Three phase transformer is divided into four main groups according to the phase difference between the corresponding line voltage on the high voltage side and the low voltage sides. The phase difference is the angle by which the low voltage line lags the high line voltage, and is measured in units of 30° in clockwise direction. These groups are

• Group number 1 – no phase displacement
• Group number 2 – 180° phase displacement.
• Group number 3 – (-30°) phase displacement.
• Group number 4 – (+ 30°) phase displacement.

The connection Y d 11 gives the following information – Y indicates that the high voltage is connected to star and d indicates the low voltage is connected in delta. The 11 indicates that the low line voltage lag, high line voltage by 11 Χ 30° = 330° measured from higher voltage phasor in a clockwise direction.

The phasor differences can also be measured by using the clock methods. Consider the minute hand of the clock shown the high voltage and the low voltage winding is represented by the hour hand. The angle of 30° is the angle between two adjacent figures on the clock dial and is taken as the unit of dial shift.

When the hour hand of the clock is at 12, then the phase displacement is zero. When the hour hand is at 1 then the phase shift -30° degree. At 6 the phase shift is 6 Χ 30º = 180º. Similarly, when the hour hand is at 11 the phase shift is 11 Χ 30º  = 330º.

The number 0, 6, 1, and 11 in the group reference number indicates the primary to secondary phase shift regarding the hours of the clock. The connection designated by D y 11 is the delta-star transformer in which the low voltage line phasor is at 11 and is a phase advanced of +30° on the corresponding line voltage on the high voltage side.

Note: The only transformer in the same group may be connected in parallel. For example, a star-star, 3-phase transformers can be parallel with another three phase transformer whose windings are either connected in Y-Y or ∆-∆. The ∆-∆ transformer cannot be parallel with Y-∆ transformer.

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1. Three separate single phase transformer is suitably connected for three phase operation.
2. A single three-phase transformer in which the cores and windings for all the three phases are merged into a single structure.

The three single-phase transformer can be used as a three-phase transformer when their primary and secondary winding are connected to each other. The three phase transformer supply has many advantages as compared to three single phase units like it requires very less space and also very lighter smaller and cheaper in size. The three phase transformer is mainly classified into two types, i.e., the core type transformer and the shell type transformer.

Core Type Three Phase Transformer

Consider a three single phase core type transformer positioned at 120° to each other as shown in the figure below. If the balanced three-phase sinusoidal voltages are applied to the windings, the fluxes φ_a, φ_b and φ_c will also be sinusoidal and balanced. If the three legs carrying these fluxes are combined, the total flux in the merged leg becomes zero. This leg can, therefore, be removed because it carries the no flux. This structure is not convenient for the core.

The core of the three phase transformer is usually made up of three limbs in the same plane. This can be built using stack lamination. The each leg of this core carries the low voltage and high voltage winding. The low voltage windings are insulated from the core than the high voltage windings.

The low windings are placed next to the core with suitable insulation between the core and the low voltage windings. The high voltage windings are placed over the low voltage windings with suitable insulation between them. The magnetic paths of the leg a and c are greater than that of leg b, the construction is not symmetrical, and there is a resultant imbalance in the magnetising current.

Shell type Three Phase Transformer

The shell type 3-phase transformer can be constructed by stacking three single phase shell transformer as shown in the figure below. The winding direction of the central unit b is made opposite to that of units a and c. If the system is balanced with phase sequence a-b-c, the flux will also be balanced

The magnitude of this combined flux is equal to the magnitude of each of its components. The cross section area of the combined yoke is same as that of the outer leg and top and bottom section of the yoke. The imbalance in the magnetic path has very little effect on the performance of the three shell-type transformers. The windings of the shell type three phase transformer are either connected in delta or star as desired.

Also see; Three-Transformer Connections.

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1. Δ – Δ (Delta – Delta) Connection
2. Υ – Υ (Star – Star) Connection
3. Δ – Υ (Delta – Star) Connection
4. Υ – Δ (Star – Delta ) Connection

The choice of connection of three phase transformer depends on the various factors likes the availability of a neutral connection for grounding protection or load connections, insulation to ground and voltage stress, availability of a path for the flow of third harmonics, etc. The various types of connections are explained below in details.

1. Delta-Delta (Δ-Δ) Connection

The delta-delta connection of three identical single phase transformer is shown in the figure below. The secondary winding a₁a₂ is corresponding to the primary winding A₁A₂, and they have the same polarity. The polarity of the terminal a connecting a₁ and c₂ is same as that connecting A₁ and C₂. The figure below shows the phasor diagram for lagging power factor cosφ.

The magnetising current and voltage drops in impedances have been neglected. Under the balanced condition, the line current is √3 times the phase winding current. In this configuration, the corresponding line and phase voltage are identical in magnitude on both primary and secondary sides.

The secondary line-to-line voltage is in phase with the primary line-to-line voltage with a voltage ratio equal to the turns ratio.

If the connection of the phase windings is reversed on either side, the phase difference of 180° is obtained between the primary and the secondary system. Such a connection is known as an 180º connection.

The delta-delta connection with 180º phase shift is shown in the figure below. The phasor diagram of a three phase transformer shown that the secondary voltage is in phase opposition with the primary voltage.

The delta-delta transformer has no phase shift associated with it and problems with unbalanced loads or harmonics.

Advantages of delta–delta connection of transformer

The following are the advantages of the delta-delta configuration of transformers.

1. The delta-delta transformer is satisfactory for a balanced and unbalanced load.
2. If one transformer fails, the remaining two transformers will continue to supply the three-phase power. This is called an open delta connection.
3. If third harmonics present, then it circulates in a closed path and therefore does not appear in the output voltage wave.

The only disadvantage of the delta-delta connection is that there is no neutral. This connection is useful when neither primary nor secondary requires a neutral and the voltage are low and moderate.

2. Star-Star (Υ-Υ) Connection of Transformer

The star-star connection of three identical single phase transformer on each of the primary and secondary of the transformer is shown in the figure below.The phasor diagram is similar as in delta-delta connection.

The phase current is equal to the line current, and they are in phase. The line voltage is three times the phase voltage. There is a phase separation of 30º between the line and phase voltage.The 180º phase shift between the primary and secondary of the transformer is shown in the figure above.

Problems Associated With Star-Star Connection

The star-star connection has two very serious problems. They are

1. The Y-Y connection is not satisfactory for the unbalance load in the absence of a neutral connection. If the neutral is not provided, then the phase voltages become severely unbalance when the load is unbalanced.
2. The Y-Y connection contains a third harmonics, and in balanced conditions, these harmonics are equal in magnitude and phase with the magnetising current. Their sum at the neutral of star connection is not zero, and hence it will distort the flux wave which will produce a voltage having a harmonics in each of the transformers

The unbalanced and third harmonics problems of Y-Y connection can be solved by using the solid ground of neutral and by providing tertiary windings.

3. Delta-Star (Δ-Υ) Connection

The ∆-Y connection of the three winding transformer is shown in the figure below. The primary line voltage is equal to the secondary phase voltage. The relation between the secondary voltages is V_LS= √3 V_PS.

The phasor diagram of the ∆-Y connection of the three phase transformer is shown in the figure below. It is seen from the phasor diagram that the secondary phase voltage V_an leads the primary phase voltage  V_AN by 30°. Similarly, V_bn leads V_BN by 30º and V_cn leads V_CN by 30º.This connection is also called +30º connection.

By reversing the connection on either side, the secondary system voltage can be made to lag the primary system by 30°. Thus, the connection is called -30° connection.

4. Star-Delta (Υ-Δ) Connection

The star-delta connection of three phase transformer is shown in the figure above. The primary line voltage is √3 times the primary phase voltage. The secondary line voltage is equal to the secondary phase voltage. The voltage ratio of each phase is

Therefore line-to-line voltage ratio of Y-∆ connection is

The phasor diagram of the configuration is shown in the figure above. There is a phase shift of 30 lead exists between respective phase voltage. Similarly, 30° leads exist between respective phase voltage. Thus the connection is called +30º connection.

The phase shows the star-delta connection of transformer for a phase shift of 30° lag. This connection is called – 30° connection. This connection has no problem with the unbalanced load and thirds harmonics. The delta connection provided balanced phase on the Y side and provided a balanced path for the circulation of third harmonics without the use of the neutral wire.

Open delta or V-V Connection

If one transformer of delta-delta connection is damaged or accidentally opened, then the defective transformer is removed, and the remaining transformer continues to work as a three phase bank. The rating of the transformer bank is reduced to 58% of that of the actual bank. This is known as the open delta or V-V delta. Thus, in open winding transformer, two transformers are used instead of three for the 3-phase operation.

Let the V_ab, V_bc and V_ca be the voltage applied to the primary winding of the transformer. The voltage induced in the transformer secondary or on winding one is V_ab. The voltage induced on the low voltage winding two is V_bc. There is no winding between points a and c. The voltage may be found by applying  KVL around a closed path made up of point a, b, and c. Thus,

Let, 

Where V_p is the magnitude of the line on the primary side.

On substituting the value of V_ab and V_bc in equation, we get

The V_ca is equal in magnitude from the secondary terminal voltage and 120º apart in time from both of them. The balanced three phase line voltage produced balanced 3-phase voltage on the secondary side.

If the three transformers are connected in delta-delta configuration and are supplying rated load and if the connection becomes V-V transformer, the current in each phase winding is increased by √3 times. The full line current flows in each of the two phase windings of the transformer. Thus the each transformer in the V-V system is overloaded by 73.2%.

It should be noticed that the load should be reduced by √3 times in case of an open delta connected transformer. Otherwise, serious overheating and breakdown of the two transformers may take place.

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Construction of an Electrical Transformer

The primary winding, secondary winding and the magnetic core are the three important of the transformer. These coils are insulated from each other. The main flux is induced in the primary winding of the transformer. This flux passes through the low reluctance path of the magnetic core and linked with the secondary winding of the transformer.

Transformer Working

Consider the T₁ and T₂ are the numbers of the turn on the primary and the secondary winding of the transformer shown in the figure above. The voltage is applied to the primary winding of the transformer because of which the current is induced in it. The current causes the magnetic flux which is represented by the dotted line in the above figure.

The flux induces in the primary winding because of self-induction. This flux is linked with the secondary winding because of mutual induction. Thus, the emf is induced in the secondary winding of the transformer. The power is transferred from the primary winding to the secondary winding. The frequency of the transferred energy also remains same.

EMF Equation of an Electrical Transformer

The emf induced in each winding of the transformer can be calculated from its emf equation. 

The linking of the flux is represented by the faraday law of electrmagentic induction. It is expressed as,

The above equation may be written as,

where E_m = 4.44ωΦ_m = maximum value of e. For a sine wave, the r.m.s value of e.m.f is given by 

The emf induced in their primary and secondary winding is expressed as,

The secondary RMS voltage is

Where φ_m is the maximum value of flux in Weber (Wb), f is the frequency in hertz (Hz) and E₁ and E₂ in volts.

If, B_m = maximum flux density in the magnetic circuit in Tesla (T)

A = area of cross-section of the core in square meter (m²)

The winding which has the higher number of voltage has high voltage while the primary winding has low voltage.

Voltage Ratio and Turns Ratio

The ratio of E/T is called volts per turn. The primary and secondary volts per turns is given by the formula 

The equation (1) and (2) shows that the voltage per turn in both the winding is same, i.e.

The ratio T₁/T₂ is called the turn ratio. The turn ratio is expressed as 

The ratio of primary to secondary turn which equals to primary to secondary induced voltage indicates how much the primary voltage lowered or raised. The turn ratio or induced voltage ratio is called the transformation ratio, and it is denoted by the symbol a. Thus,

The any desired voltage ratio can be obtained by shifting the number of turns.

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Let the primary resistance R₁ be transferred to the secondary side, and the new value of this resistance be R’₁. The R’₁ is called the equivalent resistance of primary referred to secondary side as shown in the figure below. I₁ and I₂ are the full loads primary and secondary current respectively.

Then,

Total equivalent resistance referred to secondary,

Now consider resistance R₂, when it is transferred to primary, the value of the new resistance is R’₂. The R’₂ is called the equivalent resistance of the of secondary referred to as primary as shown in the figure below.

Then,

Total equivalent resistance referred to primary,

If the winding of the transformer is connected in star then their resistance will be measured between the neutral and line.

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A transformer in which the output (secondary) voltage is greater than its input (primary) voltage is called a step-up transformer. The step-up transformer decreases the output current for keeping the input and output power of the system equal.

Considered a step-up transformer shown in the figure below. The E₁ and E₂ are the voltages, and T₁ and T₂ are the number of turns on the primary and secondary winding of the transformer.

The number of turns on the secondary of the transformer is greater than that of the primary, i.e., T₂ > T₁.Thus the voltage turn ratio of the step-up transformer is 1:2. The primary winding of the step-up transformer is made up of thick insulated copper wire because the low magnitude current flows through it.

Applications – Step-up transformer is used in transmission lines for transforming the high voltage produced by the alternator.The power loss of the transmission line is directly proportional to the square of the current flows through it.

Power = I²R

The output current of the step-up transformer is less, and hence it is used for reducing the power loss. The step-up transformer is also used for starting the electrical motor, in the microwave oven, X-rays machines, etc.

Step-down Transformer

A transformer in which the output (secondary) voltage is less than its input (primary) voltage is called a step-down transformer. The number of turns on the primary of the transformer is greater than the turn on the secondary of the transformer, i.e., T₂ < T₁. The step-down transformer is shown in the figure below.

The voltage turn ratio of the step-down transformer is 2:1. The voltage turn ratio determines the magnitude of voltage transforms from primary to secondary windings of the transformer.

Step-down transformer is made up of two or more coil wound on the iron core of the transformer. It works on the principle of magnetic induction between the coils. The voltage applied to the primary of the coil magnetise the iron core which induces the secondary windings of the transformer. Thus the voltage transforms from primary to the secondary winding of the transformer.

Applications – It is used for electrical isolation, in a power distribution network, for controlling the home appliances, in a doorbell, etc.

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The figure below shows the Scott-T transformer connection. The main transformer is centre tapped at D and is connected to the line B and C of the 3-phase side. It has primary BC and secondary a₁a₂. The teaser transformer is connected to the line terminal A and the centre tapping D. It has primary AD and the secondary b₁b₂

The identical, interchangeable transformers are used for Scott-T connection in which each transformer has a primary winding of T_p turns and is provided with tapping at 0.289T_p , 0.5T_pand 0.866 T_p.

Phasor Diagram of Scott Connection Transformer

The line voltages of the 3-phase system V_AB, V_BC, and V_CA which are balanced are shown in the figure below. The same voltage is shown as a closed equilateral triangle.The figure below shows the primary windings of the main and the teaser transformer.

The D divides the primary BC of the main transformers into two halves and hence the number of turns in portion BD = the number of turns in portion DC = T_p/2.The voltage V_BD and V_DC are equal, and they are in phase with V_BC.

The voltage between A and D is

The teaser transformer has the primary voltage rating that is √3/2 or 0.866 of the voltage ratings of the main transformer. Voltage V_AD is applied to the primary of the teaser transformer and therefore the secondary of the voltage V_2t of the teaser transformer will lead the secondary terminal voltage V_2m of the main transformer by 90º as shown in the figure below.

Then, 

For keeping the voltage per turn same in the primary of the main transformer and the primary of the teaser transformer, the number of turns in the primary of the teaser transformer should be equal to √3/2T_p.

Thus, the secondaries of both transformers should have equal voltage ratings.The V_2t and V_2m are equal in magnitude and 90º apart in time; they result in the balanced 2-phase system.

Position of Neutral Point N

The primary of the two transformers may have a four wire connection to a 3-phase supply if the tapping N is provided on the primary of the teaser transformer such that

The voltage across AN = V_AN = phase voltage = V_l/√3.

Since the voltage across the portion AD.

the voltage across the portion ND

The same voltage turn in portion AN, ND and AD are shown by the equations,

The equation above shows that the neutral point N divides the primary of the teaser transformer in ratio.

AN : ND = 2 : 1

Applications of Scott Connection

The following are the applications of the Scott-T connection.

1. The Scott-T connection is used in an electric furnace installation where it is desired to operate two single-phase together and draw the balanced load from the three-phase supply.
2. It is used to supply the single phase loads such as electric train which are so scheduled as to keep the load on the three phase system as nearly as possible.
3. The Scott-T connection is used to link a 3-phase system with a two–phase system with the flow of power in either direction.

The Scott-T connection permits conversions of a 3-phase system to a two-phase system and vice versa. But since 2-phase generators are not available, the converters from two phases to three phases are not used in practice.

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The voltage ratings of all the three windings of the transformer are usually unequal.The primary winding has the highest voltage rating; the tertiary has the lowest voltage rating, and the secondary has the intermediate voltage rating.

The chief advantages of the three winding transformers is an economy of construction and their great efficiency. The schematic diagram of a three-phase transformer is shown in the figure below.

For an ideal transformer,

The most significant advantage of the third winding is are that the harmonic generated by the primary and secondary winding extinguish by the third winding. The third winding is connected in delta.

The voltage of the tertiary winding differs than the primary and secondary winding. Thus, it is used for supplying the power to the auxiliary appliances like the fan, tube light, etc. of the substations. The tertiary winding is used for following applications.

• The reactive power is supplied to the substations with the help of the tertiary winding.
• The tertiary winding reduces the impedance of the circuit so that the fault current easily passes to the ground.
• It is used for testing the high rating transformer.

Equivalent Circuit of a Three Winding Transformer

The equivalent circuit diagram of the three-phase transformer is shown in the figure. Consider the R₁, R₂ and R₃ are the resistance and the X₁, X₂ and X₃ are the impedance of their windings.

The V₁, V₂, V₃ are the voltages and the I₁, I₂, I₃ are current flows through their windings.

Determination of Parameters of Three-Windings Transformers

The parameters of the equivalent circuit can be determined from the open circuit and the three short-circuit tests.

Short Circuit Test

Consider the Z₁, Z₂ and Z₃ are the impedances of the three winding transformers. These impedances are considered as the base for performing the short circuit test. For the short-circuit test, the two winding is short circuit and the third winding is kept open.

In the first step, consider the winding 1 and 2 are short-circuited. The low voltage winding is applied to the winding 1 due to which the full load current flows through the winding 2. The Z₁₂ indicates the impedance of winding 1 and 2 and it measured as

Equivalent resistance,

Equivalent leakage reactance,

The Z₁₂ is the series combination of Z₁ and Z₂ respectively,

In the second step, the third winding is short-circuited with the second winding and the first winding is kept open. The low voltage source is applied across the third winding so that the full load current flows through the second winding. The Z₂₃ represents the impedance of the winding 2 and 3 and the below equation expresses it

In the third step, the second winding is opened and first and third winding are short-circuited. The low voltage is supplied to the third winding, and full load current flows through the first windings. The Z₁₃ is the impedance of the first and third winding.

Solving equation (1), (2) and (3) we get the leakage impedance Z₁, Z₂ and Z₃ all referred to as primary,

Open-Circuit Test

The open circuit test is carried out to determine the core loss, magnetising impedance and turn ratios. In open circuit test the voltmeter, ammeter and wattmeter are connected in low voltage winding. The secondary side is kept open, and the voltmeter is connected.

Since the high voltage side is opened the current drawn by the primary is no load current and I₀ measured by the ammeter A. The magnetising impedance may be found by exciting current  winding 1 with both winding 2 and 3 be open circuit. Then we have,

The voltage regulation of a three-winding transformer is defined as the ratio of the magnitude of the actual kVA loading of the winding to the base kVA used in determining the network parameters.

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In a three-phase transformer, the non-sinusoidal nature of magnetising current produces sinusoidal flux which gives rise to the undesirable phenomenon. The phase magnetising currents in transformer should contain third harmonics and higher harmonics necessary to produce a sinusoidal flux.

If the phase voltage across each phase is to remain sinusoidal, then the phase magnetising currents must be of the following form.

It is seen from equation (1), (2), and (3) that the third harmonics in the three currents are co-phase, that is they have the same phase. The fifth harmonics have different phases.

Delta Connection

Let the I_AO, I_BO and the I_CO represent the phase magnetising current in a delta connection. The line currents can be found by subtracting two phases current. For examples,

The third harmonic present in the phase magnetising current of three phase transformer is not present in the line current. The third harmonic components are co-phase and hence cancel out in the line. The third harmonic components are flows rounds the closed loop of the delta.

The delta connection only allows a sinusoidal flux and voltage with no third harmonic current in the transmission line. For this reason majority of the 3-phase transformer has delta connected windings and in places where it is not convenient to have an either primary or secondary connected in delta, a tertiary winding is provided. The tertiary windings carry the circulating third harmonics current required by the sinusoidal flux in each limb of the core.

In a delta connection, the voltage acting around the closed delta is,

This is a third harmonic voltage and it will circulate a third harmonic current round the closed loop of the delta

Star connection

If I_AO, I_BO and I_CO, represents the phase magnetising current in a star connection,

Where I_n is the current in the neutral wire.

The harmonics above the seventh be neglected. The equation (6) shows that under the balanced condition the current flow in the neutral wire is the third harmonic current. The magnitude of the third harmonics current is thrice the magnitude of each third phase current. The thirds harmonic current produced inductive interference with communication circuit. If the supply to the star connection is three wires, neutral current must be zero and therefore

Thus, it is seen that the three wire star connection suppresses the flow of harmonic and magnetising currents. For a four wire star-connected system, the in phase third harmonic current flow in the neutral wire.

Similarly, the third balance phase voltage containing harmonics can be written as

The equation (7), (8) and (9) shows that the third harmonics in the three phase voltage have the same phase. The line voltage in a star connection can be obtained by subtracting two phase voltages. For example

From equation (10) it is seen that the third harmonics is not present in the line to line voltage of a star connection. This applies to all triplers harmonics.

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Let a sinusoidal voltage

 V₁ be applied to a transformer, the secondary of which is an open circuit. Here α the angle of the voltage sinusoid at t = 0. Suppose the core loss and primary resistance be neglected, then Where T₁ is the number of turns and Φ is the flux in the core. In the steady state

From equation (1) and equation (2), we get,

From equation (3) and (4)

Integration of the equation (5) gives

Where Φ_c is the constant or integration to be found from an initial condition at t = 0. Considered that when the transformer is last disconnected from the supply line, a small residual flux Φ_r remained in the core. Thus, at t = 0, Φ = Φ_r.

Substituting this value in equation (6) we get

Equation (6) then becomes

The equation (8) shows that the flux consists of two components, the steady state component Φ_ss and the transient component Φ_c. The magnitude of the transient component

Φ_c is a function of α, where α is the instant at which the transformer is switched on to the supply.If the transformer is switched on at α = 0, then cosα = 1.

Under this condition

At ωt = π,

Thus the core flux attains the maximum value of flux equal to (2φ_m+φ_r) which is over twice the normal flux. This is known as double effecting. Due to this double effect, the core goes into deep saturation. The magnetising current required for producing such a large flux in the core may be as large as ten times the normal magnetising current.

Sometimes the RMS value of magnetising current is larger than the primary rated current of the transformer. This current may produce an electromagnetic force which is about twenty-five times the normal value. Therefore the winding of the transformer is strongly braced. The improper operation of protective devices like unwarranted tripping of relays, momentary large voltage drops and large humming due to magnetostriction of the core.

To obtain no transient inrush current, Φ_c should be zero.

Since Φ_r is usually very small cosα ≅ 0 and α ≅ nπ/2

In other words, if the transformer is connected to the supply line near a positive or negative maximum voltage, the inrush current will be minimised. But usually, it is impractical to connect a transformer at a predetermined time in the voltage cycle

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