Table I

Peak Per-Cent LifeKva Lost per Cycle

60 . . ............. 0.3127055 . . .............

0.0320550 . . ............. 0.0034845 . . ............. 0.0003540 . . ............. 0.00004

Table 11

Peak Frequency Per-Cent LifeKva per Year Lost per Year

60. .............. 1 0.3127055 . 2............... 0.0641050 . 3............... 0.0104445 . 6............... 0.0021040 ... . ...........10.0.00040

Total: 0. 38974

indicate that not only the magnitude ofthe annual peak load, but also the shapeof the annual peak load cycle, is importantin determining the thermal aging of theinsulating system.

It was found that the significant por-tion of the thermal aging occurs over arelatively small portion of the annualload cycle. For example, Table IIshows that 80% of the total annualaging occurred during the annual peakload eycle. Table IV shows that forthe annual load cycle based on thesample load eycle having a larger loadfactor, 85% of the total annual agingoccurred during the annual peak loadcycle. A more detailed breakdown of

Table Ill

Peak Per-Cent LifeKva Lost per Cycle

60 . .7.1170355 . .0.5450350 ..0.0422445 . .0.0036540 . .0.00033

Table IV

Peak Frequency Per-Cent LifeKva per Year Lost per Year

60 .1. 7.1170355. 2. 1.0900650. 3. 0.12672456. 0.0219040 . 1. 0.00330

Total: 8.35901

the occurrence of significant aging chowsthat significant thermal aging occurredover 0.7% of the annual load cycle shownin Table II and over 2.1% of the annualload cycle shown in Table IV.

It was also found that the daily loadcycle having a peak load of 60 kva anda load factor of 0.521 would result inthe same degree of thermal aging as adaily load cycle having a peak loadof 54 kva and a load factor of 0.648.Even though the second load cycle has apeak load 10% less than the peak load ofthe first load cycle, both load cycles re-sult in the same degree of thermal aging.This further illustrates the point that themagnitude of the peak load alone cannot

be used as a measure of the degree ofthermal aging; the shape of the load cyclemust also be taken into consideration.

Conclusion

Using a method such as that outlinedin this paper, it is possible to estimatethe per-cent loss of life (and hence thethermal life expectancy) of a distribu-tion transformer from actual load cycledata and to evaluate the effect of heavierloading schedules on the thermal life ofthe transformer insulating system.

In the course of developing this methodof calculating transformer thermal lifeexpectancy, it was found that:

1. The significant portion of the thermalaging occurring during an annual loadcycle takes place over a relatively smallportion of the annual cycle.

2. A large portion of this significant agingoccurs during the annual peak load cycle.

3. Both the magnitude of the annual peakload and the shape of the annual peak loadcycle must be considered in life expectancycalculations.

References

1. THERMAL AGING PROPERTIES OF CELLULOSEINSULATING MATERIALS, G. Malmlow. ActaPolytechnica, Stockholm, Sweden, (Electrical Engi-neering Series), vol. 2, 1948, pp. 7-67.

2. ELECTRICAL INSULATION DETERIORATIONTREATED AS A CHEMICAL RATE PHENOMENON,T. W. Dakin. AIEE Transactions, vol. 67, pt. I,1948, pp. 113-22.

3. LIFE EXPECTANCY OF OIL-IMMERSED INSULA-TION STRUCTURES, W. A. Sumner, G. M. Stein,A. M. Lockie. Ibid., pt. III (Power Apparatusand Systems), vol. 72, Oct. 1953, pp. 924-30.4. GUIDING PRINCIPLES IN THE THERMAL EVALUA-~TION OF ELECTRICAL INSULATION, L. J. Berberich,T. W. Dakin. Ibid., vol. 75, Aug. 1956, pp.752-61.

Economics and Power Transrormer

DesignT. H. PUTMAN

MEMBER IEEE

The function to be performed is, of course,the transformation of electric energy.Quantitatively a measure of this is theannual volt-ampere hours, which may beexpressed as:

Annual volt-ampere hours = ET1-IMID 8,760(1)

Summary: This paper describes theconstraints which economics places on thedesign of power transformers. The mathe-matical analysis shows how the size, losses,reactances, and power output are relatedwhen the transformer is optimally designed.Because of the number of simplifyingassumptions which have been made, thispaper is not a treatise on how to designpower transformers. On the contrary,it is a broad view of the design problemwhich yields findings which are new and

which seem to be borne out by theexperience of transformer design engineers.The paper confirms the validity of proposi-tions which previously had only empiricalbacking.

G ENERALLY speaking, the problemj of designing a power transformer is

one of designing a transformer which willperform its function at minimum cost.

Paper 63-918, recommended by the IEEE Trans-formers Committee and approved by the IEEETechnical Operations Department for presentationat the IEEE Summer General Meeting and NuclearRadiation Effects Conference, Toronto, Ont.,Canada, June 16-21, 1963. Manuscript sub-mitted January 2, 1963; made available for print-ing April 8, 1963.

T. H. PUTMAN is with the Westinghouse ElectricCorporation, Pittsburgh, Pa.

The author wishes to acknowledge the valuablesuggestions which he received during the courseof this work from Dr. Clarence Zener, Director ofthe Westinghouse Research Laboratories, and E.C. Wentz of the Westinghouse TransformerDivision.

Pu0tmannEconomnnics and Power Transformer Design1018 DECE-MBER 1963

where

E=volts, output voltage

IM-=amperes, maximum output currentLD = load factor, ratio of average current

to maximum current

Fig. 1. Geometry consideredfor the analysis

A

The cost of a transformer includes atleast three terms which should be con-sidered. The first of these is the cost ofmanufacturing, installation, taxes, andmaintenance-which may be expressed asa single annual cost as follows:

$T-dollars, present worth of cost of manu-facture, installation, taxes, and main-tenance

FT= $/$-yr (dollars per dollar-year), capitalrecovery factor

FT$T =$/yr (dollars per year), annual cost

The second annual cost is the cost ofcore loss, which may be expressed asfollows:

WNL = watts, no-load loss at rated voltage$E = $/watt-yr, cost of electric energy

$s w = $/watt, present-worth cost of systemto supply power to transformer

FS= $/$-yr, capital recovery factor forpresent worth of system required tosupply power

WTNL ($E+FS$S1W)=$/yr, annual cost ofsupplying no load loss

The third component is the cost ofcopper loss, which is computed in a man-ner similar to the core loss.

-A amperes, maximum value of rms loadcurrent during year

LS = loss factor, ratio of average value ofsquared current to ',2

RT=ohms, transformer winding resistancereferred to winding in which IM flows

LSIM2RT= watts, average valuie of copperloss

FS$s;wIM2RT+$SBLSIM2RT= $/yr, annualcost of copper loss

The total annual cost of the trans-former is the sum of the three cost termsdefined above, or

FT$T+ WNL($E+FS$SIW)+IM2RT(FS$s/W+LS$E) (2)

The specific cost of the transformer, C,can now be defined as the ratio of theannual cost as given by equation 2 tothe annual volt-ampere hours as given byequation 1:

C= [FT$T+($B+FS$SiW)WNL+(LS$E+ FS$S,W)IM2RT] lETIMLD 8,760

(3)

It is the specific cost given by equation3 which designers try to minimize. Theremainder of this paper is concerned withan analytical approach to the designproblem.

Specific Cost as a Function ofTransformer Size and Load

Equation 3 is the expression for thespecific cost of the transformer. If oneevaluates the economic factors in equa-tion 3, which relate to the cost of utilizingand supplying electric power, and if healready has the transformer he intends touse, then it is a simple matter to deter-mine what load would produce the mini-mum annual cost per volt-ampere hour.It is now desired to remove the constraintthat the transformer has already beendesigned; this introduces additional de-grees of freedom in equation 3, because$T, WNL, and RT (all of which appear inequation 3) depend upon the design.These quantities must now be expressedin terms of design parameters. In orderto make the problem tractable, severalassumptions are necessary which will be-come evident in the following develop-ment.

Fig. 1 shows the idealized transformerconsidered here and the dimensions whichdetermine its physical size. Appendix Ishows that the core and coil areasand volumes can be expressed in terms ofone dimension which is taken to be thewidth of the tongue t and several otherdimensionless numbers or geometric fac-tors which are functions of the propor-tions of the transformer and are generallydesignated by the symbol Sj. The dimen-sion t is a measure of the size of the trans-former. All transformers with the same

Section B-B

Section A-A

geometric factors but with different valuesof t will be geometrically similar. Al-though the relationships given in Appen-dix I have been derived for the trans-former shown in Fig. 1, they are equallyvalid for any transformer.

It now remains to express $T, WNL, andRT in terms of the design parameters andmaterial properties. In Appendix II theinstalled cost of the transformer, $T, iSgiven as follows:

ST= (k3diS1 k2S2)t3+$etc (4)

The essential feature of this expressionis that the installed cost of a transformercan be expressed approximately as aconstant term, $etc, plus a term whichvaries as the transformer volume, P. Onecan check with Appendix II to find outthe exact significance of the variousconstants.

In Appendix III the iron and copperlosses of a transformer are expressed by:

Pi=wdiVi=wdiS1t3

p VPch / 4W A22 \A C2 / CBA i

= .SW I2B2

(5)

( )t-I (6)S32S42}

In equation 5 w is the iron loss in watts/kg(kilogram), dt equals the density of theiron, and Vi is the volume of the iron. Itis assumed, therefore, that the iron lossis proportional to the volume of the trans-former, P.

Equation 6, which expresses the copperor load loss, looks a bit more formidable,but in reality is easily understood. Thefactor (pVI/A,2) is simply the resistance ofa single turn which occupies the same

Putman-Economics and Power Transformer Design 1019DECEMBER 1963

IM.

c0< c,<c2<C3

Fig. 2. Load current versus size with thespecific cost as the parameter

space as the actual transformer windingsand which has a resistivity p. The totalvolume of the transformer coils is V,,and the window cross-section area is A,.In the factor (4W/wBAi), the angularfrequency is wo radian-seconds-', peakflux density is B webers/meter2, the corecross section is A im2, and the transformeroutput is W volt-amperes. The peakvoltage per turn of the transformer iswBA j, and therefore (4W/cwBA i) can beshown by simple arithmetic to be twicethe current which passes through thetransformer window in either winding.Consequently the expression for P, issimply the loss of a hypothetical single-turn winding which has the same averagecurrent density as the actual trans-former windings.

It is assumed that the no-load loss,WNL, equals the core loss, Pi. There-fore,

TVVL=Pi=WddjSlt3 (7)Equation 6 may be used to evaluate RL:

R=PC 1 8pW2 S2RL'2 IM2 \w2B2 (S2S42St (8)

The expressions for $T, WNL, and RLgiven by equations 4, 7, and 8, respec-tively, may now be substituted into equa-tion 3 to yield an expression for the spe-cific cost, which depends both upon theload IM and design of the transformer.

C= {FT[(k3diSi +k2S2)t3+$etc] +

wdiS1t( FS$s,w+$E)+

(8pET2IM2)\( S2 \t-5(Fs$s+w02B2 / k32S42) -(ssw

LS$E)}/{ETIM 8,760 LD (9)

C= { [FT(k3diSl+k2S2)+

(FS$s,w+$E)wdiSolt3+ FT$etc+

(Fs$siw+Ls$SE) ( 2B2-) X

(S32942) t~ / {ETIM8,760LD (10)

The numerator of equation 10 shows thatthere are two types of costs: one depend-ent, the other independent of the load.The cost which depends on the load is

a result of copper loss and shall becalled "copper-loss cost." Those whichare independent are of two types. Thefirst is dependent upon the volume of thetransformer and shall be called "fixed-volumetric cost," while the second is in-dependent of volume and shall be called"fixed costs."Assuming that the geometric factors of

the transformer are kept fixed, we wishto see how the transformer size, t, and theload, IM, influence the specific cost ofoperation. One way to discover this isto plot the relationship defined by equa-tion 10 between IM and t for fixed valuesof C as shown in Fig. 2. Fig. 2 showsthat there is no one combination oft and IM which produces minimumspecific cost. However, it is seen that ifthe transformer size is fixed then there is avalue of IM which gives minimum specificcost. However, at that value of IM thereis another, larger, transformer which willhave even lower specific cost.

Evidently there are two points of viewfrom which to choose. Assuming thatthe size is fixed, the load which will giveminimum specific cost can be found if

1C-=0If the indicated mathematics is carriedout, the condition for minimum specificcost is found to be

copper-loss cost } = { fixed-volumetric cost+fixed costs} (11)

The other point of view is to assume theload is known and to find the transformersize which results in minimum specificcost by requiring that

0=0

The result of this calculation shows thecondition for minimum specific cost is

{copper-loss cost } = 3/5fixed-volumetric cost } (12)

Significance of the MinimizingConstraintFrom the above discussion the relation-

ship between transformer size, load, andspecific cost should be clear. In thefollowing it is assumed that the trans-

former size chosen will minimize itsspecific cost for a specified load. This re-quires imposition of the constraint givenby equation 12. By identifying the fixed-volumetric and copper-loss costs in thenumerator of the right-hand member ofequation 10, one can write the minimizingconstraint in detail.

(Fs$s,w+Ls$E) (8pET ( s2 5

w02B2 S32S42/

=3/5{ FT(k3diSl+k2S2)+(Fs$s,w+$E)wdtSI}t3 (13)

Inspection of equation 13 shows that theconstraint is one which relates the trans-former size, t, to the load, to certain mate-rial properties, to operating conditions,and to economic factors. From this con-straining equation a number of interestingfacts concerning power transformers canbe deduced.

VARIATION OF TRANSFORMER VOLUMEWITH RATING

If one imagines equation 13 to be solvedfor t, then it is evident that

t a (ETIm)114 cc W1/4

The volume of the transformer varies as t3,and therefore the volume of a transformermust increase as its rating to the three-fourths power.

INFLUENCE OF CHANGES UPON DESIGN

If the size, t, is considered the measureof the design, equation 13 shows the in-fluence of various factors. Generally,any change which tends to increase thecopper-loss cost is counteracted by anincrease in the size, which tends to reducethe cost again. Also, any change whichtends to increase the fixed-volumetriccost is counteracted by a reduction in size,which tends to counteract the increase.

COPPER LosS AND IRON Loss

From equation 13 and the expressionsobtained for the copper and iron lossesin Appendix III, we can easily show thatthe ratio of copper loss to iron loss is

PC

Pi

3/5 1FT(k3dS0 +k2S2)+(FS4SW+$E) wdiS1JwdiSo[Fs$s,w+LS$EI

If the loss factor is unity, then

Pc 3 fixed-volumetric cost

Pi 5 core-loss cost

It is apparent, therefore, that the copperloss to iron loss ratio is a constant (in-dependent of transformer size) and de-

Putman-Economics and Power Transformer Design1020 DECEMBER 1963

pends upon economic and geometricfactors and the watts/kg loss of the iron.

INFLUENCE OF THE VOLTAGE LEVEL

In the derivation of the copper loss ofAppendix III it was assumed that thetranformer coils were made of a homo-geneous material with a resistivity p andthat the current was uniformly distributedacross the cross section of the windings.In reality, part of the cross-section area isinsulation; therefore the current does notflow uniformly. However, as far aslosses are concerned,the real situation canbe represented by a single-turn homo-geneous coil with an equivalent resistivity,which is related to the actual resistivityof the conductor by the expression p/-(1-f) where p is the actual conductorresistivity and f is the fraction of thewinding cross-section area which is insula-tion. This can easily be seen on an intui-tive basis by considering the limiting caseswhen f is zero and one.As the effect of increasing the voltage

level of a winding is always to increase thefraction of the area used for insulation,the equivalent resistivity of the windingmust increase also. Since this tends toincrease the copper-loss cost, the designmust react in such a way as to reduce thecopper loss; that is, the transformer sizemust increase. The over-all effect ofincreasing the voltage level will be toproduce a larger transformer with a higherspecific cost.

WINDING CURRENT DENSITY

It has been shown that the ratio ofcopper loss to iron loss is fixed. Since theiron loss varies with the volume of thetransformer, the copper loss must do soalso. The only way this can happen isfor the winding current density to be con-stant. This explains why the coolingproblem becomes more critical in largetransformers. The amount of heat tobe dissipated increases as the volume, t3,while the surface area to dissipate theheat increases as t2. Therefore, the heatdissipation per unit area must increasewith transformer size.

VARIATION OF PER-UNIT REACTANCESWITH POWER RATING

The base ohms of a transformer referredto any winding varies as the square ofthe number of turns of that winding.One can imagine that the magnetic

field produced by the winding consists of anumber of tubes of flux and that the totalinductance of the winding is the sum ofthe inductance resulting from each tubeof magnetic flux. The inductance pro-

duced by a single tube varies as the squareof the number of turns of the winding andas the permeance of the path formed bythe tube. Since the base ohms varies asthe square of the number of turns of thewinding, the contribution to the reactance,expressed in per unit, resulting from anytube of flux varies only as the permeanceof the tube. As the size of the trans-former increases, the permeance of thetubes of flux increases in direct proportionto the size, t, because permeance variesdirectly as area and inversely as length.Therefore, the per-unit reactance resultingfrom a tube increases in proportion to t.Some of the tubes may be considered

to contribute to the exciting reactanceand others to the leakage reactance.Since the per-unit impedances of the tubeswhich make up these reactances vary as t,the reactances themselves must vary as t.Equation 13 shows that the transformersize, t, varies as the power rating to theone-fourth power. Therefore, the re-active components of the per-unit excitingand leakage impedance must also varyas one-fourth power of the rating.

It is well known that the per-unitexciting current of transformers dimin-ishes as transformer size increases. How-ever, it is not so well known that the per-cent leakage reactance will also increaseunless design changes are made to counter-act this undesirable effect.

OPTIMUM TRANSFORMER PROPORTIONS

In the foregoing analysis it was assumedthat the geometric proportions of thetransformer were fixed. It is of interestto note that, although the optimum sizedepends upon the loss factor, LS, and isindependent of the load factor, LD, theoptimum geometric proportions are in-dependent of both the load and loss fac-tors. This is most easily shown by start-ing with equation 10 and computing theminimum value of the specific cost byinserting the value of t given by equation13. The result has the following form:

Cmin= {( ... )(S2S3-2S4-2B-2)3.8[FT(k3diSl +±k2S2) +( FS$S/W+$E)

wdtSl 6/8 +FT$,etc } /{ ... ( 14)

where the factors which have beenomitted involve the load and loss factorsbut do not involve any geometric factors.Now it is clear that the set of values forthe geometric factors which produce theoptimum value of the specific cost isindependent of load or loss factors.

OPTIMUM VALUE OF FLUX DENSITY

It was also assumed in the analysisthat the flux density, B, was held fixed.One might possibly decrease the specific

cost of a transformer by using a value of Bother than the maximum possible for thecore material as determined by excitingcurrent or noise. Equation 14 may beused to look into this problem. Firstof all,it isnoticed that increasing the valueof B would reduce the specific cost exceptfor the fact that as B increases the coreloss, w, also increases, which tends toincrease the specific cost. It is apparentthat the flux density should be increasedup to the point that any further improve-ment in the specific cost by increased fluxdensity is undone by increased core loss.The location of this point can be deter-mined by examination of the expression

U= B -4[FT(k3diSl+k2S2)+(Fs$sIw+$E)diS1wI5/8 (15)

which comes from equation 14. Theflux density should be as high as possiblewithout violating the constraint d U/dB <0. Differentiation of equation 15 showsthat this constraint is equivalent to

6< FT(k3diSl4-k2S2) wdw/dB. 1 1+- (16)5 \B( Fs$s,w+$E)diSl / B

As an example of the use of equation 16,the optimum flux density for a typicaldesign will be deternmined.

FT=0.15 $/$-yr

FS$S8W +$E 0.006 $/kilowatt-hour= 0.0525 $/watt-yr

di= 7.8X 103 kg/mneter3k2= 0.075 $/inch3=4.58X 103 $/meter3

k3 = 0.30 $/pound = 0.661 $/kgS1= 5.57 (assume a1= a2= a3= 1)

S2= 7.14 (assume al = a2=a3 = 1)

Using these values, equation 16 be-comes

dw 4.84 w-< +1.2-dB B B (17)

For a particular modern transformersteel, equation 17 is satisfied as anequality for B = 1.67 webers/meter2, w=2.27 watts/kg, and dw/dB=4.53 watt-meter2/kg-weber. Therefore, B = 1.67webers/meter2 is optimum for this par-ticular example. Furthermore, becausedw/dB changes so rapidly in the range1.60<1.75 webers/meter2, it is likely thatthe optimum flux density will lie in thisrange for almost any design.

Results

The results of this paper can be sum-marized by a series of statements of thefacts which have been analytically shownto be true about power transformers:

Putman-Economics and Power Transformer Design 1021DECEMBER 1963

1. Given the load to be supplied, a trans-former size can be determined which has aminimum specific cost. An increase inthe load reduces the specific cost up to acertain point. However, for this newload there is a still larger transformer whichwill be more economical.2. The volume of the optimum trans-former should increase as its power ratingto the three-fourths power.

3. Any change which tends to increasethe fixed-volumetric cost will be counter-acted by a reduction in size. Any changewhich tends to increase the copper-losscost is counteracted by an increase in size.4. The copper loss and iron loss bear afixed ratio independent of transformerrating.5. High-voltage transformers have a higherspecific cost than low-voltage transformersof the same rating.

6. The winding current density is inde-pendent of transformer size.7. All per-unit reactances of transformerstend to increase as the rating to the one-fourth power. Apparently the leakagereactance is kept in bounds by changes ingeometry and winding design in largertransformers.8. The optimum proportions of a trans-former are independent of its load.9. The optimum flux density for mosttransformer designs probably lies between1.60 and 1.75 webers/meter.2

Appendix 1. Geometry ofTransformer

Fig. 1 shows the transformer consideredin this paper. The important geometricconstants of the transformer in terms ofits dimensions are as follows:Mean turn length:

aIt -2(d 'rt) +r27r 2 = 2(d+t)+ 7ra ( 18)2

Mean flux path:

If= 2(b+a)+27r - =2(b +a)+ - t (19)4 2

Volume of coils:

V, = abl t= ab [2(d+t)+ ra] (20)

Volume of iron:

Vi =2d -, 1J= dt 2(b+a)+ 2 t] (21 )

Coil area:

A,c-ab (22)

Iron area:

At=dt (23)

The following are the dimensionlessgeometric factors:

a =d/t

a2 = b/a

a3= a/t (24)

By usinig equations 24, the coil and ironvolumes and areas can be expressed interms of t and geometric factors.

-VC2a32[2(al+1)+7ra31t3= S2t0

Vz= a, [2(a2+1)a3+7r/2]t3= Slt'

A c= a2a32t2 S3t2

A s=at2S4t2 (25)

Several new geometric constants are intro-duced in equations 25 whose definition isself-evident.

Appendix 11. Evaluation of theInstalled Transformer Cost

The installed cost of the transformer is$T. This cost can be expressed as the sumof the cost of the core, the cost of thewinding, and certain miscellaneous costs.

(1) Cost of core:

k3=$/kg, cost of iron

di=kg/meter3, density of iron

Vi=tmeter3, volume of iron

We assume that the cost of the core variesdirectly with the core volume. Therefore,

Core cost = k3djV-=k3d jSt3where Vi is expressed in terms of SI and tas shown in Appendix 1.

(2) Cost of coils:It is assumed that the cost of the trans-

former windings varies in proportion totheir total volume.

k2= $/meter3, per-unit volume cost of coils

V= meter3, volume of coils

Therefore,

Coil cost = k2 V,= k2S2t3

where Vc is expressed in terms of S2 and tas shown in Appendix I.The installed cost of the transformer

can now be expressed in terms of the costcomponents which have been defined.

$T=k3d1SiI3±k2S2t3±$etcwhere $ete is the miscellaneous charge.

Appendix Ill. Calculation ofLosses

Iron LossLet w equal the watts/kg loss in the iron,

and Pi equal the total watts loss in the iron.Then

Pi=wd1 Vi-=wdiSt3 (26)

where the expression for Vi is obtained fromequation 21 of Appendix I.

Copper LossLet

j=amperes/meter2, current density in thecoil (peak value)

p=ohm-meters, resistivity of coil material

P,= watts, copper lossV= meter3, volume of copper

then

1PC== -j2pVC2 (27)

The current density, j, can be related tothe volt-ampere rating, W, in the followingmanner. LetW= volt-ampere ratingAi=meter2, cross section of iron

B=webers/meter-2, maximum flux density(peak value)

= radian-second -1, angular frequencyAe=meter2, cross section of copper

W= j-ccA iB2 2

j =4W/wBA cA i= 4Wl/wBS4S3t4 (28)Therefore,

Pc= (8p12/co2B2)( VC/AC2A i2)

Pc= (8pW2/c,2B2)(S2/S32S42)t-5

(29)

(30)where the volumes and areas are expressedin terms of t and the geometric factors aredefined in equations 25 of Appendix I.

Discussion

Paul H. Jeynes (Public Service Electricand Gas Company, Newark, N. J.): Theprocedure described in this paper maywell be accepted as the classical approachto economic transformer design. It pro-poses no drastic new departures, but doesorganize the reasoning better than has beendone before.

Because it is thus authoritative, it isdesirable to have the numerous formulasexpressed in unmistakable dimensionalunits, so they can be conveniently quanti-fied in practical applications of the analysis.The author has recognized this, and hastaken pains to indicate the units of measure-ment along with the definition of each termin words.However, his presentation starts with

definitions of three basic concepts:

(1) $T=dollars, present worth of cost ofmanufacture, installation, taxes,and maintenance

(2) Fr=$/$-yr, capital recovery factor

(3) F7$T=$/yr, annual cost

One's first reaction is that: (1) is incorrect,(2) is inadequate, and (3) therefore is notvery helpful. This situation could beremedied, but it demands some explanation.

It is implicit in this approach, though notspecifically stated, that the user is goingto make a choice between proffered designswhich, at the price purchasable frommanufacturer, will accomplish two simul-taneous financial objectives:

1. Maximize his profit margin out ofgiven sales and revenues from his customers.

Putman-Economics and Power Transformer Design1022 DECEMBER 1963

2. Permit minimizing the price he mustcharge his customers (revenues) for givensales, to obtain a given profit margin.

In other words, the user of a transformerlooks to the effect of his purchase of aunit on the revenues he must obtain fromhis customers, whether he is a publicutility or a nonregulated business. Ac-cordingly, he does not care what it coststo manufacture the unit; he needs to knowwhat he must pay the manufacturer.

His purchase price may be a function ofthe cost to manufacture, but it is not oneand the same figure, as the definition of$T seems to say.

Also, purchase of a transformer meansthat the user must collect from his cus-tomers a good deal more than: its purchaseprice, installed, plus present worth oflifetime taxes, plus present worth of life-time maintenance. He also incurs engi-neering, accounting, administrative, super-visory, and storeroom expense. He hascosts of advertising, billing, and collecting.And he has working capital requirements.All this is to be obtained via revenues fromhis customers.We all know the answer-FT$T is not

"annual costs," as defined in the paper.It is the difference in revenue requirementsresulting from purchase and use of trans-former. The items omitted from thedefinition of FT$T are assumed to beidentical regardless of transformer design.Why be stuffy about this?Because sometimes items conveniently

ignored, being assumed "the same in anyevent," are not really the same in anyevent; which points up the fact that FT,capital-recovery factor, is inadequatelydefined.The ultimate net salvage of a trans-

former is typically large, whether the unitis sold for future use by another owner oris scrapped for its large metal content.Net salvage value is not a revenue require-ment; it does not come from customers,but from the junkman. Accordingly, thedefinition of FT must indicate whether ornot adjustment has been made for ultimatenet salvage. It may amount to 25% of theunadjusted figure, which makes a differ-ence, not only in the present-worthingcalculation, but also in the estimate ofincome tax.Revenue requirements, to come from

customers, include:

1. Minimum acceptable return on capitalinvestment (i.e., on purchase price, includ-ing installation cost).2. Depreciation (i.e., recovery of thatinitial investment less ultimate net salvage).3. Taxes.

4. Maintenance.

The present worth of items 1+2 (therevenue requirement for return and de-preciation) is exactly equal to the initialcapital investment if (1+2) is divided bythe capital-recovery factor adjusted forsalvage.The present worth of items 3+4 is equal

to (3+4) divided by the capital-recoveryfactor not adjusted for salvage.

Note also that item 1 is neither averagenor actual nor desired nor permitted per-centreturn. It is the minimum acceptable"bare-bones" cost of mnoney, "thresholdof confiscation" rate of return; whetherthe user is a public utility or not. Thatmay make a difference of 16% or more(1% out of a typical figure near 6%) initem 1.

Another slightly confusing usage is thesymbol $etc. In Appendix II it is referredto as "the miscellaneous charge;" but itis not what accountants refer to as amiscellaneous expense. It is said (inequation 4) to be a specific fixed componentof $T, which is in terms of present worth.The quantity $s,w, described as "present

worth of system to supply power to trans-former" cannot be said to be completelywrong; but it is an inadequate definitionof the author's exact intent, which defiespractical evaluation. It would seem de-sirable to include at least a reference,indicating exactly how the author intendedit to be quantified.

In general, this paper is so valuablethat it would seem worthwhile to makesuch small but important refinements.

T. H. Putman: Mr. Jeynes' discussioinserves to bring to the forefront a numberof points which may not be clear to allreaders.

In the sale and application of a powertransformer there are two problems: onesolved by the user and the other solved bythe transformer manufacturer. The usermust decide from among the many trans-formers offered for sale by different manu-facturers which will be the most economical.This is a standard type of engineeringeconomy problem, and the general methodof attack on such problems can be foundin reference 1. Generally one preparespayment schedules for the various alter-native transformers. These schedules maydiffer both in the timing and amount ofpayments. The salvage value is; as Mr.Jeynes indicates, an important factor, butmay be considered as a negative paynment.Some method must be used to reduce the

various payment schedules to a commonbasis of comparison in order to determinewhich is the most attractive. This canbe done by finding the present worth ofeach schedule of payments or by reducingeach payment schedule to a series of equalannual payments to be made over the lifeof the equipment. If the lives of thetransformers can be considered to be thesame, then a direct comparison of presentworths is adequate. If the transformers'lives are different, then the equal annualpayments should be used for comparison.This is the transformer user's problem,and the solution yields the transformerwhich he should purchase.The problem of the manufacturer is as

follows: There is not just one transformerwhich he can build which will satisfy thetechnical requirements of his customer.Transformers can be built with high loadlosses and small physical size or low loadloss and large physical size and all grada-tions between these extremes. In other

words, a manufacturer can build a wholeseries of transformers to supply a specificload, and his problem is to determine whichtransformer fromi the series will be mosteconomically attractive to the customer.This paper shows a sound, scientific meansof arriving at the optimum design.As one might expect, certain modifica-

tions must be introduced to adapt themethod to any specified situation, and thesecan be easily worked out. However, thereare several requirements if the method isto be meaningful. First of all, powercompanies must be able to evaluate the costof losses-both the energy component andthe demand component. These cost com-ponents are designated by the symbols$E and Fs$s1w, respectively, and havethe units $/watt-yr. Some of the literatureavailable on this is indicated as references2 and 3.

Second, it is assumed there is a closerelationship between the price of a trans-former to a customer and its cost of manu-facture. The present worth of all costsof a transformer to a user exclusive of thecosts of losses is $T; and FT, when multipliedinto $T, converts $T into a series of equalannual payments over the life of thetransformer. $T is adjusted for salvageby subtracting from the present worth ofall costs the present worth of the salvagevalue. In other words, the salvage valuemay be considered a negative cost. Nowone part of $T is well defined: the corecost and the coil cost, which areevaluated in Appendix II. Although itis not so stated in the paper, these costsmay reflect their salvage value as well asthe cost of manufacture.

All other costs which make up $T arelumped into $,t,, and evaluation of thisis of no great importance since it does notenter the actual computation of the opti-mum transformer design.

In designing a power transformer amanufacturer can vary its physical sizeand thus influence the load losses, the corelosses, and the cost of the core and coils.As the paper shows, a variation in the sizeresults in a change in the cost of the trans-former, whether it is evaluated on anannual charge basis or on a present-worthbasis. The proper size or design is theone with minimum cost since this particulardesign will be the most attractive to apotential customer.The optimization assures the manu-

facturer that he has the best design possiblewithin his particular manufacturing coststructure. It, of course, does not assurehim of successfully meeting competitionsince this depends upon the manufacturingcost structure, which is different for differentmanufacturers.

REFERENCES1. PRINCIPLES OF ENGINEERING ECONOMY (book),E. L. Grant. The Ronald Press Company, NewYork, N. Y., 1950.

2. EVALUJAT10N OF CAPACITY DIFFERENCES IN

ECONOMIC COMPARISON OF ALTERNATIVE FACILI-TIES, Paul H. Jeynes. AlEE Transactions, pt,III (Power Apparatus and Systems), vol. 71,Jan. 1952, pp. 62-80.

3. A FURTHER LOOK AT COST OF LOSSES, C. J.Baldwin, C. H. Hoffman, P. H. Jeynes. Ibid.,vol. 80, 1961 (Feb. 1962 section), pp. 1001-08.

Putman-Economics and Power Transformer Design 1023DiECIRMBIER 1963