02_3phtransformers

ECE 3650 / Dr. Athula Rajapakse 1

2. Three-Phase Transformers All major power generation, transmission and distribution systems are three-phase ac systems. These systems need three phase transformers. There are two different ways to construct three-phase transformers. 2.1 Methods of Construction 2.1.1 Using a bank of three-single phase transformers

There are two types of core used in single phase transformer construction: Core-type and Shell-type. Core-Type Transformer Shell-Type transformer Advantages of using three single-phase transformers:

Each unit in the bank can be replaced in the event of failure.

2.1.2 Using three sets of windings wrapped on a common core

A1 A2 B1 B2 C1 C2

φa φc φb


When a three-phase balanced set of voltages is applied, the fluxes in each limb of the core are: )cos(max ta ωφφ = ; )120cos(max °−= tb ωφφ ; )240cos(max °−= tc ωφφ

Therefore, 0=++ cba φφφ

Therefore, it is possible to have a three limbed transformer core as shown above. However, the above arrangement causes difficulties when the voltages are unbalanced: the resultant flux due to unbalance is forced to pass through high reluctance air paths. Four limbed or five limbed transformers can be used to rectify this situation. Three-limbed core Four-limbed core Five-limbed core Advantages of three-phase transformers with a common core are:

Lighter Smaller in size and less foot print Cheaper Slightly more efficient

Three-phase power transformers: windings and appearance


2.2 Three-phase transformer connections Regardless of the construction type (three-separate cores or single three-phase core), primary and secondary windings can be independently connected in either a Wye (Y) or a Delta (Δ). This gives four possible connections.

Wye-Wye (Y-Y) connection Wye-Delta (Y-Δ) connection Delta-Wye (Δ-Y) connection Delta-Delta (Δ-Δ) connection

2.2.1 Wye-Wye (Y-Y) connection In three-phase transformer voltage ratio specified is usually the ratio between line-to-line voltages at the primary and secondary. For Y-Y connected transformer: Relationship between the primary and secondary winding voltages:

aVV

=2

1

φ

φ

Line voltages of the primary and secondary sides:

11 .3 φVVL = 22 .3 φVVL = Ratio between the primary and secondary line voltages:

aVV

VV

L

L ==2

1

2

1

33

φ

φ

Phasor diagrams of the primary and secondary side voltages show that the corresponding phase voltages on the primary and secondary sides are in phase.

a : 1 a1

b1

c1

n1

a2

b2

c2

n2

+ Vφ2 −

+ Vφ1 −

+ VL2

−

+ VL1 −

Vab1

Vbc1

Vca1

Vb1 Vc1

Va1

primary secondary

Vab2

Vbc21

Vca2

Vb2 Vc2

Va21


Advantages of Y-Y connection: Neutral point is available on both primary and secondary sides, and can be grounded if

necessary. Electrical insulation required for each winding is 58% (or 1/√3) of the respective line

voltage. However, there are two serious drawbacks in the Y-Y connections.

If loads are unbalanced, then the voltages on the transformer can be seriously unbalanced. Third harmonic voltage can be large

Transformer exciting currents are not exactly sinusoidal and contain significant third harmonic component. Therefore, the exciting currents in three phases do not add up to zero, even if they have the same magnitude and are displaced by 120o from each other. If the neutral is not grounded, the exciting currents are forced to add up to zero at the neutral point. This results in a distorted flux waveform, and hence a distorted voltage waveform that contain third harmonics. Although fundamental frequency voltages are 120o phase shifted from each other, third harmonic component of the each phase will be in phase. These components add up resulting in large third harmonic voltages. Both the unbalance problem and third-harmonic problem can be solved by using one of two techniques:

Solidly grounding the neutrals of the transformer This allows additive third harmonic currents to flow into the neutral instead of building large voltages. Neutral also provides return path for unbalanced currents. Add a third (tertiary) winding connected in Δ The third-harmonic voltages in the Δ winding will add up, causing a circulating current flow within the winding. This suppresses the third-harmonic components of voltage in the same manner as grounding the transformer neutral.

One of the above correction techniques MUST be used any time a Y-Y transformer is installed. In practice, Y-Y transformers are rarely used because of the same job can be done with one of the other types.


2.2.2 Wye-Delta (Y-Δ) connection Relationship between the primary and secondary winding voltages:

aVV

=2

1

φ

φ


11 .3 φVVL = 22 φVVL = Ratio between the primary and secondary line voltages:

aV

VVV

L

L 33

2

1

2

1 ==φ

φ

Phasor diagrams of primary and secondary voltages show that the secondary sides line voltages lags behind the corresponding voltages on the primary side by 30o. Advantages of Y-Δ connection:

No serious problems with unbalanced loads. The Δ winding partially redistributes any imbalance that occurs.

No problem of third-harmonic voltages. The third harmonic components in voltage are

consumed in a circulating current formed on the Δ winding.

a1

b1

c1

n1

a2

b2

c2

+ Vφ2 −

+ Vφ1 −

+ VL2

−

+ VL1 −

a : 1

Vab2

Vbc2

Vca2

Vb2

Vc2 Va2

primary secondary

Vab1

Vbc1

Vca1

Vb1 Vc1

Va1


Drawbacks in the Y-Δ connection Voltage on the secondary is phase shifted with respect to the primary side. If two

transformer secondary windings are paralleling, care must be taken to make sure both secondary windings are phase shifted by the same amount.

Electrical insulation required for secondary winding is equal to the line voltage. Therefore, Y-Δ transformers are mainly used for step-down applications.

2.2.3 Delta-Wye (Δ-Y) connection Relationship between the primary and secondary winding voltages:

aVV

=2

1

φ

φ


11 φVVL = 22 3 φVVL = Ratio between the primary and secondary line voltages:

aV

VVV

L

L

31

3 2

1

2

1 ==φ

φ

According to the connection diagram shown, the secondary sides phase voltages lead the corresponding primary side voltages by 30o. Δ-Y transformers have the same characteristics as Y-Δ transformers. Winding insulation requirements make Δ-Y transformers more suitable for step-up applications. However, Δ-Y transformers are commonly used as step-down transformers in distribution systems where the voltages are not very high. The neutral point on the secondary side allows supplying both three-phase and single phase loads, while Δ primary reduces the effects of unbalanced loading, which is common in distribution systems.

Vab2

Vbc2

Vca2 Vb2 Vc2

Va2 Vab1

Vbc1

Vca1

Vb1 Vc1

Va1

a2

b2

c2

n2

a1

b1

c1

+ Vφ2 −

+ Vφ1 −

+ VL2

−

+ VL1 −

a : 1

primary secondary


2.2.4 Delta-Delta (Δ-Δ) connection Relationship between the primary and secondary winding voltages:

aVV

=2

1

φ

φ

Line voltages of the primary and secondary sides: 11 φVVL = 22 φVVL =

Ratio between the primary and secondary line voltages:

aVV

VV

L

L ==2

1

2

1

φ

φ

According to the phasor diagrams of primary and secondary voltages, the corresponding line voltages on the primary and secondary sides are in phase. Advantages of Δ-Δ connection:

No serious problems with unbalanced loads. No problem of third-harmonic voltages. No phase shift in the voltages. One transformer can be removed for maintenance while the other two continue to supply

three phase power at a reduced rating (58%). Drawbacks in the Δ-Δ connection

Electrical insulation required for each winding is equal to the respective line voltage. No neutral point is available on both sides of the transformer.

a2

c2

b2

a1

b1

c1

+ Vφ2 −

+ Vφ1 −

+ VL2

−

+ VL1 −

a : 1

Vab1

Vbc1

Vca1

Vb1

Vc1 Va1

Vab2

Vbc2

Vca2

Vb2

Vc2 Va2

primary secondary


2.3 Zig-Zag Transformer A zigzag winding is a series connection of two windings whose voltages are 600 out of phase. The two windings are typically the same voltage magnitude, but custom phase shifts can be created if the voltage magnitude of the two windings differs. There are two basic ways to create a zigzag winding:

• Connect the A leg in series with B leg (called a ZAB) • Connect the A leg in series with C leg (called a ZAC)

The polarity marks of the two windings either face toward one another or face away from one another. The connection diagram for a ZAC zig-zag winding is shown below.

The Zig-Zag transformers which have only a primary windings but no secondary winding are used to derive an earth reference point for an ungrounded electrical system. Another application is to control harmonic currents.

2.4 Transformer Vector Groups It is possible to connect three-phase transformers to achieve different phase shifts between the primary and secondary sides. For example, Δ-Y transformer, which usually has a +30o phase shift in the secondary corresponding to primary side, can be connected to achieve a -30o phase shift as shown below.


According to IEC standards, this connection is denoted as Dy1 connection. The ‘D’ indicates delta primary, ‘y’ indicates wye connected secondary and ‘1’ indicates a -30o phase shift in secondary phase voltages relative to primary phase voltages. The number ‘1’ comes from the fact that primary and secondary phase-A voltage vectors when plotted together indicate clock position ‘1’. The following table gives different vector groups of Δ-Y and Y-Δ transformers.

a2

b2

c2

n2

a1

b1

c1

+ Vφ2 −

+ Vφ1 −

+ VL2

−

+ VL1 −

a : 1

Vab2

Vbc2

Vca2

Vb2 Vc2

Va2 Vab1

Vbc1

Vca1

Vb1 Vc1

Va1

primary secondary

Va1

Va2 -30o

Dy1


Common transformer connections and their vector groups


In North America, it is customary to make the secondary voltage lag the primary voltage. According to ANSI Standards (ANSI/IEEE Std C57.70).:

• High voltage terminals are marked with H1, H2 and H3 (or A,B, and C ). • Low voltage terminals are marked with X1, X2 and X3 (or a, b, and c ).

The American Standards for labeling of the windings states that “In either a Y-Δ or ∆-Y transformer, positive-sequence quantities on the HV side shall lead their corresponding quantities on the low voltage side by 30o.” Thus in the three-phase Y-Δ or ∆-Y transformers manufactured according to American Standards, the HV side voltages are always leading the LV side voltages, regardless of the HV winding connection type. Instead of the vector group, the name plate provides a vector diagram such as the one shown below.

The above transformer example has the vector group Dy1. 2.5 Three-Phase Transformation Using Two Transformers 2.5.1 Open-Delta or V-V Connection Suppose that a Δ-Δ transformer bank composed of separate transformers has a damaged phase that must be removed for repair. If the two remaining secondary voltages are °∠= 022 Lab VV and

°−∠= 12022 Lbc VV , then the voltage across the gap, Vca2 is given by

°−∠−°∠=−−= 1200 22222 LLbcabca VVVVV )886.05.0()866.05.0( 2222 jVjVVV LLLca +−=−−−=

°∠= 12022 Lca VV

a2

b2

c2

One phase removed

a1

b1

c1

Vca2 Vab2

Vbc2


This is exactly the same voltage that would be present if the third winding were still present. Thus the open delta connection lets transformer bank get by with only two transformers, allowing some power flow to continue even with a damaged phase removed.

How much apparent power can the bank supply with one of its three transformers removed? At first it seems that it could supply 2/3 of its rated apparent power, since two out of three transformers are present. Things are not quite that simple. When all three transformers are present, the total power supplied to the resistive load

φφφ IVP 33 = With open delta connection, transformer current is equal to the line current. The currents and voltages are no longer in phase; there is a 30o phase shift. In a standard delta configuration, the line current is √3 times greater than the current flowing in the phase winding. When one of the transformers is absent, full line current flows through the phase windings, since line and phase currents are the same in an open-delta configuration. The large increase in current will cause the phase windings to overheat and will damage the transformer unless load power is reduced. The line current must therefore be reduced by √3. Power delivered by the first transformer: Power delivered by the second transformer

φφ

φφ

φφφ

IV

IVIVP

23

)30cos()120150cos(1

=

°=

°−°=

φφ

φφ

φφφ

IV

IVIVP

23

)30cos()6030cos(1

=

°=

°−°=

Therefore, the total power supplied φφφ IVP 33 =


Since the rated current is the same for each transformer, and the voltage is the same on each transformer; so the ratio of the output power available from the open-delta bank to the output power available from normal three-phase transformer bank is

%7.573

133

===Δ−Δ

−

φφ

φφ

IVIV

PP VV

The lpower capacity in the open-delta configuration is limited to 57.7% of the power capacity normal delta-delta configuration. The following example illustrates the calculation of maximum power. When three 50-kVA transformers are connected in delta-delta configuration, the total capacity of the bank is their sum, or 150 kVA. For two transformers in an open-delta configuration, the capacity is 150 kVA/√3, or 86.6 kVA, which is the same as 86.6% of the total capacity of two transformers (0.866 x 100 kVA = 86.6 kVA). 2.5.2 Open-Wye - Open-Delta Connection Open-Wye-Open-Delta connection is very similar to open-delta connection except that the primary voltages are derived from two phases and neutral. A major disadvantage is that a very large return current must flow in the neutral of the primary circuit. 2.5.3 Scott-T Connection Scott-T connection is a way to derive two phases 90o apart from a three-phase power supply. Scott-T connection was developed to interconnect two-phase and three-phase power systems at early days when both systems were common. However, today two phase power is primarily used for certain specific control applications and electric railways. Scott-T connection needed two single phase transformers with identical ratings, with one having a tap on its primary winding at 86.6% of full load voltage, and the other having a tap on its primary winding at 50%. The connection shown below results in two voltages at the secondary windings to have a 90o phase difference. Phase relationships of the voltages are as shown in the phasor diagrams.

a2

c2

b2

One phase removed

a1

b1

c1

Vφ1 Vφ2

VL2 n1 VL1


Three-phase source Primary side secondary side 2.5.4 Three-Phase T Connection In three-phase T connection, both primary and secondary sides are connected in the same way as Scott-T connection primary. In this way, three-phase voltages are reconstructed at the secondary side. The advantage of this connection is that, if needed a neutral point can be obtained at the secondary side. This connection is sometime used in distribution transformers, as it can be constructed at a lower cost.

T2

T1

a1

86.6% tap

c1

Vp2 Vy

Vx

b1

Vp1

50% tap

Vab1

Vbc1 = Vp1

Vca1 Vp2

Vab1

Vbc1

Vca1

Vb1 Vc1

Va1

Vy

Vx

T2

T1

86.6% tap Vp2

Vp1

50% tap

a1

c1

b1

86.6% tap

50% tap

a2

c2

b2

n2

50% tap


2.6 Analysis of Three-Phase Transformers Circuit computations involving three-phase transformers under balanced conditions can be made by dealing with only one transformer or one phase, and recognizing the that conditions in the other phases are the same except for the phase displacements. It is convenient to carryout computations using a single phase basis (usually know as per phase basis). In dealing with Δ-Y, Y-Δ and Δ-Δ transformers, all impedances on Δ sides can be transformed to equivalent Y connected impedances. For a balanced Δ- connected system,

Δ= ZZY 3

1 Example – 1: Three identical single-phase transformers are each rated at 30 kVA, 200/40 kV, 60 Hz. They are connected to form a three-phase, step-down, Y-Δ transformer. The bank is energized with a 345 kV three-phase source. A 60 MVA, 0.9 pf lagging three phase load is connected to the secondary of the transformer bank. Neglecting exciting currents and voltage drops across the transformer, determine the primary and secondary voltages and currents. Example – 2: Three single-phase 100 kVA, 2400/240 V, 60 Hz transformers are connected to form a three-phase, 4160/240 V transformer bank. The equivalent impedance of each transformer referred to its low voltage side is 0.045+j0.16 Ω. The transformer is connected to a three-phase source through a three-phase feeder with an impedance of 0.5+j1.5 Ω/phase. The transformer delivers 250 kW at 240 V and 0.866 lagging power factor. Determine the transformer winding currents and the sending end voltage at the source. Example – 3: Three identical single-phase transformers, each rated at 12 kVA, 120/240 V, 60 Hz are connected to form a three-phase, step-up, Y-Δ connection. The parameters of the transformers are RmL= 240 Ω, XmL= 290 Ω, RL=39.5 mΩ, XL=1.5 mΩ, RH=133.5 mΩ and XH=201 mΩ (where the subscripts L and H denote the low and high voltage windings respectively). What are the nominal voltage, current, and power ratings of the three-phase transformer? When it delivers the rated load at the rated voltage and 0.8 pf lagging, determine the line voltages, line currents, and the efficiency of the transformer.


2.7 Use of Per Unit System for Transformer Analysis Solving circuits containing transformers is a somewhat tedious task, as it needs to refer all the different voltage levels on different sides of transformers to one common level. This need for explicit voltage level conversion can be eliminated by using per unit system. Definition of per unit (pu) quantities

quantity of valueBase valueActualpuin Quantity =

In definition of base quantities for a particular circuit, two base quantities are selected arbitrarily. Then all other base values are found by using electrical laws. The common practice is to select Apparent Power Base (Sbase) and Voltage Base (Vbase) arbitrarily, and then calculate Current Base (Ibase) and Impedance Base (Zbase). For a single phase system basebasebase IVS = and basebasebase SQP == )

base

basebase I

VZ = also ( )base

basebase S

VZ2

=

base

basebase V

IY =

In a power system, Sbase and Vbase are selected for a specific point. Voltage changes when going through a transformer. Therefore, Vbase also changes at every transformer. Transformer has no effect on apparent power, and thus Sbase remain unchanged. When one device such as a transformer or a generator is being analyzed, its own ratings are used as the base of the per-unit system. This has an advantage: when expressed in pu (based on its own ratings), parameters of the electrical machine or transformer falls within a narrow range. This is a very useful check in problem solution. For example, for a power transformer Req(pu) ≈ 0.01 pu Xeq(pu) ≈ 0.02 – 0.1 pu Rc(pu) ≈ 50-200 pu Xm(pu) ≈ 10 – 40 pu If more than one machine or transformer is involved, the entire system must have a common base.


Per Unit System for Three-Phase Transformers Sbase = Apparent power rating of the three-phase transformer bank

31base

baseSS =−φ

base

basebase V

SI

−

−− =

φ

φφ

1 base

basebase V

SI−

− ⋅=

φφ 3

( )

base

basebase S

VZ

−

−=φ

φ

1

2

( )

base

basebase S

VZ

23 −= φ

For a Y connected winding, VL = √3 Vφ, and IL = Iφ, basebaseL VV −− ⋅= φ3 basebaseL II −− = φ

baseL

basebaseL V

SI−

−⋅

=3

For a Δ connected winding, VL = Vφ, and IL =√3Iφ, basebaseL VV −− = φ

basebaseL II −− ⋅= φ3

baseL

basebaseL V

SI−

−⋅

=3

Example – 4: A 100 MVA, 230/115 kV, Y-Y connected, three-phase power transformer has a series resistance of 0.02 pu and a series reactance of 0.055 pu. The excitation branch elements are Rc=110 pu and Xm=20 pu. This transformer supplies 80 MVA load at 0.85 pf lagging, with load voltage equal to the rated secondary voltage.

a) Determine the efficiency of the transformer bank under the above conditions using the per unit basis for the calculation.

b) Sketch the per-phase equivalent circuit referred to the low voltage side. Indicate the actual values of all voltages, currents and impedances on the diagram.


2.8 Instrument Transformers Instrument transformers facilitate the measurement of high voltages and currents using very accurate standard low range voltmeters and ammeters. They also provide the safety in making measurements by electrically isolating meters from the primary circuits. 2.8.1 Potential (Voltage) transformers (PT, VT)

Potential transformer is a step down type transformer: Many turns on the primary winding connected to the HV circuit and few turns on the secondary winding, which is connected to the measuring instrument. Magnetic core of a potential transformer is usually shell type to provide better accuracy. One end of the secondary winding is usually grounded to provide adequate protection to the operator. Since the voltmeter behaves like an open circuit, the output current is almost zero: the power rating of the potential transformer is very low. 2.8.2 Current transformers (CT)

CT with a wound primary Clip-on type CT


Current transformer is designed to measure high currents in power systems. Primary has few winding of heavy wire; Secondary has many turns of fine wire. In clamp-on type current transformers, the current carrying conductor it self act as one-turn primary. Low range ammeter is connected across the secondary winding. Ammeter has very low impedance, and practically acts as a short circuit. Magnetizing current is almost negligible, and flux density is relatively low. Consequently, CT core is never saturated under normal operating conditions. A CT is designed to operate with a short circuited (or very low impedance) secondary winding. It should never be left open. If the secondary is left open, the primary winding is still carrying a current (primary circuit current do not depend on the CT burden). Since there is no secondary current to counteract its emf, core flux may increases to very high level. As a result, a dangerously high voltage can induced on the secondary side. Ratio and phase angle errors introduced by the instrument transformers must be minimized. Therefore, they are designed to approximate the ideal transformers as closely as practical. 2.9 Inrush Currents

)sin()()( tVtvte m ω==

∫= dttVt mN p)sin()( 1 ωφ )cos()( t

NVt

p

m ωω

φ −=

Under steady state:

If the switch S is closed at voltage peak, and if the initial flux in the core is zero,

tt

p

m tN

Vt °=−=− 90)][cos(0)( ωωω

φ

)cos(]0)[cos()( tN

VtN

Vtp

m

p

m ωω

ωω

φ −=−−= p

m

NVω

φ =max

φ(t)

+ v(t) _

S

+ e(t) _


The maximum flux in the core is equal to the normal value of φmax under steady-state. If the switch S is closed at voltage zero (assume that the initial flux in the core is zero)

tt

p

m tN

Vt °=−=− 0)][cos(0)( ωωω

φ

)]cos(1[]1)[cos()( tN

VtN

Vtp

m

p

m ωω

ωω

φ −=−−= p

m

NV

ωφ 2

max =

Core flux in this case, which could be twice the normal flux, will result in sever core

saturation, and hence a large magnetizing current, which could even exceed the rated current of the transformer.

φ

Im1

p

m

NV

ω2

p

m

NVω

Im2


The resulting magnetizing current

Closing at voltage peak: best situation Closing at voltage zero: worst situation Any other point on the voltage waveform: gives some inrush current below the peak inrush current

If there is a residual flux in the core, the situation may become worse or better depending

on the direction of residual flux. For example, when closed at voltage zero

)]cos(1[)( tN

Vtp

mres ω

ωφφ −+=

Inrush currents will die down eventually due to circuit resistance and core losses. However, the circuit equipment must be designed to withstand the inrush currents. Inrush current can affect the operation of transformer protection equipment.

2.10 Transformer Ratings

When specifying the ratings of a three-phase transformer, the power rating normally refers

to the three-phase power and the voltage ratings refer to line-to-line voltages of the primary and secondary sides.

2.10.1 Voltage and frequency ratings Voltage rating serves two functions:

• Protect the winding insulation from beak down due to excessive voltage applied to it • Limit the magnetizing current magnitude, which could be quite large if the core is

saturated )sin()( tVtv m ω=

)cos()( tN

Vtp

m ωω

φ −=


If Vm is increased by 10%, the resulting φmax also increases by 10%. If the operating point is shifted to saturation region, 10% increase in φmax will cause much higher increase in the magnetizing current. Maximum value of flux also depends on the frequency.

p

m

NVω

φ =max

If the frequency is reduced, φmax increases. Therefore, if the frequency is reduced, the voltage must also be simultaneously reduced so that the ratio

tconsf

Vm tan=

is maintained at a constant value. That is if a 60 Hz transformer is to be operated on 50Hz, its applied voltage must also be reduced by a factor of 5/6. 2.10.1 Apparent power rating When taken with voltage rating, the apparent power rating specifies the current flow through the transformer. Usually, the output apparent power is specified. Apparent power rating limits the I2R losses in the windings so that the transformer is not damaged due to overheating. If the voltage rating is reduced due to some reason, apparent power rating must also be reduced by the same percentage.

02_3phtransformers

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