POWER SUPPLIES FOR
PARTICLE ACCELERATORS
7TH Semester Project
Antonio Prados Vílchez
Supervisor: Frede Blaabjerg
April 2011
Department of Energy Technology
Power Supplies for Particle Accelerators
Prologue
This project has been developed in the Institute of Energy Technology belonging to Aalborg
University during my stay as an Erasmus student. It is part of my degree in Electronics and
Automation Engineering, which has occupied most of my time during the last two years in
Technical University of Madrid. The project is a response to a problem presented by
Danfysik related to power supplies for particle accelerators.
The working plan for the project is to compare two different solutions for the same problem
and decide which one is better. However, due to scheduling problems, this report only
covers the design and analysis of one of the solutions. The second part is expected to be
finished in the next month and it will complete the project.
I want to thank my supervisor for all the help he has provided me and my friends and family
for all their support, which I can feel every day despite the distance.
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Power Supplies for Particle Accelerators
Table of contents
PROLOGUE ............................................................................................................................................. 2
TABLE OF CONTENTS .............................................................................................................................. 3
INTRODUCTION ..................................................................................................................................... 4
1.1.
1.2.
1.3.
1.4.
1.5.
MAGNETS IN ACCELERATORS .............................................................................................................. 6
PROBLEM DESCRIPTION ..................................................................................................................... 8
OBJECTIVES AND GOALS..................................................................................................................... 9
POWER SUPPLY SPECIFICATION.......................................................................................................... 10
REFERENCES.................................................................................................................................. 11
STATE OF THE ART ............................................................................................................................... 13
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
INTRODUCTION.............................................................................................................................. 13
CLASSICAL SOLUTION ...................................................................................................................... 13
HIGH-FREQUENCY SOLUTION WITH MEDIUM-FREQUENCY GALVANIC ISOLATION ........................................ 16
HIGH-FREQUENCY SOLUTION WITH HIGH-FREQUENCY GALVANIC ISOLATION ............................................... 19
CONTROL TOPOLOGIES .................................................................................................................... 24
DISCUSSION AND SELECTION............................................................................................................. 28
REFERENCES.................................................................................................................................. 32
DESIGN OF THE HIGH-FREQUENCY ISOLATED AC-DC CONVERTER ........................................................ 34
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
SYSTEM OVERVIEW ......................................................................................................................... 34
FORWARD- FLYBACK CONVERTER....................................................................................................... 36
VALUE OF ΔQ .............................................................................................................................. 42
CALCULATIONS .............................................................................................................................. 46
FILTER DESIGN ............................................................................................................................... 48
SYSTEM MODELING AND CONTROL..................................................................................................... 51
DECOUPLING................................................................................................................................. 57
PWM PATTERN GENERATION ........................................................................................................... 59
REFERENCES.................................................................................................................................. 61
SIMULATION OF THE HIGH-FREQUENCY ISOLATED AC-DC CONVERTER ............................................... 62
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
DESCRIPTION ................................................................................................................................ 62
CURRENT LOOP.............................................................................................................................. 62
VOLTAGE LOOP .............................................................................................................................. 65
POWER FACTOR AND THD ............................................................................................................... 67
POWER AND EFFICIENCY .................................................................................................................. 69
CONDUCTION LOSSES ...................................................................................................................... 71
REFERENCES.................................................................................................................................. 72
CONCLUSIONS...................................................................................................................................... 73
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Power Supplies for Particle Accelerators
Introduction
Since the 1920’s, particle accelerators have helped scientists understand the behavior of
subatomic particles in controlled situations. The first magnetic induction accelerator was
built at the University of Illinois in 1940 and was based on the thesis proposed in 1927 by
Rolf Widerøe for an experimental betatron accelerator. Although new concepts have been
applied to particle accelerators and many improvements have been made, the concept is still
the same: to accelerate a particle by using a controlled magnetic field.
These devices are used today in many fields, such as research, healthcare and industry. In
the last years, the most famous particle accelerator has been the LHC (Large Hadron
Collider). This collider is the world’s largest and highest energy particle accelerator. It is
expected to shed some light on the existence of the hypothesized Higgs boson. With a
budget of 3 billion euros [1], it is one of the most expensive scientific instruments ever built.
To deflect the charged particles in the accelerator, a magnetic field is used. According to the
Lorenz force, the particle is deflected in proportion to the magnitude of the magnetic field
and to the particle charge. The deflecting force is produced by magnets, which are mostly
electromagnets. The field is created when an electric current flows in a coil, typically made
from copper. As the appropriate currents are very high, a significant amount of heat needs to
be dissipated. For instance, in the Large Hadron Collider, one single corrector magnet which
deflects the beam needs to be supplied with a nominal current of 550A [2].
Fig. 1. Large Hadron Collider in Switzerland.
In our days, people and governments are becoming more and more aware of environmental
problems. New laws are introduced and initiatives are carried out to promote environmental
friendly behaviors. One of the biggest worries of mankind is the production of energy and
the way we consume it. We have used fossil fuels for hundreds of years to obtain the energy
we need to cook, to keep ourselves warm and to make our machines work. The gasses
produced in the combustion of the fuels have caused the global warming we are now
4
Power Supplies for Particle Accelerators
suffering. Researchers are working on alternative ways to obtain energy, such as wind
power or solar power.
But researching is not only done in the side of the energy producer. Companies and research
groups are making a great effort to design and produce devices which are more efficient and
can do the same work consuming less energy. Energy efficiency is being applied in many
fields, such as electronics, mechanics or the automotive industry. A part of the energy
consumed by an electronic device is turned into heat. Power supplies are the stage which
delivers the power from the grid to the device, in order to make a useful work. The
efficiency of the power supply should be very high to deliver the most part of the energy to
the rest of the equipment. In a particle accelerator, the power spent is very high even for a
single magnet. The efficiency of these devices is usually low, so if one of these devices has
a hundred magnets, there is a large amount of energy which could be saved from being
turned into heat.
In an accelerator, experiments are divided into different periods which make particles
accelerate and collide between them. Firstly, the particle is injected into the accelerator.
Then it is accelerated to be stored in the ring. Finally, the particle is extracted from the ring.
The magnets of each stage provide the magnetic field necessary to make the particle
behavior in a certain way. Since these magnets are electromagnets, each period needs
different currents to be delivered. Reference [3] presented a power supply specially adapted
to deliver the current for the electromagnets in every period of the experiment. This
topology is divided into modules which operate or are biased depending on the present stage
of the experiment. Current ripples delivered to electromagnets must be considered, as they
propagate in the magnet string affecting the beam motion inside the beam tube. The power
supply is suitable for applications which operate in two quadrants and require low ripple,
high current and large dynamic range. Some modules of this topology are built with
thyristors, which are difficult to control and have been progressively substituted by IGBTs.
To create the magnetic field in the accelerator, a high current must be delivered to the
magnets. It is very important to keep a low ripple in this current, to avoid fluctuations of the
magnetic field. The harmonic content of the current must be low. Passive filters to attenuate
harmonic components in the output current are not suitable because of the lack of control in
harmonics of the magnet voltage and current. In [4], a low ripple power supply for highcurrent magnets is proposed. It consists on a switch mode ripple regulator in series with a
rectifier power supply. This regulator is a pulse-width-controlled two-quadrant chopper and
can achieve a harmonic content in the magnet current less than 10 ppm.
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Power Supplies for Particle Accelerators
1.1. Magnets in accelerators
In some special applications permanent magnets are being used in accelerators. For
example, in devices such as undulators and wigglers approximately 100 permanent magnets
are used to deflect electrons periodically in synchrotron radiation sources. Other beam
applications include ion sources, radio frequency sources (klystrons) and linac quadrupoles.
A few permanent magnet systems have been built for special beamline applications, e. g.
where space is limited. Permanent magnets can provide magnetic fields up to 2 Tesla, as
large as those practically achievable in resistively driven electromagnets with iron cores [5].
Most magnets for particle accelerators are electromagnets and mostly based on normal
conducting (resistive) magnet coils. In some special applications the magnet coils are
superconductors with the purpose of creating very high magnetic fields unobtainable by
room temperature resistive magnets [6]. Since the electrical resistance of a superconducting
coil is zero, the power deposition in such a coil is zero. However, such magnets are
operated at a few Kelvin and since cooling to this temperature requires a large amount of
electrical power, this technology is not a suitable candidate for “green” magnet solutions.
Even the use of high-temperature superconductors is not expected to be “green”, although if
such magnets reduce the cooling requirements significantly.
Pure permanent magnets are used only in very special accelerator applications, namely in
undulators and wigglers. The main arguments for the use of permanent magnet material here
is the possibility to have many small magnets of opposite field directions over a finite
distance.
Fig. 2. Dipole magnets for a particle acclerator.
The most obvious idea hence appears to be the use of permanent magnets to replace
“resistive” magnets in beam transport, reducing the overall power consumption in principle
to zero. However, many applications often call for a magnetic field that can be varied, as the
system has to be used for different particle energies or different particle mass/charge ratios.
Furthermore, most applications require some fine tuning of the magnetic field. In addition,
the magnetic field from permanent magnet materials is temperature dependent and some
6
Power Supplies for Particle Accelerators
compensation of the field changes would be needed. Finally, permanent magnet materials
might age on long time scales, e.g. due to radiation damage, which again calls for some
tune-ability. Magnetic fields can today be measured with very high accuracy and if needed a
feedback loop can be foreseen in the adjustment of the magnetic field.
The magnetic field can be produced by a hybrid dipole magnet consisting of permanent
magnets and a resistive coil with an iron yoke to shape the magnetic field. In such a hybrid
magnet, the electromagnetic part can be used to vary the magnetic field, whereas the
permanent magnet blocks are statically providing a major part of the field. Combining these
two elements, half of the magnetic field is intended to be produced by the permanent
magnets (PM) and the other half by the electromagnetic part (EM). In this case the power
consumption can be reduced in a significant amount. By inversion of the current in the
resistive coil, the field will have its full range of variation between zero and maximum field.
The magnitude of the magnetic field is often required to be very precise with stability
around 1 ppm and hence the associated current power supply has to be designed and built to
correspondingly high precision. The electromagnetic part of a hybrid magnet can be used to
adjust precisely the magnetic field from the permanent magnet, which is difficult to
manufacture with sufficiently high precision. The field from permanent magnets is
temperature dependent, but if monitored with magnetic field probes, e. g. Hall or NMR
probes, the field can be adjusted by the resistive part of the magnet. Permanent magnet
material is somewhat radiation sensitive and hence it should be placed as far as possible
from the charged particle beam. However, the resistive part of the hybrid magnet will be
able to compensate for the reduction in the magnetic field from the permanent magnet
material.
In a hybrid magnet system, a part of the field originates from the resistive part and hence the
current and/or the number of turns in the coil can be significantly reduced as compared to a
traditional resistive magnet (in which the whole field is generated by the resistive part). For
pulsed magnets or magnets where fast ramping of the field is required, the inductive voltage
is furthermore drastically reduced as it scales with the number of turns squared. This leads
to the possibility of developing more compact magnet power supplies, as smaller power
devices can be used, which in turn opens for the possibility of higher switching frequencies
[7]. For example, IGBTs can be replaced with FETs, transformer iron in chokes can be
exchanged with Soft Magnetic Composites (SMC).
The magnetic structures for the Advanced Light Source are of hybrid permanent magnet
design [8]. The coils are used to correct the magnetic field created by the permanent
magnets and therefore control the particle beam with high precision. These electromagnets
are powered by a 13V, 120A, bipolar power supply and the coils are wired in series to
achieve maximum magnetic symmetry. The power supply can achieve a voltage ripple
7
Power Supplies for Particle Accelerators
lower than 1% and a current regulation of ±100ppm. The coil inductance is 0.5 mH and the
resistance is 35.8 mΩ.
In the last years, magnetic fields in some particle accelerators are provided by
superconducting electromagnets. As magnets need high-currents, the resistance of the coils
consumes power in a useless way, reducing the total efficiency of the magnet. If the coil
behaviors as a superconductor, the resistance is significantly reduced and the efficiency is
increased. For superconducting magnets, Nb3Sn or Nb3Ti are normally used and MgB2 is
becoming a competitor [9]. Different groups are investigating the potential of MgB2 for
application to magnetic resonance imaging, space magnets, etc. Critical current densities of
1.7 * 109 A/m2 at 4.2 K and 5 T in multi-filamentary SiC doped MgB2 wires have been
reported [10].
1.2. Problem description
The power electronics converter has to handle the power delivered to the accelerator
electromagnets from the power grid. Two parameters besides the investment costs are
relevant, namely the weight/size and the energy efficiency of the power conversion. The
present solution for the power supply, which is shown in Fig. 3, is a converter connected to
the grid using a voltage-shifting transformer, with two secondary windings and a 12-pulse
series-type diode rectifier generating a raw DC-voltage. The transformer frequency is 50/60
Hz and therefore the transformer is relatively large and heavy.
50 Hz
Grid
AC / DC
DC / DC
Electromagnets
Fig. 3. Present solution for the electromagnets power supply.
The second stage in this standard converter converts the relatively high “raw” DC-voltage
into a well defined controllable output DC-voltage, high current and high precision. A
closed loop system should be in charge of delivering a stable current to the coils of the
electromagnets as shown in Fig. 4. This current could be measured using a Hall Effect
sensor.
AC / DC
Feedback
DC / DC
Electromagnets
Sensor
Fig. 4. Closed loop system to control the current in the electromagnets.
8
Power Supplies for Particle Accelerators
In the present application the grid transformer is identified to contribute significantly to the
weight and size and in the second power conversion stage, the high current converter
contributes significantly to the power losses and hence decreases the overall efficiency.
Therefore the goal is to reduce the weight and size of the grid transformer and to increase
the efficiency in the total power conversion.
The size of the transformer depends on the frequency. A possible solution is to reduce
volume and weight by increasing the frequency. The challenge here is then to maintain a
high efficiency taking a high frequency AC-AC converter into consideration (Fig. 5). The
energy efficiency of the second stage DC-DC converter is limited by the high output current
at low voltages, as the converter is operating point sensitive. This solution is particularly
promising and challenging since it includes an AC to AC direct conversion.
High frequency
Grid
AC / AC
AC / DC
DC / DC
Electromagnets
Fig. 5. Alternative high-frequency solution for the electromagnets power supply.
There is another alternative topology to the present solution. Instead of using a direct AC to
AC conversion, a rectifier and an inverter can be used to increase the frequency in the first
stage (Fig. 6).
High frequency
Grid
AC / DC
DC / AC
AC / DC
DC / DC
Electromagnets
Fig. 6. Alternative high-frequency solution for the electromagnets power supply.
Although other disadvantages might be found, the controlled rectifier-inverter topology
must be considered because compared to the matrix converter (Fig. 5), the number of
switches of the converter is lower. Therefore, the reliability of a matrix topology is lower
than the reliability of a rectifier-inverter topology [11].
1.3. Objectives and goals
A high-precision highly efficient power supply will be necessary to produce the currents
needed for the resistive part of the magnet. This project will consist of analyzing possible
topologies for the power supplies of the coils. One of the two alternative solutions
previously discussed will be chosen to improve the performance and the dimensions of the
power supply. The power converters to be developed should include compactness, modular
9
Power Supplies for Particle Accelerators
design with highly effective switch-mode technology, digital control to the ppm level and
high efficiency.
As compactness is one of the requirements, is important to put emphasis on avoiding large
components. For example, for the AC-AC converter, a solution without a capacitor in the
DC-link should be analyzed. The number of switches of the modules should be optimal to
avoid the converters to be very large.
The last stage of the power supply, the DC to DC converter, must deliver a high current to
the electromagnet. A wide operating range with high efficiency will be the target. The DCDC converter should be designed according to the ppm accuracy required.
The project goals are:
 To understand the state-of-the-art power supplies for the magnets of the particle
accelerators.
 To evaluate the efficiency of the alternative topologies in power supplies for the
magnets.
 To design the necessary converters to reduce the volume and weight of the grid
transformers, increasing the efficiency as possible. Simulations will be done in order
to evaluate the design.
 To demonstrate operation of a highly effective AC-DC power converter based on
state-of-the-art technology, optimized to the demands and possibilities related to the
magnet.
1.4. Power supply specification
The power supply to be designed must meet the same specifications as the Danfysik
magnet power supply model 9100.
Fig. 7. Danfysik magnet power supply model 9100[12].
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Power Supplies for Particle Accelerators
The specifications of the unit are [12]:
Input voltage
Current stability:
- Short (30 min.)
- Long (8 hours)
Max. output current
Max. output voltage
Max output power
Current setting
Air cooled version
Std. remote control
Technology
Options:
- Polarity reversal sw
- Bipolar version
Application example
400 VAC ± 10%, 3 phase 47-63 Hz
±10 ppm
200 A
60 V
12 kW
18 bit
No
RS-422
Switch mode with power MOSFET
Yes
Yes
Dipoles and quadrupole
Table 1. Power supply specifications.
1.5. References
[1] CERN. Website.
http://askanexpert.web.cern.ch/AskAnExpert/en/Accelerators/LHCgeneral-en.html
[Consulted: 11 - 2 - 2010].
[2] “The LHC main ring: Main magnets in the arcs”. CERN, LHC Design report, vol. 1,
chapter 7.
[3] K. M. Smedley, Department of Electrical and Computer Engineering, University of
California, “Hybrid Power Supplies for Particle Accelerators”, Nuclear Science
Symposium and Medical Imaging Conference, November 1994, vol. 1, pp. 473 - 476.
[4] R. Liang, S. B. Dewan, “A Low Ripple Power Supply for High-Current Magnet Load”,
Conference Record of the 1992 IEEE Industry Applications Society Annual Meeting,
October 1992, vol. 1, pp. 888 – 893.
[5] K. Halbach, Lawrence Berkeley Laboratory, “Permanent Magnet Undulators”, Journal de
Physique, vol. 44, supplement of no. 2, February 1983, pp. 211-216.
[6] R. B. Palmer, “Superconducting Accelerator Magnets: A Review of their Design and
Training”, American Institute of Physics, 1993.
11
Power Supplies for Particle Accelerators
[7] T. Takaku, T. Isobe, J. Narushima, R. Shimada, “Power supply for pulsed magnets with
magnetic energy recovery current switch”, IEEE Transactions on Applied
Superconductivity, vol. 14, no. 2, June 2004, pp. 1794-1797.
[8] D. Humphries, J. Akre, E. Hoyer, S. Marks, Y. Minamihara, P. Pipersky, D. Plate, R.
Schlueter, Lawrence Berkeley Laboratory, University of California, “Design of End
Magnetic Structures for the Advanced Light Source Wigglers”, Proceedings of the 1995
Particle Accelerator Conference, May 1995, vol.3, pp. 1447 - 1449.
[9] G. Bellomo, R. Musenich, M. Sorbi, and G. Volpini, “MgB2 Coils for Particle
Accelerators”, IEEE Transactions on Applied Superconductivity, June 2006, vol. 16, no. 2,
pp. 1439 – 1441.
[10] M. D. Sumption, M. Bhatia, X. Wu, M. Rindfleisch, M. Tomsic, E.W. Collings,
“Multifilamentary, in-situ Route, Cu-stabilized MgB2 Strands”, Applied Physics Letters,
September 2004.
[11] P. W. Wheeler, J.C. Clare, L. de Lillo, K.J. Bradley, M. Aten, C. Whitley, G. Towers,
School of Electrics & Electronics Engineering, Nottingham University, “A comparison of
the reliability of a matrix converter and a controlled rectifier-inverter”, European
Conference on Power Electronics and Applications, pp. – 7.
[12] Danfysik, “System 9100 Power Converter”, Products catalogue.
12
Power Supplies for Particle Accelerators
State of the Art
2.1. Introduction
Power supplies are devices specifically designed for every application. Depending on the
characteristics of the load to which the power is delivered, the power supply must meet
different requirements. When supplying power to a regenerative load, the design of the
converter must take into account the necessity of handling reverse power flow. In addition,
the parameters of the components must be calculated according to the currents and voltages
delivered to the load. Compactness is also a desirable characteristic of every power supply,
as electronic systems are becoming smaller. Users of power supplies show a lot of interest
in low cost and highly reliable devices, as they are important economic parameters.
Currently, a designer can find many alternative designs to meet the specific requirements of
a power supply.
One of the goals of this project is to reduce the size of the power supply. This can be
achieved by reducing the volume of the galvanic isolation, which is necessary to meet some
safety requirements. The size can be reduced by increasing the frequency at which the
transformer will operate. However, the skin effect and proximity effects are major problems
in high frequency transformer design, because of induced eddy currents. Leakage
inductance and the unbalanced magnetic flux distribution are two further obstacles for the
development of high frequency transformers. For that reason, solutions including mediumfrequency transformers must be considered.
In this chapter, a review of the state of the art in the field of power supplies for particle
accelerators is presented. The chapter is divided into different sections according to the
three alternative topologies shown in chapter 1. Firstly, a state of the art review of the
conventional solution is made. Then, the state of the art of the two high-frequency solutions
is analyzed. This chapter also includes a review of the state of the art of control techniques
and finally, a discussion and a selection are made among the analyzed designs.
2.2. Classical solution
As showed in chapter 1, the conventional solution to implement the power supply for the
particle accelerator is a topology composed of a 50 Hz transformer followed by a threephase AC-DC converter. As the transformer is connected to the three-phase grid voltage, it
is a simple solution, since the frequency of the voltage is not increased in the primary side
of the galvanic isolation. However, using a 50 Hz transformer results on a heavy and large
13
Power Supplies for Particle Accelerators
power supply because the volume of a transformer increases as the frequency decreases.
The absence of AC-AC converters to change the frequency of the primary side voltage of
the transformer makes this solution not as complex as high-frequency solutions.
Compactness and simplicity are one of the desired requirements when designing a power
supply. One-stage AC-DC converters are commonly used and they provide a regulated DC
voltage output performing the conversion in only one stage. In [1], a single-stage, threephase AC-DC converter is presented. The topology is shown in Fig. 1. It is divided into two
semi-stages: the front semi-stage, which is a three-phase AC-DC buck-boost converter, is
operated in discontinuous conduction mode (DCM) to achieve unity power factor and low
THD and the rear semi-stage is a DC-DC converter for step-down voltage conversion.
Experimental results show that the proposed converter can achieve purely sinusoidal
waveforms of three input currents, unity power factor and low THD.
Fig. 1. Topology of the single-stage three-phase AC-DC converter [1].
In [2], a three-phase AC-DC boost converter is introduced. It is a unidirectional power
converter which achieves almost unity power factor and reduction of harmonics distortion.
As shown in Fig. 2, the topology consists of three legs (one for each phase) and every leg
includes a boost inductor, two power diodes and two active switches. The conversion is
performed in one stage.
14
Power Supplies for Particle Accelerators
Fig. 2. Three-phase AC-DC converter [2].
An example of this type of converters is presented in [3]. In this work, a topology based on
the Vienna Rectifier is proposed. The converter shown in Fig. 3 can handle an output power
of up to 35 kW and the efficiency is higher than 97% at full load. The power factor is above
0.990 when the power is higher than 20% of the maximum power.
Fig. 3. Unidirectional three-phase AC-DC converter based on the Vienna Rectifier [3].
15
Power Supplies for Particle Accelerators
2.3. High-frequency solution with medium-frequency galvanic
isolation
Conventional matrix converters (CMC) or direct matrix converters (DMC) are AC-AC
converters which can change the frequency or amplitude of a voltage or current signal
without any intermediate energy storage element. These frequency changers can connect
any of the three input phases to any of the three output phases to provide the desired output
waveform. Usually AC-AC converters are used as motor drives to supply power to the
motor at a desired frequency. These converters need to handle bidirectional power, as the
normal power flow is sometimes inverted when driving a motor. These converters provide a
medium-frequency solution which reduces the eddy currents and leak inductance in the
transformer.
Despite of intensive research for decades, matrix converters have low market diffusion. The
most AC-AC converter used in industry is the 2-level, voltage DC-link, back-to-back
converter (VLBBC). In [4] three-phase AC-AC matrix converters and VLBBC are
compared regarding to the characteristics and performance of both converters. Proponents
of matrix converters argue that direct converters have an increased lifetime because of the
absence of a DC-link capacitor. In addition, it can be implemented in a more compact
package due to the absence of bulky capacitors. However, one of the main disadvantages is
the limited maximum output voltage of 86.6% of the input voltage, which means that a
400V input will result in a 346V output. Fig. 4 shows these topologies.
Fig. 4. a) Back-to-back converter. b) Conventional matrix converter. c) Indirect Matrix Converter with DC-link [4].
The matrix converter switch cell needs to provide bidirectional power flow and has to be
able to block reverse voltages. In [5], the conventional matrix converter topology shown in
Fig. 5 is proposed.
16
Power Supplies for Particle Accelerators
Fig. 5. Simplified matrix converter topology with the nine switch cells and the bidirectional IGBT solution [5].
The switch cells proposed in [5] are SiC-JFETs, which have an intrinsic (instead of bonded,
as in an IGBT) diode that can provide reverse conducting capability. The authors of this
work give a first impression of the advantages of using SiC-JFETs instead of Si-IGBTs for
the switching cells of a conventional matrix converter as shown in Fig. 6. Theorically, SiC
can handle 100 times higher power density than Si [5]. SiC-JFETS show less switching
losses than the Si-IGBTs. The conventional matrix converter realized in SiC and without
energy storage element promises to continue the growth of power density.
Fig. 6. Simplified commutation cell using SiC switches [5].
AC-DC-AC converters are a group of AC-AC converters which implement a DC-link
between two stages (rectifier and inverter). Some of them include an energy storage element
in the DC-link. Removing this element results on higher reliability and lower maintenance
of the converter. The role of the DC capacitor is to serve as storage of the energy when it
flows back from the inverter during regenerating cycles. In [6], an AC-DC-AC converter
without DC-link elements is proposed. It consists of two stages (rectifier and inverter) as
shown in Fig. 7. As this topology has no DC-link capacitor, harmonics appear from the
output to the DC-link and the input currents. To filter this harmonics and meet EMI
regulations, grid AC filters are required. This multilevel topology provides reduced
harmonic distortion of the output waveforms, reduced stress of the semiconductors and
17
Power Supplies for Particle Accelerators
higher voltage and power levels. In the previously mentioned work, a way to switch the
rectifier is presented in order to reduce the losses of the input converter.
Fig. 7. Capacitor-less DC-link rectifier-inverter topology [6].
One of the disadvantages of AC-AC converters is the high number of switches involved in
the topology. Researchers in [7] propose a new topology which requires only 9 switches
instead of the 18 of the conventional matrix converter. The Ultra Sparse Matrix Converter
(USMC) has a simplified input configuration which restricts the converter to unidirectional
power flow. This converter can achieve sinusoidal input and output waveforms with an
efficiency of up to 94%. The USMC is an indirect matrix converter, divided into a rectifier
and an inverter with an intermediate DC-link but no energy storage elements as shown in
Fig. 8. In this work, a novel clamp circuit is presented to protect the converter from
overvoltages under regeneration conditions.
Fig. 8. Ultra Sparse Matrix Converter topology [7].
Indirect matrix topologies need bidirectional switches in order to provide a path for reverse
currents and to block reverse voltages. There are several ways to implement a bidirectional
switch, but they involve the use of two IGBTs and two diodes. In [8] a matrix converter
which uses reverse blocking IGBTs is proposed, using one of these elements for each switch
instead of two as shown in Fig. 9. This converter implements a rectifier stage with RB18
Power Supplies for Particle Accelerators
IGBTs and an inverter stage with IGBTs. RB-IGBTs ensure low rectifier stage conduction
losses and high power conversion efficiency as less conduction resistance is in the current
path (less semiconductors are used). However, RB-IGBTs tend to degrade the system
efficiency unless low switching frequencies are employed.
Fig. 9. RB-IGBT based Indirect Matrix Converter [8].
2.4. High-frequency solution with high-frequency galvanic
isolation
AC-DC converters are widely used in applications in which a regulated DC output voltage
or current is needed. The input of these converters is a single-phase or three-phase voltage
from the grid. This voltage is rectified and smoothed to achieve a low-ripple DC voltage.
Single-phase rectifiers are used in low power applications and therefore, when systems have
high-power requirements, three-phase AC-DC converters must be used. In these power
converters is very important to implement a galvanic isolation between the grid side and the
load side to avoid currents flowing from one side to another. As said in previous sections,
the use of high-frequency transformers results on an increase of eddy currents and leakage
inductance, which are undesirable effects. These converters provide a high-frequency
solution, which results on smaller transformers than the ones in the medium-frequency
solutions.
In [9], a three-phase AC-DC converter is presented. It consists of three full-bridge
converters with a DC regulated output as shown in Fig. 10. As galvanic isolation is required,
they include a high-frequency transformer which is smaller and lighter than low-frequency
isolation. It provides high power factor and low THD. This converter avoids current loops to
control the power factor as it is corrected by a polyphase autotransformer. The switches of
this topology are commutated using the ZVS-PWM technique. The simplicity, robustness
and high power density suggest the proposed converter as a strong candidate for modern
19
Power Supplies for Particle Accelerators
solutions to three-phase supply systems used in telecommunication, motor drive, UPS and
others.
Fig. 10. Three-phase AC-DC converter with high-frequency isolation [9].
Reduced number of switches is a desired requirement for a converter, as it increases the
reliability and reduces cost and volume. In [10], a single-stage, high-frequency isolated,
three-phase AC/DC converter is presented. It is composed by a forward/flyback converter
as shown in Fig. 11. The forward subconverter operates in continuous conduction mode
(CCM) and the flyback sub-converter in discontinuous conduction mode (DCM). This is
defined so that the forward sub-converter processes practically all of the power delivered to
the load. The rectifier stage only uses three switches to perform the conversion. An
appropriate design of the input filter is needed to achieve high power-factor (higher than
0.98) and low T.H.D of line current (up to 5%).
Fig. 11. High-frequency isolated three-phase AC-DC converter [10].
20
Power Supplies for Particle Accelerators
Three-phase AC-DC converters are usually formed by a rectifier and a DC-DC converter.
The rectifier stage of the converter typically implies the use of six switches to perform the
AC-DC conversion. In [11], a three-phase AC-DC rectifier with reduced switch count is
presented. The input stage is a three-phase diode rectifier with a LC filter and the output
stage is a zero-voltage, zero-current switching full-bridge converter (ZVZCS) as shown in
Fig. 12. It also implements power factor correction. However, this approach is usually
power limited.
Fig. 12. Three- phase single-stage AC-DC ZVZCS full-bridge converter [11].
Sometimes, power flows from the load side to the grid side. In this case, bidirectional
converter topologies are required. The use of bidirectional switches is necessary to assure
bidirectional current paths. In [12], an AC–DC–DC isolated converter with bidirectional
power flow capability is presented. It consists on a three-phase AC-DC converter, a highfrequency transformer and a DC-DC converter. Based on the space voltage vector
modulation method and the equivalent circuit of the proposed converter, a simple and
efficient control strategy is suggested in this work. As shown in Fig. 13 and Fig. 14, this
topology allows bidirectional power flowing.
Fig. 13. Direct power flow in the AC-DC converter [12].
21
Power Supplies for Particle Accelerators
Fig. 14. Reverse power flow in the AC-DC converter [12].
To feed a low-energy correction magnet of a particle accelerator, an AC-DC matrix
converter is presented in [13]. It is composed of a bidirectional three-phase to one-phase
cycloconverter and a bidirectional synchronous rectifier as shown in Fig. 15. The
cycloconverter increases the frequency in order to include a high-frequency galvanic
isolation and reduce the weight and size of the converter. The proposed converter is capable
to regenerate energy back to the utility and to control the power factor at the mains up to the
unity. In the rectifier side, the AC waveform is rectified and filtered to provide a bipolar
current in order to feed the magnet load with the desired precision.
Fig. 15. Four quadrant AC-DC matrix converter with high-frequency isolation [13].
In [14], a three-phase single stage AC-DC converter is presented. It is a boost PWM
converter with high-frequency isolation. The single stage topology shown in Fig. 16 has the
advantage of being less complex than the two-stage solution. It only uses five switches and
the power factor is almost unity. The efficiency and the power density of this converter are
high.
22
Power Supplies for Particle Accelerators
Fig. 16. Unidirectional three-phase single stage AC-DC converter [14].
An alternative to previously discussed topologies is the use of a three-phase rectifier stage
and a DC-DC converter with high-frequency galvanic isolation. In this case, the rectifier
should include the possibility of power factor correction. AC-DC converters discussed in
section 1.2 should be taken into account to implement this high-frequency solution (rectifier
+ DC-DC converter).
There are plenty of DC-DC converters with high-frequency isolation in the literature. In
[15], a three-phase DC-DC resonant converter is proposed. Resonance in converters offers
reduced semiconductor stress and a reduction of the switching losses. The converter shown
in Fig. 17 is divided into two stages (inverter and rectifier) separated by the three-phase
high-frequency galvanic isolation. Capacitors and stray inductances are used as resonant
components in this converter. Zero current switching is achieved by switching the IGBTs of
the inverter stage at resonant frequency. This converter is designed to handle 5 kW and
performs an efficiency higher than 97%. The power flow is unidirectional.
Fig. 17. Unidirectional DC-DC converter with three-phase isolation [15].
Reduction of switching losses can be achieved by implementing a zero voltage and zero
current switching. A DC-DC converter with high-frequency isolation is presented in [16].
This converter can handle 50 kW. As shown in Fig. 18, the topology is composed of two
23
Power Supplies for Particle Accelerators
bridges and depending on the power flow, one of them is switching and the other is a
conventional rectifier. In this way, bidirectional power flow can be handled. The transistors
are switched in zero-voltage and quasi-zero-current mode.
Fig. 18. Bidirectional DC-DC converter with single-phase isolation [16].
2.5. Control topologies
When designing an AC-DC converter, two parameters must be controlled in order to meet
the desired requirements of this kind of devices: output DC voltage and input AC current.
As the output of the converter is required to be a well regulated DC-voltage, this must be
measured in order to compensate variations or to change to another operating point. Sensing
this variable and implementing a suitable control technique, switches are commutated to
achieve the voltage regulation. Another important requirement of this converter is to provide
unity power factor in the first stage. To achieve this goal, a measure of the input currents
must be provided to the control topology in order to commutate the switches in the proper
way.
Sensors are needed to measure currents and voltages in the converter and to implement the
closed-loop control. This fact increases the cost and reduces reliability of the converter. In
[17], a sensorless nonlinear control technique for a three-phase PWM AC-DC converter is
proposed. An input-output feedback linearization control strategy is implemented in order to
control both the line current and the DC output voltage. In addition, an estimator of the AC
mains voltage is implemented so as to reduce the number of the requested sensors, to avoid
the undesired noise and to minimize the converter’s cost. The topology is shown in Fig. 19.
24
Power Supplies for Particle Accelerators
Fig. 19. Nonlinear control block diagram of the PWM AC-DC converter [17].
Another approach to sensorless control is proposed in [18]. In this work, a predictive control
algorithm for three-phase AC-DC converters is presented. In this case, the control proposed
does not have current feedbacks and the input current levels are controlled by the DC
voltage loop. The predictive algorithm used in this paper is base on a non-causal system
modeling, where the current i[k+1] must be known. This idea is based on the fact that in
power electronic devices with power factor correction the future samples can be known,
because the input current must be a sinusoidal function. The topology is shown in Fig. 20.
Fig. 20. Topology of the control technique proposed in [18].
Simplicity is also a desired requirement of a control technique. In [19], a simplified control
algorithm for synchronous rectifiers is presented. The proposed control algorithm is simple,
25
Power Supplies for Particle Accelerators
robust and inexpensive and can be used for unity and variable power factor operation. It
provides DC output voltage control and power factor correction. The controller is divided
into sections as shown in Fig. 21: DC voltage regulation, equivalent load conductance
calculation, hysteresis current controller, variable power factor algorithm and switching
algorithm. Fig. 22 shows a detailed diagram of the control topology.
Fig. 21. Sections of the control strategy proposed in [19].
Fig. 22. Control topology proposed in [19].
In section 1.4, a single-stage high-frequency isolated three-phase AC/DC converter was
reviewed [10]. In this work, a control topology for this converter is proposed, as well as a
modeling technique. The modeling applied is based on phase variables in the dqo
coordinates system. An AC input model and a DC load model are analyzed. The closed-loop
topology is shown in Fig. 23 and the detailed block diagram of the control strategy is shown
in Fig. 24.
26
Power Supplies for Particle Accelerators
Fig. 23. Closed-loop topology presented in [10].
Fig. 24. Detailed diagram of the control strategy presented in [10].
The work presented in [12] also proposes a control topology for the three-phase AC-DC
converter. it is decoupled into a DC-DC controller and a AC-DC controller as shown in Fig.
25. It allows switching the operating mode depending on the power flow (direct power flow
or reverse power flow).
27
Power Supplies for Particle Accelerators
Fig. 25. Block diagram of the control strategy presented in [12].
2.6. Discussion and selection
As said in previous sections, medium-frequency galvanic isolation solutions are composed
of an AC-AC converter, galvanic isolation and then a three-phase rectifier. This topology
requires a large number of components and switches, as it is divided into two different
converters. Number of switches should be an important parameter when choosing an ACAC converter. In [5], a convention matrix converter is presented. This converter requires
nine switches which can be bidirectional. However, the control of this topology is more
complex than the one of new matrix converters. In this work, is also proposed the use of
SiC-JFETs switching devices instead of Si-IGBTs. These devices can handle more power
than Si-IGBTs and the have an intrinsic diode (instead of bonded) so they must be
considered when choosing the switching devices for a converter. Medium-frequency
galvanic isolation solutions found in [6] and [8] propose topologies with more switches than
the one proposed in [7]. Although the Ultra Sparse Matrix Converter allows only
unidirectional power flow, it has only nine switching devices and the efficiency is 94%. The
best option to analyze the medium-frequency solutions is the Ultra Sparse Matrix Converter
proposed in [7].
28
Power Supplies for Particle Accelerators
Regarding to the high-frequency solution with high-frequency galvanic isolation, many
works and papers have been found. Simplicity and low number of switching components are
very important parameters which must be taken into account. Solution presented in [9]
meets the requirements as it performs a low THD, high power factor and it has the
advantage of the absence of current loops to control the power factor. However, this
topology requires a polyphase transformer which can be bulky and redundant. In [11] we
can also find a suitable topology for the purpose of this project. The simplicity of this
converter is its main advantage, but the fact of using diodes instead of transistors in the
rectifier stage can result on high harmonics in the current from the mains. The converter
proposed in [12] is one of the best alternatives, as it performs high efficiency and low
harmonics. This topology, as well as the one presented in [13], allows bidirectional power
flow and it would be the best option if the application is required to work in four quadrants.
However, this input stage of the power supply only requires unidirectional power flow,
leaving the reversal power flow handling to the output stage. The converter presented in
[10] is the best option to implement a single-stage AC-DC converter with high frequency
galvanic isolation. The THD in this case is lower than 5%, the power factor is higher than
0.98 and it only allows unidirectional power flow. Another advantage of this topology is the
availability of the information about the control strategy and the components used in the
converter, what would make it easier to simulate and build a prototype. The disadvantage of
this topology is that it has been built into a prototype of only 2.5 kW, so it is uncertain how
it will behave when trying to build a prototype of higher power density.
As they are more complex and they require more components, two stages topologies
(rectifier + DC-DC converter) are rejected. Another reason to reject this option is that the
efficiency of the two stages separately should be very high in order to achieve a high
efficiency of the whole system when these parts are together.
Table 1, 2 and 3 show a comparative of the most important parameters when choosing the
converters for the power supply.
29
Power Supplies for Particle Accelerators
Title
Reference
Study and implementation of a singlestage three-phase AC-DC converter
[1]
An improved high performance three
phase AC-DC boost converter with
input power factor correction
[2]
35 kW active rectifier with
integrated power modules
[3]
Classical solution
Topology
Efficiency
Single-stage, unidirectional,
three-phase, buck-boost AC-DC
converter
Three-phase, unidirectional, ACDC converter
Three-phase, unidirectional, ACDC converter
Trans. / Diodes
Power level
PF
THD
74% @ Full load
4 / 12
50 – 500 W
≈1
<5%
98 % @ Full load
6 / 12
4 kW
≈1
<5%
97 % @ Full load
3 / 18
35 kW
≈1
Table 1. Comparison of the topologies discussed for the conventional solution
High-frequency solution with medium-frequency galvanic isolation
Bidirectional Rectifier-Inverter
Multilevel Topology without DC-link
Passive Components
An Ultra Sparse Matrix Converter with
a Novel Active Clamp Circuit
A High Efficiency Indirect Matrix
Converter Utilizing RB-IGBTs
[6]
[7]
[8]
Bidirectional, multilevel,
rectifier-inverter AC-AC
converter
Unidirectional matrix converter
18 / 18
93 % @ Full load
9 / 18
Bidirectional, indirect matrix
95 % @ Full load
18 / 6
converter
Table 2. Comparison of the topologies discussed for the medium-frequency solution
5%
5.5 kW
6.8 kW
1%
≈1
30
Power Supplies for Particle Accelerators
High-frequency solution with high-frequency galvanic isolation
A 12 kW three-phase low THD rectifier
with high-frequency isolation and
regulated DC output
A Single-Stage High-Frequency
Isolated Three-Phase AC/DC Converter
A Three-Phase Ac-Dc Rectifier with
Reduced Switch Count
AC-DC-DC isolated converter with
bidirectional power flow capability
Four-Quadrant AC-DC Matrix
Converter with High-Frequency
Isolation
VIENNA rectifier II-a novel singlestage high-frequency isolated threephase PWM rectifier system
A novel three-phase DC/DC converter
for high-power applications
Concept of 50kW DC/DC converter
based on ZVS, Quasi-ZCS topology and
integrated thermal and electromagnetic
design
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Unidirectional, three-phase,
three full-bridge AC-DC
converter
Unidirectional, three-phase,
buck AC-DC converter
Unidirectional, three-phase,
buck AC-DC converter
Bidirectional, rectifier + DCDC converter
Bidirectional matrix converter
Three-phase,
Unidirectional, boost AC-DC
converter
Unidirectional, resonant, threephase isolation DC-DC
converter
Bidirectional, H-bridge,
resonant DC-DC converter
12 / 32
12 kW
≈1
8.6 %
3 / 15
6 kW
≈1
<5%
4 / 12
2 kW
85 % @ Full load
15 / 25
10 kW
1
< 3%
88.6 % @ 1.2 kW
16 / 16
1.5 kW
≈1
5 / 20
2.5 kW
98 %
6/6
5 kW
-
-
97 %
8/8
50 kW
-
-
90 % @ Full load
Table 3. Comparison of the topologies discussed for the high-frequency solution
31
Power Supplies for Particle Accelerators
2.7. References
[1] Y. Lung-Sheng, L. Mao-Shun, L. Tsorng-Juu, C. Jiann-Fuh, Deptartment of Electronics
Engineering, Nat Cheng Kung University, “Study and implementation of a single-stage
three-phase AC-DC converter”, IEEE 6th International Power Electronics and Motion
Control Conference, July 2009, pp. 683 – 688.
[2] V. Jaikumar, G. Ravi, V. Vijayavelan, M. Kaliamoorthy, RGMCET, Nandyal, A.P.,
“An improved high performance three phase AC-DC boost converter with input power
factor correction”, International Conference on Information and Communication
Technology in Electrical Sciences, December 2008, pp. 221 – 228.
[3] P. Wiedemuth, S. Bontemps, J. Miniböck, “35 kW active rectifier with integrated power
modules”, May 2007.
[4] T. Friedli, J. W. Kolar, “Comprehensive Comparison of Three-Phase AC-AC Matrix
Converter and Voltage DC-Link Back-to-Back Converter Systems”, 2010 International
Power Electronics Conference, June 2010, pp. 2789 – 2798.
[5] D. Domes, W. Hofmann, J. Lutz, “A First Loss Evaluation using a vertical SiC-JFET
and a Conventional Si-IGBT in the Bidirectional Matrix Converter Switch Topology”, 2005
European Conference on Power Electronics and Applications, August 2006, 10 pp. - P.10.
[6] G. S. Konstantinou, V. G. Agelidis, School of Electrical Engineering and
Telecommunications, the University of New South Wales, “Bidirectional Rectifier-Inverter
Multilevel Topology without DC-link Passive Components”, 2010 IEEE Energy Conversion
Congress and Exposition (ECCE), November 2010, pp. 2578 – 2583.
[7] J. Schönberger, T. Friedli, S. D. Round, J. W. Kolar, ETH Zurich, Power Electronic
Systems Laboratory, “An Ultra Sparse Matrix Converter with a Novel Active Clamp
Circuit”, Power Conversion Conference, June 2007, pp. 784 – 791.
[8] T. Friedli, M. L. Heldwein, F. Giezendanner, J. W. Kolar, ETH Zurich, Power Electronic
Systems Laboratory, “A High Efficiency Indirect Matrix Converter Utilizing RB-IGBTs”,
37th IEEE Power Electronics Specialists Conference, June 2006, pp. 1 – 7.
[9] F. J. Mendes de Seixas, I. Barbi, Sao Paulo State University, “A 12 kW three-phase low
THD rectifier with high-frequency isolation and regulated DC output”, IEEE Transactions
on Power Electronics, March 2004, vol. 19, pp. 371 – 377.
32
Power Supplies for Particle Accelerators
[10] D. S. Greff, I. Barbi, Power Electronics Institute, Federal University of Santa Caratina,
“A Single-Stage High-Frequency Isolated Three-Phase AC/DC Converter”, 32nd Annual
Conference on IEEE Industrial Electronics, November 2006, pp. 2648 – 2653.
[11] D. Wijeratne, G. Moschopoulos, Department of Electrical and Computer Engineering,
University of Western Ontario, “A Three-Phase Ac-Dc Rectifier with Reduced Switch
Count”, 32nd International Telecommunications Energy Conference (INTELEC), July
2010, pp. 1 – 8.
[12] J. F. Zhao, J. G. Jiang, X. W. Yang, Deptartment of Electronics Engineering, Shanghai
Jiao Tong University, “AC-DC-DC isolated converter with bidirectional power flow
capability”, IET Power Electronics, July 2010, vol. 3, pp. 472 – 479.
[13] R. Garcia-Gil, J. M. Espi-Huerta, R. de la Calle, E. Maset, J. Castelló, “Four-Quadrant
AC-DC Matrix Converter with High-Frequency Isolation”, Second International Conference
on Power Electronics, Machines and Drives, November 2004, vol. 2, pp. 515 – 520.
[14] J. W. Kolar, U. Drofenik, F. C. Zach, Deptartment of Electrical Machines & Drives,
Technical University of Wien “VIENNA rectifier II-a novel single-stage high-frequency
isolated three-phase PWM rectifier system”, Thirteenth Annual Applied Power Electronics
Conference and Exposition, February 1998, vol. 1, pp. 23 – 33.
[15] J. Jacobs, A. Averberg, R. De Doncker, RWTH Aachen, “A novel three-phase DC/DC
converter for high-power applications”, 35th Annual Power Electronics Specialists
Conference, June 2004, vol. 3, 1861 – 1867.
[16] M. Pavlovsky, S. W. H. de Haan, J. A. Ferreira, Deptartment of Electronic Power
Processes, Delft University of Technology, “Concept of 50kW DC/DC converter based on
ZVS, Quasi-ZCS topology and integrated thermal and electromagnetic design”, 2005
European Conference on Power Electronics and Applications, August 2006, pp. – 9.
[17] A. Marzouki, M. Hamouda, F. Fnaiech, SICISI Unit, ESSTT, “Sensorless nonlinear
control for a three-phase PWM AC-DC converter”, 2010 IEEE International Symposium on
Industrial Electronics (ISIE), July 2010, pp. 1052 – 1057.
[18] A. Roman-Loera, L. A. de J. Flores, F. Rizo-Diaz, L.E. Arambula-Miranda,
Universidad Autónoma de Aguascalientes, “Current sensorless predictive algorithm control
for three-phase power factor correction”, 34th Annual Conference of IEEE Industrial
Electronics, November 2008, pp. 653 – 658.
[19] S. Bhowmik, R. Spee, G. C. Alexander, J. H. R. Enslin, Deptartment of Electronics &
Computer Engineering, Oregon State University, “New simplified control algorithm for
synchronous rectifiers”, 21st International Conference on Industrial Electronics, Control,
and Instrumentation, November 1995, vol. 1, pp. 494 – 499.
33
Power Supplies for Particle Accelerators
Design of the high-frequency isolated
ac-dc converter
3.1. System overview
According to the conclusions stated in the previous chapter, the best option to implement
the high-frequency AC/DC converter is the solution presented in [1]. The converter, shown
in Fig. 1, is composed by a three-phase unidirectional buck rectifier and a forward-flyback
output stage, including the high-frequency galvanic isolation. The three-phase rectifier
provides rectified voltage modulated in high-frequency to the primary winding of the
transformer. The secondary winding ns is part of the forward subconverter and is
responsible for the greatest amount of power transferred to the load. The secondary winding
nd is part of the flyback subconverter and it provides a way for the demagnetizing power
through the load. Sine pulse width modulation technique is recommended in [1] by the
authors as the best way to provide the control signals to the switches of the rectifier. An
input filter is necessary to perform unity power factor and to reduce total harmonic
distortion in the currents from the mains.
Fig. 3. High-frequency isolated three-phase AC-DC converter [1].
The forward-flyback stage was presented in [2] and consists on a forward converter which
allows the transformer to demagnetize through the load working on continuous conduction
mode and a flyback converter working on discontinuous conduction mode. A simplified
equivalent circuit of the rectifier and forward-flyback stage is proposed in [1] and it is
shown in Fig. 2.
34
Power Supplies for Particle Accelerators
Fig. 4. Equivalent circuit of the rectifier and forward-flyback stage [1].
The operating stages of the forward-flyback converter are shown in Fig. 3 and described
below:
Fig. 5. Operating states of the forward-flyback converter [1].
a) The switch is closed in t0 and the core of the transformer is magnetized.
Simultaneously, power is transferred from the source E to the load through the
secondary winding ns.
35
Power Supplies for Particle Accelerators
b) The switch is opened in t1 and the current stored in Lo flows through the freewheeling diode DRL. The transformer’s core demagnetizes through the secondary
winding nd and flows through the load.
c) The switch is still opened and the core has demagnetized completely in t2, but the
current stored in Lo is still flowing through the load and the free-wheeling diode for
the rest of the period.
3.2. Forward- flyback converter
In [3], a DC-DC equivalent topology of a three-phase isolated rectifier is presented. This
circuit is shown in Fig. 4 and although it is a dual converter (buck+boost), it can be used as
the model for the three-phase buck converter.
Fig. 6. Equivalent dc-dc model of three-phase isolated buck+boost rectifier [3].
In order to explain the calculations and some of the most important current and voltage
waveforms, a simulation of this circuit has been run on PLECS. In Fig. 5, the simplified
circuit is depicted.
+
vaux
-
Fig. 7. Equivalent circuit for the design of the forward-flyback stage.
The simulations results presented in [1] were obtained using the following values of the
circuit components:
36
Power Supplies for Particle Accelerators
̂
𝑉
𝑁 = 380 𝑉
𝐿𝑆 = 175 µ𝐻
𝐶𝑓 = 23 µ𝐹
𝐿𝑚 = 3.3 𝑚𝐻
𝐿𝑜 = 130 µ𝐻
𝐶𝑜 = 3000 µ𝐹
𝑅𝑜 = 0.6 Ω
𝑁𝑝 = 28 𝑁𝑠 = 12 𝑁𝑑 = 3
Using the equations presented in [3] and shown in Fig. 4, the resulting values for the
equivalent DC input voltage and filter components are:
̂
𝑉
𝑁 (𝑒𝑞) = 1.5 ∗ 380 = 570 𝑉
𝐿𝑆 (𝑒𝑞) = 1.5 ∗ 175 𝜇𝐻 = 262.5 𝜇𝐻
𝐶𝑓 (𝑒𝑞) = 1.5 ∗ 23 𝜇𝐹 = 34.5 𝜇𝐹
After running this simulation with a fixed duty cycle of 0.5, a switching frequency of 30
kHz and assuming ideal components, the resulting steady-state waveforms are shown in Fig.
6:
37
Power Supplies for Particle Accelerators
Fig. 8. Voltage and current waveforms obtained from simulation.
In section 1.1 the three operation states of the converter were explained. According to this,
while the switch is closed, the source is supplying power to magnetize the transformer core.
Between t0 and t1, also the load is receiving power from the source through the secondary
winding ns. The primary side of the transformer and therefore the magnetizing inductance
are connected to the output of the input filter in this first state. The mean voltage in the
output of the filter is the same as the voltage source (as the mean voltage in the filter
inductance is zero), so it can be considered that in the first state, the primary side of the
transformer is connected to the voltage source. Consequently, the magnetizing current rises
with a slope which depends on this voltage and the value of the magnetizing inductance.
From this slope, the peak value of this current can be calculated, assuming that it is zero at
the beginning of the period.
𝑣𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = 𝑉𝑁 (𝑒𝑞)
𝑡0 < 𝑡 < 𝑡1
38
Power Supplies for Particle Accelerators
∆𝑖𝑚𝑎𝑔 𝑉𝑁 (𝑒𝑞)
=
∆𝑡
𝐿𝑚
𝑖𝑚𝑎𝑔 (𝑚𝑎𝑥) =
∆𝑖𝑚𝑎𝑔
∆𝑡
∗ (𝑡1 − 𝑡0 ) =
∆𝑖𝑚𝑎𝑔
∆𝑡
∗𝑇∗𝐷 =
𝑉𝑁 (𝑒𝑞)
𝐿𝑚
∗𝑇∗𝐷
(1)
In the second state, when the switch is opened, the transformer core is demagnetized
through the secondary winding nd and the current flowing to the load through the diode Dd.
Assuming that the output voltage Vo keeps constant, the voltage in the primary side of the
transformer can be calculated:
𝑣𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = −𝑉𝑂 ∗
𝑁𝑝
𝑁𝑑
𝑡1 < 𝑡 < 𝑡2
The current flowing in the magnetizing inductance falls until it reaches zero (to assure
discontinuous conduction mode) with a different slope than in the first state.
∆𝑖𝑚𝑎𝑔
𝑁𝑝 1
= −𝑉𝑂 ∗
∗
∆𝑡
𝑁𝑑 𝐿𝑚
From the volt·second balance applied in the primary side of the transformer, the ratio of the
demagnetizing turns per primary turns can be obtained. This ratio assures the
demagnetization of the transformer’s core and the operation of the flyback subconverter in
discontinuous conduction mode (DCM). The interval in which the demagnetization occurs
must be less than or equal to the interval in which the switch is opened.
1 𝑇
∫ 𝑣 𝑑𝑡 = 0
𝑇 0 𝐿𝑚
𝑁
𝑝
𝑉𝑁 (𝑒𝑞) ∗ 𝐷 ∗ 𝑇 = 𝑉𝑂 ∗ 𝑁 ∗ (𝑡2 − 𝑡1 ) → (𝑡2 − 𝑡1 ) =
𝑑
𝑉𝑁 (𝑒𝑞)∗𝐷∗𝑇
𝑁𝑝
𝑁𝑑
𝑉𝑂 ∗
(2)
𝑉𝑁 (𝑒𝑞) ∗ 𝐷 ∗ 𝑇
≤ (1 − 𝐷) ∗ 𝑇
𝑁𝑝
𝑉𝑂 ∗ 𝑁
𝑑
𝑁𝑑
𝑁𝑝
≤
(1−𝐷)
𝐷
𝑉𝑂
𝑁 (𝑒𝑞)
∗𝑉
(3)
From the circuit shown in Fig. 5, more current and voltages are relevant for further
calculations. They are shown in Fig. 7:
39
Power Supplies for Particle Accelerators
v(primary)
500
0
-500
-1000
v(aux)
200
100
0
v(Lo)
100
0
-100
i(Lo)
205
200
195
i(Co)
20
0
8.939
8.939
8.94
8.94
8.94
8.94 × 1e-1
Fig. 9. Voltage and current waveforms obtained from simulation.
Fig. 6
The relation between the input and output voltage of the converter also depends on the turns
ratio of the primary and secondary sides of the transformer. The voltage in the inductance Lo
is defined as:
𝑣𝐿𝑜 = 𝑣𝑎𝑢𝑥 − 𝑣𝑜
(4)
Applying the volts·second balance in this inductance and assuming that Vo is constant in the
whole period, the voltage in the inductance can be defined as:
1 𝑇
∫ 𝑣 𝑑𝑡 = 0 = ⟨𝑣𝐿𝑜 ⟩
𝑇 0 𝐿𝑜
⟨𝑣𝐿𝑜 ⟩ = ⟨𝑣𝑎𝑢𝑥 ⟩ − ⟨𝑣𝑜 ⟩ → ⟨𝑣𝑎𝑢𝑥 ⟩ = 𝑉𝑜
(5)
40
Power Supplies for Particle Accelerators
As can be seen in Fig. 7, voltage vaux is zero when the switch is opened (t1<t<T), as the
current stored in Lo is free-wheeling through DRL to the load. When the switch is closed, the
value of vaux is:
𝑁
𝑣𝑎𝑢𝑥 = 𝑉𝑁 (𝑒𝑞) ∗ 𝑁𝑠
𝑡0 < 𝑡 < 𝑡1
𝑝
(6)
Therefore, the mean value of this voltage can be calculated as:
⟨𝑣𝑎𝑢𝑥 ⟩ =
𝑇
𝑡1
1
1
𝑁𝑠
1
𝑁𝑠
∗ ∫ 𝑣𝑎𝑢𝑥 𝑑𝑡 = ∗ ∫ 𝑉𝑁 (𝑒𝑞) ∗
𝑑𝑡 = ∗ 𝑉𝑁 (𝑒𝑞) ∗
∗𝐷∗𝑇
𝑇 0
𝑇 0
𝑁𝑝
𝑇
𝑁𝑝
Substituting the value obtained in equation (5):
𝑁
(7)
𝑉𝑁 (𝑒𝑞) ∗ 𝑁𝑠 ∗ 𝐷 = 𝑉𝑜
𝑝
𝑁𝑠
𝑁𝑝
=𝑉
𝑉𝑜
1
𝑁 (𝑒𝑞)
∗𝐷
(8)
As stated in equation (4) and assuming that the output voltage V o is constant during all the
period, the instant value of the voltage in the inductance Lo can be written as follows:
𝑣𝐿𝑜 = 𝑣𝑎𝑢𝑥 − 𝑉𝑜
In equation (6) the value of the auxiliary voltage is calculated when the switch is closed.
The rest of the period, this voltage is zero. Consequently, the instant voltage of the
inductance Lo is defined as:
𝑁𝑠
− 𝑉𝑜
𝑁𝑝
𝑡1 < 𝑡 < 𝑇
𝑣𝐿𝑜 = 𝑉𝑁 (𝑒𝑞) ∗
𝑣𝐿𝑜 = −𝑉𝑜
𝑡0 < 𝑡 < 𝑡1
The increment of the current in the inductance is determined by the value of the inductance
and the voltage applied to it. Therefore, it can be written:
∆𝑖𝐿𝑜
∆𝑡
=
𝑖𝐿𝑜 (𝑚𝑎𝑥) −𝑖𝐿𝑜 (𝑚𝑖𝑛)
𝑡1 −𝑡0
∆𝑖𝐿𝑜 = 𝑉𝑁 (𝑒𝑞) ∗
𝑁𝑠
𝑁𝑝
=
𝑁
𝑉𝑁 (𝑒𝑞)∗ 𝑠 −𝑉𝑜
𝑁𝑝
𝐿𝑜
∗ (1 − 𝐷) ∗
𝑁
𝑝
1
𝐿𝑜
1
= 𝑉𝑁 (𝑒𝑞) ∗ 𝑁𝑠 ∗ (1 − 𝐷) ∗ 𝐿
∗𝐷∗𝑇
𝑡0 < 𝑡 < 𝑡1
𝑜
(9)
(10)
As can be observed in Fig. 7, the charging and discharging current of the capacitor Co is the
result of the contributions from the forward and flyback subconverters. Assuming that all
the ripple component of the currents flowing through Dd and Lo is absorbed by the capacitor
Co, the average component of this addition flows through the load and therefore it can be
considered that the output current of the converter is constant:
41
Power Supplies for Particle Accelerators
𝑖𝐶𝑜 = 𝑖𝐷𝑑 + 𝑖𝐿𝑜 − 𝐼𝑜
The output voltage ripple is a function of the output capacitor value. It can be defined as:
∆𝑣𝑜 =
∆𝑄
𝐶𝑜
(11)
Where ΔQ is the variation of charge in the capacitor. As depicted in Fig. 8, this value can be
described as the area of the current iDd + iLo (green line) above the output current Io (red
line).
ΔQ
Fig. 10. Output capacitor current (green), output current (red).
The value of ΔQ is calculated in the next section, adding the areas of to triangles which
compose the whole area.
3.3. Value of ΔQ
To calculate the value of ΔQ is necessary to calculate the area between the green line and
the red line as shown in Fig. 9. This area can be split into two smaller triangular areas, A1
and A2. To calculate these two areas is necessary to declare some points in the signal,
defined by a time value and a current value (x-y axis values). It is also necessary to define
the gradient of the signal in the interval t1<t<t2.
Fig. 11. Detailed Output capacitor current (green) and output current (red).
42
Power Supplies for Particle Accelerators
As calculated in equation (2), the interval in which the transformer’s core is demagnetized
can be defined as:
(𝑡2 − 𝑡1 ) =
𝑉𝑁 (𝑒𝑞)∗𝐷∗𝑇
𝑁𝑝
𝑁𝑑
𝑉𝑂 ∗
Substituting the value of the output voltage calculated in equation (7):
(𝑡2 − 𝑡1 ) =
𝑉𝑁 (𝑒𝑞)∗𝐷∗𝑇
𝑁𝑝
𝑁
𝑉𝑁 (𝑒𝑞)∗ 𝑠 ∗𝐷∗
𝑁𝑝
𝑁𝑑
=
𝑇∗𝑁𝑑
𝑁𝑠
(12)
According to the gradient of current calculated in equation (9):
∆𝑖𝐿𝑜
𝑁𝑠
1
= 𝑉𝑁 (𝑒𝑞) ∗
∗ (1 − 𝐷) ∗
∆𝑡
𝑁𝑝
𝐿𝑜
−𝑖𝑏 + 𝑖𝐿𝑜(𝑚𝑎𝑥) = (𝑡2 − 𝑡1 ) ∗ 𝑉𝑁 (𝑒𝑞) ∗
𝑖𝑏 = 𝑖𝐿𝑜(𝑚𝑎𝑥) − (𝑡2 − 𝑡1 ) ∗ 𝑉𝑁 (𝑒𝑞) ∗
𝑁𝑠
1
∗ (1 − 𝐷) ∗
𝑁𝑝
𝐿𝑜
𝑁𝑠
1
∗ (1 − 𝐷) ∗
𝑁𝑝
𝐿𝑜
From the value of ib, the gradient h can be calculated as:
ℎ=
𝑖𝑏 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥)
𝑡2 − 𝑡1
This gradient can be used to calculate the interval tb-t1 as follows:
ℎ=
𝑖𝑏 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥) 𝐼𝑜 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥)
=
𝑡2 − 𝑡1
𝑡𝑏 − 𝑡1
𝑡𝑏 − 𝑡1 =
𝑡𝑏 − 𝑡1 =
(𝐼𝑜 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥) ) ∗ (𝑡2 − 𝑡1 )
𝑖𝑏 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥)
(𝐼𝑜 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥) ) ∗ (𝑡2 − 𝑡1 )
𝑁
1
−(𝑡2 − 𝑡1 ) ∗ 𝑉𝑁 (𝑒𝑞) ∗ 𝑠 ∗ (1 − 𝐷) ∗ − 𝑖𝐷𝑑(𝑚𝑎𝑥)
𝑁𝑝
𝐿𝑜
43
Power Supplies for Particle Accelerators
Using equation (12):
(𝐼𝑜 − 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝑖𝐷𝑑(𝑚𝑎𝑥) ) ∗
𝑡𝑏 − 𝑡1 =
−𝑇 ∗ 𝑉𝑁 (𝑒𝑞) ∗
𝑇 ∗ 𝑁𝑑
𝑁𝑠
𝑁𝑑
1
∗ (1 − 𝐷) ∗ − 𝑖𝐷𝑑(𝑚𝑎𝑥)
𝑁𝑝
𝐿𝑜
The maximum value of the current flowing through the diode Dd can be written as:
𝑁
𝑖𝐷𝑑(𝑚𝑎𝑥) = 𝑖𝑚𝑎𝑔 (𝑚𝑎𝑥) ∗ 𝑁𝑝 =
𝑑
𝑉𝑁 ∗𝐷∗𝑇
𝐿𝑚
𝑁
∗ 𝑁𝑝
𝑑
(13)
The maximum value of the current flowing through the output inductance can be written as
follows:
𝑖𝐿𝑜(𝑚𝑎𝑥) = 𝐼𝐿𝑜 +
∆𝑖𝐿𝑜
2
Using equation (10), where the variation of current in the output inductance while the switch
is closed is calculated, the maximum value of the current is:
𝑉𝑁 (𝑒𝑞) ∗
𝑖𝐿𝑜(𝑚𝑎𝑥) = 𝐼𝐿𝑜 +
𝑁𝑠
1
∗ (1 − 𝐷) ∗ ∗ 𝐷 ∗ 𝑇
𝑁𝑝
𝐿𝑜
2
To calculate the value of ILo it is necessary to apply Kirchhoff’s Law to the circuit:
𝑖𝐿𝑜 = 𝑖𝑜 − 𝑖𝐷𝑑
⟨𝑖𝐿𝑜 ⟩ = ⟨𝑖𝑜 ⟩ − ⟨𝑖𝐷𝑑 ⟩
𝐼𝐿𝑜 = 𝐼𝑜 − 𝐼𝐷𝑑
The average current flowing through diode Dd needs to be calculated. To do it is necessary
to use equation (1) and (12):
𝑇
1
⟨𝑖𝐷𝑑 ⟩ = ∗ ∫ 𝑖𝐷𝑑 𝑑𝑡
𝑇 0
⟨𝑖𝐷𝑑 ⟩ =
⟨𝑖𝐷𝑑 ⟩ =
𝑁𝑝 1
1
∗ (𝑡2 − 𝑡1 ) ∗ 𝑖𝑚𝑎𝑔 (𝑚𝑎𝑥) ∗
∗
𝑇
𝑁𝑑 2
𝑁𝑝 1 1 𝑇 ∗ 𝑉𝑁 (𝑒𝑞) ∗ 𝐷 ∗ 𝑁𝑝
1 𝑇 ∗ 𝑁𝑑 𝑉𝑁 (𝑒𝑞)
∗
∗
∗𝑇∗𝐷∗
∗ = ∗
𝑇
𝑁𝑠
𝐿𝑚
𝑁𝑑 2 2
𝑁𝑠 ∗ 𝐿𝑚
And consequently:
1
2
𝑖𝐿𝑜(𝑚𝑎𝑥) = 𝐼𝑜 − ∗
𝑇∗𝑉𝑁 (𝑒𝑞)∗𝐷∗𝑁𝑝
𝑁𝑠 ∗𝐿𝑚
+
𝑁
1
𝑉𝑁 (𝑒𝑞)∗ 𝑠 ∗(1−𝐷)∗ ∗𝐷∗𝑇
𝑁𝑝
𝐿𝑜
2
(14)
44
Power Supplies for Particle Accelerators
Finally, the value of the interval tb-t1 can be rewritten:
𝑡𝑏 − 𝑡1 =
𝐷∗𝑁𝑝
𝑇∗𝑁
𝑁 ∗(1−𝐷)∗𝐷 𝐷∗𝑁𝑝
(−2∗𝑁 ∗𝐿 + 𝑠𝑁 ∗2∗𝐿 +𝐿 ∗𝑁 )∗ 𝑁 𝑑
𝑠 𝑚
𝑝
𝑜
𝑚 𝑑
𝑠
𝑁𝑑 ∗(1−𝐷) 𝐷∗𝑁𝑝
+
𝑁𝑝 ∗𝐿𝑜
𝐿𝑚 ∗𝑁𝑑
(15)
Now it is necessary to calculate the interval t1-ta, using the gradient ΔiLo:
∆𝑖𝐿𝑜 𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜
=
∆𝑡
𝑡1 − 𝑡𝑎
According to the current gradient calculated in equation (9):
𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜
∆𝑖𝐿𝑜
𝑁𝑠
1
= 𝑉𝑁 (𝑒𝑞) ∗
∗ (1 − 𝐷) ∗
=
∆𝑡
𝑁𝑝
𝐿𝑜
𝑡1 − 𝑡𝑎
𝑡1 − 𝑡𝑎 =
𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜
𝑁
1
𝑉𝑁 (𝑒𝑞) ∗ 𝑠 ∗ (1 − 𝐷) ∗
𝑁𝑝
𝐿𝑜
Using equation (14) and simplifying:
𝑡1 − 𝑡𝑎 =
𝑇∗𝐷∗𝑁𝑝 𝑁 ∗(1−𝐷)∗𝐷∗𝑇
−2∗𝑁 ∗𝐿 + 𝑠 𝑁 ∗2∗𝐿
𝑠 𝑚
𝑝
𝑜
(16)
𝑁𝑠 ∗(1−𝐷)
𝑁𝑝 ∗𝐿𝑜
Now, to calculate the total area, it is necessary to calculate the areas of the two triangles as
depicted in Fig. 9:
∆𝑄 = 𝐴1 + 𝐴2 =
(𝑡1 − 𝑡𝑎 ) ∗ (𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜 ) (𝑡𝑏 − 𝑡1 ) ∗ (𝑖𝐿𝑜(𝑚𝑎𝑥) + 𝐼𝐷𝑑(𝑚𝑎𝑥) − 𝐼𝑜 )
+
2
2
Applying the values calculated in equations (13), (14), (15) and (16) results:
𝑡1 − 𝑡𝑎 =
𝑡𝑏 − 𝑡1 =
𝑇 ∗ 𝐷 ∗ 𝑁𝑝 𝑁 ∗ (1 − 𝐷) ∗ 𝐷 ∗ 𝑇
−2 ∗ 𝑁 ∗ 𝐿 + 𝑠 𝑁 ∗ 2 ∗ 𝐿
𝑠
𝑚
𝑝
𝑜
𝑁𝑠 ∗ (1 − 𝐷)
𝑁𝑝 ∗ 𝐿𝑜
𝐷 ∗ 𝑁𝑝
𝑁 ∗ (1 − 𝐷) ∗ 𝐷 𝐷 ∗ 𝑁𝑝
𝑇∗𝑁
(− 2 ∗ 𝑁 ∗ 𝐿 + 𝑠 𝑁 ∗ 2 ∗ 𝐿
+𝐿 ∗𝑁 )∗ 𝑁 𝑑
𝑠
𝑚
𝑝
𝑜
𝑚
𝑑
𝑠
𝑁𝑑 ∗ (1 − 𝐷) 𝐷 ∗ 𝑁𝑝
𝑁𝑝 ∗ 𝐿𝑜 + 𝐿𝑚 ∗ 𝑁𝑑
𝑁
1
𝑉𝑁 ∗ 𝑁𝑠 ∗ (1 − 𝐷) ∗ 𝐿 ∗ 𝐷 ∗ 𝑇
1 𝑇 ∗ 𝑉𝑁 (𝑒𝑞) ∗ 𝐷 ∗ 𝑁𝑝
𝑝
𝑜
𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜 = − ∗
+
2
𝑁𝑠 ∗ 𝐿𝑚
2
45
Power Supplies for Particle Accelerators
𝑖𝐿𝑜(𝑚𝑎𝑥) + 𝐼𝐷𝑑(𝑚𝑎𝑥) − 𝐼𝑜 = −
𝑇 ∗ 𝑉𝑁 ∗ 𝐷 ∗ 𝑁𝑝 𝑉𝑁 ∗ 𝑁𝑠 ∗ (1 − 𝐷) ∗ 𝐷 ∗ 𝑇 𝑉𝑁 ∗ 𝐷 ∗ 𝑇 ∗ 𝑁𝑝
+
+
2 ∗ 𝑁𝑠 ∗ 𝐿𝑚
2 ∗ 𝐿𝑜 ∗ 𝑁𝑝
𝐿𝑚 ∗ 𝑁𝑑
3.4. Calculations
To calculate the values of the elements of the circuit, it is necessary first to define the
characteristics of the output voltage and current, as described in Table 1. The output power
of the converter is expected to be enough to allow the next stage of the power supply (DCDC converter) to deliver the appropriate power to the magnet load. Some parameters (such
as switching frequency or magnetizing inductance) have been chosen according to the
proposed values in [1]:
PARAMETER
VALUE
Vn(eq)
380 * 1.5 = 570 V
Fs
30 kHz
Dmax
50 %
Dmin
30 %
Vo
100 V
Ro
0.7 Ω
Lm
3.3 mH
Δvo/<vo>
< 1%
ΔiLo/<io>
< 10%
Table 2. Parameters and characteristics of the forward-flyback stage.
From equation (3), the turns ratio between nd and np can be calculated for the maximum
value of the duty cycle, in order to demagnetize the transformer’s core completely:
Nd (1 − D)
VO
≤
∗
Np
D
VN (eq)
Nd
≤ 0.1754
Np
The values proposed in [1] match this requirement:
Nd = 3
Np = 28
Equation (8) gives the turns ratio of ns and np from the input and output voltage ratio,
assuming a minimal duty cycle of 30 %:
𝑁𝑠 = 𝑁𝑝 ∗
𝑉𝑜 1
∗
𝑉𝑁 𝐷
46
Power Supplies for Particle Accelerators
𝑁𝑠 = 𝑁𝑝 ∗
𝑉𝑜 1
∗
𝑉𝑁 𝐷
𝑁𝑠 = 16
Now, to calculate the value of the output inductance Lo, it is necessary to use the equation
(10):
∆𝑖𝐿𝑜 = 𝑉𝑁 (𝑒𝑞) ∗
𝑁𝑠
1
∗ (1 − 𝐷) ∗ ∗ 𝐷 ∗ 𝑇
𝑁𝑝
𝐿𝑜
As defined in the requirements:
∆𝑖𝐿𝑜
≤ 10%
𝐼𝑜
∆𝑖𝐿𝑜 ∆𝑖𝐿𝑜 ∗ 𝑅𝑜
𝑁𝑠
1
𝑅𝑜
=
= 𝑉𝑁 ∗
∗ (1 − 𝐷) ∗ ∗ 𝐷 ∗ 𝑇 ∗
≤ 10%
𝐼𝑜
𝑉𝑜
𝑁𝑝
𝐿𝑜
𝑉𝑜
For the maximum value of the duty cycle, the inductance Lo is:
𝐿𝑜 ≥
1
𝑁𝑠
𝑅𝑜
∗ 𝑉𝑁 ∗
∗ (1 − 𝐷) ∗ 𝐷 ∗ 𝑇 ∗
0.1
𝑁𝑝
𝑉𝑜
𝐿𝑜 = 200 µ𝐻
Finally, to obtain the value of the output capacitor C o, the value of ΔQ must be calculated as
explained in section 1.3. Substituting the values of the components and requirements in the
equations, the variation of charge in the capacitor has the value:
𝑡1 − 𝑡𝑎 = 11.42 µ𝑠
𝑡𝑏 − 𝑡1 = 6.09 µ𝑠
𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜 = 4.2668 𝐴
𝑖𝐿𝑜(𝑚𝑎𝑥) + 𝐼𝐷𝑑(𝑚𝑎𝑥) − 𝐼𝑜 = 31.1355 𝐴
∆𝑄 =
(𝑡1 − 𝑡𝑎 ) ∗ (𝑖𝐿𝑜(𝑚𝑎𝑥) − 𝐼𝑜 ) (𝑡𝑏 − 𝑡1 ) ∗ (𝑖𝐿𝑜(𝑚𝑎𝑥) + 𝐼𝐷𝑑(𝑚𝑎𝑥) − 𝐼𝑜 )
+
= 119.79 µ𝐶
2
2
And finally, according to equation (11), the value of the output capacitor can be calculated
as:
∆𝑣𝑜
∆𝑄
=
≤ 1%
𝑉𝑜
𝑉𝑜 ∗ 𝐶𝑜
𝐶𝑜 ≥
∆𝑄
= 119.79 µ𝐹
𝑉𝑜 ∗ 0.01
𝐶𝑜 = 130 µ𝐹
47
Power Supplies for Particle Accelerators
To sum up, the chosen values for the components of the converter are:
Nd = 3
Np = 28
𝑁𝑠 = 16
𝐿𝑜 = 200 µ𝐻
𝐶𝑜 = 130 µ𝐹
3.5. Filter design
PWM rectifiers perform a good elimination of low order harmonics in the currents taken
from the mains. This advantage allows reducing the size and cost of input filters in the
rectifier. By using the PWM technique, the unwanted harmonics components are moved to a
higher frequency, resulting in a dead band where there are no unwanted harmonics.
However, some harmonics can appear in this dead band due to imperfections in the PWM
pattern or parasitic characteristics of the components. Special attention must be paid to this
dead band, as the resonance band of the filter could fit into it and result in amplification of
the residual harmonics. In this converter, a low-pass LC filter is included to satisfy the
requirements of low THD in the currents and in [4], a systematic approach to input filter
design is presented.
After running an open-loop simulation with the values calculated in the previous section and
without input filter for the rectifier (Vo = 100 V), the Fourier spectrum of the three-phase
input currents (taken from the mains) has been obtained. The result is shown in Fig. 9.
Fig. 12. Fourier spectrum of the input currents without using input filter.
It can be observed from the frequency spectrum that there are harmonics components
(multiples of the switching frequency) that need to be filtered. The largest undesired
component is located in 2*fsw (60 kHz). As explained in [4], the equivalent diagram for a
single phase in a current source rectifier is the one shown in Fig. 11.
48
Power Supplies for Particle Accelerators
Fig. 13. Equivalent circuit for a single phase in a current source rectifier.
Where Rs and Ls are the parasitic resistance and inductance of the mains line, Cf and Rf are
the filtering components Vac is the mains voltage, Iline is the current taken from the mains
and I1 is the fundamental component of the current demanded by the rectifier. The transfer
function (for every harmonic component of the spectrum) of this circuit is:
1 + 𝑅𝑓 ∗ 𝐶𝑓 ∗ 𝑠
𝐼𝑙𝑖𝑛𝑒,ℎ
(𝑠) =
𝐼𝑟𝑒𝑐𝑡,ℎ
𝐿𝑠 ∗ 𝐶𝑓 ∗ 𝑠 2 + (𝑅𝑓 + 𝑅𝑠 ) ∗ 𝐶𝑓 ∗ 𝑠 + 1
First of all, it is necessary to know the parameter α, the attenuation of the filter at the
switching frequency. As observed in Fig. 8, the amplitude of the current component at 30
kHz is 5.8 A. The desired amplitude of this component in the line current is 0.05 A.
𝛼=
𝐼𝑙𝑖𝑛𝑒,𝑓𝑠𝑤 0.05
=
= 0.086
𝐼𝑟𝑒𝑐𝑡,𝑓𝑠𝑤
5.8
The break frequency of the filter is given by:
𝑓𝑏 = √𝛼 ∗ 𝑓𝑠𝑤 = 2785 𝐻𝑧
In [4], different ways to calculate the values of the filter components are presented. In this
case, the constraint used to calculate the values is the displacement factor. This factor is the
angle between the line current and voltage phasors. The power factor can be defined as:
𝑃𝐹𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑃𝐹𝑑𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛 ∗ cos 𝜃
Being θ the displacement angle. Although the distortion power factor is increased by using
the input filter, the displacement power factor is reduced, resulting in a decrease of the
overall power factor. For that reason, is necessary to limit the displacement angle in the
filter. It can be expressed as:
𝜃 = tan−1
𝑉𝐶
𝐼 ∗ 𝑋𝐿
− tan−1
𝑋
𝐼 ∗ 𝑋𝐶
𝑉𝐶 ∗ (1 − 𝑋𝐿 )
𝐶
Where Vc is the input voltage of the rectifier, I is the input current of the rectifier and X L,
XC are (assuming that the filter capacitor has no parasitic resistance):
𝑋𝐶 =
1
𝜔𝐶𝑓
𝑋𝐿 = √𝐿2𝑠 𝜔 2 + 𝑅𝑠2
49
Power Supplies for Particle Accelerators
The relation between the inductance value and the capacitance value is given by the break
frequency:
𝐿𝑠 =
1
2
𝐶𝑓 ∗𝜔𝑏
(17)
Substituting this value and assuming that the desired displacement factor is zero, the
equations can be operated and grouped in a sixth order polynomial:
𝐶𝑓6 𝜔𝑏2 𝑅𝑠2 𝑎2 𝜔2 + 𝐶𝑓4 (𝑎2 𝜔6 + 2𝜔𝑏4 𝑅𝑠2 𝑎𝜔2 ) − 𝐶𝑓3 𝜔𝑏4 𝑎𝜔 + 𝐶𝑓2 (2𝑎𝜔4 + 𝜔𝑏4 𝑅𝑠2 ) + 𝜔2 = 0
𝑎=
𝑉𝑐2
𝐼2
Now, substituting the following values of the parameters:
𝜔𝑏 = 2𝜋 ∗ 2785 𝑟𝑎𝑑/𝑠
𝑅𝑠 = 0.001 Ω
𝑉𝑐 = 380 𝑉
𝐼 = 32 𝐴
𝜔 = 2𝜋 ∗ 50 𝑟𝑎𝑑/𝑠
The six roots of the polynomial are:
−0.0306 ± 0.0530𝑗
−1.4372 ∗ 10−6 ± 2.4897 ∗ 10−6 𝑗
𝐶𝑓 = (
)
2.8751 ∗ 10−6
0.0611
Between the two real roots, the chosen value for the filter capacitor is the smallest. The
value of the filter inductance can be calculated using equation (17):
𝐶𝑓 = 3 𝜇𝐹
𝐿𝑠 = 1.1 𝑚𝐻
As it can be seen in Fig. 12, if the simulation is run in the same conditions as before but
including the filter (Vo = 100 V), in the frequency spectrum of the currents taken from the
mains the undesired harmonic components depicted in Fig. 10 have been eliminated. The
fundamental component is not affected by the filter, and the current THD is improved.
Fig. 14. Fourier spectrum of the input currents using input filter.
50
Power Supplies for Particle Accelerators
3.6. System modeling and control
The complete control diagram used in this converter is shown in Fig. 13. It includes the
controllers, the decoupling of the dq current components and the PWM pattern generator.
Fig. 15. Complete control loop block diagram [1].
As explained in [1], the control and modeling techniques are based on the dqo coordinate
system (Park’s Transform) due to its simplicity when working with three-phase signals. To
control the output voltage and the dq components of the current taken from the mains it is
necessary to implement two control loops. The internal loop will be used to control the
current and the external loop will be used to control the output voltage through the current
loop. This topology is depicted in Fig. 14:
𝑖𝑞(𝑟𝑒𝑓) (𝑠) = 0
𝑣𝑜(𝑟𝑒𝑓) (𝑠)
+
Cv(s)
𝑖𝑑(𝑟𝑒𝑓) (𝑠)
&
+
Ci(s)
Decoupling &
PWM
generator
𝑚𝑑 (𝑠)
Gi(s)
𝑖𝑑 (𝑠)
𝑖𝑞 (𝑠)
Gv(s)
𝑚𝑞 (𝑠)
Fig. 16. Detailed current and voltage loops.
51
𝑣𝑜 (𝑠)
Power Supplies for Particle Accelerators
According to [1], the transfer function Gi(s) is the same for the d and q components of the
current so it is not necessary to implement two different controllers for the current loop. The
reference value of the q component of the current must be set to zero, as it is a requirement
to keep reactive power as low as possible (to keep power factor as high as possible).
In this paper, using the fundamental concepts of small-signal modeling, linearization and
decoupling, in addition to some algebraic manipulations, the small-signal transfer function
of the current components with respect to the modulation signals has been obtained:
𝐺𝑖 (𝑠) =
𝑖̃𝑞 (𝑠)
𝑖̃𝑑 (𝑠)
𝐼′𝑜
=
= 2
𝑚
̃𝑑 (𝑠) 𝑚
̃𝑞 (𝑠) 𝑠
𝜔𝑐2 − 𝜔 2
𝑅𝑠 ∗ 𝑠
+
+
𝜔𝑐2 𝐿𝑠 ∗ 𝜔𝑐2
𝜔𝑐2
Where:
𝐼′𝑜 = 𝐼𝑜 ∗
𝑁𝑠
= 81.63 𝐴
𝑁𝑝
𝜔𝑐 = 2 ∗ 𝜋 ∗ 𝑓𝑐 = 17500 𝑟𝑎𝑑/𝑠
However, as can be noticed in the transfer function, the gain depends on the output current
Io. And the output current of the converter depends on the value of the dq components of
the input current, so the characteristics of the system depend on the input values.
According to this, the transfer function is:
𝐺𝑖 (𝑠) =
𝑖̃𝑞 (𝑠)
𝑖̃𝑑 (𝑠)
81.63
=
=
−9
2
𝑚
̃𝑑 (𝑠) 𝑚
̃𝑞 (𝑠) 3.3 ∗ 10 ∗ 𝑠 + 3 ∗ 10−9 ∗ 𝑠 + 0.9997
Fig. 15 shows the step response of this transfer function. As it can be observed, the response
is oscillatory and the settling time very long.
Step Response
160
140
120
Amplitude
100
80
60
40
20
0
0
2
4
6
8
10
12
Time (sec)
Fig. 17. Open-loop impulse response of the current system.
The poles of this system are located in −0.45455 ± 17405j as shown in Fig. 16.
52
Power Supplies for Particle Accelerators
Fig. 18. Pole-zero diagram of the current system.
The compensator chosen to control the current components in a closed-loop control system
will be a PI controller. The integral action is necessary to eliminate the steady-state error of
the current components with respect to their references. Derivative action is considered not
necessary as the noise and ripple of the measured values of the components could result on
oscillations and instability. The transfer function of the continuous PI controller is:
𝐾𝑖
𝐶𝑖 (𝑠) = 𝐾𝑝 + = 𝐾𝑝
𝑠
𝑠+
𝐾𝑖
𝐾𝑝
𝑠
This controller places a pole in the origin and a real zero in −
Ki
Kp
. To place the zero, the
Matlab root-locus graphical tool has been used. The main goals are the stability of the
closed-loop system and the decrease of the response’s overshoot. By using a PI controller is
not possible to avoid underdamping of the closed-loop system response, as there will be
complex poles in the transfer function. When placing the zero of the PI controller in -1, the
root locus is the one shown in Fig. 17:
53
Power Supplies for Particle Accelerators
Fig. 19. Root locus of the current loop using a PI controller.
By changing the closed-loop gain and observing the impulse response, the best value for the
closed-loop poles can be obtained and therefore the best value for the gain. The poles of the
system are located in:
−0.3
−0.3 ± 20800𝑗
The loop gain is set to 0.005. The closed-loop impulse response of the system is shown in
Fig. 18:
Fig. 20. Current loop impulse response.
54
Power Supplies for Particle Accelerators
The settling time (time until the response reaches 95% of the final value) is 10.2 seconds.
The response is slow due to the characteristic of the open-loop system, but the overshoot is
eliminated, as well as the steady-state error. The transfer function of the PI controller is:
𝐶𝑖 (𝑠) = 0.005
𝑠+1
𝑠
And therefore, the current loop transfer function is:
𝑀𝑖 (𝑠) =
𝐶𝑖 (𝑠) ∗ 𝐺𝑖 (𝑠)
0.4082 𝑠 + 0.4082
=
−9
3
1 + 𝐶𝑖 (𝑠) ∗ 𝐺𝑖 (𝑠) 3.3 ∗ 10 𝑠 + 310−9 𝑠 2 + 1.408 𝑠 + 0.4082
Now it’s time to calculate the controller for the external voltage loop. For the same reasons
as for the current loop, the chosen controller is a PI. The external voltage loop should be
slower than the internal current loop, to give time to the current for reaching the current
reference commanded by the voltage loop. As explained in [1], the transfer function for the
output voltage with respect to the d component of the current is as follows:
𝑠2
𝑅 ∗𝑠
𝜔2 − 𝜔2
+ 𝑠 2+ 𝑐 2
2
𝑣
̃(𝑠)
𝑁𝑠 3 𝑉𝑁 𝜔𝑐 𝐿𝑠 ∗ 𝜔𝑐
𝜔𝑐
𝑜
√ ∗
𝐺𝑣 (𝑠) =
=
∗
𝐿′
𝑖̃𝑑 (𝑠) 𝑁𝑝 2 𝐼′𝑜
𝐿′𝑜 ∗ 𝐶′𝑜 ∗ 𝑠 2 + 𝑜 ∗ 𝑠 + 1
𝑅′𝑜
Where:
𝐼′𝑜 = 𝐼𝑜 ∗
𝑁𝑠
= 81.63 𝐴
𝑁𝑝
𝑁𝑝 2
𝐿′𝑜 = 𝐿𝑜 ∗ ( ) = 612.5 𝜇𝐻
𝑁𝑠
𝜔𝑐 = 2 ∗ 𝜋 ∗ 𝑓𝑐 = 17500
𝑟𝑎𝑑
𝑠
𝑉𝑁 = 380 𝑉
2
𝑁𝑝 2
𝑁𝑠
𝑅′𝑜 = 𝑅𝑜 ∗ ( ) = 2.1 Ω 𝐶′𝑜 = 𝐶𝑜 ∗ ( ) = 42.45 𝜇𝐹
𝑁𝑠
𝑁𝑝
Consequently, the transfer function is:
𝐺𝑣 (𝑠) =
𝑣
̃(𝑠)
1.075 ∗ 10−8 𝑠 2 + 9.773 ∗ 10−9 𝑠 + 3.257
𝑜
=
𝑖̃𝑑 (𝑠)
2.6 ∗ 10−8 𝑠 2 + 0.0002857 𝑠 + 1
As was shown in Fig. 14, the voltage controller must provide the d component of the current
reference to the current loop. The output of the current loop is the input of the voltage
55
Power Supplies for Particle Accelerators
system Gv(s). Therefore, the system controlled by the voltage controller is the current loop
in series with Gv(s):
𝑀𝑖 (𝑠) ∗ 𝐺𝑣 (𝑠) =
𝑣
̃(𝑠)
𝑜
=
𝑖𝑑(𝑟𝑒𝑓)
̃ (𝑠)
4.388 ∗ 10−9 𝑠 3 + 8.377 ∗ 10−9 𝑠 2 + 1.329 𝑠 + 1.329
=
8.58 ∗ 10−17 𝑠 5 + 9.429 ∗ 10−13 𝑠 4 + 3.99 ∗ 10−8 𝑠 3 + 0.0004023 𝑠^2 + 1.408 𝑠 + 0.4082
The location of the poles and zeros of this transfer function is shown in Fig. 19:
Fig. 21. Root locus of the current loop and the voltage system.
As explained before in the case of the current controller, the controller will be a PI and the
technique used to tune it is the Matlab root-locus graphical tool. Once again, the main goals
are the stability of the voltage loop and the settling time. The external loop must be slower
than the internal loop, so the settling time of the voltage loop must be larger than the settling
time of the current loop. By placing the zero of the PI controller in -1 and changing the loop
gain in the Matlab tool, the desired impulse response can be obtained. In Fig. 20, the
impulse response of the closed loop system can be observed. For the chosen loop gain, the
poles of the system are:
−0.118
−0.215
−5.07 ± 20700𝑗
−5490 ± 3050𝑗
56
Power Supplies for Particle Accelerators
Fig. 22. Voltage loop impulse response.
The settling time of the response is 31 seconds. The PI controller calculated for the voltage
loop is:
𝐶𝑣 (𝑠) = 0.0276
𝑠+1
𝑠
3.7. Decoupling
As explained in section 1.6, in [1] the equations for modeling the converter are obtained.
The equations which relate modulation signals md(t) and mq(t) with input currents id(t) and
iq(t) are:
𝑑𝑖𝑞 (𝑡)
𝑑𝑖𝑑 (𝑡)
𝑑 2 𝑖𝑑 (𝑡)
− 𝜔𝑅𝑠 𝐶𝑓 𝑖𝑞 (𝑡) + 𝐿𝑠 𝐶𝑓
−
2𝜔𝐿
𝐶
− 𝜔2 𝐿𝑠 𝐶𝑓 𝑖𝑑 (𝑡 )
𝑠
𝑓
2
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑣𝑑 (𝑡)
= 𝐶𝑓
− 𝜔𝐶𝑓 𝑣𝑞 (𝑡) + 𝑚𝑑 (𝑡)𝐼′𝑜
𝑑𝑡
𝑖𝑑 (𝑡 ) + 𝑅𝑠 𝐶𝑓
𝑑𝑖𝑞 (𝑡)
𝑑 2 𝑖𝑞 (𝑡)
𝑑𝑖𝑑 (𝑡)
+ 𝜔𝑅𝑠 𝐶𝑓 𝑖𝑑 (𝑡) + 𝐿𝑠 𝐶𝑓
+ 2𝜔𝐿𝑠 𝐶𝑓
− 𝜔2 𝐿𝑠 𝐶𝑓 𝑖𝑞 (𝑡 )
2
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑣𝑞 (𝑡)
= 𝐶𝑓
+ 𝜔𝐶𝑓 𝑣𝑑 (𝑡) + 𝑚𝑞 (𝑡)𝐼′𝑜
𝑑𝑡
𝑖𝑞 (𝑡 ) + 𝑅𝑠 𝐶𝑓
Transforming these equations into the Laplace’s domain, assuming zero initial conditions
and assuming that vq(t) = 0 (balanced voltages from the mains) results in:
57
Power Supplies for Particle Accelerators
𝐼𝑑 = 𝐼𝑞
𝜔𝑅𝑠 𝐶𝑓 + 2𝜔𝐿𝑠 𝐶𝑓 𝑠
𝑉𝑑 𝑠
𝑀𝑑 𝐼′𝑜
+
+
1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 2 − 𝜔 2 𝐿𝑠 𝐶𝑓 1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 2 − 𝜔 2 𝐿𝑠 𝐶𝑓 1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 2 −
𝐼𝑞 = −𝐼𝑑
𝜔𝑅𝑠 𝐶𝑓 + 2𝜔𝐿𝑠 𝐶𝑓 𝑠
𝑀𝑞 𝐼′𝑜
𝑉𝑑 𝜔
+
+
2
2
2
2
1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 − 𝜔 𝐿𝑠 𝐶𝑓 1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 − 𝜔 𝐿𝑠 𝐶𝑓 1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 2
The topology of the system is depicted in Fig. 21 and it can be observed that both
components of the input currents are coupled between them and also depend on other
variables of the system.
Md(s)
Id(s)
Gi(s)
G2(s)
G1(s)
G3(s)
G1(s)
Vd(s)
Mq(s)
Iq(s)
Gi(s)
Fig. 21. Coupled topology of the converter’s current system.
Where:
Gi (s) =
𝐼′𝑜
1 + 𝑅𝑠 𝐶𝑓 𝑠 + 𝐿𝑠 𝐶𝑓 𝑠 2 − 𝜔 2 𝐿𝑠 𝐶𝑓
G1 (s) =
ωR s Cf + 2ωLs Cf s
1 + R s Cf s + Ls Cf s2 − ω2 Ls Cf
G2 (s) =
s
1 + R s Cf s + Ls Cf s2 − ω2 Ls Cf
58
Power Supplies for Particle Accelerators
G3 (s) =
ω
1 + R s Cf s + Ls Cf s2 − ω2 Ls Cf
In [5], a method to decouple the two components of the input currents in a current source
PWM rectifier is proposed. It also includes a technique to compensate the power demanded
by the load, but it will not be implemented in this converter. The proposed decoupling
topology is shown in Fig. 22:
Vd(s)
PI(s)
Id(ref)(s)
Md(s)
𝜔𝐿𝑠
Id(s)
Iq(s)
𝜔𝐿𝑠
PI(s)
Iq(ref)(s)
Mq(s)
Fig. 22. Decoupling technique presented in [5].
3.8. PWM pattern generation
There are many suitable techniques to command the switching devices of a three-phase
rectifier. Space vector modulation (SVPWM) is commonly used in this type of converters
and there is a wide range of information about it in literacy. However, in [1] a different
technique is recommended to generate the PWM pattern in the high-frequency isolated
three-phase AC-DC converter. This method is called sinusoidal pulse width modulation
(SPWM) and is based on the comparison of a modulating sinusoidal signal with a triangular
carrier, as depicted in Fig. 23:
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Power Supplies for Particle Accelerators
Fig. 23. Sinusoidal pulse width modulation (SPWM).
As explained in [6], there are some important parameters to define when using this
technique. The modulation index m and the frequency modulation ratio p are:
𝑚=
𝑉𝑀𝑂𝐷
𝑉𝑇𝑅𝐼𝐴𝑁𝐺
𝑝=
𝑓𝑇
𝑓𝑠
Where 𝑉𝑀𝑂𝐷 and 𝑉𝑇𝑅𝐼𝐴𝑁𝐺 are the amplitudes of the modulating signal and the triangular
carrier respectively and 𝑓𝑇 and 𝑓𝑠 are the frequencies of the carrier and the mains supply
respectively. Choosing p as an odd integer results in eliminating subharmonics and even
harmonics. If p is a multiple of 3, the modulation of the three phases will be identical.
Thanks to the input filter of the three-phase rectifier, larger p numbers generate cleaner
currents. Overmodulation occurs when m > 1, and it results on more harmonics appearing in
the input currents.
The three modulating signals are obtained calculating the Inverse Park’s Transform of the
signals Md(t) and Mq(t), result of the decoupling of the PI command signals.
After comparing the triangular carrier and the sinusoidal modulating signals, three PWM
signals are obtained (one for each leg of the rectifier). These two-level signals need to be
converted into appropriate signals to command the switches of the converter. This
conversion is shown in Table 2, as proposed in [1]:
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Power Supplies for Particle Accelerators
ma
+1
+1
+1
-1
-1
-1
Two-level signals
mb
+1
-1
-1
+1
+1
-1
mc
-1
+1
-1
+1
-1
+1
S1
0
1
1
1
1
0
Switches
S2
1
1
0
0
1
1
S3
1
0
1
1
0
1
Table 2. Switching states.
The command signals can be obtained as the XOR logic operation of the three two-level
signals obtained from the comparison of the modulating signal and the triangular carrier
(assuming they are bit logic values 0 and 1):
𝑆1 = 𝑚𝑎 ⊕ 𝑚𝑏
𝑆2 = 𝑚𝑏 ⊕ 𝑚𝑐
𝑆3 = 𝑚𝑎 ⊕ 𝑚𝑐
3.9. References
[1] D. S. Greff, I. Barbi, Power Electronics Institute, Federal University of Santa Caratina,
“A Single-Stage High-Frequency Isolated Three-Phase AC/DC Converter”, 32nd Annual
Conference on IEEE Industrial Electronics, November 2006, pp. 2648 – 2653.
[2] J.N. Park, T.R. Zaloum, “A Dual Mode Forward/Flyback Converter”, IEEE Power
Electronics Specialists Conference, PESC’82 Record, 1982, pp. 3-13.
[3] A. Mohammadpour, M. R. Zolghadri, School of Electronic Engineering, Sharif
University of Technology, “Control of Three-Phase Single-Stage Isolated Buck+Boost
Unity Power Factor Rectifier for Unbalanced Input Voltages”, International Conference on
Electric Power and Energy Conversion Systems, November 2009, pp. 1 - 6.
[4] N. R. Zargari, G. Joos, P. D. Ziogas, Department of Electronics & Computing
Engineering, Concordia University, “Input Filter Design for PWM Current -Source
Rectifiers”, Eigth Annual Applied Power Electronics Conference and Exposition, March
1993, p. 824.
[5] M. Knapczyk, K. Pienkowsky, “High Performance Decoupled Control of PWM
Rectifier with Load Compensation”, 2007.
[6] M. H. Rashid, “Power Electronics Handbook”, Academic Press, 2001.
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Power Supplies for Particle Accelerators
Simulation of the high-frequency
isolated ac-dc converter
4.1. Description
In order to validate the converter designed in chapter 3, a simulation of the circuit and
control topology has been performed. The software used to obtain the most relevant
waveforms has been Matlab-Simulink and PLECS. The simulation has been done in the
continuous domain and it includes all the blocks depicted in Fig. 13 (section 3.6).
To transform three-phase currents and voltages into dq components, Park’s Transform (and
Inverse Park’s Transform) has been used. In this simulation, specialized blocks have been
used and they can be found in PLECS components library.
The solver method chosen for this simulation has been ode23t. In order to improve the
simulation conditions and to avoid unaccurate results, a variable step solver is required.
Current and voltage probes have been placed in the circuit to sample the most relevant
waveforms of the converter, as well as to obtain further parameters (such as load power or
efficiency).
In this chapter, different simulations have been run to test the operation of some parts of the
converter separately. Firstly, the internal current loop has been simulated and after that a
complete simulation of the converter including the voltage loop has been run.
4.2. Current loop
As explained in section 3.6, it is necessary to implement two control loops to control
sepparately the output voltage and the input currents of the converter. In this section, the
internal current loop has been simulated according to the controller designed in section 3.6,
the decoupling method explained in section 3.7 and the PWM pattern generator explained in
section 3.8. The current loop controls the dq components of the input currents of the
converter and given that the dynamic behavior is the same for the two components, the two
current controllers are exactly the same. The reference value of the q component of the input
currents is set to zero to reduce reactive power and obtain a nearly unity power factor. Based
on empirical observations and assuming a value of zero for Iq, a value of 27 in Id results on
an output voltage of 100 V. So the reference value of Id is set to 27. The resulting
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Power Supplies for Particle Accelerators
waveforms of the dq input currents components and the reference values are shown in Fig.
1.
Fig. 1. Reference and measured waveforms of d (pink) and q (green) components.
As can be observed, the impulse response of the current loop when using the controller
calculated in section 3.6 is unstable. The reason of this instability is the imprecise modeling
of the current system explained in [1]. To deal with this problem, it is necessary to retune
the current controller manually. By rising the values of Kp and Ki, the response of the
current loop becomes more stable and faster. When setting the current controller parameters
to:
K p = 0.05
K i = 30
The response is completely stable, as depicted in Fig. 2:
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Power Supplies for Particle Accelerators
Fig. 2. Reference and measured waveforms of d (pink) and q (green) components.
Although the overshoots of Id and Iq are extremely high, the response of the system is stable
and the settling time is 250 ms. The overshoots of the signals shown in Fig. 2 cause
overshoots in the input currents of the converter. As the steady-state error is completely
eliminated, the value of Iq is zero after the transient time, resulting in input currents in phase
with the voltages from the mains (nearly unit displacement factor), as can be seen in Fig. 3:
Fig. 3. Three-phase input voltages and currents.
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Power Supplies for Particle Accelerators
4.3. Voltage loop
Assuming that the current controller is different than tha one calculated in section 3.6 and
that the model of the converter taken from [1] is not accurate enough, it is necessary to tune
the voltage controller manually. The parameters for the voltage controller which make the
system response stable are:
Kp = 1
Ki = 1
Setting the output voltage reference value to 100 V, the input current components (reference
and measured value) are shown in Fig. 4:
Fig. 4. Reference and measured waveforms of d (pink) and q (green) components.
The overshoot in the input currents still occurs when closing the voltage loop. This response
is stable but the high input currents in the transient time are transmitted to the load, as can
be observed in the output voltage shown in Fig. 5:
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Power Supplies for Particle Accelerators
Fig. 5. Output voltage response in closed-loop.
An important parameter to measure the quality of the output voltage is its ripple. As defined
in section 3.4, this ripple is desired to be less than 1%. A detail of the output voltage in
steady-state is shown in Fig. 6 and can be observed that the ripple requirement is met:
Fig. 6. Detail of output voltage ripple.
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Power Supplies for Particle Accelerators
The settling time of the voltage loop (instant time in which the output voltage reaches 95 %
of reference voltage) is 1.8 seconds. As shown in Fig. 4, the dq current components reach
their reference values, resulting in low harmonics and nearly unit power factor in the input
currents, as can be seen in Fig. 7:
Fig. 7. Three-phase input voltages and currents.
4.4. Power factor and THD
The input power factor of the converter evaluates the quality of the power taken from the
mains. Two parameters have a direct influence on the overall power factor: distortion and
displacement factor.
𝑃𝐹𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑃𝐹𝑑𝑖𝑠𝑡𝑜𝑟𝑡𝑖𝑜𝑛 ∗ cos 𝜃
Assuming that input voltages and currents are perfectly sinusoidal (as seen in Fig. 7), the
overall power factor can be calculated from the active and reactive power measured in the
converter as:
𝑃𝐹𝑜𝑣𝑒𝑟𝑎𝑙𝑙 =
𝑃
𝑃
=
𝑆 √𝑃2 + 𝑄 2
Where P, Q and S are the active, reactive and apparent power respectively. Steady-state
input power factor of this converter is shown in Fig. 8. After a transient period, the power
factor reaches a value very close to unity:
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Power Supplies for Particle Accelerators
Fig. 8. Input power factor.
Due to the input filter and the high frequency of the switching pattern, the harmonic
distortion of the input currents is very low. In Fig. 9, the frequency spectrum of the input
currents can be observed and in Fig. 10, a detail of residual harmonics is shown:
Fig. 9. Frequency spectrum of the input currents.
Fig. 10. Detail of the frequency spectrum of the input currents.
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Power Supplies for Particle Accelerators
Using the Total Harmonic Distortion Block in Simulink, the THD of the three-phase input
currents has been calculated as:
THD (%) = 100 ∗ √∑
h≠1
Ih
I1
Where Ih is the amplitude of the h-order harmonic component of the input current and I1 is
the amplitude of the fundamental component. In Fig. 11 can be seen that after the transient
time, the THD in the three current lines is less than 5%:
Fig. 11. Total Harmonic Distortion of the input currents.
4.5. Power and efficiency
When running the simulation in the same conditions as before and with the parameters
described in Chapter 3 (Table 1), the active and reactive power have been measured, as well
as the load power. When reaching the steady-state, the power taken from the mains is:
𝑃 = 15640 𝑊
𝑄 = −150 𝑉𝐴𝑟
The load consumes a total power of 14 kW, as can be seen in the detailed waveform in Fig.
12:
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Power Supplies for Particle Accelerators
Fig. 12. Detailed view of the load power in steady-state.
The efficiency of the converter can be obtained as:
ŋ = 100 ∗
𝑃𝑜
𝑃𝑖𝑛
Where Po is the load power and Pin is the active power taken from the mains. The
efficiency of the converter in the steady-state is depicted in Fig. 13. It varies between 89.8%
and 91.3%.
Fig. 13. Detailed view of the efficiency in steady-state.
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Power Supplies for Particle Accelerators
4.6. Conduction losses
Given that this converter has been simulated assuming an ideal transformer (except for the
magnetizing inductance) and ideal components (without parasitic resistance), the main loss
of effective power is produced in the semiconductors in two ways: switching losses and
conduction losses. The instantaneous value of the IGBT conduction losses p CT(t) and the
diode conduction losses pCD(t) is:
𝑝𝐶𝑇 (𝑡 ) = 𝑉𝐶𝐸 ∗ 𝑖𝑐 (𝑡 ) + 𝑟𝐷𝑇 ∗ 𝑖𝑐 2 (𝑡 )
𝑝𝐶𝐷 (𝑡 ) = 𝑉𝐷 ∗ 𝑖𝐹 (𝑡 ) + 𝑟𝐷𝐷 ∗ 𝑖𝐷 2 (𝑡)
Where VCE is the on-state zero-current collector-emitter voltage, ic(t) is the collector current,
rDD and rDT are the on-resistance of the diode and the IGBT, VD is the voltage across the
diode, iF(t) is the forward current and iD(t) is the direct current. In the simulation, the
following parameters for the semiconductors have been defined:
𝑉𝐶𝐸 = 1 𝑉 𝑟𝐷𝑇 = 0.004 Ω
𝑉𝐷 = 0.6 𝑉 𝑟𝐷𝐷 = 0.04 Ω
Fig. 14 shows the instantaneous conduction losses of the three IGBTs in the steady-state
time:
Fig. 14. Conduction losses of the three IGBTs in the steady-state.
The three mean values of the conduction losses in this steady-state period are 26.15 W,
34.77 W and 34.63 W respectively.
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Power Supplies for Particle Accelerators
4.7. References
[1] D. S. Greff, I. Barbi, Power Electronics Institute, Federal University of Santa Caratina,
“A Single-Stage High-Frequency Isolated Three-Phase AC/DC Converter”, 32nd Annual
Conference on IEEE Industrial Electronics, November 2006, pp. 2648 – 2653.
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Power Supplies for Particle Accelerators
Conclusions
The high-frequency isolated three-phase AC-DC converter is a compact solution to convert
AC voltage from the three-phase mains into a well regulated DC output voltage including
the galvanic isolation. The components which might be a drawback for the compact design
are the inductors and capacitors of the input filter. The reduced number of switches is an
advantage to reduce the size of the converter, as only three switches are necessary.
As explained in section 4.2, the model of the converter used to calculate the current and
voltage controller is not accurate enough, resulting in unstable impulse response of the
converter. One point to improve is the modeling technique in order to obtain a more precise
way to calculate the controllers. By that way, stability could be achieved reducing the
overshoot produced in the input currents (which is transmitted to the output voltage).
Special attention must be paid to overcurrents and overvoltages in the circuit components.
As shown in the simulation results, in the steady-state time of the response the dq
components of the input currents reaches the reference values eliminating the error
completely due to the integral component of the current controller. That also happens to
output voltage, which reaches the reference voltage after 1.8 seconds. As defined in the
design specifications, the ripple of the output voltage is less than 1%, which makes easier
for the next stage of the power supply to regulate the current delivered to the magnet more
precisely.
The appropriate PWM pattern generation and the input filter improve the quality of the
currents taken from the mains. By choosing a high switching frequency, total harmonic
distortion in the currents has been reduced (less than 4% in steady-state time), resulting in a
nearly unity power factor as shown in the simulation results of Chapter 4. This is very
important in order to meet the norm requirements for power supplies.
The efficiency of the converter at full load is above 90%, which is among the highest
efficiencies presented in Chapter 2 for the high-frequency isolated AC-DC converters.
However, this efficiency will be lower when the losses in the transformer are taken into
account. In the simulations presented in Chapter 4, only losses in the semiconductors have
been introduced.
There are different challenges between the further goals of this converter. Improving the
transient response and taking into account the losses in the transformer are some of them. It
is also necessary to model in detail the transformer in order to introduce the power losses in
the simulation and obtain a more realistic value of the efficiency of the converter. Finally,
the design and simulation of the medium-frequency isolated AC-DC converter (as explained
in section 2.6) will be carried out.
73