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### Summary

### Summary

Complete with equations, illustrations, and tables, this book covers the basic theory of electric power transformers, its application to transformer designs, and their application in utility and industrial power systems. The author presents the principles of the two-winding transformer and its connection to polyphase systems, the origins of transformer losses, autotransformers, and three-winding transformers and compares different types of transformer coil and coil construction. He describes the effects of short circuits on transformers, the design and maintenance of ancillary equipment, and preventative and predictive maintenance practices for extending transformer life.

### Author Notes

John J. Winders, Jr. is Senior Engineer at PPL Electric Utilities, Allentown, Pennsylvania.

### Table of Contents

Series Introduction | p. iii |

Preface | p. v |

1 Basic Transformer Theory | p. 1 |

1.1 Definition of a Transformer | p. 1 |

1.2 Magnetic Units and Conversion Factors | p. 1 |

1.3 Currents and Magnetic Fields | p. 3 |

1.4 Magnetic Induction | p. 4 |

1.5 Constructing a Simple Transformer | p. 5 |

1.6 The Magnetic Circuit | p. 6 |

1.7 The B-H Curve | p. 10 |

1.8 The B-H Curve and Hysteresis | p. 11 |

1.9 Magnetizing Currents and Harmonics | p. 15 |

1.10 Transformer Core Design and Construction | p. 16 |

1.11 Magnetostriction | p. 20 |

1.12 Completing the Transformer by Adding a Second Winding | p. 20 |

1.13 Placement of the Windings | p. 22 |

References | p. 24 |

2 Two-Winding Transformer Connections | p. 25 |

2.1 Introduction | p. 25 |

2.2 The Y-Y Connection in Three-Phase Systems | p. 26 |

2.3 Advantages of the Y-Y Connection | p. 29 |

2.4 Disadvantages of the Y-Y Connection | p. 30 |

2.5 The Y-[Delta] Connection and the [Delta]-Y Connection | p. 33 |

2.6 Phase Angle Displacement and Phase Rotation | p. 36 |

2.7 The Y-[Delta] Grounding Bank | p. 40 |

2.8 The Zigzag Connection | p. 42 |

2.9 Comparisons of Economy of the Different Winding Configurations | p. 44 |

2.10 Trade-Off Between Steel and Copper in the Design of a Transformer | p. 47 |

2.11 Connecting Three-Phase Banks Using Single-Phase Transformers | p. 48 |

2.12 Transforming Three-Phase Voltages into Two-Phase Voltages | p. 50 |

2.13 The Scott Transformer Connection | p. 51 |

2.14 Three-Phase Transformer Designs | p. 56 |

2.15 Standard Terminal Markings for Transformers | p. 62 |

References | p. 67 |

3 Transformer Impedance and Losses | p. 69 |

3.1 Leakage Flux and Leakage Reactance | p. 69 |

3.2 Conductor Losses | p. 74 |

3.3 No-Load Losses | p. 76 |

3.4 Magnetizing Reactance | p. 79 |

3.5 Equivalent Circuit of a Two-Winding Transformer | p. 80 |

3.6 A Brief Tutorial on Symmetrical Components | p. 81 |

3.7 Transformer Equivalent Circuits Modeled in Zero Phase Sequence Networks | p. 85 |

3.8 Series Impedance and Regulation | p. 92 |

3.9 Matching Transformers for Parallel and Bank Operations | p. 93 |

3.10 Impedance Mismatch in Three-Phase Transformer Banks | p. 95 |

3.11 Temperature Rise and the Thermal Capability | p. 98 |

3.12 Interpreting Transformer Test Reports | p. 99 |

3.13 Calculating the Hot-Spot Temperature Using the IEEE Method | p. 102 |

3.14 Calculating the Loss of Life | p. 105 |

References | p. 116 |

4 Autotransformers and Three-Winding Transformers | p. 117 |

4.1 Autotransformer Connections | p. 117 |

4.2 Impedance of an Autotransformer | p. 122 |

4.3 Limitations of the Autotransformer Connection | p. 128 |

4.4 Autotransformer Voltages with Short Circuits Applied | p. 129 |

4.5 Impulse Voltages Applied to Autotransformers | p. 131 |

4.6 Autotransformer Core and Coil Designs and Terminal Configurations | p. 131 |

4.7 Advantages and Disadvantages of the Autotransformer Connection | p. 132 |

4.8 Three-Winding Transformers | p. 133 |

4.9 Modification of Transformer Laws with Three Windings | p. 134 |

4.10 Equivalent Circuit of a Three-Winding Transformer | p. 136 |

4.11 Core and Coil Construction of Three-Winding Transformers | p. 138 |

4.12 Thermal Capability of Three-Winding Transformers | p. 141 |

4.13 The Stabilizing Effect of a [Delta] Tertiary Winding | p. 141 |

Reference | p. 146 |

5 Short Circuits, Inrush Currents, and Other Phenomena | p. 147 |

5.1 Effects of Short Circuits on Transformers | p. 147 |

5.2 Comparisons of Short-Circuit Currents for Various Faults | p. 148 |

5.3 Mechanical Forces in Transformers | p. 155 |

5.4 Forces between Transformer Windings | p. 158 |

5.5 Short-Circuit Forces in Three-Winding Transformers | p. 163 |

5.6 Exciting Current Inrush | p. 165 |

5.7 Tank Overheating from Zero-Sequence Currents | p. 169 |

5.8 Primary Fuse Misoperations | p. 172 |

5.9 Ferroresonance | p. 175 |

5.10 Voltage Surges | p. 177 |

References | p. 187 |

6 Ancillary Equipment | p. 189 |

6.1 Bushings | p. 189 |

6.2 Oil Preservation Systems | p. 192 |

6.3 Cooling Equipment | p. 196 |

6.4 Temperature, Oil Level, and Pressure Gauges | p. 201 |

6.5 Tap-Changing Equipment | p. 202 |

6.6 Load Tap Changers | p. 204 |

6.7 Voltage Regulating Transformers | p. 209 |

6.8 Tap Changer Automatic Controls | p. 210 |

6.9 Variable Phase Shifting Transformers | p. 212 |

Reference | p. 220 |

7 Reading and Applying Nameplate Information | p. 221 |

7.1 Minimum Nameplate Requirements | p. 221 |

7.2 Manufacturer's Information | p. 223 |

7.3 Cooling Class, Number of Phases, and Operating Frequency | p. 223 |

7.4 Voltage Ratings | p. 224 |

7.5 KVA or MVA Ratings | p. 225 |

7.6 Winding Connection Diagram | p. 226 |

7.7 Phasor or Vector Diagram | p. 229 |

7.8 Weights and Oil Capacity | p. 229 |

7.9 Operating Pressure Range | p. 230 |

7.10 Impedance | p. 231 |

7.11 Basic Insulation Level | p. 232 |

7.12 Nameplate Layout | p. 233 |

References | p. 233 |

8 Maintenance, Testing, Troubleshooting, and Reliability | p. 235 |

8.1 Good Utility Practices | p. 235 |

8.2 Preventative Maintenance versus Predictive Maintenance | p. 252 |

8.3 Factory Tests | p. 252 |

8.4 Ratio Test | p. 252 |

8.5 Other Factory Tests | p. 255 |

8.6 Field Tests | p. 258 |

8.7 Gas-in-Oil Analysis | p. 261 |

8.8 Water-in-Oil Analysis | p. 265 |

8.9 Drying Transformers | p. 268 |

8.10 Oil Dielectric Test | p. 268 |

8.11 Reliability Calculations | p. 269 |

8.12 The Chi-Squared Distribution | p. 269 |

8.13 The Poisson Distribution | p. 271 |

8.14 Statistical Economics | p. 273 |

References | p. 275 |

Index | p. 277 |