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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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

This page is left intentionally blank.

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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