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MODELING OF MULTI-PULSE TRANSFORMER
RECTIFIER UNITS IN POWER DISTRIBUTION SYSTEMS
Carl T. Tinsley, III
Thesis submitted to the Faculty of
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
in
Electrical Engineering
Dr. Dushan Boroyevich, Chair
Dr. Jason Lai
Dr. William Baumann
August 5, 2003
Blacksburg, Virginia
Keywords: Average Model, Multi-pulse transformer, Small-Signal Stability
Copyright 2003, Carl T. Tinsley, III
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Modeling of Multi-Pulse Transformer Rectifier Units in Power
Distribution Systems
by
Carl T. Tinsley, III
Dushan Boroyevich, Chairman
Electrical Engineering
(ABSTRACT)
Multi-pulse transformer/rectifier units are becoming increasingly popular in power
distribution systems. These topologies can be found in aircraft power systems, motor
drives, and other applications that require low total harmonic distortion (THD) of the
input line current. This increase in the use of multi-pulse transformer topologies has led
to the need to study large systems composed of said units and their interactions within
the system. There is also an interest in developing small-signal models so that stability
issues can be studied.
This thesis presents a procedure for developing the average model of multi-pulse
transformer/rectifier topologies. The dq rotating reference frame was used to develop the
average model and parameter estimation is incorporated through the use of polynomial
fits. The average model is composed of nonlinear dependent sources and linear passive
components. A direct benefit from this approach is a reduction in simulation time by
two orders of magnitude. The average model concept demonstrates that it accurately
predicts the dynamics of the system being studied. In particular, two specific topolo-
gies are studied, the 12-pulse hexagon transformer/rectifier (hex t/r) and the 18-pulse
autotransformer rectifier unit (ATRU). In both cases, detailed switching model results
are used to verify the operation of the average model. In the case of the hex t/r, the
average model is further validated with experimental data from an 11 kVA prototype.
The hex t/r output impedance, obtained from the linearized average model, has also
been verifified experimentally.
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ACKNOWLEDGMENTS
I graciously thank my advisor, Dr. Dushan Boroyevich, for the time and effort that
he has devoted to all his students over the last two years. I am very grateful to Dr.
Boroyevich, who afforded me the opportunity to start my research in power electronics,
while I was still an undergraduate student. I also extend my gratitude to him for the
generous guidance that he has provided to me over the last three years as my research
and graduate advisor.
Thanks to my other committee members, Dr. Jason Lai and Dr. William Baumann,
for their commitment to serving as dedicated committee members. Dr. Lai’s undergrad-
uate courses initially sparked my interest in power electronics. Dr. Baumann’s controls
course gave me a strong foundation in classical control systems.
I would like to take this time to thank the many students that I have worked with
during my time at CPES. Thanks and appreciation is given to my team members on the
Thales project: Rolando Burgos, Chong Han, Frederic Lacaux, Konstantin Louganski,
Xiangfei Ma, Sebastian Rosado, Alexander Uan-Zo-li, and Dr. Fred Wang. I also want
to thank my other friends at CPES: Julie Zhu, Bing Lu, Bass Sock, Joe Barnette, Jerry
Francis and Josh Hawley. I have enjoyed spending time with you guys inside and outside
of the lab. I would like to thank Steve Chen, Jaime Evans, Marianne Hawthorne, Dan
Huff, Bob Martin, Trish Rose, Theresa Shaw, Elizabeth Tranter, and the rest of the
CPES staff for their support during the last two years. Their dedication makes CPES
what it is today.
Special thanks goes to my family and friends for the support that they have provided
to me during my educational career. Your love, encouragement and motivation has been
a godsend to me during the last two years. To my mom - Sheila Tinsley, my dad - Carl
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Tinsley, Jr., my brother - DeAnthony Tinsley, my nephew, my grandparents, my aunts,
and my uncles: thank you for having faith in me and being there for me as I pursued my
goals.
I would like to acknowledge that there is a power greater than me that made all of
this possible. Thank God for all his wonderful blessings, without Him, none of what I
have achieved would exist.
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TABLE OF CONTENTS
CHAPTER PAGE
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Multi-pulse transformer/rectifier overview . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Different types of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Switching models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Average models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 HEX T/R SWITCHING MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Operation of the hexagon transformer and rectifier . . . . . . . . . . . . 92.1.1.1 Transformer configuration . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Development of the switching model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Simulation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Switching model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 HEX T/R AVERAGE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Average model concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Definition of average model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Hex t/r average model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.1 Average model equation formulation . . . . . . . . . . . . . . . . . . . . . . . 243.4.1.1 Initial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.1.2 Revised model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 1st harmonic assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.3 Switching model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.4 Parameter extraction and estimation . . . . . . . . . . . . . . . . . . . . . . 31
3.4.4.1 Parameter extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
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3.4.4.2 Commutation inductor value estimation . . . . . . . . . . . . . 323.5 Average model verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.1 Steady-state results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5.2 Transient results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6 Average Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 EXPERIMENTAL VERIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Experimental hardware/test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Description of hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.3 Description of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.3.1 Time-domain measurements . . . . . . . . . . . . . . . . . . . . . . 474.2.3.2 Output impedance measurements . . . . . . . . . . . . . . . . . . 48
4.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Time-domain results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.2 Output impedance results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 MODELING OF AN 18-PULSE AUTOTRANSFORMERAND RECTIFIER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Operation of autotransformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.1 Transformer configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Switching model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3.1 Switching model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.2 Switching model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Average model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.2 Equation formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.3 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.4 Average model verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
APPENDIX A 11 kVA HEX T/R SWITCHING MODELOPERATING POINT DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
APPENDIX B STATISTICAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . 85B.1 MATLAB files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.1.1 The α polynomial fit m-file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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B.1.2 The kv polynomial fit m-file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86B.1.3 The ki polynomial fit m-file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86B.1.4 Linear approximation of the variables α, kv, and ki m-file . . . . . . . 87
APPENDIX C SABER SCHEMATIC MODELS . . . . . . . . . . . . . . . . . 91C.1 SABER schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C.1.1 Hex t/r SABER schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.1.2 ATRU SABER schematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C.2 SABER MAST code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.2.1 The α polynomial saber mast file . . . . . . . . . . . . . . . . . . . . . . . . . 99C.2.2 The kv polynomial saber mast file . . . . . . . . . . . . . . . . . . . . . . . . 99C.2.3 The α linear saber mast file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100C.2.4 The kv linear saber mast file . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
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LIST OF FIGURES
Figure Page
1.1 Simplified aircraft power system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Aircraft maintenance frequency changer . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 12-pulse transformer rectifier system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Hexagon transformer/rectifier topology: (a) hexagon transformer and (b)12-pulse rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Switching model schematic of hexagon transformer/rectifier . . . . . . . . . . . 12
2.3 Hex t/r input voltage and current at 10.6 kVA . . . . . . . . . . . . . . . . . . . . 15
2.4 Hex t/r output voltage and current at 10.6 kVA . . . . . . . . . . . . . . . . . . . 15
2.5 Hex t/r output current at 10.6 kVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Harmonic spectrum of hex t/r input current, ia, at 10.6 kVA . . . . . . . . . . 16
2.7 Hex t/r input voltage and current at 4.9 kVA . . . . . . . . . . . . . . . . . . . . . 17
2.8 Hex t/r output voltage and current at 4.9 kVA . . . . . . . . . . . . . . . . . . . . 17
3.1 Black box model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Generator/rectifier space vector diagram . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Hexagon transformer/ rectifier space vector diagram . . . . . . . . . . . . . . . . 25
3.4 Initial average model schematic (steady state) . . . . . . . . . . . . . . . . . . . . . 26
3.5 Initial average model schematic (steady state) with cross-coupling terms . 27
3.6 Average model schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Hex T/R abc to dq transformation: (a) hex t/r dq voltages and (b) hext/r dq currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.8 Average model parameters plotted over the operating range of the hex t/r:(a) α vs. the load current, idc, (b) kv vs. the load current, idc, and (c) kivs. the load current, idc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.9 α vs. the load current, idc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.10 kv vs. the load current, idc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.11 α v. the load current, idc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.12 kv v. the load current, idc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.13 Transient response of output voltage, vdc: (a) without Commutation In-ductance and (b) with Commutation Inductance . . . . . . . . . . . . . . . . . . . 37
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3.14 Transient Response of Output Current, idc: (a) without CommutationInductance and (b) with Commutation Inductance . . . . . . . . . . . . . . . . . 38
3.15 Hex t/r output voltage, vdc, at 10.6 kVA under steady-state conditions . . . 403.16 Hex t/r output voltage, vdc, at 5.1 kVA under steady-state conditions . . . 403.17 Hex t/r output current, idc, at 10.6 kVA under steady-state conditions . . . 413.18 Hex t/r output current, idc, at 5.1 kVA under steady-state conditions . . . . 413.19 Hex T/R Output Voltage, vdc, under transient conditions . . . . . . . . . . . . . 423.20 Hex T/R Output Current, idc, under transient conditions . . . . . . . . . . . . . 42
4.1 11 kVA hex t/r hardware prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Hex t/r 5 kW ac experimental waveforms, 200 V/div, 10 A/div . . . . . . . . 484.3 Hex t/r 5 kW dc experimental waveforms, 50 V/div, 5 A/div . . . . . . . . . . 494.4 Hex t/r 8 kW ac experimental waveforms, 200 V/div, 10 A/div . . . . . . . . 494.5 Hex t/r 8 kW dc experimental waveforms, 50 V/div, 10 A/div . . . . . . . . . 504.6 Output impedance block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7 Output impedance test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.8 Output impedance measurement board . . . . . . . . . . . . . . . . . . . . . . . . . . 524.9 Experimental output impedance at 5 kW . . . . . . . . . . . . . . . . . . . . . . . . 534.10 Comparison of hex t/r output current, idc, at 5 kW . . . . . . . . . . . . . . . . . 544.11 Comparison of hex t/r output voltage, vdc, at 5 kW . . . . . . . . . . . . . . . . . 554.12 Comparison of hex t/r output current, idc, at 8 kW . . . . . . . . . . . . . . . . . 554.13 Comparison of hex t/r output voltage, vdc, at 8 kW . . . . . . . . . . . . . . . . . 564.14 Comparison of hex t/r output impedance at 5 kW . . . . . . . . . . . . . . . . . . 58
5.1 18-pulse ATRU topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 18-pulse autotransformer vector diagram . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Switching model schematic of ATRU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4 ATRU input current, ia, and input voltage, va, at 100 kVA this is a test
to make this really long i hope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.5 ATRU input line current, ia, harmonic spectrum at 100 kVA . . . . . . . . . . 645.6 ATRU output voltage, vdc, and output current, idc, at 100 kVA . . . . . . . . 655.7 ATRU output voltage rails with respect to the input voltage neutral,
vdc,plus and vdc,minus, at 100 kVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.8 ATRU bridge rectifier dc currents at 100 kVA . . . . . . . . . . . . . . . . . . . . . 665.9 abc to dq transformation of ATRU rectifier bridge to currents: (a) abc
currents and (b) dq currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.10 ATRU space vector diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.11 18-Pulse ATRU average model block diagram . . . . . . . . . . . . . . . . . . . . . 715.12 Average model breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.13 Average model circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.14 ATRU output voltage, vdc, and output current, idc, at 100 kVA . . . . . . . . 745.15 ATRU ±270 V output voltage rails, vdc,minus and vdc,plus, at 100 kVA . . . . 745.16 ATRU bridge current, idc,Br, at 100 kVA . . . . . . . . . . . . . . . . . . . . . . . . . 75
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5.17 ATRU Bridge 1 output voltage rails, vdcplus,Br1 and vdcminus,Br1, at 100 kVA 75
C.1 Hex t/r switching model SABER schematic . . . . . . . . . . . . . . . . . . . . . . . 92C.2 Hex t/r average model SABER schematic . . . . . . . . . . . . . . . . . . . . . . . . 93C.3 ATRU switching model SABER schematic . . . . . . . . . . . . . . . . . . . . . . . . 94C.4 ATRU average model SABER schematic . . . . . . . . . . . . . . . . . . . . . . . . . 95C.5 ATRU average model block SABER schematic . . . . . . . . . . . . . . . . . . . . 96C.6 ATRU bridge rectifier average model SABER schematic . . . . . . . . . . . . . . 96C.7 ATRU Bridge 1 average model SABER schematic . . . . . . . . . . . . . . . . . . 97C.8 Average model circuit SABER schematic model . . . . . . . . . . . . . . . . . . . . 98
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LIST OF TABLES
Table Page
2.1 12-pulse hex t/r switching model parameter values at 10.6 kVA . . . . . . . . 14
3.1 The dq rotating coordinates average values . . . . . . . . . . . . . . . . . . . . . . . 313.2 The α polynomial terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 The kv polynomial terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 The α linear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 The kv linear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Hex t/r average model circuit parameter values . . . . . . . . . . . . . . . . . . . . 39
4.1 11 kVA hex t/r hardware prototype specifications . . . . . . . . . . . . . . . . . . 464.2 Audio amplifier, Jensen XA2150, specifications . . . . . . . . . . . . . . . . . . . . 514.3 Comparison of hex t/r results at 5 kW . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4 Comparison of hex t/r results at 8 kW . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 Output impedance measurement system characterization . . . . . . . . . . . . . 57
5.1 Input/output specifications for 100 kVA 18-pulse ATRU . . . . . . . . . . . . . 605.2 100 kVA 18-pulse ATRU switching model parameter values . . . . . . . . . . . 635.3 ATRU average model circuit parameter values . . . . . . . . . . . . . . . . . . . . . 73
A.1 Hex t/r switching model operating point data . . . . . . . . . . . . . . . . . . . . . 84
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xvi
CHAPTER 1
INTRODUCTION
1.1 Motivation
This work was motivated by the need to simulate large power distribution systems
and to study interactions between the individual subsystems. Aircraft power system is
one of the applications in which these large scale power distribution simulation models
prove useful [1]. As the move towards the More Electric Aircraft (MEA) continues, there
is a desire to effectively model these systems in both the time domain and the frequency
domain [1], [2], [3]. A simplified aircraft power distribution system is presented in Figure
1.1 [2]. This system differs from the traditional aircraft power system in that the aircraft
starter/generator supplies a variable frequency. One of the main components of this
system is the AC/DC converter that provides the 270 V dc bus voltage. It has been
shown in literature that one possible solution is the 18-pulse autotransformer rectifier
unit (ATRU) [4]. This ATRU topology has the advantages of reduced kVA ratings and
improved line current harmonics [4].
Aircraft maintenance frequency changers, such as the one shown in Figure 1.2, are
commonly connected in parallel at the input to service several aircrafts at one time. This
frequency changer converts the 60 Hz ground supply to 400 Hz so that the avionics in
the aircraft can be serviced [5]. The step-down isolation transformer and 12-pulse diode
rectifier, commonly referred to as the 12-pulse hexagon transformer/rectifier (hex t/r), is
the front-end to the system. The hex t/r rectifies the ac input and provides a stable dc
1
AircraftEngine
Starter/Generator
AC/DCConverter
270 VMain DC Bus
270 VLoad
270 VLoad
Figure 1.1 Simplified aircraft power system
supply for the 4-leg inverter. In order to study the stability of the entire system, small-
signal models of the individual subsystems are needed. For this particular application,
the system being studied is the hex t/r. This topology greatly reduces the harmonics in
the input line current [6].
Step-DownIsolationHexagon
Transformer
12-pulseDiode
Rectifier
4-Leg Inverter
Controller
DC Link
3-phaseVariable Frequency
3-phase60 Hz
Figure 1.2 Aircraft maintenance frequency changer
Two examples of multi-pulse transformer/rectifier units operating in power distribu-
tion systems have been presented. One of the aims of this work is to effectively develop
models of these multi-pulse topologies that can be used in the large system simulation
models. It should be noted that the term stability refers to small-signal stability. In
most cases, the small-signal model is obtained by linearizing about an operating point.
In this thesis, the average model of the multi-pulse transformer topologies is derived for
use in small-signal analysis. The derived model is continuous and non-linear in nature.
Software is used to linearize the system about an operating point. While literature shows
that average models of 6-pulse bridge rectifiers have been developed which are suitable
2
for small-signal analysis, [7] and [8], there is little information on the average models of
multi-pulse transformer/rectifier units.
The next sections will describe the operation of multi-pulse transformer/rectifier units,
and will discuss the different types of models that could be used in the simulation of large
power distribution systems.
1.2 Multi-pulse transformer/rectifier overview
This section will provide an overview of multi-pulse transformer/rectifier systems.
The operation of the 12-pulse transformer rectifier will be reviewed, and some applications
of these circuits will be provided.
1.2.1 Operation
Literature on multi-pulse transformer/rectifier topology has existed for several years
[9]. The term multi-pulse is defined as any number of n 6-pulse bridge rectifiers con-
nected in series or parallel, where n is greater than 1. The two main advantages to
using multi-pulse transformer/rectifier topologies are a reduction in the ac input line
current harmonics and a reduction in the dc output voltage ripple [6]. The input current
harmonics are reduced through the use of phase-shifting transformers.
The expressions
Harm = 6kn± 1 (1.1)
Mag =1
6kn± 1(1.2)
k = any positive integer (1.3)
n = number of six pulse converters (1.4)
provide a simple way to calculate the frequency and magnitude of harmonics that will be
present in the ac input line current when multi-pulse topologies are implemented [6]. The
3
frequencies at which the harmonics will appear for an n-pulse converter are computed
by multiplying 1.1 by the fundamental frequency of the system. The magnitude of
the harmonics are calculated by multiplying 1.2 by the amplitude of the signal at the
fundamental frequency. For example, if a 12-pulse transformer/rectifier system were to be
implemented, the first harmonics to contribute to the total harmonic distortion (THD)
of the input line current would be the 11th and 13th. It can be seen that by using a
phase-shifting transformer and adding and additional 6-pulse diode rectifier bridge, the
harmonic content in the supply current is attenuated up to the 11th harmonic as opposed
to the 5th for the traditional 6-pulse bridge rectifier.
The 12-pulse transformer/rectifier system shown in Figure 1.3 is a topology com-
monly found in existing literature [10] - [11]. The system shown in Figure 1.3 has the
two diode bridges connected in parallel. For this particular type of connection, inter-
phase transformers are required. The interphase transformers absorb the difference in
the instantaneous voltage produced by the two 6-pulse rectifiers [9]. The interphase
transformers prevent the two bridges from interacting with one another and allow the
conduction angle of the diode to remain at 120. For this particular topology, each bridge
rectifier processes 50% of the load power.
The cancellation of harmonics in the ac input line current is achieved through the
use of phase-shifting transformers. For this example, the phase shift employed by the
transformer is 30. The primary side of the transformer is connected in a delta, while
the secondary is connected in delta and wye. The phase shift produced by the delta
and wye secondary voltages is what allows for the cancellation of the current harmonics.
One of the issues associated with this topology is that the turns ratio in the secondary
must approximate an irrational number (√
3). This approximation can lead to voltage
imbalance between the two bridges which will reduce the attenuation of harmonics in the
ac input line current.
4
Load
Bridge 1
Bridge 2
InterphaseTransformer
idc
idc,Br1
idc,Br2
+
-
vdc
+
+
-
-
vdc,Br2
vdc,Br1
ia
ib
ic
ia,Br1
ib,Br1
ic,Br1
ia,Br2
ib,Br2
ic,Br2
Figure 1.3 12-pulse transformer rectifier system
5
1.2.2 Applications
The multi-pulse transformer topology generally acts as an interface between the power
electronics load and the utility supply. Some of the most common applications for multi-
pulse transformer/rectifier systems include motor drives, interruptible power supplies
(UPS) systems, aircraft variable speed constant frequency (VSCF) systems, and fre-
quency changer systems [10], [12]. In other work [13], the author develops an 18-pulse
autotransformer rectifier system that does not require the use of interphase transform-
ers. He instead takes advantage of the unequal current sharing in the three bridges. He
is able to reduce the system size by eliminating the interphase transformers, and also
achieves a harmonic current that is reduced as compared with that of a 12-pulse system.
Some other applications of multi-pulse transformer topologies involve adding switching
circuitry to the interphase transformers to improve the pulse number of the line current
[11], [14].
1.3 Different types of models
This section will provide some background information on the different types of models
available for analyses. For this work, the focus will be directed toward switching models
and average models.
1.3.1 Switching models
Simulation models that account for the turning on and off of semiconductor switches
are commonly referred to as switching models. These detailed computer simulation mod-
els are used to observe the operation of the converter during steady-state and transient
operation. Computer simulation programs such as SABER and MATLAB can be used to
simulate these complex circuits [15], [16], [17], [18]. Some of the disadvantages to using
switching models include numerical instability, long simulation time, convergence errors,
6
and huge computational loads [19] - [20]. These issues, along with the need for a model
that can be used for small-signal analyses, has led to the development of average models.
1.3.2 Average models
Some of the advantages of average models include reduced simulation time, ability to
simulate transient conditions, and the ability to perform small-signal analyses [20]. Some
of the functions related to small-signal analysis include assessment of stability and design
of closed-loop controllers. Average models have been used to simulate large dc power
systems [20]. There, the results have been compared with test data to demonstrate that
the modeling approach is valid.
1.4 Objectives
This thesis presents a detailed procedure for developing the average model of multi-
pulse transformer/rectifier units. An average model of the 12-pulse hex t/r is presented
and verified through simulation and experimental data. The average model concept
developed for the 12-pulse hex t/r is then extended to the more complex 18-pulse ATRU.
The ATRU results are validated through a comparison with the detailed switching model.
The presented procedures and concepts are adopted from previous results for 6-pulse
transformer/rectifiers and are applied here to 12-pule and 18-pulse units for the first
time.
Chapter 2 will discuss the switching model of the 12-pulse hex t/r. A general review
of the topology is presented, and the issues encountered during simulation are discussed.
Simulation results obtained under steady-state conditions are provided for reference.
The average model of the hex t/r is the main focus of Chapter 3. The development
of the average model from conception to implementation is discussed. Issues such as
accounting for commutation inductance and accounting for the variation in parameters
is presented. Results are compared with the switching model under steady-state and
transient conditions. Chapter 4 presents experimental results that were collected from a
7
11 kVA hex t/r hardware prototype. The data from the hardware testing is compared
with the average model and switching model simulations under steady-state conditions.
The small-signal validity of the average model is verified by experimentally measuring
the output impedance. The average model concept is extended to the 18-pulse ATRU
topology in Chapter 5. A detailed switching model is provided to verify the results
generated by the 18-pulse ATRU average model. Finally, Chapter 6 concludes this work
by summarizing the main points covered and providing a few final comments.
8
CHAPTER 2
HEX T/R SWITCHING MODEL
2.1 Introduction
This chapter will provide detailed information about the switching model of the hex
t/r. The switching model is defined as the detailed computer simulation that models
the commutation and conduction of the diodes in the rectifier bridge. The hexagon
transformer windings are also included in the switching model. Since diodes are used in
this topology, the switching is uncontrolled. The following sections will provide insight
into the operation of the hex t/r as well as some simulation results. Issues encountered
during the development of the switching model will also be discussed.
2.1.1 Operation of the hexagon transformer and rectifier
The hex t/r topology, shown in Figure 2.1, is used as the front-end to the frequency
changer. The hex t/r function is similar to that of the standard wye-delta-wye 12-pulse
transformer/rectifier. As mentioned in the introduction, the harmonics in the line current
are reduced due to the multi-pulse concept. Some other advantages gained by using the
hex t/r topology include well-matched voltage and leakage reactances and the elimination
of the interphase reactor [21]. The windings of the hexagon transformer are discussed in
the following section.
9
A
BC
A1
A2
B1
B2
C1
C2
X10
X1
X7
X5
X11
X3
X9
X2
X8
X6
X12
X4
Taps
Secondary Winding
Virtual Neutral
PrimaryWinding
(a)
X10
X1 X7 X5 X11X3 X9
X2 X8 X6 X12 X4
LcLcLc
Lc Lc Lc
Lfilter
Cfilter R
+
-
vdc
(b)
Figure 2.1 Hexagon transformer/rectifier topology: (a) hexagon transformer and (b)12-pulse rectifier
10
2.1.1.1 Transformer configuration
The primary of the hexagon transformer is connected in delta. The hexagon shape is
formed by connecting the secondary windings end to end. Each primary winding has two
associated secondary windings. The secondary windings are tapped such that the output
voltages are phased 30 apart with respect to the virtual neutral, , which is located in
the center of the hexagon. The twelve taps are connected to diodes, where two 6-pulse
midpoint converters are formed. One of the converters provides the positive dc voltage
potential, while the other provides the negative dc voltage potential [21].
The turns ratio between the secondary windings and the tap windings is tan 15.
Although this number is difficult to reproduce, it can be approximated easier than the√
3 in the delta-wye-delta configuration, particularly at low voltages. The leakage reac-
tances in the system are well matched due to the fact that each primary winding can be
sandwiched between two secondary windings [21].
In this particular application, commutation inductors are used to improve the cur-
rent harmonics. The commutation inductors adjust the commutation overlap angle of
the diodes by interacting with the leakage inductances of the transformer [22]. By com-
pensating the line reactance in the transformer, the 11th harmonic can be reduced to less
that 3% [6].
2.2 Development of the switching model
The switching model of the hex t/r consists of a three-phase delta connected power
supply, a transformer core consisting of twelve secondary legs, a rectifier bridge containing
12 diodes, and an output filter. The switching model was constructed using the SABER
simulation program [15], [17]. The model as it appears in SABER is shown in Figure 2.2.
The transformer models used in the switching model are linear and only account for
magnetizing inductance. The switching model does not take into account the parasitics
such as leakage inductance and winding resistance. The diode models that are used are
piecewise linear models. An on and off conductance, as well as an on voltage, can be
11
Thr
ee-P
hase
Inpu
t Sou
rce
Hex
agon
Tra
nsfo
rmer
12-p
ulse
B
ridge
Rec
tifie
r
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10
X11
X12
L cL c
L c
L cL c
L c
L filt
er Cfil
ter
Rv d
c
+ -
Vb
Va
Va
Vb
Vb
Vc
Vb
Vc
Vc
Vc
Va
Va
X2
X1
X3
X4
X5
X6
X7
X8
X12
X11
X10
X9
c1 c2
c2
c1c3
c3
c4
c4 c5
c5 c6
c6
i dc
Va
Vb
Vc
Figure 2.2 Switching model schematic of hexagon transformer/rectifier
12
specified for the diode model. A series resistance is included in both the filter inductor
and filter capacitor to make the circuit more realizable.
2.2.1 Simulation issues
Some of the issues encountered while simulating the switching model include conver-
gence and numerical instability. Due to the complexity of the topology, ramp functions
were used to soft start the system to aid with convergence. These soft starts are required
partly due to the commutation inductors used in the topology. The commutation induc-
tors are connected in series with the switching elements (diodes) and in series with the
output filter inductor. This configuration makes it difficult for the solver to calculate the
steady-state operating point. Initial conditions, other than zero, are unhelpful because
it is practically impossible to precisely match the values of all the inductor currents and
all the diode conductor states.
Due to the complex circuitry, the range of the time step required for convergence is
generally very large. The maximum time step is on the order of hundreds of microseconds
while the smallest time step is in the nanosecond to picosecond range. As the operating
point of the hex t/r moves into the light load range, the range of the time step becomes
more reasonable.
There is significant numerical instability that shows up the in the switching model
waveforms. This numerical instability is addressed in the next section.
2.3 Switching model results
The switching model of the hex t/r is simulated at various load conditions (full load
and half load) in order to illustrate some the issues listed in the previous section and to
verify the operation of the hex t/r. The parameters used in the simulation are shown
in Table 2.1. The on voltage and the on and off resistance of the diodes are represented
as von, Ron and Roff , respectively. The magnitizing inductance used in the hex t/r
simulation is labeled as Lmag. The line frequency of the system is listed as fline. The full
13
load operating point of the hex t/r corresponds to an output power of 10.6 kVA (R =
4.05 Ω), while the half load operating point is 5.1 kVA (R = 10 Ω). The results shown
in Figures 2.3 - 2.8 demonstrate the operation of the hex t/r system.
Table 2.1 12-pulse hex t/r switching model parameter values at 10.6 kVA
Parameter Value
Vab (rms) 440.0 VLfilter 1124 µHRLfilter,esr 200 mΩCfilter 2400 µFRCfilter,esr 50 mΩLc 675.0 µHR 4.050 ΩVon 1.25 VRon 1 µΩRoff 1 TΩLmag 3 Hfline 60 Hz
The input voltage and current waveforms are shown in Figures 2.3 and 2.7 at full
load and half load, respectively. The clean input voltage waveforms can be attributed to
the ideal voltage source used. The input current, ia, has nearly sinusoidal shape. This
nice waveform can be attributed to the 12-pulse topology. The output voltage and the
output current of the hex t/r at full load and half load are shown in Figures 2.4 and 2.8.
The output voltage, vdc, has very little ripple, due to the large filter capacitor used in
the simulation. The output current, idc, has been plotted over one 60 Hz line cycle in
Figure 2.5. It can be observed that there are 12 ripples in one line cycle. The harmonic
spectrum of the line current, ia is shown in Figure 2.6. The magnitude of the harmonics
is plotted as a percentage of the fundamental. The bottom half of the figure zooms in on
the harmonic content, specifically focusing on the high orders such as the 11th, 13th, 23rd,
25th and so on. It can be observed that the harmonic content is extremely low. This is
in agreement with the the claims of the hex t/r topology.
14
0.04 0.05 0.06 0.07 0.08 0.09 0.1
−600
−400
−200
0
200
400
600
Inpu
t Vol
tage
(V
olts
)
Time (sec)
vab
0.04 0.05 0.06 0.07 0.08 0.09 0.1−30
−20
−10
0
10
20
30
Time (sec)
Inpu
t Cur
rent
(A
mps
) ia
Figure 2.3 Hex t/r input voltage and current at 10.6 kVA
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08205
206
207
208
209
210
Out
put V
olta
ge (
Vol
ts)
Time (sec)
vdc
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.0845
50
55
Time (sec)
Out
put C
urre
nt (
Am
ps)
idc
Figure 2.4 Hex t/r output voltage and current at 10.6 kVA
15
0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.05645
46
47
48
49
50
51
52
53
54
55
Time (sec)
Out
put C
urre
nt (
Am
ps)
idc
Figure 2.5 Hex t/r output current at 10.6 kVA
0 500 1000 1500 2000 2500 30000
20
40
60
80
100
Frequency (Hz)
% M
agni
tude
ia
500 1000 1500 2000 2500 30000
1
2
3
4
5
Frequency (Hz)
% M
agni
tude
11th
13th
23th
25th 35th
37th
ia
Figure 2.6 Harmonic spectrum of hex t/r input current, ia, at 10.6 kVA
16
0.04 0.05 0.06 0.07 0.08 0.09 0.1
−600
−400
−200
0
200
400
600
Inpu
t Vol
tage
(V
olts
)
Time (sec)
vab
0.04 0.05 0.06 0.07 0.08 0.09 0.1−15
−10
−5
0
5
10
15
Time (sec)
Inpu
t Cur
rent
(A
mps
) ia
Figure 2.7 Hex t/r input voltage and current at 4.9 kVA
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09225
226
227
228
229
230
Out
put V
olta
ge (
Vol
ts)
Time (sec)
vdc
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.0920
21
22
23
24
25
Time (sec)
Out
put C
urre
nt (
Am
ps) i
dc
Figure 2.8 Hex t/r output voltage and current at 4.9 kVA
17
Some of the convergence issues associated with this model can be seen in Figure 2.7.
Large spikes can be observed in the line current, ia. These spikes can be attributed to
the complexity of the hex t/r topology, as discussed in the previous section. In Chapter
4, it will be shown that this error is nothing more than numerical instability.
18
CHAPTER 3
HEX T/R AVERAGE MODEL
3.1 Introduction
For the analyses of a large system, the average model satisfies three purposes. First,
the average model is needed to provide accurate steady-state and transient results so
that the computational expenses and numerical instabilities can be eliminated. Second,
the average model of the hex t/r is required so that the stability of the system can be
assessed on both global and local scales. Finally, the average model can be used to
perform parametric studies.
3.2 Average model concept
The concept of the average model of the hex t/r will be described in the following
sections.
3.2.1 Definition of average model
The switching model waveforms presented in Chapter 2, such as those given in Figures
2.3 and 2.4, contain high-order harmonics in both the ac and dc variables. In terms
of assessing the steady-state operation of the system, the higher-order terms can be
neglected since only the average and root mean square (rms) values of the first harmonic
are of interest. Because of this, the average model needs only to consider the fundamental
19
frequency. The assumptions made when working with the fundamental frequency will be
formally presented in Section 3.4.2.
In most switching systems, the average model is obtained by calculating the average
over one switching interval. During this switching interval, the high-frequency switching
content is removed or averaged out. This averaging function, sometimes referred to
as a ’moving average’ is described in other work [23] and is shown in (3.1). The value
computed by (3.1) will change from switching period to switching period as low-frequency
perturbations are encountered by the system.
x(t) =1
Ts
∫ t
t−Tsx(τ)dτ (3.1)
The multi-pulse transformer/rectifier diode commutations occur 6n times every line
period, so that Ts = 6n/fline, where n is the number of 6-pulse rectifiers in the unit. The
average models of three-phase bridge rectifiers have already been proposed [7], [24], [25],
and [8]. In all of these cases, the topology of the bridge rectifier differs from the one
described in this application. In particular, the topologies studied were 6-pulse, not 12-
pulse as is the case of the hex t/r. Also, there are no commutation inductors immediately
following the diodes in the bridge. It will be shown in Section 3.4.4.2 that the value of
the commutation inductance used in the average model greatly affects transient response.
One attribute that is similar in all the proposed solutions is that the average model is
derived in the dq0 reference frame, so the three-phase abc input can be directly related
to the dc output through the use of scaling constants. This concept will be used in the
derivation of the hex t/r average model.
3.2.2 General approach
The switching model provides information important to the creation of a reduced-
order model with which the system can be modeled using a black box approach, as
shown in Figure 3.1. By linearizing about several operating points, equations do not need
to be developed that detail the exact relationship between the input and the output.
20
Therefore, there is sufficient experimental and simulation data to describe the system
empirically. In this way, any system can be described as long as its basic operating
principles are understood. Naturally, for the system shown in Figure 2.2 and other similar
configurations, the basic principle is to relate the magnitude of the input vector to the
output. This methodology can produce accurate models, but these models are only valid
for a specific operating range. However, more general approaches normally do not take
into account second order effects such as operating temperature, core saturation, etc. The
switching model automatically includes these because it is based upon experimental data.
Likewise, the resulting average model should also capture these dynamics. The trade-off
is between a mathematical model that is accurate throughout all possible operating points
and an empirical model that is very accurate but only for a certain range of operating
points.
Inputs Outputs
Experimentaland Simulation Data
System
Figure 3.1 Black box model
3.3 Previous work
The average model of the hex t/r evolved from previous research conducted by Ivan
Jadric [26]. Jadric was interested in designing a dc-link controller for a synchronous
generator set. This generator set included two separate 6-pulse diode rectifiers. In order
to design the controller, Jadric needed small-signal models of all of the subsystems in
the generator set, which included the two diode rectifiers. This led him to develop an
average model of a 6-pulse diode rectifier.
21
In the derivation of the 6-pulse diode rectifier model, Jadric develops a relationship
between the fundamental frequency of the ac variables and the average value of the dc
variables. The magnitudes of the fundamental harmonics of the generator’s voltage and
current are assumed to be proportional to the dc components of the rectified voltage and
current, vdckv
and idcki
, respectively. The space vector diagram for this system is shown in
Figure 3.2.
q
d
vd
δ
φ
id
iqvq
vdc/kv
idc/ki
Figure 3.2 Generator/rectifier space vector diagram
Using the information shown in Figure 3.2, equations can be derived that describe the
operation of the average model of the diode rectifier. The equations shown in (3.2) - (3.7)
can be used to model the 6-pulse diode rectifier. The angle δ represents the generator’s
rotor angle, while φ accounts for the phase shift between the fundamental harmonic of
the generator’s voltage and current. The quantities vd, vq, id, and iq are the generator’s
voltage and current transformed to the dq reference frame.
22
vdc = kv(vdsinδ + vqcosδ) (3.2)
idc = ki(idsin(δ + φ) + iqcos(δ + φ)) (3.3)
id =idckisin(δ + φ) (3.4)
iq =idckicos(δ + φ) (3.5)
δ = tan−1
(vdvq
)(3.6)
φ = tan−1
(idiq
)− tan−1
(vdvq
)(3.7)
The average model presented by Jadric requires some general comments. During the
development of the model, a first harmonic assumption is made. This means the model
is only valid at the fundamental frequency of the generator’s voltage and current. The
second assumption that is made is that the energy transfer occurs at the fundamental
frequency. The equations governing the power balance of the system are shown in (3.8)
and (3.9). Equation (3.10) is valid only when the diode rectifier is assumed to be lossless.
pin = vdid + vq iq (3.8)
pout = vdcidc (3.9)
pin = pout (3.10)
3.4 Hex t/r average model development
The average model of the hex t/r is an extension of the model developed by Jadric.
This model differs from his in that the diode bridge includes 12 diodes in one bridge in-
stead of six in Jadric’s case. The hex t/r topology also contains commutation inductors
that can not be neglected. These factors must be taken into account during model devel-
opment. The development of the average model of the hex t/r in the dq0 reference frame
23
will be discussed in this section. The proposed model is divided into three subsections:
equation formulation, first harmonic assumptions, and parameter extraction.
3.4.1 Average model equation formulation
The development of the hex t/r average model can be broken down into several steps.
The initial model that was developed was revised several times, leading to its present
form. This section will describe in detail the initial model and the subsequent revisions.
3.4.1.1 Initial model
The first step in developing the hex t/r average model involves generating a space
vector diagram similar to the one depicted in Figure 3.2. The space vector diagram
in Figure 3.3 differs from Jadric’s in that the input voltage is aligned solely with the
d-channel. In this case, the Park’s transform is aligned with the line-to-neutral voltage
vector vln to force the vq component to zero. A direct result of a zero value for vq is that
the angle δ is zero and can removed from the diagram. The vector |v ′dc| represents the
output voltage of the rectifier prior to filtering. The angle α represents the phase shift
between the fundamental harmonic of the hex t/r’s input voltage and input current. The
vectors id and iq represent the input current of the hex t/r in the dq coordinate system.
The output current is represented by idc. The variables kv and ki are used to develop a
relationship between the ac and dc quantities of the voltages and currents.
The next step in the development process of the hex t/r average model is to write
equations that describe the geometry of the space vector diagram. These equations,
shown in (3.11) - (3.15), are continuous in nature and describe the operation of the
hex t/r. These equations are valid at any operating point. Now that the space vector
diagram and resulting equations have been explained in detail, the circuit model can be
introduced.
24
q
dα
vd=vin-abc=vdc,Br/kv
iin-abc=idc,Br/ki
id
iq
Figure 3.3 Hexagon transformer/ rectifier space vector diagram
|v′dc| = kv
√v2d + v2
q (3.11)
vq = 0 (3.12)
id =idcki
cos(α) (3.13)
iq =idcki
sin(α) (3.14)
α = tan−1
(iqid
)(3.15)
The third step in developing the average model of the hex t/r involves developing
a circuit model. This average model circuit is composed of dependent sources and a
passive filter. The circuit of the average model of the hex t/r is presented in Figure 3.5.
The resistor Rw is used to approximate the losses. This value is computed by using the
efficiency data from the switching model or measurements. The filter components, Lfilter
and Cfilter have the same value as in the real circuit. The dq currents and dc link voltage,
vdc, are represented by dependent current and voltage sources, respectively.
Using the set of continuous equations provided in (3.11) - (3.15) and the circuit model
shown in Figure 3.4, the average model can be used to simulate the steady-state operation
of the hex t/r at any operating point. For this model to work properly, operating point
25
+−
+−
-+
+
-
vd
vq
id
iq
vdc’ RC vdc
Rw
idc
Lfilter
Figure 3.4 Initial average model schematic (steady state)
data must be collected from either a detailed switching model or actual hardware. When
considering the validity of this average model during transient conditions, other factors
must be considered, such as variations in the parameters α, kv and ki. This is discussed
in detail in Section 3.4.4.
One of the key features of the hex t/r topology is the commutation inductance on the
dc side of the unit. This commutation inductance is used to adjust the leakage reactance
in the transformer. It is known that these commutation inductors affect the dynamics
of the system and incur a voltage drop. In an effort to include the voltage drop in the
average model circuit, cross-coupling terms were added to the circuit. The circuit model
shown in Figure 3.5 uses the product βωLc multiplied by the current to account for
the voltage drop. The term ω represents the electrical frequency of the rectifier’s input
voltage, β, which is a variable that is computed at each operating point to adjust the
output voltage to its correct value, and Lc represents the value of one of the coils in the
commutation inductor. New equations governing the operation of the hex t/r must be
derived due to the inclusion of the cross-coupling terms.
The equations shown in (3.13) - (3.14) and (3.16) - (3.18) describe the operation of
the hex t/r circuit presented in Figure 3.5. For this system, there are more equations
than unknowns, therefore at any operating point all of the unknown variables can be
solved for. When comparing this model at steady state and in its transient period, it is
26
+−
+−
+
-
+
-
-+
+
-
vd
vq
vd’
vq’
id
iq
vdc’ RC vdc
Rw
idc
Lfilter
-++-
βωLcid
βωLciq
Figure 3.5 Initial average model schematic (steady state) with cross-coupling terms
observed that the transient results are not within desirable limits. The output current
transients match very closely, yet there is an appreciable steady-state error in the voltage
transient simulations. To correct this phenomenon, the inclusion of inductance on the ac
side of the average model is considered.
v′d = vd − βωLciq (3.16)
v′q = vq + βωLcid (3.17)
|v′dc| = kv
√v′
2d + v′
2q (3.18)
Adding inductance to the ac side of the average model circuit did not improve the
dynamic response of the output voltage during transient periods. Convergence errors
were encountered in software, and this approach was abandoned.
3.4.1.2 Revised model
It has previously been discussed that the dynamic response of the average model of
the hex t/r requires modifications. The revisions are executed in order to improve the
response and to simplify some of the mathematics. This revised system simplifies the
equations and strictly follows the space vector diagram shown in Figure 3.3.
27
The first step in revising the model involves dropping the cross-coupling terms from
the ac side of the hex t/r average model. These cross-coupling terms did not provide any
insight into the system. Initially, the goal was to model the voltage drop associated with
the commutation inductance. Due to the complexity of the design, this voltage drop can
not be measured in simulation or in the hardware, so it was decided to remove these
terms from the average model. This reduces the number of equations from six to four,
and (3.11)-(3.15) can be used to describe the operation of the hex t/r. With the reduction
in equations, there are now three variables, α, kv and ki, that must be calculated at every
operating point. It will be shown that some of these variables vary over the entire load
range and require some polynomial fits to improve the overall accuracy of the hex t/r
model. The use of the polynomial fits will be discussed in detail in section 3.4.4.1.
The second improvement that was made to the average model involved adding induc-
tance on the dc side of the circuit model to account for the commutation inductance.
This addition of inductance greatly improved the transient response of the system, and
will be discussed in further detail in Section 3.4.4.2. The revised average model of the
hex t/r is shown in Figure 3.6. The equations governing the operation of the revised
hex t/r average model are presented for completeness in (3.19) - (3.23). The use of the
inductor 23Lc is discussed in Section 3.4.4.2.
+−
+−
-+
+
-
vd
vq
id
iq
vdc’ RC vdc
Rw
idc
2/3*LcLfilter
Figure 3.6 Average model schematic
28
|v′dc| = kv
√v2d + v2
q (3.19)
vq = 0 (3.20)
id =idcki
cos(α) (3.21)
iq =idcki
sin(α) (3.22)
α = tan−1
(iqid
)(3.23)
3.4.2 1st harmonic assumption
In section 3.2.1 it was stated that a 1st harmonic assumption was taken in developing
the model. This implies that power transfer occurs only at the fundamental frequency.
It is also assumed that the average model is valid only at the fundamental frequency.
Based on the previous two assumptions the Park’s transformation is used to eliminate
the time-varying nature of the ac voltages and currents at the fundamental frequency.
3.4.3 Switching model analysis
In the next section, the estimation of three parameters α, kv and ki will be discussed.
Prior to calculating these parameters, certain operating point data must be extracted
from the switching model. At this time, the only operating point data from the switch-
ing model that has not been discussed is the transformation of the hex t/r’s input voltage
and input current into the dq rotating reference frame. The d-channel of the Park’s trans-
formation is aligned with the line-to-neutral voltage vector. A result of this alignment
is that the voltage in the q-channel is zero. The abc-to-dq0 transformation is shown in
(3.24). Since this is a balanced three-phase system, the 0-channel does not exist. The
waveforms of the hex t/r input voltages and currents obtained from the switching model
in Section 2.3 at 10.6 kVA and transformed to rotating coordinates are shown in Figure
3.7.
29
Tabc/dq0 =
√2
3
sin(θ) sin(θ − 2π3
) sin(θ + 2π3
)
− cos(θ) − cos(θ − 2π3
) − cos(θ + 2π3
)
1√2
1√2
1√2
(3.24)
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.080
50
100
150
200
250
300
350
400
450
Time (sec)
dq V
olta
ge (
Vol
ts)
vd
vq
(a)
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.085
10
15
20
25
30
Time (sec)
dq C
urre
nt (
Am
ps)
idiq
(b)
Figure 3.7 Hex T/R abc to dq transformation: (a) hex t/r dq voltages and (b) hex t/rdq currents
It can be observed that a ripple only exists in the dq currents and not the dq voltages.
This can be explained by the fact that the voltage sources used in the simulation are
nearly ideal, therefore all harmonics other than the fundamental are negligible. Since
only the first harmonic is present, there is no ripple in the dq voltages. This also ex-
plains why a ripple appears in the dq current. When viewing the time-domain and
frequency-domain waveforms of the line current, ia, in Figures 2.3 and 2.6, respectively,
it is observed that harmonics other than the fundamental are present and can not be
neglected. Since Park’s Transformation only considers the fundamental frequency, the
additional harmonics generate the ripple that is seen in the Figure 3.7(a). The average
values of the waveforms presented in Figure 3.7 are listed in Table 3.1 for reference. This
information will be used in the following chapter to aid in computing the parameters α,
kv and ki.
30
Table 3.1 The dq rotating coordinates average values
Parameter Average Value
vd 440.0 Vvq 0.000 Vid 25.75 Aiq 11.74 A
3.4.4 Parameter extraction and estimation
As mentioned previously, there are three parameters, α, kv and ki, that must be
calculated at each operating point. The average model of the hex t/r needs to be valid over
the entire operating range of the hex t/r. This requirement exists since the average model
needs to also be valid during transient conditions. Due to variations in the parameters,
polynomial fits are required to improve the accuracy of the model. This section will
describe the methods used to calculate the parameters and the polynomial fits used
during transient periods.
3.4.4.1 Parameter extraction
The parameters extracted from the switching model are α, ki and kv. They are
extracted by post-processing the operating point data, vd, vq, id, iq, idc, vdc and v′dc.
The average value of the operating point data is applied to equations (3.11) - (3.15) to
compute α, kv and ki. This process is repeated at each desired operating point.
One of the goals of the average model is for it to be valid during transient periods.
For this to be true, the parameters at each operating point are calculated to determine
how they vary with the load. The results are shown in Figure 3.8. It can be observed that
α and kv vary greatly over the load range. In order to remedy this problem, polynomial
fits of these parameters are generated. These polynomial fits are third order in nature,
and are used to improve the accuracy of the average model during transient periods.
31
In observing the data shown in Figure 3.8(c), it is clear that the variation in ki over
the load range is very small compared to the other parameters. Using this information,
ki is replaced with a constant value.
The polynomial fits used to map the parameters α and kv are given by the equations
α = α3i3dc + α2i
2dc + α1idc + α0 (3.25)
kv = kv,3i3dc + kv,2i
2dc + kv,1idc + kv,0 (3.26)
and the coefficient values are shown in Tables 3.2 and 3.3. A comparison between the
polynomial fit and the original data is displayed in Figures 3.9 and 3.10. The MATLAB
script used to compute the polynomial terms is provided in Appendix B. If accuracy is
the goal, then computational time will increase in proportion to accuracy. By relaxing
the accuracy constraint, the polynomial fits can be reduced.
In order to determine the sensitivity of the hex t/r average model, the parameters α
and kv were also fitted with linear polynomials. The idea is that additional computational
time can be reduced if the parameters are fitted with simple approximations rather than
the more complicated polynomial equations. A comparison between the two average
models is discussed in Section 3.5. The linear approximation and the original data
for α and kv can be compared in Figure 3.11 and 3.12. The terms used in the linear
approximation are shown in Tables 3.4 and 3.5.
Table 3.2 The α polynomial terms
Term Value
α3 -.00000025876793α2 0.00001196916142α1 0.00578832204852α0 0.12933829713806
3.4.4.2 Commutation inductor value estimation
In order to assess the transient response of the average model, the circuit shown in 3.4
was simulated under transient conditions (load-step) for comparison with the switching
32
10 20 30 40 50 60 70 800.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Load Current, idc
(Amps)
α (R
adia
ns)
(a)
10 20 30 40 50 60 70 800.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
0.54
0.55
0.56
Load Current, idc
(Amps)
k v
(b)
10 20 30 40 50 60 70 800.548
0.55
0.552
0.554
0.556
0.558
0.56
Load Current, idc
(Amps)
k i
(c)
Figure 3.8 Average model parameters plotted over the operating range of the hex t/r:(a) α vs. the load current, idc, (b) kv vs. the load current, idc, and (c) ki vs. the load
current, idc
Table 3.3 The kv polynomial terms
Term Value
kv,3 0.00000007802294kv,2 -0.00001258358603kv,1 -0.00055547151970kv,0 0.54735932166799
33
0 10 20 30 40 50 60 70 800.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Load Current, idc
(Amps)
α (r
adia
ns)
Original DataPolynomial Fit
Figure 3.9 α vs. the load current, idc
0 10 20 30 40 50 60 70 800.46
0.48
0.5
0.52
0.54
0.56
Load Current, idc
(Amps)
k v
Original DataPolynomial Fit
Figure 3.10 kv vs. the load current, idc
34
Table 3.4 The α linear terms
Term Value
α1 0.00501938363468α0 0.14266271303755
Table 3.5 The kv linear terms
Term Value
kv,1 -0.00113560905579kv,0 0.55323646705100
10 20 30 40 50 60 70 800.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Load Current, idc
(Amps)
α (R
adia
ns)
Original DataLinear App.
Figure 3.11 α v. the load current, idc
35
10 20 30 40 50 60 70 800.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
0.54
0.55
0.56
Load Current, idc
(Amps)
k v
Original DataLinear App.
Figure 3.12 kv v. the load current, idc
model. This model neglects to account for the commutation inductance, which must be
known so that its effect on the dynamic operation of the system can be determined. The
polynomial fits discussed in Section 3.4.4.1 are used for the parameters kv and α. The
output voltage and output current are shown in Figures 3.13(a) and 3.14(a). It can be
seen that the average model poorly tracks the transient response of the switching model
for both the output voltage and the output current.
In examining the results shown in Figures 3.13(a) and 3.14(a), it is apparent that the
dynamic response of the average model needs to be improved. This can be achieved by
changing the values of the energy-storage elements in the circuit. In comparing the circuit
in Figure 3.4 to the circuit shown in Figure 2.2, it can be seen that the commutation
inductance in the switching model is not represented in the average model. This discovery
justifies increasing the inductance on the dc side of the average model.
Difficulty arises in modeling the commutation inductors because they are connected
in parallel and series in the rectifier bridge as shown in Figure 2.2. At any given time, it
is known that the diodes will have two inductors in series: one inductor in the positive
36
rail and one inductor in the negative rail. Each inductor represents one-third of the
total commutation inductance. By adding these two together, the 23Lc ratio is produced.
These two inductors in series represent two-thirds of the total commutation inductance in
the circuit. This information can be translated to the average model in order to produce
the circuit schematic shown in Figure 3.6. The results from a load-step simulation for the
output voltage and output current, using the circuit in Figure 3.6, are shown in Figures
3.13(b) and 3.14(b). It can be observed that the transient response of the average model
tracks more accurately than the switching model.
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24200
205
210
215
220
225
230
235
240
Time (sec)
Out
put V
olta
ge (
Vol
ts)
vdc
(Switching)v
dc (Average)
(a)
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24200
205
210
215
220
225
230
235
240
Time (sec)
Out
put V
olta
ge (
Vol
ts)
vdc
(Switching)v
dc (Average)
(b)
Figure 3.13 Transient response of output voltage, vdc: (a) without Commutation In-ductance and (b) with Commutation Inductance
3.5 Average model verification
The hex t/r average model circuit in Figure 3.6 is verified by comparing its steady-
state and transient responses with those of the detailed switching model. The comparison
involves simulating the models at different load points and verifying the average value
of the dc output voltage and the dc load current. The circuit parameters that were
used to simulate the average and the switching models are shown in Tables 2.1 and 3.6,
37
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.2410
15
20
25
30
35
40
45
50
55
60
Time (sec)
Out
put C
urre
nt (
Am
ps)
idc
(Switching)idc
(Average)
(a)
0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.2410
15
20
25
30
35
40
45
50
55
60
Time (sec)
Out
put C
urre
nt (
Am
ps)
idc
(Switching)idc
(Average)
(b)
Figure 3.14 Transient Response of Output Current, idc: (a) without CommutationInductance and (b) with Commutation Inductance
respectively. The results of the simulations are shown in Figures 3.15 - 3.19. In each of
the figures, there are three curves plotted. One curve is the response of the switching
model, while the two other curves represent the response of two different average models.
The difference in the two models lies in the method used to compute the parameters kv
and α. One of the average models uses a linear approximation while the other uses a
polynomial fit.
3.5.1 Steady-state results
The results show good agreement between the average model and the switching model
during steady-state and transient conditions. In observing the dc voltages in Figures 3.15
and 3.16, it can be seen that the difference between the three models is less than 1 V. If
greater accuracy is required at steady state, the mathematical expressions can be replaced
with the actual value of the parameter. The output current under steady-state conditions
is shown in Figures 3.17 and 3.18. In both figures, it can be seen that the average models
accurately predict the steady-state value of the switching model current. Based on the
results presented, the linear approximation works just as well as the polynomial fit.
38
Table 3.6 Hex t/r average model circuit parameter values
Parameter Value
Vd 440.0 VVq 0 VId 23.90 AIq 10.22 ALc 430.0 µHRw 0.1960 ΩLfilter 1125 µHC 2400 µFR 4.050 Ωα 23.17
kv 0.5018ki 1.807
Depending on the accuracy desired from the model, the more complex polynomial fit can
be replaced with the simpler linear approximation.
3.5.2 Transient results
The switching and average models are simulated under transient conditions at var-
ious load points. A comparison of the results for the dc output voltage, vdc, and the
dc output current, idc, are shown in Figures 3.19 and 3.20. In both plots, the average
model accurately predicts the transient response of the switching model. Transient char-
acteristics such as rise time, overshoot, and settling time are almost identical between
the switching and average models. There is a small error that appears in the voltage
transient waveform in Figure 3.19. If this error is undesirable then more terms can be
added to the polynomial fit to improve the accuracy.
3.6 Average Model Summary
The development of the average model of a hexagon transformer/rectifier has been
presented. The process of revising the model to improve the transient response has
39
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08206
206.5
207
207.5
208
208.5
209
209.5
210
Time(sec)
Out
put V
olta
ge (
Vol
ts)
Linear ApproximationPolynomial FitSwitching Model
Figure 3.15 Hex t/r output voltage, vdc, at 10.6 kVA under steady-state conditions
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08227
227.5
228
228.5
229
229.5
230
Time(sec)
Out
put V
olta
ge (
Vol
ts)
Linear ApproximationPolynomial FitSwitching Model
Figure 3.16 Hex t/r output voltage, vdc, at 5.1 kVA under steady-state conditions
40
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.0848
48.5
49
49.5
50
50.5
51
51.5
52
52.5
53
Time(sec)
Out
put C
urre
nt (
Am
ps)
Linear ApproximationPolynomial FitSwitching Model
Figure 3.17 Hex t/r output current, idc, at 10.6 kVA under steady-state conditions
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.0821
21.5
22
22.5
23
23.5
24
24.5
25
Time(sec)
Out
put C
urre
nt (
Am
ps)
Linear ApproximationPolynomial FitSwitching Model
Figure 3.18 Hex t/r output current, idc, at 5.1 kVA under steady-state conditions
41
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5180
190
200
210
220
230
240
250
Time (sec)
Out
put V
olta
ge (
Vol
ts)
vdc
(Switching)v
dc (Average)
vdc
(Lin. App.)
Figure 3.19 Hex T/R Output Voltage, vdc, under transient conditions
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510
20
30
40
50
60
70
80
Time (sec)
Out
put C
urre
nt (
Am
ps)
idc
(Switching)idc
(Average)idc
(Lin. App.)
Figure 3.20 Hex T/R Output Current, idc, under transient conditions
42
been discussed in detail. A set of continuous-time equations have been provided, relat-
ing directly to the dynamics of the actual system. The use of polynomial fits and the
representation of the commutation inductance have also been presented.
43
This page is left intentionally blank.
44
CHAPTER 4
EXPERIMENTAL VERIFICATION
4.1 Introduction
In order to further verify the concept of the average model, experimental data is col-
lected from an 11 kVA hexagon transformer/rectifier hardware prototype. This chapter
will describe the hardware under test and the procedure used to collect the experimental
data. Results from the experimental prototype are directly compared with simulation
results obtained from the average and switching models.
4.2 Experimental hardware/test setup
4.2.1 Description of hardware
The average and switching models of the hex t/r were further validated by comparing
their results to experimental data collected from the 11 kVA hex t/r hardware setup
shown in Figure 4.1. The hex t/r hardware prototype was designed and assembled by
a project sponsor. The specifications for the 11 kVA hardware prototype are shown in
Table 4.1. The hexagon transformer is located inside the wooden box, and the 12-pulse
diode rectifier is located above it in Figure 4.1. The actual hardware includes RC snubber
circuits that are placed across each diode to limit the dv/dt when the diode turns off.
The 11 kVA hex t/r prototype requires water-cooling for the transformer core and the
45
diode bridges. The transformer was designed in such a way that one of the windings of
the transformer is also used for cooling.
A three-phase switching power supply is used to provide a balanced three-phase input
voltage to the transformer, and a 30 kW air-cooled resistor bank is used as the load. The
load can be configured for various resistor values ranging from 1 Ω to 11 Ω. A 2,400
µF capacitor was connected in parallel with the resistive load. The output capacitor
had a measured equivalent series resistance (esr) value of approximately 50 mΩ. A 3 µF
polypropylene capacitor was used to attenuate any high-frequency noise that might be
present on the dc bus.
Figure 4.1 11 kVA hex t/r hardware prototype
Table 4.1 11 kVA hex t/r hardware prototype specificationsSpecification Value
AC rms Input Voltage 440 V, 3-φDC Output Voltage 217 VDC Output Current 49 APower Rating 10.6 kVA
46
4.2.2 Test setup
The test equipment used to collect experimental data from the hex t/r prototype
included a digital oscilloscope, four digital multimeters, two current probes, two differ-
ential voltage probes, and a three-phase power analyzer. A current shunt was added to
the experimental setup so that the dc current could be accurately measured. Eight- and
12-guage wires were used to make all of the connections for the hex t/r experimental test
setup.
Both time-domain and frequency-domain data were collected from the hex t/r during
the experimental testing. The next section will describe the procedure used for collecting
the different types of data.
4.2.3 Description of measurements
4.2.3.1 Time-domain measurements
In order to make a side-by-side comparison, the same variables that were measured
in simulation are measured experimentally. The quantities that are measured include
both line-to-line and and line-to-neutral input voltages, input current, input current har-
monics and THD, output voltage, output current, voltage drop across the filter inductor,
and input power. The data collected from the hex t/r can be processed and directly
compared to the simulation results of the hex t/r average and switching models. The
results collected from the steady-state measurements are presented in Section 4.3.
Experimental data from the hex t/r were collected at two different load points, 5 kW
and 8 kW. Due to the limitations of the power supply, it was not possible to run the
hex t/r at its full-load operating point of 10.6 kVA. The experimental input and output
waveforms from both the 5 kW and 8 kW tests are shown in Figures 4.2 - 4.5.
In both Figures 4.2 and 4.4, it can be observed that the input current, ia, has a
sinusoidal wave shape with very low THD. The line-to-line input voltage, vab, is shown for
reference. It can be seen that the input voltage is nearly ideal. The 12-pulse characteristic
of the hex t/r can be verified in both Figures 4.3 and 4.5 by counting twelve pulses over
47
one 60 Hz line cycle. There is some low-frequency ripple in the output current. This
can be attributed to a possible imbalance in the transformer windings. It can also be
observed that as the power level of the hex t/r increases, so does the ripple in the output
current. This is an expected characteristic of the topology. The output voltage displayed
in Figures 4.3 and 4.5, has a very small ripple due to the use of the large filter capacitor
connected to the output of the diode rectifier.
Figure 4.2 Hex t/r 5 kW ac experimental waveforms, 200 V/div, 10 A/div
4.2.3.2 Output impedance measurements
In order to verify the small-signal modeling accuracy of the hex t/r average model, the
output impedance of the 11 kVA hardware prototype was experimentally measured using
a concept similar to the one described in other work [27]. In theory, the output impedance
is measured by perturbing the output current and measuring the output voltage of the
system being studied. The general definition of the output impedance is given in (4.1).
Due to the configuration of the network analyzer, a voltage source (instead of a current
source) generates the perturbation. Some modifications to the experiment are required
48
Figure 4.3 Hex t/r 5 kW dc experimental waveforms, 50 V/div, 5 A/div
Figure 4.4 Hex t/r 8 kW ac experimental waveforms, 200 V/div, 10 A/div
49
Figure 4.5 Hex t/r 8 kW dc experimental waveforms, 50 V/div, 10 A/div
due to the high power level of the hex t/r hardware prototype. A block diagram describing
the approach used to measure the output impedance is shown in Figure 4.6.
Zo,gen =voio
(4.1)
AudioAmplifervtest
vref+ -
+
-Cfilter R
Cblock
Hex T/RNetworkAnalyzer
+
-
DUT
Output ImpedanceBoard
Lwire Rwire
1 Ω
Figure 4.6 Output impedance block diagram
The maximum output voltage generated by the network analyzer is 1.25 V. In order
to perturb the dc bus of the hex t/r, a larger perturbation signal is needed. An audio
amplifier is used in this experiment to increase the magnitude of the perturbation signal.
50
A dc blocking capacitor, rated at twice the output voltage is used to prevent the dc
voltage generated by the hex t/r from harming the network analyzer. A low inductive 1
Ω resistor is used to sense the current in the return path. The impedance of the hex t/r
hardware prototype is computed by measuring the output voltage, labeled vtest in Figure
4.6, and dividing it by the voltage measured by the 1 Ω shunt resistor, labeled vref , which
is essentially the current io. The output impedance as measured on the 11 kVA hex t/r
prototype is defined in (4.2).
Zo,meas =vtestvref
(4.2)
A picture of the hardware setup used to measure the output impedance is shown in
Figure 4.7. High-voltage differential probes are used to measure the signals vtest and
vref . The network analyzer performs the calculation given in (4.2) and plots the output
impedance. Due to the limited band range of the audio amplifier, the frequency range of
interest for the output impedance measurements is 10 Hz to 10 kHz. The board that was
added to the hex t/r test setup to measure the output impedance, along with the audio
amplifier, is shown in Figure 4.8. The specifications of the audio amplifier are shown in
Table 4.2. Eight-gauge wire is used to connect the output impedance board to the load
of the hex t/r. This wire has an associated inductance, Lwire, that will greatly affect the
measured results. The inductance of this wire will be discussed in more detail in Section
4.3.2.
Table 4.2 Audio amplifier, Jensen XA2150, specificationsSpecification Value
DC Input Voltage 14.4 VFrequency Response 20 Hz - 20 kHz ± 3 dBRMS Power Rating 200 W Bridged
The network analyzer generated a perturbation signal of 34 mV that was multiplied
by the audio amplifier. The audio amplifier produced an output voltage of 3.4 V that
was used to perturb the dc bus of the hex t./r. The output impedance experiment was
51
Figure 4.7 Output impedance test setup
Figure 4.8 Output impedance measurement board
52
conducted with the hex t/r operating at 5 kW. This corresponds to a resistive load of
10.69 Ω. The measured output impedance plot of the hex t/r prototype is shown in
Figure 4.9.
Figure 4.9 Experimental output impedance at 5 kW
4.3 Experimental results
This section will compare the experimental data presented in the previous sections
with simulation data from the average and switching models.
4.3.1 Time-domain results
The switching and average models of the hex t/r were simulated at 5 kW (R =
5.92 Ω) and 8 kW(R = 10.69 Ω), respectively, to compare their data with those of the
experimental hex t/r. The output voltage, vdc, and output current, idc, are plotted for
both cases in Figures 4.10 - 4.13. For both operating points, there is good agreement
between the average model, switching model and experimental data. In Figures 4.10 and
53
4.12, the average model accurately predicts the average value of the output current, idc.
In comparing the experimental output current to the switching model results, it can be
seen that the switching model does not capture the imbalance present in the experimental
output current. This is due to the fact that the switching model does not account for
imbalance in the transformer windings. If it was desired to include this imbalance in the
model, a less ideal transformer model could be developed for the switching model. The
output voltages in Figures 4.11 and 4.13 show that there is good correlation between the
average model, switching model and experimental data. As mentioned in Chapter 3, the
accuracy of the average model can be improved by adding more terms to the polynomial
fits. The rms and average dc values of the results presented in Figures 4.10 - 4.13 are
shown in Tables 4.3 and 4.4. It can be observed that the average model has an error of
less than 1%. The THD levels of the line current at 5 kW and 8 kW were 6.170% and
4.043%, respectively.
−0.02 −0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002 015
20
25
Am
plitu
de (
Am
ps)
Time (sec)
idcexp
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.0815
20
25
Am
plitu
de (
Am
ps)
Time (sec)
idcswitchidcavg
Figure 4.10 Comparison of hex t/r output current, idc, at 5 kW
54
−0.02 −0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002 0220
225
230
235
240
Am
plitu
de (
Vol
ts)
vdcexp
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08220
225
230
235
240
Am
plitu
de (
Vol
ts)
Time (sec)
vdcswitch
vdcavg
Figure 4.11 Comparison of hex t/r output voltage, vdc, at 5 kW
−0.02 −0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002 020
25
30
35
40
45
50
Am
plitu
de (
Am
ps)
idcexp
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.0820
25
30
35
40
45
50
Am
plitu
de (
Am
ps)
Time (sec)
idcswitchidcavg
Figure 4.12 Comparison of hex t/r output current, idc, at 8 kW
Table 4.3 Comparison of hex t/r results at 5 kWExperimental Switching Average
DC Voltage (V) 230.0 228.7 229.0DC Current (A) 21.50 21.39 21.42
AC rms Current (A) 6.815 6.788 6.865
55
−0.02 −0.018 −0.016 −0.014 −0.012 −0.01 −0.008 −0.006 −0.004 −0.002 0210
215
220
225
Am
plitu
de (
Vol
ts)
vdcexp
0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08210
215
220
225
Am
plitu
de (
Vol
ts)
Time (sec)
vdcswitch
vdcavg
Figure 4.13 Comparison of hex t/r output voltage, vdc, at 8 kW
Table 4.4 Comparison of hex t/r results at 8 kWExperimental Switching Average
DC Voltage (V) 219.2 218.8 218.6DC Current (A) 37.00 36.95 36.92
AC rms Current (A) 11.99 11.95 11.83
56
4.3.2 Output impedance results
The output impedance of the average model of the hex t/r was simulated in SABER
for comparison with the experimental measurements presented in the previous section.
A few comments need to be made about the experimental output impedance presented
in Figure 4.9. The plot shown in Figure 4.9 clearly exhibits behavior indicative of an
inductor at high frequencies. Upon careful examination of the test setup, it was deter-
mined that the inductance present in the experimental output impedance existed due
to the wire used to connect the output impedance measurement board to the hex t/r
hardware prototype. An impedance analyzer then measured the loop inductance of the
wire so that this information could be added to the simulation model. The impedance of
the wire was found to have a resistance of 18 mΩ and 2.7 µH. To improve the accuracy
of the average model, the impedance of each of the components in the test setup was
measured so that the information could be added to the average model simulation. The
components that were characterized and their impedance values are shown in Table 4.5.
After characterizing the test setup, the results in Figure 4.14 were obtained. There is a
good match between the magnitude and the phase at all frequencies. There is a slight
difference in the first resonant frequency. The dc gains and slopes of both the average
model and the experimental output impedance are close to one another.
Table 4.5 Output impedance measurement system characterizationComponent Impedance
Cblock 2400 µFRblock,esr 50 mΩCfilter 2400 µFRfilter,esr 50 mΩRload 10.69 ΩLload,ind 95.4 µHLwire 2.70 µHRwire 18.00 mΩ
Now that it has been shown that the average model can also be used for small-
signal analysis, some parametric studies can be performed to improve the accuracy of
the model. In Chapter 3, one of the issues that was discussed was the representation
57
101
102
103
104
105
−30
−20
−10
0
10
Mag
nitu
de (
dBΩ
)
ExperimentalSimulation
101
102
103
104
105
−100
−50
0
50
100
Freq (Hz)
Pha
se (
deg)
ExperimentalSimulation
Figure 4.14 Comparison of hex t/r output impedance at 5 kW
of the commutation inductance in the average model. Now that it has been shown that
the experimental output impedance is a good approximation of the simulation model,
the commutation inductance in the average model can be adjusted until the resonant
frequencies of the experimental output impedance and simulated output impedance are
nearly identical.
This validation of the output impedance permits the hex t/r average model to be used
to study stability. In terms of power distribution systems, several of these average models
can be lumped together to simulate a large power network. The proposed average model
of the hex t/r would allow for the study of both time-domain and frequency-domain
characteristics.
58
CHAPTER 5
MODELING OF AN 18-PULSE
AUTOTRANSFORMER
AND RECTIFIER
5.1 Introduction
The ATRU is one of many subsystems in an aircraft power system. It is desired to
study issues such as transient response, system interactions, and stability for the entire
aircraft power system. In order to perform these various analyses, small-signal models
of the individual subsystems are needed. This is one of the main driving factors for
developing an average model of the 18-pulse ATRU.
In this chapter, the average model concept presented in Chapter 3 is extended to the
more complex 18-pulse ATRU. A review of the 18-pulse ATRU topology is included, as
is a discussion of the issues encountered during the development of the switching and
average models.
5.2 Operation of autotransformer
The 18-pulse ATRU topology is shown in Figure 5.1. The ATRU topology is composed
of three 6-pulse diode bridges, and uses phase-shifting of the secondary voltages in the
autotransformer to effectively attenuate harmonics below the 17th. Ideally, the diode
bridges should equally share the power handled by the system. Due to the parallel
59
connection of its diode bridges, this topology requires the use of interphase transformers.
The interphase transformers absorb instantaneous voltage differences between the diode
rectifiers, and ensure that the conduction angle of the diodes remains at 120 [6]. The
load of the ATRU is resistive. The system is rated at 100 kVA. A three-phase 400 Hz
voltage source is used to supply power to the ATRU. The input and output specifications
of the ATRU are listed in Table 5.1. This unit is designed to provide a dc output voltage
of ±270 V.
va’ va’’
va
vb’
vb’’vbvc
vc’’
vc’
Bridge 1
Bridge 2
Bridge 3
R vdc
+
-
+
-vdc,Br1
idc
idc,Br1
idc,Br2
idc,Br3
+
-vdc,Br2
+
-vdc,Br3
InterphaseTransformer
vb vcva
Figure 5.1 18-pulse ATRU topology
Table 5.1 Input/output specifications for 100 kVA 18-pulse ATRUSpecification Value
Input Voltage (RMS) 3-φ 231.0 VInput Current (RMS) 167.5 AOutput Voltage ±270.0 VOutput Current 214.6 AR 2.5 Ωfline 400 Hz
5.2.1 Transformer configuration
The autotransformer uses a phase-shifting method to reduce the harmonics in the
input line current. The autotransformer has three primary windings and three secondary
windings. The secondary windings are tapped in such a way to produce six phase-shifted
60
voltages. These six voltage vectors are connected to two 6-pulse diode bridge rectifiers. In
order to make the system 18-pulse, another 6-pulse converter is required. This six-pulse
converter, labeled Bridge 1 in Figure 5.1, is connected directly to the ac mains.
The autotransformer produces two secondary voltages per input line-to-neutral volt-
age. The two secondary voltages are phase-shifted 40 with respect to the primary voltage
vector. A vector diagram depicting the phase shift is shown in Figure 5.2. The windings
on the autotransformer are tapped in such a way that the secondary voltage is phase-
shifted with respect to the primary voltage vector. Based on the literature, the minimum
phase shift required for an 18-pulse converter is 20 [6]. The vectors k1 and k2, shown in
orange in Figure 5.2, represent the secondary tap windings of the autotransformer.
The reduction in size of an autotransformer, as compared to other topologies can
be attributed to its unique winding structure. The autotransformer design of the 18-
pulse ATRU allows for reduced kVA sizing as compared to an equivalent multi-pulse
transformer topology that employs galvanic isolation [6]. The secondary voltages are
produced by continuing to wind the primary windings on the same core and and tapping
the windings according to the values of k1 and k2. By using the same winding, the kVA
rating of the entire autotransformer system can be reduced [4] .
5.3 Switching model
The switching model of the 18-pulse ATRU was developed using the SABER simula-
tion program. A schematic of the topology is shown in Figure 5.3. The voltage sources
used to provide power to the ATRU are ideal and the autotransformer is constructed
using ideal transformer models. The diode models are piecewise linear functions whose
on and off conductances, as well as on voltage, can be specified. In order to accurately
simulate the interphase transformers, mutual coupling is used in the model. Each in-
ductor has a series resistance of 1 mΩ. Other than the interphase transformers used to
ensure equal current sharing, this topology does not include a filter at the output.
61
va
va’
va’’vbvb’
vb’’
vc
vc’
vc’’
k1
k1
k1
k1
k1
k1
k2
k2
k2
k2
k2
k2
40 deg.
Figure 5.2 18-pulse autotransformer vector diagram
The simulation issues listed in Chapter 2 for the 12-pulse hex t/r are do not exist
for this topology. The maximum time step that is used to simulate the system ranges
from 10 to 20 microseconds. Soft starts are not required to help with convergence, and
numerical instability has not been observed in any of the waveforms. It should be noted
that this topology is very sensitive to asymmetries, which generate imbalances. These
asymmetries were not taken into account in the model.
InterphaseTransformers
18-PulseAutotransformer
Voltage Sources
va vb vc
R
Bridge 1
Bridge 2
Bridge 3
idc
idc,Br1
idc,Br2
idc,Br3
+
-
vdc,Br1
+
-
vdc,Br2
+
-
vdc,Br3
+
-
vdc
ia,Br1
ib,Br1
ic,Br1
ia,Br2
ic,Br2
ia,Br3
ib,Br3
ib,Br2
ic,Br3
Figure 5.3 Switching model schematic of ATRU
62
5.3.1 Switching model results
The switching model was simulated at full load (100 kW, R = 2.5 Ω) to demonstrate
the 18-pulse characteristics of the topology. The circuit parameters are listed in Table
5.2. The steady-state results of the input and output are shown in Figures 5.4 - 5.8.
The input current and input voltage of the ATRU are shown in Figure 5.4. The
input voltage is nearly ideal, and the input current has a sinusoidal shape. The harmonic
spectrum of the line current, ia, is shown in Figure 5.5. The harmonics are plotted as
a percentage of the fundamental. It can be observed that all harmonics below the 17th
have been effectively attenuated. The output voltage and output current, vdc and idc, are
presented in Figure 5.6. In both waveforms, the ripple voltage and ripple current are less
than 1V and 1A, respectively. The 18-pulse characteristic can be verified in Figure 5.6
by counting 18 pulses in idc over one 400 Hz line cycle. The output voltage rails, vdc,plus
and vdc,minus, are shown in Figure 5.7. It can be observed in Figure 5.8 that nearly equal
current sharing exists among the three rectifier bridges. This slight imbalance can be
attributed to the transformer topology chosen for the ATRU.
Table 5.2 100 kVA 18-pulse ATRU switching model parameter valuesParameter Value
Va,rms 231.0 VLinterphase 1.500 mHCoupling of Linterphase 0.8500R 2.500 Ωfline 400.0 HzVon 0.700 VRon 1 mΩRoff 1 MΩ
5.3.2 Switching model analysis
Prior to developing an average model of the 18-pulse ATRU, some operating point
data must be extracted from the switching model. So far, steady-state ac and dc wave-
forms from the ATRU have been presented. As in Chapter 3, the average model will
63
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02
−300
−200
−100
0
100
200
300
Inpu
t Vol
tage
(V
olts
) va
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02
−200
−100
0
100
200
Time (sec)
Line
Cur
rent
s (A
mps
) ia
Figure 5.4 ATRU input current, ia, and input voltage, va, at 100 kVA this is a test tomake this really long i hope
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
0
20
40
60
80
100
Frequency (Hz)
% M
agni
tude
ia
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
0
2
4
6
8
10
Frequency (Hz)
% M
agni
tude 17th
19th
35th37th
ia
Figure 5.5 ATRU input line current, ia, harmonic spectrum at 100 kVA
64
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02534
535
536
537
538
Out
put V
olta
ge (
Vol
ts)
vdc
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02214
214.5
215
215.5
216
Time (sec)
Out
put C
urre
nt (
Am
ps) i
dc
Figure 5.6 ATRU output voltage, vdc, and output current, idc, at 100 kVA
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02255
260
265
270
275
280
285
290
Pos
itive
DC
Rai
l (V
olts
) vdc,plus
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02−290
−285
−280
−275
−270
−265
−260
−255
Time (sec)
Neg
ativ
e D
C R
ail (
Vol
ts)
vdc,minus
Figure 5.7 ATRU output voltage rails with respect to the input voltage neutral, vdc,plusand vdc,minus, at 100 kVA
65
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.0260
62
64
66
68
70
72
74
76
78
80
Time (sec)
Rec
tifie
r B
ridge
Cur
rent
(A
mps
)
idc,Br1idc,Br2idc,Br3
Figure 5.8 ATRU bridge rectifier dc currents at 100 kVA
be developed in the dq0 rotating reference frame. Unlike the case of the hex t/r, each
6-pulse rectifier bridge will be modeled independently. This approach is taken so that
the current-sharing principle can be validated. Three different Park’s transformations
are required to transform the abc input of the bridge rectifiers to the dq rotating refer-
ence frame. The Park’s transformations are phase shifted by 40, corresponding to the
primary and secondary voltages of the autotransformer. The alignment of the Park’s
transformation is chosen so that the d-channel is aligned with the line-to-neutral input
voltage vector. The result is a vq vector with a magnitude of zero. The three Parks
transformation matrices, along with their inverses, are:
Tabc/dq0,Br1 =
√2
3
sin(θ) sin(θ − 2π3
) sin(θ + 2π3
)
− cos(θ) − cos(θ − 2π3
) − cos(θ + 2π3
)
1√2
1√2
1√2
(5.1)
Tabc/dq0,Br2 =
√2
3
sin(θ + 2π9
) sin(θ − 2π3
+ 2π9
) sin(θ + 2π3
+ 2π9
)
− cos(θ + 2π9
) − cos(θ − 2π3
+ 2π9
) − cos(θ + 2π3
+ 2π9
)
1√2
1√2
1√2
(5.2)
66
Tabc/dq0,Br3 =
√2
3
sin(θ − 2π9
) sin(θ − 2π3− 2π
9) sin(θ + 2π
3− 2π
9)
− cos(θ − 2π9
) − cos(θ − 2π3− 2π
9) − cos(θ + 2π
3− 2π
9)
1√2
1√2
1√2
(5.3)
T−1abc/dq0,Br1 =
√2
3
sin(θ) − cos(θ) 1√2
sin(θ − 2π3
) − cos(θ − 2π3
) 1√2
sin(θ + 2π3
) − cos(θ + 2π3
) 1√2
(5.4)
T−1abc/dq0,Br2 =
√2
3
sin(θ + 2π9
) − cos(θ + 2π9
) 1√2
sin(θ − 2π3
+ 2π9
) − cos(θ − 2π3
+ 2π9
) 1√2
sin(θ + 2π3
+ 2π9
) − cos(θ + 2π3
+ 2π9
) 1√2
(5.5)
T−1abc/dq0,Br3 =
√2
3
sin(θ − 2π9
) − cos(θ − 2π9
) 1√2
sin(θ − 2π3− 2π
9) − cos(θ − 2π
3− 2π
9) 1√
2
sin(θ + 2π3− 2π
9) − cos(θ + 2π
3− 2π
9) 1√
2
(5.6)
As an example, the ac currents and voltages in Bridge 2 of Figure 5.3 are transformed
to the dq rotating reference frame and are plotted in Figures 5.9. The dq currents of
Bridge 2 have a ripple due to the shape of the input current shown in 5.9(a).
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02−80
−60
−40
−20
0
20
40
60
80
Time (sec)
Rec
tifie
r B
ridge
2 C
urre
nt (
Am
ps)
ia,Br2ib,Br2ic,Br2
(a)
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02
−100
−80
−60
−40
−20
0
20
40
60
80
100
Time (sec)
Rec
tifie
r B
ridge
2 d
q C
urre
nt (
Am
ps)
id,Br2iq,Br2
(b)
Figure 5.9 abc to dq transformation of ATRU rectifier bridge to currents: (a) abccurrents and (b) dq currents
67
5.4 Average model
5.4.1 Approach
As mentioned in Section 5.1, an average model of the ATRU is needed so that several
of the units can be simulated together as part of a large system model of an aircraft
power system. The theory used to develop the average model of the hex t/r is extended
to the 18-pulse ATRU topology. There are some fundamental differences in the two
topologies that must be addressed prior to presenting the average model of the ATRU.
In the case of the hex t/r, it was decided to couple together the 12-pulse diode bridge and
the hexagon transformer into one average model. The hex t/r average model effectively
accounts for the transformer and the 12-pulse diode bridge rectifier through the constants
ki and kv. For the 18-pulse ATRU, the transformer core is not included in the average
model. Instead of lumping the entire circuit into one big block, each 6-pulse diode bridge
rectifier is represented by a set of average model equations. This is done for two reasons.
First, one of the main characteristics of the this 18-pulse topology is equal current sharing
between the diode bridges. In order to verify that the system is operating correctly, it is
necessary to observe the output current in all three rectifier bridges. The second reason
for choosing this structure for the average model is that parasitics affect the operation
of the ATRU. In order to observe the effect of the parasitics, the transformer core is not
lumped into the average model.
5.4.2 Equation formulation
The first step in developing the average model of the ATRU involves creating a space
vector diagram. Since we are generating one average model for each bridge rectifier, there
will be three space vector diagrams created. The difference between the space vector
diagrams is the phase shift added to the Park’s transformation to account for the phase-
shifted secondary voltages produced by the autotransformer. The space vector diagram
for Bridge 1 is shown in Figure 5.10. As was discussed in Chapter 3, the d-channel of the
68
Park’s transformation is aligned with the line-to-neutral voltage vector. This alignment
produces a vq with a magnitude of zero. The angle α is again used to represent the phase
difference between the input voltage and the input current. The output voltage and
output current of the rectifier are represented by vdc and idc, respectively. The vectors
id, iq and vd represent the dq equivalents of the abc voltages and currents.
q
dα
vd=vin-abc=vdc,Br/kv
iin-abc=idc,Br/ki
id
iq
Figure 5.10 ATRU space vector diagram
The next step in developing the average model involves writing equations that define
the geometry of the space vector diagram. These equations are similar to the ones
presented in Chapter 3. The equations listed in (5.7) - (5.11) describe the operation of
one diode rectifier. These same equations can be used for all three diode rectifiers. The
only difference that may be encountered is that the value of parameters kv, ki and α may
vary slightly between the three rectifier bridges.
69
|vdc,Br| = kv
√v2d + v2
q (5.7)
vq = 0 (5.8)
id =idc,Brki
cos(α) (5.9)
iq =idc,Brki
sin(α) (5.10)
α = tan−1
(iqid
)(5.11)
For the 18-pulse ATRU average model, constant values are used for the parameters
α, kv and ki. At the time of development, only an average model that operated at full
load was needed. As presented in Chapter 3, the switching model of the ATRU can be
simulated at several operating points. If there is a large variation in the parameters, then
polynomial fits can be developed, similar to those described in Section 3.4.4. The next
step in the average model process involves discussing the actual circuit model.
5.4.3 Model description
The average model of the 18-pulse ATRU has a hierarchal structure. A general block
diagram of the 18-pulse ATRU average model is shown in Figure 5.11. As mentioned in
Section 5.4.1, the same transformer model used in the switching model is included in the
average model. Each 6-pulse bridge rectifier is represented by an average model. The
average model block is hierarchal in nature and is composed of several subsystems. This
structure of the average model is necessary due to the complex topology of the ATRU.
The subsystems in each average model block perform different functions, such as Park’s
transformations, evaluating average model equations, and converting phase currents to
line currents, as shown in Figure 5.12.
In order to enable the mathematical (equations) model of the 6-pulse bridge rectifier in
dq coordinates to be connected to the circuit model of the rest of the system in stationary
coordinates, the line currents, iab, ibc, and ica are calculated and and fed back into the
70
InterphaseTransformers
18-PulseAutotransformer
Voltage Sources
va vb vc
R
AverageModel
Bridge 1
Bridge 2
Bridge 3
AverageModel
AverageModel
idc
idc,Br1
idc,Br2
idc,Br3
+
-
vdc,Br1
+
-
vdc,Br2
+
-
vdc,Br3
+
-
vdc
ia,Br1
ib,Br1
ic,Br1
ia,Br2
ic,Br2
ia,Br3
ib,Br3
ib,Br2
ic,Br3
Figure 5.11 18-Pulse ATRU average model block diagram
transformer model. Figure 5.12 shows the block diagram of this concept. Each bridge
rectifier average model uses a modeling concept similar to the one presented in other
work [28].
Average ModelEquations
abcto
dq0Trans.
abcto
dq0
Trans.
Lineto
PhaseTrans.
va vb vc
iab
ica
ibc
ia
ib
ic
+-
+-
vd
vq
idc,Br
+vdc,Br-
id
iq
MATHEMATICAL MODEL
Figure 5.12 Average model breakdown
The average model block inputs voltages va, vb and vc. A Park’s transformation is
used to compute the value of vd and vq. These voltages are then used as the input to
the average model circuit shown in Figure 5.13. The average model circuit computes the
values of vdc,Br, id and iq using the equations presented in (5.7) - (5.11). The output,
vdc,Br, is connected to the interphase transformers. An inverse Park’s transformation is
applied to the currents id and iq to compute their abc phase current equivalents. The
71
phase currents are then transformed to line currents using the equations shown in (5.12)
- (5.14). This calculation of the line currents uses the assumption given in (5.15).
-+
+
-
vq
id
iq
vdc,Br
idc,Br
vdc,Br
+
+
-
-
vd
Figure 5.13 Average model circuit
iab =1
3(ia − ib) (5.12)
ibc =1
3(ib − ic) (5.13)
ica =1
3(ic − ia) (5.14)
iab + ibc + ica = 0 (5.15)
After the line currents have been calculated, they are connected across the phase
terminals as depicted in Figure 5.12. This assures correct loading of the autotransformer.
The process described above is identical in all of the average models. The only
difference between the three bridge rectifier models is the phase shift of the voltages and
currents. Now that the average model has been described in detail, the average model
results can be presented.
5.4.4 Average model verification
The average model of the 18-pulse ATRU shown in Figure 5.11 is verified by comparing
its response to that of the switching model under steady-state conditions. The parameters
72
shown in Tables 5.2 and 5.3 are used in the ATRU switching model and average model
simulations.
Table 5.3 ATRU average model circuit parameter valuesParameter Value
Vd 398.7 VDCVq 0.000 VDCId 97.44 AIq 0.8820 ALinterphase 1.500 HR 2.500 Ωα −0.3030
kv 1.347ki 1.390
The average model results are shown in Figures 5.14 - 5.16. The results of the steady-
state simulations show good correlation between the average model and switching model
results. It can be observed in Figure 5.14 that the average model accurately predicts the
value of the output voltage, vdc, and output current, idc. The output voltage rails, vdc,plus
and vdc,minus, are correctly predicted by the ATRU average model as shown in Figure
5.15. The dc currents in the three rectifier bridges are plotted in Figure 5.16. It can be
observed that the average model accurately predicts the value of the current for all three
bridges. The accuracy of the average model can be improved by recalculating α, kv, and
ki for each bridge instead of using the same value for all three models. The switching
and average waveforms of the output voltage rails, vdcr,plus and vdcr,minus, of Bridge 1 are
plotted in Figure 5.17. It can be seen that the average model correctly predicts the value
of the voltages.
5.5 Summary
The modeling of an 18-pulse ATRU has been presented. Simulation results have been
provided to validate the operation of the proposed average model. The issues encountered
73
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02535
536
537
538
539
540
Out
put V
olta
ge (
Vol
ts) v
dc,switchv
dc,avg
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02214
214.5
215
215.5
216
Out
put C
urre
nt (
Am
ps)
Time (sec)
idc,switchidc,avg
Figure 5.14 ATRU output voltage, vdc, and output current, idc, at 100 kVA
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02255
260
265
270
275
280
285
290
Pos
itive
DC
Rai
l (V
olts
) vdcplus,switch
vdcplus,avg
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02−290
−285
−280
−275
−270
−265
−260
−255
Neg
ativ
e D
C R
ail (
Vol
ts)
Time (sec)
vdcminus,switch
vdcminus,avg
Figure 5.15 ATRU ±270 V output voltage rails, vdc,minus and vdc,plus, at 100 kVA
74
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02
70
80
90
Cur
rent
(A
mps
) idc,Br1,switchidc,Br1,avg
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02
70
80
90
Cur
rent
(A
mps
) idc,Br2,switchidc,Br2,avg
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02
70
80
90
Cur
rent
(A
mps
)
Time (sec)
idc,Br3,switchidc,Br3,avg
Figure 5.16 ATRU bridge current, idc,Br, at 100 kVA
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02100
200
300
400
500
Brid
ge 1
Pos
. DC
Rai
l (V
olts
)
vdcplus,Br1,switch
vdcplus,Br1,avg
0.015 0.0155 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02−500
−400
−300
−200
−100
Time (sec)
Brid
ge 1
Neg
. DC
Rai
l (V
olts
)
vdcminus,Br1,switch
vdcminus,Br1,avg
Figure 5.17 ATRU Bridge 1 output voltage rails, vdcplus,Br1 and vdcminus,Br1, at 100 kVA
75
during model development have been discussed. This average model is now ready to be
linearized so that is can be used to study small-signal stability in aircraft power systems.
76
CHAPTER 6
CONCLUSIONS
6.1 Conclusions
The research presented in this thesis has focused on the modeling of multi-pulse
transformer/rectifier units in power distribution systems. Both detailed switching models
and reduced-order average models have been analyzed and validated with experimental
data. The issues that occur in simulation due to the complex topologies have been
addressed, and solutions have been presented.
As the role of multi-pulse transformer/rectifier units increases in power distribution
systems, more attention is directed toward the approach used to model the topologies
in large scale systems. The need for particular models is a direct result of the types
of analysis that must be performed. Due to the need to study stability, there is a
great benefit in developing an average model that accurately models the time-domain
and frequency-domain characteristics of the actual system. The average models of such
systems are of great interest, with the main driving factors being a reduction in simulation
time, transient analysis, parametric studies and stability analysis.
A general procedure for developing the average model of two multi-pulse trans-
former/rectifier topologies has been presented. The average model develops a relationship
between the system’s 1st harmonic ac variables and average dc variables. This relation-
ship is made possible through the use of scaling constants, namely α, ki and kv. The
average models are derived in the dq0 rotating reference frame. A set of continuous
77
equations can be written that describe the operation of a multi-pulse converter from
an input/output perspective. These continuous equations can be used to describe the
operation of an n-pulse diode rectifier.
The proposed average model of the 12-pulse hexagon transformer/rectifier was veri-
fied with detailed switching model data and experimental data. The time-domain and
frequency-domain characteristics of the average model were validated with experimental
data from an 11 kVA hardware prototype. The time-domain measurements were col-
lected under steady-state conditions. The small-signal properties of the hex t/r average
model were verified experimentally by measuring the output impedance and comparing
it with the simulation results. Good correlation was shown between the average model
and the experimental data for all test cases.
The average model concept was extended to the more complex 18-pulse ATRU. For
this particular topology, each 6-pulse bridge rectifier is represented by an average model.
The results are verified against an ATRU switching model under steady-state conditions.
Average models can be developed that accurately predict the steady-state and tran-
sient responses of actual systems. For the average models presented in this thesis, the
error between the average model and the switching model and/or experimental results
was less than 1%. The accuracy of the average model is dependent on the constants α, ki
and kv. The validity of the average model over the entire load range can be achieved by
developing polynomial fits that map the variation of the parameters as the load changes.
The detail to which the polynomial fits are developed will greatly affect simulation time
and accuracy. This research provides the groundwork for developing average models of
complex multi-pulse transformer/rectifier topologies. The validity of the average model
has been verified, and can now be used as a subsystem in the analysis of large-scale power
distribution systems.
78
REFERENCES
[1] S. Mollov, A. Forsyth, and M. Bailey, “System modeling of advanced electric powerdistribution architectures for large aircraft,” in Proceedings of the SAE Power Sys-tems Conference, no. P-359, 2000.
[2] A. Emadi and M. Ehsani, “Aircraft power systems: Technology, state of the art,and future trends,” IEEE AES Systems Magazine, pp. 28–32, Jan. 2000.
[3] J. Richard E. Quigley, “More electric aircraft,” in Applied Power Electronics Con-ference and Exposition, 1993, pp. 906–911.
[4] S. Choi, P. N. Enjeti, and I. J. Pitel, “Polyphase transformer arrangements withreduced kVA capacities for harmonic current reduction in rectifier-type utility inter-faces,” IEEE Trans. on Power Electronics, vol. 11, no. 5, Sept. 1996.
[5] C. Tinsley, C. Papenfuss, R. Gannett, E. Hertz, D. Cochrane, D. Chen, andD. Boroyevich, “Modeling and control of PEBB-based aircraft electrical service sta-tion: Final report,” Center for Power Electronics, Tech. Rep., May 2002, preparedfor the Office of Naval Research.
[6] D. A. Paice, Power Electronic Converter Harmonics: Multipulse Methods. IEEEPress, 1995.
[7] I. Jadric, D. Borojevic, and M. Jadric, “Modeling and control of a synchronousgenerator with an active dc load,” IEEE Trans. Power Electronics, vol. 15, no. 2,pp. 303–11, March 2000.
[8] S. Sudhoff, K. Corzine, H. Hegner, and D. Delisle, “Transient and dynamic average-value modeling of synchronous machine fed load-commutated converters,” IEEETrans. Energy Conversion, vol. 11, no. 3, pp. 508–514, Sept. 1996.
[9] J. Schaefer, Rectifier Circuits: Theory and Design. John Wiley & Sons, 1965.
[10] D. Rendusara, A. V. Jouanne, P. Enjeti, and D. Paice, “Design considerations for12-pulse diode rectifier systems operating under voltage unbalance and pre-existingvoltage distortion with some corrective measures,” IEEE Trans. on Industry Appli-cations, vol. 32, no. 6, pp. 1293–1303, Nov. - Dec. 1996.
[11] Y. Nishida and M. Nakaoka, “A new harmonic reducing three-phase diode rectifierfor high voltage and high power applications,” in Industry Applications Conference,vol. 2. Industry Applications Society, 1997, pp. 1624–1632.
79
[12] S. Choi, P. Enjeti, H. Lee, and I. Pitel, “A new active interphase reactor for 12-pulserectifiers provides clean power utility interface,” in Industry Applications Conference,vol. 3. Industry Applications Society, 1995, pp. 2468–2474.
[13] G. R. Kamath, D. Benson, and R. Wood, “A novel autotransformer based 18-pulserectifier circuit,” in Applied Power Electonics Conference and Exposition, 2002, pp.795–801.
[14] S. Choi, B. S. Lee, and P. N. Enjeti, “New 24-pulse diode rectifier systems for utilityinterface of high-power ac motor drives,” IEEE Trans. on Industry Applications,vol. 33, no. 2, pp. 531–541, April/May 1997.
[15] S. Chwirka, “Using the powerful SABER simulator for simulation, modeling, andanalysis of power systems, circuits, and devices,” in 7th Workshop on Computers inPower Electronics. COMPEL, July 2000, pp. 172–176.
[16] O. Ustun, M. Yilmaz, and R. Tuncay, “Simulation of power electronic circuits usingvissim software: A study on toolbox development,” in 7th Workshop on Computersin Power Electronics. COMPEL, July 2000, pp. 183–187.
[17] “Saberbook version 2.8,” Avant! Corporation, 2001, Electronic Help Files.
[18] D. Hanselman and B. Littlefield, The Student Edition of MATLAB: Version 5 User’sGuide. Prentice Hall, 1997, the MathWorks Inc.
[19] A. B. Yildiz, B. Cakir, E. Ozdemir, and N. Abut, “An analysis method for the sim-ulation of switched-mode converters,” in 9th Mederterranean Electrotechnical Con-ference, vol. 1. MELECON, 1998, pp. 570–574.
[20] B. R. Needham, P. H. Eckerlin, and K. Siri, “Simulation of large distributed dc powersystems using averaged modeling techniques and the saber simulator,” in AppliedPower Electronics Conference and Exposition, vol. 2, Feb 1994, pp. 801–807.
[21] J. Rosa, “U.S. patent no. 4,255,784,” March 1981.
[22] ——, “U.S. patent no. 4,683,527,” July 1987.
[23] R. W. Erickson, Fundamentals of Power Electronics. Kluwer Academic Publishers,1997.
[24] J. Alt and S. Sudhoff, “Average value modeling of finite inertia power systems withharmonic distortion,” in Proceedings of SAE Power Systems Conference 2000, no.P-359, 2000, pp. 1–15.
[25] S. Sudhoff and O. Wasynczuk, “Analysis and average value modeling of line-commuted converter-synchronous machine system,” IEEE Trans. Energy Conver-sion, vol. 8, no. 1, pp. 92–99, March 1993.
[26] I. Jadric, “Modeling anc control of a synchronous generator with electronic load,”Master’s Thesis, Virginia Tech, Jan. 1998.
80
[27] D. Boroyevich, Modeling and Control of DC/DC Converters Short Course Lab Man-ual, Center for Power Electronics Systems, June 2003.
[28] K. Louganski, “Modeling and analysis of a dc power distribution system in 21st
century airlifiters,” Master’s thesis, Virginia Tech, Sept. 1999.
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82
APPENDIX A
11 kVA HEX T/R SWITCHING MODEL
OPERATING POINT DATA
This appendix provides a table of steady-state data collected from the 11 kVA hex
t/r SABER simulation model.
83
Table
A.1
Hex
t/rsw
itchin
gm
od
elop
erating
poin
td
ataR
(Ω)
Load
(W)
Ia ,rm
s(A
)V′dc
(V)
Vdc
(V)
Idc
(A)
Id
(A)
Iq
(A)
α(rad
ians)
1ki
kv
2.50014510
24.11205.2
190.476.20
36.2120.98
0.52510.5491
0.46673.500
1180018.57
214.2203.1
58.0728.78
14.100.4556
0.55190.4871
4.5009926
15.10220.6
211.246.99
23.9010.22
0.40410.5532
0.50185.500
856412.73
224.7217.0
39.4620.46
7.6200.3566
0.55320.5110
6.5007504
11.01227.6
220.833.98
17.836.030
0.32640.5539
0.51777.500
66749.690
229.7223.7
29.8415.78
4.9700.3048
0.55450.5223
8.5005993
8.660230.9
225.626.57
14.134.220
0.29040.5550
0.52519.500
54397.820
232.1227.4
23.9212.81
3.4900.2661
0.55510.5279
10.504988
7.150232.8
228.521.83
11.713.160
0.26310.5556
0.529411.50
46336.520
233.5229.6
20.1810.76
2.7800.2532
0.55060.5311
12.504254
6.070234.2
230.618.45
9.9502.520
0.24800.5564
0.532813.50
39705.650
234.8231.5
17.159.300
2.2500.2378
0.55810.5341
14.503726
5.450253.8
232.316.04
8.7101.900
0.21500.5560
0.535715.50
34975.090
235.8232.9
15.018.170
1.7400.2096
0.55650.5364
16.503306
4.810236.1
233.414.16
7.7201.600
0.20420.5565
0.537017.50
31194.44
236.1233.6
13.367.290
1.4900.2019
0.55750.5370
18.502978
4.24236.9
234.412.71
6.9301.480
0.21010.5574
0.538919.50
28314.03
237.2234.8
12.066.580
1.3700.2051
0.55750.5395
20.502701
3.86237.5
235.211.49
6.2801.270
0.20030.5576
0.5402
84
APPENDIX B
STATISTICAL ANALYSIS
The MATLAB m-files used to compute the polynomial fits of the parameters α, kv
and ki for the hex t/r are provided in this section. The data that were used to develop
the polynomial fits is listed in Table A.1.
B.1 MATLAB files
B.1.1 The α polynomial fit m-file
%This m-file applies a curve fit to the datapoints listed for alpha and
%list the polynomials of the function.
clear all;
close all;
%this section of the script reads the alpha.ascii
%file and places the data in arrays
load alpha.asc;
%alpha=alpha1;
x=alpha(:,1);
y=alpha(:,2);
%curve-fitting
%the n is the order of the polynomial
n=3;
p_alpha=polyfit(x,y,n)
xi=linspace(0,80,10);
z=polyval(p_alpha,xi);
%plot the original data and calculated polynomial
85
plot(x,y,’-o’,xi,z,’r--’)
grid
xlabel(’Load Current, i_dc (Amps)’)
ylabel(’\alpha’)
%title(’Polynomial Fit of \alpha with 2 degrees of freedom’)
legend(’original data’, ’polynomial fit’);
print -depsc2 polyalpha.eps
B.1.2 The kv polynomial fit m-file
%This m-file applies a curve fit to the datapoints listed for alpha and
%list the polynomials of the function.
clear all;
close all;
%this section of the script reads the kv.ascii
%file and places the data in arrays
load kv.asc;
%alpha=alpha2;
x=kv(:,1);
y=kv(:,2);
%curve-fitting
%the n is the order of the polynomial
n=3;
p_kv=polyfit(x,y,n)
xi=linspace(0,80,10);
z=polyval(p_kv,xi);
%plot the original data and calculated polynomial
plot(x,y,’-o’,xi,z,’r--’)
grid
xlabel(’Load Current, i_dc (Amps)’)
ylabel(’k_v’)
%title(’Polynomial Fit of k_v with 3 degrees of freedom’)
legend(’original data’, ’polynomial fit’);
print -depsc2 polykv.eps
B.1.3 The ki polynomial fit m-file
%This m-file applies a curve fit to the datapoints listed for alpha and
%list the polynomials of the function.
86
clear all;
close all;
%this section of the script reads the ki.ascii file
%and places the data in arrays
load ki.asc;
%alpha=alpha2;
x=ki(:,1);
y=ki(:,2);
%curve-fitting
%the n is the order of the polynomial
n=3;
p_ki=polyfit(x,y,n)
xi=linspace(0,80,10);
z=polyval(p_ki,xi);
%plot the original data and calculated polynomial
plot(x,y,’-o’,xi,z,’r--’)
grid
xlabel(’Load Current, i_dc (Amps)’)
ylabel(’k_i’)
%title(’Polynomial Fit of k_i with 3 degrees of freedom’)
legend(’original data’, ’polynomial fit’);
print -depsc2 polyki.eps
B.1.4 Linear approximation of the variables α, kv, and ki m-file
%Linear Approximation 11kw
%This m-file calculates the linear approximations,
%ax + b, of the variables alpha, kv, and ki.
clear all;
close all;
load alpha.asc;
load kv.asc;
load ki.asc;
%linear approximation for alpha
slope_inta=alpha(1,:)-alpha(19,:);
slope_alpha=slope_inta(1,2)/slope_inta(1,1);
87
y_intera=-1*slope_alpha*alpha(1,1) + alpha(1,2);
x=0:10:80;
y_alpha=slope_alpha*x+y_intera;
figure(1);clf;
plot(alpha(:,1),alpha(:,2),’b-o’);
hold on;
plot(x,y_alpha,’r--’);
grid;
axis ([10 80 0.15 0.55]);
xlabel(’Load Current, i_dc (Amps)’);
ylabel(’\alpha (Radians)’);
%title(’\alpha vs. I_dc (11 kVA)’);
legend(’original data’, ’linear app.’,2);
print -depsc2 alphalin.eps
%linear approximation for kv
slope_intkv=kv(1,:)-kv(19,:);
slope_kv=slope_intkv(1,2)/slope_intkv(1,1);
y_interkv=-1*slope_kv*kv(1,1) + kv(1,2);
x=0:10:80;
y_kv=slope_kv*x+y_interkv;
figure(2);clf;
plot(kv(:,1),kv(:,2),’b-o’);
hold on;
plot(x,y_kv,’r-.’);
grid;
axis ([10 80 0.46 0.56]);
xlabel(’Load Current, i_dc (Amps)’);
ylabel(’k_v’);
%title(’k_v vs. I_dc (11 kVA)’);
legend(’original data’, ’linear app.’);
print -depsc2 kvlin.eps
%linear approximation for ki
slope_intki=ki(1,:)-ki(19,:);
slope_ki=slope_intki(1,2)/slope_intki(1,1);
y_interki=-1*slope_ki*ki(1,1) + ki(1,2);
x=0:10:80;
y_ki=slope_ki*x+y_interki;
figure(3);clf;
plot(ki(:,1),ki(:,2),’b-o’);
hold on;
plot(x,y_ki,’r-.’);
88
grid;
axis ([10 80 0.548 0.56]);
xlabel(’Load Current, i_dc (Amps)’);
%xlabel(’Load Current, i_dc (Amps)’,’FontAngle’,’italic’);
ylabel(’k_i’);
%title(’k_i vs. I_dc (11 kVA)’);
legend(’actual Data’, ’linear App.’);
print -depsc2 kilin.eps
89
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90
APPENDIX C
SABER SCHEMATIC MODELS
This appendix provides a list of all the SABER schematics used for the switching
model and average model simulations. The switching model and average model SABER
schematics of the hex t/r and ATRu are presented in this section. The MAST Files that
were used in the SABER simulations has also been included in this appendix.
C.1 SABER schematics
The SABER schematics used to simulate the hex t/r and ATRU are presented in this
section.
C.1.1 Hex t/r SABER schematics
The hex T/R SABER schematics are shown in Figures C.1 - C.2.
C.1.2 ATRU SABER schematics
The ATRU SABER schematics are shown in Figures C.3 - C.8.
C.2 SABER MAST code
The SABER MAST code used in the hex t/r average model SABER schematics are
presented in this section. A brief description is provided with the code, as is information
stating with which schematic model the file is associated.
91
Cur
rent
to
Con
trol
Inte
rfac
e
i2va
r
[0,0
,25m
,1,5
00m
,1]
sym
1
0
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
V
vmul
t
[0,0
,25m
,1,5
00m
,1]
sym
3
359
sym
4
V
vmul
t
[0,0
,25m
,1,5
00m
,1]
sym
5
3−ph
ase
Sou
rce
and
abc/
dqo
Coo
rdin
ate
Tra
nsfo
rmat
ion
0
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
359
sym
7
V
vmul
t
0
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Cur
rent
to
Con
trol
Inte
rfac
e
i2va
r
359
sym
9
Cur
rent
to
Con
trol
Inte
rfac
e
i2va
r
0
1meg
1meg
DC
/DC
n1:1
30
n2:8
n3:1
4
n4:8
p1 m1
p2 m2 p3 m3 p4 m4
1meg
DC
/DC
n1:1
30
n2:8
n3:1
4
n4:8
p1 m1
p2 m2 p3 m3 p4 m4
1meg
1meg
1meg
1meg
1.1m
eg
Hex
Tra
nsf
orm
er
1meg
1meg
DC
/DC
n1:1
30
n2:8
n3:1
4
n4:8
p1 m1
p2 m2 p3 m3 p4 m4
1meg
1meg
1meg
1meg
DC
/DC
n1:1
30
n2:8
n3:1
4
n4:8
p1 m1
p2 m2 p3 m3 p4 m4
1meg
DC
/DC
n1:1
30
n2:8
n3:1
4
n4:8
p1 m1
p2 m2 p3 m3 p4 m4
1meg
1meg
DC
/DC
n1:1
30
n2:8
n3:1
4
n4:8
p1 m1
p2 m2 p3 m3 p4 m4
1meg
1meg
675u
675u
675u
675u
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Cur
rent
to
Con
trol
Inte
rfac
e
i2va
r
562u
675u
675u
562u
2400
u
Rec
tifi
er a
nd
DC
Lo
ad
pwld
dqo
abc
Con
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2
ref:c
onta
bc2d
qo1
freq
:60
oqda b c
dqo
abc
Con
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2 ref:s
ym17
freq
:60
oqda b c
pwld
pwld
pwld
pwld
pwld
pwld
pwld
pwld
pwld
pwld
pwld
0.1
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
0.1
10.6
9
1meg
Figure C.1 Hex t/r switching model SABER schematic
92
439.
68V
d
0V
q
0 0
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Co
ntr
ol
to
Vo
ltag
e
+ −
var2
v
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
.196
562u
lfilte
r1
Current
to
Control
Interface
i2var
Co
ntr
ol
to
Cu
rren
t
var2
i
Co
ntr
ol
to
Cu
rren
t
var2
i
2400
u
cos
k2=
k1=
Cos
ine
Mul
tiplie
r
mco
s
.555 1.0
sin
k2=
k1=
Sin
e M
ultip
lier
msi
n
.5551.0
kv map
prim
itive
:kvm
ap
vin
vout
idc
abc
dqo
Con
trol
Mod
el
prim
itive
:con
tdqo
2abc
freq
:60
ocba
qd
abc
dqo
Con
trol
Mod
el
prim
itive
:con
tdqo
2abc
freq
:60
ocba
qd
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
rVol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
alp
ha
map
prim
itive
:alp
ham
ap
angl
eid
c
562u
lfilte
r2
225u
lcom
m2
225u
lcom
m1
4.5
i0
Figure C.2 Hex t/r average model SABER schematic
93
Vdc
min
us
Vdc
plus
ampl
itude
:231
*1.4
1
freq
uenc
y:40
0
Vb
phas
e:−
120
ampl
itude
:231
*1.4
1
freq
uenc
y:40
0
Vc
phas
e:12
0
ampl
itude
:231
*1.4
1
freq
uenc
y:40
0
Va ph
ase:
*opt
*
pwldpwld
pwld
pwld
pwld
pwld
1m
1mpwld
pwld
pwldpwld
pwld pwld
pwld
pwld
pwldpwld
pwld pwld
ml
1st_
indu
ctor
_to_
coup
le:l.
l2
2nd_
indu
ctor
_to_
coup
le:l.
l4
m:1
.5m
*0.8
5
ml
1st_
indu
ctor
_to_
coup
le:l.
l5
2nd_
indu
ctor
_to_
coup
le:l.
l11
m:1
.5m
*0.8
5
1.5m
1.5m
1.5m
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l8
2nd_
indu
ctor
_to_
coup
le:l.
l9
m:1
.5m
*0.8
5
1.5m
1.5m
1.5m
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l7
2nd_
indu
ctor
_to_
coup
le:l.
l10
m:1
.5m
*0.8
5
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l1
2nd_
indu
ctor
_to_
coup
le:l.
l12
m:1
.5m
*0.8
5
1.5m
1.5m
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l3
2nd_
indu
ctor
_to_
coup
le:l.
l6
m:1
.5m
*0.8
5
1m
2.5
AT
RU
( 2
31 V
AC
/ 54
0VD
C )
AT
RU
Win
ding
s
Rec
tifie
rs
Inte
rpha
se s
elfs
Load
Alim
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e+−
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2
ref:c
onta
bc2d
qoLt
oL2_
2
freq
:400
oqda b c
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2
ref:c
onta
bc2d
qoLt
oL2_
3
freq
:400
oqda b c
Current
to
Control
Interface
i2var
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2
ref:c
onta
bc2d
qoLt
oL2_
5
freq
:400
oqda b c
dqo
abcC
ontr
ol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
280
ref:c
onta
bc2d
qoLt
oL2_
9
freq
:400
oqda b c
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2
ref:c
onta
bc2d
qoLt
oL2_
4
freq
:400
oqda b c
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
280
ref:c
onta
bc2d
qoLt
oL2_
8
freq
:400
oqda b c
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
240
ref:c
onta
bc2d
qoLt
oL2_
7
freq
:400
oqda b c
dqo
abcCon
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
240
ref:c
onta
bc2d
qoLt
oL2_
6
freq
:400
oqda b c
DC
/DC
n1:N
p
n2:N
k2
n3:N
k2
p1 m1
p2 m2 p3 m3
DC
/DC
n1:N
p
n2:N
k1
n3:N
k1
p1 m1
p2 m2 p3 m3
DC
/DC
n1:N
p
n2:N
k2
n3:N
k2
p1 m1
p2 m2 p3 m3
DC
/DC
n1:N
p
n2:N
k1
n3:N
k1
p1 m1
p2 m2 p3 m3
DC
/DC
n1:N
p
n2:N
k2
n3:N
k2
p1 m1
p2 m2 p3 m3
DC
/DC
n1:N
p
n2:N
k1
n3:N
k1
p1 m1
p2 m2 p3 m3
SA
BE
R
1meg
1meg
Figure C.3 ATRU switching model SABER schematic
94
_n21
DC
/DC
n1:1
n2:1
/3.4
137
n3:1
/3.4
137
p1 m1
p2 m2 p3 m3
DC
/DC
n1:1
n2:1
/3.4
137
n3:1
/3.4
137
p1 m1
p2 m2 p3 m3
DC
/DC
n1:1
n2:1
/3.4
137
n3:1
/3.4
137
p1 m1
p2 m2 p3 m3
DC
/DC
n1:1
n2:1
/6.3
87
n3:1
/6.3
87
p1 m1
p2 m2 p3 m3
DC
/DC
n1:1
n2:1
/6.3
87
n3:1
/6.3
87
p1 m1
p2 m2 p3 m3
DC
/DC
n1:1
n2:1
/6.3
87
n3:1
/6.3
87
p1 m1
p2 m2 p3 m3
ampl
itude
:231
*1.4
1
freq
uenc
y:40
0
Vb
phas
e:−
120
ampl
itude
:231
*1.4
1
freq
uenc
y:40
0
Vc
phas
e:12
0
ampl
itude
:231
*1.4
1
freq
uenc
y:40
0
Va ph
ase:
*opt
*
AT
RU
( 2
31 V
AC
/ 54
0VD
C )
AT
RU
Win
ding
sA
lim
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
Current
to
Control
Interface
i2var
2.5
ml
1st_
indu
ctor
_to_
coup
le:l.
l21
2nd_
indu
ctor
_to_
coup
le:l.
l18
m:1
.5m
*0.8
5
1.5m
1.5m
1.5m
1.5m
Inte
rpha
se s
elfs
ml
1st_
indu
ctor
_to_
coup
le:l.
l19
2nd_
indu
ctor
_to_
coup
le:l.
l15
m:1
.5m
*0.8
5
1.5m m
l
1st_
indu
ctor
_to_
coup
le:l.
l16
2nd_
indu
ctor
_to_
coup
le:l.
l14
m:1
.5m
*0.8
5
Load
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l20
2nd_
indu
ctor
_to_
coup
le:l.
l24
m:1
.5m
*0.8
5
1.5m
1.5m
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l13
2nd_
indu
ctor
_to_
coup
le:l.
l22
m:1
.5m
*0.8
5
1.5m
1.5m
1.5m
ml
1st_
indu
ctor
_to_
coup
le:l.
l23
2nd_
indu
ctor
_to_
coup
le:l.
l17
m:1
.5m
*0.8
5
1meg1meg
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e+−
v2va
rVol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
atru
avg
.
mo
del
va vb vc vap
vbp
vcp va
pp
vbpp
vcpp
vdcp
lus1
vdcm
inus
1
vdcp
lus2
vdcm
inus
2
vdcp
lus3
vdcm
inus
3
Current
to
Control
Interface
i2var
0
0 0
Figure C.4 ATRU average model SABER schematic
95
atru
avg.
model
va
vb
vc
vap
vbp
vcp
vapp
vbpp
vcpp
vdcplus1
vdcminus1
vdcplus2
vdcminus2
vdcplus3
vdcminus3
0
0
0
Figure C.5 ATRU average model block SABER schematic
avg. rect1va
vb
vc
vdcplus1
vdcminus1
avg. rect2vap
vbp
vcp
vdcplus2
vdcminus2
avg. rect3vapp
vbpp
vcpp
vdcplus3
vdcminus3
va
vb
vc
vap
vbp
vcp
vapp
vbpp
vcpp
vdcplus1
vdcminus1
vdcplus2
vdcminus2
vdcplus3
vdcminus3
Figure C.6 ATRU bridge rectifier average model SABER schematic
96
rect
ifie
r av
g.
mo
del
vdpl
us
vdm
inus
vqpl
us
vqm
inus
vdcp
lus
vdcm
inus
id iqC
on
tro
l
to
Vo
ltag
e
+ −
var2
v
Co
ntr
ol
to
Vo
ltag
e
+ −
var2
v
dqo
abc
Con
trol
Mod
el
prim
itive
:con
tabc
2dqo
LtoL
2re
f:con
tabc
2dqo
LtoL
2_1
freq
:400
oqda b c
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
sum
sum
Vol
tage
to
Con
trol
Inte
rfac
e
+ −
v2va
r
sum
k:1/
3
k:1/
3
abc
dqo
Con
trol
Mod
el
prim
itive
:con
tdqo
2abc
freq
:400
ocba
qdk:
1/3
va vb vc
vdcp
lus1
vdcm
inus
1
Co
ntr
ol
to
Cu
rren
t
var2
i
Co
ntr
ol
to
Cu
rren
t
var2
i
Co
ntr
ol
to
Cu
rren
t
var2
i
Figure C.7 ATRU Bridge 1 average model SABER schematic
97
0 0
Vol
tage
toC
ontr
olIn
terf
ace
+ −
v2va
r
Vol
tage
toC
ontr
olIn
terf
ace
+ −
v2va
r
mul
tsu
m
in
out
Squ
are
Roo
t
sqrt
mul
t
Co
ntr
ol
toV
olt
age
+ −
var2
v
vcvs
k:kv
vmvp
0
Currentto
ControlInterface
i2var
sin
k2=
k1=
Sin
e M
ultip
lier
msi
n
ki1.0
Co
ntr
ol
toC
urr
ent
var2
i
Co
ntr
ol
toC
urr
ent
var2
i
cos
k2=
k1=
Cos
ine
Mul
tiplie
r
mco
s
ki 1.0
cons
tant
alp
1meg
Figure C.8 Average model circuit SABER schematic model
98
C.2.1 The α polynomial saber mast file
# Polynomial fit for the alpha variable
element template alphamap idc angle
input nu idc
output nu angle
var nu alpha
val nu cons3, cons2, cons1, cons0
values
cons3 = -0.00000025876793
cons2 = 0.00001196916142
cons1 = 0.00578832204852
cons0 = 0.12933829713806
alpha = cons3*idc*idc*idc + cons2*idc*idc + cons1*idc + cons0
angle = alpha
C.2.2 The kv polynomial saber mast file
# Polynomial fit for the kv variable
element template kvmap vin idc vout
input nu vin,idc
output nu vout
var nu kv
val nu cons3, cons2, cons1, cons0
values
cons3 = 0.00000007802294
cons2 = -0.00001258358603
cons1 = -0.00055547151970
cons0 = 0.54735932166799
99
kv = cons3*idc*idc*idc + cons2*idc*idc + cons1*idc + cons0
vout = kv*vin
C.2.3 The α linear saber mast file
# Polynomial fit for the alpha variable
element template alphamaplin idc angle
input nu idc
output nu angle
var nu alpha
val nu cons1, cons0
values
cons1 = 0.00501938363468
cons0 = 0.14266271303755
alpha = cons1*idc + cons0
angle = alpha
C.2.4 The kv linear saber mast file
# Polynomial fit for the kv variable
element template kvmaplin vin idc vout
input nu vin,idc
output nu vout
var nu kv
val nu cons1, cons0
100
values
cons1 = -0.00113560905579
cons0 = 0.55323646705100
kv = cons1*idc + cons0
vout = kv*vin
101
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102
VITA
Carl Terrie Tinsley, III was born in Camp Lejeune, NC on March 7, 1978. He received
his bachelor of science degree from Virginia Tech in May 2001. In 1998 and 1999, he
worked as an engineering co-op student with Duke Power Company in Charlotte, North
Carolina. In August 2001, he began working as a graduate student at the Center for Power
Electronics Systems (CPES) at Virginia Tech. Upon completion of his M.S. degree, the
author will begin full-time employment with Lockheed-Martin Corporation in Manassas,
VA.
He is a member of Eta Kappa Nu Honor Society. His research interests include three-
phase inverters, control of power electronics, and modeling of multi-pulse transformer
rectifier systems.
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