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APPENDIX 5B. SCALING RELATIONSHIPS IN TRANSFORMER MANUFACTURING

TABLE OF CONTENTS

5B.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-1

5B.2 THEORY AND BASIS FOR SCALING RULES . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-4

5B.2.1 Power and Voltage Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-4

5B.2.2 Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-5

5B.2.3 Output and Winding Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-10

5B.2.4 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-11

LIST OF TABLES

Table 5B.1 Common Scaling Relationships in Transformers . . . . . . . . . . . . . . . . . . . . . . . . . 5B-2

Table 5B.2 Nominal 60 Hz, Core-Type, Liquid-Filled, 12 kV Distribution Transformers . . 5B-11

LIST OF FIGURES

Figure 5B.1 Size and Performance Relationships by kVA Rating . . . . . . . . . . . . . . . . . . . . . 5B-3

Figure 5B.2 Basic Three-Phase Transformer Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 5B-12

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APPENDIX 5B: SCALING RELATIONSHIPS IN TRANSFORMER

MANUFACTURING

5B.1 INTRODUCTION

There exist certain fundamental relationships between the ratings in kVA of transformers

and their physical size and performance. A rather obvious such relationship is the fact that large

transformers of the same voltage have lower percentage losses than small units, i.e., large

transformers are more efficient. These size-versus-performance relationships arise from

fundamental equations describing a transformer's voltage and kVA rating. For example, by fixing

the kVA rating and voltage frequency, the product of the conductor current density, core flux

density, core cross-sectional area, and total conductor cross-sectional area is constant. 

To illustrate this point, consider a transformer with frequency, magnetic flux density,

current density, and BIL all fixed. If one enlarges (or decreases) the kVA rating, then the only

free parameters are the core cross-section and the core window area through which the windings

pass. Thus, to increase (or decrease) the kVA rating, the dimensions for height, width and depth

of the core/coil assembly may be scaled equally in all directions. Careful examination reveals that

linear dimensions vary as the ratio of kVA ratings to the 1⁄4 power. Similarly, areas vary as the

ratios of kVA ratings to the 1⁄2 power and volumes vary as the ratio of the kVA ratings to the 3⁄4

or 0.75 power. Hence, there is the term "0.75 scaling rule." Table 5B.1 depicts the most common

scaling relationships in transformers.

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Table 5B.1 Common Scaling Relationships in Transformers

Parameter Being Scaled Relationship to kVA Rating 

(varies with ratio of kVAx

)

Weight (kVA1/kVA0)

3/4

Cost (kVA1/kVA0)

3/4

Length (kVA1/kVA0)

1/4

Width (kVA1/kVA0)

1/4

Height (kVA1/kVA0)

1/4

Total Losses (kVA1/kVA0)

3/4

No-load Losses (kVA1/kVA0)

3/4

Exciting Current (kVA1/kVA0)

3/4

% Total Loss (kVA1/kVA0)

-1/4

% No Load Loss (kVA1/kVA0)

-1/4

% Exciting Current (kVA1/kVA0)

-1/4

% R (kVA1/kVA0)

-1/4

%X (kVA1/kVA0)

1/4

Volts/Turn (kVA1/kVA0)

1/2

The following three elements are true as the kVA rating increases or decreases if the

following conditions are met: holding constant the type of transformer (distribution or power

transformer, liquid-filled or dry-type, single-phase or three-phase), the primary voltage, the core

configuration, the core material, the core flux density, and the current density (amperes per square

inch of conductor cross-section) in both the primary and secondary windings.

1. The physical proportions are constant (same relative shape),

2. The eddy loss proportion is essentially constant, and

3. The insulation space factor (voltage or BIL) is constant.

In practical applications it is rare to find that all of the above are constant over even

limited ranges; however, over a range of one order of magnitude in both directions (say from

50kVA to 5kVA or from 50kVA to 500kVA), the scaling rules shown in Table 5B.1 can be used

to establish reasonable estimates of performance, dimensions, costs and losses. In practice, these

rules can be applied over even wider ranges to estimate general performance levels. The same

quantities are depicted graphically in Figure 5B.1 for reference.

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Size/Performance Relations vs. Rating

0.10

1.00

10.00

0.1 1 10

Ratio of kVA's

Ratio of Quantit

y

Weight, Cost, Total

Loss, No-load loss,

Exciting Current -

0.75 power

Volts/ turn - 0.5

power

Dimensions - Percent

Reactance 0.25 power

Percentage: No-load

Loss, Total Loss,

Resistance - -0.25

power

Figure 5B.1 Size and Performance Relationships by kVA Rating

.

0.75 0.75

1 1

1 0 1 0

0 0

S S NL NL and TL TL

S S

⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

0.75 0.75

1 1

10 0

0 0

S S LL = TL - NL

S S

⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠

To illustrate how the scaling laws are used, consider two transformers with kVA ratings of

S0 and S1. The no-load losses (NL) and total losses (TL) of these two transformers would be

depicted as NL0 and TL0, and NL1 and TL1. Then the relationships between the NL and TL of the

two transformers could be shown as follows:

These two equations can be manipulated algebraically to show that the load loss also

varies to the 0.75 power. Starting with the concept that total losses equals no-load losses plus

load losses, we can derive the relationship for load loss (LL), and show that it also scales to the

0.75 power. Specifically:

 LL1 = TL1 - NL1

Plugging the TL1 and NL1 terms into this equation:

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( )

0.75

1

0 0

0

S TL NL

S

⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠

0.75

1

1 0

0

S LL = LL .

S

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

v N

t

∂φ = − ∂

t

∂φ

∂

That is,

In this way, the 0.75 scaling rule can be used to derive the losses of a transformer knowing

the losses of a reference unit, if the specified type of transformer is held constant and key

parameters are fixed such as the type of core material, core flux density, and conductor current

density in the high and low voltage windings.

5B.2 THEORY AND BASIS FOR SCALING RULES 

In order to understand the origins of winding and output coefficients and related scaling

laws, it is necessary to review some basic equations and definitions. Most are lifted freely or

derived from similar material in the text, Modern Power Transformer Practice, Wiley 1979,

edited by R. Feinberg.1 No mathematics beyond elementary algebra is required, but a good deal

of implied physics and electrical engineering is required to fully appreciate these derivations.

5B.2.1 Power and Voltage Equations

The machine equation relates the induced volts, V, per phase to the number of turns (N)

the frequency (f) in Hertz, the peak core flux density Bm in Tesla, and the cross-sectional area of

the core steel (AFe) in square meters. The units are mixed to simplify the basic equations, a

common practice in transformer design texts. The machine equation is derived from Faraday's

law, which is expressed as 

 

,

where v is the instantaneous value of V, and is the derivative of changing magnetic flux

with respect to time. 

Considering V as the root-mean-square (rms) value of a sine-wave alternating current

voltage, the above equation can be converted into:

V/N = 4.44 f Bm AFe Eq. 1

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The voltage and turns may apply to either primary or secondary winding and, for the ideal

transformer with no losses and no-leakage flux, 

V1/V2 = N1/N2 = n = I2/I1

where V1 and V2 represent primary and secondary voltages respectively, N1 and N2 primary and

secondary turns, and I1 and I2 primary and secondary currents in amperes (amps). The quantity n

is referred to as the “turns ratio.” With the parameters defined, and using equation (1), the output

or transformer capacity (S) in MVA per phase can be expressed as:

S = 4.44 f Bm AFe N I Eq. 1a

The overall cross-section of primary plus secondary conductors in square meters is 

ACu = (N1a1 + N2a2) x 10-6

and, assuming current densities for primary and secondary windings to be equal, then

ACu = 2 x 10-6 Na

where “a” is the conductor cross-section in square millimeters (mm2

) of an individual turn referred

to the winding with N turns and a1 and a2 are conductor cross-sections of primary and secondary

turns respectively. As long as the winding current densities are equal, either winding may be used

as reference, provided the choice of primary or secondary is consistent. Starting with (equation

1a), using the ACu relationship explained above, and letting J represent current density in amps per

mm2

:

S = 2.22 f Bm J AFe ACu Eq. 2

Let Aw be the core window area in square meters, and kw the window space factor, as

given by 2 ACu /Aw. (Refer to Figure 5B.2 and note that in a three-phase transformer there are two

coil phases occupying a given core window). This fraction is indicative of the insulation and

cooling channel requirements. For distribution transformers, kw is found to be about 0.3-0.4 for

nominal 12 kV systems. Using these definitions,

S = 1.11 f Bm J AFe kw Aw Eq. 2a

Note that, for a given MVA rating, and specified flux and current densities, the product of

conductor and core cross-section is constant and inversely related; i.e. AFe " 1/ACu.

5B.2.2 Losses

Ideally, if the values of energy loss in Watts per kilogram (W/kg) of unit mass of the core

and windings are known, the total core and load losses (PFe and PCu) can be readily obtained. 

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i

i R

%P P = P

100

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

i i

R R iR

%P %P P 1 P kP.

100 100

⎛ ⎞⎛ ⎞ =+ = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

IZ 100

V

×

( )( ) 2 2 %R %X +

These results are accomplished by multiplying the W/kg for both core and windings by the core

mass and the conductor mass respectively (or by their volumes times material densities). 

The Department uses the convention that lower case corresponds to per-unit quantities

and upper case corresponds to total or to total-per-phase quantities. Load losses consist of

resistive (pR) and eddy (pi

) components. Expressions can be derived that express each in terms of

the conductor properties and geometry. The fraction of eddy losses plays an important role and

can be expressed as 

%Pi = 100 Pi/PR, or 

Ignoring stray loss, (which is associated with eddy losses), let Pt represent total load loss

for a three-phase transformer. That is,

Pt

 = 3PCu

Also assume the same eddy loss fraction in primary and secondary windings.

PCu = PR+ Pi = PR +

Closely associated with the load loss of a transformer is its impedance. When the load

loss of a given transformer is determined by test (the wattmeter reading in the test circuit), that

same test also provides the value of the impedance (the voltmeter reading in the test circuit). 

Impedance in a transformer is expressed in terms of the “impedance voltage,” which is defined as

“the voltage required to circulate rated current through one of two specified windings of a

transformer when the other winding is short-circuited, with the windings connected as for rated

voltage operation” (IEEE C57.12.80).

For convenience, “percent impedance,” %Z, is used to describe the impedance voltage of

a transformer. In accordance with the definition given above,

%Z = 

that is, when related to the primary or secondary winding of a transformer, the percent impedance

is the percent voltage drop due to impedance when rated current flows through the respective

primary or secondary winding of the transformer. 

The %Z may be represented by its resistive and reactive components, %R and %X, as

%Z = 

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2 22 2

6

I R 10 I R 10

I V 3S 10

× × = × ×

4

t 10 P %R

3S

−

∴ =

2 IR 10 %R .

V

× =

Therefore, we can express percent resistance (%R) as follows: 

Note that R in the numerator must represent the total resistance in the transformer

windings. Therefore, if the transformer is being viewed from the primary terminals, the value of R

would be the total resistance of the primary winding, plus the total resistance of the secondary

winding referred to the primary winding, (R2(N1/N2)

2

).

Where the percent impedance, percent reactance, and percent resistance are related to the

voltage across the primary or secondary winding of a transformer, the percent load loss (%I2

R) is

related to the MVA capacity of the transformer, stray loss being ignored as stated previously. 

Multiplying numerator and denominator in the above equation by I, and letting Pt

represent total load loss in watts and S represent the megavoltamperes (MVA per phase) rating,

we determine the percent load loss as:

Percent load loss = 

 

Thus, an expression of %R is equivalent to indicating the transformer's load loss.

From equation (2), it is evident that once the core flux density and current density are

fixed, the transformer rating is dependent upon the core cross-section and window area. Next, we

will derive information about the window shape.

In a detailed discussion of the reactance, the electrical characteristics would depend on: 

• the ratio of winding height (h) to the winding mean turn (s), and

• the ratio of the cross-sectional areas of the core and conductor (AFe /ACu).

The mean value of s (a linear measurement, recording the circumference), is given by the

equation s = (s1 + s2)/2, where s1 is the mean turn of the primary winding and s2 is the mean turn

of the secondary winding.

These ratios, together with the necessary space factors for insulating and cooling

clearances, establish the relative volumes of the core and conductor. Consequently, if fixed values

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i

R iR

%P 1 P kP

100

⎛ ⎞ ⎜ ⎟ + = ⎝ ⎠

2 2 P (I R I R )k , Cu 1 1 2 2 i = +

1 1

1

1

N s R ohms,

a

ρ =

2 2

1 11 2 2 2

Cu i

1 2

I Ns I Ns P k

a a

⎛ ⎞ ρ ρ ∴= + ⎜ ⎟ ⎝ ⎠

11 2 2

Cu i

1 2

Is Is P IN k

a a

⎛ ⎞

∴= + ρ ⎜ ⎟ ⎝ ⎠

Cu

i

P J

2INs k

∴ =

ρ

for the specific loadings and therefore specific losses for core and conductor can be assumed, the

ratios of core loss and load loss are established.

The following application of relationships derive an expression relating the flux and

current densities. The expression starts with:

PCu =

where subscripts 1 and 2 indicate primary and secondary windings, respectively. The resistance

per phase of the primary winding is given by 

where a1 is the cross-sectional area of the primary copper conductor, and D is the resistivity at full

load operating temperature of the conductor, 21.4 x 10-3 ohm - meters. The value of R2 is

similarly obtained:

where IN is the ampere-turns in either winding. As before, the assumption of equal current

densities in the windings is made, driven by the condition for minimum I2R loss. Accordingly, 

 PCu = 2INJsD ki

 ,the current density equation. , 

Multiplying equation (1) by I and rearranging algebraically we get

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m Fe

VI IN

4.44fB A

= .

6

m Fe

10 S IN

4.44fB A

∴ =

6

m Fe Cu

i

104 10 fB A P J

k sS

− × = .

2

Cu

Cu

P 10 %P

S

× =

2

Cu Cu

3

P 10 0.1P

S 10 S

× = = × .

6 6

m Fe Cu m Fe

Cu

i i

104 10 fB A S %P 1040 10 fB A J %P

ksS 0.1 ks

− − × ×

= ×= ×

It was established earlier that S is the rating per phase in MVA, that is VI = 106S. Thus:

Using the current density equation, substituting the resistivity value for D, and the above

value for IN, we can derive that:

The watts of conductor loss (for copper) can be expressed as a percentage of the

transformer MVA rating:

or in kilowatts:

%PCu

By substituting in the revised equation for J (amperes per square meter), we get

 Eq. 3

If aluminum windings were used instead of copper, a value of 655 would be substituted

for 1040. The expression assumes equal J in both windings and that both windings are made of

the same material. The losses are expressed at operating temperature.

If J and Bm are chosen independently, the transformer will have a natural value of

conductor loss depending upon the ratio AFe/s. Conversely, if losses are specified, the choice of J

is determined by Bm and AFe/s. Note that this relationship gives no information about the other

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m Cu

S

2.22 f B J A

( )

2

2

m Cu

S

2.22f B J A ( )

m Fe Cu

2

m Cu

2.22f B J(A )(A ) S

2.22f B JA

Fe

m Cu

A S

(2.22f B J)(A )

S

2

m Fe

m Fe

m Cu

V (4.44fB ) SA 4.44fB A

N 2.22fB JA = =

K S VS ( ) m Fe

Cu

8.88fB A S

J A

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

transformer dimensions. The impedance, voltage, and other space requirements provide the

majority of this information.

5B.2.3 Output and Winding Coefficients

Starting with the output or power equation (2), we can write:

S = 2.22 f Bm J AFe ACu or AFe = 

Then, without changing the value, we can state:

AFe = = or 

AFe = Eq. 4

Use KAS to represent the portion of Equation (4) to the right of . 

The expression KAS is essentially constant for a wide range of transformer classes and is

called the output coefficient. For three-phase liquid filled distribution transformers at 60 Hz, the

value of KAS ranges from 0.050-0.055 with a nominal median value of 0.052. For single-phase,

wound core, liquid filled units at 60 Hz the median value is about 0.040.

In a similar fashion, making use of equation (4), we can restate equation (1) as follows:

 = = Eq. 5

The expression KVS is also essentially constant for a wide range of transformer classes and

is called the winding coefficient. We can also express KVS in terms of KAS:

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VS

V K S

N =

A KS Fe AS =

A KS Cu CS = CS

AS

1 K

K=

b w

4

KVS= 4.44 f Bm KAS

For 60 Hz systems this may be rewritten as KVS = 266.4 Bm KAS. Thus the median values

for KVS become 21.5 for three-phase and 17.0 for single-phase wound core distribution

transformers at 60 Hz with Bm = 1.55 Tesla. Equations (4)-(5) provide initial estimates for

transformer dimensions in studies. They are the starting basis for the scaling laws used to scale

designs and performance. Typical values are given in Table 5B.2 for core type, liquid filled, 60

Hz distribution transformers at 12 kV, 95 kV BIL.

Table 5B.2 Nominal 60 Hz, Core-Type, Liquid-Filled, 12 kV Distribution Transformers

Class

of Dist. J(A/mm2

)

Bm

(Tesla) AFe/ACu KAS KVS %X

Transf. Range Nominal Nominal Range Nominal Range Nominal

3-Phase 2.4-

3.2

2.7 1.55 1.4-2.8 1.6 0.050-

0.055

0.052 21.5 4.75

1-Phase 2.0-

2.5

2.3 1.55 0.65-0.85 0.8 0.038-

0.043

0.041 17.0 4.75

5B.2.4 Scaling Laws

Having established the output and winding coefficients, it is instructive to examine the

origin of the 0.75 rules for scaling transformer losses. To illustrate, first of all, we need to set

relationships as follows:

 

 , (where ) 

 s ~ (AFe

0.5 + ) ~ S0.25

The shape of the window is set by voltage and the ratio h/s, which is essentially constant

for a given voltage and size, thus setting bw. Refer to Figure 5B.2 for dimensional definitions.

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2

I R

1000

2

S R

V 1000

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

2

S R

V 1000

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

17 2 2

2

Cu

4.28 10 S sN

A V

− × 0.75 K S s K'S . × =

bcan

S

2π

S1

2π

S2

2π

1

4 bw

1 2

n

1

2 bcan

1

2 bw

1 2

d

–

b01 b1 b02

t0

b2

4 n01 n02

3

nw

bw

2bFe

AFeA

Note: This diagram is not to scale

bcan

S

2π

S1

2π

S2

2π

1

4 bw

1 2

n

1

2 bcan

1

2 bw

1 2

d

–

b01 b1 b02

t0

b2

4 n01 n02

3

nw

bw

2bFe

AFeA

Note: This diagram is not to scale

Source: R. Feinberg, Modern Power Transformer Practice

Now, we consider the load losses, PCu (in kW/phase):

PCu = = = 

= =

The other scaling laws are derived in a similar fashion.

Figure 5B.2 Basic Three-Phase Transformer Dimensions

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1. Modern power transformer practice, Edited by R Feinberg, Wiley Publishers, New York, NY,

1979. ISBN: 047026344X 

REFERENCES

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