APPENDIX 5B. SCALING RELATIONSHIPS IN TRANSFORMER MANUFACTURING

TABLE OF CONTENTS

5B.1 INTRODUCTION

................................................................................................... 1 5B.2 THEORY AND BASIS FOR SCALING RULES

................................................... 3 5B.2.1 Power and Voltage Equations

.................................................................................. 3 5B.2.2 Losses

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

............................................................................ 8 5B.2.4 Scaling Laws

.......................................................................................................... 10

LIST OF TABLES

Table 5B.1 Common Scaling Relationships in Transformers

........................................... 1 Table 5B.2 Nominal 60 Hz, Core-Type, Liqui d-Filled, 12 kV Distribution Transformer

............................................................................................................................... ............ 10

LIST OF FIGURES

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

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

.............................................. 11

5B-iAPPENDIX 5B. SUPPLEMENTARY ENGINEERING ANALYSIS RESULTS

5B.1 INTRODUCTION

There exist certain fundamental relationships between the ratings in kilovolt-amperes (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 mo re efficient. These size versus performance relationships arise from fundamental equations describing a transfor mer's voltage and kVA rating. For example, by fixing the kVA rati ng and voltage frequency, the product of the conductor current density, core flux density, core cross sectional area, an d total conductor cross sectional area is constant.

To illustrate this point, consider a transformer with frequency, magnetic flux density, current density, and basic impulse insulation le vels (BIL) all fixed. If one enlarges (or decreases) the kVA rating, then the only free parame ters are the core cross section and the core window area through which the windings pass. T hus, to increase (or decrease) the kVA rating, the dimensions for height, width, and depth of th e 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 the term "0.75 scaling rule." Table 5B.1 depicts the most common scaling relationshi ps in transformers.

Table 5B.1 Common Scaling Rela tionships in Transformers Parameter Being Scaled Rela tionship to kVA Rating (varies with ratio of kVA x ) Weight (kVA 1 /kVA 0 ) 3/4

Cost (kVA 1 /kVA 0 ) 3/4
Length (kVA 1 /kVA 0 ) 1/4

Width (kVA 1 /kVA 0 ) 1/4
Height (kVA 1 /kVA 0 ) 1/4

Total Losses (kVA 1 /kVA 0 ) 3/4

No-load Losses (kVA 1 /kVA 0 ) 3/4
Exciting Current (kVA 1 /kVA 0 ) 3/4

% Total Loss (kVA 1 /kVA 0 ) -1/4

% No Load Loss (kVA 1 /kVA 0 ) -1/4
% Exciting Current (kVA 1 /kVA 0 ) -1/4
% Resistance (R) (kVA 1 /kVA 0 ) -1/4
% Reactance (X) (kVA 1 /kVA 0 ) 1/4

Volts/Turn (kVA 1 /kVA 0 ) 1/2

The three elements listed below are true as the kVA rating increases or decreases, if the following factors are held constan t: the type of transformer (dis tribution or power transformer, liquid filled or dry-type, singl e-phase or three-phase), th e primary voltage, the core

5B-1configuration, the core material, the core flux density, and the current density (amperes per square inch of conductor cr oss section) in both the pr imary and secondary windings.

1. The physical proportions are constant (same relative shape), 2. The eddy loss proportion is e ssentially constant, and 3. The insulation space factor (voltage or BIL) is constant.

In practical applications, it is rare to find that all of th e above are constant over even limited ranges; however, over a range of one orde r of magnitude in both directions (e.g., from 50kVA to 5kVA or from 50kVA to 500kVA), the scal ing 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 ev en wider ranges to estimate general performance levels. The same quantities are depicted graphi cally in Figure 5B.1 for reference.

Figure 5B.1 Size and Performanc e Relationships by kVA Rating

To illustrate how the scaling laws are used, consider two transformers with kVA ratings of S 0 and S 1 . The no-load losses (NL) and total losses ( TL) of these two transformers would be depicted as NL 0 and TL 0 , and NL 1 and TL 1 . Then the relationships between the NL and TL of the two transformers could be shown as follows:

଴.଻ହ ଴.଻ହ NL ଵ ൌ NL ଴ ቀ S భ ቁ and TL ଵ ൌ TL ଴ ቀ S భ ቁ

S బ S బ

5B-2These two equations can be manipulated algebr aically 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, one can derive the rela tionship for load loss (LL), and show that it also scales to the 0.75 power. Specifically:

LL ଵ ൌ TL ଵ െ NL ଵ
Plugging the TL1 and NL1 terms into this equation:

଴.଻ହ ଴.଻ହ LL ଵ ൌ TL ଴ ൬ S ଵ ൰ െ NL ଴ ൬ S ଵ ൰
S ଴ S ଴

଴.଻ହ

ൌ ሺ TL ଴ െ NL ଴ ሻ ൬
S S ଵ

଴ ൰

That is,

଴.଻ହ LL ଵ ൌ LL ଴ ൬
S S ଵ

଴ ൰

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 specifi ed 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 FO R SCALING RULES

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 lift ed freely or derived from similar material in Modern Power Transformer Practice , Wiley 1979, edited by R. Feinberg. i No mathematics beyond elementary algebr a is required, but a good deal of implied physics and electrical engineer ing is required to fully ap preciate 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 B

m in Tesla, and the cross-sectional area of the core steel (A Fe ) in square meters. The units are mixed to simplify the basic equations, a common practice in transformer design texts. Th e machine equation is derived from Faraday's law, which is expressed as

׎߲ ܰ ൌെݒ ݐ߲

5B-3where v is the instantaneous value of V, and డ׎
డ௧ 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 ൌ 4.44fB ୫ A Fୣ
Equation 5B.1

The voltage and turns may apply to either the primary or the secondary winding and, for the ideal transformer with no losses and no-leakage flux,

V

ଵ ⁄ V ଶ ൌ N ଵ ⁄ N ଶ ൌnൌI ଶ ⁄ I ଵ

where V 1 and V 2 represent primary and sec ondary voltages respectively, N 1 and N 2 primary and secondary turns, and I 1 and I 2 primary and secondary currents in amperes (amps). The quantity n is referred to as the “turns ratio.” With th e parameters defined, and using Equation 5B.1, the output or transformer capacity (S) in megavolt-amperes (MVA) pe r phase can be expressed as:

S ൌ 4.44fB ୫ A Fୣ NI

Equation 5B.2

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

A C୳ ൌ ሺ N ଵ a ଵ ൅ N ଶ a ଶ ሻ ൈ 10 ି଺

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

A C୳ ൌ 2 ൈ 10 ି଺ Na

where “a” is the conductor cross-se ction in square millimeters (mm 2 ) of an individual turn referred to the winding with N turns, and a 1 and a 2 are conductor cross-sections of primary and secondary turns, respectively. As long as the wi nding current densities are equal, either winding may be used as reference, provi ded the choice of primary or s econdary is consistent. Starting with Equation 5B.2, using the A Cu relationship explained above, a nd letting J represent current density in amps per mm 2 :

S ൌ 2.22 f B ୫ JA Fୣ A C୳
Equation 5B.3

Let A w be the core window area in square meters, and k w the window space factor, as given by 2 A C୳ ⁄ A ୵ . (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

5B-4channel requirements. For distribution transformers, k w is found to be about 0.3–0.4 for nominal 12 kV systems. Using these definitions,

S ൌ 1.11 f B ୫ JA Fୣ k ୵ A ୵ Equation 5B.4

Note that, for a given MVA rating, and specified flux and current densit ies, the product of conductor and core cross-section is cons tant and inversely related; i.e. A Fe α 1/A Cu

.

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 to tal core and load losses (P Fe and P Cu ) can be readily obtained. These results are accomplished by multiplying th e W/kg for both core and windings by the core mass and the conductor mass respect ively (or by their volumes tim es material densities).

The Department uses the convention that lower case corresponds to per-unit quantities and upper case corresponds to total or total-per-phase quantities. Lo ad losses consis t of resistive (P R ) and eddy (P i ) components. Expressions can be derive d 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

%P ୧ ൌ 100 P ୧ ⁄ P R , or P ୧ ൌ P R ቀ %P ౟ ቁ

ଵ଴଴
Ignoring stray loss, (which is associated with eddy losses), let P t represent total load loss for a three-phase transformer. That is,

P ୲ ൌ 3P C୳
Also assume the same eddy loss fraction in primary and secondary windings.

P

C୳ ൌ P R ൅ P ୧ ൌ P R ൅ P R ൬ %P ୧ 100 ൰

ൌ ൬ 1 ൅ %P ୧ 100 ൰ P R ൌ k ୧ P R

Closely associated with the load loss of a tr ansformer is its impedance. When the load loss of a given transformer is determined by test (t he wattmeter reading in the test circuit), that same test also provides the value of the impedan ce (the voltmeter reading in the test circuit). Impedance in a transformer is expressed in term s of the “impedance voltage,” which is defined as “the voltage required to circulate rated cu rrent through one of two specified windings of a transformer when the other winding is short-circ uited, 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,

5B-5%Z ൌ IZ ൈ 100
V

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 resistiv e and reactive component s, %R and %X, as

%Z ൌ ඥ ሺ %R ሻ ଶ ൅ ሺ %X ሻ ଶ

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

%R ൌ IR ൈ 10 ଶ

V

Note that R in the numerator must repres ent the total resistance in the transformer windings. Therefore, if the transformer is being vi ewed from the primary terminals, the value of R would be the total resistance of the primary wi nding, plus the total resi stance of the secondary winding referred to the primary winding, (R 2 (N 1 /N 2 ) 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 (%I 2 R) is related to the MVA capacity of th e transformer, stray loss being ignored as stated previously.

Multiplying numerator and denominator in the above equation by I, and letting P t represent total load loss in watts and S represent the MVA per pha se rating, one can determine the percent load loss as:

ൌ I ଶ R ൈ 10 ଶ Percent load loss ൌ I ଶ R ൈ 10 ଶ

IൈV 3S ൈ 10 ଺

୲ P ିସ 10 R ൌ ׵ % 3S

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

From Equation 5B.3, it is evident that, once th e core flux density and current density are fixed, the transformer rating is dependent on the core cross-section and window area. Next, one can derive information about the window shape.

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

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

5B-6• The ratio of the cross-sectional areas of the core and conductor (A Fe /A Cu ).

The mean value of s (a linear measurement, recording the circumfere nce), is given by the equation s = (s 1 + s 2 )/2, where s 1 is the mean turn of the primary winding and s 2 is the mean turn of the secondary winding.

These ratios, together with the necessar y space factors for insulating and cooling clearances, establish the relative volumes of th e core and conductor. Consequently, if fixed values 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 de rives an expression relating the flux and current densities. The ex pression starts with:

P

C୳ ൌ ൬ 1 ൅ %P ୧ 100 ൰ P R ൌ k ୧ P R

P

C୳ ൌ ൫I ଵ ଶ R ଵ ൅ I ଶ ଶ R ଶ ൯k ୧ ,
where subscripts 1 and 2 indica te primary and secondary windi ngs, respectively. The resistance per phase of the primary winding is given by

R ଵ ൌ ρN ଵ s ଵ ohms,
a ଵ

where a 1 is the cross-sectional area of the primary copper conductor, and ρ is the resistivity at full load operating temperature of the conductor, 21.4 x 10 -3 ohm-meters. The value of R 2 is similarly obtained:

ଶ ρN ଵ s ଵ ൅ I ଶ ଶ ρN ଶ s ଶ ׵ P C୳ ൌ ቆ I ଵ ቇ k

୧ a ଵ a ଶ

൅ I ଶ s ଶ ׵ P C୳ ൌ IN ൬ I ଵ s ଵ ൰ ρk ୧ a ଵ a ଶ
where IN is the ampere-turns in either windi ng. As before, the assumption of equal current densities in the windings is made, driven by the condition for minimum I 2 R loss. Accordingly,

P

C୳ ൌ 2INJsρk ୧

׵Jൌ P C౫

, the current density equation. ଶINୱ஡୩ ౟
Multiplying Equation 5B.1 by I and rearranging algebraically, one gets:

5B-7VI IN ൌ 4.44fB ୫ A Fୣ

It was established earlier that S is th e rating per phase in MVA, i.e., VI = 10 6 S. Thus:

10 ଺ S ׵ IN ൌ 4.44fB ୫ A Fୣ
Using the current density equation, su bstituting the resistivity value for ρ , and the above value for IN, one can derive that:

J ൌ 104 ൈ 10 ି଺ fB ୫ A Fୣ P C୳ k ୧ sS

The watts of conductor loss (for copper) ca n be expressed as a percentage of the transformer MVA rating:

%P C୳ ൌ P C୳ ൈ 10 ଶ

S

or, in kilowatts:

%P C୳ ൌ P C୳ ൈ 10 ଶ

ൌ 0.1P C୳

. S ൈ 10 ଷ S

By substituting in the revised equation for J (amperes per square meter), one gets

J ൌ 104 ൈ 10 ି଺ fB ୫ A Fୣ S ൈ %P C୳ 0.1 ൌ 1040 ൈ 10 ି଺ fB ୫ A Fୣ ൈ %P C୳ k ୧ sS k ୧ s

Equation 5B.5

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 expr essed at operating temperature.

If J and B m are chosen independently, the transformer will have a natural value of conductor loss depending on the ratio A Fe /s. Conversely, if losses are specified, the choice of J is determined by B m and A Fe /s. Note that this relationship gi ves no information about the other 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 5B.3, one can write:

5B-8S S ൌ 2.22fB ୫ JA Fୣ A C୳ or A Fୣ ൌ 2.22fB ୫ JA C୳
Then, without changing the value, one can state:

S ଶ A Fୣ ൌ ඨ

ሺ 2.22fB ୫ JA C୳ ሻ ଶ ൌ √ S ඨ 2.22fB ୫ J ሺ A Fୣ ሻሺ A C୳ ሻ

or ሺ 2.22fB ୫ JA C୳ ሻ ଶ

A Fୣ A Fୣ ൌ √ S ඨ
ሺ 2.22fB ୫ J ሻሺ A C୳ ሻ

Equation 5B.6

Use K AS to represent the portion of E quation 5B.6 to the right of √ S

The expression K AS is essentially constant for a wide range of transformer classes and is called the output coefficient. For three-phase, liqui d-filled distribution tran sformers at 60 Hz, the value of K AS ranges from 0.050 to 0.055, with a nominal median value of 0.052. For singlephase, wound-core, liquid-filled units at 60 Hz, the median value is about 0.040.

In a similar fashion, making use of Equation 5B.6, we can restate Equation 5B.1 as follows:

V
N ൌ 4.44fB ୫ A Fୣ ൌ ඨ ሺ 4.44fB ୫ ሻ ଶ SA Fୣ 2.22fB ୫ JA C୳

ൌ ඨ ൬ 8.88fB ୫ ൰ ൬ A Fୣ ൰ ሺ S ሻ ൌ K VS √ S J A C୳

Equation 5B.7

The expression K VS is also essentially constant for a wide range of transformer classes and is called the winding coefficient. One can also express K VS in terms of K AS :

K VS ൌ 4.44fB ୫ K AS
For 60 Hz systems, this may be rewritten as K VS = 266.4 B m K AS . Thus the median values for K VS become 21.5 for three-phase and 17.0 for single-phase, wound-core distribution transformers at 60 Hz with B m = 1.55 Tesla. Equation 5B.6 a nd Equation 5B.7 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.

5B-9Table 5B.2 Nominal 60 Hz, Core-Type, Liqui d-Filled, 12 kV Distribution Transformer

5B.2.4 Scaling Laws

Having established the output and winding coeffici ents, it is instructive to examine the origin of the 0.75 rules for scaling transformer losses. To illustrate, first of all, one needs to set relationships as follows:

N V ൌ K VS √ S

A Fୣ ൌ K AS √ S

1 A C୳ ൌ K CS √ S

, ൬ where K CS ൌ ൰

s~ ൬ A Fୣ ଴.ହ ൅ b

4 ୵ ൰ ~s ଴.ଶହ K AS

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 b

w . Refer to Figure 5B.2 for dimensional definitions.

Now, one considers the load losses, P Cu (in kW/phase):

ଶ R P

C୳ ൌ I ଶ R
V ൰
1000 ൌ ൬ S
1000

ൌ 4.28 ൈ 10 ିଵ଻ S ଶ sN ଶ

A C୳ V ଶ ൌ K √ S ൈ s ൌ K ᇱ S ଴.଻ହ

The other scaling laws are de rived in a similar fashion.

5B-10Figure 5B.2 Basic Three-Phase Transformer Dimensions

i Modern power transformer practice, Edited by R Fei nberg, Wiley Publishers, New York, NY, 1979. ISBN: 047026344X

5B-11