Cover
Title Page
Preface
About the Companion Website
1 Introduction to Harmonic Balance Finite Element Method (HBFEM)
1. 1.1 Harmonic Problems in Power Systems
2. 1.2 Definitions of Computational Electromagnetics and IEEE Standards 1597.1 and 1597.2
3. 1.3 HBFEM Used in Nonlinear EM Field Problems and Power Systems
4. References
2 Nonlinear Electromagnetic Field and Its Harmonic Problems
1. 2.1 Harmonic Problems in Power Systems and Power Supply Transformers
2. 2.2 DC‐Biased Transformer in High‐Voltage DC Power Transmission System
3. 2.3 Geomagnetic Disturbance and Geomagnetic Induced Currents (GIC)
4. 2.4 Harmonic Problems in Renewable Energy and Microgrid Systems
5. References
3 Harmonic Balance Methods Used in Computational Electromagnetics
1. 3.1 Harmonic Balance Methods Used in Nonlinear Circuit Problems
2. 3.2 CEM for Harmonic Problem Solving in Frequency, Time and Harmonic Domains
3. 3.3 The Basic Concept of Harmonic Balance in EM Fields
4. 3.4 HBFEM for Electromagnetic Field and Electric Circuit Coupled Problems
5. 3.5 HBFEM for a DC‐Biased Problem in High‐Voltage Power Transformers
6. References
4 HBFEM for Nonlinear Magnetic Field Problems
1. 4.1 HBFEM for a Nonlinear Magnetic Field with Current‐Driven Source
2. 4.2 Harmonic Analysis of Switching Mode Transformer Using Voltage‐Driven Source
3. 4.3 Three‐Phase Magnetic Frequency Tripler Analysis
4. 4.4 Design of High‐Speed and Hybrid Induction Machine using HBFEM
5. 4.5 Three‐Dimensional Axi‐Symmetrical Transformer with DC‐Biased Excitation
6. References
5 Advanced Numerical Approach using HBFEM
1. 5.1 HBFEM for DC‐Biased Problems in HVDC Power Transformers
2. 5.2 Decomposed Algorithm of HBFEM
3. 5.3 HBFEM with Fixed‐Point Technique
4. 5.4 Hysteresis Model Based on Neural Network and Consuming Function
5. 5.5 Analysis of Hysteretic Characteristics Under Sinusoidal and DC‐Biased Excitation
6. 5.6 Parallel Computing of HBFEM in Multi‐Frequency Domain
7. References
6 HBFEM and Its Future Applications
1. 6.1 HBFEM Model of Three‐Phase Power Transformer
2. 6.2 Magnetic Model of a Single‐Phase Transformer and a Magnetically Controlled Shunt Reactor
3. 6.3 Computation Taking Account of Hysteresis Effects Based on Fixed‐Point Reluctance
4. 6.4 HBFEM Modeling of the DC‐Biased Transformer in GIC Event
5. 6.5 HBFEM Used in Renewable Energy Systems and Microgrids
6. References
Appendix
1. Appendix I & II
2. Appendix III
Index
End User License Agreement

List of Tables

Chapter 01
1. Table 1.1 Sources and problems of harmonics
Chapter 02
1. Table 2.1 Current distortion due to nonlinear load and power electronics
2. Table 2.2 Harmonic components in 6‐pulse and 12‐pulse rectifier transformers
Chapter 03
1. Table 3.1 Comparison of time‐periodic steady‐state nonlinear EM field analysis method
Chapter 05
1. Table 5.1 Different DC bias conditions specified by quantity in the magnetic field
2. Table 5.2 Errors between calculated and measured results in exciting current
3. Table 5.3 Each harmonic component of magnetic induction in one element in the silicon steel region under DC bias conditions
4. Table 5.4 Scheme of setting the relaxation factor for the convergence of the harmonic solution
5. Table 5.5 Calculation results under different DC bias excitations
6. Table 5.6 Comparison of the computational cost between the two methods
7. Table 5.7 Harmonic components of B_x and reluctivity in point B
8. Table 5.8 DC flux density (B_x0) under different DC bias and AC excitations
9. Table 5.9 AC flux density (B_x1) under different DC bias and AC excitations
10. Table 5.10 Data of magnetization curve under DC‐biased magnetization (H_dc = 107 A/m)
11. Table 5.11 Comparison of the two methods
12. Table 5.12 Comparison of the exciting current
13. Table 5.13 Comparison of the flux density
14. Table 5.14 Comparison of the two methods (I_dc = 0.426 A)
15. Table 5.15 Comparison of the two methods (I_dc = 0.847 A)
16. Table 5.16 Simulated results of hysteresis loops under sinusoidal excitation (I_dc = 0 A)
17. Table 5.17 Simulated results of hysteresis loops under sinusoidal excitation (I_dc = 0.426 A)
18. Table 5.18 Simulated results of hysteresis loops under sinusoidal excitation (I_dc = 0.847 A)
19. Table 5.19 Iron loss under sinusoidal magnetization
20. Table 5.20 Iron loss in different areas under sinusoidal magnetization
21. Table 5.21 Iron loss in different areas under DC‐biased magnetization (I_dc = 0.426 A)
22. Table 5.22 Comparison between measured and predicted results
Chapter 06
1. Table 6.1 Comparison of the RMS value of the magnetizing currents under different DC‐biased conditions
2. Table 6.2 Comparison of the peak value of magnetizing currents under different DC‐biased conditions

List of Illustrations

Chapter 01
1. Figure 1.1 (a) Sine wave. (b) Distorted waveform or non‐sinusoidal
2. Figure 1.2 Distorted waveform and number of harmonics by Fourier series
3. Figure 1.3 Harmonic distortion of the electrical current waveform, where the distorted waveform is composed of fundamental and 3rd harmonics
4. Figure 1.4 Creation of distorted current
Chapter 02
1. Figure 2.1 (a) Characteristics of magnetic impedance associated with a B‐H curve and permeability. (b) Excitation current corresponding to a sinusoidal voltage excitation associated with a hysteresis B‐H curve
2. Figure 2.2 Excitation current corresponding to a sinusoidal voltage excitation
3. Figure 2.3 Nonlinear magnetic and nonlinear dielectric materials, (a) the B‐H hysteresis loop of the magnetic material, and (b) the direction of the polarization (D‐E) hysteresis loop of ferroelectric material
4. Figure 2.4 Harmonics in time domain presentation
5. Figure 2.5 Frequency domain graphs – frequency spectrums
6. Figure 2.6 Examples of load current waveforms with harmonics
7. Figure 2.7 (a) Waveforms of flux; (b) voltage for sinusoidal magnetizing current in nonlinear magnetics
8. Figure 2.8 (a) Magnetic flux waveforms; (b) current waveform with a square excitation voltage
9. Figure 2.9 The full bridge transformer‐isolated buck converter
10. Figure 2.10 A two‐transistor version of the forward converter
11. Figure 2.11 The flyback converter, a single‐transistor isolated buck‐boost converter
12. Figure 2.12 A half‐bridge LLC converter
13. Figure 2.13 Magnetic triplers with three‐phase input at 50Hz and single phase output at 150Hz. (a) magnetic frequency tripler with three input magnetizing coils and two secondary coils connected in a series as an output. (b) magnetic frequency tripler using three secondary coils connected in a series
14. Figure 2.14 Three phase rectifier transformer (6‐pulse transformer). (a) three phase transformer with Y connection at secondary side. (b) waveforms of 6‐pulse output voltage for three phase rectifier transformer with delta connection at the secondary side
15. Figure 2.15 Waveforms of 6‐pulse input voltage and current
16. Figure 2.16 Three phase with 12‐pulse rectifier transformer with Y and delta connections at the secondary side
17. Figure 2.17 Waveforms of input voltage and current for 12‐pulse rectifier transformer
18. Figure 2.18 Schematic diagram of the DC bias phenomenon
19. Figure 2.19 Inpouring reverse current for compensation
20. Figure 2.20 The changing geomagnetic field (φ_gm) induces a geoelectric field (E_ge) that drives currents in conductor loops
21. Figure 2.21 A small DC content in the AC may cause a power transformer to enter half‐cycle saturation
22. Figure 2.22 Magnetizing current and hysteresis loop under different direct magnetizing current (single phase transformer)
23. Figure 2.23 Three‐phase DC/AC inverter connected to a building transformer
24. Figure 2.24 Output voltage waveform
25. Figure 2.25 Frequency spectrum
26. Figure 2.26 Microgrid coordinator/coordinated control system, conceptual commercial‐level microgrid architecture
27. Figure 2.27 Basic concept of the microgrid connected through a PCC to the grid
28. Figure 2.28 (a) Voltage waveform at the PCC; (b) VSC current waveforms from the inverter
Chapter 03
1. Figure 3.1 Distorted waveform and spectrum. (a) Distorted waveforms; (b) Spectrum
2. Figure 3.2 The concept of harmonic balance for a non‐linear circuit
3. Figure 3.3 The circuit diagram for using harmonic balance in a non‐linear circuit
4. Figure 3.4 Magnetic core for a 2D transformer structure, and its typical B‐H cure. (a) Transformer with nonlinear magnetic core. (b) B‐H curve and permeability
5. Figure 3.5 B‐H curve with hysteresis characteristics (a), and without hysteresis characteristics (b)
6. Figure 3.6 B‐H curve with hysteresis characteristics and DC‐biased condition. (a) H‐B curve with hysteresis; (b) H‐B curve with hysteresis and DC‐biased case
7. Figure 3.7 Size of the system matrix. (a) Static FEM; (b) HBFEM
8. Figure 3.8 Simulation model of LLC converter with resonant tank with idea transformer
9. Figure 3.9 Coupling between the electric circuit and the magnetic field
10. Figure 3.10 HVDC power transmission system
11. Figure 3.11 B‐H curve with hysteresis characteristics and a DC biased condition. (a) DC‐biased hysteresis loop; (b) Magnetizing current
12. Figure 3.12 The block diagram of the three phase HVDC transformer including neutral points
Chapter 04
1. Figure 4.1 Current‐source excitation to magnetic field. (a) Switch‐mode push‐pull converter; (b) Zero‐voltage switched resonant converter
2. Figure 4.2 Magnetic system with current‐source excitation. (a) Magnetic configuration; (b) Hysteresis characteristic
3. Figure 4.3 The magnetic flux distribution for fundamental and third harmonic components. (a) Fundamental component; (b) Third harmonic component
4. Figure 4.4 The analysis model with an air gap and two slots at the central leg, and B‐H curve
5. Figure 4.5 The experimental results compared with numerical computation results in the case of current source excitation, where … indicates an experimental result and — indicates a numerical result. (a) Magnetic density B₁; (b) Magnetic density B₂; (c) Magnetic density B₃; (d) Excitation current density J
6. Figure 4.6 The magnetic flux distribution for fundamental and third harmonic components. (a) Fundamental harmonic component; (b) Third harmonic component
7. Figure 4.7 Voltage‐source to the magnetic system used for switch mode transformers
8. Figure 4.8 Generalized model of voltage‐source to the magnetic system
9. Figure 4.9 Magnetic core for a 2‐D transformer structure and its B‐H cure. (a) Magnetic core (b) B‐H curve
10. Figure 4.10 Magnetic flux distribution for their harmonic components at phase of zero degree. (a) fundamental component; (b) third harmonic component; (c) fifth harmonic component
11. Figure 4.11 Comparison between computation and measurement. (a) Input voltage source; (b) Current caused by voltage source
12. Figure 4.12 A circuit diagram of a magnetic frequency tripler
13. Figure 4.13 B‐H curve with (a) hysteresis; and (b) without hysteresis characteristics
14. Figure 4.14 A configuration of the magnetic frequency tripler with a voltage driven source connected to the magnetic system
15. Figure 4.15 A three‐phase magnetic tripler problem as a voltage‐driven source connected to the magnetic system
16. Figure 4.16 Geometric size ½, a configuration of the magnetic frequency tripler ½ for numerical computation model
17. Figure 4.17 Output voltage waveform: (a) experimental result, (b) simulation result
18. Figure 4.18 Flux distribution of fundamental, third and fifth harmonics.
19. Figure 4.19 Comparison between computation and experiment results of waveforms of input and output currents, and input voltage (phase U) and neutral voltage V_NN″
20. Figure 4.20 The waveforms of the magnetic flux density distribution for each phase of magnetic leg and output side of magnetic legs
21. Figure 4.21 Characteristics of output current against load and phase U input current against input voltage. (a) Output current against load; (b) Phase U input current against input voltage
22. Figure 4.22 High‐speed hybrid induction motor consists of three‐phase to two‐phase magnetic frequency tripler
23. Figure 4.23 The compact structure of high‐speed and hybrid induction motor and three‐phase magnetizing windings and four additional coils connected with the capacitors
24. Figure 4.24 Numerical model of electric machine taking account of motion
25. Figure 4.25 The distributions of magnetic flux in the core at excitation voltage 213 V. (a) Magnetic flux distributions of different harmonic component; (b) The impact of slip on the distributions of magnetic flux
26. Figure 4.26 The waveform of the flux density at different legs of the magnetic tripler
27. Figure 4.27 The fundamental and third harmonic components of magnetic flux distributions at different rotating angles
28. Figure 4.28 Magnetic flux distributions of high‐speed and hybrid induction motor: (a) at normal rotating case; (b) at stopping case
29. Figure 4.29 Input voltage vs output voltage for both fundamental and third harmonics
30. Figure 4.30 Various waveforms, including input voltages, output voltage and voltage accrued at neutral point
31. Figure 4.31 (a) Simulation model of a switching mode transformer. (b) B‐H curve of the magnetic core
32. Figure 4.32 Magnetic flux distribution [4]. (a) DC flux; (b) First harmonic flux; (c) Second harmonic flux
33. Figure 4.33 Flux density of the transformer with DC biased excitation. (a) Experimental result; (b) Simulation result
34. Figure 4.34 The 2‐D axi‐symmetrical numerical model of voltage excitation and distributions of eddy current in the windings. (a) Axi‐symmetrical winding configuration; (b) Eddy current at Z = 0 mm, (c) Eddy current at Z = 2.25 mm
Chapter 05
1. Figure 5.1 Operation modes of the high‐voltage direct current transmission system
2. Figure 5.2 Epstein frame‐like core model
3. Figure 5.3 Magnetizing curve of the silicon‐steel sheet
4. Figure 5.4 Exciting current under different DC bias (U_m = U_m,2 = 133 V; B_m = B_m,2 = 0.49 T)
5. Figure 5.5 Exciting current under different DC bias (U_m = U_m,3 = 240 V; B_m = B_m,3 = 0.88 T)
6. Figure 5.6 Exciting current under different DC bias (U_m = U_m,4 = 370 V; B_m = B_m,4 = 1.37 T)
7. Figure 5.7 Exciting current under different DC bias (U_m = U_m,6 = 495 V; B_m = B_m,6 = 1.82 T)
8. Figure 5.8 Each harmonic component of exciting current under different DC bias (U_m = U_m,3 = 240 V; B_m = B_m,3 = 0.88 T)
9. Figure 5.9 DC bias effect on different harmonics (U_m = U_m,3 = 240 V; B_m = B_m,3 = 0.88 T)
10. Figure 5.10 Each harmonic component of exciting current under different DC bias (U_m = U_m,6 = 495 V; B_m = B_m,6 = 1.82 T)
11. Figure 5.11 AC voltage (peak value) effect on each harmonic component under 50% DC bias (I_dc = I_dc,2 = 0.847 A; H_dc = H_dc,2 = 213.12 A/m)
12. Figure 5.12 DC bias effect on different harmonics (U_m = U_m,6 = 495 V; B_m = B_m,6 = 1.82 T)
13. Figure 5.13 One quarter of the computational model
14. Figure 5.14 DC component of flux density (B_x,0) under different DC bias conditions
15. Figure 5.15 Magnetic flux density (B_x) under different DC bias when the AC excitation is constant (U_m = U_m,3 = 240 V)
16. Figure 5.16 AC voltage effect on waveforms of magnetic flux density (B_x) under 50% DC bias (I_dc = I_dc,2 = 0.847 A)
17. Figure 5.17 AC voltage effect on DC and AC components of the magnetic flux density (B_x,0 and B_x,1) under 25% DC bias
18. Figure 5.18 AC voltage effect on DC and AC components of the magnetic flux density (B_x,0 and B_x,1) under 50% DC bias
19. Figure 5.19 AC voltage effect on DC and AC components of the magnetic flux density (B_x,0 and B_x,1) under 75% DC bias
20. Figure 5.20 DC bias effect on DC and AC components of the magnetic flux density (B_x,0 and B_x,1) with alternating voltage of 133 V
21. Figure 5.21 DC bias effect on DC and AC components of the magnetic flux density (B_x,0 and B_x,1) with alternating voltage of 240 V
22. Figure 5.22 DC bias effect on DC and AC components of the magnetic flux density (B_x,0 and B_x,1) with alternating voltage of 370 V
23. Figure 5.23 The total flux distribution
24. Figure 5.24 Flux distribution of harmonic components (ωt = π/3). a. DC component; b. Fundamental component; c. Second harmonic component; d. Third harmonic component; e. Fourth harmonic component; f. Fifth harmonic component; g. Sixth harmonic component; h. Seventh harmonic component; i. Eighth harmonic component; j. Ninth harmonic component
25. Figure 5.25 An HVDC transmission system
26. Figure 5.26 LCM and the B‐H curve. (a) Laminated core model (σ_steel = 2.22 × 10⁶ S/m); (b) Exciting coil and search coil (σ_coil = 5.71 × 10⁷ S/m); (c) B‐H curve of the LCM
27. Figure 5.27 Schematic circuit diagram of the core model in experiment
28. Figure 5.28 Exciting currents under different DC bias conditions (I_dc = 0.847 A; U_ac = 320 V)
29. Figure 5.29 Computational region: one quarter of the square laminated core model
30. Figure 5.30 Calculated flux density, exciting current and reluctivity (I_dc = 1.27 A, U_ac = 370 V). (a) Waveform of flux density B_x in point B; (b) Exciting current; (c) Waveform of reluctivity
31. Figure 5.31 Harmonic flux distributions (I_dc = 1.27A, U_ac = 370V). (a) DC flux; (b) Fundamental (ωt = π/2); (c) Second order (2ωt = π/2); (d) Third order (3ωt = π/2); (e) Fourth order (4ωt = π/2); (f) Fifth order (5ωt = π/2)
32. Figure 5.32 Hysteresis loops under DC‐biased and sinusoidal excitations
33. Figure 5.33 The SLC and computational region
34. Figure 5.34 Measured hysteresis loops (solid line) and magnetization curve (dashed line) under DC‐biased magnetization (H_dc = 107 A/m)
35. Figure 5.35 Calculated and measured exciting current (H_dc = 213 A/m, U_ac = 320 V)
36. Figure 5.36 Waveforms of calculated ν_FP (dashed line) and magnetic flux density (solid line) in point E in SLC (H_dc = 107 A/m, U_ac = 368 V)
37. Figure 5.37 Evolution of the relative error of the harmonic solution in two methods (H_dc = 107 A/m, U_ac = 368 V, N_c = 11)
38. Figure 5.38 Comparison of calculated exciting currents (H_dc = 107 A/m, U_ac = 495 V); i₁ and i₂ are exciting currents calculated by basic and DC‐biasing magnetization curves respectively, and i_m is the measured result
39. Figure 5.39 Comparison of calculated magnetic flux density (U_ac = 240 V, 368 V, 495 V; H_dc = 107 A/m). B₁ and B₂ are computed by the basic and DC‐biasing magnetization curves respectively, B_av is the average value obtained in (5‐47)
40. Figure 5.40 Magnetizing current under sinusoidal excitation
41. Figure 5.41 Hysteresis model based on consuming function
42. Figure 5.42 Laminated core model and electric schematic diagram
43. Figure 5.43 Simulated and measured hysteresis loop under sinusoidal magnetization. (a) Symmetrical hysteresis loops. (b) Simulated and measured results
44. Figure 5.44 Computed and measured magnetizing currents under different DC‐biased magnetizations. (a) U_ac = 80 V, I_dc = 0.425 A; (b) U_ac = 370 V, I_dc = 0.425 A; (c) U_ac = 452 V, I_dc = 0.425 A
45. Figure 5.45 Simulated and measured hysteresis loops under DC‐biased magnetization. (a) DC‐biasing hysteresis loops (H_dc = 107 A/m); (b) DC‐biasing hysteresis loops (H_dc = 213 A/m); (c) Simulated and measured results
46. Figure 5.46 Hyperbolic tangent transfer function
47. Figure 5.47 Training performance (three hidden layers with ten hidden neurons in each layer)
48. Figure 5.48 Training performance (four hidden layers with 20 hidden neurons in each layer)
49. Figure 5.49 Comparison between the measured and simulated results (I_dc = 0 A)
50. Figure 5.50 Comparison between the measured and simulated results (I_dc = 0.426 A)
51. Figure 5.51 Comparison between the measured and simulated results (I_dc = 0.847 A)
52. Figure 5.52 Geometric model of the laminated core for computation
53. Figure 5.53 Magnetizing current under sinusoidal magnetization (U_ac = 420 V, I_dc = 0 A)
54. Figure 5.54 Magnetizing current under DC‐biased magnetization (U_ac = 420 V, I_dc = 0.426 A)
55. Figure 5.55 H_y‐B_y hysteresis loop on point B under sinusoidal magnetization (U_ac = 420 V, I_dc = 0 A)
56. Figure 5.56 Hysteresis loop on point A under sinusoidal magnetization (U_ac = 420 V, I_dc = 0 A).(a) H_x‐B_x hysteresis loop. (b) H_y‐B_y hysteresis loop
57. Figure 5.57 H_y‐B_y hysteresis loop on point B under DC‐biased magnetization (U_ac = 420 V, I_dc = 0.426 A)
58. Figure 5.58 Hysteresis loop on point A under DC‐biased excitation (U_ac = 420 V, I_dc = 0.426 A). (a) H_x‐B_x hysteresis loop; (b) H_y‐B_y hysteresis loop
59. Figure 5.59 The NN DC‐biasing hysteresis model in the limb‐yoke region
60. Figure 5.60 The training performance based on NN
61. Figure 5.61 The predicted and measured DC‐biasing hysteresis loops (H_dc = 213 A/m, B_acm = 0.5065 T)
62. Figure 5.62 Comparison of exciting current between measurement and calculation (H_dc = 107 A/m, U_ac = 470 V)
63. Figure 5.63 The calculated hysteresis loops in point A (H_dc = 107 A/m, U_ac = 470 V)
64. Figure 5.64 The calculated hysteresis loops in point A (H_dc = 107 A/m, U_ac = 470 V)
65. Figure 5.65 The calculated hysteresis loops in point B (H_dc = 107 A/m, U_ac = 470 V)
66. Figure 5.66 Size of the equation matrices for HBFEM method (b) compared with the traditional FEM (a)
67. Figure 5.67 Domain decomposition approach for the FEM mesh
68. Figure 5.68 Domain Decomposition approach for (a) the finite difference mesh and (b) by block
69. Figure 5.69 A sample of (a) Red‐black ordering of a 6 × 4 grid and (b) Associated matrix
70. Figure 5.70 Domain decomposition and frequency domain divided for the system matrix
Chapter 06
1. Figure 6.1 A circuit diagram of a three‐phase transformer. (a) Δ/Y; (b) Δ/Δ
2. Figure 6.2 B‐H curve with hysteresis
3. Figure 6.3 Three‐phase transformer with three input magnetizing coils and three secondary coils as an output winding in Y/Y connection. (a) transformer circuit diagram; (b) 2D cross section of transformer
4. Figure 6.4 A simplified power system wiring diagram under no‐load conditions
5. Figure 6.5 Equivalent magnetic circuit model structure
6. Figure 6.6 Scheme circuit for iron core structure and coil arrangement
7. Figure 6.7 Scheme circuit for iron core structure and coil arrangement
8. Figure 6.8 The laminated core model
9. Figure 6.9 Experimental circuit of the DC‐biased test
10. Figure 6.10 Exciting current under 25% I_dc (U = 293 V)
11. Figure 6.11 Exciting current under 50% I_dc (U = 293 V)
12. Figure 6.12 The magnetic flux density under different DC‐biased magnetization with an alternating voltage of 293 V
13. Figure 6.13 The reluctance under different DC bias (U = 159 V)
14. Figure 6.14 Magnetic circuit model of a single‐phase three limb transformer
15. Figure 6.15 U_ac = 95 V, I_dc = 0
16. Figure 6.16 U_ac = 95 V, I_dc = 0.2708 A
17. Figure 6.17 U_ac = 95 V, I_dc = 0.5091 A
18. Figure 6.18 U_ac = 95 V, I_dc = 1.5214 A
19. Figure 6.19 Waveforms of magnetic flux in the central core
20. Figure 6.20 Waveforms of magnetic flux in the side yoke
21. Figure 6.21 (a) DC‐biased voltage and current waveform against magnetizing curve; (b) the problems caused by combined DC and AC excitation
22. Figure 6.22 GIC or HVDC DC‐biased transformer, including internal equivalent circuit (leakage inductance and winding resistance) and external circuits (transmission line impedance)
23. Figure 6.23 (a) B‐H curve with hysteresis characteristics and DC‐biased condition; (b) excitation current during DC bias
24. Figure 6.24 Block diagram of harmonics current flowing through a transformer in the microgrid with nonlinear load and DG and DS
25. Figure 6.25 Transformer with Y/Δ connected to the grid
26. Figure 6.26 Waveforms of voltage and current in the LV side due to power electronic devices
Appendix
1. Figure A1.1 (a) The laminated core model; (b) geometric model for numerical computation
2. Figure A1.2 The flowchart of the Matlab program for 2‐D numerical computation
3. Figure A3.1 (a) Simulation model of a switching mode transformer; (b) B‐H curve of the magnetic core
4. Figure A3.2 Axi‐symmetrical structure in the z and r domain
5. Figure A3.3 Example program flowchart of a three‐dimensional axi‐symmetrical transformer with DC‐biased excitation

Preface

In writing this book on the Harmonic Balance Finite Element Method (HBFEM): Applications in Nonlinear Electromagnetics and Power Systems, two major objectives were borne in my mind. Firstly, the book intends to teach postgraduate students and design engineers how to define quasi‐static nonlinear electromagnetic (EM) field and harmonic problems, build EM simulation models, and solve EM problems by using the HBFEM. Secondly, this book will delve into a field of challenging innovations pertinent to a large readership, ranging from students and academics to engineers and seasoned professionals.

The art of HBFEM is to use Computational Electromagnetics (CEMs) with harmonic balance theories, and CEM technologies (with IEEE Standard 1597.1 and IEEE Standard 1597.2) to analyze or investigate nonlinear EM field and harmonic problems in electrical and electronic engineering and electrical power systems. CEM technologies have been significantly developed in the last three decades, and many commercially available software packages are widely used by students, academics and professional engineers for research and product design. However, it takes untrained engineers or users several months to understand how to use those packages properly, due to a lack of knowledge on CEMs and EM modeling, and computer simulation techniques. This is particularly true for the harmonic analysis technique, which has not been fully presented in any CEM textbook or used in any commercially available packages. Although a number of CEM‐related books are available, these books are normally written for experts rather than students and design engineers. Some of these books only cover one or a few areas of CEMs, and many common CEM techniques and real‐world harmonic problems are not introduced. This book attempts to combine the fundamental elements of nonlinear EM, harmonic balance theories, CEM techniques and HBFEM approaches, rather than providing a comprehensive treatment of each area.

This book covers broad areas of harmonic problems in electrical and electronic engineering and power systems, and includes the basic concepts of CEMs, nonlinear EM field and harmonic problems, IEEE Standards 1597.1 and 1597.2, and various numerical analysis methods. In particular, it covers some of the methods that are very useful in solving harmonic‐related problems – such as the HBFEM – that are not mentioned in any other numerical calculation books or commercial software packages. In relation to computational technology, this book introduces high‐performance parallel computation, cloud computing, and visualization techniques. It covers application problems from component level to system level, from low‐frequency to high‐frequency, and from electronics to power systems.

This book is divided into six chapters and three appendices. Chapter 1 provides a short introduction to the HBFEM used for solving various harmonic problems in nonlinear electromagnetic field and power systems. This chapter will also discuss definitions of CEM techniques and the various methods used for nonlinear EM problem solving. It also describes high‐performance computation, visualization and optimization techniques for EMs, and CEM standards and validation (IEEE Standard 1597.1 and IEEE Standard 1597.2, 2010).

Chapter 2 highlights some fundamental EM theory used in nonlinear EM fields, harmonic problems in transformer power supplies, DC‐biased phenomenon in High Voltage Direct Current (HVDC) power transformers, harmonic problems in geomagnetic disturbances (GMDS), geomagnetic induced current (GIC), harmonic problems in distributed energy resource (DER) systems and microgrids, and future smart grids with electric vehicles (EV) and vehicle to grid (V2G).

Chapter 3 covers: the fundamental theory of harmonic balance methods used in nonlinear circuit problems; CEM for nonlinear EM field and harmonic problems; basic concepts of HBFEM used in nonlinear magnetic field analysis; HBFEM for electric circuits and magnetic field coupled problems; HBFEM for three‐phase electric circuits coupled with magnetic field; and HBFEM for DC‐biased HVDC power transformers.

Chapter 4 investigates HBFEM and its applications in nonlinear magnetic fields and harmonic problems. Several case study problems are presented, such as: HBFEM for a nonlinear magnetic field with current driven (inductor and single phase transformer); HBFEM for a nonlinear magnetic field with voltage‐driven (switch mode power supply transformer); three‐phase magnetic tripler transformer (electric circuit and magnetic field coupled problems); three‐phase high speed motor based on frequency tripler using HBFEM; DC‐biased 3D asymmetrical magnetic structure transformer using HBFEM.

Chapter 5 is devoted to the advanced numerical approaches of HBFEM. These include: the decomposed algorithm of HBFEM; HBFEM with a fixed‐point technique; hysteresis model based on a neural network and consuming function; and analysis of hysteretic characteristics under sinusoidal and DC bias excitation, parallel computing techniques for multi‐frequency domain problem.

Chapter 6 discusses: three‐phase power supply transformer model; magnetically controlled shunt reactors (MCSR); computation taking account of hysteresis effects based on fixed‐point reluctance; harmonics analysis in HVDC transformers (three phase model) with geo‐magnetics and geomagnetic induced current (GIC); HBFEM used for low‐voltage network transformers in renewable energy and microgrid grid systems with distributed energy resource (DER); and electric vehicle (EV) charging systems and vehicle to grid (V2G).

There are three appendices included in this book: MATLAB Program 1 (magnetic circuit analysis of a single phase transformer) and MATLAB Program 2 (main program for 2D magnetic field analysis in current driven); and Fortran program 3 (3D Asymmetrical magnetic structure transformer using HBFEM).

Junwei Lu

HARMONIC BALANCE FINITE ELEMENT METHOD

APPLICATIONS IN NONLINEAR ELECTROMAGNETICS AND POWER SYSTEMS

Preface

About the Companion Website