The electrical engineering handbook CH064

The electrical engineering handbook

Gross, C.A “Power Transformers”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

© 1999 by CRC Press LLC

64 Power Transformers

64.1 Transformer Construction The Transformer Core • Core and Shell Types • Transformer Windings • Taps

64.2 Power Transformer Modeling The Three-Winding Ideal Transformer Equivalent Circuit • A Practical Three-Winding Transformer Equivalent Circuit • The Two-Winding Transformer

64.3 Transformer Performance 64.4 Transformers in Three-Phase Connections Phase Shift in Y– D Connections • The Three-Phase Transformer • Determining Per-Phase Equivalent Circuit Values for Power Transformers: An Example

64.5 Autotransformers

64.1 Transformer Construction

The Transformer Core

The core of the power TRANSFORMER is usually made of laminated cold-rolled magnetic steel that is grain oriented such that the rolling direction is the same as that of the flux lines This type of core construction tends

to reduce the eddy current and hysteresis losses The eddy current loss P e is proportional to the square of the product of the maximum flux density B M (T), the frequency f (Hz), and thickness t (m) of the individual steel lamination

K e is dependent upon the core dimensions, the specific resistance of a lamination sheet, and the mass of the core Also,

In Eq (64.2), P h is the hysteresis power loss, n is the Steinmetz constant (1.5 < n < 2.5) and K h is a constant dependent upon the nature of core material and varies from 3210–3m to 20210–3m, where m = core mass in kilograms

The core loss therefore is

Charles A Gross

Auburn University

Core and Shell Types

Transformers are constructed in either a shell or a core structure The shell-type transformer is one where the windings are completely surrounded by transformer steel in the plane of the coil Core- type transformers are those that are not shell type A power transformer is shown in Fig 64.1

Multiwinding transformers, as well as polyphase transformers, can be made in either shell- or core-type designs

Transformer Windings

The windings of the power transformer may be either copper or aluminum These conductors are usually made

of conductors having a circular cross section; however, larger cross-sectional area conductors may require a rectangular cross section for efficient use of winding space

The life of a transformer insulation system depends, to a large extent, upon its temperature The total temperature is the sum of the ambient and the temperature rise The temperature rise in a transformer is intrinsic to that transformer at a fixed load The ambient temperature is controlled by the environment the transformer is subjected to The better the cooling system that is provided for the transformer, the higher the

“kVA” rating for the same ambient For example, the kVA rating for a transformer can be increased with forced air (fan) cooling Forced oil and water cooling systems are also used Also, the duration of operating time at high temperature directly affects insulation life

FIGURE 64.1 230kVY:17.1kV D 1153-MVA 3 f power transformer (Photo courtesy of General Electric Company.)

Other factors that affect transformer insulation life are vibration or mechanical stress, repetitive expansion and contraction, exposure to moisture and other contaminants, and electrical and mechanical stress due to overvoltage and short-circuit currents

Paper insulation is laid between adjacent winding layers The thickness of this insulation is dependent on the expected electric field stress In large transformers oil ducts are provided using paper insulation to allow a path for cooling oil to flow between coil elements

The circuit current in a transformer creates enormous forces on the turns of the windings The short-circuit currents in a large transformer are typically 8 to 10 times larger than rated and in a small transformer are 20 to 25 times rated The forces on the windings due to the short-circuit current vary as the square of the current, so whereas the forces at rated current may be only a few newtons, under short-circuit conditions these forces can be tens of thousands of newtons These mechanical and thermal stresses on the windings must be taken into consideration during the design of the transformer The

current-carrying components must be clamped firmly to limit

move-ment The solid insulation material should be precompressed and

formed to avoid its collapse due to the thermal expansion of the

wind-ings

Taps

Power transformer windings typically have taps, as shown The effect

on transformer models is to change the turns ratio

64.2 Power Transformer Modeling

The electric power transformer is a major power system component which provides the capability of reliably and efficiently changing (transforming) ac voltage and current at high power levels Because electrical power

is proportional to the product of voltage and current, for a specified power level, low current levels can exist only at high voltage, and vice versa

The Three-Winding Ideal Transformer Equivalent Circuit

Consider the three coils wrapped on a common core as shown in Fig 64.2(a) For an infinite core permeability (m) and windings made of material of infinite conductivity (s):

(64.4)

where f is the core flux This produces:

(64.5)

For sinusoidal steady state performance:

(64.6)

where V, etc are complex phasors

The circuit symbol is shown in Fig 64.2(b) Ampere’s law requires that

(64.7)

d

d dt

v v

N N

v v

N N

v v

N N

1

2 1

2 2

3 2

3 3

1 3

1

N

N

1 1

2

2

3

3

1 1

ˆ ˆ

0 = N1i1 + N2i2 + N3i3 (64.8)

Transform Eq (64.8) into phasor notation:

(64.9)

Equations (64.6) and (64.9) are basic to understanding transformer operation Consider Eq (64.6) Also note that –V1, –V2, and –V3 must be in phase, with dotted terminals defined positive Now consider the total input complex power –S

(64.10)

Hence, ideal transformers can absorb neither real nor reactive power

It is customary to scale system quantities (V, I, S, Z) into dimensionless quantities called per-unit values The basic per-unit scaling equation is

The base value always carries the same units as the actual value, forcing the per-unit value to be dimensionless Base values normally selected arbitrarily are Vbase and Sbase It follows that:

FIGURE 64.2 Ideal three-winding transformer (a) Ideal three-winding transformer; (b) schematic symbol; (c) per-unit equivalent circuit.

N I1 1+ N I2 2+ N I3 3= 0

S = V I1 1* + V I2 2* + V I3 3* = 0

actual value base value

V

I

V S

base base

base

base

base

base

base

base

=

2

When per-unit scaling is applied to transformers Vbase is usually taken as Vrated as in each winding Sbase is

common to all windings; for the two- winding case Sbase is Srated, since Srated is common to both windings

Per-unit scaling simplifies transformer circuit models Select two primary base values, V1base and S1base Base

values for windings 2 and 3 are:

(64.11)

and

(64.12)

By definition:

(64.13)

It follows that

(64.14)

Thus, Eqs (64.3) and (64.6) scaled into per-unit become:

(64.15) (64.16)

The basic per-unit equivalent circuit is shown in Fig 64.2(c) The extension to the n-winding case is clear

A Practical Three-Winding Transformer Equivalent Circuit

The circuit of Fig 64.2(c) is reasonable for some power system applications, since the core and windings of

actual transformers are constructed of materials of high m and s, respectively, though of course not infinite

However, for other studies, discrepancies between the performance of actual and ideal transformers are too

great to be overlooked The circuit of Fig 64.2(c) may be modified into that of Fig 64.3 to account for the

most important discrepancies Note:

R1,R2,R3 Since the winding conductors cannot be made of material of infinite conductivity, the windings must

have some resistance

X1,X2,X3 Since the core permeability is not infinite, not all of the flux created by a given winding current will

be confined to the core The part that escapes the core and seeks out parallel paths in surrounding

structures and air is referred to as leakage flux

R c,X m Also, since the core permeability is not infinite, the magnetic field intensity inside the core is not zero

Therefore, some current flow is necessary to provide this small H The path provided in the circuit

for this “magnetizing” current is through X m The core has internal power losses, referred to as core

loss, due to hystereses and eddy current phenomena The effect is accounted for in the resistance R c

Sometimes R c and X m are neglected

N

1

1 1

base base base base = =

S1base base base= S2 = S3 = Sbase

S

S V

1

1

2

2

3

3

base

base

base

base

base

base

base base base

N

2

3 1

base base base base = =

V1pu pu pu= V2 = V3

I1pu pu pu+ I2 + I3 = 0

The circuit of Fig 64.3 is a refinement on that of Fig 64.2(c) The values R1, R2, R3, X1, X2, X3 are all small

(less than 0.05 per-unit) and R c , X m, large (greater than 10 per-unit) The circuit of Fig 64.3 requires that all values be in per-unit Circuit data are available from the manufacturer or obtained from conventional tests It must be noted that although the circuit of Fig 64.3 is commonly used, it is not rigorously correct because it does not properly account for the mutual couplings between windings

The terms primary and secondary refer to source and load sides, respectively (i.e., energy flows from primary

to secondary) However, in many applications energy can flow either way, in which case the distinction is

meaningless Also, the presence of a third winding (tertiary) confuses the issue The terms step up and step

down refer to what the transformer does to the voltage from source to load ANSI standards require that for a

two-winding transformer the high-voltage and low-voltage terminals be marked as H1-H2 and X1-X2,

respec-tively, with H1 and X1 markings having the same significance as dots for polarity markings [Refer to ANSI

C57 for comprehensive information.] Additive and subtractive transformer polarity refer to the physical posi-tioning of high-voltage, low-voltage dotted terminals as shown in Fig 64.4 If the dotted terminals are adjacent,

then the transformer is said to be subtractive, because if these adjacent terminals (H1-X1) are connected together, the voltage between H2 and X2 is the difference between primary and secondary Similarly, if adjacent terminals X1 and H2 are connected, the voltage (H1-X2) is the sum of primary and secondary values.

The Two-Winding Transformer

The device can be simplified to two windings Common two-winding transformer circuit models are shown

in Fig 64.5

(64.17)

FIGURE 64.3 A practical equivalent circuit.

FIGURE 64.4 Transformer polarity terminology: (a) subtractive; (b) additive.

Ze = Z1 + Z2

Circuits (a) and (b) are appropriate when –Z m is large enough that magnetizing current and core loss is negligible

64.3 Transformer Performance

There is a need to assess the quality of a particular transformer design The most important measure for performance is the concept of efficiency, defined as follows:

(64.19)

where Pout is output power in watts (kW, MW) and Pin is input power in watts (kW, MW)

The situation is clearest for the two-winding case where the output is clearly defined (i.e., the secondary winding), as is the input (i.e., the primary) Unless otherwise specified, the output is understood to be rated power at rated voltage at a user-specified power factor Note that

The transformer is frequently modeled with the circuit shown in Fig 64.6 Transformer losses are made up of the following components:

FIGURE 64.5 Two-winding transformer-equivalent circuits All values in per-unit (a) Ideal case; (b) no load current negligible; (c) precise model.

= +

P

out in

Magnetic (core) loss: Pc = Pe + Ph = V1/Rc (64.21)

Hence:

A second concern is fluctuation of secondary voltage with load A measure of this situation is called voltage

regulation, which is defined as follows:

(64.23)

where V 2FL = rated secondary voltage, with the transformer supplying rated load at a user-specified power

factor, and V 2NL = secondary voltage with the load removed (set to zero), holding the primary voltage at the full load value

A complete performance analysis of a 100 kVA 2400/240 V single-phase transformer is shown in Table 64.1

64.4 Transformers in Three-Phase Connections

Transformers are frequently used in three-phase connections For three identical three-winding transformers, nine windings must be accounted for The three sets of windings may be individually connected in wye or delta

in any combination The symmetrical component transformation can be used to produce the sequence equiv-alent circuits shown in Fig 64.7 which are essentially the circuits of Fig 64.3 with Rc and X m neglected The positive and negative sequence circuits are valid for both wye and delta connections However, Y–D connections will produce a phase shift which is not accounted for in these circuits

FIGURE 64.6 Transformer circuit model.

FIGURE 64.7 Sequence equivalent transformer circuits.

V

NL FL FL

2

TABLE 64.1 Analysis of a Single-Phase 2400:240V 100-kVA Transformer

Voltage and Power Ratings

HV (Line-V) LV (Line-V) S (Total-kVA)

Test Data Short Circuit (HV) Values Open Circuit (LV) Values Voltage = 211.01 240.0 volts Current = 41.67 22.120 amperes Power = 1400.0 787.5 watts

Equivalent Circuit Values (in ohms) Values referred to HV Side LV Side Per-Unit Series Resistance = 0.8064 0.008064 0.01400 Series Reactance = 4.9997 0.049997 0.08680 Shunt Magnetizing Reactance = 1097.10 10.9714 19.05 Shunt Core Loss Resistance = 7314.30 73.1429 126.98

Power Factor Efficiency Voltage Power Factor Efficiency Voltage

Rated load performance at power factor = 0.866 lagging.

Secondary Quantities; LOW Voltage Side Primary Quantities; HIGH Voltage Side

Current 416.7 amperes 1.0000 Current 43.3 amperes 1.0386 Apparent power 100.0 kVA 1.0000 Apparent power 109.9 kVA 1.0985

Reactive power 50.0 kvar 0.5000 Reactive power 64.6 kvar 0.6456 Power factor 0.8660 lag 0.8660 Power factor 0.8091 lag 0.8091 Efficiency = 97.43%; voltage regulation = 5.77%.

The zero sequence circuit requires special modification to account for wye, delta connections Consider winding 1:

1 Solid grounded wye — short 1¢ to 1¢¢

2 Ground wye through –Z n — connect 1¢ to 1¢¢ through 3–Zn

3 Ungrounded wye — leave 1¢ to 1¢¢ open

4 Delta — short 1¢¢ to reference

Winding sets 2 and 3 interconnections produce similar connection constraints at terminals 2¢–2¢¢ and 3¢–3¢¢, respectively

terminals as follows:

Winding set 1 wye, grounded through –Z n

Winding set 2 wye, solid ground

Winding set 3 delta

The zero sequence network is as shown

Phase Shift in Y–D Connections

The positive and negative sequence networks presented in Fig 64.7 are misleading in one important detail For Y–Y or D–D connections, it is always possible to label the phases in such a way that there is no phase shift between corresponding primary and secondary quantities However, for Y–D or D–Y connections, it is impos-sible to label the phases in such a way that no phase shift between corresponding quantities is introduced ANSI standard C57.12.10.17.3.2 is as follows:

For either wye-delta or delta-wye connections, phases shall be labeled in such a way that positive sequence quantities on the high voltage side lead their corresponding positive sequence quantities on the low voltage side by 30 o The effect on negative sequence quantities may be the reverse, i.e., HV values lag LV values by 30 o

This 30o phase shift is not accounted for in the sequence networks of Fig 64.7 The effect only appears in the

positive and negative sequence networks; the zero sequence network quantities are unaffected

The Three-Phase Transformer

It is possible to construct a device (called a three-phase

trans-former) which allows the phase fluxes to share common magnetic

return paths Such designs allow considerable savings in core

material, and corresponding economies in cost, size, and weight

Positive and negative sequence impedances are equal; however,

the zero sequence impedance may be different Otherwise the

circuits of Fig 64.7 apply as discussed previously

Determining Per-Phase Equivalent Circuit

Values for Power Transformers

One method of obtaining such data is through testing Consider the problem of obtaining transformer equiv-alent circuit data from short-circuit tests A numerical example will clarify per-unit scaling considerations