Columns 14 and 15 are used to indicatethe appropriate rate to be applied to re-quirement data for use in rate studies andpay-out analyses. Each rate has beenassigned an identifying code number.Some of the BPA rates are 1-part andsome are 2-part, with the bill dependingon both the peak and energy amounts.But since both peak and energy loadscan be machine-sorted and tabulated byrate schedules the calculation of revenueestimates is facilitated. Some of theBPA 2-part rates provide that themonthly billing demand shall be thehighest measured or computed demandexperienced during any of the preceding11 months. By printing all months it isa simple matter to pick out the "ratch-eted" demand. In the same way an-nual summaries of energy loads can beobtained by the machine and punchedinto the "total" space on the card(columns 77 through 80) to facilitate thedetermination of total energy charges.Column 16, entitled "service," is used

to indicate the physical and contractualarrangements made for the delivery ofpower. When loads are served directlyover BPA facilities the major substationarea is also entered into the columns re-served for that purpose. When loadsare served by transfer or wheeling ar-rangements over the facilities of otherutilities the area column, as has beenpreviously mentioned is also used to

indicate the particular transfer agent.Four basic types of service arrange-

ments are coded; direct; coincidentaltransfer; noncoincidental transfer; andtransfer service for other utilities, alsoknown as reverse wheeling arrangements.When a customer is served on a coinci-dental transfer basis, BPA has the re-sponsibility for simultaneously supplyingthe utility making the ultimate delivery,the transfer agent, with an equivalentamount of power. Therefore in relatingsystem loads to generation the require-ments of coincidental transfers must beincluded in the federal load. In otherreports these loads would be omitted fromthe total EPA load and included in theload of the utility making the ultimatedelivery. XVhen loads are served on anoncoincidental basis, the transferringagent is responsible for coincidentallyproviding the generation. BPA has theoption of later repayment on the basis of 2kilowatt-hours for 1 delivered. There-fore, in relating total system load to gen-erating capability, these loads are omitted.On the other hand, they are included forrevenue analysis. Transfer service loadsprovided for other utilities (reversewheeling), while not included in sum-maries for revenue purposes except forcalculation of service charges, or in load-resource analyses, are required in tabula-tions used for planning transmission andcustomer service facilities.

The remaining columns require littleexplanation. Column 17 is used toclassify into peak and average all load,resource, and requirement data. Columns18 and 19 are used to indicate the year.Space is then provided to record the basicdata for each of the 12 months of the op-erating year from July through the fol-lowing June. The remaining columns,previously described, are reserved for re-cording the average of the energy loadsover both the minimum and medianmonth storage periods and the operatingyear totals.

Conclusion

The method described, which has beensuccessfully employed over a period ofyears, makes it possible to conduct with ahigh degree of accuracy many investiga-tions and studies which otherwise couldnot be considered with a limited staff. Aprimary object in designing a coding sys-tem is flexibility. With this in mind, itis desirable as a first step to determine allthe purposes to which the system is to beapplied. The broadest application of themethod is possible when a single basicestimate is used for all purposes, so thatinformation required for a particular re-port or study, Nrhich may range from arevenue analysis to an a-c analyzer study,can be readily machine-selected andtabulated.

Effects of Corona on Traveling Waves

C. F. WAGNERFELLOW AIEE

THE EFFECT' of corona on travelingwaves is to retard any given point on

a voltage wave above the corona thresholdvalue by an amount proportional to thedistance traveled. This suggests thatthe effect is equivalent to a reduction invelocity. If a linear circuit, with negli-gible series and shunt resistance, can becharacterized by assigning to it a certaininductance L per unit length and capaci-tance C per unit length, then a wave im-pressed upon such a circuit will propagatealong the circuit with a velocity v, suchthat

VV=I (1)

This relation indicates the possibility ofexplaining corona effects by an increase

B. L. LLOYDASSOCIATE MEMBER AIEE

in capacitance. Most previous investi-gators based their analysis of wave dis-tortion upon the energy loss associatedwith the voltage wave and used theavailable power-frequency loss data as afoundation upon which to build. Boehne,2however, suggested that with increasingvoltage the corona envelope and hencethe effective capacitance increases. Withincreasing voltage the velocity decreases,which accounts for the increasing re-tardation.There has been some attempt to deter-

mine the increase in capacitance of con-ductors under corona at power frequenciesby measuring the increase in chargingcurrent. However, the space charge sur-rounding the conductor is probably quitedifferent under this condition than if

a unidirectional surge were applied.Prompted by a desire to explain the re-sults obtained during the tests with travel-ing waves at the Tiddl power station, theauthors undertook laboratory tests inparallel with the field tests by stringing aconductor of small diameter along thecenter line of a long metallic barrel. Theresults of these tests showed conclusivelythat an increase in capacitance took placewhen the conductor was surged. En-couraged by these tests, other tests weresubsequently made on full-scale conduc-tors adjacent to a large flat screen. The

Paper 55-453, recommended by the AIEE Trans-mission and Distribution Committee and approvedby the AIEE Committee on Technical Operationsfor presentation at the AIEE Summer GeneralMeeting, Swampscott, Mass., June 27-July 1,1955. Manuscript submitted March 28, 1955;made available for printing April 19, 1935.

C. F. WAGNER and B. L. LLOYD are with theWestinghouse Electric Corporation, East Pitts-burgh, Pa.

The authors wish to acknowledge the valuableassistance of C. P. Saalbach for performing thedigital computer studies, J. W. Skooglund andC. A. DeSalvo for assisting with the analyticalwork, and the personnel of the Engineering Labora-tories for assisting with the laboratory tests.

Wagner, Lloyd Effects of Corona on Traveling Waves OCTOBER 19558358

x

z

LUJ<D

01-J0

z

LU]

-J0

z

w1-J0

800

700

600

500

400

300

200

IX

tests under these conditions also showthat the corona characteristics can bedescribed adequately as an increase incapacitance.Having demonstrated the increase in

capacitance occasioned by corona, this isonly a part of the problem. While equa-

2.8 MILES

I

BOCKMAN, HYLTEN-CAVALLIUS a RUSCK

FIG. 9

00,i2 4 t 6 8 10

tion 1 is correct, it must be observed thatit is rigorously correct on a circuit of dis-tributed constants only when the mediumis homogeneous, such as an aerial linebelow corona in which the medium is airor in a cable in which the medium is rub-ber or some other suitable dielectric. In

Fig. 1. Distor-tion of negativeimpulses as afunction of dis-tance traveled.All time scalesin microseconds.Numbers oncurves representdistance traveledfrom point of

origin

2 t 3 4

applying the increase in capacitance tothe interpretation of its effect upon atraveling wave, consideration must begiven to the following:

1. Suppose that two parallel conductorsisolated from ground were loaded by shuntcapacitors in such a manner that the dis-tributed capacitance were increased withoutchanging the inductance. One might applyequation 1 to such a system and concludethat the propagation velocity is decreasedcorrespondingly. However, it is funda-mental that any electrical disturbance inair propagates with the velocity of light.It would be expected then that as a waveof charge travels along the line some dis-

Wagner, Lloyd-Effects of Corona on Traveling WavesOCTOBER 1955 859

Fig. 2. Sampling ofdata from Tidd tests,'showing effect of volt-age, distance, conduc-tor, and polarity.One conductor ener-gized. Height of con-ductor at tower 86 feetand at mid-span 50feet. Line terminatedby resistance equal tonatural surge impedance

z

5(0

Ir-0

0

D0z00)

TIME IN MICROSECONDS

turbance precedes it in space. This canbe shown analytically to be true.

2. Neglecting the previous effect such asmight be the case in an artificial line havinga large number of elements, then thepropagation of the wave along such a linewould follow equation 1 rigorously. How-ever, if the artificial line were altered sothat at a predetermined voltage additionalcapacitors were switched into the circuit(simulating the increase in capacitance dueto corona), then it might be inferred thatthe wave propagates at one velocity belowthis point and a lower velocity above thispoint. There are certain limits withinwhich the waves follow the change incapacitance.3. Digital computer studies would be ofconsiderable value in exploring the limitsto which equation 1 applies.

w

°- AT 7e,AT--

POSITIVE

4200'2040'

) 2 3 4 5

TIME IN MICROSECONDS

In view of the foregoing discussion,this paper falls into several natural parts.First field data are presented from pre-vious investigations. Aside from theirown intrinsic value these data will beused as a basis for verification and con-firmation of the analytical deductions.Second is the laboratory data, tests onsmall lengths of conductor. Third is theanalytical deductions based upon thecharacteristics of the conductor obtainedin the laboratory. Fourth is the correla-tion of the first three parts. Fifth is adiscussion of related subjects, such as thedetermination of the capacitance charac-teristics from the field data, typical dis-tortion curves, and a discussion of the

Fig. 3 (left). Illustrationof detailed steps fordetermining (AT)/d

Fig. 4 (right). Average(AT)/d curves for surgeson single conductors for

_______________,the three conductors3 4 shown in Fig. 2. Data

;ONDS/IOOO FT from Tidd tests'

2 3 4

TIME IN MICROSECONDS

relation between the impulse and the 60-cycle characteristics of a conductor.

Field Data

A number of investigators,',I-' both inthis country and abroad, have obtaineddata by actual tests in the field on the dis-tortion and attenuation of electric waveson transmission lines. A typical sam-pling of these data is shown in Fig. 1 fornegative waves. The most recent work ofthis character is that of Wagner, Gross,and Lloyd.' Taking advantage of theavailability of two 1'/2-mile test lines

IN MICROSECONDS PER THOUSAND FEETd

Wagner, Lloyd Effects of Corona on Traveling Waves

NEGATIVE

z

wCD

I

0

z0U

0 2

TIME IN MICROSECONDS OR 4-T IN MICROSEC

t

860 OCTOBER 1955

65- HHPOS-2PH ..

. ,6o5s-3HH

0 0.1 0.2 Q3 0.4 0.5 0.6

AT IN MICROSECONDS PER THOUSAND FEETd

Fig. 5. Average (AT)/d curves for surges

on 2-conductor and 3-conductor combina-tions. Data from Tidd tests1

having unusually high insulation and al-ready equipped with conductors of thetype that would be used for extra-highvoltage transmission, a 2,000-kv surge

generator was set up in the field for ap-

plication of surges to the line. Fig. 2shows a sampling of these test results.As previously observed by other inves-

tigators, it was found that for each set oftest conditions the wave for any particularvoltage above the corona threshold volt-age appeared to be retarded in time by an

amount proportional to the distancetraveled by the wave. To isolate thisfactor the Tidd test data were replottedin terms of a new quantity (AT/d). Thesignificance of this quantity is shown inFig. 3. Both the voltage at the sending

14~~~~~~~~~~~~~~~~~~~~~~~J

17

.0

0

03

TX2

end and at 4,000 feet are plotted. Thehorizontal distance at any voltage, suchas el, is indicated as AT. These distancesare shown by the dashed line and indicatethe time lag of the wave for any voltage.The time AT was then divided by the dis-tance d in thousands of feet giving a new

quantity, (AT)ld, which is also plotted.These curves were then determined foreach of the conductors and for eachpolarity.

Fig. 4 shows the average values of(AT) d obtained by this method for thethree conductors tested, namely thoseshown in cross section in Fig. 2, each ofwhich was surged singly. These curves

represent the mean of a large numiber ofcurves, the spread being approximately4-0.05 from the plotted curves. Theywere determined without previous benefitof the analytical work presented here.Fig. 5 shows similar data when two or

three phases were surged simultaneously.

Laboratory Tests

It has been postulated that the effectof corona in distorting a traveling wave

can be explained by an increase of shuntcapacitance. The field data discussedin the foregoing show the distortion ofvoltage waves due to corona, but they donot provide a fundamental understandingof the manner of variation of this shuntcapacitance. To aid in interpreting andunderstanding the field data, laboratorytests were undertaken to determine thevariation of shunt capacitance whencorona exists on the conductor. Thejustification for representing the effectof corona as a varying capacitance was

also established in these tests.

20 40 60 80 100 120 140 0 0.5 10 1.5 2.0 2.5 3A0 3.5

CONDUCTOR VOULTAGE, KV e TiME, MICROSECONDS

A 8

Fig. 6. Corona characteristics of 6 feet of 0.1 56-inch-diameter copper tube inside a 24-inchcylinder. Positive polarity

(A) q-e nest for different voltages(B) e-time and q-time curves for the maximum q-e curve of (A)

Fig. 7. q-e curves for positive and negativesurges and for 60-cycle voltage. 6 feet of0.1 56-inch-diameter copper tube inside a

24-inch cylinder

To determine the surge corona charac-teristics of a conductor, it was necessary

to find and measure some quantity whichcompletely describes these characteristics.If corona can be represented as a shuntcapacitance which varies with voltagealone, the charge on this variable capaci-tor can be used to describe the corona

characteristic. This charge-voltage (q-e)relation was the quantity which was

measured in the laboratory.It was planned to obtain tests on prac-

tical transmission conductors at prac-

tical spacings to ground. Hovw ever, it

was felt that tests should first be run on

smaller scale setups to develop some

fundamental concepts and to developmeasuring techniques.

CYLINDRICAL GROUND PLANE

The first group of tests was made in theImpulse Laboratory in East Pittsburghon a conductor which wx as strung inside a

24-inch diameter cylinder. The cylinderwas tied to ground through a mica capac-

itor Cs, see insert of Fig. 6. The voltageacross this capacitor is proportional tothe integral of current and thus measures

the charge q in the portion of the cylinderbetween the guard rings.The conductor voltage e wxas measured

w-ith a conventional double-layer, wire-wound resistance divider, 2,000 to 6,000ohms, 2 to 6 feet in length. Oscillo-graphic measurements %vere made using a

cold-cathode oscillograph.Data were obtained in two different

forms, charge versus time and voltageversus time, and charge versus voltage.In the latter form, vhich proved more use-

ful, the voltage across Cs was applied to

Wagner, Lloyd Effects of Corona on Traveling Vaves

1600

1400

1.65* HHNEG-2P`--2.00 ACSRNEG-2PHI /

21000|AEXT10t b3HZ

z

r0

0

z

0

lquu

200 i i

1- ---I

or I -1-- -I-tj

S61OCTOBE-R 1 955

:ONOUGTOR VOLTAGE N KV

Fig. 8. Effect of time to crest upon q-e curves. 0.1 56-inch diameter copper tube 1 2 inchesfrom flat ground plane

the vertical plates of the oscillograph bymeans of a short wire down the center of ahollow metal tube. This concentric-lead minimized extraneous pickup. Ashort cable was connected from the volt-age divider to the horizontal plates of theoscillograph. The resulting trace pro-vided a plot of charge versus voltage with-out the necessity of replotting this q-erelation from charge-time and voltage-time oscillogram s.

Considerable difficulty was encounteredin the laboratory in making the chargemeasurements. These difficulties can bereadily appreciated by those who haveattempted to measure impulse voltagesby means of an air-capacitor divider.Considerable laboratory time was re-quired to rid the charge-measuring circuitof its tendency to oscillate and its affinityfor pickup so that adequate results couldbe obtained. It is necessary to keep thelength of the charge and voltage measur-ing leads down to an absolute minimumwhen making a direct q-e measurement.The length of the charge-measuring leadsmust be kept to not more than a fewfeet to prevent the inductance of theleads from oscillating with the variablecapacitance or CG. The length of thevoltage-measuring lead must be kept veryshort to prevent voltage being applied tothe horizontal plates at a later time thanvoltages applied to the vertical plates.

Typical data obtained from this seriesof tests are shown in Figs. 6 and 7. Fig.6(A) shows a number of q-e traces at dif-ferent voltages which were made on thesame film. With increasing voltage(going in a counterclockwise direction)all traces start with a slope equal to thenatural capacitance. After the coronavoltage has been exceeded, the individual

traces follow essentially the same pathuntil the crest voltage of the particularsurge is reached. As the voltage decayson the tail of the wave, the q-e trace ap-proaches zero voltage along a line with aslope approximately equal to the naturalcapacitance until a relatively low voltageis obtained. Fig. 6(B) shows a volt-timeand a charge-time trace corresponding tothe q-e curve of Fig. 6(A) having thehighest voltage.

If i, is defined as the current into theconductor-to-cylinder capacitance, theenergy stored in this capacitor is

W=f eicdt=f edq (2)

Hence, the energy removed from thevoltage wave by corona is the area en-closed by the q-e curve.

Fig. 7 shows a comparison of the q-echaracteristics obtained with a positive-polarity surge, a negative-polarity surge,and a 60-cycle applied voltage, all withapproximately the same crest voltage. Acomparison of areas shows that muchgreater energy is involved in 1 cycle ofthe 60-cycle wave than is involved in thesum of the positive and negativeimpulses.Fig. 7 indicates that the space-chargerelations for oscillatory waves are con-siderably different from those for a uni-polarity wave. Extreme caution shouldbe used in attempting to derive coronacharacteristics for unipolarity waves fromcorona data which were obtained at 60cycles.

Perhaps the most important observa-tion to be obtained from Fig. 6(A) is thatthe same q-e path is traced by all the dif-ferent voltage waves so long as the volt-age on the conductor continues to in-crease. This suggests the possibility ofdetermining a single q-e curve for in-

creasing voltages on a particular con-ductor. If this q-e relation can be shownto hold for all practical surge wave fronts,this relation can be used in the predeter-mination of distortion of the front of awave traveling on a conductor when thecorona voltage is exceeded.The voltage waves used in obtaining

the q-e relation shown in Fig. 6(A) rose tocrest in approximately 1.5 microseconds.Since all the voltage waves used in deter-mining Fig. 6(A) had the same time tocrest, the rate of rise varied over a rangeof about 2 to 1 from the maximum to theminimum voltage applied. The fact thatthe fronts of the different voltage wavestrace essentially the same q-e path forthese different rates of rise is an encourag-ing indication that the q-e curve is a func-tion of voltage alone and not of rate ofchange of voltage. This was further in-vestigated by obtaining tests similar toFig. 6(A) with surge waves having about0.5 microsecond to crest. Except for theminor oscillations which were still pres-ent in the test setup, the q-e traces for thetwo different times to crest for increasingvoltage were identical. Hence, at leastfor the cylindrical ground plane arrange-ment, the q-e characteristics were shownto be dependent on voltage alone and notrate of change of voltage for a considerablerange of rates of rise.

VERTICAL GROUND PLANE IMPULSELABORATORY

The tests just described were madewith a cylindrical configuration andmeasurements were made by means of acold-cathode oscillograph. It was even-tually desired to obtain q-e characteristicson practical transmission conductors atthe Trafford High-Voltage Laboratory.Due to the physical arrangement of theTrafford Laboratory, it was known that it

Fig. 9. Test setup for Trafford High-VoltageLaboratory

Wagner, Lloyd-Effects of Corona on Traveling Waves862 OCTOBER 1955

would be necessary to use a portable hot-cathode oscillograph in order to keep themeasuring leads very short. To gainsome experience with the flat ground planeconfiguration and to develop suitablemeasuring techniques using the hot-cathode oscillograph, another small scalesetup was made in the Impulse Labora-tory. An 8-foot by 12-foot ground plane,made up of wire mesh, was built in theImpulse Laboratory. The conductor wasstrung parallel to the screen with a spac-ing to the screen of from 6 to 18 inches.The ground screen was again divided intothree sections to eliminate end effects,and only the center section was used forcharge measurements. The charge meas-urements were made in a manner similarto that described previously, and thevoltage measurements were again madeby use of a resistance divider.

Typical results obtained in this seriesof tests are shown in Fig. 8. The oscillo-grams shown in Fig. 8 are not as good inquality as those shown in Fig. 6 for threereasons:

1. The large ground screen was moresusceptible to oscillations and pickup thanwas the cylindrical ground screen.

2. The hot-cathode oscillograph was notas suitable for these measurements as wasthe cold-cathode oscillograph.3. The oscillographs were recorded on35-millimeter film.

The oscillograms shown in Fig. 8 are in-cluded because they again illustrate theindependence of the q-e characteristicupon the front of the wave. Fig. 8 showsthe q-e characteristics obtained for posi-tive- and negative-polarity surges withthree different times to crest: 5 micro-seconds, 1.3 microseconds, and 0.5 micro-second. It will be noted that for eachpolarity the same charge-voltage charac-teristic is traced on the rising part of thevoltage wave for all the different timesto crest. This provides further evidencethat the charge-voltage characteristicobtained with increasing voltage is afunction of voltage alone.

VERTICAL GROUND PLANE TRAFFORDHIGH-VOLTAGE LABORATORY

After the preliminary testing in the Im-pulse Laboratory, the final laboratorytests were preformed at the Trafford High-Voltage Laboratory using a practicaltransmission conductor and practicalspacings from the conductor to the groundplane. The test setup is shown in Fig. 9.The ground plane was made of wire meshscreen, and was 40 feet high and 25 feetwide. Again the screen was divided inthree sections to eliminate end effects,and only the center 10-foot width was

Fig. 10. Corona P01TJIVEcharacteristics oft0.927-inch steel-reinforced alumi. 4num-cable conduc- a.-tor. Configurationshown in Fig. 9 ~

used for the charge measurements.The conductor was strung in a hori-

zontal position parallel to the screen, 10feet from the screen and 20 feet above thefloor. The conductor used in these testswas a 0.927-inch steel-reinforced alumi-num-cable (556,500-circular-mil) conduc-tor. It was planned to obtain the coronacharacteristics of a large number of trans-mission conductors, but the laboratorytime did not permit.The measuring techniques were similar

to those described earlier for the smallerscale setups. The charge in the variablecapacitance from the conductor to theground screen was determined by measur-ing the voltage across the screen-to-ground capacitor C,. The conductorvoltage was measured by use of a specialcapacitance-compensated resistance-typevoltage divider.'0 Measurements weremade with the same hot-cathode oscillo-graph which was used in the previoustests. Considerable modifications hadbeen made in the timing circuits, sweepcircuits, and beam-control circuits beforethese tests were begun. A 1 by 1-oscillo-graph camera was also available for thesetests.

Typical q-e curves obtained in thisseries of tests are shown in Fig. 10 forpositive- and negative-polarity appliedwaves. Because of the width and thedarkness of the return trace on the q-ecurve, it was not practical to superimposea number of q-e plots on the same film.The dashed lines which have been drawnon the oscillograms in Fig. 10 representthe natural capacitance from the conduc-tor to the ground screen. The slope ofthis line was verified by measuring a q-echaracteristic of a surge wave below thecorona level and greatly increasing thesensitivity in both the charge and voltagedirections. By determining the slope ofthe natural capacitance line with differentvalues of screen-to-ground capacitance,it was also possible to determine the straycapacitance from the screen to ground,which was included in the charge calibra-tion.The q-e characteristics for a number of

different crest voltages have been re-plotted in Fig. 11. Fig. 11 again showsthat essentially the same q-e characteristicis traced by the different waves as longas the voltage continues to rise. Thereplot shown in Fig. II will later be usedin establishing a correlation of field andlaboratory data and will also be used inestablishing a surge corona analogue.The tests in the laboratory were re-

markably consistent as contrasted withsimilar a-c tests in which either coronaloss or radio influence was measured.This statement applies to both the im-pulse tests in the field and to the impulsetests in the laboratory. Perhaps thiscomparison is more apparent than realbecause in the impulse tests much higherratios of applied voltage to corona thresh-old voltage were used than in the corona-loss and ratio-influence tests. Thevariance in a-c corona loss decreased asthe ratio of applied voltage to coronathreshold voltage increased.The effect of water on the conductor

was investigated in the screen tests withsmall configurations. Water had no ef-fect upon the loss loop in the q-e curves.This is further substantiation of a sinmilarresult obtained in the field tests' whichindicated that rain had no effect upon theattenuation and distortion of travelingwaves.

Theoretical Analysis

ELIMINATION OF ELECTROMAGNETICEFFECTS MIOVING XVITH TIIE X ELOCITYOF LIGHIT

Any analytical approach to the solutionof waves on transmission lines that travelwith a velocity less than that of lightmust take into consideration that thevelocity of any electrical effect in freespace equals that of light. The conven-tional treatment of traveling waves fre-quently supposes, without specificallysaying so, that rectangular waves ofcharge and current, traveling with thevelocity of light, have electric and magneticfields associated with them that rise in-stantly from zero to a constant value as

Wagner, Lloyd-Effects of Corona on Traveling WavesOCTOBER 1955 <863

40 20 0 20 40 60 80 100 m IN FT

0.04 0 0.04 0.08 1.02 X 10-6 t (70 %)

7}r / 0.08 0.04 0 0.04 0.08 0.12 0.16 X Q-~6 t (50°%)= -- ____ 0 *0.05 FT 0C2 99.95 FT

r 10 FT 02a 90 FT

Fig. 11. Replotted nest of q-e curves for a 0.927-inch steel-reinforced Fig. 12. Electrostatic potential of two rectangular waves of chargealuminum-cable conductor. Configuration shown in Fig. 9 moving with velocities less than that of light

the heads of the waves pass a particularpoint. By application of the classicalmethods of treatment this can be shownto be the case.Wthen thewaves travel with a velocity less

than that of light recourse must be madeto the so-called "retarded potentials" tocalculate the effects. This method isillustrated in Fig. 12. The insert in thisfigure shows two parallel geometric lines(as contrasted with electric conductors).Upon a is a positive rectangular wave ofcharge whose position at a particular in-stant is shown by A. A similar negativecharge resides on b. Both of these wavesare moving with a velocity v expressed asa fraction of that of light, and both havecorresponding currents associated withthem. Suppose for the moment thatthis velocity is 70 per cent. If the positionof the charge on a at any instant is A',then by the time that the presence of thecharge at A' is felt at point p the wave hasmoved along the line to A, where A'A is0.7 (A'p). The electrostatic potential ata point due to a charge q at a distance rfrom the point is q'r. By summing upsuch contributions of all the charges tothe left of A', the total potential ½ atpoint p is obtained for the head of thewave in the position given by A. Thishas been done for both the positive andnegative waves, for a, = 0.05 feet and a2=99.95 feet and the results are plotted bythe solid line. The potential of a corre-sponding point adjacent to line b wouldhave a value of ; just the negative of thevalue at point p. These potentials areplotted as a fraction of the potentialseventually obtained after the wave hasmoved to the right a great distance. Thepotential between these two points adja-cent to the two lines is the quantity that

is frequently referred to as the "voltage be-tween conductors." It can be seen thatthe significant rise in potential occurs in avery small distance, 80 per cent of thechange taking place in a distance of 20feet.

If one wished to obtain the transverseelectric field, one could determine a simi-lar curve for slightly different values of a,and a2 and then take the limit of (AS)/(ha) as Aa becomes very small. For thepresent case the transverse field is not ofparticular interest although the potentialis. The longitudinal field is of someconcern. In Fig. 13, & is plotted for a, =10 feet, a2 = 90 feet, and v = 0.7. Whilethis gives the potential at point p for dif-ferent positions of the wave, it should beevident that for any given position of thewave this should also be the potential ofdifferent points along the line for whichthe al's and a2's are the same. It wouldonly be necessary to reverse the positiveand negative abscissas. The slope orgradient of this curve E,s then gives thelongitudinal component of the electricfield.The magnetic field can be obtained in a

somewhat analogous fashion. For thispurpose the retarded vector potential A,is obtained. The potential at a pointdue to a current i a distance r from thepoint is equal to i/r. By summing upsuch contributions for all of the current tothe left of A', the total potential A at pointp is obtained for the head of the wave inthe position given by A. It is to be ob-

Fig. 13. Longitudinal electricfield due to two waves of charge =-moving with a velocity of 70per cent of light. a1=10 feet 40

and a2=90 feet

served first that since all elements of ithat contribute to A are parallel to linesa and b, A will have a longitudinal com-ponent only. There will be no trans-verse component. Next it will be ob-served that since the limits of integrationare the same as for the charges and sincethe distance r enters into the relation inan identical manner, A is proportional to31, and has the same wave form. Thusthere are two identical waves, ; and A,traveling along the line. The longi-tudinal electric field due to the currentsEmx is proportional to - (dA) (dt). Thenegative sign means that it is directedoppositely to the currents and is opposedto E,,. First, because Emx is a timederivative of A and, second, because theamplitude of the current wave I is equalto the product of charge and v, Emx isproportional to the square of the velocityof the wave:- When the actual numericalwork is carried out it is found that

Emx = V2E,,Thus the total longitudinal electric field is

Ex = Ex +Emx= (I -V^2)E,x'

E sx

20 0 20 40 60 80 100

m IN FT

l8Jit(gner, Lloyd-Effects of Corona on Traveling ITav-es86S4 OCTOBER 1 9,50

q

c G-

AC

Ce C2 - e-̂G3 - ^ Ce2v/

/ C ~~~~~+AC,

_ /SLOPE //LOPEC+A~~C+AC

When the wave travels w-ith the speed oflight v= 1 and E1 is zero, the electric fielddue to the charge just annuls the electricfield due to the current. At very lowvelocities, E1 approaches the field arisingfrom the charge alone. This is not to saythat {t, when expressed in terms of dis-tance along the line, does not changewith speed. The extent to which itchanges is illustrated in Fig. 12. For al =0.05 and a2= 99.95 feet the difference for vequal to 0 and v = 0.7 is negligible (see thecircle which is plotted for v = 0), and fora,=- 10 feet and a =90 feet the differencebetween v=0 and v=0.7 is indicated bythe two dashed curves.

It was originally postulated that thecharges and the currents were flowing ongeometric lines without making any refer-ence tox the nature of the conductivity.Now if it is assumed that the rectangularwaves of charge and current are flowingon metallic conductors, an inconsistencyarises in that a longitudinal electric fieldcannot exist on a good conductor. Ac-tually a charge separation would occur ofsuch nature as to tend to annul this field.On the a conductor, a positive chargewould be pushed slightly ahead of thefront of the wave and a negative chargejust behind the front of the wave. Thenet effect would be to round the cornersand slope the fronts of the charge and thecurrent waves and at the same time makethe front of the /' wave more steep. Inother words, the wave shape of 3V and qwould be brought into closer agreement.Certainly it can be said that the differencebetween the two waves would be lessthan the departure of the Qt curves of Fig.12 from a rectangular front. For thepractical configurations of transmissionlines this difference is negligible. Onecan proceed with assurance and assumethat the curve of V, is of the same waveshape as the charge and that the usual

L/2 L L L/2.. r w~~~~~~~-YN

_._ _AC __ _ AG L _ ACi

f/,,,,,,

Fig. 14 (left). AnalogueNATURAL approximation of charge-CAPACITANCE voltage characteristics of

short length of conductor-SLOPE

I el Fig. 1 5 (above). Analogueof a long line using the ap-proximation of corona con-

,GE e ditions indicated in Fig. 14

line constants on a per-unit length can beused.

ANALOGUES

From the laboratory experiments onshort lengths of conductor that have beendiscussed previously, it has been shownthat the charge on a conductor under im-pulse conditions for the rising portion ofthe curve is a function of the voltage only.The charge is independent of the rate ofrise of voltage or the wave shape of theimpulse. As the voltage applied to theconductor increases from zero, the chargeincreases linearly with voltage along aslope corresponding to the "naturalcapacitance" C of the conductor as indi-cated in Fig. 14. When the coronathreshold voltage eo is exceeded, the slopeof the curve becomes steeper and thecurve increases until the maximum volt-age of the applied wave is reached. Asthe voltage decreases, the curve followsan essentially straight line. For thelower values of crest voltage the slope ofthe return path corresponds quite closelyto the natural capacitance but at thehigher crest voltages the slope becomesprogressively larger.As a first approximation, the curved

portion of the rising characteristic can berepresented by a straight line whose slopewill be designated C+-AC and whose in-tercept with the straight line representingthe natural capacitance will be desig-nated ei. The downward portion of thecurve can be represented by anotherstraight line having a slope equal to C.These approximations can be representedfor both increasing and decreasing volt-ages by the analogue shown in the lowerright-hand insert. Because of the greaterimportance of the rising part of thecharacteristic and the complexity of ob-taining an analogue having closer corre-spondence to the actual characteristics

for decreasing voltage, this simplified ana-logue is regarded as sufficiently satis-factory for most purposes. For a moreaccurate representation of the rising por-tion, the characteristic can be representedby the two dashed straight lines. Theanalogue for this approximation is thatshown by the upper left-hand insert.For more accurate approximations, alarger number of shorter straight linescan be used and additional elements addedto the analogue. Fig. 15 shows how thisanalogue can be inserted to form theequivalent for a long line.

DIGITAL COMPUTER CALCULATIONS

The laboratory tests showed that thesurge corona characteristics can be pre-sented by a shunt capacitance whichvaries with voltage alone. It remains todevelop methods for interpreting a knownq-e characteristic in terms of distortionof a traveling wave.

If was felt that calculations performedon a digital computer would be most help-ful in providing an understanding of theeffect of a nonproportional q-e relation.A sample traveling wave problem wasprogrammed for an International Busi-ness Machines 701 digital computer. Thetransmission line selected for study was4,800 feet in length with a natural surgeimpedance of 400 ohms and a naturalvelocity of propagation of 1,000 feet permicrosecond. The line was representedby 480 equal pi sections such as shown inFig. 15, each section representing 10 feetof conductor length. For one group ofcalculations the corona capacitors weredeleted.The voltage wave applied to the sam-

ple line rose linearly to crest in 1 micro-second, and was flat beyond 1 micro-second. A more detailed description ofthe digital representation is given inAppendix I.The results of the digital computer cal-

culations are shown in Figs. 16 and 17.The data in Fig. 16 were obtained withthe corona capacitor removed. The pur-pose of these calculations was to verifythe ability of the computer to make ac-curate calculations on a traveling waveproblem whose solution was alreadyknown. If the line were composed ofuniformly distributed inductance andcapacitance, the applied wave wouldtravel undistorted until it reached the

Wagner, Lloy'd-Effects of Corona on Traveling Waves 865OCTOBER 1955

FT2.0 - . 4800'

r-yr---,.APPLIEDTz a ?IE I I 1

1.5 -- -10 FT INTERVALS

a.z APPLIED 3200'

1.0

00.5

C0 2 3 4

TIME IN MICROSECC

open end of the line where a reflectionwould occur. The voltage at the openend of the line would be twice the appliedvoltage and have exactly the same waveform. The voltage at an intermediatepoint along the line would be exactly thesame as the applied voltage until the re-flection from the open end reached thepoint. Fig. 16 shows that the digitalcomputer reproduces the expected resultwithin plotting accuracy except for a slightrounding off when the wave first beginsto rise and a very small oscillation near thecrest of the wave. It was concluded thatthe digital computer was capable of mak-ing the desired calculations wNith sufficientaccuracy.

Fig. 17 shows the digitally calculatedcurrents and voltages obtained when theshunt capacitance was allowed to varyin a manner to produce the q-e curveshown in the insert. Referring to theinsert, with increasing voltage, the chargeincreases along a line representing thenatural capacitance until the conductorvoltage reaches the corona threshold volt-age of 1.0 per unit. Above this voltagethe slope of the q-e curve is twice thenatural capacitance. Otherwise the rep-resentation of the transmission line wassimilar to that described previously.The voltage wave indicated by the waveform at 0 feet was applied. Fig. 17 showsthe conductor voltages and line currentsat the sending end, at 800, 1,600, 2,400,and 3,600 feet. To eliminate the possi-bility of confusion arising from the simi-larity of the wave shapes in Figs. 16 and17, it should be noted that in Fig. 16 thestepped character of the wave at 3,200feet is due to a reflection from the openend whereas in Fig. 17 it is due to corona.The significance of these calculations isdiscussed in the following section.

VELOCITY IN TERMS OF Q-F RELATION

In the discussion of field data, it hasheen suggested that the effect of corona isto retard each point on the front of thewave by an amount proportional to thedistance traveled. This is equivalent to

5 6 7 8ONDS

Fig. 16 (above). Re-sults of digital com-puter calculations of atraveling wave on a line

without coronaz

zw

a.

Fig. 17 (right). Resultsof digital computer cal-culations of a travelingwave on a line with

corona

postulating a velocity for each point onthe wave which is a function of voltagealone. It has already been shown thatthis relation holds for the Tidd test data.'Fig. 17 shows that this same concept ap-plies for the digitally calculated voltagewaves.

Fig. 17 shows that above corona thetime required for a given level of voltageto propagate a specified distance along theline is greater than the propagation timefor a level below corona by a factor of a-2.Doubling the slope of the q-e curve resultsin a velocity which was decreased by /2.This suggests the possibility of rewritingequation 1 in the following manner

VIL(C+ACc)=vL AC (3)1

C

where VL iS the velocity of light and C+ AvCis the slope of the q-e curve at any pointas distinguished from the slope of thestraight-line approximation that has beenpostulated in the analytical considerationsto this point. It is shown in Appendix IIthat equation 3 is indeed rigorously cor-rect under the following conditionsabe->0 (4)

be>a>0 (5)

dq>de->(6)de

TIME IN MICROSECONDS

d2> (7)

de2 -

Equations 4 and 5 indicate that the ve-locity concept expressed in equation 3 isapplicable only for the front of the volt-age wave and not for the tail where thevoltage is dropping. Equation 6 im-poses this same limitation that the ve-locity concept does not hold for the tail ofthe wave. Equation 7 imposes the limi-tation that the slope of the q-e curve mustnot decrease at any point with increasingvoltage, i.e., the curve must not have apoint of inflection. This last limitationis necessary to prevent the mathematicaloccurrence of multivalued voltage wavesas a function of time or distance.

Fortunately the laboratory data inFig. 11 and the test data in Figs. 4 and 5indicate that equation 7 is satisfied forpractical transmission conductors andpractical configurations. The velocityconcept in equation 3 can then be appliedto predict the distortion of the front of avoltage wave traveling on a conductorwhose corona characteristics are known.

DISTORTION OF VOLTAGE WAVES FROMQ-E CURVESIn Fig. 18(A) the fundamental q-e

curve characterizing the conductor isshown. The slope of this curve is givenby the C+ AC curve. The C+AC curvein turn permits the determination of thevelocity characteristics from equation 3.

Wagner, Lloyd-Effects of Corona on Traveling Waves866 OCTOBIER 1955

()(C) X=O x-d

]

0

C,

eO

CONDUCTOR VOLTAGE

Now turning to Fig. 18(C) an arbitraryvolt-time curve is shown which is appliedat x= 0. Assuming that the travelingwave has moved a distance d, the undis-torted portion of the wave below thecorona threshold voltage eo requires atime tl=d/vL to travel this distance. Apoint on the voltage wave above eo willrequire the time ti+ AT. This total timeis related to the instantaneous velocitycorresponding to the particular point onthe voltage wave by the relation

dtl+AT= (8)

v

Substituting the value for v from equation3, there results

d 'AtCti +AT:= 1+)

VL C

or

d C 'AjC \dAvT=-* 1+ C -ti +-C Ia L

or

AT I AC \- - 1+ C l (9)

Fig. 18. Method oftermining distortiontraveling waves from

curves

de-of

q-e

: e

The curve for AT/d is plotted in Fig.18(A). For convenience this curve isreplotted in Fig. 18(B). To determinethe retardation AT for any voltage it ismerely necessary to multiply the value inFig. 18(B) by the distance traveled. Thisprovides the curve shown in Fig. 18(C)for x= d. Thus a method is available fordetermining the distortion of a wave trav-eling on a line whose q-e characteristic isknown.

VOLTAGE-CURRENT RELATIONS

Once the distortion of the voltage wavehas been determined, it is important toknow the distortion of the associated cur-

rent wave. It is well known that, for a

line with uniformly distributed inductanceand capacitance that does not changewith voltage, there exists a constant ratioof voltage to current associated with a

particular wave

e LZe= 1- (10)i sC

where Z is the surge impedance.If the capacitance varies in a manner

such that C+ AC is dependent upon volt-age alone, it might be suspected that theconcept of surge impedance as definedin equation 10 might still be meaningfulif C is replaced by C+ AC. Intuitively,one might also expect this concept to bevalid in the ratio of elements of e and i,i.e.

di 1 1/ ACde_- 1 +C

(11)

It is shown in Appendix III that equa-tion 11 is rigorously correct within thesame limits which were imposed uponequation 3. Integration of equation 11with respect to e yields

.lfe AC

0z 1 de (12)

It is interesting to apply equations 11and 12 to the digital computer resultsshown in Fig. 17. Analysis of the cur-rent waves in Fig. 17 shows that abovethe corona level the rate of current rise isgreater than that below the corona levelby a factor of -2. The current in Fig.17 has been plotted against voltage, andthe results are shown in Fig. 19. Thecurve in Fig. 19 applies for any combina-tion of voltage and distance co-ordinatesfor the assumed q-e variation. The cur-rent is dependent upon the voltage alone.

t= t, t=o

Ae

di = y(e)Fe

q

e

VOLTAGE IN PER UNIT

Fig. 19 (left). Current-voltagerelations from digital computer

calculations shown in Fig. 1 7

Fig. 20 (right). Illustration ofdifference between volt-time and

volt-distance curves

AX =t, (VL-V)

=X,C-gVC+AC

X

XZO

AT - X V- )

X,V,C-sAC-VF=

VL ,-c

t=o

Tlragner, Lloyd-Effects of Corona on Traveling Waves

0.2 0.4 0

p SEC/IOOO FT. TIME IN MICROSECONDS

2.4

2.0

i

i.6-cr

K-1.2

z

rcro) 0.8

0.4

(B)

OCTOBER 1955 867

z

4

-i0

cr0

0

z0

z

w

C,

z

0.4

-J

z

cr4

0 0.1 0.2 0.3 0.4 0.5 0.6

AT IN MICROSECONDS PER THOUSAND FEET

Fig. 21. Comparison of retardation curves obtained inthe field with similar curves computed from laboratory data

200 400 600 800 1000 1200 1400 160CCONDUCTOR VOLTAGE IN KV

Fig. 22. Retardation curves of Fig. 4 converted to q-e curves

The slope of the current-voltage curve

above corona is greater than the slopebelow corona by V/2. This checks withequation 11 since 1+AC C==2. It isapparent then that the use of equation 11or 12 and the volt-time curves shown inFig. 17 makes possible the calculation ofthe associated current waves. This ap-

plies generally to any q-e relation. Hav-ing found the voltage distortion of a trav-eling wave the corresponding current can

be obtained through the application ofequation 12.

DIFFERENCE BETWEEN VOLT-TIMECURVE AND VOLT-DISTANCEDISTRIBUTION

It is customary to think of a travelingwave as being a function of time or dis-tance interchangeably. For example, a

volt-time curve will result in exactly thesame volt-distance distribution where thedistance and time co-ordinates are re-

lated by the velocity of light. As anotherexample, if a wave required 1 micro-second to reach crest voltage, the frontof the wave will encompass a length ofline equal to the velocity of light times 1

microsecond. However, when a wave

travels on a line which causes distortion,it is no longer possible to think of thewave in terms of time and distance inter-changeably.With corona effects included, Fig. 20

illustrates the difference between thevolt-time curve at a distance xi and thevolt-distance distribution at time ti=Xl/VL for a square wave applied to a lineat x=0. The equations for the retarda-tion in time AT and the retardation in

distance Ax are shown in the figure.These equations make it clear that ATand Ax are no longer related by a constantbat are related by the variable AC/C.Extreme care must be used in relatingthe volt-time curve and the volt-distancedistribution.

Correlation of Laboratory andField Data

Fig. 4 shows the AT/d curves obtainedduring the Tidd tests for the 0.927-inchsteel-reinforced aluminum-cable conduc-tor for both polarities. These curves

have been replotted in Fig. 21. The totalvariance' is indicated by the shaded areas.

Fig. II shows the q-e curves obtained inthe laboratory for the same conductor.By methods that have just been pre-

sented, the front portions of the curves

for both polarities having the maximumcrest voltage were converted to' AT/dcurves. These are also plotted in Fig. 21for comparison with the field data. Con-sidering the vagaries of corona and the dif-ference between field and laboratory setupand measurements, it is felt that the cor-

relation is quite satisfactory.

Conversion of Field Data to q-eCurves

The field data shown in Fig. 4 have beenconverted to the more fundamental q-e

characteristic, and the results are shownin Fig. 22. This form of the data is lessuseful than the retardation curves, butwill be quite helpful to those doing fur-ther fundamental work on this subject.

Effect of Corona on Tail of VoltageWave

The preceding discussions explain thedistortion of the front of a traveling wavein terms of theoretical considerations andfundamental corona characteristics ob-tained in the laboratory. If the wave isflat at its crest voltage, the distortion ofthis flat tail can also be determined. Nomention has been made of distortion ac-

companying a wave tail which decreaseswith time.

Fig. 23 is included to provide an under-standing of the effects present on the tailof the wave. A transmission line is as-

sumed to have the q-e characteristic shownin (A). The volt-time curve e shown bythe solid lines in (B) is applied to the lineat x =0. The crest voltage of e is 2eo.Consider the wave to be resolved intotwo components. The component ep is a

wave which consists of the linearly risingfront and is flat at the crest voltage. Thewave en is a linearly rising wave, alsoflat at the crest, but is opposite in polarityto ep and originates at the time when e

begins to drop.Fig. 23(C) is a volt-distance curve and

shows the manner in which these two com-

ponents, hence the resulting wave e,

propagate along the line. At time t4 thecomponent en has not originated, and theep and e waves are the same. The wave

has been retarded along the front in thesame manner shown in Fig. 17. The cur-

rent wave is likewise retarded and risesto a higher crest value than would haveexisted if corona were not present. Cal-culation of the current by means of equa-

Wagner, Lloyd-Effects of Corona on Traveling Waves

_0.927" ACSR _i- l 7

120TIDD BANDS (NEG)--,f X \\_

1000 111 M 2-

TIDD AVERAGE 9 T \ DAERG

600 - \- I-____ __ [ LABORATORY(POS)

v

OCTOIBER 1955868

tion 12 shows the crest current to be

ip (crest ) = i (crest ) =eo( +A2)z

Crest current occurs at the same time ascrest voltage.At time t2, the entire voltage wave has

moved out onto the line. The com-ponent ep is subjected to additional retar-dation along the front. However, enoriginates at some time after ep hasreached its flat crest. The effect of en isto lower the voltage e on the conductor,hence the capacitance effective for en isthe slope C, the return trace of the q-ecurve, rather than the slope 2C which iseffective for increasing voltage. Thecomponent en then travels with the veloc-ity of light, and has an associated current

en1n =_

and a crest value

2e0in (crest)= __z

It will be seen that adding currents ipand in results in a current which doesnot drop to zero along the tail but dropslinearly to the constant value.

eoJis = 2 (V\/2-1)

This current exists all along the line fromthe origin to the point just behind the tailof the wave e. It should be noted thatthe voltage along this section of the lineis zero. At a later time t3 it is apparentthat this same current is exists from theorigin to the tail of e at its new position.A current existing along the line with-

out an associated voltage mav seem con-

q

tradictory until it is recognized that thecurrent is constant through the seriesinductance back to the source, and nochange in voltage lesults. It is true thatthe total inductance from the origin tothe tail of the wave varies as the wavetail moves along the line. However, be-cause the elements of inductance whichare added already have the current isflowing, no change in flux linkages occurs.It is important to note that the energyput into the line at x = 0 is exactly equalto the energy stored in the inductance andcapacitance in the line for all positions ofthe wave shown in Fig. 23(C). As thevoltage wave moves along the line, storedenergy is left behind in the corona capaci-tor -AC and in the line inductance. Theseenergies arise from the distortion of thevoltage and current waves as they pro-gress along the line.Up to the time t3 the retarded front of

ep has not overlapped en. When thisoverlap occurs, the resolution suggestedearlier is no longer valid. A portion ofthe component en tends to travel into aregion of ep where the conductor voltageis actually rising. Hence, a portion of entends to travel in a region where thesurge impedance is determined by theslope 2C of the q-e curve, and the re-mainder, in a region where C determinesthe surge impedance. A complex formof continuous reflections probably occursin the overlapping region, and the authorshave not yet been able to reach an ana-lvtical solution to this problem. Thisis a fruitful area for further investigation.The field tests shown in Figs. 1 and 2

do provide an evaluation of the effect ofoverlapping of the tail and the retardedfront. If the tail is relatively long com-

pared to the front, the tail can be assumedto propagate without retardation or dis-tortion. The wave front is set back asshown in Fig. 3 until the retarded frontintersects the original tail. This point ofintersection can be taken as the crest ofthe distorted wave, and beyond this pointthe wave follows the original tail. Itshould be noted that this procedure is notrigorously correct, since the energy rela-tions are not satisfied. However, thisprocedure is a sufficiently good approxima-tion for most purposes.

Reflections

At first thought one would expect thata wave in the corona region on reachingan open-ended line would double, be-cause the abrupt change at this point isfrom the surge impedance of the line(whatever it might be) to an infinitelylarge impedance. Actually this does notoccur. A similar modification must bemade in the theory when the line is ter-minated by a resistance equal to itsnatural surge impedance.To analyze these conditions considera-

(A)

ep(B)x'

/x=ot

I' tI \ t2 t3

x'O

SURGE IMPEDANCE Z/ A

e e

FICTITIOUS GROUND

0 eF

- - ~~~4800 FT

5 6 7

TIME IN MICROSECONDS

/ n /e nn / In

Fig. 23. Illustration of corona effects on the tail of a voltage wave

Fig. 24. Comparison of digital computercalculations and analytical calculations of thevoltage appearing at the end of an open-

circuited line under corona

Wagner, Lloyd-Effects of Corona on Traveling Waves

2eo-

eO-

116

-eO

OCTOBE-R 1955 869

2000

q

W I200 VOLIAE AlAENDeD / ,- O~~~~~~~~~~~FTERMIN-

ATED LINE0

800

400-

O~~~~~4O 2 3 4 5 6TIME IN MICROSECONDS

Fig. 25. Voltages measured at the end of an open-circuited line and a line terminated by its natural surgeimpedance together with a computed comparison between

them

tion must be given to the current-voltagerelations at the point of reflection. Theessential features of the phenomenon willbe brought out by considering the specialtype of line that had previously been dis-cussed in connection wx ith the digitalcomputer studies, a line in wvhich AC= C.The current-voltage relationship for thisline is plotted in Fig. 19. This curve isreplotted in Fig. 24(B). It can be seenthat, since the slope of this curve above euis V/2, then the projected intercept withthe abscissa is 1-1/V/2 or 0.293. It fol-lows then that an alternate analogue tothat which had beeni presented previouslyis that shown in Fig. 24(A), in which theimpedance of the line is changed to Z/V\/2and a fictitious ground is inserted forwhich a battery maintains its potential totrue ground to a value ea. With respectto propagation and reflection effects, theline with respect to the fictitious groundperforms just like a conventional line.The voltage to true ground e is then equalto e'+ea. This condition is true only onthe rising part of the voltage wave andfor voltages above eo. For analyticalpurposes it is more convenient to use thanthe analogue shown in Fig. 14, but thatof Fig. 14 is to be preferred for analoguecomputer studies because it representsthe true current-voltage relations overthe entire portion of the rising and fall-ing voltage wave.

Fig. 24(C) shows the results of adigil al computer study for the voltage atthe end of an open-circuited line of thetype under consideration. Rather thandraw the results of an analytical studythat would almost superpose upon thiscurve, it will be shown how the significantpoiilts of this curve can be computed.The line was 4,800 feet long and the con-stants were adjusted so that below coronathe velocity of propagation was exactly

GE + AC

SLOP C

eo eA

Fig. 26 (above). Ana-logue for a reflectionlesstermination (B) for aline having the q-erelationship shown in

(A)

1,000 feet per microsecond. The voltageOBCD was applied at x=(. As the wavetravels along the line two levels of volt-age develop, that of e= 1 and at a latertime the value e= 2.0. As the first levelreaches the end of the line only the com-ponent e'= 1-0.293=0.707 doubles. Thereflected voltage to true ground is then (2)(0.707)+0.293=1.707. This checks thelevel of HJ. Similarly when the flatportion CD reaches the end, the part thatreflects is 2-0.293 =1.707. Again thevoltage to true ground is (2) (1.707)+0.293=3.707, which also checks exactly.The curve FNPQ is the form that the

original wave would have had at x=4,800feet had the line been continuous. PointF is at 4.8 microseconds. Because of thedoubling effect below eo point G occurs attime corresponding to Ml and is 4.8+0.25 = 5.05 microseconds. That portionof the curve between MN will be delayedbecause all of the line is now above eo andtherefore the time between G and H isV2(0.25)=0.334microsecond. Thismakesthe point HI equal to 5.05+0.354 micro-second. Now consider point K. This issimply the discontinuity represented bypoint C, whichis -/2(4.8)+ 1.0 = 7.8micro-seconds. The difference in time betweenJ and K corresponds to that between Band C which is 0.5 microsecond.

During the progress of the Tidd testsconsiderable discussion developed re-garding the proper method of terminatingthe line to prevent reflections. Fig. 25,taken from a description of these tests,ishows the voltage at the end of an open-circuited line and the voltage at the endof the same line when terminated with aresistance equal to the natural surge im-pedance of the line. By assuming thatthe corona threshold voltage is 300 kvand that AC= C, one can compute, bymethods just described, the volt-time

B

curve of the wave approaching the endof the line if there were no reflection.This is the dashed curve ef. When theline is terminated by a resistance equal tothe natural surge impedance, the voltageat the end of the line e, is

2Rez= 2 (ef-300)+300

2= 1.707 (ef_300)-+300

This curve likewise is indicated in Fig. 25.Considering the crude approximationsmade, the agreement is quite close. Thisillustrates that under corona conditionsthe natural surge termination is not quitecorrect.

Fig. 26(B) shows the correct reflection-less termination for a line having the q-echaracteristic shown in Fig. 26(A). Thisrelationship is developed in Appendix INT.This termination will also be useful inanalogue computer studies to representan infinitely long line, but is rigorouslycorrect only for the rising portion of thewave.

Conclusions

1. As a result of the application of impulsewaves to short lengths of conductor in thelaboratory it has been demonstrated that:

a. The effect of corona is to increase thecapacitance of the conductor.

b. On the rising front of the wave, thecharge is a function of voltage only and isindependent of the rate of rise of voltagefor a wide range of surge fronts.

c. As the voltage decreases the q-erelationship can be approximated by thenatural capacitance of the conductor.

2. Analytical methods supported by digitalcomputer studies show that the increase incapacitance occasioned by corona retardsthe front of the wave on a transmissionline by a calculable amount that is pro-portional to the distance traveled.

3. For practical purposes the tail of avoltage wave can be assumed to travelundistorted. The crest voltage will beobtained where the retarded front intersectsthe tail.

4. Water droplets on the conductor donot affect its q-e characteristics, which

87 lragner, Lloyd- -Effects of Corona on Traveling Waves81S70 OCTOBE-R 1 955

LINE SECTION LINE SEGTION2

LINE SECTION480

1 1~~~~~~~~~~~~~

o,Ii A 1479, 480

es e e2fT e480 T

(A)

Fig. 27 (left). Elementsof a transmission line undercorona used in the digital

computer studies

AiX I Ai 1i

el; q Ax Ax(q+-sAql e + Ae

x x + AX

(C)es 4

,/ 4 VOLTSq

i =q __dq bebx bi de ot

(18)

where q = q(e). Differentiating equation 17with respect to x, equation 18 with respectto t, and eliminating the term (-2i)/(bxbt)

0 2e IN VOLTS

.02 VOLTS t4 0 I SEI

Fig. 28 (right). Schematicdiagram of an element ofa transmission line under

corona

b2e dq b2e d2q obe\2=L- L _

8X2 de bt2 de2 \at (19)

Assume as a trial solution to equation 19

e =f(x-vtI) (20)

verifies the field tests that rain does notinfluence the effect of corona in distortingtraveling waves.

5. Various types of analogues to representthe effect of corona have been presented.These are:

a. A device to represent the increase incapacitance which is inserted in the incre-mental length of conductors.

b. A combination of resistors, a battery,and a rectifier to represent a reflectionlessinfinitely long line.

c. A somewhat simplified adaptation of(b) consisting of a single resistor and batterythat can be used for analytical work onlyabove the corona threshold voltage.

Appendix 1. Digital ComputerAnalysis

The transmission-line representationshown in Fig. 27(A) consists of 480 linesections, each section representing 10 feetof line. The q-e characteristic of thecapacitors is shown in (B). The constantswere selected so the velocity of propagationis 1,000 feet per microsecond and Z=400ohms. The triangular-front wave appliedat the sending end is approximated in thestepwise manner shown in Fig. 27(C).Each voltage step is 0.02 volt high; 200steps are required to represent the wavefront. The crest of the applied wave is4 volts which is twice the corona voltage.Equations 13 through 16 are used to de-termine the voltages and currents for eachlitle section, where N is the number of theparticular line section of concern and AC isthe capacitance increase due to corona.

AiN I,N L Al (13)

2 LAeN=

ex-1 eAt (14)

AeN-

'Atf\r (15 )

AeN=,-,NA- N+ I

C+±AC

were zero. Once a nonzero current changehas been obtained for the line section,equation 14 is used for all subsequentcurrent calculations. With equations 15and 16 the voltage at the capacitor of eachline section is calculated. Equation 15 isused below corona and equation 16 abovecorona. The time interval chosen was0.005 microsecond and the computer printedvoltage and current every 0.05 microsecondfor line sections 0, 40, 80, 120, etc.The first time interval calculation

consists of raising e, to the value of the firstvoltage step and determining the incre-mental current change in every linie sectionusing equation 13. Only line section 1 willcontain current since el, e2,. . . e480 willremain at zero potential. From theseline currents the voltage changes arecalculated using equation 15.The incremental currents in the next

interval are obtained by raising e8 to thesecond voltage step and assuming all otherline voltages remain at the values cal-culated in the first interval. These incre-mental currents are added to the previouscurrents to obtain the currents in the secondinterval. From these currents the secondinterval voltage increments are determined.These increments are added to the voltagesof the preceding interval to obtain thevoltages in the second interval. This pro-cedure continues until es reaches the crestof the wave; for subsequent intervals es iskept constant at crest voltage.

Appendix 11. Velocity in Termsof Charge-Voltage Relation

Consider an elemental length of trans-mission line shown in Fig. 28 with uni-formly distributed inductance, and shuntcapacitance which varies with voltagealone. The differential equations relatingcurrent and voltage can be readily written

biAe =-L- Axatbe

(16) ox

Equation 13 is used to calculate thecurrent in the series inductance if allpreceding current increments in the section

bt

bqAi=- Ax

bt

or

Wagner, Lloyd-Effects of Corona on Tra2

where v= v(e), x-vt-u. The first and sec-ond partial derivatives of equation 20,when substituted into equation 19, mustyield an identity if equation 20 is a solu-tion to the differential equation. Per-forming these operations yields

[d2f df \3 d2v I dqd2-I --t 2~ _v2XLdu2 \du/ de2J de

d2f df 3 d2xvdf 2

du2 du de2 j du/

i+df dv\\ dq dvL d2qA( du de/\ de de de2

(21)For equation 21 to be an identity

dq 1

de v2

dq dv d2q2L- v-+L v0Ode de de2

From equation 22

dq_ 1de LV2

Differenitiatinigd2q -2 d7)de2 Lv3 de

Substituting23 gives

(22)

(23)

(24)

(25)

equations 24 and 25 into

1 dv -2 dv\V02L-v ±e+L deJLv2 de Lv3 d

(26)

Equation 21 is an identity, and equation20 is a solution of the differential equation19 when

v = v(e)==L q

de

(27)

The foregoing equations are rigorouslycorrect if dq/de. 0 and d2q/de2. 0. Thefirst limitation is required to prevent v beinigimaginary. In passing from the rising to

(17) the falling portion of a q-e curve, it isnecessary to pass through a region wheredq/de is negative. The velocity conceptsdeveloped in this Appendix do not applyfor the tail of the wave, i.e., where be/btor be/bx is negative.

veling Waves 871OCTOBE-R 1955

The second limitation requires that theq-e curve not have a point of inflection.If this occurs, the same v will exist for twolevels of voltage and a multivalued voltagewave will result. Such a wave is physicallyimpossible, hence equation 20 cannot be asolution if d1q/de2 is negative.

It is useful to rewrite equation 27 in asomewhat different form

V= L(C+AC) +//AC X'LC

VL

,I C (28)1+ C

where C is the natural capacitance belowcorona and

C C dqde

Appendix Ill. Current-VoltageRelations

If a transmission line consists of uniformlydistributed inductance L and constantcapacitance C, the relation between voltageand current is known as the surge impedance

(29)

If the capacitance varies with voltage,let it be assumed that

di- = yde

where

Y= Y(e)

dfoi di be du-=- -= Y- (32)

lx de ox df dvdu de

Substituting equations 31 and 32 into18 yields

/dqdq _ dede L

1 L /cy=YC C+IAC

Hence if Z, represents the surge impedancebelow corona

di 1 1 / ACde-=- c+ (33)de Z Zn, C

Equation 33 provides a concept of surgeimpedance when the shunt capacitancevaries. For a known q-e characteristic andknown voltage wave, the current can becalculated analytically or graphically by

1 f/ Aci=zI I+ AC de (34)Again it should be emphasized that theforegoing relations are not valid for dq/de<Oor d2q/de2<0.

Appendix IV. ReflectionlessTermination of a Line Under

Corona

Assume a line has the q-e characteristicshown in Fig. 26. Below eo the surgeimpedance is

(30)

Equation 30 will be tested to determinewhether it satisfies equations 17 and 18.

If q=q(e) and equation 20 is correct

dq dfaq dq df bu de duZt de du bt dv df

de du

(31)

Above eO the surge impedance is

Z CA-C (36)

Fig. 26 shows the reflectionless termina-tion to consisi of two resistors Z-HZn and a

second resistor z, which is in series with arectifier and battery. Above eo the parallelcombination of the two resistors mustsatisfy equation 36. Hence

Zz, LZ+ZC C+AC

Solution of this equation for Z, shows thatthe value of this second resistor is

ZnAC

cc

References1. HIGH-VOLTAGE IMPULSE TESTS ON TRANS-MISSION LINES, C. F. Wagner, I. W. Gross, B. L.Lloyd. AIEE Transactions, vol. 73, pt. III-A,Apr. 1954, pp. 196-210

2. Discussion by E. W. Boehne. Ibid, vol. 50,June 1931, pp. 538-59.

3. SURGE PHENOMENA (book), Electrical ResearchAssociation. The British Electrical and AlliedIndustries Research Association, London, England,1941

4. PROPAGATION OF SURGE GENERATOR WAVESUP TO 850 KV ON A 132-Kv LINE, M. Bockman, N.Hylten-Cavallius, S. Rusck. Paper Vo. 314,CIGRE, Paris, France, 1950.

5. EXPERIMENTAL STUDIES IN THE PROPAGATIONOF LIGHTNING SURGES ON TRANSMISSION LINES,0. Brune, J. R. Eaton. AIEE Transactions,vol. 50, Sept. 1931 pp. 1132-38.

6. LIGHTNING ARRESTERS AS A CRITERION FORINSULATION LEVELS, Harold L. Rorden. Ibid.,vol 69, pt. I, 1950, pp. 84-96.

7. DISTORTION OF TRAVELING WAVES BY CORONA,H. H. Skilling, P. de K. Dykes. AIEE Trans-actions (Electrical Engineering), vol 56, July 1937,pp. 850-57.

8. LIGHTNING LABORATORY AT STILLWATER, NEWJERSEY, R. N. Conwell, C. L. Fortescue. AIEETransactions, vol 49, July 1930, pp. 872-76.

9. TRAVELING WAVES ON TRANSMISSION LINESWITH ARTIFICIAL LIGHTNING, K. B. McEachron,J. G. Hemnstreet, W. J. Rudge. General ElectricReview, Schenectady, N. Y., Apr. 1930, p. 204.

10. VOILTAGE DIVIDER FOR MEASURING IMPULSEVOLTAGES ON TRANSMISSION LINES, S. B. Griscom,B. L. Lloyd, A. R. Hileman. AIEE Transactions,vol 73, pt. III-A, Apr. 1954, pp. 228-37.

11. MEASURING EQUIPMENT AND TECHNIQUESUSED FOR HIGH-VOLTAGE IMPULSE TESTS ONLINES AND SUBSTATIONS, J. W. Skooglund, W. H.Kolb, T. L. Dyer, Jr. Ibid., pp. 223-28

Wagner, Lloyd-Effects of Corona on Traveling Waves872 OCTOBER 1 955

t I ce Z L