Modeling of parasitic elements in high voltage multiplier modules

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Modeling of parasitic elements

in high voltage multiplier

modules

Jianing Wang

王佳宁

Electrical Power Processing (EPP) Group Electrical Sustainable Energy Department Delft University of Technology

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Modeling of parasitic elements

in high voltage multiplier

modules

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 30 juni 2014 om 15.00 uur

door

Jianing Wang

Master of Engineering, Power Electronics and Renewable Energy Center

Xi’an Jiaotong University, China

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Dit proefschrift is goedgekeurd door de promotor:

Prof. Dr. J.A. Ferreira

Copromotor Ir. S.W.H. de Haan

Copromotor Dr. Ir. M.D. Verweij

Samenstelling promotiecommissie:

Rector Magnificus

voorzitter, Technische Universiteit Delft

Prof. Dr. J.A. Ferreira

Technische Universiteit Delft, promotor

Ir. S.W.H. de Haan

Technische Universiteit Delft, copromotor

Dr. Ir. M.D. Verweij

Technische Universiteit Delft, copromotor

Prof. Dr. J.A. La Poutre

Technische Universiteit Delft

Prof. Dr. Ir. F.B.J. Leferink

Universiteit Twente

Prof. Dr. J.J. Smit

Technische Universiteit Delft

Prof. Dr. A. Yaravoy

Technische Universiteit Delft

Prof. Ir. L. Van der Sluis

Technische Universiteit Delft,

reservelid

Copyright © 2014 by Jianing Wang

All rights reserved. No part of the material protected by this copyright notice may be

reproduced or utilized in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval system,

without the prior permission of the author.

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Acknowledgement

Life is short.

Life is to experience the necessary experiences.

I sincerely appreciate anyone who makes a mark in the travel of my life.

The research presented in this thesis was carried out at Delft University of Technology in the Netherlands, in the research group of Electrical Power Processing (EPP), cooperated with Philips Solid State Lighting. Here, I would like to express my deep appreciation for the people who directly contributes to the thesis.

First of all, I would like to thank you, my promoter Professor Jan A. Ferreira. You always have positive attitude to my research and give me confidence when I am dispirited. Thank you very much.

I would like to thank you, my daily supervisors, Associated Professor Sjoerd W.H. de Haan and Martin D. Verweij. Prof. de Haan, you taught me the importance of the writing and presentation of the research work, which I ignored before. Martin, you helped me greatly with your abundant knowledge on the EM fields and your friendship on my personal problems. Thank you both very much.

I would like to thank you also, Dr. Peter Luerkens from Philips. You always gave me detailed and valuable comments on my research all the way along, without which I can hardly finish my work on time. Thank you very much.

Furthermore, I would like to thank you, Frans Pansier from NXP. Your door is always open to me and your rich knowledge is always helpful to solve my strange questions. Thank you very much.

I would like to thank you, Aniel Shri for the dutch translation of the summary and propositions of this thesis.

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In the end, I would like to thank you, my colleagues, Rob Schoevaars, Kasper Zwetsloot, Harrie Olsthoorn and Bart Roodenburg for the helps on the experiments and computers. I would like to thank you, Marcelo, Ilija, Yeh and Prasanth for the frequent discussions, which push my work forward step-by-step. Thank you very much.

Last but not least, maybe more importantly, I would like to express all my appreciation to all the colleagues, secretaries, friends, families and people I met. You touched my heart, let me fell what life is and gave me joy and courage to go further and further. Thank you all very much.

Especially, I would like to thank you, Prof. Xu Yang, my supervisor in master study, who took me into the world of power electronics and introduced EPP group. Without you, I can not join EPP group for the Ph.D study. Thank you very much.

I would like to express my acknowledgement to the European Commission and the Dutch Ministry of Economic Affairs for supporting this project under envelope of the ENIAC project SmartPM with the Grant Agreement no. 120008

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Abbreviations and Symbols

3D Three dimensional

A, B ··· Node number A, B ···

AC Alternating current

AC+ Upper AC side of multiplier

AC- Lower AC side of multiplier

a, b, m, p Parameters

aCstru Ratio of the new added structural capacitance Cdt to the original structural

capacitance Cdg

C Capacitance

CT Computed tomography

C-V Capacitance voltage curve

C.W. Cockcroft Walton

C0 Per-unit-length capacitance of transmission line

C1, C2 ··· Capacitor number 1, number 2 ···

CDcht Total equivalent parasitic capacitance of diode chains

CTS Stray capacitance of transformer windings

Cdd Structural capacitance between diodes in multiplier module

Cdg Structural capacitance between diode and ground in multiplier module

Cdpp Structural capacitance between diode and push-pull capacitors in

multiplier module

Ceac Equivalent capacitance of the rectifier in LCC by fundamental frequency

analysis

Cem Equivalent parasitic capacitance of multiplier

Cj Junction capacitance of diode

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Co Output capacitor in rectifier or multiplier

Co1, Co2 ··· Output capacitor number 1, number 2 ···

Cp Parallel resonant capacitance

Cpa Parasitic capacitance, in general

Cpp Push-pull capacitors in multiplier

Cppg Structural capacitance between push-pull capacitor and ground in

multiplier module

Cppgt Total structural capacitance between push-pull capacitor and ground in

multiplier module

Cs Series resonant capacitance

Cstru Structural capacitances between the diode and other components in

multiplier module, including the capacitances Cdg, Cdpp, Cppg, Cppgt

D Diode

DC Direct current

Dch Diode chain

d Distance

D1, D2 ··· Diode number 1, number 2 ···

Dch1, Dch2 ··· Diode chain number 1, number 2 ···

E Vector indicating magnitude and direction of electric field

E Magnitude of electric field

EM Electromagnetic

EMI Electromagnetic interference

E0, E1 ··· Zero-order electric field, first-order electric field ···

Ek kth-order electric field

Ent Electric field on the surface of the diode in the multiplier module without

shielding trace

Et Electric field on the surface of the diode in the multiplier module with

shielding trace

F Function

FE Finite element

f Frequency

F Fourier transform

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H Vector indicating magnitude and direction of magnetic field

H Magnitude of magnetic field

HV High voltage

H0, H1 ··· Zero-order magnetic field, first-order magnetic field ···

Hk kth-order magnetic field

I Time invariant current

IGBT Insulated gate bipolar transistor

i Unit vector

ix, iy, iz Unit vector at x direction, y direction and z direction respectively

i Time-varying current

iCp Current through parallel resonant capacitance

iCo1, iCo2 ··· Current through capacitor Co1, Co2 ···

iD1, iD2 ··· Current through diode D1, D2 ···

iLs Current through series resonant inductance

iTLin Input current of transmission line

io Output current

irect Input current of multiplier

irect(1) Fundamental element of input current of multiplier

irect+ Input current of multiplier at AC+ side

irect- Input current of multiplier at AC- side

is Source current

is 0,1 Source current correct up to and including the first-order quantity

is 0..k Source current correct up to and including the kth-order quantity

j Imaginary unit

K Surface current density

k Number

Kr Surface current density as reference

L Inductor

LCC/SPRC Series parallel resonant converter

LME Lumped multi element

LV Low voltage

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L0 Per-unit-length inductance of transmission line

Lpa Parasitic inductance, in general

Lpae Effective parasitic inductance

Ls Series resonant inductance

MOSFET Metal-oxide-semiconductor field effect transistor

N Turn ratio of transformer

nd Diode number per chain

nst Number of stages in multiplier

PCB Printed circuit board

PRC Parallel resonant converter

PWM Pulse width modulation

Q Charge

RL Load

Rdp Damping resistor

Reac Equivalent resistance of the rectifier in LCC by fundamental frequency

analysis

rCstru Reduction factor of the structural capacitance Cdg by adding the trace

rE Reduction factor of the electric field

rv Ratio of the voltage vdt to vdg

S Spatial surface vector

S Spatial surface scalar

Si Silicon

SiC Silicon carbide

T Thickness

t Time

t1 Time instant 1

tVch1, tVch2 Time when voltages across diode chain are Vch1, Vch2

tpulse Width of a pulse

tr Rise time of a pulse

U0 Initial voltage

V Time invariant voltage

VBR Breakdown voltage of diode

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k

Vo Output voltage

Vpk Peak input voltage of multiplier

v Time varying voltage

vC1, vC2 ··· Voltage across capacitor C1, C2 ···

vCo1 Voltage across Co1

vDch1, vDch2 ··· Voltage across the diode chain Dch1, Dch2 ··· in the multiplier

vTLin Input voltage of transmission line

vac+ Unipolar input voltage of multiplier at AC+ side

vac- Unipolar input voltage of multiplier at AC- side

vcc1+, vcc2+ ··· Voltage on node cc1+, cc2+ ···

vcp Voltage across the parallel resonant capacitor

vdd Voltage across the capacitance Cdd

vdg Voltage across the capacitance Cdg

vdt Voltage across the capacitance Cdt

vin Input voltage

vrect Input voltage of multiplier

vrect(1) Fundamental element of input voltage of multiplier

vs Source voltage

vs 0,1 Source voltage correct up to and including the first-order quantity

vs 0..k Source voltage correct up to and including the kth -order quantity

vtg Voltage across the shielding trace and the ground

∆vco1 Voltage ripple across Co1 in multiplier

δvco1 Voltage drop on Co1 compared in multiplier to the principle voltage in

multiplier

w Width

ZCS Zero current switching

ZVS Zero voltage switching

Zin Input impedance

Zin 1 First-order input impedance

x,y,z Spatial variables

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ε Permittivity εr Relative permittivity ε_o Permittivity in vacuum λ Wavelength µ Permeability µr Relative permeability µ_o Permeability in vacuum φ Magnetic flux

φbi Built-in potential of diode

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Chapter 1 Introduction

In this thesis, the parasitic related issues in the voltage multiplier module of the high voltage (HV) generator in medical X-ray machines are described. The focus is on the modeling of the parasitics at modular level and the subsequent study on improving the effect of the parasitics on the module and the generator.

In this chapter, medical X-ray machines are introduced and this triggers the exploration of the high voltage generator and multiplier. After that, the motivation and objectives of the work are presented, followed by the structure of the thesis.

1.1 Background

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Fig.1–1: Examples of medical X-ray machines, (a) computed tomography (CT) machine; (b) mobile C-arm (Image source: Philips Healthcare)

Medical X-ray machines intended for medical diagnosis are used during the treatment of patients. The X-rays produce medical images. The different modes of making images are referred to as modalities, such as computed tomography (CT), mammography and fluoroscopy

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[Bus02]. Consequently, different machines were invented, such as the CT machine, the mammography machine and the fluoroscopy machine.

Fig.1–1 shows examples of the medical machines. Medical X-ray machines are utilized for the imaging of different body parts, like bone, breast, teeth, as well as different forms of imaging, such as two dimensional (2D), three dimensional (3D) and movement monitoring.

High voltage (HV) power supply (HV generator) for accelerating the electrons Low voltage (LV) power supply for cathode heating

e-X-rays X-ray tube Cathode Anode High voltage cable Power supplies

Fig.1–2: Schematic representation of main power supplies for X-ray tubes in medical X-ray machines

To generate X-rays, two basic power supplies are required, the low voltage (LV) and the high voltage (HV) direct current (DC) power supplies. Fig.1–2 shows how they are connected to the X-ray tube. The LV power generates a current of several amperes in the cathode filament of the X-ray tube, heating it up to emit the electrons, which spread as a cloud around the cathode. The HV power supplies high voltage, ranging from 20 kilovolt (kV) to 160 kV across the anode and the cathode to establish a strong electric field to accelerate the electrons [Sob02]. The electrons gain sufficient energy through the acceleration and collide with the anode material, generating X-rays. The flow of electrons forms the output current of the HV power supply, which is in the range from 10 to 5000 milliampere (mA) [Sob02]. This HV power supply is also named the HV generation in many publications [Tay32][Bel05]. The shorter phrase HV generator is used in this thesis.

To make the machine compact, the HV generator is required to have a small volume. Its circuit topology has been developing for almost a century with the continuous improvement of power switches. They developed from mechanical switches in the early stage of 20 century [Tay32], to low frequency semiconductor power devices in the middle of 20 century [Ras01], such as

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the thyristor, then to high frequency semiconductor power devices since the 1980s , such as IGBTs and MOSFETs. Thanks to the rapid development of the HF HV semiconductor power devices, the state-of-the-art topology can cause the HV generator to operate at a switching frequency of around hundreds of kilohertz (kHz) [Cav03]. The high switching frequency and usage of HV devices significantly reduce the volume of the HV generator compared to that in the early equipment. In the next section, the state-of-the-art circuit configuration of the HV generator is introduced and two methods to reduce volume are described.

1.2 Volume reduction of the generator

In this section, the state-of-the-art circuit configuration of the HV generator is presented, which gives insight into how the switching frequency and power devices influence the volume. The HV multiplier is a key component and consumes much volume. Thus, after the introduction of the HV generator as a system, description focuses on the HV multiplier. The parasitics of the multiplier is the main topic of the research.

Fig.1–3: Typical circuit configuration of today’s HV generator

The HV generator in medical X-ray machines is a device that supplies regulated, HV, DC power to the X-ray tube. Today’s HV generator is typically a high frequency switched mode power supply, which operates in the tens of kHz range. The output voltage of the generator can be as high as 160kV. The main power circuit of the HV generator consists of a DC-AC inverter, a step-up transformer and a rectifier, as shown in Fig.1–3. The DC input is obtained by rectifying the 3 phase AC from the electrical grid. The inverter inverts the DC input to a high frequency AC output at the primary side of the transformer. The transformer steps up the low

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AC voltage to a high value. Then, the AC voltage is rectified to a DC output to power the X-ray tube. The function of the feedback control circuit is to make the output precise, stable and adjustable. The passive components and the rectifier are the main contributors to the volume of the HV generator.

The main passive components are the inductor, capacitor and the transformer. Of these the HV transformer is the largest because of the space that is required for the isolation. By increasing the switching frequency of the converter, the value and size of the passive components can be reduced [Per09]. Most HV generators have a moderate volume because the switching frequency is high. Some researchers have reported the feasibility of the generator with a switching frequency over a hundred kHz, which means that it is possible to reduce the system size further [Lue11] [Cav03].

Fig.1–4: A typical HV tank in the HV generator

Besides the passive components, the rectifier is the other major contributor to the volume of the HV generator. The rectifier is typically a voltage multiplier circuit in the HV generator, such as the Cockcroft Walton circuit [Coc32], multiplying a moderately high AC voltage from the transformer to the final high DC voltage in the range 20kV to 160kV. The turn ratio of the transformer can be reduced by utilizing the multiplier compared to the normal bridge rectifier. The parasitics and cost of the transformer can be reduced in turn. The multiplier circuit comprises capacitors and diodes. It is usually assembled in a tank together with the transformer because of the high voltage they have. The tank is grounded for safety in the X-ray machine and filled with insulation oil for higher electrical breakdown voltage. Fig.1–4 shows an example of the HV tank from Philips. It can be seen that the HV multiplier module fills the biggest part of the tank. .Besides the capacitors, diodes also contribute to the volume because a

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large number of diodes are required to block high voltage. For example, a HV generator with output voltage rating of 150kV can have 960 1kV Si diodes [Lue11]. Sufficient space should be available around the diodes to avoid electrical breakdown of the insulation oil and to dissipate heat. One efficient way to o reduce the volume caused by the diodes, is to use diodes with high breakdown voltage while leading to low individual losses. This can lead to a reduced number of didoes, which decreases the volume of the HV tank.

The silicon carbide (SiC) diode, a wide bandgap device, is well suited to the multiplier. Benefiting from their larger critical electric field, sufficiently high carrier mobility and high thermal conductivity, SiC based devices offer ultra high voltage blocking capability and low conduction and switching losses [Mat05]. Due to these properties, SiC diodes should be a better candidate in HV applications. Using 4.5kV SiC diodes, the number of diodes in the HV multiplier can be reduced by four compared to popular commercial 1kV Si diodes. Correspondingly, the volume of the multiplier module with SiC diodes can be reduced to a quarter of the volume of a Si diode unit.

In addition, the SiC diode has a faster switching transient than a Si diode, which can reduce the switching losses. Consequently, the generator can operate at a higher frequency for given losses. Thus, it is possible to increase the switching frequency of the generator with the application of a SiC diode, which can lead to a further reduction of the volume of the multiplier module.

frequency SiC

Volume

Parasitics

Fig.1–5: The two approaches and the problem of volume reduction in the multiplier module In conclusion, the HV tank in the generator is responsible for the biggest part of the volume. It is feasible to reduce the volume of the tank by increasing the switching frequency and by applying high voltage power devices like SiC diodes. Fig.1–5 shows the relations of the two

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approaches and the volume reduction. However, parasitic related problems arise, which are introduced in the next section.

1.3 Problem definition

When the volume of the HV generator becomes small and the switching frequency becomes high, parasitic elements may become important to the system operation [Dai96]. The reduced volume usually leads to the reduction of the distance between the metals as well as to reducing the loop of the circuit layout. Consequently, the parasitic capacitances become larger, although the inductances may become smaller. The increased time derivative of voltage or current (dv/dt or di/dt) leads to a higher current in parasitic capacitances or a larger voltage across parasitic inductances. This makes the parasitics more significant in circuit design.

In the generator, the parasitics in the primary side of the transformer are still small compared to the circuit components, such as the resonant capacitor or inductor. Thus, they don’t have a significant effect on the circuit operation. The transformer in Fig.1–3 usually has large turn ratio in order to achieve a large voltage gain. The parasitic capacitance in the secondary side windings, which is also named as self-capacitance [Dal07], is comparable to the circuit components when they are referred to the primary side, and heavily influence the circuit operation. However, their significant effects have been extensively noticed, well analyzed and utilized in previous work [Dal07] [Bie08]. In the last bulky part, namely the HV multiplier module, the parasitic capacitances are not as large as those in the transformer mentioned earlier. However, as the volume shrinks and the switching frequency rises, the parasitic capacitances can become larger and play a role in the circuit steady operation. Besides, the parasitic inductance can also influence the circuit transient operation as the tube arcs [Bel05]. The parasitic issues in the multiplier module are not well documented and investigated, leading to the following questions:

How much does the parasitic capacitance influence the circuit steady operation?

Usually, the inverter in the HV generator employs a resonant topology [Mar08]. The resonant components, such as the capacitor and inductor, determine the resonant frequency of the generator, which further determines characteristics of the circuit. If the switching frequency of the generator is increased, the value of resonant components can become low enough to be comparable to parasitic capacitances in the multiplier module. In this case, the parasitic capacitances become relevant to the circuit operation and can influence the circuit

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characteristics. It is crucial to have a model of parasitic capacitances in the multiplier module and to take this into account in circuit design. However, until now, the role or the modeling of the parasitic capacitances in the operation of the generator is not mentioned in literature.

How can the electric field strength be contained?

Another issue is the strong spatial electric fields inside the module. Due to the high voltage, the strength of the electric field can be so high as to cause breakdown of the insulation oil. As the module becomes smaller than before, the field potentially becomes stronger. It is crucial to know the distribution of the strength of the field in the module and to reduce it as much as possible to avoid the breakdown. The spatial electric field can be expressed as a function of parasitic capacitances, thus the distribution and reduction is parasitic related as well.

How can the transient behavior be modeled?

During the operational life of an X-ray tube, the possibility of arcing inside the tube increases due to the deposition of tungsten [Bel05]. The arcing leads to short circuit to the output of the HV multiplier and can induce abrupt current pulses in the multiplier circuit, which induces a fast transient EM field in the module. The fast current pulses may result in voltage overshoot in the multiplier and the resultant breakdown of components. To perform the circuit analysis the values of the parasitics should be determined. On the one hand, because the maximum frequency in the bandwidth of the EM field is above quasi-static range, a high-order circuit model is required for accurate circuit analysis. On the other hand, if the frequency is still lower than full-wave range, a continuous model is not necessary. Most approaches of circuit modeling of the parasitics are either for quasi-static field or full-wave field. Thus, a novel approach for the modeling the parasitics in the intermediate frequency range is needed for development.

1.4 Objectives and approaches

Because of the problems listed above, the researcher had three objectives when doing this research, namely to determine the influences of the parasitic capacitances on the steady-state circuit operations, to contain the electric field strength in the module, and to model the parasitics in the case of a fast transient EM field. The objectives can be divided into two aspects, which are listed as follows:

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• The role of the parasitic capacitances in the circuit operation of the HV generator should be determined.

• The complete model of parasitic capacitances in the multiplier module should be determined.

• A solution to reduce the electric field in the multiplier module to avoid breakdown of the insulation oil should be determined.

In transient operation:

• A simple approach for the circuit modeling of the parasitics in the intermediate frequency range should be determined.

• High-order circuit model of the parasitics in the multiplier module should be developed in case of the fast transient EM field.

The following approaches are adopted in order to achieve the aforementioned objectives:

In steady operation:

• The role of the parasitic capacitances is determined by the steady-state circuit analysis of the HV generator.

• The parasitic capacitances in the multiplier module are obtained through 3D finite element (FE) field simulation.

• The complete model is constructed by circuit reduction of the obtained parasitic capacitances.

• The electric field distribution in the multiplier module is obtained through 3D FE field simulation.

• A field shielding technique is proposed based on the expression of the electric field as a function of parasitic capacitances.

In transient operation:

• A power series approach is employed for the circuit modeling of the parasitics in the intermediate frequency range.

1.5 Thesis layout

• In Chapter 2, the circuit topologies of the HV generators are reviewed. Firstly, the topologies for the overall resonant converter are reviewed. Then, the focus is on the

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rectifier part of the resonant converter, namely the HV multiplier. After the review, a typical and popular topology for today’s HV generator is chosen as the example for the parasitic analysis in this thesis, which is the series parallel resonant converter (LCC) with symmetrical Cockcroft Walton (C.W.) multiplier.

• In Chapters 3 and 4, the parasitic capacitances and electric field distribution in the multiplier module in circuit steady operation are investigated.

• In Chapter 3, the parasitic capacitances in the multiplier are introduced. The role of the parasitic capacitances in steady circuit operation is introduced, followed by a presentation of the complete model of the parasitic capacitances. Next, an analytical analysis to the complete model is introduced, leading to guidelines for optimization of the parasitic capacitances.

• In Chapter 4, the reduction of the electric field in the multiplier is discussed. Firstly, the distribution of the field strength is analyzed. The electric field can be expressed as a function containing parasitic capacitances. A simple shielding technique based on the expression is proposed to reduce the field.

• In Chapter 5, a simple procedure to model the parasitics in the multiplier module in fast transient EM fields is introduced. There are two steps in this chapter. The first step is the theoretical development of a procedure for applying the power series approach in a continuous spectrum system. The second step is to apply the procedure to the multiplier module.

• In Chapter 6, conclusions of the thesis and recommendations for future researches are given.

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Chapter 2 Overview of circuit topologies of the

HV generator

As addressed in Chapter 1, the parasitics in the HV multiplier module can play a significant role in the circuit operation of the current HV generator. Before the investigation of the parasitics starts, the circuit operation of the current HV generator is introduced to clarify the possible influence of the parasitics.

2.1 Introduction

In this chapter, the circuit topologies for the HV generator are reviewed. The state-of-the-art topology with a detailed introduction of its circuit operation is presented. The introduction of the circuit operation is divided into three parts due to the relatively complex operating principle of the generator. Firstly, the operation of the whole circuit, which is a resonant converter with a normal full-bridge rectifier, is introduced. After that, the introduction moves to the operation of the rectified voltage multiplier. In the end, the combination of the converter and the multiplier for steady and transient operation is introduced.

2.2 Evolution of the HV generator

In this part, there is a brief introduction of three kinds of power circuits for the HV generator widely used in the last century.

The HV generator supplies DC voltage pulses to the X-ray tube. The exposure time, which is the time of one pulse at the output, depends on the modalities. X-ray machines should produce high quality images of different parts of the body. The following are required if the HV generator is to supply high quality images [Bus02] [Kre90]:

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2. The output voltage and load should have wide adjustable ranges.

3. The output voltage should have a fast dynamic response with a small overshoot. The tolerance for the exposure time should be as small as possible.

4. The generator should have a small volume and weight with low cost. 5. The generator should have a long lifetime.

Conventional generators

Constant voltage generators High frequency generators

1960s 2010s 1910s High power Low frequency Electro-mechanical switches High power High frequency Power semiconductor switches

Vacuum tube diodes Power semiconductor diodes

Fig.2–1: Evolution of the HV generators driven by development of power switches

In last century, there were mainly three kinds of power circuits for the HV generator, which are the conventional rectified circuit from single-pulse to 12-pulse, the constant voltage generator (also named the direct current generator [Kre90]) and the high frequency generator (frequency much higher than 60Hz). The first two were widely used in the early stage of last century but have seldom been manufactured in the last 30 years. The third type is the contemporary state-of-the-art configuration for the HV generator because of its superior performance. The main driving force of the change to the new type is the development of the fast power semiconductor switches, which emerged since 1960s and quickly developed to high power high frequency field in 1980s [Ras01]. The features of the three kinds of power circuit are briefly introduced in the following sections.

2.2.1 Early Stage

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Auto transformer _Exposure timer HV transformer _Rectifier X-ray tube 220V line voltage

Fig.2–2: Diagram of a single-phase full-wave rectified circuit for the HV generator

Fig.2–2 shows the diagram of the power circuit for a conventional generator in a fairly early stage. It consists of an autotransformer, a switch device, a HV transformer and a bridge rectifier. No output capacitor is added as a filter. The possible reasons are high cost and volume of the capacitor that filters low frequency high voltage waves. Single-phase 220V line voltage from the grid is fed as the input. The adjustable high output voltage is achieved through the autotransformer and the HV transformer. This can be controlled through an autotransformer with motorized adjustment. However, such control mechanisms operate in the 100ms range, which can eliminate the rapid disturbance on the line voltage. The exposure time is controlled by the timer, which generates electronic signals to command the electro-mechanical switches. The response time of the control is in the range of tens of milliseconds, which prevents the generation of fast accurate output pulses [Kre90]. The full-wave rectifier can convert the sinusoidal waveform into a waveform with two pulses in a cycle [Ras01]. The single-phase two-pulse rectified circuit can be easily adjusted to different similar patterns, such as single-phase one-pulse, three-single-phase six-pulse and three-single-phase twelve-pulse [Bus02]. More pulses in a cycle at the output lead to higher delivery power and smaller voltage ripple, at the price of higher cost and larger volume of the circuit. Among the conventional generators, the three-phase twelve-pulse rectified circuit gives the best performance, with maximum achievable power to 150kW and minimum output voltage ripple around 3%.

The conventional generator was invented and widely used before the emergence of power semiconductor devices. At that time, besides the electro-mechanical switches, the vacuum tubes were the popular devices as the switches, such as tetrodes. Vacuum tubes like vacuum tube diodes were also used as rectifiers. Although vacuum tubes have fast switching process, they have several major limitations for high power applications, such as a relatively short life

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time, maximum current in the order of several hundreds of mA, high cost and large volume. The limitations of the tube switches prevent the invention of generators with operating frequencies much higher than 60Hz, which were made possible by the emergence of the technology of power semiconductor switches and pulse width modulation (PWM). Moreover, the low operating frequency leads to a large volume of the passive components and large output voltage ripple. With the development of the power semiconductor switches, the conventional generators have been gradually replaced by new topologies that utilize the new switches. This generator has been almost abandoned since 1980s.

2.2.1.2 Constant voltage generators

Regulator Regulator Uref Three phase power supply HV transformer electron tube electron tube X-ray tube HV control circuit Fig.2–3: Power circuit of a constant voltage generator [Kre90]

A constant voltage generator provides nearly constant voltage over the X-ray tube with negligible ripple, as shown in Fig.2–3. It is an improved configuration for the three-phase conventional generator. The main difference is that the output voltage and exposure time are controlled at the secondary side through HV electron tubes. The autotransformer is neglected. The line voltage is fed as the input, boosted by the three-phase transformer and rectified into six or twelve pulses. Two HV electron tubes, which can be triode, tetrode or pentrode, are connected in series in the output loop. A comparator circuit is added to measure the difference between the output voltage and the set reference voltage and to adjust the grid electrodes (as the gate of the transistor) of the electron tubes. The close loop control ensures fast adjustment of the voltage magnitude and exposure time. This configuration allows extremely constant voltage at the output. The response time for this control method can be as fast as 20µs, which is superior compared to that of the conventional generator [Kre90].

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However, the advantage of this type is at the expense of expensive, bulky devices and operational cost. Besides, the operating frequency of the generator is still low. Currently, this method of control is only used in applications with a stringent requirement of constant voltage over the X-ray tube.

2.2.2 The contemporary high frequency generator

Thanks to the invention of power semiconductor switches, especially the power transistors such as insulated gate bipolar transistors (IGBT) and metal-oxide-semiconductor field effect transistors (MOSFET), the HV generator changed from conventional low frequency circuits to high frequency topologies. The operating frequency of the generator is usually at tens of kHz, and can even rise to hundreds of kHz with state-of-the-art technologies [Cav06].

DC AC Rectifier HV generator AC DC DC AC AC Three/single phase AC from the electrical grid

X-ray tube DC

HV transformer Inverter

Fig.2–4: Power circuit of a high frequency generator (modern type)

Fig.2–4 shows the block diagram of the power circuit of a high frequency generator. As already introduced in Chapter 1, the high frequency generator is a switched mode DC-DC converter. The three or single-phase line voltage is rectified into a DC output, which feeds as the input of the HV generator. The generator consists of a HV transformer and a rectifier and an inverter that chops the DC into AC through fast power transistors. The high frequency of the AC voltage, which is rectified into DC output, leads to much reduced volume of the passive components and minimized voltage ripple at the output. Besides, it can lead to fast response time through control circuit, which is approximately hundreds of microseconds. The fast close loop control, which is not shown in the figure, ensures a stable output voltage that is immune to the disturbance on the line voltage or change of the tube current. Further, the output voltage and load can be regulated over a wide range without much effort.

Compared to the constant voltage generators, the high frequency generators have relatively larger voltage ripple and slower response time. However, the cost, weight and volume of the

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high frequency generators are significantly reduced. With state-of-the-art technologies, the volume is more than 80% less, and the weight is more than 75% less than those of the constant voltage generators [Kre90] [Lue11].

Since the 1980s, resonant topologies have been invented and widely used in switched mode DC-DC converters [Ran82][Ste85]. Compared to hard-switching topologies, resonant converters can realize soft switching, which means almost no losses are generated during the switching on or/and off transitions of the switches. The soft switching is usually in forms of zero voltage switching on (ZVS) or zero current switching off (ZCS). Due to the great reduction of the switching losses, resonant converters allow much higher switching frequency than hard-switching converters. The high switching frequency in turn reduces the volume of passive components in converters, which further leads to the much shrunk volume of converters. In addition, the parasitics often induce unexpected behaviors in hard-switching converters, which can cause problems such as low efficiency, electromagnetic interference (EMI). However, resonant converters can incorporate the parasitics as part of the circuit and utilize them to generate expected waveforms. This feature makes resonant topologies quite suitable in the applications where large parasitic elements appear, such as the HV generator. Due to the superior performance, resonant DC-DC converters have been widely used in the application of HV generators. In the next section, several popular resonant topologies will be reviewed in order to find the most suitable one for the HV generator.

2.3 Series parallel LCC resonant converter with capacitive

output

During the last 30 years, the only topologies for HV generators found in the literature are resonant DC-DC converters. A considerable number of topologies have been proposed, covering circuits from two-element resonance [Joh88] to four-element resonance [Zha00], and from single-level [Mar08] to multi-level converters [Wu99]. Among them, three topologies are widely used in practice and fundamental to all the others, these are series resonant converters (SRC), parallel resonant converters (PRC) and series-parallel LCC resonant converters (SPRC-LCC, abbreviated as LCC). In this section, the pros and cons of these converters are briefly reviewed for the application of the HV generator and the best is selected for the following parasitic research.

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2.3.1 Resonant topologies for the generator

Resonant DC-DC converters have been extensively and thoroughly investigated in the last 30 years. Due to the main advantage, namely the reduced switching losses, there have been attempts to adapt them to any applications of power converting. The resonant converters are also utilized in the application of the HV generator. All the resonant converters contain a resonant tank, which can filter the high harmonics of the squared wave and consequently generate a sinusoidal voltage and current that achieve the soft switching of the transistors. Different tanks lead to different resonant converters. The most fundamental and widely used resonant converters for the HV generator are still SRC, PRC and LCC. These are briefly reviewed next to check the pros and cons for the HV generator application. The three resonant converters reviewed are equipped with a classic full-bridge inverters and rectifiers.

2.3.1.1 SRC

.

+ -Ls Cs

Fig.2–5: Schematic diagram of SRC

The steady state performance of SRC is well analyzed in [Ste88]. The main pros and cons of SRC regarding its use in applications of the HV generator are listed as follows.

Advantages:

• The current in the switches and resonant tank decreases as the load decreases. This feature leads to reduced conduction losses and other losses, which consequently results in good light load efficiency in applications with a wide load range, such as the HV generator.

• A capacitive output filter is employed in the topology rather than an inductive filter. This feature is quite attractive in HV applications, because the output inductor can be very large, heavy and expensive due to high isolation requirements. Although in the circuit with capacitive output filter, large current ripple presents in the capacitor,

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which leads high losses. However, in the HV generator, the output current is quite low, which eliminates the losses.

Disadvantages:

• The maximum voltage gain of SRC can only be 1. In the case of the HV generator, the high output voltage needs a large step-up voltage gain. The limited voltage gain of SRC places the burden of voltage-boosting on the transformer and the rectifier. The large turn-ratio of the HV transformer, which realizes a high voltage gain for boosting, is likely to lead to large parasitic inductance and capacitance as well as large volume and cost.

• SRC has good voltage regulation at heavy load but a poor regulation at light load, particularly at no load. This feature prevents SRC from managing the wide load range of the HV generator.

• SRC can incorporate the parasitic inductance of the transformer into the series resonant inductance Ls. However, the parasitic capacitance of the transformer cannot

be utilized by SRC and can deteriorate the circuit behavior.

2.3.1.2 PRC

.

+ -Ls Cp (a)

.

+ -Ls Cp (b)

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Normally, there are two types of PRC, the circuit with inductive output filters as shown in Fig.2–6 (a), and the circuit with capacitive output filters, as shown in Fig.2–6 (b). The steady state operations of the two types are well analyzed and compared in [Joh88]. With regard to the HV generator, the capacitive output filter is preferred. The reason has been mentioned in subsection 2.3.1.1. The capacitive PRC is capable of almost the same performance as the inductive PRC. The removal of the output inductor results in a reduced value of the resonant components with the same transistor and tank current, which in turn leads to smaller tank volume. Thus, the capacitive PRC is selected for application in the HV generator, rather than the other one. The pros and cons of the capacitive PRC are listed as follows.

Advantages:

• The voltage gain of PRC can be larger than 1, which is quite useful for the HV generator.

• The parasitic inductance and capacitance of the HV transformer can both be incorporated into the resonant tank.

• The output inductor is removed, which saves much volume and cost. .

• PRC has a natural proof for output short circuit. The short circuit current is limited by the impedance of the resonant inductance Ls. This feature adds another advantage of

PRC to the application of the HV generator, of which the output is sometimes shorted because of the arcing in the X-ray tube.

Disadvantages:

• PRC has good voltage regulation at light load but a poor regulation at heavy load. This feature prevents the SRC from managing the wide load range of the HV generator.

• The current in switches and the resonant tank does not decrease with a decreased load. The feature leads to poor light efficiency of the PRC in the generator application with a wide load range.

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2.3.1.3 LCC

.

+ -Ls Cs Cp (a)

.

+ -Ls Cs Cp (b)

Fig.2–7: Schematic diagram of LCC, (a) inductive output filter; (b) capacitive output filter The LCC is intended to combine the advantages of the SRC and the PRC and eliminate their disadvantages [Ste88, Bha91]. By good design, usually done with proper selection of the ratio of the capacitance Cs to Cp, the goal can be met. Similarly, the LCC also has two types of

output filters, the inductive filter and capacitive filter. The capacitive LCC is likewise preferred in the HV generator, the characteristics of which are nearly the same as those of the inductive LCC, although the operating modes are different. The good combination of the best characteristics the SRC and the PRC makes the LCC the most suitable for the application of the HV generator, and indeed the LCC is widely used in practice for this purpose. The pros and cons of the LCC are listed below.

Advantages:

• The voltage gain of the LCC can be larger than 1, which is quite useful for the HV generator.

• The parasitic inductance and capacitance of the HV transformer can both be incorporated into the resonant tank.

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• The current in the switches and resonant tank decreases as the load decreases, which ensures a good light load efficiency.

• The LCC can eliminate the short circuit current easily by increasing the switching frequency above the series resonant frequency, which is 1/(LsCs)0.5. Actually, when

the parallel resonant capacitance Cp is shorted the LCC works as a SRC.

The price paid for the advantages of the LCC is a somewhat wider frequency range of operation.

2.3.2 Comparison of the topologies

The characteristics of the SRC, PRC and LCC for the HV generator application are briefly reviewed. The pros and cons are summarized here for a clear comparison.

Table 2–1: Comparison of the SRC, PRC and LCC for application of the HV generator

SRC Capacitive PRC Capacitive LCC

voltage gain maximum is 1 can be larger than 1 can be larger than 1 voltage

regulation good at heavy load but poor at light load good at light load but poor at heavy load good in a wide load range parasitic

utilization parasitic capacitance of the HV transformer

parasitic capacitance and inductance of the HV transformer

parasitic capacitance and inductance of the HV transformer Light load

efficiency good poor good

2.3.3 Basic circuit operation of the LCC

The circuit diagram of the LCC with capacitive full-bridge rectifier is shown in Fig.2–7 (b). The DC input and the full bridge switches can generate a square wave as the input of the resonant tank. By taking 50% duty cycle for the switches as an example, the circuit can be redrawn as in Fig.2–8.

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0

.

Cp irect iLs Ls Cs vCp A B -+ Co R vrect -+ N D1 D2 D3 D4 Vo -+

Fig.2–8: LCC with square wave as input

The purpose of the resonant tank, which consists of Ls, Cs and Cp, is to suppress the harmonic

elements of the input square wave, leaving the fundamental sinusoidal wave [Kaz95]. Usually, for the LCC, the current iLs in the resonant tank is approximately sinusoidal. The current will

charge the capacitor Cp and the load respectively in a cycle. Usually the output capacitor Co

can be assumed to be large enough, which results in a constant output voltage Vo in steady state.

Cp Ls 0 B A Vo/N

+

-Cs (a) Cp Ls B A Cs (b)

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Cp Ls 0 B A Vo/N

-+

Cs (c)

Fig.2–9: Equivalent circuits of the LCC in different intervals, (a) when D1 and D4 conduct; (b)

all diodes in the rectifier are blocked; (c) when D2 and D3 conduct

Fig.2–9 shows the equivalent circuits of the LCC in different intervals in one cycle. They are the simplification of the rectifier, and depend on whether the diodes conduct or not. When the current iLs flows into the rectifier and charges the load, the capacitor Cp is clamped to the

voltage Vo divided by the turn ratio of the transformer N. The polarity of the clamped voltage

vcp is inversed in the two half cycles, as shown in Fig.2–9 (a) and (c). In the rest of the cycle

when the rectifier is blocked, the capacitor Cp is charged or discharged between the positive

and negative clamped voltage, as shown in Fig.2–9 (b). The LCC can run in three modes, based on which the circuit behaviors can be well analyzed by state-space method [Bha91].

0

vrect vrect(1)

irect

irect(1)

Fig.2–10: Input voltage and current waveform of the rectifier and their fundamental elements Another simple way to analyze the LCC is fundamental frequency analysis. Fig.2–10 shows the input voltage and current of the rectifier. The rectifier is nonlinear due to the fact that it can be turned on and off. However, the rectifier can be replaced by a RC network if the

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fundamental elements of the input voltage and current are taken into consideration. [Ive99]. There is a phase shift between the fundamental wave of the voltage vrect(1) and the current

irect(1). That is the reason why a RC circuit is used to replace the rectifier rather than a pure AC

resistor [Ste88]. By using the RC circuit, the LCC can be simplified as follows:

Cp Ls Cs Ceac Reac vab(1) Equivalent RC circuit

Fig.2–11: Equivalent circuit of the LCC with capacitive output by fundamental frequency analysis

Based on the equivalent circuit, the voltage transfer function can be easily derived as well as other steady voltages and currents in the circuit[Ive99]. The voltage gain is shown in Fig.2–12.

Fig.2–12: Voltage gain of LCC

In the figure, fs is the switching frequency and fsr is the series resonant frequency, which is

defined as 1/2π(LsCs)0.5. The LCC can work in two regions, which are below resonance and

above resonance. When the switching frequency of the converter is below resonance, zero-current switching condition can be created for the power switches. In contrast, if the converter

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works above resonance, zero-voltage switching (ZVS) condition can be created for the power switches. For power MOSFET, ZVS condition is preferred [Kaz95].

2.4 Symmetrical Cockcroft-Walton (C.W.) voltage multiplier

The rectified voltage multiplier is a type of rectifier, which converts the AC input to DC output with a boosted voltage level. In HV applications, there are a series of rectified voltage multipliers which were developed from the one invented by John Douglas Cockcroft and Ernest Thomas Sinton Walton in 1932 [Coc32]. With a stepping-up of the input voltage, the multiplier lightens the voltage-boosting burden of the HV transformer in the HV generator. Consequently, the turn ratio and voltage stresses of the transformer are reduced, which further reduces the volume and cost as well as parasitic elements of the transformer. As a result, the overall volume of the generator is expected to be reduced and the effect of the parasitics on the circuit operation is weakened.

In this section, the principle of the rectified voltage multiplier is introduced, followed by a brief review of the different types of the C.W. multipliers. In the end, a circuit operation is introduced for the symmetrical C.W. multiplier, which is selected as an optimum topology among the multipliers.

2.4.1 The rectified voltage multiplier

The principle of the rectified voltage multiplier is introduced by analyzing a one-stage C.W. voltage multiplier, which is also known as a voltage doubler.

C1 vo Co1 vin 2Vpk GND AC vcc vac + vc1 -+ vco1 -D1 D2

Fig.2–13: Schematic diagram of a one-stage C.W. voltage multiplier

Fig.2–13 shows the circuit of a one-stage C.W. voltage multiplier. It consists of one push-pull capacitor C1, also named a coupling/transfer/transition capacitor [Wei69][Lue07] [Kob10], one

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output capacitor Co1, also named a smoothing capacitor and a pair of diodes. A sinusoidal

voltage with peak value of Vpk is fed as the input. The output voltage vo is generated through

the multiplier as a DC voltage of 2Vpk in steady state.

~

t0 t1 t2 t3 t4 t5 t7 1st_cycle ₂nd_cycle vin vc1 vco1 t6 0 1.5kV -1.5kV

Fig.2–14: Building up of the voltage in the one-stage C.W. voltage multiplier

Fig.2–14 exhibits how the voltage is built up in the one-stage C.W. multiplier. The peak value of the input Vpk is set to 1kV and the two capacitors are set the same. The diodes are assumed

ideal. The process is explained as follows, which can clarify the essential principle of the steady-state operation of the multiplier.

In the 1st_cycle:

• t0-t1: In the beginning of the first positive half-cycle, the diode D1 has a forward

voltage bias. The push-pull capacitor C1 is charged till its voltage vc1 reaches Vpk, as

shown in Fig.2–15 (a). Then the diode D1 starts to be blocked and the charging of C1

stops.

• t1-t2: In the first cycle, the initial voltage of Co1 is zero. Thus, as the input voltage vin

decreases from the peak, the diode D2 has a forward bias. Charge flows through Co1,

D2 and C1, as shown in Fig.2–15 (c). The capacitor Co1 is charged until its voltage vco1

reaches Vpk. At this moment, the capacitor C1 is discharged to zero. It seems that in

this period, the charges are moved from the capacitor C1 to Co1. A similar charge

movement can be observed in subsequent cycles, thus C1 is also named the push-pull,

or transfer capacitor. At t2, the diode D2 starts to be blocked when the voltage vin

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• t2-t3: As the input voltage vin starts to increase, both diodes are blocked. In this period,

the two capacitors are both not charged, as shown in Fig.2–15 (b). The voltage across them remains constant.

C1 vin + -C1 Vo Co1 vin + -+vc1 - ₊vc1 -+vco1 -C1 Vo Co1 vin + -+ vc1 -+vco1 -vcc (a) (b) (c)

Fig.2–15: Equivalent circuits of the one-stage C.W. voltage multiplier in different intervals, (a) when the push-pull capacitor is charged; (b) when no capacitors are charged; (c) when the output capacitor is charged

In the 2nd_cycle:

• t3-t4: In the beginning of the second positive half-cycle, the diode D1 again has an

immediate forward bias because the voltage vc1 was zero before . The same as the

period t0-t1, the capacitor C1 is charged to Vpk again when the voltage vin reaches Vpk

as happened in the period to t1.

• t4-t5: Because the capacitor Co1 has been charged to a certain level in the previous

cycle, the diode D2 needs to wait for some time before it has a forward bias. In the 2nd

cycle, the diode D2 starts to conduct when the input voltage vin decreases to zero.

Thus, in this period, both diodes are blocked and the voltage on the capacitors remains unchanged.

• t5-t6: The input voltage decreases to zero, and D2 starts to conduct. The capacitor Co1

begins to be charged again. As in the period t1-t2, the capacitor C1 is discharged and it

seems that some charges are moved from C1 to Co1. However, the amount of moved

charges decreases in each cycle in the building-up transition until the circuit runs steady. In this period, the amount is half that moved in period t1-t2.

• t6-t7: As the input voltage vin reaches the negative peak again, both diodes are blocked

and the voltage on the capacitors remains unchanged until the next positive half-cycle, which is not shown in Fig.2–14.

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In the subsequent cycles, the capacitor C1 is charged to Vpk in the positive half-cycles, and the

charges are moved to Co1 in the negative half-cycles. The voltage vco1 builds up and goes into

steady state after several cycles. Then, the diodes are always blocked and the voltage on both capacitors remains constant. The steady voltage on C1 is Vpk and on C2 is 2Vpk.

If load is added in parallel to the output capacitor Co1, voltage fluctuation presents on the

output voltage vco1 in steady state, as shown in Fig.2–16. The voltage ripple δvco1 is caused by

charge assumption in the load. In the positive half-cycles of the input voltage vin, the charges

are moved to the push-pull capacitor C1. Meanwhile, the load absorbs energy from the output

capacitor Co1, thus the charge and voltage on the capacitor decreases. In the negative

half-cycles, once the diode D2 conducts, the charges are moved from the capacitor C1 to Co1,

consequently the voltage vco1 increases. In total, the capacitor Co1 is charged and discharged in

the whole cycle, resulting in voltage ripple.

vin vco1 δ vco1 ∆ vco1 Vpk -Vpk 2Vpk

Fig.2–16: Output voltage waveform of the one-stage C.W. voltage multiplier in steady state Additionally voltage drop ∆vco1 appears as soon as load current is drawn from the multiplier.

The output voltage will be lower than that in no-load condition. The voltage drop reflects the efficiency of energy conversion of the multiplier. Thus, the ratio between the real output voltage to the no-load voltage is defined as the efficiency of a multiplier.

As mentioned before the output voltage of the HV generator is required to be stable with as low a ripple as possible. Consequently, the voltage drop and ripple are two important criteria for the performance of the multiplier. They are influenced by the capacitors, the operating frequency, the load and the stage amount as well as different topologies. In the next section, different multipliers will be compared on these two criteria to evaluate their performance.

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2.4.2 Comparison of the voltage multipliers

The rectified voltage multiplier was first invented and the specifications published by a Swiss physicist, Heinrich Greinacher, in 1920 and 1921 [Gre20][Gre21]. However, the discovery remained unnoticed until the publication by Cockcroft and Walton in 1932, which was widely quoted [Coc32]. After that, the circuit was widely called the C.W. multiplier.

C2 C1 vo GND AC Cn Co1 Co2 Con (a) C2 C1 GND AC Cn Co1 Co2 Con GND Co(n-1) vo (b) C2 C1 GND AC+ Cn Co1 Co2 Con C2_1 C1_1 AC-Cn_1 vo (c)

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C2 C1 GND AC+ Cn Co1 Co2 Con C2_1 C1_1 AC-Cn_1 Co0 symmetrical C.W. multiplier full-bridge rectifier vo (d)

Fig.2–17: C.W. voltage multiplier and its derivatives, (a) original C.W. voltage multiplier; (b) C.W. voltage multiplier with one output capacitor; (c) symmetrical C.W. voltage multiplier; (d) hybrid symmetrical C.W. voltage multiplier.

Fig.2–17 (a) shows the original C.W. multiplier. It was also known as the half-wave C.W. multiplier, because the output capacitors are charged once per cycle. It is a cascade of the unit multiplier that was shown in the last subsection. The operating principle the multistage multiplier is similar to that of the single stage multiplier. The output voltage of the C.W. multiplier is 2nstVpk in no-load condition, in which nst is the stage amount of the multiplier. As

load is added the voltage drops and ripple appears on the output voltage Vo. Many researchers

have investigated the mathematical formulas for the voltage drop and ripple caused by the resistive load current [Coc32], [Wei69]. Different researchers made different assumptions, which led to different formulas [Wei69]. Regarding the application of HV generators in medical X-ray machine, in which the stage amount is usually lower than 5, the following assumptions are valid [Wei69], [Kob10].

• The total charge flowing in i stage is i times smaller than that flowing in the first stage. • The charging time of the push-pull and output capacitors is much shorter than the

period of the cycle.

• The load resistor across the output and GND can be regarded as nst resistors

respectively connected in parallel to each output capacitor.

Based on the three assumptions, [Coc32] and [Bou40] present the formulas for voltage drop and ripple, which are shown in Table B–1. In the early stage, some authors [Eve53] [Wei69]

Modeling of parasitic elements in high voltage multiplier modules

Modeling of parasitic elements

in high voltage multiplier

modules

Jianing Wang

王 佳 宁

Modeling of parasitic elements

in high voltage multiplier

modules

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 30 juni 2014 om 15.00 uur

door

Jianing Wang

Master of Engineering, Power Electronics and Renewable Energy Center

Xi’an Jiaotong University, China

Dit proefschrift is goedgekeurd door de promotor:

Prof. Dr. J.A. Ferreira

Copromotor Ir. S.W.H. de Haan

Copromotor Dr. Ir. M.D. Verweij

Samenstelling promotiecommissie:

Rector Magnificus

voorzitter, Technische Universiteit Delft

Prof. Dr. J.A. Ferreira

Technische Universiteit Delft, promotor

Ir. S.W.H. de Haan

Technische Universiteit Delft, copromotor

Dr. Ir. M.D. Verweij

Technische Universiteit Delft, copromotor

Prof. Dr. J.A. La Poutre

Technische Universiteit Delft

Prof. Dr. Ir. F.B.J. Leferink

Universiteit Twente

Prof. Dr. J.J. Smit

Technische Universiteit Delft

Prof. Dr. A. Yaravoy

Technische Universiteit Delft

Prof. Ir. L. Van der Sluis

Technische Universiteit Delft,

Copyright © 2014 by Jianing Wang

All rights reserved. No part of the material protected by this copyright notice may be

reproduced or utilized in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval system,

without the prior permission of the author.

Acknowledgement

Abbreviations and Symbols

Contents

Chapter 1

Introduction

1.1 Background

1.2 Volume reduction of the generator

1.3 Problem definition

1.4 Objectives and approaches

1.5 Thesis layout

Chapter 2

Overview of circuit topologies of the

HV generator

2.1 Introduction

2.2 Evolution of the HV generator

2.2.1 Early Stage

2.2.2 The contemporary high frequency generator

2.3 Series parallel LCC resonant converter with capacitive

output

2.3.1 Resonant topologies for the generator

.

.

.

.

.

.

.

.

.

.

2.3.2 Comparison of the topologies

2.3.3 Basic circuit operation of the LCC

.

王佳宁