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J. H. NEHERMEMBER A! EE

IN 1932 D. M. Sim mo ns1published aseries of articles entitled, Calculation

of the Electrical Problems of UndergroundCables.* Over the intervening 25 yearsthis work has achieved the status of ahandbook on the subject. During this

period , however, there have been nu mer-ous developments in the cable art, andmuch theoretical and experimental workhas been done with a view to obtainingmore accurate methods of evaluating theparameters involved. The ad ve nt of thepip etype cable system has emphasized

the d esirability of a more rational m ethodof calculating the performance of cablesin duct in order tha t a realistic comparisonmay be made between the two systems.

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In this paper the authors have endeavored to extend the work of Simmons

by presentin g un de r one cover the basicprinciples involved , toge ther with morerecently developed procedures for han-dling such problems as the effect of theloading cycle and the temperature riseof cables in various types of duct struc-tures. Includ ed as well are expressions

required in the evaluation of the basicparameters for certa in specialized alliedprocedures. I t is thou gh t th a t a w ork ofthis type will be useful not only as a guideto engineers entering the field and as areference to the more experienced, butpa rticu lar ly as a basis for se tting up com-pu tation me thods for the prep aration ofindustry load capability and ac/dc ratiocompilations.

The calculation of the temperature riseof cable systems unde r essentially steadystate conditions, which includes the effect

of operation und er a repe titive load cycle,as opposed to transient temperature risesdue to the sudden application of largeamounts of load, is a relatively simpleprocedure and involves only the applica-tion of the thermal equivalents of Ohmsand Kirchoffs Laws to a relatively simplethermal circuit. Because this circuitusually has a number of parallel pathswith h eat flows entering a t several points,however, care must be exercised in themethod used of expressing the heat flowsand thermal resistances involved, anddiffering methods are used by various en-gineers. The method employed in this

paper has been selected af ter careful con

M.H. McGRATHMEMBER AIEE

sideration as being the most consistentand most readily handled over the fullscope of the problem.

All losses will be developed on the basisof w atts per conductor foot. The heatflows and temperature rises due to dielec-tric loss and to currentpro duced losses will

be treated separately , and, in the la ttercase, all heat flows will be expressed interms of the current produced loss originat-ing in one foot of conductor by means ofmultiplying factors which take into account the added losses in the sheath and

conduit.In general, all thermal resistances willbe developed on the basis of the per con-ducto r he at flow throug h them. In thecase of underground cable systems, it isconvenient to utilize an effective thermal

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resistance for the earth portion of thethermal circuit which includes the effect

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of the loading cycle and the mutual heat-ing effect of the other cable of the system.

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All cables in the system will be consideredto carry the same load currents and to beoperating u nder the same load cycle.

The system of nomenclature employedis in accordance with that adopted by theInsulated Conductor Committee as stand-ard, and differs appreciably from th at usedin many of the references. This systemrepresents an attem pt to utilize in so faras possible the various symbols appearingin the American Standards AssociationStandards for Electrical Quantities, Me-chanics, Heat and ThermoDynamics,and Hydrau lics, when these symbols can

be used withou t am biguity . Cer tainsymbols which have long been used by

cable engineers have been retained, eventhough they are in direct conflict withthe abovementioned standards.

N omenclature

(A F)= attainment factor, per unit (pu)A s= crosssection area of a shielding tape

or skid wire, square inchesa thermal diffusivity, square inches per

hourClconductor area, circular inches

= distance, inches12etc. = from center of cable no. 1to center

of cable no. 2 etc.

dn etc. = from center of cable no. 1 toimage of cable no. 2 etc.duetc. = from center of cable no. 1 to a

point of interference

du etc. = from image of cable no. 1 to apoint of interference

D diameter, inchesD0 inside of annular conductorDc outside of conductorDi outside of insulationD$~ outside of sheathDSmmean diameter of sheathDl= outside of jack etD$' effective (circumscribing circle) of

several cables in contactDp inside of duct wall, pipe or conduitDe =diameter at s tart of the earth portion

of the thermal circuitDx fictitious diameter at which the effectof loss factor commences

2= line to neutral voltage, kilovolts (kv)tcoefficient of surface emissivityr specific inductive capacitance of insula-

tion/ = frequency, cycles per secondF, Fint products of ratios of distancesF{x) derived Bessel function of x (Table

III and Fig. 1)G geometric factorGi applying to insulation resistance (Fig. 2

of reference 1)Gzapplying to dielectric loss (Fig. 2 of

reference 1)

Gj>=applying to a duct bank (Fig. 2)I= conductor current, kiloamperesks= skin effect correction factor for annular

and segmental conductorskp=relative transverse conductivity factor

for calculating conductor proximityeffect

I= lay of a shielding tape or skid wire, inchesi

L depth of reference cable below earthssurface, inches

Z,6= depth "to center of a duct bank (orbackfill), inches

(//)load factor, per unit(LF) loss factor, per unitn number of conductors per cable

n f number of conductors within a stateddiameterN=number of cables or cable groups in a

system__

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P perimeter of a duct bank or backfill,inches

cos power factor of the insulationq$ ratio of the sum of the losses in the

conductors and sheaths to the lossesin the conductors

qeratio of the sum of the losses in theconductors, sheath and conduit tothe losses in the conductors

R electrical resistance, ohmsRdc = dc resistance of conductorRae tota l ac resistance per conductor

Rs dc resistance of sheath or of theparallel paths in a shieldskid wireassembly

Rthermal resistance (per conductor losses)thermal ohmfeet

Ri ofinsulationRj af jacketRsd= between cable surface and surrounding

enclosure

Paper 57-660, recommended by the AIEE InsulatedConductors Committee and approved by the AIEETechnical Operations Department for presentationat the AIEE Summer General Meeting, Montreal,Que., Canada, June 24-28, 1957. Manuscriptsubmitted March 20, 1957; made available forprinting April IS, 1957.

J. H. N e i i e r is with the Philadelphia ElectricCompany, Philadelphia, Pa., and M . H. M c G r a t his with the General Cable Corporation, PerthAmboy, N. J.

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