Three-Phase Transformer Connections
Types of Three-Phase Transformer Connections
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Star-Star (Y-Y) Connection
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Delta-Delta (\(\Delta\)-\(\Delta\)) Connection
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Star-Delta (Y-\(\Delta\)) Connection
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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:
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\(V_L = \sqrt{3} V_{ph}\) (both sides)
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\(I_L = I_{ph}\) (both sides)
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Neutral available
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Third harmonic problems
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Requires neutral grounding
Relations:
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:
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\(V_L = V_{ph}\) (both sides)
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\(I_L = \sqrt{3} I_{ph}\) (both sides)
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No neutral point
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Good for unbalanced loads
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Open delta possible
Relations:
GATE Tip: Open delta operation at 57.7% capacity. Vector group: Dd0 or Dd6.
Star-Delta (Y-\(\Delta\)) Connection
Characteristics:
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Step-down configuration
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Primary: \(V_L = \sqrt{3} V_{ph}\)
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Secondary: \(V_L = V_{ph}\)
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Phase shift: \(\pm 30^{\circ}\)
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Good for stepping down
Relations:
GATE Important: Vector groups: Yd1 (lag \(30^{\circ}\)) or Yd11 (lead \(30^{\circ}\)).
Delta-Star (\(\Delta\)-Y) Connection
Characteristics:
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Step-up configuration
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Primary: \(V_L = V_{ph}\)
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Secondary: \(V_L = \sqrt{3} V_{ph}\)
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Phase shift: \(\pm 30^{\circ}\)
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Good for stepping up
Relations:
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
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X: Primary connection (Y, D, Z)
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x: Secondary connection (y, d, z)
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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:
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Clock Rule: Secondary voltage vector position on clock
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\(30^{\circ}\) Rule: Y-\(\Delta\) and \(\Delta\)-Y always have \(\pm 30^{\circ}\) shift
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Lag/Lead Rule:
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Yd1 and Dy11: Secondary lags by \(30^{\circ}\)
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Yd11 and Dy1: Secondary leads by \(30^{\circ}\)
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GATE Formula: Phase displacement = Clock number \(\times 30^{\circ}\)
Parallel Operation
Conditions for Parallel Operation
Essential Conditions (Must be satisfied):
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Same voltage ratio (primary and secondary voltages)
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Same vector group (identical phase displacement)
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Same phase sequence (RYB or RBY)
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Same frequency (50/60 Hz)
Desirable Conditions (for optimal operation):
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Similar percentage impedance
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Similar X/R ratio
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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:
For equal current sharing: \(Z_1 = Z_2\) (equal % impedance)
kVA sharing formula:
GATE Tip: Lower impedance transformer carries more load.
Circulating Current
Causes:
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Unequal voltage ratios
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Different vector groups
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Phase sequence mismatch
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Unequal frequencies
Circulating current magnitude:
Where \(\vec{E_1}, \vec{E_2}\) are secondary side EMF phasors.
GATE Important: Circulating current exists even at no-load, causing:
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Additional copper losses
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Heating of transformers
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Reduced efficiency
Per-Unit System and Equivalent Circuits
Per-Unit System for Three-Phase Transformers
Base quantities:
Per-unit impedance:
GATE Advantage: Per-unit values are same on both primary and secondary sides.
Equivalent Circuit
Single-phase equivalent circuit parameters:
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\(R_1, R_2\): Primary and secondary resistances
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\(X_1, X_2\): Primary and secondary reactances
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\(R_c\): Core loss resistance
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\(X_m\): Magnetizing reactance
Referred to primary side:
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:
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:
GATE Tip: Copper loss varies as square of current ( \(I^2R\)).
Efficiency and Regulation
Efficiency:
Maximum efficiency occurs when:
Voltage regulation:
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:
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Single winding with taps
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Lower cost and losses
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Higher efficiency
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No electrical isolation
Key relations:
GATE Tip: Auto-transformers are economical for transformation ratios close to 1.
Tap Changing
Types:
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Off-load tap changer: Manual, transformer de-energized
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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’:
Where n = +ve for taps above nominal, -ve for below nominal.
Harmonics in Three-Phase Transformers
Third harmonic issues:
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Present in magnetizing current
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In-phase in all three phases
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Cannot flow in delta-connected windings
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Causes distortion in Y-Y transformers
Solutions:
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Delta-connected tertiary winding
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Grounded neutral in Y-Y connection
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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:
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Yd1, Dy11: Secondary lags by \(30^{\circ}\)
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Yd11, Dy1: Secondary leads by \(30^{\circ}\)
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Same letter connections (Yy, Dd): No phase shift
Important GATE Formulas
Parallel Operation:
Regulation:
Efficiency:
Auto-transformer:
Common GATE Mistakes to Avoid
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Forgetting \(\sqrt{3}\) factor in Y-\(\Delta\) and \(\Delta\)-Y connections
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Wrong phase angle calculations for vector groups
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Incorrect impedance referral in equivalent circuits
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Mixing up lag and lead in phase displacements
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Not considering all parallel operation conditions
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Wrong regulation formula for leading power factor
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Confusing turns ratio with voltage ratio
Problem-solving tip: Always draw phasor diagrams for phase displacement problems.