Three-Phase Transformers GATE EE Exam Quick Notes

Three-Phase Transformer Connections

Types of Three-Phase Transformer Connections

  • Star-Star (Y-Y) Connection

  • Delta-Delta (\(\Delta\)-\(\Delta\)) Connection

  • Star-Delta (Y-\(\Delta\)) Connection

  • Delta-Star (\(\Delta\)-Y) Connection

GATE Focus: Each connection has specific voltage/current ratios, phase relationships, and applications. Understanding these is crucial for problem-solving.

Star-Star (Y-Y) Connection

Characteristics:

  • \(V_L = \sqrt{3} V_{ph}\) (both sides)

  • \(I_L = I_{ph}\) (both sides)

  • Neutral available

  • Third harmonic problems

  • Requires neutral grounding

Relations:

\[\begin{aligned} \dfrac{V_{L1}}{V_{L2}} &= \dfrac{N_1}{N_2} \\ \dfrac{I_{L1}}{I_{L2}} &= \dfrac{N_2}{N_1} \\ \text{Phase shift} &= 0^{\circ} \end{aligned}\]

GATE Tip: Y-Y connection is rarely used in practice due to harmonic issues. Vector group: Yy0 or Yy6.

Delta-Delta (\(\Delta\)-\(\Delta\)) Connection

Characteristics:

  • \(V_L = V_{ph}\) (both sides)

  • \(I_L = \sqrt{3} I_{ph}\) (both sides)

  • No neutral point

  • Good for unbalanced loads

  • Open delta possible

Relations:

\[\begin{aligned} \dfrac{V_{L1}}{V_{L2}} &= \dfrac{N_1}{N_2} \\ \dfrac{I_{L1}}{I_{L2}} &= \dfrac{N_2}{N_1} \\ \text{Phase shift} &= 0^{\circ} \end{aligned}\]

GATE Tip: Open delta operation at 57.7% capacity. Vector group: Dd0 or Dd6.

Star-Delta (Y-\(\Delta\)) Connection

Characteristics:

  • Step-down configuration

  • Primary: \(V_L = \sqrt{3} V_{ph}\)

  • Secondary: \(V_L = V_{ph}\)

  • Phase shift: \(\pm 30^{\circ}\)

  • Good for stepping down

Relations:

\[\begin{aligned} \dfrac{V_{L1}}{V_{L2}} &= \sqrt{3} \times \dfrac{N_1}{N_2} \\ \dfrac{I_{L1}}{I_{L2}} &= \dfrac{1}{\sqrt{3}} \times \dfrac{N_2}{N_1} \\ \text{Phase shift} &= \pm 30^{\circ} \end{aligned}\]

GATE Important: Vector groups: Yd1 (lag \(30^{\circ}\)) or Yd11 (lead \(30^{\circ}\)).

Delta-Star (\(\Delta\)-Y) Connection

Characteristics:

  • Step-up configuration

  • Primary: \(V_L = V_{ph}\)

  • Secondary: \(V_L = \sqrt{3} V_{ph}\)

  • Phase shift: \(\pm 30^{\circ}\)

  • Good for stepping up

Relations:

\[\begin{aligned} \dfrac{V_{L1}}{V_{L2}} &= \dfrac{1}{\sqrt{3}} \times \dfrac{N_1}{N_2} \\ \dfrac{I_{L1}}{I_{L2}} &= \sqrt{3} \times \dfrac{N_2}{N_1} \\ \text{Phase shift} &= \pm 30^{\circ} \end{aligned}\]

GATE Important: Vector groups: Dy1 (lead \(30^{\circ}\)) or Dy11 (lag \(30^{\circ}\)).

Vector Groups and Phase Displacement

Vector Group Notation (IEC Standard)

Notation Format: XxN

  • X: Primary connection (Y, D, Z)

  • x: Secondary connection (y, d, z)

  • N: Clock number (0-11, each step = \(30^{\circ}\))

Common Vector Groups for GATE:

Vector Group Phase Shift Application
Yy0, Dd0 \(0^{\circ}\) Distribution/Transmission
Yd1, Dy11 \(-30^{\circ}\) (lag) Step-down/Step-up
Yd11, Dy1 \(+30^{\circ}\) (lead) Step-down/Step-up
Yy6, Dd6 \(180^{\circ}\) Special applications

Phase Displacement Rules

Memory Tricks for GATE:

  • Clock Rule: Secondary voltage vector position on clock

  • \(30^{\circ}\) Rule: Y-\(\Delta\) and \(\Delta\)-Y always have \(\pm 30^{\circ}\) shift

  • Lag/Lead Rule:

    • Yd1 and Dy11: Secondary lags by \(30^{\circ}\)

    • Yd11 and Dy1: Secondary leads by \(30^{\circ}\)

GATE Formula: Phase displacement = Clock number \(\times 30^{\circ}\)

\[\begin{aligned} \text{For Yd1:} \quad \phi = 1 \times 30^{\circ} = -30^{\circ} \text{ (lag)} \\ \text{For Dy11:} \quad \phi = 11 \times 30^{\circ} = -30^{\circ} \text{ (lag)} \end{aligned}\]

Parallel Operation

Conditions for Parallel Operation

Essential Conditions (Must be satisfied):

  1. Same voltage ratio (primary and secondary voltages)

  2. Same vector group (identical phase displacement)

  3. Same phase sequence (RYB or RBY)

  4. Same frequency (50/60 Hz)

Desirable Conditions (for optimal operation):

  1. Similar percentage impedance

  2. Similar X/R ratio

  3. Similar kVA ratings

GATE Fact: Violation of essential conditions prevents parallel operation or causes circulating currents.

Load Sharing in Parallel Operation

Current sharing based on impedances:

\[\begin{aligned} I_1 &= \dfrac{Z_2}{Z_1 + Z_2} \times I_{total} \\ I_2 &= \dfrac{Z_1}{Z_1 + Z_2} \times I_{total} \end{aligned}\]

For equal current sharing: \(Z_1 = Z_2\) (equal % impedance)

kVA sharing formula:

\[\begin{aligned} \text{kVA}_1 &= \dfrac{Z_2}{Z_1 + Z_2} \times \text{Total kVA} \\ \text{kVA}_2 &= \dfrac{Z_1}{Z_1 + Z_2} \times \text{Total kVA} \end{aligned}\]

GATE Tip: Lower impedance transformer carries more load.

Circulating Current

Causes:

  • Unequal voltage ratios

  • Different vector groups

  • Phase sequence mismatch

  • Unequal frequencies

Circulating current magnitude:

\[\begin{aligned} I_c = \dfrac{|\vec{E_1} - \vec{E_2}|}{Z_1 + Z_2} \end{aligned}\]

Where \(\vec{E_1}, \vec{E_2}\) are secondary side EMF phasors.

GATE Important: Circulating current exists even at no-load, causing:

  • Additional copper losses

  • Heating of transformers

  • Reduced efficiency

Per-Unit System and Equivalent Circuits

Per-Unit System for Three-Phase Transformers

Base quantities:

\[\begin{aligned} S_{base} &= \text{kVA rating} \\ V_{base} &= \text{Line voltage} \\ I_{base} &= \dfrac{S_{base}}{\sqrt{3} \times V_{base}} \\ Z_{base} &= \dfrac{V_{base}^2}{S_{base}} \end{aligned}\]

Per-unit impedance:

\[\begin{aligned} Z_{pu} = \dfrac{Z_{actual}}{Z_{base}} = \dfrac{\% Z}{100} \end{aligned}\]

GATE Advantage: Per-unit values are same on both primary and secondary sides.

Equivalent Circuit

Single-phase equivalent circuit parameters:

  • \(R_1, R_2\): Primary and secondary resistances

  • \(X_1, X_2\): Primary and secondary reactances

  • \(R_c\): Core loss resistance

  • \(X_m\): Magnetizing reactance

Referred to primary side:

\[\begin{aligned} R_2' &= a^2 R_2 \\ X_2' &= a^2 X_2 \\ V_2' &= a V_2 \\ I_2' &= \dfrac{I_2}{a} \end{aligned}\]

Where \(a = \dfrac{N_1}{N_2}\) is the turns ratio.

Testing and Losses

Open Circuit Test

Purpose: Determine core loss and magnetizing parameters

Test setup: LV side energized, HV side open

Measurements: \(V_0, I_0, W_0\)

Calculations:

\[\begin{aligned} R_c &= \dfrac{V_0^2}{W_0} \\ |Z_0| &= \dfrac{V_0}{I_0} \\ X_m &= \dfrac{V_0^2}{\sqrt{V_0^2 I_0^2 - W_0^2}} \end{aligned}\]

GATE Tip: Core loss is constant and independent of load.

Short Circuit Test

Purpose: Determine copper loss and leakage parameters

Test setup: HV side energized, LV side short-circuited

Measurements: \(V_{sc}, I_{sc}, W_{sc}\)

Calculations:

\[\begin{aligned} R_{eq} &= \dfrac{W_{sc}}{3 I_{sc}^2} \\ |Z_{eq}| &= \dfrac{V_{sc}}{\sqrt{3} I_{sc}} \\ X_{eq} &= \sqrt{Z_{eq}^2 - R_{eq}^2} \end{aligned}\]

GATE Tip: Copper loss varies as square of current ( \(I^2R\)).

Efficiency and Regulation

Efficiency:

\[\begin{aligned} \eta = \dfrac{\text{Output}}{\text{Input}} = \dfrac{V_2 I_2 \cos\phi_2}{V_2 I_2 \cos\phi_2 + \text{Losses}} \end{aligned}\]

Maximum efficiency occurs when:

\[\begin{aligned} \text{Copper loss} = \text{Core loss} \end{aligned}\]

Voltage regulation:

\[\begin{aligned} \text{Regulation} = \dfrac{V_2(\text{no load}) - V_2(\text{full load})}{V_2(\text{full load})} \times 100\% \end{aligned}\]

GATE Formula: \(\text{Regulation} = \dfrac{I_2 R_{eq} \cos\phi \pm I_2 X_{eq} \sin\phi}{V_2} \times 100\%\)

(+ for lagging, - for leading power factor)

Special Topics

Auto-transformers

Characteristics:

  • Single winding with taps

  • Lower cost and losses

  • Higher efficiency

  • No electrical isolation

Key relations:

\[\begin{aligned} \text{Transformation ratio} &= \dfrac{V_1}{V_2} = \dfrac{N_1}{N_2} = K \\ \text{Current ratio} &= \dfrac{I_1}{I_2} = \dfrac{1}{K} \\ \text{Copper saving} &= \left(1 - \dfrac{1}{K}\right) \times 100\% \end{aligned}\]

GATE Tip: Auto-transformers are economical for transformation ratios close to 1.

Tap Changing

Types:

  • Off-load tap changer: Manual, transformer de-energized

  • On-load tap changer (OLTC): Automatic, under load

Purpose: Voltage regulation under varying load conditions

Typical range: \(\pm 10\%\) to \(\pm 15\%\) in steps of 1.25% or 2.5%

GATE Formula: For tap position ’n’:

\[\begin{aligned} V_2 = V_{2,rated} \times \left(1 + \dfrac{n \times \text{step size}}{100}\right) \end{aligned}\]

Where n = +ve for taps above nominal, -ve for below nominal.

Harmonics in Three-Phase Transformers

Third harmonic issues:

  • Present in magnetizing current

  • In-phase in all three phases

  • Cannot flow in delta-connected windings

  • Causes distortion in Y-Y transformers

Solutions:

  • Delta-connected tertiary winding

  • Grounded neutral in Y-Y connection

  • Use Y-\(\Delta\) or \(\Delta\)-Y connections

GATE Tip: Third harmonic current = 0 in delta connection due to closed loop.

Quick Revision

Connection Summary Table

Connection Voltage Ratio Current Ratio Phase Shift Vector Group
Y-Y \(\dfrac{N_1}{N_2}\) \(\dfrac{N_2}{N_1}\) \(0^{\circ}\) Yy0, Yy6
\(\Delta\)-\(\Delta\) \(\dfrac{N_1}{N_2}\) \(\dfrac{N_2}{N_1}\) \(0^{\circ}\) Dd0, Dd6
Y-\(\Delta\) \(\sqrt{3}\dfrac{N_1}{N_2}\) \(\dfrac{1}{\sqrt{3}}\dfrac{N_2}{N_1}\) \(\pm 30^{\circ}\) Yd1, Yd11
\(\Delta\)-Y \(\dfrac{1}{\sqrt{3}}\dfrac{N_1}{N_2}\) \(\sqrt{3}\dfrac{N_2}{N_1}\) \(\pm 30^{\circ}\) Dy1, Dy11

Phase Shift Memory:

  • Yd1, Dy11: Secondary lags by \(30^{\circ}\)

  • Yd11, Dy1: Secondary leads by \(30^{\circ}\)

  • Same letter connections (Yy, Dd): No phase shift

Important GATE Formulas

Parallel Operation:

\[\begin{aligned} I_1 &= \dfrac{Z_2}{Z_1 + Z_2} \times I_{total} \\ I_c &= \dfrac{|\vec{E_1} - \vec{E_2}|}{Z_1 + Z_2} \end{aligned}\]

Regulation:

\[\begin{aligned} \text{Reg} = \dfrac{I R_{eq} \cos\phi \pm I X_{eq} \sin\phi}{V_2} \times 100\% \end{aligned}\]

Efficiency:

\[\begin{aligned} \eta_{max} \text{ when } W_c = W_{cu} = I^2 R_{eq} \end{aligned}\]

Auto-transformer:

\[\begin{aligned} \text{Copper saving} = \left(1 - \dfrac{1}{K}\right) \times 100\% \end{aligned}\]

Common GATE Mistakes to Avoid

  1. Forgetting \(\sqrt{3}\) factor in Y-\(\Delta\) and \(\Delta\)-Y connections

  2. Wrong phase angle calculations for vector groups

  3. Incorrect impedance referral in equivalent circuits

  4. Mixing up lag and lead in phase displacements

  5. Not considering all parallel operation conditions

  6. Wrong regulation formula for leading power factor

  7. Confusing turns ratio with voltage ratio

Problem-solving tip: Always draw phasor diagrams for phase displacement problems.