1. Introduction
Three-phase transformers are essential components in electrical power systems for stepping up or stepping down voltages in three-phase AC systems. They can be constructed either as a bank of three single-phase transformers or as a single three-phase unit with a common magnetic circuit.
Advantages of Three-Phase Transformers
- More economical and requires less space than three single-phase units
- Lighter weight and lower cost
- Higher efficiency
- Better utilization of core material
2. Construction Types
2.1 Core-Type Construction
Three single-phase transformers with separate cores, or a single unit with three independent magnetic circuits connected at the yokes.
2.2 Shell-Type Construction
The flux paths are partially in common, providing better mechanical strength and magnetic coupling.
3. Three-Phase Transformer Connections
Three-phase transformers can be connected in various configurations. The four most common connections are:
3.1 Wye-Wye (Y-Y) Connection
Voltage Relationships:
\[V_{L,primary} = \sqrt{3} \, V_{ph,primary}\] \[V_{L,secondary} = \sqrt{3} \, V_{ph,secondary}\]Turns Ratio:
\[a = \frac{V_{ph,primary}}{V_{ph,secondary}} = \frac{N_1}{N_2}\]Line Voltage Ratio:
\[\frac{V_{L,primary}}{V_{L,secondary}} = a\]Advantages: Neutral available for grounding, economical for high voltage ratings
Disadvantages: Third harmonic issues, requires neutral grounding
3.2 Delta-Delta (Δ-Δ) Connection
Voltage Relationships:
\[V_{L,primary} = V_{ph,primary}\] \[V_{L,secondary} = V_{ph,secondary}\]Current Relationships:
\[I_{L,primary} = \sqrt{3} \, I_{ph,primary}\] \[I_{L,secondary} = \sqrt{3} \, I_{ph,secondary}\]Line Voltage Ratio:
\[\frac{V_{L,primary}}{V_{L,secondary}} = a\]Advantages: No phase shift, can operate with one phase open (V-V connection)
Disadvantages: No neutral point, higher insulation requirements
3.3 Wye-Delta (Y-Δ) Connection
Line Voltage Ratio:
\[\frac{V_{L,primary}}{V_{L,secondary}} = \frac{\sqrt{3} \, N_1}{N_2} = \sqrt{3} \, a\]Phase Shift:
\[\theta = 30° \text{ (secondary lags primary)}\]Applications: Step-down transformers, eliminates third harmonics
3.4 Delta-Wye (Δ-Y) Connection
Line Voltage Ratio:
\[\frac{V_{L,primary}}{V_{L,secondary}} = \frac{N_1}{\sqrt{3} \, N_2} = \frac{a}{\sqrt{3}}\]Phase Shift:
\[\theta = 30° \text{ (secondary leads primary)}\]Applications: Step-up transformers at power stations, neutral available on secondary
4. Power and kVA Ratings
Three-Phase Apparent Power:
\[S_{3\phi} = \sqrt{3} \, V_L \, I_L = 3 \, V_{ph} \, I_{ph}\]Three-Phase Real Power:
\[P_{3\phi} = \sqrt{3} \, V_L \, I_L \, \cos\phi = 3 \, V_{ph} \, I_{ph} \, \cos\phi\]Three-Phase Reactive Power:
\[Q_{3\phi} = \sqrt{3} \, V_L \, I_L \, \sin\phi = 3 \, V_{ph} \, I_{ph} \, \sin\phi\]Where \(V_L\) is line voltage, \(I_L\) is line current, \(V_{ph}\) is phase voltage, \(I_{ph}\) is phase current, and \(\phi\) is the power factor angle.
5. Equivalent Circuit Parameters
The per-phase equivalent circuit of a three-phase transformer is identical to that of a single-phase transformer.
Equivalent Resistance (referred to primary):
\[R_{eq} = R_1 + a^2 R_2\]Equivalent Reactance (referred to primary):
\[X_{eq} = X_1 + a^2 X_2\]Equivalent Impedance:
\[Z_{eq} = R_{eq} + jX_{eq} = \sqrt{R_{eq}^2 + X_{eq}^2} \angle \tan^{-1}\left(\frac{X_{eq}}{R_{eq}}\right)\]Magnetizing Branch
Magnetizing Reactance:
\[X_m = \frac{V_1}{I_m}\]Core Loss Resistance:
\[R_c = \frac{V_1^2}{P_{core}}\]6. Voltage Regulation
Voltage regulation indicates how well a transformer maintains constant secondary voltage with varying load.
Percentage Voltage Regulation:
\[\text{VR} = \frac{V_{2,nl} - V_{2,fl}}{V_{2,fl}} \times 100\%\]Approximate Formula:
\[\text{VR} \approx \frac{I_2(R_{eq} \cos\phi_2 + X_{eq} \sin\phi_2)}{V_2} \times 100\%\]Per Unit Formula:
\[\text{VR}_{pu} = I_{2,pu}(R_{eq,pu} \cos\phi_2 + X_{eq,pu} \sin\phi_2)\]Where \(V_{2,nl}\) is no-load secondary voltage, \(V_{2,fl}\) is full-load secondary voltage, and \(\phi_2\) is the load power factor angle.
7. Efficiency
Transformer Efficiency:
\[\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{losses}} \times 100\%\]For Three-Phase Transformers:
\[\eta = \frac{3V_{2,ph}I_{2,ph}\cos\phi_2}{3V_{2,ph}I_{2,ph}\cos\phi_2 + P_{core} + 3I_{2,ph}^2 R_{eq}} \times 100\%\]Condition for Maximum Efficiency:
\[P_{copper} = P_{core}\] \[3I_{2,ph}^2 R_{eq} = P_{core}\]Load at Maximum Efficiency:
\[x = \sqrt{\frac{P_{core}}{P_{cu,fl}}}\]Where \(x\) is the fraction of full load at which efficiency is maximum, \(P_{cu,fl}\) is copper loss at full load.
8. Per-Unit System
The per-unit system simplifies three-phase transformer calculations by normalizing values.
Base Values:
\[S_{base,3\phi} = \text{Three-phase kVA rating}\] \[V_{base,L} = \text{Line-to-line voltage rating}\] \[V_{base,ph} = \frac{V_{base,L}}{\sqrt{3}} \text{ (for Y connection)}\] \[V_{base,ph} = V_{base,L} \text{ (for Δ connection)}\]Base Currents:
\[I_{base,L} = \frac{S_{base,3\phi}}{\sqrt{3} \, V_{base,L}}\]Base Impedance:
\[Z_{base} = \frac{V_{base,ph}}{I_{base,ph}} = \frac{(V_{base,L})^2}{S_{base,3\phi}}\]Per-Unit Quantity:
\[\text{Value}_{pu} = \frac{\text{Actual Value}}{\text{Base Value}}\]9. Open-Delta (V-V) Connection
When one transformer of a delta-delta bank fails, the system can continue to operate with two transformers in an open-delta configuration.
Power Capacity:
\[S_{V-V} = \sqrt{3} \, V_L \, I_L\]Comparison with Delta-Delta:
\[\text{Power delivered} = \frac{\sqrt{3}}{2} \times \text{Power of 3 transformers}\]Utilization Factor:
\[\text{UF} = \frac{S_{V-V}}{2S_{transformer}} = \frac{\sqrt{3}}{2} = 0.866 = 86.6\%\]The V-V connection delivers 57.7% of the power that would be delivered by a complete delta-delta bank using three transformers of the same rating.
10. Parallel Operation
For proper parallel operation of three-phase transformers, the following conditions must be satisfied:
- Voltage Ratings: Same voltage ratings (primary and secondary)
- Voltage Ratio: Same voltage transformation ratio
- Percent Impedance: Same percentage impedance on their own ratings
- Impedance Angle: Same X/R ratio (impedance angle)
- Phase Sequence: Same phase sequence and angular displacement
- Polarity: Correct polarity connections
Load Sharing (if impedances are different):
\[\frac{S_1}{S_2} = \frac{Z_2}{Z_1} \times \frac{S_{rated,1}}{S_{rated,2}}\]Circulating Current (if voltage ratios differ):
\[I_{circ} = \frac{|V_{2a} - V_{2b}|}{Z_{eq,a} + Z_{eq,b}}\]11. Tap Changing
Tap changers are used to maintain secondary voltage within specified limits despite variations in primary voltage and load.
Voltage with Tap Position:
\[V_2 = V_1 \times \frac{N_2}{N_1} \times \left(1 + \frac{\text{tap}\%}{100}\right)\]Types of Tap Changers:
| Type | Operation | Application |
|---|---|---|
| Off-Load (OLTC) | Changed when de-energized | Fixed voltage adjustment |
| On-Load (LTC) | Changed under load | Automatic voltage regulation |
12. Harmonics in Three-Phase Transformers
Magnetizing current in transformers is non-sinusoidal and contains harmonics, primarily the third harmonic.
Third Harmonic Behavior
- Y-Y Connection: Third harmonics are trapped, causing distortion and overvoltage. Requires neutral grounding.
- Δ-Δ Connection: Third harmonics circulate within the delta, preventing distortion in line voltages.
- Y-Δ and Δ-Y: Delta winding provides path for third harmonic currents.
Third Harmonic Frequency:
\[f_3 = 3f_1\]Fifth Harmonic (also present):
\[f_5 = 5f_1\]13. Testing of Three-Phase Transformers
13.1 Open-Circuit Test
Performed on secondary (LV) side to determine core loss and magnetizing parameters.
13.2 Short-Circuit Test
Performed on primary (HV) side to determine copper loss and equivalent impedance.
13.3 Polarity Test
Ensures correct phase relationships between primary and secondary windings.
13.4 Phase Displacement Test
Verifies the angular displacement between primary and secondary line voltages (0° or 30°).
14. Protection Schemes
Three-phase transformers require comprehensive protection against various faults.
| Protection Type | Purpose | Device |
|---|---|---|
| Overcurrent Protection | Overload and external faults | Fuses, Circuit Breakers |
| Differential Protection | Internal faults | Differential Relays |
| Buchholz Protection | Incipient faults, oil level | Buchholz Relay |
| Overtemperature Protection | Excessive heating | Temperature Sensors, RTDs |
| Earth Fault Protection | Ground faults | Earth Fault Relays |
| Overvoltage Protection | Lightning, switching surges | Surge Arresters |
15. Cooling Methods
Transformer cooling is designated by a four-letter code indicating the cooling medium and circulation method.
| Designation | Description | Application |
|---|---|---|
| ONAN | Oil Natural, Air Natural | Small to medium transformers |
| ONAF | Oil Natural, Air Forced | Medium transformers with fans |
| OFAF | Oil Forced, Air Forced | Large power transformers |
| OFWF | Oil Forced, Water Forced | Very large transformers |
Temperature Rise:
\[\Delta T = \frac{P_{losses}}{K \cdot A}\]Where \(K\) is heat transfer coefficient and \(A\) is cooling surface area.
16. Applications
Power System Applications
- Generation Step-up: Δ-Y connection at power plants (typically 11 kV to 132/220/400 kV)
- Transmission: Y-Y connection for EHV transmission networks
- Distribution: Y-Δ or Δ-Y for medium voltage distribution (33 kV to 11 kV)
- Utilization: Δ-Y or Y-Y for final distribution (11 kV to 415 V)
Special Applications
- Grounding transformers (zigzag connection)
- Phase shifting transformers for power flow control
- Scott-T connection for three-phase to two-phase conversion
- Autotransformers for voltage regulation
17. Key Points Summary
- Choice of connection affects voltage ratio, phase shift, and harmonic behavior
- Y-Δ and Δ-Y introduce 30° phase shift
- Delta windings provide path for third harmonic currents
- Per-unit system simplifies calculations
- Parallel operation requires matching impedances and phase relationships
- Maximum efficiency occurs when copper loss equals core loss
- Proper protection is essential for reliable operation