quired test equipment is much less expen-sive and far less costly and cumbersometo transport. Also, insulation leakage athigh stresses may be measured. There-fore d-c overpotential tests were recom-mended for maintenance tests.

During the last 2 years, many gener-ators have had d-c overpotential main-tenance tests applied to the stator wind-ings. For 13.8-kv machines with moderninsulation, the usual d-c overpotentialtest was 30 kv for I minute. In no casehas this test failed to pick out seriouscases of insulation weaknesses. In addi-tion to stator insulation faults, other in-cipient weaknesses have been detected inbuswork and cables. During maintenanced-c overpotential tests, 10-minute di-electric absorption curves are made in therange of 1 to 15 kv direct current. Over-potential tests usually have been made onall three phases to ground with directcurrent; separating of phases usually in-volves considerable labor and time and isnot considered necessary unless troubledevelops. On windings indicating low in-sulation resistance or failure on overpo-tential test, phases are separated to assistin locating the trouble. However, mostcases of winding faults during overpoten-tial tests are readily visually detected.The ability of direct current to detect andlocate insulation faults with minimumdamage to the insulation is now well es-tablished for maintenance testing.

In several instances d-c tests have in-

dicated very low insulation resistance.One such case involved a thermocouple inthe neutral which, with the winding iso-lated from ground, caused very high leak-age at relatively low d-c test voltage.This thermocouple, not used since thefirst few days of operation, was removedand the machine had a satisfactory in-sulation resistance at 30 kv direct current.In another instance, a generator cablepassing near a concrete wall had beensprayed with aluminum paint, whichpossibly could cause trouble if not re-moved. The d-c test quickly indicatedhigh leakage. In only one case has a high-voltage machine indicated high leakageduring d-c test which could be attributedto the machine insulation.The usual range of insulation resistance

after 1-minute voltage application duringd-c overpotential test (25 to 30 kv directcurrent) encountered in high-voltagesynchronous machine windings is as fol-lows:Hydrogen-cooled sync hronous generators,100 to 500 megohms.Air-cooled synchronous generators, 50 to200 megohms.These values are average values for manymachines tested and do not necessarily in-dicate that an insulation resistance lowerthan these limits should cause concern.Insulation resistance, especially at highdirect voltages, depends upon manyfactors. For an individual generatorwinding, such factors as type of insula-

tion, age, atmospheric conditions, clean-liness, and size of winding must be takeninto account when considering insulationresistance of a particular machine.

Summary

To date about 4,000,000 kva of high-voltage generating equipment have beeninspected in the field and d-c overpoten-tial tests have been made to give assur-ance of adequate insulation level. About1,000,000 kva of generating equipmenthave been tested with direct current inthe Westinghouse factory. The successand general acceptance of this test methodhas been widespread both in the shop andin the field.

While much fundamental investigationof d-c versus a-c testing will be accom-plished by many individuals in the nextfew years, there can be little doubt thatd-c overpotential testing will be ex-panded where applicable to take advan-tage, in addition to fundamental advan-tages, of its lower initial cost and loweroperating cost, with more informativetest results.

References

1. ALTERNATlNG AND DIRECT VOLTAGE ENDLJR-ANCE STUDIES ON MICA INSULATION FOR ELECTRICMACHINERY, Graham Lee Moses. AIEE T'rans-actionts, volume 70, part I, 1951, pages 763-69.

2. A MIAINTENANCE INSPECTION PROGRAM FORLARGE ROTATING MACHINES, John S. Johnson.AIEE Transactions, volume 70, part 1. 1951, pages749-55.

No Discussion

Synopsis: Following a method due origi-nally to Stalhane and Pyk,1 the thermalresistivity of soil may be measured in situby thrusting a long heated needle into theground to the required depth and measuringits temperature every 30 seconds for 15 to 30minutes. The slope of the resulting tem-perature versus time curve on semilog paperis proportional to the thermal resistivity ofthe soil, and the constant of proportionalityis derivable theoretically. The finite diam-eter of the needle is taken into account byadjusting the zero time to a value found byplotting dtlde against t. The effects of thefinite length of the needle are difficult toassess but appear to be unimportant for the

duration of a typical determination. Re-sistivity values obtained agree with thoseobtained by the steady-state method. Acrew of three can make determinations at therate of one to three per hour.

THE effect of the thermal resistivityof the soil on the temperature rise of

a buried cable is well known. It is of par-ticular importance for the newer directlyburied and pipe-type installations inwhich the temperature rise through thesoil forms a larger part of the total rise

than is the case with the older concreteduct types.

In the autumn of 1949, plans of ThleHydro-Electric Power Commission ofOntario to install some 25,000 feet ofunderground high-voltage cable necessi-tated the immediate determination of thethermal resistivity of the soil along theproposed route with a reasonable degreeof accuracy.

There are two possible procedures to befollowed in making such measurements.Samples of the soil may be conveyed tothe laboratorv or the measurements max'

Paper 52-158, recommended by the AIEE InsulatedConductors Committee and approved by the AIEETechnical Program Committee for presentation atthe AIEE Summer General Meeting, Minneapolis,Minn., June 23-27, 1952. Manuscript submittedMarch 24, 1952; made available for printing April16, 1952.

V. V. MASON and M. KURTZ are with The Hydro-Electric Power Commission of Ontario, Toronto,Ont., Canada.

Mason, Kurtz-Measurement of the Thermal Resistivity of Soil

Rapid Measurement of the ThermalResistivity of SoilV. V. MASON M. KURTZ

ASSOCIATE MEMBER AIEE ASSOCIATE MEMBER AIEE

5a70 AUGUST 1 959

be made in situ. Since the thermal re-sistivity of the soil is very dependent uponthe degree of compaction and the moisturecontent and since these are bound to suf-fer change while the sample is being dugup and conveyed to the laboratory, an insitu method is greatly to be preferred.

Laboratory Methods

-The classical laboratory procedure is touse a hot plate with a guard ring. This isan absolute method in that it measuresthe thermal resistivity directly in tenns ofits definition as the temperature drop perunit thickness per unit rate of heat flowper unit area. For homogeneous, simplesubstances, this device provides the basicmethod of thermal resistivity measure-ment against which all other methods maybe compared. For complex, heterogeiie-ous materials such as soil, however, thepicture is somewhat different. The largethermal gradient involved in the usualhot-plate determination, and the longtimes required to reach equilibrium, causethe moisture in the soil to migrate to thecool plate. The resulting variations inmoisture content are bound to give er-roneous results. Also, the apparatus re-quired for this method is very costly andmust be used with great care to obtainsignificant results. Obviously then, it isnot suited to the investigation of thethermal resistivity of the soil along a pro-posed cable route.

Classical in situ Method

Any method designed to measure thethermal resistivity of an undisturbedsample of soil must employ some buried,heated body and must provide for themeasurement of the temperature rise ofthis body, or of an adjacent measuringpoint, over the temperature of some re-mote point. The buried, heated bodymay be either a cylinder or a sphere, butthe latter is the more usual choice as itavoids the troublesome end effects. Thevery act of burying such a body results insome disturbance of the soil, but this canbe overcome by careful recompaction andby allowing time for the soil to regain itsoriginal conditions before measurementsare made. The heated body must be sup-plied with a heat inflow which nmust bemaintained constant until equilibriumconditions are reached. This may takemany days.The buried-sphere method was used at

five positions along the proposed cableroute and values of thermal resistivityfrom 52 to 860 thermal ohm-centimeters(degrees Celsius-centimeters per watt)

were obtained. This abnormally largevalue turned out to be due to a bed ofbroken porcelain insulators. However, anumber of actual soil resistivities wentwell over 100 thermal ohm-centimeters.The large range of these results indicatedthe need for a more detailed survey of theroute. This was clearly impractical withthe buried spheres because of the longtimes involved and especially because thestorage batteries supplying them must bechanged daily.The type of sphere used for these steady-

state measurements may be of interest.Those described in the literature2 consistof an air-filled, hollow metal sphere con-taining a heater which is carefully de-signed and expensively constructed toproduce an approximately isothermal sur-face. This type of construction wasclearly impractical for the large numberof determinations to be made in a shorttime. To achieve an isothermal surfaceinexpensively so that a large number ofspheres could be constructed and in-stalled at reasonable cost, a solid, cast-aluminum-alloy sphere was developed.It was made approximately 4 inches indiameter with a 11/4,inch diameter hole21/2 inches long cast into it. This holewas plugged with an aluminum cylinderon which was wound a heater. A ther-mocouple was embedded in the outer sur-face of the sphere on its equator. Labora-tory tests indicated that the outer sur-face was isothermal to within a small frac-tion of a degree Celsius.

Transient Methods

While searching for a more rapidiliethod of making these thermal measure-ments, it was discovered that F. C.Hooper of the University of Toronto hadin his possession copies of some Europeanpapers1'3'4 which described a rapid, tran-sient method of measuring thermal re-sistivity and which gave the mathemat-ical derivation of the necessary equations.This is based on a suggestion made origi-nally by Stalhane and Pyk.1

This transient method is based on thefact that not only the ultimate teiiipera-ture rise of the heated body but also therate of temperature rise will depend uponthe thermal constants of the material inwhich it is immersed. Whereas it isusual to use a sphere for the steady-statemethod to avoid the end effects, the tran-sient method can avoid these simply byusing a time short enough to preventthem becoming important. The use of acylinder in the form of a long needle hasthe advantage that it may be buriedsimply by thrusting it into the soil to the

required depth. The most difficult thiiiigabout this method is the mathematicalbasis. However, the practical calcula-tions are simple and may be reduced to aneasy routine which can be followed suc-cessfully without need for mathematicaltraining.

This transient needle method was usedsuccessfully and checked experimentallyagainst the steady-state, buried-spheremethod during the fall and winter of1949-1950. Further laboratory exl)eri-ments are planned to check the needleoperation under a range of controlledsoil conditions and to answer certain ques-tions about sample size, accuracy, andmoisture migration. In addition, furthertheoretical work is being carried out in anattempt to put the measurements oii aless empirical basis. The Chemical Re-search Department of The Hydro-Elec-tric Power Commission of Ontario also isusing the transient principle, but with amuch smaller needle, to measure the re-sistivities of thermal insulating materials.

Construction of Needle

All the nieedles built to date by the pres-ent authors have used a 1/4-inch out-side-diameter steel tube as the outercover. Since this is fairly flexible, it isnecessary to make the heater assemblyflexible also. A convenient way of doingthis and at the same time obtaining anelectrical resistance which can be matchedconveniently by storage batteries is towind number 26 gauge enameled, double-cotton-covered copper wire on a length ofvarnished cambric or polyethylene tubingwhich is then pulled into the steel tube.The lower end of the heater wire is sold-ered to the steel tube which acts as onelead. The lower end of the tube is pluggedwith a pointed steel rod to assist in in-serting the needle into the soil. The up-per end of the tube has a handle brazedto it to facilitate its removal after thetest.When constructing the earlier needles,

great care was taken to ensure that thethermocouple was in contact with the steeltube so that the temperature would bemeasured at a definite radius and at apoint outside the actual source of heat.Further experience, however, indicatedthat it was quite as good and much nioreconvenient to put the thermocouple in-side the hollow heater coil form, and soavoid aliy heater asymmetries and risksof short circuits.

These needles use a heater at least 2feet long. The latest one has a totalheater length of 5 feet with four tiuniber30, Copper-Constantan thermocouples,

AU15ason, Kurtz-Measurement of the Thermal Resistiwity of SoilAUGUST 1 952 571

Figure 1. Guide plate forauger

I,

~I

itII

placed at 1, 2, 3, and 4 feet from the endof the needle so that each test gives fourvalues for the thermal resistivitv at fourdifferent depths.The leads from the heater and the ther-

mocouples are brought out to convenientsockets mounted on the handle, and con-necting cords are furnished to allow ofeasy connection to the instruments.

If insulated resistance wire with a neg-ligible temperature coefficient and a con-venient resistivity were available, itcould be used with advantage for theheater as it would then be necessary tomeasure only the current input ratherthan the power as at present.

Method of Use

For most soils it is adv-isable to preparea hole by using a solid steel rod with anauger welded to its end. To avoid en-larging the hole near the surface, a plate,see Figure 1, is used to guide the rod.Once the needle is in place, the tem-

peratures of the thermocouples are readat intervals until they have reached thetemperature of the earth as shown by nofurther change in the temperatures indi-cated. This usually takes about 10minutes. The power is then turned onand is maintained at a constant value bythe use of a variable resistor and a watt-meter. A second operator takes readingsof the temperatures indicated by thethermocouples at exactly 30-second in-tervals from the instant of switching on.They are taken for approximately 25minutes unless the indicated temperaturesreach dangerously high values (of theorder of 100 degrees Celsius) earlier.If the thermal resistivity is abnormallyhigh, it may be necessary to allow theneedle to cool to ambient earth tempera-ture, and then repeat the run with a re-duced power input. For most soils, apower input of 15 to 25 watts per foot ofneedle is satisfactory.A crew of three men with two needles

can make one to three resistivity deter-minations per hour. The third man goes

on ahead and buries the second needle sothat it has reached equilibrium with thesoil by the time the other two get toit.The temperature readings obtained are

plotted on the linear scale of a sheet ofsemilog paper against the times from theinstant of switching on the power. Atypical curve obtained forms Figure 2.Neglecting the readings for the first 3 to 5minutes, it is usually possible to draw astraight l ne through the rest of the pointsso plotted. The thermal resistivity ismeasured by reading off the rise in tem-perature (in degrees Celsius) over onecycle of the logarithmic scale, and mul-tiplying this by the factor 4irL/2.303 p,where p is the power input to the needlein watts, and L is the length of the needlein centimeters. This gives the thermalresistivity of the soil in thermal ohm-

80

Q: 70

L~~~~~~~~~J~~.

ci:

a-7-5C0Lu

centimeters (degrees Celsius-centim-eters per watt). For a needle with a 2-foot-long heater, this needle constant re-duces to 332/p.

It is tempting to simplify the measure-ment procedure still further by takingonly two spot readings at say 5 and 10minutes after switching on. If this isdone, curves can be prepared so that thethermal resistivity is read off directly, ora simple slide-rule calculation can be used.Experience has indicated, however, thatthis is not sufficiently reliable. If it isworth making the measurement, it isworth taking readings every 30 secondsand plotting the temperature versus log-time curve. Any unusual behavior thenshows up, and grossly inaccurate resultscaused by one incorrect reading can beavoided.For the usual run of soils, a plot of the

temperature rise 0 -versus log t for timesof a to 25 minutes gives an obviouslystraight portion. For some soils, par-ticularly for those with high resistivity,there may be no straight portion. In suchcases, the assumption that the time t canbe measured from the instant of switch-ing on must be discarded for a more ac-curate one. As will be shown in detail,this corrected time t' is measured from atime -to so that t' = t+to. This arti-ficial zero time -to may be determinedby plotting a curve of dtIdO versus t andextending the straight portion of this

Mason, Kurtz Measurement of the Thermal Resistivity of Soil

2 5 10T I M E - M/INUTES

Figure 2. A typical curve of temperature rise versus time on semilog paper

572 AUGUST 1902

Figure 3 (left).Uncorrectedcurve of Figure 2on linear timescale, showingone tangent at

t=4

Figure 5 (right).Temperature riseversus time at a

distance r

curve to cut the time axis at -to. Figure4 is such a plot for the data of Figure 2.To obtain the values of dtIdO, it is neces-

sary to plot a curve of 0 versus t on linearpaper, see Figure 3, and then to draw a

number of tangents. Unfortunately, thisis a rather laborious process and themethod of drawing tangents yields a largescatter of the points on the dtIdO versus tcurve.

The corrected times t' obtained by ad-ding to to each value of t are then plottedon semilog paper against 0. As shown inFigure 2, this yields a straight line, theslope of which gives a considerably more

accurate value of R than the original plotof 0 versus log t. Usually this correctedvalue will be slightly lower than the un-

corrected one. The values found for towith this type of apparatus normallyrange up to 3 minutes.The slope of the dt dO curve may also

be used to obtain a value of R as describedlater. This has the theoretical advantageover the use of the slope of the 0 versus

log t curve in that a lower value of t is al-lowable without correction. Due to thescattering of the dtIdO values, however,the slope of this curve is sometimes ratherdifficult to obtain. Nevertheless, it doesgive a check on the value obtained fromthe 0 versus log t curve.

1.2

0.8UAw

D

0.4~~~~~

TIME

Accuracy

Comparison of the thermal resistivityvalues measured by the transient needlemethod with those obtained by the steady-state buried-sphere method show agree-ment to within the accuracv of measure-ment with either method.The accuracies obtainable depend to

some extent upon the time which can bespent plotting the points. If the appar-ent zero time to found from the dtId6curve is used to plot a corrected curve of0 versus log t', the points so obtained willusually lie on a very good straight line,the slope of which may be measured tothree figures. For most soils, however,sufficient accuracy is obtainable from theinitial plot of 0 versus log t. The value ofresistivity obtairis usually high bis a small error c

of the resistivitythe changing mo

Theory of Tran

The long hea1/4 inch in diam

0 -to 2 4 6 8TIME - MINUTES

dtFigure 4. dt versus t for the data of Figures 2 a

used for measuring soil resistivity is ap-proached mathematically by consideringthe temperature rise 0 at a distance r froman infinitely long, infinitely thin, heatedfilament supplied with a constant heat in-flow of q watts per centimeter length(q= p/L). As is shown in Appendix I, thesolution for this involves an exponentialintegral of the form £x

-

(E- X x)dx. TheEuropean literature kindly made avail-able by Mr. Hooper deals with this byusing an approximation valid for suffi-ciently small distances r and sufficientlylong times t.

Within these limits on r and t, the tem-perature rise is given by the approximaterelation

a=ARq logiot+B (1)

aed from this initial plot where A and B are constants. This ap-

y a few per cent, but this proximate equation for 0 cain be renderedompared to the variation more accurate by measuring the timewith the seasons due to from an instant prior to the actual switch-

isture content. ing on, that is, by adding a constant to' tothe measured value of time t before plot-

isient Method ting a curve of 0 versus log t. As is provedin Appendix I, to' can be evaluated by

ited cylinder or needle, plotting a curve of dtI/do versus t andeter and 2 to 5* feet long, finding the t-axis intercept of the straight

portion. The intercept will be at thepoint -to' so long as the requirements ofan infinitely long, infinitely thin filamentin an infinite medium are met. As will beshown, for this apparatus, to' is usually

000 Ksmall compared to to", the correction forthe finite diameter of the needle. Themethod of obtaining this zero correctionactually yields the total correction towhich is the sum of the two correctionsto' and to".We can replace the infinitely long fila-

ment with an infinitely long cylinder so________ long as we realize that this may result in

l0D 2. 14 the temperature rise not being measuredfrom the initial temperature of the needle,the time not being measured from the in-

and 3 *Total length is 5 to 8 feet, but only 2 to 5 feet isheated.

1Mason, Kurtz Mleasurement of the Thermal Resistivity of Soil

wv)r-

w

tL

cr

a.

40 X 7

0 --2 4 6 ,,8_ 10 12 14

w(I)cr

wcr2ZI-

w0-

wT M E - MINUTES

AUGUST 1952 5-J73

q =IwATT/cm30 R 80 OC-cm/WATT

wCl)cc 20

crI0 INC

wa.

w

0

5 l 15 20 25 30TIME - M/NUrES

Figure 6 (above). Theoretical temperature rise curves for severalvalues of r, and typical soil constants

Figure 7 (right). Equipotential and flow lines for electrical model ofneedle

A. Mid-point of heater (thermocouple location)B. End of needle (4 inches from A in model, corresponding to 12

inches on needle)Voltdge values indicdte potential with respect to infinity (actually 90-

degree arc, radius 14 inches, center A)

stant of switching on, and possibly, thedistance r not being measured from theaxis of the cylinder. Fortunately, thedistance r does not enter into the aboveequation, and it is apparent that thethermal resistivity R is proportional to0/log t, so that only the rate of change ofO with respect to log t is required. Also,for many soils it is accurate enough to as-sume that t is measured from the instantof switching on the power. The initialvalue of 0 is avoided by using the changein 0 in an interval starting some minutesafter the heating power is switched on.

Effect of Finite Radius of Needle

There are two effects to be evaluatedso far as the finite radius is concernedThese are:

1. The effect on the temperature rise at anypoiilt in the infinite medium at any timeafter switching on.

2. The effect of measuring the temperaturerise of the needle rather than of the infinitemedium.

The first of these effects has been evalu-ated by Van der Held3 for a needle of uni-form homogeneous material and shownmore or less intuitively to be equivalentto a heat production prior to t= 0. Thatis, it may be included as an addition toto'. To avoid confusion, let this change inzero time be accounted for by a timeto" to be added to t+to' to give the totalcorrected time, t'=t+to=t+to'+tof.The second effect seems to have es-

INCHES FROM AXIS NEEDLE18 16 14 12 10 8 6 4

caped specific mention in the literature.Because of these apparent gaps in thereasoning, it was thought best to ap-proach the measurements empirically bycomparing the result of a transient needledetermination with that obtained by thesteady-state sphere method. This ap-proach having l)roved successful, it is nowpossible to formulate a more theoreticalbasis for these effects.

Intuitively, it is reasonable that oln asufficiently gross scale, a 1/4-inch diam-eter rod will approximate an infinitelythin needle. Or, on a more nearly quan-titative basis, when the rate of heat flowthrough the surface of the needle is verynearly equal to that being dissipatedwithin it, that is, after the initial heatingtransient of the needle itself is practicallyover, the difference in its rate of tempera-ture rise from that which would occur atthe same radius for a filamentary sourcemust be very small. After this time therate of rise of temperature of the needlemust for all practical purposes be equalto that of the infinite medium at the sameradius and due to an ideal filamentarysource. Thus the only difference here be-tween the ideal and the practical cases isthat the practical needle will appear tohave been turned on before or after theideal one depending upon whether the ef-fective diffusivity of the needle is greateror less than that of the soil.

2 0

To evaluate the effect of measuriing thetemnperature of the needle rather than of apoint in the medium, consider that evenwith a holimogeneous needle there is boundto be somle thermal resistance at the in-terface between the needle and the me-dium. For a soil needle, made and used asdescribed herein, there is the additionalthermal resistance between the heater andthe shell and the inevitably imperfectcontact between the needle and the earth.Thus even when very little of the heateroutput is being used to raise the tempera-ture of the needle, there still will be an al-most constant temperature drop from theinside of the needle where the thermmo-couple is located to the soil just outsidethe shell. Thus the measured rise over theinitial or ambient temperature will differfrom the rise in the ideal theoretical caseby ani approximately constant amountafter the initial needle heating transient isover.These two effects tend to give a curve

such as B, Figure 5, rather thain A which isdrawn for the ideal case. Curve B is for aneedle with an effective diffusivity andresistivity greater than those of the soil.It is apparent from this discussion thatthe parts cd and ef of the two curves arepractically identical in shape but are dis-placed both horizontally (in time) andvertically (in temperature). The timecorrection to' may be taken care of in the

54ason, Kurtz-Measurement of the Thermal Resistivity oJ Soil AUGUST 1,952574

sanme way as was tot, the correctioni to iii-prove the fit of the approximate equlation(equation 1) for 0. That this procedureis effective is proved in Appendix II.The temperature difference in the twocurves A and B may be taken care of nmosteasily by dealing only with rates of temn-perature rise or with temperature dif-ferences over time intervals starting afterthe initial needle-heating transient isover. In practice, this time may be rec-ognized by the fact that the 0 versuslog t plot is curved for times less than thisand straight thereafter.

This approach is intuitive and shouldbe replaced when possible by a completetheory which takes into account the finitediameter of the needle and its thermalcharacteristics in more detail. Similarremarks apply with even more force to thefollowing discussion of the effects of thefinite length of the needle.

Effects of Finite Length

One of the effects of the finite length ofthe needle is that the heat loss from theends of the needle-and from the un-heated parts near the end-will cause asmall temperature gradient along theneedle. This will lead to an inaccuracydue to an incorrect value being taken forthe net watts per centimeter entering thesoil opposite the thermocouple. Otherauthors3 have considered this and shownit to be small.A second effect of the finite length is the

spreading out of the heat-flow lines asthey get farther and farther away fromthe needle. In a plane normal to theheater, the flow lines diverge because theyare radial; but in a plane containing theheater axis, the heat-flow lines for aTn in-finite needle are parallel and do not di-verge. With the finite heater, the linesfrom the central part of the heater Illust

bend outtward to fill up the otherwiseemiipty semicircles at the ends. Hence theheat leaving one centimeter of the heaterwill have to spread out over a greater andgreater distance in this plane as it getsfarther from the heater. Therefore, thetemperature gradient along these flowlines will decrease with distance morerapidly for the finite needle than for an in-finite one. The result is that the rate oftemperature rise of a finite needle will de-crease more rapidly with time than itwould if the needle were infinite in length.If continued for long enough times, allpractical 6 versus log t curves will eventu-ally bend gradually downwards from thestraight part of the curve indicating thatthe end effect is becoming apparent. Insome cases this makes the drawing of thebest straight line very difficult.

In an effort to assess the magnitude ofthis effect theoretically, the family ofcurves shown in Figure 6 was prepared.These curves show the calculated tem-perature rise versus tillme relation at vari-ous distances from a transient needle fortypical soil and power input conditions.To get some indication of the spreading ofthe heat-flow lines with distance from theneedle, a very approximate steady-state,electrical model to a one-third scale wasset up using a sheet of Teledeltos paperbounded by a 90-degTee arc of metal witha radius of approximiiately 14 inches(equivalent to 42 inches) to represent aquarter-infinite mediunm and a small stripof brass approximately 4 inches long torepresent the bottom 12 inches of a needlewith a 24-inch heater. The resulting plotof equipotential lines which represent iso-thermals and the approximate heat-flowlines are shown in Figure 7.

Fromn Figure 6 it is apparent thatnothing happening at much more than 2inches from the needle (r= 1/8 inch) willhave anvy noticeable effect on the needle

for at least the first 13 minutes. FromrFigure 7 it is apparent that there is neg-ligible divergence near the mid point ofthe needle for the first 2 or , inches.Therefore, if the shape of the steady-stateequip)otential lines of Figure 7 are reason-

ably close to the shape of the transientisothermials of the needle, the end effectsshould certainly not become importanit forthe first 15 mninutes and probably will iiotbecome too serious for the first 30 minutesafter turning on the power. However, itmust be remiiembered that the ratio of theelectrical resistivities of the needle andsoil models is very much greater than theratio of the thermal resistivities of theactual needle and soil, and this has theeffect of crowding the electric flow lines tothe end of the needle more than the heatflow linies would be expected to crowd.This will cause less divergence in theregion of interest in the electrical miodelthan in the actual thermal case. It is alsoto be noted that a uniformly conductingsheet is Inot an exact model of a sector of auniformly conducting solid medium.

Finite Nonhomogeneous Medium

If the heated portion of the needle isembedded deeply enough, the effects ofthe earth's surface will not affect the re-sults until some time after the end ef-fects previously described have becomeexcessive.Any inhomogeneity of the medium will

cause the heat-flow lines to bend awayfrom the paths they would follow in ahomogeneous medium. Depending uponthe shape and nature of the irregularities,they may cause unexpected kinks in the 0versus log t curve or may cause a generalbending of the otherwise straight line.This may make it more difficult to de-cide upon the best straight line to draw,but with care it is usually possible to ob-

0.

wI-

SEPT 28 OCT 6 14 22 30DATE

Figure 8. Steady-state sphere thermal resistivity test

w

c:

w

I-w

w

a.-

0.4 05 0.7 1 2 3 5 7 10 20 30 50TIM E - MINUTES

Figure 9. Theoretical temperature rise 1/8 inch from infinite filamentary source

A S9ason, Kurtz-Mleasurement of the Thermal Resistivity of SoilAUGUST 1952 0-75

tain a good average value for any soilcondition.The size of the sample measured by this

method is not accurately definable be-cause the material adjacent to the needle-will have more effect on the first part of-the curve than that farther away. The-lata listed previously from Figure 6 and7, however, indicate that the average*size of the sample measured is of theorder of 3 inches diameter and 6 incheslong.

Results

Figure 8 shows the variation of thermalresistivity with time for one installationof a steady-state sphere. At this par-ticular location, the thermal resistivityvaried verv considerably with time dueapparently to the initial recompaction ofthe soil and to variations in moisture con-tent. Other installations, however, ap-peared to have resistivities almost com-pletely independent of rainfall. One ofthe latter locations was used to comparethe transient with the steady-statemethod.With the transient needle method there

will be no moisture migration effects and,therefore, an allowance should be madefor this when making cable temperaturecalculations if its effect is likely to be sig-nificant.A number of the transient 0-log I heat-

ing curves obtained appear to curve downfrom a straight line after 12 to 15 minutes.This may be due to the end effect be-coming significant. While this is not inagreement with the data deduced fromFigures 6 and 7, the approximations inthe latter figure are so gross that it wouldbe more surprising if there were goodnumerical agreement.To date, over 100 soil thermal resis-

ti\itv determinations have been madewith the transient needle equipment.These include ones made on the earth sur-rounding a heat-pump ground coil forwhich purpose the multipoint needle hasgiven results of particular interest.

Efforts are being made to achieve a

much more complete theory by consider-ing explicitly the finite size of the needle,the nonzero thermal resistivity at theneedle surface, and the actual location ofthe thermocouples within the needle. Itis hoped it will be possible soon to reportthe results of this theoretical work andthe attendant experiments.

Conclusions

The transient needle method of measur-ing the thermal resistivity of soil is a con-

venient, rapid method which is capableof sufficient accuracy for the purpose ofcable temperature calculations. Whilethere are some parts of the theory not yetcomplete, the agreement found in practicebetween this and the steady-state spheremethod is sufficient to allow the newmethod to be used with confidence.

Appendix 1. The MathematicalTheory of the Transient Needle

The temperature rise 0 in degrees Cel-sius at a point a distance r centimetersfrom an infinite filamentary heat source re-leasing heat at the rate of q watts per centi-meter is as shown in section 9.8 of reference 5

qR co -S2 qR@=2 y $1d1= 2wI(d) (2)

where

R = thermal resistivity of the medium inthermal ohm-centimeters

fl=r/2 V/ata =diffusivity of the medi im in square

centimeters per hourt=time from start of release of heat in hours

This i-unction I(f3) is tabulated in Appen-dix F of reference 5. Equation 2 may alsobe written in the form

qRfCOe-x qRqR/-dx= [_Ei(-x)] (3)47r J x 47r

where x=-2 = r2/(4at)The function -Ei( -x) is very fully

tabulated in reference 6. For small valuesof x, these integrals may be evaluated in theform of a series, giving

C-iRxl-x - 2

4r=q[CI (1!1) (2!2)

X3(4)

where C=Euler's constant=0.5772If x is small (that is, for small distances r

and long times t), only the first two terms ofthe preceding series are significant andtherefore

n t-In -C] (5)

The accuracy with which this approxima-tion holds for a typical soil is illustrated byFigure 9 which shows the actual tabulatedvalues of -Ei(-x) plotted on semilogpaper. It is seen that these curves are verynearly straight lines for t greater than 3minutes.The improved method of approximation

given by Van der Held3 is obtained by sub-stituting for the value of x in equation 3 andobtaining

qR rt er2/4atO= I dt (6)

47rO t

Whence, by differentiating both sides withrespect to time t, and inverting

dt 4te--teXdO qR

47r Sx2 x3= -- t 2+x+ +±..3!47[ 2! 3!

47F r2 t / r2 \2 1it-F +1 J-) ...iqRL 4a 2! \4atI (7)

Again in this series, the third term is verysmall for small r and large t and so may beneglected. Putting r2/4a -to,* we get

dt 4r_ = (t +to)do qR

The value of to can, therefore, be obtainedby plotting dtIdO against t and finding the t-axis intercept of the linear portion. Thisintercept is to the left of the origin for posi-tive to because for dt/dO=O, t= -to.

Appendix IL. Proof of Construc-tion for to"

Following Figure 5 and the accompanyingdiscussion, consider that the equation ofcurve A is given by equation 3, while that ofcurve B is

qRr -2/4attqR f ' dt'+--(t) (8)

where t'-=t-fto', and o'(t) is the temperaturedrop from the thermocouple to the surfaceof the soil around the needle due to the in-ternal thermal resistance in the needle andthe resistance at the needle-soil interface.After a very few minutes, 6' will be constant.

Differentiating equation 8 with respect tothe time from turn on t, and since to' con-stant, we have for times t large enough tomake o'(t) a constant

do qR c--7 /4att= (9)

dt 47r t'

Hence, as before

dt 47rw r2/4at'= t e/4ido qR

=4t'[I+rr24at' + / ) .. .]qR L2! i

4wr= (t'+r2 4a) neglecting small termsqR

= R (t +to' +to" )qR(10)

Hence, plotting dt/do against t gives the x-intercept of value -to = -(to' + to "). Sinceto" is negative for cases where the thermaldiffusivity of the needle is less than that ofthe soil and since in most cases to" is greaterthan to' the x-intercept can lie to the right ofthe origin.

References1. NEW METHOD FOR DETERMINING THE CO-EFFICIENTS OF THERMAL CONDUCTIVITY, B. Stal-hane, S. Pyk. Tekniisk Tidskrift (Stockholm,Sweden), volume 61, 1931, pages 389-93.

*Others3,4 have used 1-2/4a =-to, but this appears tobe only a needless complication of signs since r2 anda are both positive.

AUGUST 19527l6ason, Kurtz Measurement of the Thermal Resistivity of SoilZ)7

2. DlRECTIONS FOR THE DETERMINATION OF

THERMAL RESISTIVITY OF THE GROUND. TechnicalReport F/S5, The British Electrical and AlliedIndustries Research Association (London, Eng-land), 1937.

3. A METHOD TO MEASURE THE HEAT CONDUC-TIVITY IN LIQUIDS, BASED ON THE WARM-UPPHENOMENA. E. F. M. Van der Held, T. G. Van

Drumen. Congress on Applied Mechanics (London,England), 1948.

4. TRANSIENT HEAT FLOW APPARATUS FOR THE

DETERMINATION OF THERMAL CONDUCTIVlTIES,F. C. Hooper, F. R. Lepper. Journal, AmericanSociety of Heating and Ventilating Engineers (NewYork, N. Y.), June 1950.

5. HEAT CONDUCTION (book), L. R. Ingersoll,0. J. Zobel, A. C. Ingersoll. McGraw-Hill BookCompany, Inc., New York, N. Y., 1948.

6. TABLES OF SINE COSINE AND EXPONENTIALINTEGRALS, VOLUME I. (New York MathematicsProject), United States Bureau of Standards,Washington, D. C., 1940.

No Discussion

THE usual assumption that the con-stant voltage busses of a 2-machine

stability problem are connected by purereactances results in a number of approxi-mations which may or may not be valid.On the other hand, if complicating factorssuch as losses, intermediate loads, trans-former taps, and so forth are included,the process of obtaining a solution maybecome very laborious indeed.The method of solution described in

this paper preserves accuracy by permit-ting the consideration of all system param-eters-including the inertia of bothmachines without approximation. Yetthe procedure is relatively simple, evenwhen line reclosure is to be considered inthe switching schedule. Part of the sim-plification is due to the utilization of gen-eralized swing curves to determine rela-tive angle and velocity at any time duringthe swing. Similar curves have been de-scribed previously,1 but the presentmethod of derivation removes the formerrestrictions as to type of system disturb-ance for which they are applicable.

This method is well adapted to the ac-curate determination of transient powerlimits because the computations involvesets of composite general circuit con-stants, 2 which need not change as thepower transfer in the system is varied.Consequently, they and many auxiliaryquantities need be computed only once.An auxiliary curve provides a quantita-tive measure of the degree of stability orinstability of the system.

Paper 52-160, recommended by the AIEE Trans-mission and Distribution Committee and approvedby the AIEE Technical Program Committee forpresentation at the AIEE Summer General Meet-ing, Minneapolis, Minn., June 23-27, 1952. Manu-script submitted March 20, 1952; made availablefor printing April 16, 1952.

RALPH D. GOODRICH, JR., is with the Bureau ofReclamation, Denver, Colo.

General Theory

To utilize the swing equation1'3'4

d2b/dt2 = K'(Po-P)

where

=phase angle between the two machines

K' (rf) system kilovolt-ampere base (2)(H) machine kilovolt-ampere base

PO = steady-state or initial machine power,in per unit

P =machine power at time t

(note that these quantities are normallypositive for both receiving- and sending-end machines)it is necessary to find a relation betweena and P. In section A of Appendix I it isshown that

PS=PM sin (6- ±)+PS' (3)

PR PM sin (6±+0) -PR' (4)

in which

the subscripts S and R indicate sending-and receiving-end quantities respectivelyPM = eseR/b (5)

0=90-0

Ps' = (es/b)2(DB)Rp

(6)

(7)PR = (eR/b)2(AB)Rp (8)

es= absolute value of sending-machinevoltage, in per unit

eR = absolute value of receiving-machinevoltage, in per unit

A =Ar+jAiB=Br+jBi=bI d

D =Dr+jDiAB = (AB)Rp +j(AB)1p

= (Ar-jA )(Br+jBi)DB = (DB)RP +j(DB)Ip

= (Dr-jDi)(Br +jB,)Hence it is possible to write two swing

(1)

equations, one for the sending-end ma-chine and the other for the receiving-endmachine. These are added to give a com-plete description of the angle 6. After in-tegrating once and rearranging terms, seesection B of Appendix I, the result can beexpressed in the concise fonn

X =dO/dt = dl/dt= V2K'PMV\RO+ cosO-C (9}

in which

co is the relative machine velocity

K' = V\(Ks')2+(KR')2+2Ks'KR' cos 2q(10)

R = (Ks'Pso'+KR'PRO')IK'P-i (11)

(12)

tan = [(Ks'- KR')/(Ks'+KR')] Xtank0 (13)

Ps ' =Pss-Ps' (14)

PRO'= PRO+PR' (15)

C is a constant of integration, determinedfrom initial boundary conditions

It is convenient to let

W=X2K'PM (16)

o"'==VRO+ cosO-C (17)

so that equation 9 becomes

wWXw' (18)

If the system is stable, then there mustbe a value of 0 greater than the equilib-rium value 6e, for which co is zero. It isevident that solving the equation

RO+cos O-C=0 (19)

will yield the desired value of 0 if it exists.If it does not exist, the system is un-stable.Another deduction from equations 9and

17 is that the maximum and minimumvalues of w and w' occur when

R = sin 0 (20)

The maximum corresponds to fe in thefirst quadrant; the minimum correspondsto OM in the second quadrant. But if cwhas a minimum value, it does not becomezero, and the system must be unstable.A borderline case occurs when cw' and its

Goodrich, Jr. A ccurate Computation of 2-Machine Stability

Accurate Computation of 2-MachineStability

RALPH D. GOODRICH, JR.MEMBER AIEE

AUGUST 1 95=2 a'5r77