (as used in this calculation) to3.0 feet, the contributions from a1 and a2 areunaffected and the contributions from a1'and a2' are 1.1894 and 0.0725 respectively.

These observations are quite helpfulwhen examining the effects of radius anddepth on the capacitance of the grid. Aset of coefficients is first obtained for aparticular area, radius of the conductor,and depth of the grid. Then, when it isdesired to obtain a new set of coefficientscorresponding to the same area but for adifferent radius of the conductor and adifferent depth of the grid, coefficientsKaa', Kbb,' and K,,' are obtained fromthe original set by recalculating the con-tributions from a, and a,' (for Kaa'), biand bl' (for Kbb'), and cl and cl' (for K,,1).There would be some change due to con-tributions from other subareas, but it isso small that it can be neglected. Simi-larly, the mutual coefficients remain un-altered for the same reason. Thus anew set of simultaneous linear equationsis obtained. This is solved for qa, qb, andq, and the new value of capacitance results.

It was observed in the case of gridswith one mesh that the charge densityalong the wire is almost constant. If thisis assumed, then it facilitates the solu-tion of the simultaneous linear equations.Validity of this assumption can alwaysbe checked once the solution is obtained.This assumption is used to obtain an ana-lytical expression for the capacitance of asquare grid with one mesh.

One might expect the mutual coefficientslike Kab' and Kba' to be reciprocal. Theyare almost, but not exactly, reciprocal.For example, for the sample calculationgiven in Appendix II, K,b' is equal to2.4368 while Kba' is equal to 2.4568.These coefficients depend not only on thedistance between the center of one sub-area to the center of the other but alsoon their orientation with respect to eachother. This dependence on the orienta-tion is less important when the distancebetween the subareas is much larger thanthe length L of each subarea. This can beexplained when it is noted that the equi-potential surfaces for a charge uniformlydistributed over a conductor of length Lare ellipsoids of revolution. When thedistance from this charged conductor islarge compared with its length L, theequipotential surface can be approximatedto a sphere instead of an ellipsoid of revo-

lution. Whereas the contribution fromb6 to Kba' is equal to 0.2828, that from a6to Kab' is equal to 0.2715. However,the contribution from b5 to Kab' is equalto 0.0766 and that from a8 to Kab' isequal to 0.0765.

Results of Calculations

The results are expressed as the recip-rocal of the capacitance and not as thecapacitance of the grid since this valuecan be used more conveniently in ob-taining the resistance to ground of thegrounding grid. This is evident fromequation 3. The results are tabulated inTables I, II, and III. Values that wereobtained by using the short cuts men-tioned in the previous section are dis-tinguished by an asterisk (*). However,the values thus obtained are reasonablyaccurate.

Table 1. Results of Calculations for a Square Grid with One Mesh

Radius r of (1/C) X 104Area A, conductor, Depth of grid (C in W/C

Square Feet Feet s= d/2, Feet Centimeters) rd/A (W in Feet)

10,000 ............ 1.5 ........ .. ..12.68......... 9 .0X 10-6 0.126820,000. 0.031 ............ 9.36 .........4.5X1O-6 ......... 0.132350,000 .......... .03 ..... 1.5.6.25. 1.8X10 . .........1397100,000... 0.03 ............. 1.5. ........... .59 . ........ 9.1X10-7 ......... 0.145020,000 .... ....... ............ 1..............0 * 1.5X10-6 ......... 0.141420,000 .......... .02 .. ...........62*. ........0X106. 0.136020,000... 0.04 ............ 1.5............ .21*. .0X10-6 ......... 0.130220,000. 0.05 1......... ...... 9.08* . 7.X10-6 ......... 0.128520,000........ 0.0748............ 0 . ........... ........ 8X10 t. 0.155520,000... 0.03 ............

1 . . .......... 12.0........ 4.5X160X- ......... 0.170420,000 ... 0.03. ............. 10.5.........7.5X10-7 ......... 0.147020,000... 0.03. ........... 5.50 ............ 10.02*. ...1.5X10-6 .... 0.141820,000 ... 0.03. ............ 9.62* ......... 3.0X10-6 ......... 0.136020,000 ... 0.03 .............00 ............ 9.23* ......... 6.0X10-6 ......... 0.130620,000 ... 0.03 ............ 3.00 ............ 8.99*. .. 9.OX10-6 ......... 0.1270

*,tSee text for explanation.

Table IL. Results of Calculations for a Square Grid with Four Meshes


Square Feet Feet s =d/2, Feet Centimeters) rd/A (W in Feet)

10,000... 0.03 ............ 1.5 ............ 11.0 ......... 9.OX10-6 ......... 0.110020,000 .......... 0.03 ............ 1.5 ............ 8.03. . 4.5X10-6 .........0.113650,000.... 0.03 ............ 1.5 ............ 5.33 ... 1.8X10-6 ......... 0.1192100,000.... 0.03 . ........... 3.88 ..9.OX10-7......... 0.122820,000... 0.0748......... 0 ............ 9.12 ... 2.8XI0-7t... 0.129020,000... 0.03 ......... 0. ...........9.86 .........

4.5X10-t ... 0.139520,000... 0.03 . 0.25 ............ 8.72* .7.5X10-7.. 0.123320,000... 0.03 . ........... 7.74*... 9.0X10-6.. 0.1095


Table Ill. Results of Calculations for a Square Grid with Nine Meshes


Square Feet Feet s= d/2, Feet Centimeters) rd/A (W in Feet)

10,000 .... 0.03 ............ 1.5 ............ 10.15 ......... 9.0X10-6 ......... 0.101520,000.... 0.03 ............ 1.5 ............ 7.39 . .4.5X10-6 ......... 0.104550,000 .... 0.03 ............ 1.5 ............ 4.84 ......... 1.8X10-6 ......... 0.1082100,000 .... 0.03 ............ 1.5 ............ 3.53 .. .9.OX10-7 ......... 0.111520,000 ... 0.0748 ......... 0 ............ 8.21 ......... 2.8X10 -7t . 0.116020,000. ...03.......03 . ........... 8.............75 4.5X10-8t . 0.123720,000... 0.03 . 0.25 ............ 7.91*... 7.5X10-7 0.1119




0-x

-J'

1 CONDUCTOR RADIUS

12 ___ DEPTH OF THE GRID

10 -s; | ~~~~~~~~~~2itr(C<11 _ A~~~REA- 210r(0)QAREA = 0,000 SQ.

8AREA =20,000 SQ.

6

AREA = 50,000 SQ.

_, AREA = 100,000 SQ.

2

1 4 10 100 1000 10,000NUMBER OF MESHES

Fig. 5. Influence of the number of meshes on the resistance

Fig. 5 shows a plot of the reciprocal ofthe capacitance of the grounding gridagainst the number of meshes. Theareas were arbitrarily chosen as 10,000,20,000, 350,000, and 100,000 square feet.The radius of the conductor was chosen-to be equal to 0.03 foot and the depthbelow the surface of the ground wastaken as 1.5 feet. The capacitance isexpressed in electrostatic units (that is,in centimeters): hence if the resistivity pis expressed in ohms-centimeter, then the-resistance to ground is given by equation3. The resistance of the grid decreaseswith increasing numbers of meshes. Thedecrease is quite rapid in the beginningbut slow after about 16 meshes. Thelower limit is reached when the criss-cross conductors touch each other and-form a square plate. This value for asquare plate is plotted for a number ofmeshes equal to 100,000. However, thisvalue corresponds to zero depth (that is,corresponding to a plate on the surfaceof the earth It is assumed here thatthe resistance of a plate on the surface ofground is almost equal to the resistanceof a plate buried at a depth of 1.5 feet. Itwill be evident from Fig. 7 that the effectof increasing the depth of a 1-mesh gridfrom zero to 1.. feet is to decrease theresistance by about 25 per cent. Sincethe radius of the conductor is very smallcompared to the depth of the grid, eventhis small depth is effective in reducing theresistance of the grid by some amount.However, for a square plate of the sameover-all dimensions as the grid, a depthof 1.5 feet is verv small compared to thedimensions of the plate and the reductionin the resistance would not be large.The experimental data obtained by Mc-Crocklin and W'endlandt14 show that ifthe depth of the plate is increased from

zero to 1.5 feet, the reduciis only about 6 per cent10,000 square feet, andlarger areas. Hence, Imade in drawing the cwquite reasonable.

It is evident from Fig.tion in resistance obtainethan 16 meshes is uIbetter way of reducing thincrease the area enclosec

UNIVERSAL CURVE FOR CGRIDS

It has been mentionedof a square grid with onecharge density along theOf course, in the neighcorners of the grid the cbound to be higher, butpurposes it can be assustant. On the basis of thanalytical expression fc

0.20

0.18

0.16

0.12

0.10

0.08

= 0.03 FT.= 1.5 FT.

. FT.

FT.

FT.. FT.

ship between the capacitance of the gridand its physical dimensions was obtained.The derivation is given in Appendix III,where the relationshil) is expressed as

-0= .0316-0.0189 logn ,if «1d-<<I (13)C -- I-V

Mx here1li=wxidth of the grid, feet = v AC = capacitance of the grid, centimetersr =radius of the conductor, feetd = distance betweeni the grid and its im-

age=2s, feet1 = area of the grid, square feet

When the grid is on the surface of theground and not buried, s would be equal

100,000 to zero. However, d should be put equalto r in equation 13 since the potential ofthe grid is the potential on the surfaceof the wire; these values are indicatedby a dagger (t) in Tables I, II, and III.Equation 13 is valid only for grids buried

tion in resistance very close to the surface of the eartht for an area of since d/W<<1. If the grid could beI is smaller for buried deeply in the ground, obviousthe assumption choice would be deep-driven rods forrves of Fig. 5 is ground connections rather than a ground-

ing grid buried parallel to the surface of5 that the reduc- the earth. Hence, equation 13 has prac-d by using more tical usefulness.neconomical. A Equation 13 is plotted in Fig. 6 on ale resistance is to semilogarithmic system of co-ordinates toI by the grid. give a straight line. The points plotted

are the values that were actually calcu-,ROUNDING lated; see Table I. The close agreement

is noticeable since the calculated pointsthat in the case lie on the curve representing equation 13.mesh the linear The calculated values for 4- and 9-wire is constant. mesh grids are also plotted in Fig. 6.iborhood of the They satisfy a relationship similar to thebharge density is one for the 1-mesh grid given by equationfor all practical 13. In this case a wide range of valuesmed to be con- for rd/A has been covered, hence theiis assumption an validity of this relationship can be as-Dr the relation- sumed for values of rd 'A likelv to occur

I MESHI

< ~~~~~~~~~~-a-4MESiH

__ __ _ _ _ L 1 I__ _ _ _ _ _ I _ _ _

i07 rdA

-6 io-5

Fig. 6. Resistance to ground. Universal curve for grounding grid

Gross, Chitnis, Stratton-Grounding Grids for High- TVoltage Stations

14L

i

I 'AI

I

._

lo-9 5 - IC' ICF' 10-4

AUGUST 1953-S04


1.0 2.0DEPTH IN FEET

3.0

16Fig. 7 (left). Effect of depthon the resistance

14

0x

-IC

Fig. 8 (right). Effect of con-ductor radius on the resistance

108

8~~~f >i_>-0

AREA - 20,000 SQ. FT.DEPTH = 1.5 FT.ONE MESH ONLY

0.02 0.04CONDUCTOR RADIUS IN FEET

in practice. Specifically, it must bestated that the d/W ratio should be very

much smaller than one.

EFFECT OF DEPTII AND CONDUCTORRADIUS

Although thle curves presented in Fig. 6are of great usefulness, they do not indi-cate the relationship between the individ-ual parameters and the resistance of thegrid with enough clarity. The variationof resistance with the number of meshesand with the area is already shown inFig. 5. The variation of resistance withthe depth of the grid is shown in Fig. 7and the variation of resistance with tlleradius of the conductor is shown in Fig. 8.The curves are drawn through pointsactually calculated as given in Table I.Although these curves are drawn for a

grounding grid with one mesh enclosingan area of 20,000 square feet, the conclu-sions drawn are applicable to grids withlarger number of meshes. The resistanceof a grid is very high when the grid is on

the surface of the earth. It decreaseswith increase in depth, quite rapidly atfirst and afterwards slowly. The varia-tion of resistance with the radius of theconductor is quite small for the sizes ofconductor that would be used normally.The curve is asymptotic for values of r

approaching zero. It is evident fromFig. 6, 7, and 8 that the effect of doublingthe radius of the conductor is the same as

that of doubling the depth. Normally,thermal and mechanical considerationswould determine the radius of the con-

ductor, and the nature of the soil woulddetermine the depth.

COMPARISON OF CALCULATED \ ALUESWITH AVAILABLE EXPERIMENTAL DATA

Some experimental work has been doneon this subject by McCrocklin and Wend-landt'4 and the comparison between theexperimental data and the calculatedvalues is shown in Table IV. The depthof the grid was chosen to be equal to zero

because some difficulty was encounteredin reading their curves for very smalldepth.

The experiments were made with a

model grid in a natural lake.

Conclusions

The problem of calculating the resist-ance to ground of a grounding grid is, infact, a problem of calculating the capaci-tance of the grounding grid and its imagebecause there is a simple relationshipbetween this capacitance and the resist-ance to ground. Maxwell's method ofsubareas is a very powerful tool in calcu-lating the capacitance of a symmetricalbody. The results can be obtained toany desired degree of accuracy, althoughthe numerical work involved is considera-ble for a very high degree of accuracy.

The method also gives useful informationregarding the charge distribution on thebody. Only numerical results can beobtained with this method.The method of subareas was used in

calculating the resistance to ground ofsquare grids buried in ground, parallelto the surface of the earth, to very shallowdepths. It was found that the chargedistribution, in the case of a 1-mesh grid,is very nearly uniform along the wire.Hence it is expected that the results for a

1-mesh grid are good to within slide-ruleaccuracy. This is also borne out by thefact that. the calculated values closelycheck the values obtained from the ana-

lytical expression derived for one meshgrid; this means that there is no changein the answer when going from eight sub-areas to 48 subareas in the calculations.

Calculations for 4-mesh grids were madewith 48 subareas. However, when one

of these calculations was repeated with96 subareas, the answer was only 0.2per cent higher than the one obtainedfor 48 subareas. Therefore, it is expectedthat the results for 4-mesh grids are alsogood to within slide-rule accuracy. Theratio of minimum charge density tomaximum charge density in the case of 4-mesh grids is about 0.65 to 0.70, depend-ing on the size of the area whereas thesame ratio for 9-mesh grids is about 0.58to 0.65 for the calculations made in this

investigation (charge densitv here refersto the average charge density over asubarea). This ratio gives a good indica-tion as to how many subareas should beused to give an accurate answer. Thelower the ratio, the finer ought to be thesubdivision. Since a fairlv accurateanswer was obtained for 4-mesh gridswith 48 subareas, it can be assumed thatthe same order of accuracy would be ob-tained with 9-mesh grids with 48 sub-areas because the ratio just mentioned isof the same order.The grid usually occupies a large area

and an average value of ground resistivityhas to be used in the calculations. Theseasonal variations in resistivity are verymarked at the depths to which the gridsare usually buried. Because of these tworeasons the accuracy with which theresistivity is known is not of high orderand the degree of accuracy obtained in thecalculations of the capacitance of a grid isconsidered quite adequate.The resistance of a grounding grid de-

creases with increase in the area enclosed.It also decreases with increase in thediameter of the wire used and the depthto which it is buried. When the depthof the grid is of the same order of magni-tude as the diameter of the wire the resist-ance is quite high, but as the depth isincreased the resistance decreases quiterapidly at first and then very slowly.Variation of resistance with the size of thewire is of a similar nature except for thefact that very thin wires are rarely usedand therefore ranges where variationsare rapid are seldom encountered.

Table IV. Comparison Between the Calcu-lated Values and the Available Experimental

DataAred=20,000 Square Feet; Conductor Ra-

dius=0.0748 Foot; Depth of Grid =0

ExperimentalNumber Calculated Value of

of Value of (1/C) X 104 Deviation,Meshes (1/C) X 104 (Reference 14) Per Cent

1. 11.0

4. 9.129. 8.2...

.10.5 ... -4.59.95 +9.0

23.2 +11.25

Gross, Chitnis, Stratton-Grounding Grids for High -Tloltage Stations

AREA=20,000SQ.FT.CONDUCTOR RADIUS - 0.03 FT.

- ONE MESH ONLY

II

16

14

0x 12

10

80

0.06

9

-P

ri

AUGIJST 1953 805


TabIt V. Sample Calculation for a Square Grounding Grid with One Mesh

Due to a Due to b Kij'IDue to c K1i'

Potential on al.la5.7914 . ..bi 0.4771. ....c......... 0. 2219

al .1.8115.... bi'. 0.4710.... cl'...... 0.2213a2 0.0725 .... b2 . 0.0712 c C . .. 0. 0688a2'0. .0.0725.b... 0. 0.0712..C... 2...... 0.0688

a3. 0.0873 ..... b. . 0. 1083 ..... C3 ........O.0.1461a3' .0.0873 .... b3' . 0.1083 ....I C3...... 0.1461a40. 0.0556 ...... . 0.0600 .... co...... 0.0647

a4 0 5 .0.0556.... 0..0 . 0600.LI... C4 ... ... 0.0647O.0.0784.... b5 . 0.0766 .... ..... 05 ,.O.. 0720

. ,0.0787.... b5'. 0.0763 .... c5 .. 00.0720a6.0.6269 . ..b6 0.2828. .... 06 .1725a6. 0.6151.... .b6 0.2816............0C6 .1722

a7.0.0558 ..0.0607. ..... c0.0665a7 . 0.0558.... b'. 0.0607 .. 0.0665

a80.0866 . ..b8 0.0956.... c8 . 0.1236a8.0.0866. .. b '. 0.0954 . ... C..0.1236

Kaa'. .9.7176 .... Kba' . 2.4568 .............a............. 1.8715

All throughout the calculations wheneverpotential over any linear element is calcu-lated, it is taken as the potential at the mid-point of the corresponding linear element.Whenever self-potential (that is, potentialdue to its own charge) is calculated, it istaken as the potential on the surface of thewire at the mid-point of the element.Hence x = Oand y = r, where r is the radiusof the wire. It is evident that in this case

L L( 2)-+ -I(;)+ra

2 it2)V==qln-

2~ +

='2q ln ifr<<Lr

(1)

Potential on b1a .. 0.4771 .......... 5. 7914.... ci .......... 0.4771al .0.4710 .... bi' .. 1.8115 . ..cl'. 0.4710

a2 . 0.0712 . . 0.0725 . .. c. 0.0712a2. 0.0712 .... b2'.. 0.0725 .... c2l .. 0.0712

a3 . 0.1083 ..... b... 0.1461 ........ c3 .. 0.2219a3'.. 0.1083 .... b3' .. 0. 1461 . .. cat .......... 0.2213a4 ..0.0600.... N.. 0.0647.... C4 . 0.0688

a4. 0.0600 .... b4l .. 0.0647 . .... 4 .. .0688a .. 0.0959.... b5 .. 0.0918 . .a.c .. 0.0844a6' .0.0959 .... b5' ... 0.0918 . ............. 0.0844a6 .0.2715 .... b6 .. 0.2056 .... C6 .. .1500a6 ' 0.2708.. ..b6'.. .0.2051.. .C60. 1500a7 . 0.0613 .... b .. 0.0681 .... C7 .. 0.0762a7'.. 0.0613.... b7'.. 0.0681.... C7l.. 0.0762a8 . 0.0765 . .. b8 .. 0.0916 . . c s 0.1143a8'.. 0.0765 . .. b8 ' 0.0916 .... cs '.. 0.1143

Kab'. .2.4368... Kbb'. 9.0832 . Krcb.. 2.5211

Potential on Ci

a, . 0.2219 . . 0.4771 . .. cl ....5.7914

aa'. .0.2213 .... b' .0.4710.... cl .... 1.8115a2 . .0.0688.... b2.0.0712. .... C2.... 0.0725a2' 0.0688 .... b2' .0.0712 . .. c2f .... 0.0725a3 . 0.1461 . ... 0.2219.... c3 .... 0.4771a3 .0.1461 .... b'. 0.2213 .... C3 .

....0.4710

a4 . 0.0647 . ... 0.0688. .... C4 .... 0.0712a4f 0.0647 .... b'. 0.0688. .... C4 .... 0.0712a6 .........0.1225 ... ... ba .. .. 0.1137..CD . 0.1012

a6' .0. 1223 ...... b1' .... 0.1140. c6 . 0.1009a6 . 0.1693 . .. b6 0.1486 . C6. 0.1231a6'. . 0.1692...... b6'.... 0.1486. C6f. 0.1230a7 . 0.0665 . .. b7 0.0763. C7. 0.0876a7' 0.0665 ...... b7' .... 0.0763.C. C7. 0.0876a s .0.0721 ...... b8 ..... 0.0842. s. 0.101418' . .0.0721 ...... b8' .... 0.0840 ..C.. . 0.1013

Kac'. 1.8629 ...... Kb,c ... 2.5170 . Kcc' . 9.6645

The following points should be bornein mind when considering the resistancein designing a grounding grid:

1. The area enclosed by the grid should beas large as possible. If further decrease inresistance is desired, criss-cross conductorsshould be added. However, the number ofmeshes need not exceed 16 in an economicaldesign.2. The diameter of the wire is determinedby thermal rather than by electrical con-siderations. Tubing may be advantage-ously used instead of wire.

3.X The depth to which the grid is buried isdetermined by the nature of the soil. Thegrid should be buried as deeply as possiblewithout involving too much expense inexcavation.

Fig. 6 is a very useful in determining theresistance to ground of grounding grids.It gives the ratio of the width of the grid(in feet) to the capacitance of the grid

(in centimeters) as a function of the radiusof the conductor, depth to which it isburied, and the area enclosed by the grid.When the capacitance of the grid is deter-mined, then the resistance is given byequation 3.

Appendix 1. Potential at anyPoint Due to Charge on a

Linear Element

Let the linear charge density on the wireof length L be q. Then the potential at anypoint P(x,y), where the x-axis is coaxialwith the wire, is given by'5

L IL2x+-- X+tI-I-+y2

2 l22)V=q ln

x- + Ix-I_ +y22 \2/

(10)

Appendix 11. Sample Calculation

for a Square Grounding Grid

with One Mesh

The area=20,000 square feet; the depth

of the grid= 1.5 feet; and the conductor

radius=0.03 foot.

As was explained in the theory Vij =Kijqi.Equations 11 indicate that Kij is the

natural logarithm of a dimensionless quan-

tity. Using logarithms to base ten, the

equations will be

Vij = 2.3026Kij'qi (12)

The calculations are given in Table V.The three simultaneous equations are

9.7176qa+2.4568qb+ 1.8715qc= Vo/2.30262.4368q,+9.0832qb+2.52lqc= Vo/2.30261.8629qa+2.5170qb+9.6645qC= Vo/2.3026Substituting

2.3026qa 2.3026qbXa= V, Xb= V and

2.3026q,

we obtain

9.7176Xa+2.4568Xb+ 1.8715Xc = 1 (A)2.4368Xa+9.0832Xb+2.521 lXc =1 (B)

1.8629Xa+2.5170Xb+9.6645Xc 1 (C)

Using the Doolittle method of solvingsimultaneous linear equations"8 (Table VI)the solution is

Xa = 0.0713, qa= 0.0309 VoXb = 0.0713, qb= 0.0309 VoXc= 0.0712, qc = 0.0309 Voand

1- x104= 9.36

Appendix Ill. AnalyticalExpression for the Capacitance

of a Square Grid with One Mesh

Consider a square grounding grid of thefollowing dimensions (refer to Fig. 9):

Gross, Chitnis, Stratton-Grounding Grids for High- Voltage Stations AUGUST 1953806


Area enclosed by the grid = AWidth of the grid = WRadius of the conductor = rDepth to which buried =sDistance between the grid and its image=

d =2a

Let the linear charge density be consid-ered constant along the grid and let itsvalue be equal to q. Then the potential at r(mid-point of mn) is obtained by usingequation 11.

Potential at r due to charge on mn

Wr=21 in-, if <<1

r TV

Potential at r due to charge on m'n' (m'n'being image of mn)

=qIn-22 2 W dI

=2qIn if «<1d' W

Potential at r due to charge on np

W±~WW2=q In

/W 2

2=q In (2±+ 5)

Potential at r due to charge on n'p'

W+ I'T2+±QY)2+d2=q In

-(W2+d2

=q In (2-+f-5), if <<1w

Potential at r due to pq

+ (W)2+W2

_f+ (W)2+W22~ ~ 1

2

=2q ln +.\'-2

Fig. 9. Diagram used in connec-tion with the derivation given in

Appendix Ill

Iw

m r

q

n

p

F- q- PId s 2s

sII

q

Potential at r due to p'q'

+ +n2+d

wl+ -W)+WTT2+d2

1 + d=2q ln 2

-

. if <<12

Potential at r due to charge on qrn

W+lTr2+( 2

/T72=q In

Potential at r due to charge on q'm'

Wf+lW2+(W +d2=qIn

2

= n +X),if22~~~~~~J2

Table VI. Doolittle Method of Solving Simultaneous Linear Equations

Coefficient Coefficient Coefficient Right CheckDesignation of Equation of Xa of Xb of Xc Number Sum*

A ..................................+9.718.....+2.457..... +1.871 .....+1.0 .... .+15.046A'=A/9.718.................. +1.0 .....+0.253.....+0.1925..... +0.103 .....+ 1.548B........ +2.437.....+9.083.....+2.521 .....+1.0 . +15.041

-2.437A' .-2.437.....-0.617.....-0.469 .....-0.251 - 3.770B-2.437A'......... 0 ..... +8.466.....+2.052 .....+0.749 .....+11.271B'=(B-2.437A')/8.466. 0 ... +1.0 .....+0.242 .....+0.0885 ..... + 1.331C..... +1.863.....+2.517.....+9.664.+1.0

..... +15.044-1.863A' ..... -1.863.....-0.472.....-0.359 .....-0.192 .....- 2.885C-1.863A'.......................... 0 ..... +2.045.....+9.305 .....+0.808 .....+12.159-2.045B'........................... 0

..... -2.045 .....-0.495 . -0.181 .....- 2.720C-1.863A'-2.045B'................. 0 ..... 0 ..... +8.810 . +0.627 .....+ 9.439C'=(C-1.863A'-2.045B')/8.810 .. 0 ..... 0 + +1.0 .....+0.0712.....+ 1.071

*The check sum is the sum of all coefficients and the right-hand member of each equation. The checksums are treated exactly as the coefficients of the corresponding equations. Thus, if the equation is multi-plied by a factor, the check sum is to be multiplied by the same factor. Or, if the two equations are addedor subtracted, the corresponding check sums are to be added or subtracted. The result of the operationscarried out with the check sums of the original equations has to be equal to the check sum of the final equa-tion. This fact is used as the check on the computation.

IP

w

Total potential at r is given by

V=c 4 In 2+V%4 ln (2+\W)+

2 In -+2± n-]r d

Total charge

Q=8qWand

1 VC QTherefore

1IL2n ( ++)± In (2+-\2)

4 ln-+ l1I,ifd4d

Hence

W rd d- = 0.0316-0.0189 log A , if «<<1

(13)

where W is in feet and C is in centimeters.

Appendix IV. Capacitance of aSquare Plate

The plate will have to be square in orderthat the calculations obtained by consider-ing it will be applicable to the over-allproblem. Dwight gives an equation forthe reciprocal of the capacitance of anisolated thin rectangular plate. If theequation is altered so that it is applicableonly to a square plate, the result is

1 ' 2.973c VA (14)

where -VA is the length of one of the sides



of the square plate expressed in centimeters.Dwight points out' that, because of theerror in the average potential method whenapplied to a plate, the value of (11C)' is toohigh and 8 per cent should be subtracted.7If this is done, then the corrected value forthe reciprocal of the capacitance is

1 2.736_V

(15)C VA

It is important to notice that equation 15applies to an isolated plate. Therefore, ifthe plate is deeply embedded in ground, theresistance to ground is given by

p 1R4= - (16)

If the plate is not deeply buried, but is atthe surface of the ground, the resistance istwice as great, since the image must beconsidered along with the plate. For thecase of a plate buried at the shallow depth of3/2 feet, the resistance to ground is approxi-mately the same as that of a plate at thesurface. Therefore, the expression for theresistance to ground of the plate which pro-vides the lower limit for the grid is given byequation 3 where 1 C is giveni by equation 15and p is the resistivity expressed in ohm-centimeters. An accurate computation us-ing the method of subareas to determine thecapacitance of a plate, and of its im'cge, nearthe surface of the earth is in preparation.

References1. EARTHING PROBLEMS, R. W. Ryder. Proceed-inigs, Institution of Electrical Engineers, London,England, vol. 95, 1948, pp. 175-84.

2. GROUNDING PRINCIPLES AND PRACTICE. II--ESTABLISHING GROUNDS, Claude Jensen. Elec-trical Engineering, vol. 64, Feb. 1943, pp. 68-74.

3. GROUND CONNECTIONS FOR ELECTRICAL SYS-TEMS, 0. S Peters. Technological Paper No. 108,National Bureau of Standards, Washington, D. C.,1918.

4. GROUNDING ELECTRIC CIRCUITS EFFECTIVELY,J. R. Eaton. General Electric Review, Schenectady,N. Y., vol. 44, 1941, pp. 397-404.

5. CAPACITY OF RADIO-TELEGRAPH ANTENNAE,G. W. 0. Howe. The Electrician, London, Eng-land, vol. 73, 1914, pp. 829, 859, and 906.

6. A TREATISE ON ELECTRICITY AND MAGNETISMVOLUME I (book), J. C. Maxwell. Oxford Uni-versity Press, Oxford, England, 3rd ed., 1892, pp.148-54.

7. ELECTRICAL RESEARCHES (book), H. Caven-dish. Edited by J. C. Maxwell. CambridgeUJniversity Press, Cambridge, England, 1879, pp.426-27.

8. CALCULATION OF THE RESISTANCE TO GROUNDAND OF CAPACITANCES, H. B. Dwight. Journal ofcathewnatics and Physics, Massachusetts Instituteof Technology, Cambridge, Mass., vol. 10, 1930-31,pp. 30-74

9. CALCULATION OF RESISTANCES TO GROUND,H. B. Dwight. AIEE Transactions (ElectricalEngisneering), vol. 33, Dec. 1936, pp. 1319-28.

10. AN APPROXIMATE CALCULATION OF THEEl ECTRICAL CAPACITIES OF RECTANGULAR AND

ANNULAR AREAS, D. K. Reitan. M.S Thesis,University of Wisconsin, Madison, Wis., 1949.

11. GROUNDING PRINCIPLES AND PRACTICE. IFUNDAMENTAL CONSIDERATIONS ON GROUND CUR-RENTS, Reinhold Rudenberg. Electrical Enigineer-ing, vol. 64, Jan. 1945, pp. 1-13.

12. TRANSIENT PERFORMANCE OF ELECTRICPOWER SYSTEMS (book), R. Riidenberg. Mc-Graw-Hill Book Company, New York, N, Y., 1950,pp. 316, 324-39.

13. COMPARATIVE PROPERTIES OF GRCOUNDINGELECTRODES, R. Ruidenberg. Electrical World,New York, N. Y., vol. 129, Jan. 31, 1948, p. 72.

14. DETERMINATION OF RESTSTANCE TO GROUND)OF GROUNDING GRIDS, A. J. McCrocklin, Jr, C W.Wendlandt. AIEE Tranisactions, vol. 71, pt. II,1952, pp. 1062-64.

15. THE CLASSICAL THEORY OF ELECTRICITY ANDMAGNETISM (book), M. Abraham, R. Becker.Translated by J. Dougall. Hafner PublishingCompany, New York, N. Y., 1932, p. 63.

16. TRANSPOSITION OF HIGH-VOLTAGE OVERHEADLINES AND ELIMINATION OF ELECTROSTATIC UN-BALANCE TO GROUND, Eric T. B. Gross, Andrew H.Weston. AIEE Tranisactionts. vol. 70, pt. II, 1951,pp. 1837-44.

17. GROUNDING EFFECTIVENESS AT GRANDCOULEE 230-Kv SWITCHYARDS VERIFIED BYSTAGED FAULT TESTS, A. C. Conger, R. K. Seely,W. H. Clagett. AIEE Trailsactions, vol. 70. pt. II,1951, pp. 1347--52.18. GROUNDING GRIDS FOR HIGH VOLTAGESTATIONS, L. J. Stratton. M. S. Thesis, Illinois In-stitute of Technology, Chicago, Ill.. Jan V951.

19. GROUNDING GRIDS FOR HIGH V(OLTAGESTATIONS, B. V. Chitnis. M. S. Thesis, Illinois In-stitute of Technology, Chicago, Ill.. Jan, 1932

DiscussionMartin J. Lantz (Bonneville Power Adminis-tration, Portland, Oreg.): The authors are

to be congratulated on their analysis ofgrounding grids. A number of interestingrelationships have been determined regard-ing the optimum physical layout of ground-ing grids. The conclusions state that an

economical design does not require more

than 16 meshes in the grid. It is statedelsewhere that it is evident from Fig. 5 ofthe paper that the number of 16 meshes isthe point of optimum reduction of resistanceof a mesh; however, from the curve it isnot apparent to the discusser how the value16 was determined.The subject of mat resistances is of con-

siderable interest. Obtaining the correctvalues of mat resistance is important tosystem engineering problems such as deter-mining probable substation mat potentialrise during fault and the effect of tower-footing resistances on the magnitude offault currents.1The primary variables of mat resistance

for large grids in service such as substationsare as stated: moisture content and tem-perature. An important factor for thesmaller grids such as transmission towerfootings seems to be the magnitude of faultcurrent passing through the limited area.

A limited study of actual system fault madeby the discusser indicates that the tower-footing resistance varies with the magni-tude of fault current, usually decreasingwith increase in current. The variation ofresistance in some cases is as much as 2 to21/2 to 1. Oscillographic records of faultdata have been analyzed where currentvalues and the location of the fault are

known. This prosvides two values of cur-

rent, one prior to the first circuit-breaker

openinig and the second one after. Twocalculations of fault resistance nsay bemade for the same fault location with twovalues of current magnitude. An attempthas been made to correlate meggered valuesof footing resistance with values calculatedfrom actual faults. The results have notbeen conclusive mainly because of thedifferent times of measurement and theapparent variation of resistance with cur-rent. One controlled test, however, pro-vided the results given in Table I of thediscussion.The data in Table I of the discussion are

based on two successive faults on eachtower with the meggered measurement ofresistance made before and after the tests.The successive measured fault currents werein very close agreement as well as the be-fore and after meggered values. The firsthigh-resistance tower shows substantialagreement while the second high-resistancetower shows the characteristics of a nega-tive coefficient of resistance for the footing.All three footings were of the grillage typeburied about 7 feet in the ground. Thereare no driven ground rods. Tower 72 islocated on loose gravel and dirt, tower 82is located on sand and white chalk, andtower 91 is located on broken rock and dirt.

It may also be of interest to note thecomparison of measurement of the mat re-

Table I

Tower Meggered Ohms

CalculatedBased on

Test

72......943.8.. 8282 ....... 8.0 .......12.391............76.3.. 48.8

sistance of a large substation by twvomethods. The approximate dimensions ofthe substation grid are 880 by 600 feetsurrounded by a connected metal fenceapproximately 1,170 by 850 feet. Theactual grid is irregular in shape and consistsof approximately 200 meshes of varyingsizes. In addition, three other ties are madeapproximately 500 feet to an adjacentground mat of an aluminum company. Acurve of the mat resistance determined bythe 3-point megger method out to 8,000 feetresults in a resistance of 0.60 ohm. Theaverage of five fault tests in which the matrise (261 to 281 volts average 273) wasmeasured and the mat current (527 am-peres) determined from calculations basedon the measured total fault current resultedin a mat resistance of 0.52 ohm. Twentymiles of a 230-kv unenergized transmissionline radial to the substation in an oppositedirection to ground currents flowing to themat was used as a remnote ground probe forthe test and the potential measured acrossa partially opened ground switch at thesubstation. The meggered value was meas-ured in early June and the test value meas-ured in September. The two values are inclose enough agreement for practical pur-poses.

REFERENCE

1. EFFECT OF FAULT RESISTANCE ON GROUNDCURRENT, M. J. Lantz. AIEE Transac!iomns, vol.72, pt. III, 1933 (Papep 53-331).

OhmsFault W. A. Morgan (Ebasco Services Inc., New

York, N. Y.): The authors are to be com-mended upon their solution of a complex

i problem. The analysis covers the effect ofthe different factors that enter into thedesign of a grounding rnat.

Gross, Chitnis, Stratton-Grounding Grids for hligh- Voltage Statfions AuIGUST 1953808


There is still the practical problem ofmeasuring the resistivity of the soil. Inactual practice it is not always possible tomake this measurement accurately underthe desired conditions because of variationsof moisture content and temperature andthe season of the year to which the measure-ments might be limited. Thus, even withthe tools made available by the authors, anexact determination of the resistance toground of a grounding mat may still bedifficult to achieve.

However, the main point is to know whichfactors are important in laying out a ground-ing mat where the soil is shallow and theresistivity of the soil is expected to be high.Fig. 5 of the paper shows that the area ofthe grounding mat is important, and thatit is not important to increase the numberof meshes above 16. Fig. 8 of the papershows that the diameter of the conductorhas very little effect on reducing the resist-ance to ground of the grounding mat.A question may be raised regarding the

suggestion of using tubing instead of wirefor thermal reasons. Actually, it may bedifficult to justify on a theoretical basis any-thing near the sizes of conductors ordinarilyused for grounding mats if an exact deter-mination on a thermal basis were made.Consider the short times that fault currentsmay flow with modern protective devicesand practices. A very small size wire willhandle a large current for a fraction of asecond, especially when surrounded by aheat conducting material. It is suggestedthat a reasonably large size conductoris justified for a grounding mat to givemechanical strength for the physical han-dling of the conductor and to resist abrasionby the rock cover such as when a heavytruck would be driven over the top of themat. Another reason for using larger con-ductor than theoretically justified would beto allow for a certain amount of corrosionthroughout the life of the grounding mat.In certain areas of the country, especiallyin the West, there is a high salt content inthe soil which causes corrosion of metals.It would appear that tubing would be morevulnerable to both mechanical abrasion andcorrosion.

These points are raised merely to reviewsome of the practical aspects of laying outgrounding mats beyond the scope of thispaper. It is to be emphasized that a theo-retical analysis such as this is very helpfulto the designer in laying down basic funda-mentals and in simplifying an otherwisecomplex problem for practical application.

Henry H. Plumb (United States Bureau ofReclamation, Denver, Colo.): Withoutreading any of the bibliography, but aftera careful reading of this paper, I am highlypleased to find that there is now availablean easy-to-use tool for guiding the griddesigners as to the proper land area to usefor the grid, size of wire, depth to be buriedin the soil, etc.

It has plainly demanded much originaleffort on the part of the authors to solvethis very important basic problem of eco-nomics. Every designer of grids will appre-ciate that until now there was no rationalapproach known whereby such grids couldbe intelligently designed. I am afraid weall have put much more copper into ourgrids thain we needed just to cover our

margin of ignorance, and to be sure wewould have a sufficiently low resistance grid.In these times of national emergency, it isvery necessary not to waste strategic coppersupplies or construction costs that may adddirectly or indirectly to the burden of thetaxpayers. Due to the difficulty of thisproblem, our use of too much copper couldnot be recognized, because we never got toolow a resistance such as would indicate grossoveruse of copper. Now by making eachfoot of copper grid work at maximumefficiency, as this paper makes possible forthe first time, I am convinced that somevery worth-while savings can be achieved,thanks to the authors.

Fig. 7 of the paper strongly suggests thata substantial reduction in construction costcould safely be realized by adopting ashallower burying of the conductor to, say,about 6 inches. A plow furrow would easilyembed it that deep in most soils, and thebackfill would be cheaper also. The slightloss in resistance (increase) by the shallowerburying can be more cheaply recouped bymaking the grid area slightly larger, or byother obvious means.

Lewis H. Austin (Bonneville Power Ad-ministration, Portland, Oreg.): The au-thors have supplied a much needed clari-fication of the fundamental characteristicsof grounding grids, and have provided atool for calculation which should enlightenmany who are seeking economical andeffective means of obtaining better grounds.

In the interest of clarity, it is believedthat the definition of "resistance to ground"could be improved, although the definitiongiven does faithfully agree with that ofreference 1 of the paper. It is suggestedthat three words, "of zero resistance," beinserted, to make the definition read "Theresistance to ground is the resistance be-tween the electrode system and anotherelectrode of zero resistance in the ground atinfinite spacing." It may be that theoriginal definition implied zero resistance inthe distant electrode by the term "infinitespacing." Yet the discusser was at firstled to think that it implied infinite distance,only, to the distant electrode. A correctdefinition which makes reference to anotherelectrode would require said other electrodeto be at a great distance to avoid proximityeffects, and would require it to be of zeroresistance to avoid introducing two re-sistances in series.The mathematical work and the explana-

tion of the purpose of using capacitance inthe calculations are helpful to the formationof a correct concept in the reader's mind.There are many practical applications wherethe correct concept will assist in choosingworth-while corrective measures. For ex-ample, it is of value when a designer en-counters a situation, where it is next toimpossible to drive ground rods, to picturethe benefit of horizontal buried conductors,and to see that the substitution is adequate,and the reasons why it is so.The conclusion that 16 meshes is the

optimum number seems to be supported bythe picking of a point of apparent reducedgain on a curve. Such choices generally canbe altered by plotting the curve to a differ-ent scale. However that may be, thenumber of meshes is often dictated by thelocation of equipment.

M. H. Kight (Bureau of Reclamation,Denver, Colo.): Even though a very highpercentage of the total electrical resistancesurrounding an electrode buried in the earthis generally within a radius of 10 feet fromthe electrode, the accurate calculation ofthe resistance to ground is difficult due tothe uncertainty as to the cross-sectionalarea and length of the current path forsuch ground currents. The relationshipbetween the resistance and the capacitanceto ground of a grid system, as establishedby the authors, eliminates the need fordetermining the cross-sectional area andlength of this current path. In accordancewith equation 3 of the paper, this leaves fordetermination the resistivity of the earthin the vicinity of the grid and the capaci-tance of the grid.The resistivity of the soil for the particular

location and conditions can be determinedby one of several test methods, as coveredby considerable literature on the subject.

It is noted that in deriving the equationfor the capacitance of a grounding gridseveral assumptions and approximationswere made, mainly to simplify the calcu-lations. However, it is believed that anyerrors resulting therefrom are minor, whenconsideration is given to the fact that thevalue for the resistivity of the earth isdetermined for the worst condition, whichis for dry, cold earth. In other words, fora given grid in a particular location whichis subject to large temperature and moisturechanges during the year, the variation inthe resistance of the grid to ground due tothe changes in the resistivity of the earthwill be very great as compared to any errorscaused by the various assumptions andapproximations made in connection withthe determination of the capacitance of thegrid. This is especially true for grids buriedonly 6 to 12 inches deep in normally drysoil, which is subject to seasonal rainfall.

It is well that the authors stress the im-portance of considering the magnitude ofthe fault current to ground when deter-mining the required value of the resistanceto ground, as this determines to a largeextent, for any particular soil condition, thesize of the grid required to keep the voltageduring faults down to a safe level. Refer-ence 17 of the paper gives some interestingexamples of voltage rises in various portionsof the grounding system at the GrandCoulee 230-kv switchyards due to variousvalues of staged fault currents to ground.

For a given value of ground resistancerequired, it is a matter of economics and soilconditions as to the type of groundingsystem used, that is, whether only groundrods are driven, whether a grid withoutground rods is used, or whether a combina-tion of both driven ground rods and a gridare used, the ground rods being connectedto the grid at various points.

This paper is very interesting and willundoubtedly be quite useful to designers ofgrounding systems for electric installations.

G. D. Floyd (The Hydro-Electric PowerCommission of Ontario, Toronto, Ontario,Canada): There has been some confusionin the past with respect to groundingmethods. The practice has been to a greatextent empirical and very little informationon the fundamental basis of ground resist-ance has appeared in the literature. This

Gross, Chitnis, Stratton-Grounding Grids for IHigh- Voltage StationsAUGU'S,T 1953 Soo


14l

12 : u)--_

10 AEAREA= 10,

__ __ 2C___

4~~~~~~~~~~~~~4

2

0 400 800 1200 1600

FEET 01

Fig. 1. Influence of the length of

paper fills a gap in this respect in so far asthe grounding grid is concerned. Althoughthe paper states that grounding grids areused when rocky ground makes the use ofdriven ground rods impractical, it should henoted that in high-voltage stations wheregrounding conditions are good the equiva-lent of a grounding grid is used, In thesesituations, a ground rod is driven at thebase of each steel column and these groundrods are all connected together with one ormore copper ground wires. This in effectproduces the equivalent of the groundinggrid discussed in the paper.

This paper would have been of more valueif the theoretical analysis had been sup-ported by test data. It is noted that only asingle series of tests was compared with thetheoretical analysis and Table IV of thepaper shows good agreement between thecalculated values and the experimentaldata. However, it would be wrong to base

F COPPER

f copper on the resistance to ground

definite colnclusionls on the comparison ofcalculated values and one set of experi-mental data. As this analysis forms thematerial for a graduate student thesis, Iassume that it was not possible in the timeavailable to check the theoretical analysiswith experimental observations. I wouldsuggest that those utilities interested mightdo well to check the theoretical resultsgiven in the paper with the experimentaldata on their ownI systems so that the in-dustry can be provided with verification ofthe theoretical analysis given in the paper.

E. T. B. Gross, B. V. Chitnis, and L. J.Stratton: The discussions seem to indicatethat this paper may be of help in the designof grounding grids for high-voltage stationswhich are used when rocky soil makes theapplication of driven ground rods imprac-tical. It should be emphasized that the

paper deals exclusivelv with this oie com-ponent of the grounding problem, that is, agrounding mat used primarily for establish-ing a low-resistance station ground.Some of the discussers referred in par-

ticular to a 16-mesh grid; Fig. 1 of the dis-cussion shows directly the effects of addedconductor length on the mat resistance.This figure checks other results summarizedin the paper. If a particular length of wireis to be used, the 1-mesh grid, with thelargest area, will give the lowest resistance.However, a 1-mesh grid would be impracti-cal since the grounded parts of the high-voltage equipment, as well as the structures,must be connected to the grid, and theseconnections lead automatically to meshes.The curves in Fig. 1 of the discussion leveloff considerably with the increasing numberof meshes and the added cost of copper plusinstallation for additional meshes will notbe justified. A much greater gain wouldresult from using the added copper for anincreased area. As pointed out by Mr.Austin, the number of meshes is often dic-tated by other factors, such as the locationof equipment.Mr. Floyd refers to a general type of

grid, where driven rods are also used. Theanalysis does not apply to such cases. Itappears that this investigation is based onsound assumptions and we would feel in-clined to apply the proverb "Nothing ismore practical than a good theory." Theresults check well with the results of theonly tests known to us, and made inde-pendently by others for development pur-poses at considerable cost; it does not seemthat further experimental work could addmuch to our knowledge. How-ever, itwould be of interest to compare resistancevalues measured at stations groundedthrough such mats with values computedfor the same station.The other discussions bring out many

pertinent points of considerable interest andpractical importance to design and operat-ing engineers, and their remarks are veryvaluable additions to the paper. We wishto sincerely thank all discussers for theirinterest in this paper, and for their contribu-tions.

Design Charts ror Determining OptimumGround-Rod Dimensions

ance have been subjected to very thoroughstudies3-I little has been publighed in theform of such a guide; it is the purposeof this paper to attempt to fill this void.

Reference Rod

J. ZABORSZKY JCMEMBER AIEE

M ECHANICAL ground-rod drivers,which have become quite com-

monly available in recent years, make itpossible to drive thinner ground rods togreater depths than is possible withmanual methods; for instance, the feasi-bility of driving ground rods to depthsof 100 feet has been demonstratedl2 atmany locations in the country.These developments permit wider

)SEPH W. RITTENHOUSEMEMBER AIEE

ranges from which to choose ground roddimensions; so there arises a need forsome guide which the practical engineercan use to select the optimum dimen-sions and number of ground rods requiredfor obtaining satisfactory ground resist-ance with a minimiium amount of thecritical materials of which ground rods are

composed.AWhile many aspects of ground resist-

The results of this study are presentedin simple graphs in which the resist-ances of arbitrary grounds are expressedin general as percentages, with theground resistance of a 10-foot-deep 3/4-

Paper 53-240, recommended by the AIEE Sub-stations Committee and approved by the AIEECommittee on Technical Operations for presenta-tion at the AIEE Summer General Meeting,Atlantic City, N. J., June 15-19, 1953. Manu-script submitted March 16, 1953; made availablefor printing May 3, 1953.J. ZABORSZKY and JOsEPH W. RIrrENHOUsE arewith the University of Missouri School of Minesand Metallurgy, Rolla, Mo., and consultants toJames R. Kearney Corporation.

Zaborszky, Rittenhouse-Optimum Ground-Rod Dimensions810 AuGUST 1953


grounding resistance 1

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