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Three-phase dual active bridge converters

Citation for published version (APA):Baars, N. H. (2017). Three-phase dual active bridge converters: a multi-level approach for wide voltage-rangeisolated dc-dc conversion in high-power applications. Technische Universiteit Eindhoven.

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Three-PhaseDual Active Bridge Converters

A multi-level approach for wide voltage-rangeisolated dc-dc conversion in high-power applications

N.H. Baars



N. H. Baars

This research has been performed in the ARMEVA (Advanced Reluctance Motors for Elec-tric Vehicle Applications) project. The project received funding from the European Union’sSeventh Framework Programme (FP7) for research, technological development and demon-stration under grant agreement no. 605195.

First edition, published 2017

Copyright © N.H. Baars, 2017. All right reserved. No part of this thesis may be reproducedor distributed in any form or by any means, or stored in a database or retrieval system,without the prior written permission of the author.

A catalogue record is available from the Eindhoven University of Technology Library

ISBN: 978-90-386-4368-7



PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische UniversiteitEindhoven, op gezag van de rector magnificus prof.dr.ir. F.P.T. Baaijens,

voor een commissie aangewezen door het College voor Promoties, in hetopenbaar te verdedigen op maandag 6 november 2017 om 16:00 uur

door

Nicolaas Hermanus Baars

geboren te Kamerik

Dit proefschrift is goedgekeurd door de promotor en de samenstelling van de pro-motiecommissie is als volgt:

voorzitter: prof. dr. ir. J. H. Blom

promotor: prof. dr. E. A. Lomonova MSc

copromotoren: prof. ir. C. G. E. Wijnands

dr. J. Everts MSc (Prodrive Technologies B.V.)

leden: prof. dr. ir. dr. h. c. R. W. De Doncker (RWTH Aachen)

prof. dr. P. C. Kjær MSc (Aalborg University)

dr. ir. H. Huisman

adviseur: dr. ir. B. J. D. Vermulst

Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd inovereenstemming met de TU/e Gedragscode Wetenschapsbeoefening.

Summary

Three-Phase Dual Active Bridge ConvertersA multi-level approach for wide voltage-range

isolated dc-dc conversion in high-power applications

P OWER ELECTRONIC CONVERTERS are essential in the ongoing transition to-wards a more-electric and sustainable society. This includes, for instance,the electrification of transport and renewable electricity generation. These

applications require power electronics to convert electrical energy between differ-ent voltage levels. Additionally, electrical isolation from input to output or a largevoltage conversion ratio may be required, which is traditionally realised by lowfrequency transformers. This, however, limits the power density and the flexibilityof the system. By utilising isolated dc-dc converters, the power density, flexibility,and scalability of the system can be improved. This thesis focusses on a family ofisolated dc-dc converters, considering applications with a wide voltage-range andpower levels in the range of kilowatts up to megawatts.

An attractive isolated dc-dc converter for high-power applications is the three-phase dual active bridge (DAB) converter. It features buck-boost operation, effec-tively utilises the parasitic transformer stray inductance, and requires small passivefilter components due to its interleaved structure. Furthermore, the power flow canbe regulated by a single control variable, while achieving soft-switching and lowcirculating currents for input voltages close to the primary-referred output volt-age. However, for voltage ratios other than unity, the converter partly loses soft-switching and is subjected to large circulating currents, deteriorating the efficiencyand power density.

The main goal of this research lies in the derivation of three-phase DAB convert-ers that achieve soft-switching and low circulating currents for a wide voltage and

v

vi Summary

power range. By introducing a multi-level topology and combining different trans-former winding configurations, a family of three-phase DAB converters is derived.In addition, by applying symmetric and asymmetric operating methods, multiplecontrol variables arise for fixed frequency operation. As a result, the circulatingcurrent can be minimised while complying to soft-switching constraints.

A piecewise-linear (PWL) model of the selected converters is derived to describethe converter currents and power level for steady-state operation. For convert-ers with multiple control variables, a large number of switching modes is present.Therefore, a systematic approach is introduced to identify all switching modes andtheir boundaries. Consequently, the derivation of analytical equations is presentedfor any arbitrary switching mode.

Modulation schemes for the proposed converters are derived that control the powerlevel while low circulating currents and soft-switching are achieved. With the aid ofa numerical optimiser, optimal converter behaviour is analysed and, consequently,approximated by equations describing the control variables for optimised opera-tion. By utilising these equations with the PWL model, closed-form solutions arefound that result in close-to-optimal operation for a wide voltage and power range.

The impact of the transformer connection on the performance of the symmetricallyoperated two-level topology has been investigated for a star-star and star-deltaconnection. The comparison revealed that a converter with a star-delta connectedtransformer features lower component stress for medium to high power levels. Inaddition, a wider soft-switching range, independent of the phase-shift, is found forphase-shifts below π/6. Experimental verification confirms the theoretical analysisand reveals a higher efficiency for the symmetrically operated two-level topologywith a star-delta connected transformer, thereby making this converter attractivefor applications with a limited voltage range.

A comparative evaluation of existing and the proposed converters is carried outfor a wide voltage and power range. This confirms that, for equal conditions, theconverters which have multiple control variables can achieve significantly lowercirculating currents. This result also reflects in the current stress of the componentsand the required total amount of semiconductor chip area. The latter revealed thatthe asymmetrically operated two-level topology requires approximately 25 % lesschip area compared to the conventional symmetrically operated two-level topol-ogy. For an equal amount of total chip area, the symmetrically operated three-leveltopology with a star-star connected transformer has the lowest average total semi-conductor losses.

The operation of two- and three-level topologies with a star-star connectedtransformer and their associated modulation schemes are experimentally verified.

Summary vii

For this purpose a 100 kW prototype converter was designed and built to operatewith a nominal output voltage of 750 V while the input voltage can vary between375 V and 1500 V. The corresponding measurements show good agreement withthe simulated values and thereby confirm the theoretical analysis. The proposedconverters with multiple control variables demonstrate low circulating currents,soft-switching, and high overall efficiency, exceeding 98 % for power levels up to100 kW.

Overall, it has been demonstrated that the proposed two- and three-level topologywith, respectively, asymmetric and symmetric modulation schemes are suitable forhigh-power applications with a wide voltage range. The symmetrically operatedthree-level topology can potentially achieve the highest average efficiency, whereasthe asymmetrically operated two-level topology can be realised at a lower cost dueto the reduced complexity of the phase-leg and busbar structure.

viii Summary

Contents

Summary v

1 Introduction 11.1 Electrical power conversion in transportation . . . . . . . . . . . . . . 2

1.1.1 Battery chargers . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Converters for hybrid energy storage systems . . . . . . . . . 31.1.3 Auxiliary power unit for railway applications . . . . . . . . . 41.1.4 Solid-state transformers . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Isolated dc-dc converters . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 The dual active bridge topology . . . . . . . . . . . . . . . . . 61.2.2 A family of three-phase dual active bridge converters . . . . . 7

1.3 Research goal and objectives . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

I Modelling and Modulation Schemes 13

2 Piecewise-linear modelling 152.1 Symmetrical operation of two-level three-phase DAB converters . . . 16

2.1.1 Star-star connected transformer . . . . . . . . . . . . . . . . . . 172.1.2 Star-delta connected transformer . . . . . . . . . . . . . . . . . 23

2.2 Symmetrical operation of three-level three-phase DAB converters . . 282.2.1 Star-star connected transformer . . . . . . . . . . . . . . . . . . 282.2.2 Star-delta connected transformer . . . . . . . . . . . . . . . . . 39

2.3 Asymmetrical operation of two-level three-phase DAB converters . . 432.3.1 Switching modes . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.2 PWL model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Soft-switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.4.1 Zero voltage switching principle . . . . . . . . . . . . . . . . . 512.4.2 ZVS conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

ix

x Contents

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Modulation schemes 573.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.1 Numerical optimiser . . . . . . . . . . . . . . . . . . . . . . . . 593.1.2 Operating range . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Symmetrical operation of three-level three-phase DAB converters . . 603.2.1 Optimal modulation for a star-star connected transformer . . 603.2.2 Analytical modulation for a star-star connected transformer . 623.2.3 Optimal modulation for a star-delta connected transformer . 723.2.4 Analytical modulation for a star-delta connected transformer 72

3.3 Asymmetrical operation of two-level three-phase DAB converters . . 833.3.1 Optimal modulation schemes . . . . . . . . . . . . . . . . . . . 833.3.2 Analytical modulation scheme . . . . . . . . . . . . . . . . . . 86

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

II Evaluation and Experimental Results 93

4 Impact of transformer connections on the two-level topology 954.1 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1.1 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.1.2 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1.3 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.2.1 Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2.2 Turn-off current . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.2.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Comparative evaluation of three-phase DAB converters 1115.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2 Component current stress comparison . . . . . . . . . . . . . . . . . . 113

5.2.1 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2.2 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.3 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.4 Overview of the component stress comparison . . . . . . . . . 123

5.3 Converter comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3.1 Semiconductor chip area comparison . . . . . . . . . . . . . . 1245.3.2 Semiconductor loss comparison . . . . . . . . . . . . . . . . . 1285.3.3 Capacitance comparison . . . . . . . . . . . . . . . . . . . . . . 129

Contents xi

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Experimental verification 1336.1 High-power dc-dc converter prototype design . . . . . . . . . . . . . 133

6.1.1 Three-level inverter bridge realisation . . . . . . . . . . . . . . 1346.1.2 Dc-link capacitors and busbar design . . . . . . . . . . . . . . 1366.1.3 Three-phase transformer . . . . . . . . . . . . . . . . . . . . . . 1386.1.4 Prototype isolated dc-dc converter . . . . . . . . . . . . . . . . 138

6.2 Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.2.1 Symmetrical operation of the two-level topology . . . . . . . . 1406.2.2 Symmetrical operation of the three-level topology . . . . . . . 1456.2.3 Asymmetrical operation of the two-level topology . . . . . . . 151

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

III Closing 159

7 Conclusions and recommendations 1617.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.2.1 Journal publications . . . . . . . . . . . . . . . . . . . . . . . . 1657.2.2 Conference publications . . . . . . . . . . . . . . . . . . . . . . 165

7.3 Recommendations for future work . . . . . . . . . . . . . . . . . . . . 166

A Nomenclature 169A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.2 Circuit, waveform, and converter designations . . . . . . . . . . . . . 170A.3 Scalar symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170A.4 Vector and matrix symbols . . . . . . . . . . . . . . . . . . . . . . . . . 173A.5 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

B Converter models and analytical modulation schemes 175B.1 Symmetrically operated three-level topology with a star-star con-

nected transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175B.1.1 Switching modes . . . . . . . . . . . . . . . . . . . . . . . . . . 175B.1.2 PWL equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 178B.1.3 Analytical modulation scheme . . . . . . . . . . . . . . . . . . 181

B.2 Asymmetrically operated two-level topology . . . . . . . . . . . . . . 183B.2.1 Switching modes . . . . . . . . . . . . . . . . . . . . . . . . . . 183B.2.2 PWL equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 185B.2.3 Analytical modulation scheme . . . . . . . . . . . . . . . . . . 187

xii Contents

Bibliography 189

Acknowledgement 201

About the author 205

Chapter 1

Introduction

S INCE the beginning of electricity generation, direct and alternating currentsystems are co-existing and, thereby, the need to convert electrical energyfrom one form into another emerged. In the beginning of the 20th century,

mainly alternating current (ac) was generated and distributed economically overlarge distances, while city centres and railway systems relied on direct current (dc)for historical and practical reasons. This was solved by rotary converters [112],which are basically an ac-motor and dc-generator that have a single rotating ar-mature. These converters were placed in sub-stations, located close to residential,railway, and industrial areas, to provide the required direct current [3, 19, 79].

At the same time, mercury-arc rectifiers were developed which could convert alter-nating current into direct current more efficiently and with lower operational coststhan rotary converters [21, 48]. By 1930, the mercury-arc rectifier was regarded asthe best method to convert alternating current into direct current [30]. In addition,the mercury-arc valve was developed, which enabled controllable electronic powerconversion from direct current to alternating current and vice versa. Since then,many types of converters are derived and, the term ”electronic power converter”was proposed in 1944 to indicate all electric circuits which convert electric power[2]. This can be seen as the first definition of the field of power electronics.

In the second half of the 20th century, solid-state rectifiers and switches, based ongermanium and silicon semiconductor material, were introduced [67, 95]. Thesepower semiconductor devices provided faster switching characteristics and a lowervoltage drop and, unlike the mercury-arc valves, could be operated in any orien-tation. This has led to more efficient and compact electronic power converters for

1

2 Chapter 1 Introduction

(a) 1890-1960. (b) 1910-1960. (c) 1960-present. (d) 1980-present. (e) 2010-present.

Figure 1.1: Historical overview of several well-known components essential to electri-cal power conversion and the years of their widespread use, with a rotary converter (a),mercury-arc rectifier (b), thyristor (c), insulated gate bipolar transistor (IGBT) (d), and sili-con carbide (SiC) metal-oxide-semiconductor field-effect transistor (MOSFET) (e).

existing and new applications. Recent developments in semiconductor materialshave led to wide-bandgap power devices which can be operated at higher volt-ages, frequencies, and temperatures compared to silicon-based semiconductor de-vices [18, 78].

A brief historical overview of the above mentioned components, essential to elec-trical power conversion, is displayed in figure 1.1, including the time-frame of theirwidespread use. These developments, together with the progress in microelectron-ics, enable the use of more advanced converter topologies with potentially higherefficiency and flexibility for electronic power conversion.

1.1 Electrical power conversion in transportation

To meet up with the world’s increasing population, transportation systems for bothpeople and goods need to increase their capacity. In addition, due to the Parisagreement [105] to reduce carbon emissions, the fossil-fuel burning internal com-bustion engines will have to be replaced by more efficient electric systems. Mostlypassenger cars, small trucks, and buses will be powered by energy storage systems,such as batteries or fuel-cells, whereas rail vehicles, trolley-buses, and heavy trucks[51, 69] will continue to be powered by an overhead line or a third rail. Regardingthese transport systems, a number of applications are described below to empha-sise the role of isolated dc-dc converters, which are the subject of this work.

1.1 Electrical power conversion in transportation 3

ac

dcbattery

ac-grid LF-transformer

(a)

ac

dcdc

dcbattery

ac-grid

(b)

Figure 1.2: An application of a grid-connected battery charger for electric vehicles, wherethe low-frequency (LF) transformer (a) can be omitted by utilising an isolated dc-dc con-verter (b), thereby reducing the weight and volume while galvanic isolation from the grid ismaintained.

1.1.1 Battery chargers

A typical application of isolated dc-dc converters is the charging of battery elec-trical vehicles (BEV) and plug-in hybrid electrical vehicles (PHEV) from an ac ordc-grid, or from a stationary battery [110]. For safety, galvanic isolation betweenthe vehicle and the mains grid is necessary [111]. When charging from an ac-grid, this can be realised by a low-frequency (LF) transformer on the mains side,as shown in figure 1.2a. However, the weight and volume for the system canbe reduced significantly by using an isolated ac-dc converter containing a high-frequency transformer to provide the galvanic isolation [46, 68, 113], as shown infigure 1.2b. Usually, this conversion is realised by an ac-dc converter stage, provid-ing power factor correction (PFC), followed by an isolated dc-dc converter [92, 113].When the intermediate energy storage capacitor is removed, a so-called single-stage converter is created [36, 39, 61]. This results in a lower converter volume andcost, whereas two-stage converters provide a higher efficiency at the nominal op-erating point [37, 107, 115]. Nevertheless, isolated dc-dc converters are an essentialpart in charging batteries from an ac or dc power source. With the trend towardshigher battery voltages and faster charging with powers up to 600 kW [50, 62] andbeyond, the efficiency is a key factor to maintain a high power density and lowoperational cost.

1.1.2 Converters for hybrid energy storage systems

Considering the power source of an electric vehicle, several energy storage systemsexist [42, 108]. It is often beneficial to combine two or more energy storage sys-tems to increase the lifetime, power capability, and efficiency of the total energystorage system [87, 108]. Common hybrid energy storage systems are combininga high-energy-density storage system (battery, fuel-cell) with a high-power-densitystorage system (supercapacitor, ultracapacitor) [23, 76, 87, 94, 103], as shown infigure 1.3. To couple these energy storage systems to the load, one or more bidi-


battery

dcdc

supercap.

dc

ac

electricalmachine

battery

(a)

battery

dcdc

fuelcell

ac

dcelectricalmachine

(b)

Figure 1.3: Examples of hybrid energy storage systems in electric vehicles, where an isolateddc-dc converter regulates the power of a super capacitor (a) or a fuel cell (b) to increase thelifetime and power capability of the hybrid storage system.

acdc

LF-transformer

ac-loadsoverhead line

(a)

acdc

overhead line

dcdc

ac-loads

dc-link

(b)

Figure 1.4: Auxiliary power unit (APU) for light rail vehicles using a dc electrification sys-tem. The low-frequency (LF) transformer (a) can be omitted by utilising an isolated dc-dcconverter (b), thereby reducing the weight and volume of the APU.

rectional dc-dc converters are required. These converters have to provide a largepart of the peak power and need to operate efficiently over a wide voltage-range.Furthermore, cost and weight are important requirements for mobile mass-marketapplications, such as BEVs and PHEVs.

1.1.3 Auxiliary power unit for railway applications

Most light rail vehicles, such as trams and metro trains, are using a dc-electrificationsystem with a voltage of 600 V or 750 V. All electrical systems other than traction,such as lighting, air-conditioning, pumps, etc., are called auxiliary systems and aresupplied by an auxiliary power unit (APU), which is connected to the overheadline. For safety, the APU provides galvanic isolation from the overhead line. Sincemost high-power auxiliary systems, like air-conditioning and pumps, require an acvoltage of 400 V, a common solution is to provide isolation by an LF transformerwhich is fed by a line-connected inverter [64], as shown in figure 1.4a. However, LFtransformers are heavy and bulky and, in addition, the turns ratio must be chosensuch that the output voltage can be maintained for variation of the input voltage.These variations exist due to loading and regenerative braking of all rail vehicles

1.1 Electrical power conversion in transportation 5

acdcdc

dc

ac

dc

ac-grid

dcdc

ac

dc

n modules electricalmachine

(a)

acdcdc

dc

ac

dc

ac-grid

acdcdc

dc

ac

dc

n modules

ac-grid

(b)

Figure 1.5: Utilisation of isolated dc-dc converters in a semi-modular (a) and modular (b)three-stage configuration of a solid-state transformer (SST) for traction or grid applications,respectively.

present on the same line section. For example, for 750 V systems, the line voltagecan vary between 500 V and 1270 V [35]. This requires a low transformer turns ratioand, thereby, increases the current through the primary winding and the inverter.To improve the efficiency and power density of the APU, an isolated dc-dc con-verter can be used, as shown in figure 1.4b, which allows to omit the LF transformerand regulate the intermediate dc-link voltage [29]. Since large voltage variations onthe overhead line exist, an isolated dc-dc converter with a wide input-voltage rangeis required.

1.1.4 Solid-state transformers

Heavy and bulky LF transformers can be replaced by so-called solid-state trans-formers (SST). This is an ac-ac converter with a medium- or high-frequency trans-former to provide galvanic isolation. In applications where the weight and volumeof an LF transformer is an obstacle, such as traction in rail vehicles, a SST is pre-ferred [45, 56, 68, 74, 86]. A semi-modular configuration for traction in rail vehiclesis shown in figure 1.5a. The additional controllability makes SSTs also attractive forreplacing LF transformers in smart grids [56, 74]. Among SST topologies, the mod-ular three-stage configuration, which includes an isolated dc-dc converter, is thepreferred concept concerning efficiency and flexibility [41, 56, 58, 74, 93, 116, 117].An example of this configuration is shown in figure 1.5b. Due to the isolated dc-dc converter, the input and output of the modules can be connected in parallel orseries, which improves scalability. Consequently, the SST can be optimised by thenumber of modules. Considering the module itself, a highly efficient isolated dc-dc converter, as well as the ac-dc/dc-ac converter stages, are essential to achieve ahigh total efficiency.


1.2 Isolated dc-dc converters

The previously discussed applications underpin the importance of efficient andcompact isolated dc-dc converters. Some well known isolated dc-dc converters forthis purpose are the full-bridge converter, which is member of the double-endedconverter family, and the series resonant converter, which is belonging to the fam-ily of resonant converters. However, these converters are lacking in flexibility andscalability properties, such as buck-boost operation and current-transfer character-istics for easy parallel connection of multiple converters. Alternatively, the dualactive bridge (DAB) converter, introduced in [26], features these properties and,therefore, is more favourable for high-power applications. Accordingly, this topol-ogy serves as a starting point of this work.

1.2.1 The dual active bridge topology

Within the class of isolated dc-dc converters, the dual active bridge topology,shown in figure 1.6, is recognised as an efficient isolated dc-dc converter topology[1, 26, 58]. It features galvanic isolation, soft-switching, bidirectional power flow,buck-boost operation, and current-transfer characteristics. Initially, two versionsof the topology were introduced, a single-phase and a three-phase variant. Thethree-phase DAB converter, shown in figure 1.6b, proves to be more efficient andcompact than the single-phase variant due to its interleaved structure and lowerinternal circulating currents [26, 27, 32, 53, 80, 89, 101]. Both topology variants areoriginally operated by a single phase-shift between the inverter bridge voltages,so-called single phase-shift (SPS) control, to regulate the power flow.

A major drawback of this control method is that, when the input and primaryreferred output voltage differ, soft-switching operation is lost and the circulatingcurrents increase. A solution to this problem is found for the single-phase full-bridge DAB converter by also controlling the duty-cycle of both inverter bridges[39, 71, 102, 106]. This approach results in three control variables for fixed frequencyoperation which can be utilised to achieve soft-switching and low circulating cur-rents for a wide voltage-range. Recently, the same approach has been applied tothe three-phase DAB converter [54, 55, 73]. As a result, soft-switching and low cir-culating currents can be achieved for a wide voltage-range. However, the currentin the transformer becomes asymmetric and the impact on the converter compo-nents, such as the switches, transformer, and capacitors, is unknown. Furthermore,complete modelling and analytical modulation schemes are still missing.

1.2 Isolated dc-dc converters 7

U1 U2

A

B

a

b

1

2

3

4

5 7

6 8

iaN : 1iA L

ac-link

(a)

U1 U2

A

B

C

a

b

c

1

2

3 5

4 6

7 9 11

8 10 12

LA

LB

LC

S s

iA iaN : 1

ac-link

(b)

Figure 1.6: Dual active bridge (DAB) dc-dc converter topology [26], with a single-phase (a)and a three-phase (b) intermediate ac-link.

1.2.2 A family of three-phase dual active bridge converters

The two-level inverter bridges limit the three-phase DAB converter to a single con-trol variable for symmetric operation, whereas three control variables are foundfor asymmetric operation. A possible solution is the introduction of a multi-leveltopology within the three-phase DAB converter, as it increases the number of de-grees of freedom. Consequently, three control variables and symmetric operationcan be obtained by three-level inverter bridges, as shown in figure 1.7.

Additionally, different transformer winding configurations can be implemented inthe ac-link of the converter, two common transformer connections are the star-starand star-delta winding configuration, as shown in figure 1.8. In [60], the impact ofthese winding configurations on a three-phase single active bridge (SAB) is inves-tigated. It is reported that the winding configurations shape the current throughthe main-inductor differently and, consequently, influence the current stress of the


U12

U22

A

B

C

a

b

c

iA iaU12

N n

U22Three-phase network

ia

Figure 1.7: Three-phase DAB converters utilising three-level inverter bridges.

S

A

B

C

a

b

cs

(a) Star-star connected transformer (YY).

S

A

B

C

a

b

c

(b) Star-delta connected transformer (Y∆).

Figure 1.8: Transformer winding configurations.

converter components. However, the impact of different transformer winding con-figurations on the performance of three-phase DAB converters is not found in liter-ature.

Finally, by combining different operating methods, topologies, and transformerwinding configurations, a family of three-phase DAB converters is created. As aresult, six possible combinations arise, in which at maximum three control vari-ables are considered for fixed frequency operation. These are listed in table 1.1.

1.3 Research goal and objectives

Power electronic converters play a key role in the electrification of transport, and inthe distribution and storage of electrical energy. The efficiency, power density, andcost of power electronic converters are hereby crucial. These performance indica-tors can be improved by more advanced converter topologies and higher switching

1.3 Research goal and objectives 9

Table 1.1: Family of three-phase DAB dc-dc converters for possible combinations of oper-ating methods, topologies, and transformer winding configurations which lead to, at most,three control variables.

Operating Three-phase DAB topology Transformer Abbreviationmethod connection

Symmetric Two-level topology star-star S2L-YYSymmetric Two-level topology star-delta S2L-Y∆Symmetric Three-level topology star-star S3L-YYSymmetric Three-level topology star-delta S3L-Y∆

Asymmetric Two-level topology star-star A2L-YYAsymmetric Two-level topology star-delta A2L-Y∆

frequencies, which are made possible by the evolution of power semiconductor de-vices and microelectronics. In this respect, multi-level topologies and high switch-ing frequencies are recognised to reduce the volume and cost of passive compo-nents [22, 46, 72, 77].

The research goal of this dissertation is to derive and investigate isolated dc-dc converter topologies that achieve a high efficiency and power density for awide range of voltage and power levels. As a starting point, the three-phase DABconverter is selected due to its favourable properties such as soft-switching, bidirec-tional power flow, buck-boost operation, and current-transfer characteristics. How-ever, soft-switching operation and low circulating currents are not maintained for awide voltage-range. Therefore, to overcome the limitations of the three-phase DABconverter, a multi-level topology is introduced in the three-phase DAB converter. Inaddition, by combining different operating methods, topologies, and transformerwinding configurations, a family of three-phase DAB converters is created. In thisthesis, a selection of five converters, including the existing solution, are investi-gated and compared in accordance with the research goal. The objectives to reachthe goal are defined as follows:

• Development of a generic modelling method that supports the whole fam-ily of three-phase dual active bridge converters.A converter model, preferably of low complexity, is required to simulate thecircuit behaviour accurately and provide analytical equations of, for instance,the power flow. Furthermore, the modelling method should be applicable forthe whole family of three-phase dual active bridge converters.


• Derivation of analytical modulation schemes that result in soft-switchingand low circulating currents.In order to achieve a high efficiency and power density for the three-phaseDAB converters with multiple control variables, modulation schemes haveto be established which result in soft-switching and minimal circulating cur-rents. The aim is to derive analytical equations that achieve optimal or close-to-optimal operation of the converter, and can be easily implemented on acontrol system.

• Investigation of the impact of different transformer winding config-urations on the performance of the symmetrically operated two-leveltopology.The current through the ac-link, which also flows through the inverterbridges, is affected by different transformer winding configurations, such asthe star-star and star-delta connection. To assess which transformer wind-ing configuration results in the best performance of the three-phase DABconverter with two-level inverter bridges, the impact on the soft-switchingregion and the converter components need to be investigated.

• Comparative evaluation of the proposed three-phase DAB converters forapplications with a wide voltage-range.In order to assess which of the investigated three-phase DAB converters isbest suited for wide voltage-range applications, a comparative evaluationis required. This should include an evaluation of the current stress of theswitches, transformer, and capacitors. Additionally, the total required semi-conductor chip area as well as the losses for an equal chip area should beincluded to determine which converter performs best.

• Experimental verification of the modelling method, modulation schemes,and topological concepts.The validation of the modelling method, modulation schemes, and topolog-ical concepts should be performed by measurements on a hardware proto-type. Consequently, a converter should be designed and built to verify themost promising three-phase DAB converters at realistic voltage and powerlevels in accordance with the targeted applications.

1.4 Thesis structure 11

1.4 Thesis structure

The goal of this dissertation is addressed in three parts, whereas the objectives arelinked to the chapters. Furthermore, not every chapter includes all five investi-gated three-phase DAB converters since they are not all applicable to the subjector focus of the chapter. Therefore, an overview of the existing and proposed three-phase DAB converters discussed and investigated within this thesis is presented intable 1.2.

Part I: Modelling and modulation schemes

The first part of this thesis treats the modelling and modulation of investigatedthree-phase DAB dc-dc converters. The analysis is based on piecewise-linear equa-tions and the corresponding switching modes, and is covered in chapter 2. Thisincludes a systematic approach for the identification of the switching modes, theirboundaries, and the derivation of the piecewise-linear equations. The modulationschemes for symmetric and asymmetric operation of, respectively, the three- andtwo-level topologies are introduced in chapter 3.

Part II: Evaluation and experimental results

This part addresses the evaluation and comparison of the selected three-phase DABconverters, together with the experimental verification. In chapter 4 the impact oftwo common transformer connections on the performance of the conventional two-level topology is investigated for applications with a limited voltage range. Thisincludes measurements, obtained from an experimental setup.

A comparative evaluation is carried out in chapter 5, including five different three-phase DAB converters, considering a wide voltage-range high-power application.Experimental verification of the modelling results and the modulation schemes in-troduced in Part I, are presented in chapter 6. This includes a description of thedesign and realisation of a 100 kW converter prototype.

Part III: Closing

The main conclusions, an overview of the contributions, and recommendations forfuture research are given in Chapter 7. Lastly, analytical models and the modula-tion schemes of the experimentally verified converters are given in the appendices.


Table 1.2: Overview of the existing and proposed three-phase dual active bridge convertersthroughout the thesis.

ChapterThree-phase dual active bridge converters

S2L-YY S2L-Y∆ S3L-YY S3L-Y∆ A2L-YY


2 • • • • •3 • • •


4 •1 •1

5 • • • • •6 •2 •2 •2

Part III: Closing

7 • • • • •Appedix B • •1 Including experimental verification using a 33 kW prototype with U1 = U2 = 375 V2 Including experimental verification using a 100 kW prototype with U1 = 375− 1500 V, U2 = 750 V

Part I

Modelling and ModulationSchemes

13

Chapter 2

Piecewise-linear modelling

A NALYSING and designing electronic power converters very much relies onthe existence of a, preferably, low complexity model that allows to simulatethe circuit behaviour accurately. The three-phase DAB topology, like many

switch-mode converters, exhibits nonlinear behaviour but can be considered as alinear circuit for a given state of the switches. During a switching cycle, multiplestates of the switches, each representing a linear circuit, will occur. This set of linearcircuits, representing a switched linear system, can be modelled as a piecewise-linear (PWL) system.

This chapter introduces a systematic approach to model three-phase DAB con-verters with piecewise-linear equations and derive the corresponding switching

Contributions of this chapter are published in:• N. H. Baars, H. Huisman, J. L. Duarte, and J. Verschoor, ‘A 80 kW isolated dc-dc converter for

railway applications’, in Proceedings of the IEEE European Conference on Power Electronics and Ap-plications, pp. 1–10, Aug. 2014.

• N. H. Baars, J. Everts, H. Huisman, J. L. Duarte, and E. A. Lomonova, ‘A 80-kW isolated dc-dc converter for railway applications’, IEEE Transactions on Power Electronics, vol. 30, no. 12, pp.6639–6647, Dec. 2015.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Impact of different transformer-winding configurations on the performance of a three-phase dual active bridge dc-dc converter’,in Proceedings of the IEEE Energy Conversion Congress and Exposition, pp. 637–644, Sep. 2015.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Performance evaluation of a three-phase dual active bridge dc-dc converter with different transformer winding configurations’,IEEE Transactions on Power Electronics, vol. 31, no. 10, pp. 6814–6823, Oct. 2016.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Modelling and modulation ofthree-level three-phase dual active bridge dc-dc converters’, IEEE Transactions on Power Electron-ics, under review, 2017.

15

16 Chapter 2 Piecewise-linear modelling

U12

U12

U22

U22

A

B

C

a

b

c

N n

1

2

3 5

4 6

7 9 11

8 10 12

Three-phase networkiA ia

Figure 2.1: Three-phase dual active bridge dc-dc converter topology.

modes. The models presented in this chapter are used to derive analytical modu-lation schemes in chapter 3 and to investigate the impact of different transformerwinding configurations on the performance of the two-level topologies in chapter 4.

2.1 Symmetrical operation of two-level three-phaseDAB converters

The three-phase DAB dc-dc converter topology can be modelled with linear storageelements and switches. For a given state of the switches, a linear circuit is identifiedand a continuous-time model can be derived using linear differential equations.The modelling is simplified by considering ideal switches, neglecting resistance,and replacing the input and output capacitors by ideal voltage sources. As a result,the conventional three-phase DAB topology is modelled with the circuit shownin figure 2.1. It consists of two three-phase inverter bridges which are connectedthrough a linear three-phase network, containing an ideal three-phase transformerand inductances. Considering the transformer, multiple winding configurationsare possible. However, only the two most common configurations, i.e. the star-starand star-delta winding configuration, are included in the modelling.

Traditionally, the inverter bridges of the two-level three-phase DAB converter areoperated in so-called six-step mode [26–28, 80, 100, 101]. This results in three-phasevoltages with a duty ratio of 50 %, resulting in equal on- and off-times for bothswitches in a phase-leg, as shown in figure 2.2. To maintain the odd symmetry in

2.1 Symmetrical operation of two-level three-phase DAB converters 17

uAN

uBN

uCN

uan

ubn

ucn

φ

0 π 2π

θ = ωt

π3

2π3

4π3

5π3

Figure 2.2: Idealised phase-to-neutral voltages of the two-level three-phase DAB topologyfor symmetric operation.

the voltage waveforms, the duty ratio must be fixed to 50 % and only the phase-shift between the inverter bridges, denoted by φ, remains as a control variable forfixed frequency operation.

2.1.1 Star-star connected transformer

The three-phase DAB converter is often employed with a transformer primary andsecondary winding connected in star configuration, resulting in star-points ’S’ and’s’, as shown in figure 2.3. The three-phase transformer is considered ideal and ismodelled by three single-phase transformers. The inductors LA, LB, and LC repre-sent the total series inductance per phase, including the transformer leakage induc-tance and any external inductors, further referred to as transfer-inductors. Further-more, the transformer magnetizing inductance and, optionally, external commuta-tion inductors are included by the inductances L1A,B,C and L2a,b,c.

Applying a phase-shift φ between the inverter bridges causes a voltage differenceacross the inductors LA,LB, and LC, which result in a current flow, which is of maininterest since it determines the power level and also affects the converter losses andperformance. Assuming the three-phase currents to be balanced, the current of asingle phase is sufficient to calculate the power level. This inductor current canbe found by integrating the inductor voltage. For instance, the inductor current of


U12

U12

U22

U22

A

B

C

a

b

c

N n

1

2

3 5

4 6

7 9 11

8 10 12

LA

LB

LC

S s

iA ia

L1A,B,C L2a,b,c

N : 1

Figure 2.3: Two-level three-phase dual active bridge dc-dc converter topology with a star-star connected transformer.

uAS u′as

LAiLA

Figure 2.4: Per-phase equivalent circuit of the three-phase DAB converter with a star-starconnected transformer.

phase ’A’ is given by

iLA(θ) =∫ θ

0

uLA(θ)

ωLAdθ + iLA(0), (2.1)

where θ represents the switching angle (θ = ωt), the angular frequency is given byω = 2πfsw in which fsw is the switching frequency, and uLA the voltage across theinductor LA.

Considering a balanced three-phase system, the model is reduced to a per-phaseequivalent circuit with two voltage sources and the transfer-inductor LA, as shownin figure 2.4. The voltage sources represent the phase-to-star voltages of the firstand the second inverter bridge seen from the primary side of the transformer.Voltages and currents that are referred to the primary side of the transformer are


uAS

u′as

uLA

iLA

1 2 3 4 5 62ππ0

θ = ωt

π3

2π3

φ

interval

φ φ+ π3 φ+ 2π

3

(a)

uAS

u′as

uLA

iLA

1 2 3 4 5 62ππ0

θ = ωt

π3

2π3

φ

interval

φ φ+ π3φ- π

3

(b)

Figure 2.5: Idealised voltage and current waveforms of the two-level three-phase DABconverter with a star-star connected transformer for phase-shifts 0 ≤ φ ≤ π/3 (a) andπ/3 ≤ φ ≤ 2π/3 (b), assuming a transformer winding turns ratio N = 1 and U1=U2.

marked with the accent symbol, ie u′as = uasN and i′LA= iLA /N with N the trans-

former turns ratio. The inductor voltage is given by

uLA = uAS − u′as =(uAN −

uAN + uBN + uCN

3

)−(

uan −uan + ubn + ucn

3

)N, (2.2)

where uAN, uBN, uCN, uan, ubn, and ucn present the phase-to-neutral voltages ofthe inverter bridges, as shown in figure 2.2. The phase-to-star voltage waveformsare shown in figure 2.5, together with the resulting inductor voltage and currentwaveform for two different phase-shifts φ. Here, it is visible that there are (at most)twelve intervals in a switching cycle, marked with a transition of the state of theswitches. However, it is sufficient to consider only six intervals since the currentis an odd symmetric function. In each interval, the inductor current iLA can bedescribed with a linear equation. When the phase-shift is larger than π/3, a differ-ent inductor voltage and current waveform appears, as shown in figure 2.5b. Thisindicates a different switching mode and will be further elaborated upon.


I II III IV V VI

π30 2π

3 π 4π3

5π3 2π

φ

P

0

Mode

Figure 2.6: Switching modes (I-VI) and the corresponding first harmonic approximation ofthe power for symmetric operation of the two-level three-phase DAB topology with a star-star connected transformer.

Switching modes

The state of the switches in the six intervals defines a switching mode. When inone or more intervals the state of the switches changes, a different switching modearises. This occurs due to control variables crossing specific thresholds. In the caseof the two-level three-phase DAB with a star-star connected transformer, this hap-pens when the phase-shift crosses the values π/3, 2π/3, π, 4π/3, 5π/3, and 2π. Thesetransitions can be found graphically in figure 2.2 and 2.5. In total, six switchingmodes are found for symmetric operation. These are shown together with first har-monic approximation of the corresponding power in figure 2.6. The first harmonicof the power is an approximation of the PWL power equations and is given as

P = 3UAN1U′an1

ωLAsin(φ), (2.3)

where UAN1 and Uan1 ’ are, respectively, the rms values of the first harmonic com-ponents of the phase-to-star voltages uAN and uan’. The PWL equations will bepresented in the following. Since the power is an odd function of the phase-shift(P(−φ) = −P(φ)), it is sufficient to consider only the switching modes for positivepower levels (I-III). Solutions for negative power levels can be found by solving thephase-shift for the positive value of the power and then multiply the solution by’−1’.

Transfer-inductor current

The first switching mode is valid for 0 ≤ φ ≤ π/3, the current in each interval canbe described with the equations given in table 2.1. Since the voltage waveforms areodd symmetric, six intervals are sufficient to construct the current waveform for acomplete switching cycle. The second switching mode is valid for π/3 ≤ φ ≤ 2π/3,


Table 2.1: Piecewise-linear equations of the transfer-inductor current in switching mode I forsymmetric operation of the two-level three-phase DAB converter with star-star connectedtransformer.

Interval Inductor current iLA for mode I Valid for

1 iLA(0) +U1+U′2

3ωL θ 0≤ θ≤ φ

2 iLA(φ) +U1−U′2

3ωL (θ − φ) φ≤ θ≤ π3

3 iLA(π3 ) +

2U1−U′23ωL (θ − π

3 )π3 ≤ θ≤ φ + π

3

4 iLA(φ + π3 ) +

2U1−2U′23ωL (θ − φ− π

3 ) φ + π3 ≤ θ≤ 2π

3

5 iLA(2π3 ) +

U1−2U′23ωL (θ − 2π

3 ) 2π3 ≤ θ≤ φ + 2π

3

6 iLA(φ + 2π3 ) +

U1−U′23ωL (θ − φ− 2π

3 ) φ + 2π3 ≤ θ≤ φ

the current in each interval can be described with the equations given in table 2.2.The third switching mode, valid for 2π/3 ≤ φ ≤ π, is not much of practical usesince the maximum power level is reached at φ = π/2. Higher phase-shifts onlylead to a decrease in active power and an increase in reactive power in the ac-link.Therefore, the third switching mode is omitted here.

Because the current iLA is symmetric, the current iLA(0) can be found by solving theset of equations, assuming steady-state condition iLA(0) = −iLA(π). This results in

iLA(0) =

(U′2−U1)2π−3U′2φ

9ωL ∀ 0 ≤ φ ≤ π3

(3U′2−2U1)π−6U′2φ9ωL ∀ π

3 ≤ φ ≤ 2π3

. (2.4)

Phase-leg currents

The phase-legs in each inverter bridge carry the transfer-inductor current and apart of the transformer magnetizing current. In addition, external commutationinductors may be connected to facilitate ZVS. To model the current of the phase-legsof both bridges, a ’π’-network is used as shown in figure 2.7. Here, the inductorsL1A and L2a represent the transformer magnetizing inductance and, optionally, theexternal commutation inductors of phase ’A’ and ’a’ of, respectively, the first andsecond inverter bridge.


Table 2.2: Piecewise-linear equations of the transfer-inductor current in switching mode IIfor symmetric operation of the two-level three-phase DAB converter with star-star connectedtransformer.

Interval Inductor current iLA for mode II Valid for

1 iLA(0) +U1+2U′2

3ωL θ 0≤ θ≤ φ− π3

2 iLA(φ− π3 ) +

U1+U′23ωL (θ − φ + π

3 ) φ− π3 ≤ θ≤ π

3

3 iLA(π3 ) +

2U1+U′23ωL (θ − π

3 )π3 ≤ θ≤ φ

4 iLA(φ) +2U1−U′2

3ωL (θ − φ) φ≤ θ≤ 2π3

5 iLA(2π3 ) +

U1−U′23ωL (θ − 2π

3 ) 2π3 ≤ θ≤ φ + π

3

6 iLA(φ + π3 ) +

U1−2U′23ωL (θ − φ− π

3 ) φ + π3 ≤ θ≤π

uAS u′as

LAiLAiAS

L1A L′2a

iL1A i′L2a

i′as

Figure 2.7: Lossless per-phase equivalent circuit, including inductances L1A and L2a to modelthe transformer magnetizing and commutation currents, for a two-level three-phase DABconverter with a star-star connected transformer.

According to the equivalent circuit shown in figure 2.7, the phase-leg current forphase ’A’ belonging to the first inverter bridge is defined by

iA = iAS = iLA + iL1A (2.5)

and, the current for phase ’a’ belonging to the second inverter bridge is given by

i′a = i′as = iLA − i′L2a. (2.6)

The current through L1A, representing the transformer magnetizing inductance orcommutation inductor, is calculated by

iL1A(θ) =∫ θ

0

uAS

ωL1Adθ + iL1A(0), (2.7)


whereas the current through L2a is found with

iL2a(θ) =∫ θ

0

uas

ωL2adθ + iL2a(0). (2.8)

Note that i′L2a(θ) = iL2a(θ)/N.

Power flow

Assuming a lossless converter and balanced three-phase currents, the power flowis calculated by

P =3

2π

∫ 2π

0uAN(θ)iA(θ)dθ. (2.9)

Since only iLA contributes to the active power and the voltage and current are sym-metric, (2.9) is simplified to

P =3U1

π

∫ π

0iLA(θ)dθ. (2.10)

The equation of the power is derived for two switching modes, using the piecewise-linear description given in this chapter. This results in

P(φ) =

U1U′2

ωL φ[

23 −

φ2π

]∀ 0 ≤ φ ≤ π

3U1U′2

ωL

[φ− φ2

π − π18

]∀ π

3 ≤ φ ≤ 2π3

. (2.11)

2.1.2 Star-delta connected transformer

Another common transformer winding configuration is the star-delta connection.This configuration exhibits a star connection on the primary side and a delta con-nection on the secondary side of the transformer, as shown in figure 2.8. The per-phase equivalent circuit is shown in figure 2.9 and the corresponding inductor volt-age is given by

uLA = uAS − u′ab =

(uAN −

uAN + uBN + uCN

3

)− (uan − ubn) N, (2.12)

where u′ab represents the phase-to-phase voltage of the second inverter bridge, re-ferred to the primary side of the transformer.


U12

U12

U22

U22

A

B

C

a

b

c

N n

1

2

3 5

4 6

7 9 11

8 10 12

LA

LB

LC

S

iA ia

L1A,B,C L2a,b,c

N : 1

Figure 2.8: Two-level three-phase dual active bridge converter with a star-delta connectedtransformer.

uAS u′ab

LAiLA

Figure 2.9: Per-phase equivalent circuit of the three-phase DAB converter with a star-deltaconnected transformer.

The star-delta connection causes a disparity in the voltage waveforms and, there-fore, affects the inductor voltage and current waveform, as shown in figure 2.10.Additionally, the winding configuration creates a phase-shift and an amplitude dif-ference between the voltages uAS and uab. Therefore, the control variable φ is heredefined as the phase-shift between the centers of uAS and uab. For illustration, thetransformer turns ratio is chosen to be N = 1/

√3, such that the amplitude of the

fundamental harmonic of both voltages are equal.

The idealised phase-to-star and phase-to-phase voltage waveforms are shown infigure 2.10, together with the resulting inductor voltage and current waveform fortwo different phase-shifts. This reveals that there are at most 5 intervals in half aswitching cycle. Furthermore, figure 2.10a and figure 2.10b display two differentswitching modes. The transition from one to another mode occurs when the phase-shift crosses the value π/6. More details on the switching modes is given furtheron.


uAS

u′ab

uLA

iLA

1 2 3 4 52ππ0

θ = ωt

π3

2π3

φ

π6 +φ 5π

6 +φ

interval

(a)

uAS

u′ab

uLA

iLA

1 2 3 4 52ππ0

θ = ωt

π3

2π3

φ

π6 +φφ- π

6

interval

(b)

Figure 2.10: Idealised voltage and current waveforms of the two-level three-phase DABconverter with a star-delta connected transformer for phase-shifts 0 ≤ φ ≤ π/6 (a) andπ/6 ≤ φ ≤ π/2 (b), assuming a transformer winding turns ratio N = 1/

√3 and U1=U2.

π6

3π6

5π6

7π3

9π3

11π6- π

6

φ

P

I II III IV V VI

0

Mode

Figure 2.11: Switching modes (I-VI) and the corresponding first harmonic approximationof the power for symmetric operation of the two-level three-phase DAB topology with astar-delta connected transformer.

Switching modes

Similar to the star-star winding configuration, six switching modes (I to VI) arefound for the star-delta configuration. The modes and their boundaries are shownin figure 2.11 together with the first harmonic approximation of the power. Com-pared to the star-star connected transformer, the switching modes are ’shifted’ dueto the phase-shift caused by the winding configuration.


Table 2.3: Piecewise-linear equations of the transfer-inductor current in switching mode I forsymmetric operation of the two-level three-phase DAB converter with star-delta connectedtransformer.

Interval Inductor current iLA for mode I Valid for

1 iLA(0) +U1

3ωL θ 0≤ θ≤ φ + π6

2 iLA(φ + π6 ) +

U1−3U′23ωL (θ − φ− π

6 ) φ + π6 ≤ θ≤ π

3

3 iLA(π3 ) +

2U1−3U′23ωL (θ − π

3 )π3 ≤ θ≤ 2π

3

4 iLA(2π3 ) +

U1−3U′23ωL (θ − 2π

3 ) 2π3 ≤ θ≤ φ + 5π

6

5 iLA(φ + 5π3 ) + U1

3ωL (θ − φ− 5π6 ) φ + 5π

6 ≤ θ≤π


As discussed in section 2.1.1, it is sufficient to consider only the switching modesfor positive power flow and for phase-shifts up to the value π/2. Therefore, thePWL model is derived for switching modes I and II only. The star-delta configura-tion incorporates five intervals for half a switching cycle in every switching mode,as shown in figure 2.10. For switching modes I and II the current iLA in each inter-val is described with piecewise-linear equations, which are given in table 2.3 andtable 2.4, respectively.

The current iLA(0) is found by solving the set of equations, assuming steady-statecondition iLA(0) = −iLA(π). This results in

iLA(0) =

(3U′2−2U1)π

9ωL for 0 ≤ φ ≤ π6

(9U′2−4U1)π−18U′2φ18ωL for π

6 ≤ φ ≤ π2

. (2.13)

Phase-leg currents

The phase-leg current for both inverter bridges can be derived from the per-phaseequivalent circuit, given in figure 2.12. The inductors L1A and L2a represent thetransformer magnetizing inductance and the external commutation inductor forphase ’A’ and ’a’ of, respectively, the first and second inverter bridge.


Table 2.4: Piecewise-linear equations of the transfer-inductor current in switching mode IIfor symmetric operation of the two-level three-phase DAB converter with star-delta con-nected transformer.

Interval Inductor current iLA for mode II Valid for

1 iLA(0) +U1+3U′2

3ωL θ 0≤ θ≤ φ− π6

2 iLA(φ− π6 ) +

U13ωL (θ − φ + π

6 ) φ− π6 ≤ θ≤ π

3

3 iLA(π3 ) +

2U13ωL (θ − π

3 )π3 ≤ θ≤φ + π

6

4 iLA(φ + π6 ) +

2U1−3U′23ωL (θ − φ− π

6 ) φ + π6 ≤ θ≤ 2π

3

5 iLA(2π3 ) +

U1−3U′23ωL (θ − 2π

3 ) 2π3 ≤ θ≤π

uAS u′ab

LAiLAiAS

L1A L′2a

iL1A i′L2a

i′ab

Figure 2.12: Lossless per-phase equivalent circuit, including inductances L1A and L2a tomodel the transformer magnetizing and commutation currents, for a two-level three-phaseDAB converter with a star-delta connected transformer.

Since the transformer is star-connected for inverter bridge 1, the phase-leg currentiA for the same inverter bridge equals the current iAS, as given in (2.5). The phase-leg current ia, belonging to inverter bridge 2, carries the current of two phases dueto the delta connection of the transformer. Consequently, the phase-leg current ia isgiven by

i′a = i′ab − i′ca = (iLA − i′L2a)− (iLC − i′L2c

), (2.14)

were the transformer magnetizing or commutation current iL2a for inverter bridge 2is found by

iL2a(θ) =∫ θ

0

uabωL2a

dθ + iL2a(0). (2.15)


Power flow

The power flow for the star-delta configuration is calculated by using (2.9), consid-ering a lossless converter and the phase currents to be balanced. The equation of thepower is derived for two switching modes, using the piecewise-linear descriptiongiven in this chapter. This results in

P(φ) =

U1U′2

ωL φ for 0 ≤ φ ≤ π6

U1U′2ωL

[32

(φ− φ2

π

)− π

24

]for π

6 ≤ φ ≤ π2

. (2.16)

2.2 Symmetrical operation of three-level three-phaseDAB converters

The three-level three-phase DAB topology, introduced in chapter 1, consists of in-verter bridges which are able to generate three voltage levels. This increases thenumber of switching modes considerably compared to the symmetrically operatedtwo-level topology. Therefore, this section presents a systematic and generic mod-elling method to identify the switching modes and their boundaries in order toderive piecewise-linear models. In the following, three-level three-phase DAB con-verters with star-star and star-delta winding configurations are treated, consideringsymmetric operation.

2.2.1 Star-star connected transformer

The three-level three-phase DAB topology with a star-star connected transformer isshown in figure 2.13. Compared to the two-level topology, a bidirectional switch isadded per phase-leg, which enables a third voltage level. For instance, the phase-legs of the first inverter bridge are able to generate the voltage levels -U1/2, 0,and U1/2 with respect to the neutral point N. To reduce complexity and limitthe degrees of freedom, only odd symmetric voltage waveforms are considered(uAN(0) = −uAN(π)), as shown in figure 2.14. As a consequence, two additionaldegrees of freedom arise, which are the duty-cycles d1 and d2 of, respectively, thefirst and second inverter bridge. Together with the phase-shift between voltages ofboth inverter bridges, there are three control variables for fixed frequency opera-tion, as shown in figure 2.14.

The equivalent circuit and the voltage source designation for the three-level topol-ogy remains the same as presented in figure 2.4, regarding the two-level variant.

2.2 Symmetrical operation of three-level three-phase DAB converters 29

U12

U12

U22

U22

A

B

C

a

b

c

N n

1

2

3 5

4 6

7 9 11

8 10 12

LA

LB

LC

S s

iA ia

L1A,B,C L2a,b,c

N : 1

Figure 2.13: Three-level three-phase DAB topology with a star-star connected transformer.

The voltage waveforms, however, are more complex and the number of intervalshas increased, as shown in figure 2.14. Bounded by the rising and falling edgesof the phase-leg voltages, twelve intervals can be recognised for half a switchingcycle.

Switching modes

Similar as for the two-level topology, the state of the switches in one or more inter-vals can change due to control variables crossing specific thresholds. Such an eventindicates a transition from one switching mode to another. To model the converterin the whole operating range, all switching modes need to be found.

One possible way to derive the switching modes is to code the rising and fallingtransitions of the phase-leg voltages, and, find all possible sequences. The voltagetransitions are coded with their corresponding phase symbol, for instance, the tran-sitions of phase ’A’ are labelled by A1 to A4, as shown in figure 2.14. The voltagetransitions of all other phases are coded in the same way.

Analysing a single inverter bridge reveals that there exist three possible se-quences per inverter bridge, which are only dependent on the correspondingduty-cycle. The sequences and conditions for both inverter bridges are given intable 2.5. Combining the sequences of both inverter bridges results in nine possiblecombinations, as shown in figure 2.15.

When the sequences of both inverter bridges are merged and shifted, a new se-quence arises that presents the transitions of the voltage waveforms of both bridges.One such sequence is shown in figure 2.14. For example, merging the sequences of


uAN

uBN

uCN

uan

ubn

ucn

d12π

φ

A1 A2

A3 A4

B1 B2

C1 C2

a1 a2

b1 b2

c1c2

B4 B3

C3 C4

a3 a4

b3 b4

c4c3

d22π

θ = ωt

uAS

u′as

1 2 3 4 5 6 7 8 9 10 11 122π

Interval

iLAuLA

0 θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9 θ10 θ11 π

φ

Inverter bridge 1

Inverter bridge 2

Figure 2.14: Idealised waveforms of the phase-to-neutral voltages, with the correspondingphase-to-star voltages and transfer-inductor voltage and current waveform, for symmet-ric operation of the three-level three-phase DAB topology with a star-star connected trans-former.

the input and output inverter bridge with a specific (phase) shift for, respectively,0 ≤ d1 ≤ 1/6 and 0 ≤ d2 ≤ 1/6 results in ’A1a1a2A2C3c3c4C4B1b1b2B2’ for half aswitching cycle. Multiple sequences can be found, for the same bridge sequencecombinations, by merging and shifting the inverter sequences in different ways. Asequence is only valid if the order of both bridge sequences, as given in table 2.5, ispreserved and the three-phase symmetry is maintained.

The sequences, corresponding to the switching modes, are generated by mergingand shifting the inverter sequences in different ways, following the rules givenabove, for the nine combinations shown in figure 2.15. For instance, the sequences


Table 2.5: Sequences that describe the order of rising and falling voltage transitions for theindividual inverter bridges of the symmetrically operated three-level topology, consideringa complete switching cycle.

Inverter bridge Sequence Condition

1A1A2C3C4B1B2A3A4C1C2B3B4 0≤ d1≤ 1/6

A1B4C3A2B1C4A3B2C1A4B3C2 1/6≤ d1≤ 1/3

A1C2C3B4B1A2A3C4C1B2B3A4 1/3≤ d1≤ 1/2

2a1a2c3c4b1b2a3a4c1c2b3b4 0≤ d2≤ 1/6

a1b4c3a2b1c4a3b2c1a4b3c2 1/6≤ d2≤ 1/3

a1c2c3b4b1a2a3c4c1b2b3a4 1/3≤ d2≤ 1/2

1/6 ≤ d1 ≤ 1/3

0 ≤ d1 ≤ 1/6

1/3 ≤ d1 ≤ 1/2

1/6 ≤ d2 ≤ 1/3

0 ≤ d2 ≤ 1/6

1/3 ≤ d2 ≤ 1/2

1/6 ≤ d2 ≤ 1/3

0 ≤ d2 ≤ 1/6

1/3 ≤ d2 ≤ 1/2

1/6 ≤ d2 ≤ 1/3

0 ≤ d2 ≤ 1/6

1/3 ≤ d2 ≤ 1/2

Figure 2.15: Tree diagram for possible combinations of the inverter bridge sequences, pre-sented in table 2.5, for symmetric operation of the three-level three-phase DAB converter.

for 0 ≤ d1 ≤ 1/6 and 0 ≤ d2 ≤ 1/6 are given in table 2.6. The sequences aredivided in three groups because the inverter bridge sequences can be merged inthree ways. That is, the successive rising and falling voltage transition of the sec-ond inverter bridge, e.g. a1 and a2, include zero (...a1a2...), one (...a1A2a2...), or two(...a1A2C3a2...) rising or falling voltage transitions of the first inverter bridge, asshown in the first row of table 2.6. Shifting the merged sequences until all pos-sibilities have occurred results in twelve sequences per group, thus 36 for eachcombination. Since there are nine combinations, as shown in figure 2.15, a totalof 324 unique sequences, and thus switching modes, exist. Deriving all possiblesequences by hand is a tedious task, therefore, a computer is used to generate thesequences and the corresponding interval angles.


Table 2.6: Sequences (36) that describe the order of rising and falling voltage transitions fora half a switching cycle when the sequences of the individual inverter bridges are mergedand shifted for 0 ≤ d1 ≤ 1/6, 0 ≤ d2 ≤ 1/6, and 0 ≤ φ ≤ 2π, considering the symmetricallyoperated three-level topology with a star-star connected transformer.

0 1 2

A1a1a2A2C3c3c4C4B1b1b2B2

A1A2a1a2C3C4c3c4B1B2b1b2

A1b3b4A2C3a1a2C4B1c3c4B2

A1A2b3b4C3C4a1a2B1B2c3c4

A1c1c2A2C3b3b4C4B1a1a2B2

A1A2c1c2C3C4b3b4B1B2a1a2

A1a3a4A2C3c1c2C4B1b3b4B2

A1A2a3a4C3C4c1c2B1B2b3b4

A1b1b2A2C3a3a4C4B1c1c2B2

A1A2b1b2C3C4a3a4B1B2c1c2

A1c3c4A2C3b1b2C4B1a3a4B2

A1A2c3c4C3C4b1b2B1B2a3a4

A1a1A2a2C3c3C4c4B1b1B2b2

A1b4A2a1C3a2C4c3B1c4B2b1

A1b3A2b4C3a1C4a2B1c3B2c4

A1c2A2b3C3b4C4a1B1a2B2c3

A1c1A2c2C3b3C4b4B1a1B2a2

A1a4A2c1C3c2C4b3B1b4B2a1

A1a3A2a4C3c1C4c2B1b3B2b4

A1b2A2a3C3a4C4c1B1c2B2b3

A1b1A2b2C3a3C4a4B1c1B2c2

A1c4A2b1C3b2C4a3B1a4B2c1

A1c3A2c4C3b1C4b2B1a3B2a4

A1a2A2c3C3c4C4b1B1b2B2a3

A1b4a1A2C3a2c3C4B1c4b1B2

A1A2b4a1C3C4a2c3B1B2c4b1

A1c2b3A2C3b4a1C4B1a2c3B2

A1A2c2b3C3C4b4a1B1B2a2c3

A1a4c1A2C3c2b3C4B1b4a1B2

A1A2a4c1C3C4c2b3B1B2b4a1

A1b2a3A2C3a4c1C4B1c2b3B2

A1A2b2a3C3C4a4c1B1B2c2b3

A1c4b1A2C3b2a3C4B1a4c1B2

A1A2c4b1C3C4b2a3B1B2a4c1

A1a2c3A2C3c4b1C4B1b2a3B2

A1A2a2c3C3C4c4b1B1B2b2a3

Switching mode boundaries

There are certain boundaries of the control variables d1, d2, and φ for which aswitching mode is valid. In the case of the two-level topology, the boundaries arerepresented on a single line since there is only one degree of freedom. The three-level topology, however, has three degrees of freedom and thus results in a boundedthree dimensional space. The boundaries of the switching modes can be found byrespecting the order of the corresponding sequence. This is done by selecting fivesuccessive interval angles, including at least two from each inverter bridge, andverify if the selected angles are in ascending order. This is given by

θ1

θ2

θ3

θ4

≤

θ2

θ3

θ4

θ5

. (2.17)


φ[rad]

d2 [-] d1 [-]

01/6

1/31/2

01/6

1/31/2

−π/6

0

π/6

2π/6

3π/6

4π/6

Figure 2.16: Boundaries of 108 switching modes for the symmetrically operated three-levelthree-phase DAB with a star-star connected transformer for 0 ≤ φ ≤ π/2.

The interval angles θ1 to θ12, with ascending values, are derived for every switchingmode m and are described by

θm =

θ1...

θ12

= Cmx+ dm, (2.18)

where x is the vector notation of the control variables given by [d1 d2 φ]ᵀ. Cm anddm are matrices with, respectively, a size of 12× 3 and 12× 1. Then the switchingmode boundaries are given by

θ1

θ2

θ3

θ4

−

θ2

θ3

θ4

θ5

= θ′m − θ′′m = (C ′m −C ′′m)x+ d′m − d′′m = Amx+ bm ≤ 0. (2.19)

The resulting boundaries, applying symmetric modulation for 0 ≤ φ ≤ π/2, arevisualised in figure 2.16. Each switching mode is bounded by four planes, forminga tetrahedron.

Examining the case for d1 = d2 = 0.5 (two-level operation) reveals, for 0 ≤ φ ≤ π/2,two line segments which correspond to switching mode I and II of the two-leveltopology with star-star connected transformer, as shown in figure 2.6.


Table 2.7: Transformation of the sequence ’A1a1b4A2C3c3a2C4B1b1c4B2’ into the states of thephase-legs in each interval for half a switching cycle.

Interval k θk−1 ≤ θ ≤ θk uA uB uC ua ub uc

1 0, θ1 1 0 0 0 -1 02 θ1, θ2 1 0 0 1 -1 03 θ2, θ3 1 0 0 1 0 04 θ3, θ4 0 0 0 1 0 05 θ4, θ5 0 0 -1 1 0 06 θ5, θ6 0 0 -1 1 0 -17 θ6, θ7 0 0 -1 0 0 -18 θ7, θ8 0 0 0 0 0 -19 θ8, θ9 0 1 0 0 0 -1

10 θ9, θ10 0 1 0 0 1 -111 θ10, θ11 0 1 0 0 1 012 θ11, θ12 0 0 0 0 1 0


In order to calculate the transfer-inductor current, the phase-leg voltages of bothinverter bridges need to be derived from each of the 324 switching mode sequences.Therefore, these sequences, describing the rising and falling voltage transitions, aretransformed to the states of the both inverter bridges in each interval. For instancethe single sequence (i.e. switching mode)’A1a1b4A2C3c3a2C4B1b1c4B2’ is transformed into the states given in table 2.7. Thecolumns of this table, that is uA to uc, form a unique matrix for the correspondingswitching mode m which is given by

Um =[uA uB uC ua ub uc

]. (2.20)

Consequently, the phase-leg voltages in each interval can be found by multiplyingthe columns of Um (uA to uc) with the corresponding dc-link voltage (U1 or U2).


Since the phase-leg voltages are known, the transfer-inductor current at the end ofinterval n (n = 1 : 24) for a given phase and switching mode can be described by

iL[n] =1

ωL

n

∑k=1

[uL[k](θk − θk−1)] + iL[24], (2.21)

with uL[k] the voltage across the transfer-inductance L in the kth interval and iL[24]the current at the end of the 24th interval (θ24 = 2π). The inductor voltage uL[k]corresponds to the elements of the voltage vector uL, this is given by

uL =U1uAS

2− NU2uas

2, (2.22)

where uAS and uas are given by, respectively,

uAS =2uA − uB − uC

3(2.23)

anduas =

2ua − ub − uc

3. (2.24)

The vectors for defining the state of the phase-legs, i.e. uA, uB, uC, ua, ub, and uc,are corresponding to the columns of matrixUm for switching mode m. An examplematrix is given in table 2.7.

The current iL[24] can be determined by assuming steady state condition, resultingin iL[24] = −iL[12]. Solving this results in

iL[24] = −iL[12] = −(

1ωL

12

∑k=1

[uL[k](θk − θk−1)] + iL[24]

)=

− 12ωL

12

∑k=1

[uL[k](θk − θk−1)] . (2.25)

Combining (2.25) and (2.21) gives

iL[n] =1

ωL

(n

∑k=1

[uL[k](θk − θk−1)]−12

12

∑k=1

[uL[k](θk − θk−1)]

). (2.26)

Equation (2.26) can be written more conveniently in matrix notation. Since the volt-ages and currents are odd symmetric only 12 intervals are included, which reduces


the size of the matrices and vectors. The inductor current, given in (2.26), is rewrit-ten in matrix notation and is described by

iL =

iL1

iL2......

iL12

=

1ωL

uL1 0 0 . . . 0

uL1 uL2 0 . . . 0

uL1 uL2 uL3

. . ....

......

.... . . 0

uL1 uL2 uL3 . . . uL12

−

1/2

1/2

...

...1/2

uL1

uL2......

uL12

ᵀ

θ1 − θ0

θ2 − θ1......

θ12 − θ11

=1

2ωL

uL1 −uL2 −uL3 . . . −uL12

uL1 uL2 −uL3 . . . −uL12

uL1 uL2 uL3

. . ....

......

.... . . −uL12

uL1 uL2 uL3 . . . uL12

θ1 − θ0

θ2 − θ1...

θ12 − θ11

=

12ωL

ULm

(θm − θ′m

), (2.27)

where θ′m is derived from θm which is shifted downwards with one element andθ12 is replaced by θ0 = 0. Consequently, θm − θ′m is defined as

θm − θ′m = Cmx+ dm − (C ′mx+ d′m) = Emx+ fm. (2.28)

The matrix ULm can be expanded, according to (2.22), to allow substitution for thedc-link voltages U1 and U2. This results in

ULm =U1UASm

2− NU2Uasm

2, (2.29)

where UASm and Uasm are given by, respectively,

UASm =

uAS1 −uAS2 −uAS3 . . . −uAS12

uAS1 uAS2 −uAS3 . . . −uAS12

uAS1 uAS2 uAS3

. . ....

......

.... . . −uAS12

uAS1 uAS2 uAS3 . . . uAS12


and

Uasm =

uas1 −uas2 −uas3 . . . −uas12

uas1 uas2 −uas3 . . . −uas12

uas1 uas2 uas3

. . ....

......

.... . . −uas12

uas1 uas2 uas3 . . . uas12

.

Combining (2.28) and (2.29) into (2.27) produces the following linear expression forthe inductor current:

iLm =1

2ωL(U1UASm −U2Uasm) (Emx+ fm) = Gmx+ hm. (2.30)

Phase-leg currents

The per-phase equivalent circuit for the two-level topology, presented in figure 2.7,is used to derive the phase-leg currents of both inverter bridges. Consequently, thephase-leg currents of the first phase are defined by (2.5) and (2.6) for the first andsecond inverter bridge, respectively. These equations require the transfer-inductorcurrent iLA , which is given by (2.30), and the corresponding currents through theinductors L1A and L2A. The currents iL1A and iL2A are derived in a similar way asthe transfer-inductor current (i.e. (2.21) to (2.26)). The difference is that the voltageacross L1A or L2A is defined by only one inverter bridge. Therefore, the requiredequations can be extracted from (2.30) by excluding the inverter bridge voltagewhich is not relevant. As a result, iL1A is defined by

iL1A =1

2ωL1A(U1UASm) (Emx+ fm) = G1mx+ h1m , (2.31)

and iL2a is given by

iL2a =1

2ωL2a(U2Uasm) (Emx+ fm) = G2mx+ h2m . (2.32)


Power flow

Assuming a lossless converter and the phase currents to be balanced, the powerflow of the three-level three-phase DAB converter is calculated by

P =3π

∫ π

0pAdθ =

3π

∫ π

0uANiAdθ, (2.33)

where pA is the instantaneous power and, uAN and iA the phase-leg voltage andcurrent, respectively, of phase ’A’. Considering active power only, iA is replaced byiLA for a star-star connected transformer. The average power in each interval is, inmatrix notation, written as

〈pm〉 =U1

2

uA1 0 0

0. . . 0

0 0 uA12

12

iL1...

iL12

+

−iL12

...

iL11

=U1UAm

2(Gavgm

x+ havgm) =Kmx+ lm, (2.34)

withGavgmx+ havgm

being the average current in each interval. This is given by

Gavgmx+ havgm

=Gmx+ hm +G′mx+ h′m

2, (2.35)

whereG′mx+h′m is derived fromGmx+hm which is shifted downwards with oneelement while the first element, after shifting, is multiplied with ’-1’ (-iL12 ) due tosymmetry. Finally the power for switching mode m is defined by

Pm =3π〈pm〉 • (θm − θ′m)

=3π(Kmx+ lm)

ᵀ(Emx+ fm)

=3π(xᵀKᵀ

mEmx+ (lᵀmEm + fᵀmKm)x+ lᵀmfm). (2.36)

The power Pm can be written in the standard quadratic form:xᵀQmx+ rᵀmx+ sm, with

Qm =3πKᵀ

mEm, rᵀm =3π(lᵀmEm + fᵀmKm), and sm =

3πlᵀmfm.


U12

U12

U22

U22

A

B

C

a

b

c

N n

1

2

3 5

4 6

7 9 11

8 10 12

LA

LB

LC

S

iA ia

L1A,B,C L2a,b,c

N : 1

Figure 2.17: Three-level three-phase DAB topology with a star-delta connected transformer.

2.2.2 Star-delta connected transformer

The three-level three-phase DAB topology with a star-delta connected transformeris shown in figure 2.17. To model the converter, the same equivalent circuit as forthe two-level variant, which is presented in figure 2.9, is used. Like the three-leveltopology with a star-star connected transformer, the voltage waveforms are morecomplex and the amount of intervals has increased, as shown in figure 2.18. Aswith the two-level topology with a star-delta connected transformer, a phase-shiftand an amplitude difference exist between the voltages uAS and uab. Therefore,the control variable φ is here defined from the center of uAS to the center of uab.Furthermore, a transformer turns ratio of N = 1/

√3 is assumed for illustration,

such that the amplitude of the fundamental harmonic of both voltages (uAS anduab) are equal.

Switching modes

To find all possible switching modes, the rising and falling voltage transitions ofboth inverter bridges are coded in the same way as the three-level topology witha star-star connected transformer, as explained in section 2.2.1. The number of in-dividual bridge sequences (three) and the possible combinations (nine) remain un-changed with respect to the three-level topology with a star-star connected trans-former, as presented in table 2.5 and figure 2.15, respectively.


uAN

uBN

uCN

uan

ubn

ucn

d12πA1 A2

A3 A4

B1 B2

C1 C2

a1 a2

b1 b2

c1c2

B4 B3

C3 C4

a3 a4

b3 b4

c4c3

d22π

φ + π/6

θ = ωt

uAS

u′ab

1 2 3 4 5 6 7 8 9 10 Interval

iLAuLA

0 θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9 π

φ

Inverter bridge 1

Inverter bridge 2

2π

Figure 2.18: Idealised waveforms of the phase-to-neutral voltages, with the correspondingphase-to-star and phase-to-phase voltages and transfer-inductor voltage and current wave-form, for symmetric operation of the three-level three-phase DAB topology with a star-deltaconnected transformer.

Due to the delta connection on the secondary side of the transformer, only twophase-leg voltages of the output inverter bridge need to be considered. As a re-sult, in total ten voltage transitions of both inverter bridges are included in a se-quence for half a switching cycle. For instance, the sequences for 0 ≤ d1 ≤ 1/6

and 0 ≤ d2 ≤ 1/6 are given in table 2.8. The sequences are divided in three groupsbecause the inverter bridge sequences can be merged in three ways. Shifting themerged sequences until all possibilities have occurred results in twelve sequencesper group, thus 36 for each combination. Since there are nine combinations ofbridge sequences (figure 2.15), again a total of 324 unique sequences, and thusswitching modes, exist.


Table 2.8: Sequences (36) that describe the order of rising and falling voltage transitions fora half a switching cycle when the sequences of the individual inverter bridges are mergedand shifted for 0 ≤ d1 ≤ 1/6, 0 ≤ d2 ≤ 1/6, and 0 ≤ φ ≤ 2π, considering the symmetricallyoperated three-level topology with a star-delta connected transformer.

0 1 2

A1a1a2A2C3C4B1b1b2B2

A1A2a1a2C3C4B1B2b1b2

A1b3b4A2C3a1a2C4B1B2

A1A2b3b4C3C4a1a2B1B2

A1A2C3b3b4C4B1a1a2B2

A1A2C3C4b3b4B1B2a1a2

A1a3a4A2C3C4B1b3b4B2

A1A2a3a4C3C4B1B2b3b4

A1b1b2A2C3a3a4C4B1B2

A1A2b1b2C3C4a3a4B1B2

A1A2C3b1b2C4B1a3a4B2

A1A2C3C4b1b2B1B2a3a4

A1a1A2a2C3C4B1b1B2b2

A1b4A2a1C3a2C4B1B2b1

A1b3A2b4C3a1C4a2B1B2

A1A2b3C3b4C4a1B1a2B2

A1A2C3b3C4b4B1a1B2a2

A1a4A2C3C4b3B1b4B2a1

A1a3A2a4C3C4B1b3B2b4

A1b2A2a3C3a4C4B1B2b3

A1b1A2b2C3a3C4a4B1B2

A1A2b1C3b2C4a3B1a4B2

A1A2C3b1C4b2B1a3B2a4

A1a2A2C3C4b1B1b2B2a3

A1b4a1A2C3a2C4B1b1B2

A1A2b4a1C3C4a2B1B2b1

A1b3A2C3b4a1C4B1a2B2

A1A2b3C3C4b4a1B1B2a2

A1a4A2C3b3C4B1b4a1B2

A1A2a4C3C4b3B1B2b4a1

A1b2a3A2C3a4C4B1b3B2

A1A2b2a3C3C4a4B1B2b3

A1b1A2C3b2a3C4B1a4B2

A1A2b1C3C4b2a3B1B2a4

A1a2A2C3b1C4B1b2a3B2

A1A2a2C3C4b1B1B2b2a3

This is equal to the number that was found for the three-level topology with a star-star connected transformer. The reason for this is that excluding a single phasedoes not reduce the number of bridge sequence combinations (table 2.5) and thepossible ways of merging and shifting the sequences, because the missing phase isredundant information.


As discussed in section 2.2.1, switching modes are only valid for a specific range ofcontrol variables (d1, d2, and φ). The boundaries of a switching mode can be foundby respecting the order of the corresponding sequence. This is done by selectingfive successive interval angles, including at least two from each inverter bridge,and verify if the selected angles are in ascending order. This results in the linearinequality given by

Amx+ bm ≤ 0, (2.37)

which was derived in (2.19).


φ[rad]

d2 [-] d1 [-]

01/6

1/31/2

01/6

1/31/2

-π/6

0

π/6

2π/6

3π/6

4π/6

Figure 2.19: Boundaries of 108 switching modes for the symmetrically operated three-levelthree-phase DAB with a star-delta connected transformer for 0 ≤ φ ≤ π/2.

The corresponding boundaries, applying symmetric modulation for 0 ≤ φ ≤ π/2,are visualised in figure 2.19. Each switching mode is bounded by four planes form-ing a tetrahedron, which was also found for the three-level topology with a star-starconnected transformer. However, due to the star-delta winding configuration, allswitching mode boundaries are shifted in the z-direction (φ-axis) by π/6, as is thecase for the two-level topology with a star-delta connected transformer.

Examining the case for d1 = d2 = 0.5 (two-level operation) reveals, for −π/6 ≤ φ ≤π/2, two line segments which correspond to switching mode I and II of the two-leveltopology with star-delta connected transformer, which were shown in figure 2.11.


Similar as for the three-level topology with a star-star connected transformer, the se-quences are transformed into the states of the both inverter bridges in each interval,to describe the bridge voltages. Due to the star-delta connection of the transformerten, instead of twelve, intervals exists. As an example, the sequence’A1a1A2a2C3C4B1b1B2b2’ is transformed into the states given in table 2.9. Thecolumns uA to ub make up the unique matrix Um =

[uA uB uC ua ub

], for

switching mode m (m = 1 : 324).

2.3 Asymmetrical operation of two-level three-phase DAB converters 43

Table 2.9: Transformation from the sequence ’A1a1A2a2C3C4B1b1B2b2’ into the states of thephase-legs in each interval for half a switching cycle.

Interval k θk−1 ≤ θ ≤ θk uA uB uC ua ub

1 0, θ1 1 0 0 0 02 θ1, θ2 1 0 0 1 03 θ2, θ3 0 0 0 1 04 θ3, θ4 0 0 0 0 05 θ4, θ5 0 0 -1 0 06 θ5, θ6 0 0 0 0 07 θ6, θ7 0 1 0 0 08 θ7, θ8 0 1 0 0 19 θ8, θ9 0 0 0 0 110 θ9, θ10 0 0 0 0 0

Consequently, the transfer-inductor voltage for phase ’A’ in each interval for a halfcycle is a vector with ten elements and is given by

uLA =U1uAS

2− NU2uab

2, (2.38)

where uAS is given by (2.23) and uab is found by

uab = ua − ub. (2.39)

The derivation of the transfer-inductor current, phase-leg currents, and the poweris similar to the derivation for the three-level topology with a star-star connectedtransformer, as presented in section 2.2.1. Therefore, the remaining derivation isomitted here.

2.3 Asymmetrical operation of two-level three-phaseDAB converters

The two-level three-phase DAB dc-dc converter can also be operated with duty-cycles other than 50 %, resulting in asymmetric voltage and current waveforms.In [98] it is shown that a fixed duty-cycle of 1/3 for both inverter bridges, while


applying phase-shift control, results in a good energy efficiency. However, no clearbenefits, compared to symmetric two-level operation, are reported. When the duty-cycle of both inverter bridges is kept equal and utilised as a control variable, in total,two degrees of freedom are available. As a result, the ZVS range is extended andseveral switching modes have been identified and modelled with PWL equations[55]. Separate control of the duty-cycle for both inverter bridges results in three de-grees of freedom, including the phase-shift, leading to an increased ZVS range andimproved light-load efficiency [73]. The presented modelling, however, relies onFourier series of the voltages and current, considering multiple harmonics, whichmakes analytical solutions for the control variables practically impossible. There-fore, a numerical optimizer was used in [73] to find solutions for the control vari-ables. In [54] asymmetric operation with three degrees of freedom is modelled withPWL equations, however, due to a limited range for the duty-cycles and phase-shiftonly three switching modes were identified.

In this section the approach which was used for symmetric operation is re-appliedto derive all possible switching modes and the corresponding boundaries for asym-metric operation of the two-level topology with a star-star connected transformer.This enables the derivation of the PWL models for all switching modes and allowsto analytically solve the control variables for the modulation scheme presented inchapter 3.

2.3.1 Switching modes

The asymmetrically operated two-level topology can be analysed using the con-verter circuit and per-phase equivalent circuit of the symmetrically operated two-level topology presented in section 2.1. Compared to symmetric operation of thetwo-level topology, the voltage waveforms, however, are more complex and thenumber of intervals has increased, as shown in figure 2.20. A total of twelve inter-vals can be identified, bounded by the rising and falling transitions of the bridgevoltages.

For analysis, three degrees of freedom are considered (d1, d2, and φ), assumingfixed frequency operation. The phase-shift is here defined from the center of uANto the center of uan, as shown in figure 2.20. Because the waveforms are not sym-metric any more, a complete switching cycle, instead of a half cycle, needs to beincluded in the modelling. Labelling the rising and falling transitions of the bridgevoltages results in three sequences per inverter bridge, as given in table 2.10. Simi-lar as for the three-level topology, nine combinations can be created with the bridgesequences. A tree diagram of the possible combinations is shown in figure 2.21.


uAN

uBN

uCN

uan

ubn

ucn

d12π

φ

A1 A2

B1 B2

C1 C2

a1 a2

b1 b2

c1c2

d22π

θ = ωt

uAS

u′as

1 2 3 4 5 6 7 8 9 10 Interval

iLA

uLA

0 θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9

φ

2π11 12

θ10 θ11

Inverter bridge 1

Inverter bridge 2

Figure 2.20: Idealised waveforms of the phase-to-neutral voltages, with the correspondingphase-to-star voltages and transfer-inductor voltage and current waveform, for asymmetricoperation of the two-level three-phase DAB topology with a star-star connected transformer.

Following the procedure introduced in section 2.2, the sequences, correspondingto the switching modes, are generated by merging and shifting the bridge se-quences in different ways, respecting the order of the individual bridge sequences(table 2.10). For instance, the sequences for 0 ≤ d1 ≤ 1/3 and 0 ≤ d2 ≤ 1/3 aregiven in table 2.11. The sequences are divided in three groups because the inverterbridge sequences can be merged in three ways. Shifting the sequence of the secondinverter bridge with respect to the sequence of the first inverter bridge result in sixsequences per group, thus 18 for this combination. Considering all combinations(nine), there exist 162 unique sequences which correspond to all possible switchingmodes.


Table 2.10: Sequences that describe the order of rising and falling voltage transitions for theindividual inverter bridges of the asymmetrically operated two-level topology, consideringa complete switching cycle.

Inverter bridge Sequence Condition

1A1A2B1B2C1C2 0≤ d1≤ 1/3

A1C2B1A2C1B2 1/3≤ d1≤ 2/3

A1B2B1C2C1A2 2/3≤ d1≤ 1

2a1a2b1b2c1c2 0≤ d2≤ 1/3

a1c2b1a2c1b2 1/3≤ d2≤ 2/3

a1b2b1c2c1a2 2/3≤ d2≤ 1

1/3 ≤ d1 ≤ 2/3

0 ≤ d1 ≤ 1/3

2/3 ≤ d1 ≤ 1

1/3 ≤ d2 ≤ 2/3

0 ≤ d2 ≤ 1/3

2/3 ≤ d2 ≤ 1

1/3 ≤ d2 ≤ 2/3

0 ≤ d2 ≤ 1/3

2/3 ≤ d2 ≤ 1

1/3 ≤ d2 ≤ 2/3

0 ≤ d2 ≤ 1/3

2/3 ≤ d2 ≤ 1

Figure 2.21: Tree diagram for possible combinations of the inverter bridge sequences, givenin table 2.10, for asymmetric operation of the two-level three-phase DAB converter.


The boundaries of the switching modes are derived by respecting the order of eachsequence, as discussed in section 2.2.1. When considering asymmetric modulationand 0 ≤ φ ≤ π/2, in total 72 switching modes are found. These are visualised infigure 2.22.

2.3.2 PWL model

In order to derive the converter currents in each switching mode, the correspondingsequences need to be transformed to the phase-leg states of both inverter bridges.This can be done by evaluating the sequence and change the phase-leg state ac-cording to the labelled voltage transitions, as shown in figure 2.20. For instance the


Table 2.11: Sequences (18) that describe the order of rising and falling voltage transitions fora half a switching cycle when the sequences of the individual inverter bridges are mergedand shifted for 0 ≤ d1 ≤ 1/6, 0 ≤ d2 ≤ 1/6, and 0 ≤ φ ≤ 2π, considering the symmetricallyoperated three-level topology with a star-star connected transformer.

0 1 2

A1a1a2A2B1b1b2B2C1c1c2C2

A1A2a1a2B1B2b1b2C1C2c1c2

A1c1c2A2B1a1a2B2C1b1b2C2

A1A2c1c2B1B2a1a2C1C2b1b2

A1b1b2A2B1c1c2B2C1a1a2C2

A1A2b1b2B1B2c1c2C1C2a1a2

A1a1A2a2B1b1B2b2C1c1C2c2

A1c2A2a1B1a2B2b1C1b2C2c1

A1c1A2c2B1a1B2a2C1b1C2b2

A1b2A2c1B1c2B2a1C1a2C2b1

A1b1A2b2B1c1B2c2C1a1C2a2

A1a2A2b1B1b2B2c1C1c2C2a1

A1c2a1A2B1a2b1B2C1b2c1C2

A1A2c2a1B1B2a2b1C1C2b2c1

A1b2c1A2B1c2a1B2C1a2b1C2

A1A2b2c1B1B2c2a1C1C2a2b1

A1a2b1A2B1b2c1B2C1c2a1C2

A1A2a2b1B1B2b2c1C1C2c2a1

φ[rad]

d2 [-] d1 [-]

01/3

2/31

01/3

2/31

−π/3

0

π/3

2π/3

π

Figure 2.22: Boundaries of 72 switching modes for the asymmetrically operated two-levelthree-phase DAB with a star-star connected transformer for 0 ≤ φ ≤ π/2.

single sequence (i.e. switching mode) ’A1a1a2A2B1b1b2B2C1c1c2C2’ is transformedinto the states given in table 2.12. The columns of this table, that is uA to uc, form aunique matrix for the corresponding switching mode m which is given by

Um =[uA uB uC ua ub uc

]. (2.40)

Consequently, the phase-leg voltages in each interval can be found by multiplyingthe columns of Um (uA to uc) with the corresponding dc-link voltage (U1 or U2).


Table 2.12: Transformation of the sequence ’A1a1a2A2B1b1b2B2C1c1c2C2’ into the states ofthe phase-legs in each interval for a complete switching cycle.

Interval k θk−1 ≤ θ ≤ θk uA uB uC ua ub uc

1 0, θ1 1 -1 -1 -1 -1 -12 θ1, θ2 1 -1 -1 1 -1 -13 θ2, θ3 1 -1 -1 -1 -1 -14 θ3, θ4 -1 -1 -1 -1 -1 -15 θ4, θ5 -1 1 -1 -1 -1 -16 θ5, θ6 -1 1 -1 -1 1 -17 θ6, θ7 -1 1 -1 -1 -1 -18 θ7, θ8 -1 -1 -1 -1 -1 -19 θ8, θ9 -1 -1 1 -1 -1 -1

10 θ9, θ10 -1 -1 1 -1 -1 111 θ10, θ11 -1 -1 1 -1 -1 -112 θ11, θ12 -1 -1 -1 -1 -1 -1


Since the voltage and current waveforms are asymmetric (iL(0) 6= iL(π)) no directsteady-state relation can be applied to solve the transfer-inductor current . Conse-quently, the transfer-inductor current is given by

iL(θ) =∫ θ

0

uL(θ)

ωLdθ + c, (2.41)

where c is the integration constant. When assuming steady-state conditions, theaverage current through the transfer-inductor for a complete switching cycle mustbe zero, which is defined by

12π

∫ 2π

0iL(θ)dθ = 0. (2.42)

Therefore, the integration constant can be found by

c = − 12π

∫ 2π

0

(∫ θ

0

uL(θ)

ωLdθ

)dθ. (2.43)


Since the phase-leg voltages are not continuous, (2.41) is written as a PWL equation.The transfer-inductor current at the end of interval n (n = 1 : 12) for a given phaseand switching mode can be described by

iL[n] =1

ωL

n

∑k=1

[uL[k](θk − θk−1)] + c, (2.44)

with uL[k] the voltage across the transfer-inductance L in the kth interval and c theintegration constant. The inductor voltage uL[k] corresponds to the elements of thevoltage vector uL, this is given by

uL =U1uAS

2− NU2uas

2, (2.45)

where uAS and uas are given by, respectively,

uAS =2uA − uB − uC

3(2.46)

anduas =

2ua − ub − uc

3. (2.47)

The vectors for defining the state of the phase-legs, i.e. uA, uB, uC, ua, ub, and uc,are corresponding to the columns of matrixUm for switching mode m. An examplematrix is given in table 2.12.

Equation (2.44) can be written more conveniently in matrix notation. Since the volt-ages and currents are asymmetric all of the 12 intervals are included. The transfer-inductor current, given in (2.44), is rewritten in matrix notation and is describedby

iL =

iL1

iL2...

iL12

=1

ωL

uL1 0 . . . 0

uL1 uL2

. . ....

......

. . . 0

uL1 uL2 . . . uL12

θ1 − θ0

θ2 − θ1...

θ12 − θ11

+ c

=1

ωLULm

(θm − θ′m

)+ c, (2.48)

where θ′m is derived from θm which is shifted downwards with one element andθ12 is replaced by θ0 = 0. Consequently, θm − θ′m is defined as

θm − θ′m = Cmx+ dm − (C ′mx+ d′m) = Emx+ fm. (2.49)


The matrix ULm can be expanded, according to (2.45), to allow substitution for thedc-link voltages U1 and U2. This results in

ULm =U1UASm

2− NU2Uasm

2, (2.50)

where UASm and Uasm are given by, respectively,

UASm =

uAS1 0 . . . 0

uAS1 uAS2

. . ....

......

. . . 0

uAS1 uAS2 . . . uAS12

and

Uasm =

uas1 0 . . . 0

uas1 uas2

. . ....

......

. . . 0

uas1 uas2 . . . uas12

.

Substituting (2.49) and (2.50) into (2.48) gives the following linear expression forthe inductor current

iLm =1

2ωL(U1UASm − NU2Uasm) (Emx+ fm) + c = Gmx+ hm + c. (2.51)

Then, according to (2.43) the integration constant c is found by

c =−12π〈Gmx+ hm〉 •

(θm − θ′m

)=−12π

(Gavgm

x+ havgm

)ᵀ(Emx+ fm)

=−12π

[xᵀGᵀ

avgmEmx+ (fᵀmGavgm

+ hᵀavgmEm)x+ hᵀavgm

fm

], (2.52)

with 〈Gmx+ hm〉 being the average current in each interval. This is given by

Gmx+ hm +G′mx+ h′m2

= Gavgmx+ havgm

, (2.53)

whereG′mx+h′m is derived fromGmx+hm which is shifted downwards with oneelement.

Evaluating (2.52) for all switching modes reveals that the product ofGᵀavgm

andEm

2.4 Soft-switching 51

results in a zero matrix (Gᵀavgm

Em = 0). As a consequence, the integration constantc is a linear expression and the resulting transfer-inductor current is given by

iLm = Gmx+ hm −(fᵀmGavgm

+ hᵀavgmEm)x+ hᵀavgm

fm

2π= GLmx+ hLm . (2.54)

From here, the derivation of the phase-leg currents and the power is similar aspresented in section 2.2.1 for the three-level topology with a star-star connectedtransformer. Therefore, the remaining derivation is omitted.

2.4 Soft-switching

One of the features of DAB converters is soft-switching operation. Compared tohard-switching, it reduces the switching losses and EMI emissions which, as a con-sequence, increases the efficiency or power density, or both. Therefore, it is of im-portance for designing and controlling DAB converters to have conditions to verifysoft-switching.

2.4.1 Zero voltage switching principle

Considering the half-bridge circuit shown in figure 2.23, when the current throughthe top switch is positive at the turn-off instant (figure 2.23a), the current is forcedto commutate to the (parasitic) switch capacitances Cs (figure 2.23b). Ideally, thisallows to turn off the top switch at zero volts, which is referred to as zero voltageswitching (ZVS). If the current has charged Cs1 and discharged Cs2 such that phase-to-neutral voltage uAN is lower than the negative voltage rail, D2 starts to conduct(figure 2.23c). As a result, S2 can be turned on at zero volts (figure 2.23d), finalising aZVS commutation from S1 to S2. The same principle applies for ZVS commutationfrom S2 to S1, however, this requires a negative current I.

If both commutation events are ZVS, the converter is considered to be fully soft-switching. This requires the current to change sign at least once per switching cy-cle. The corresponding voltage waveforms and conduction states of the switchesand diodes are shown in figure 2.24, assuming a constant current during commu-tation and linear switch capacitances. The time between when a switch is openedand the complementary switch is closed is the so-called dead time, denoted by tdt.This is necessary for the soft-switching procedure as described above to finalisecorrectly. Note that the anti-parallel diodes also can conduct for negative switchcurrent, depending on the transistor type and characteristics.


S1

S2

U2

U2

D1

D2

Cs1

Cs2

I

N A

(a)

S1

S2

U2

U2

D1

D2

Cs1

Cs2

I

N A

(b)

S1

S2

U2

U2

D1

D2

Cs1

Cs2

I

N A

(c)

S1

S2

U2

U2

D1

D2

Cs1

Cs2

I

N A

(d)

Figure 2.23: Current paths for ZVS commutation from S1 to S2 with a positive current I.

2.4.2 ZVS conditions

Considering a single phase-leg of a DAB converter, the current through the transfer-inductor charges and discharges the switch capacitances during the dead time. Thisis a resonant process governed by differential equations corresponding to a second-order circuit. Consequently, the minimum transfer-inductor current that result in aZVS commutation within the dead time can be determined.

In [47], the resonant transition is included in the derivation of the ZVS operatingrange for a single phase DAB converter. Two different resonant transitions werereported, that is commutation of only one phase-leg or two phase-legs simultane-ously. Applying the same approach for the three-level three-phase DAB converter,however, would result in many more resonant transitions due to the six phase-legs,requiring a large number of equivalent circuits and equations. As an alternative,the transfer-inductor current during the switching transition can be approximatedby the PWL description, as presented in this chapter, which results in non-resonantcircuit behaviour. Consequently, ZVS operation can be verified by comparing thecharge of the phase-leg current that is available during dead time, to the chargethat is required to commutate the (non-linear) capacitance of the switches [38, 39].This, however, requires to include multiple time intervals of the PWL model to cal-culated the available charge. An approach with reduced complexity, but also lessaccurately, is the current-based method. Here, the current during dead time is ap-proximated by a constant, equal to the switch turn-off current found by the PWLmodel. This will lead to an over estimation of the available charge, therefore, asafety margin should be applied to ensure ZVS operation. However, for the pur-pose of analysing and comparing three-phase DAB topologies, the current-basedmethod is chosen for its straightforward implementation.

The ZVS commutation principle, as mentioned earlier, can only be achieved if thecurrent during the dead time contains sufficient charge to reach the desired voltage

2.4 Soft-switching 53

uS1

uS2

uAN

S1

D2

S2

D1

uS1

uS2

uAN

S1

D2

t1 t2tdt

D1

S2

t3 tdt t4

t1 t2 t3 t4

Figure 2.24: Idealised waveforms of switch voltages (uS1 ,uS2 ), the phase-to-neutral voltage(uAN), and the conduction states of the switches and diodes (S1,S2,D1,D2) for ZVS commu-tations of falling (left bottom) and a rising (right bottom) transitions of uAN, assuming aconstant current during commutation and linear switch capacitances.


level. Therefore, the required charge for a phase-leg can be calculated with

QHB =∫ u2

u1

2Cs(u)du, (2.55)

were u1 and u2 are the voltage levels before and after the commutation, respectively.For linear switch capacitances, the required charge is given by 2Cs(u2 − u1). Theavailable charge, assuming a constant switch turn-off current, is given by.

QA = iA(θx) tdt, (2.56)

with iA(θx) the phase-leg current at the turn-off instant θx and tdt being the deadtime. Then the available charge must be greater than or equal to the requiredcharge, considering absolute values (|QA| ≥ |QHB|). This results in the minimumswitch turn-off current to achieve ZVS which is described by

IZVS ≥∣∣∣∣QHB

tdt

∣∣∣∣ . (2.57)

Current-based ZVS conditions

Four ZVS constraints, two per inverter bridge, are required for the three- andtwo-level topology with a star-star connected transformer operated, respectively,symmetrically and asymmetrically. The current-based ZVS conditions for inverterbridge 1 are given by

iA(A1) ≤ −IZVS1 (2.58)

iA(A2) ≥ IZVS1 , (2.59)

whereas the conditions for inverter bridge 2 are defined as

ia(a1) ≥ IZVS2 (2.60)

ia(a2) ≤ −IZVS2 , (2.61)

where the angles A1, A2, a1, and , a2 correspond to the switching instants of phase”A”. For the topologies with a star-star connected transformer, these are defined by

A1 = 0

A2 = 2d1π

a1 = φ + (d1 − d2)π

a2 = φ + (d1 + d2)π.

2.5 Summary 55

Three-phase DAB converters with a star-delta connected transformer:Considering a star-delta winding configuration, inverter bridge 2 is shifted by π/6to compensate for the phase-shift created by the transformer connection. Therefore,the angles corresponding to the ZVS conditions for inverter bridge 2 are given by

a1 = φ + (d1 − d2)π + π/6

a2 = φ + (d1 + d2)π + π/6.

The symmetrically operated two-level topologies require only two ZVS conditionsgiven by (2.58) and (2.60).

2.5 Summary

The discrete states of the switches within the three-phase DAB converter can be ef-fectively modelled by piece-wise linear (PWL) equations to describe the convertercurrents. The PWL equations are, however, only valid for a specific set of switchstates during a half or complete switching cycle, denoted by a switching mode.Due to the control variables, such as the phase-shift and duty-cycles, crossing spe-cific thresholds, the switch states can change and a different switching mode arises.Therefore, PWL modelling for topological variations of the three-phase DAB con-verter is presented together with a systematic approach to find the correspondingswitching modes and their boundaries. The two- and three-level topology variants,including different transformer winding configurations, are modelled for symmet-ric operation, whereas the two-level topology is also modelled for asymmetric op-eration. Furthermore, conditions are presented to verify ZVS operation, utilizinga current based method, for the investigated topological variants. The piecewise-linear models and ZVS conditions are used in the derivation of the modulationschemes, as brought forward in the next chapter.

Chapter 3

Modulation schemes

S WITCH-MODE dc-dc converters are electronic circuits with the purpose of pro-cessing power. Mostly, the goal is to regulate the output voltage efficiently, bycontrolling the power or current, in the presence of input voltage and output

load variations. For this reason, a modulation scheme is required which dictatesthe values of the control variables.

Within the symmetrically operated two-level three-phase DAB converter, thepower flow is controlled with the phase-shift between the inverter bridges [27].The ZVS operating range is relatively narrow and, theoretically, ZVS throughoutthe whole load range is only achieved for equal input and output voltages,considering a transformer turns-ratio of one. Because the converter contains onlyone degree of freedom to control the power, the PWL equations for the power,presented in chapter 2, are uniquely determined. Therefore, the performance of

Contributions of this chapter are published in:

• N. H. Baars, C. G. E. Wijnands, and J. Everts, ‘ZVS modulation strategy for a three-phase dualactive bridge dc-dc converter with three-level phase-legs’, in Proceedings of the IEEE EuropeanConference on Power Electronics and Applications, pp. 1–10, Sep. 2016.

• N. H. Baars, C. G. E. Wijnands, and J. Everts, ‘A three-level three-phase dual active bridge dc-dcconverter with a star-delta connected transformer’, in Proceedings of the IEEE Vehicle Power andPropulsion Conference, pp. 1–6, Oct. 2016.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Modulation strategy for wide-range ZVS operation of a three-level three-phase dual active bridge dc-dc converter’, in Proceed-ings of the IEEE Applied Power Electronics Conference and Exposition, pp. 3357–3364, Mar. 2017.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Modelling and modulation ofthree-level three-phase dual active bridge dc-dc converters’, IEEE Transactions on Power Electron-ics, under review, 2017.

57

58 Chapter 3 Modulation schemes

the converter can not be improved by solely phase-shift modulation. Differenttransformer winding configurations can increase the ZVS range to some extent, aswill be presented in the next chapter. Other options to increase the ZVS range areadding a reactive current by decreasing the transformer magnetizing inductance orincluding external commutation inductances [5, 66], operating in a discontinuousmanner [5, 81], or adding auxiliary circuits to ensure ZVS [97]. However, thesemethods deteriorate the power density and increase the cost due to the auxiliarycircuits and/or higher current stress of the converter components.

By increasing the degrees of freedom, more control variables are available to im-prove the converter performance, such as a wider ZVS operating range and lowercirculating currents. This has already been recognised for the single-phase DABconverter for which several modulation schemes are available [39, 71, 102, 106].Consequently, it is proposed to operate the three-phase DAB converter with a sin-gle phase [52], utilising the modulation scheme of a single-phase DAB converter.However, the advantages of the three-phase DAB converter, such as a small filtercomponents and lower switch currents, are lost when operated as a single phasetopology. Additionally, the maximum power is reduced because only one phasetransfers power and the transfer-inductance is increased by 50 percent.

To overcome the problems mentioned above, three-level phase-legs have beenadopted in the three-phase DAB converter, as presented in chapter 2, which resultsin three control variables for symmetric operation. Additionally, asymmetricoperation is considered for the two-level topology which also results in threedegrees of freedom, as identified in [54, 55, 73]. However, modulation schemesfor both approaches, covering a wide operating-range, are missing. This chapterpresents analytical modulation schemes, which result in ZVS operation and close-to-minimal circulating currents, for both the two- and three-level three-phase DABconverter considering asymmetric and symmetric operating methods, respectively.

3.1 Method

The two- and three-level three-phase DAB converters with, respectively, asymmet-ric and symmetric operating methods have three control variables for fixed fre-quency operation. However, the system of equations is under determined sincethere is one equation, the required power, and three unknowns (d1, d2, and φ).Therefore, additional objectives, such as minimal circulating currents and ZVS op-eration, can be included in order to obtain a uniquely determined system.

As a first step, a numerical optimiser is used to derive optimal control variablesthat result in ZVS operation and minimal circulating currents. The optimisation

3.1 Method 59

problem is described in section 3.1.1, whereas the operating range and converterparameters are given in section 3.1.2.

Based on the optimisation result with three degrees of freedom, the second step isto replace a control variable by a constant and verify if ZVS operation and close-to-minimal circulating currents are still preserved for medium to high power con-ditions. If this is the case, only two degrees of freedom remain and the amount ofswitching modes is reduced significantly. From this point, the power equality andthe active ZVS boundaries can be conveniently displayed in a two-dimensionalgraph to derive the second equation to make the system uniquely determined.

A third step is required if the previous step does not lead to ZVS operation andclose-to-minimal circulating currents. This implies that all three degrees of freedommust be utilised to achieve optimal operation. Consequently, two equations, inaddition to the power equation, must be found to solve the three control variablesanalytically. These can be found by analysing the switch turn-off currents, derivedwith the numerical optimiser.

3.1.1 Numerical optimiser

As explained earlier, a numerical optimiser is used to obtain control variables thatresult in the lowest rms current in the transfer-inductor while respecting the powerequality and the ZVS inequality constraints. This has been identified to result inthe lowest conduction and switching losses [39, 71]. The optimisation problem isgiven as

minimise IL(d1, d2, φ)

subject to P(d1, d2, φ) = P∗

iA(A1) ≤ −IZVS1

−iA(A2) ≤ −IZVS1

−ia(a1) ≤ −IZVS2

ia(a2) ≤ −IZVS2

with:

0 ≤ d1 ≤ 1/2

0 ≤ d2 ≤ 1/2

0 ≤ φ ≤ π.

(3.1)

The equality constraint ensures that the average power P equals the reference valueP∗, and the four current-based inequality constraints, representing switch turn-offcurrents, provide the minimum theoretic conditions for ZVS operation as presentedin section 2.4. Furthermore, Fourier-based models [40, 109] of the three-phase DABconverters with a high number of harmonics are used for their straightforwardimplementation and high computational speed.


3.1.2 Operating range

The control variables will be optimised for a certain voltage and power range. Thevoltage range is given by a voltage ratio U1/U2 from 1/2 to 2 for a fixed voltage U2.For this, a transformer turns ratio N of 1 and 1/

√3 is considered for, respectively, a

star-star and star-delta connected transformer.

The power is varied from 0 to 1 p.u., where 1 p.u. equals maximum power forU1/U2 = 1/2 and φ = π/3. Furthermore, it is ensured that for the selected operat-ing range the optimiser is able to find a solution that satisfies the ZVS inequalityconstraints by providing sufficient commutation current for both inverter bridges.This current is provided by the commutation inductances L1 and L2.

3.2 Symmetrical operation of three-level three-phaseDAB converters

This section presents modulation schemes for the three-level topologies with a star-star and a star-delta connected transformer considering symmetric operation, asdescribed in section 2.2.

3.2.1 Optimal modulation for a star-star connected transformer

Following the first step of the procedure described in section 3.1, optimal controlvariables are derived for the three-level three-phase DAB converter with a star-star connected transformer. The resulting three control variables are shown in fig-ure 3.1a. It can be seen that the three control variables can be approximated byd1 = 1/2 for U1/U2 ≤ 1 and d2 = 1/2 for U1/U2 ≥ 1. This results in two degreesof freedom and creates two optimisation problems, respectively, for voltage ratiosbelow and above one. The result of the optimisation with two control variables isshown in figure 3.1b.


d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2

(a) Three degrees of freedom.

d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2

(b) Two degrees of freedom.

Figure 3.1: Control variables, obtained with a numerical optimiser, of the three-level DABconverter with a star-star connected transformer for three (a) and for two degrees of freedom(b). The control variables d1, d2, and φ are displayed from top to bottom, respectively.


U1/U2[-]

P [p.u.]

64%32% 16% 8% 4% 2% 1%

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

Figure 3.2: Contour plot of ∆Iopt2,opt3, defined by (3.2), representing the difference of rmscurrent through the transfer-inductor, in percent, for the symmetrically operated three-leveltopology with a star-star connected transformer using an optimal modulation scheme withtwo and three degrees of freedom.

Comparing optimal modulation with two and three degrees of freedom

It is expected that utilising only two, instead of three, degrees of freedom does notresult in minimal rms current through the transfer-inductor for the entire operatingrange. To compare this, the difference in rms current is normalised to the results ofthe optimal modulation scheme with three control variables and is given as

∆Iopt2,opt3 =Iopt2 − Iopt3

Iopt3× 100 %. (3.2)

The contour lines of this function are presented in figure 3.2. It is visible that two de-grees of freedom lead to higher rms currents in the transformer. However, for highoutput powers there is no significant difference. The reduced degrees of freedom,therefore, only affect the efficiency at low power levels (P < 0.5 p.u.). Another im-portant result is that by using only two degrees of freedom the amount of switchingmodes is reduced from 108 to 24 modes for 0 ≤ φ ≤ π/2. The remaining 24 modesare found on the φd1 (d2 = 1/2) and φd2 (d1 = 1/2) planes visible in figure 2.16.

3.2.2 Analytical modulation for a star-star connected transformer

In the previous it is shown that two, instead of three, control variables still preservelow circulating currents for high power levels and reduce the number of switching


φ

d2

34

33

32

31

24

23

22

21

14

13

12

11

0 1/6 1/3 1/2

-π/6

0

π/6

2π/6

3π/6

4π/6

Figure 3.3: Switching modes for the three-level three-phase DAB converter with a star-starconnected transformer with d1 = 1/2.

modes significantly. This simplifies the analysis of the ZVS boundaries and allowsto derive analytical solutions using the PWL model for a wide range of voltageratios and power levels.

Analysis for U1/U2 < 1

As shown in section 3.2.1, the duty-cycle d1 can be fixed to 1/2 for voltage ratiosbelow one while a close-to-minimal rms current in the transfer-inductor is still pos-sible for P > 0.5 p.u. This results in twelve switching modes for 0 ≤ φ ≤ π/2,for which the switching mode boundaries are shown in figure 3.3. Consequently,only one ZVS inequality constraint remains for the first inverter bridge since theduty-cycle d1 is fixed to 1/2 and the current is odd symmetric (iA(0) = −iA(π)).Furthermore, it is found from the optimisation that only one inequality constraintof the second inverter bridge, i.e. ia(a2) ≤ 0, is an active constraint. Therefore, oneZVS inequality constraint of each inverter bridge will be investigated, utilising theanalytical model, for the given voltage and power range. To simplify the analysis,the magnetizing currents (iL1 , iL2 ) and the minimum ZVS commutation currents(IZVS1 , IZVS2 ) are considered zero. As a consequence, the remaining ZVS inequalityconstraints are given by

iA(A1) = iLA(A1) ≤ 0 (3.3)

ia(a2) = iLA(a2) ≤ 0, (3.4)


where ’A1’ and ’a2’ refer to the rising and falling voltage transition of, respectively,phase-leg ’A’ of inverter bridge 1 and phase-leg ’a’ of inverter bridge 2, as shownin figure 2.14.

As shown in figure 3.4, the boundaries of the ZVS inequality constraints, i.e.iL(A1) = 0 (red line) and iL(a2) = 0 (green line), are solved for a given voltageratio using the PWL model. Consequently, the control variables which resultin ZVS operation are given by the enclosed area of the green and red line. Byintroducing the expression iL(A1) = iL(a2) (blue line), the control variables arefound for operating between the ZVS boundaries line (green and red). The blueline contains solutions for the required power range since it intersects the dashedand solid black lines, representing power levels of, respectively, 1 and 0.1 p.u.,as can be seen in figure 3.4. As a result, the control variables can be found bysolving the power equality and including the expression iL(A1) = iL(a2) for theappropriate switching modes, utilising the PWL model presented in chapter 2.

However, when the ZVS boundaries (green and red lines) intersect or do not en-close a single area, as shown in figure 3.4a and figure 3.4b, no control variablescan be found for a certain power range which lead to ZVS. This can be solved byincluding L1 and L2, representing the transformer magnetizing inductance or ex-ternal commutation inductors. As an example, a magnetizing current is includedwith L1 = L2 = 10LA. This is shown in figure 3.4c and figure 3.4d. As a conse-quence, the ZVS boundaries (green and red lines) are shifted and the enclosed areahas increased. The solutions for iA(A1) = ia(a2) (blue line) remain unchanged andare now resulting in ZVS for the complete power range.

Analysis for U1/U2 > 1

The same switching modes as those used for voltage ratios below one can be usedfor analysing voltage ratios above one. This is permitted by interchanging the volt-ages (U1,U2) and the duty-cycles (d1,d2), given by

P(U1, U2, d1, d2, φ) = P(U2, U1, d2, d1, φ). (3.5)

As a consequence, similar or equivalent curves for the current expressions arefound, as shown in figure 3.5. The power curves, however, are not the same sincethe input voltage U1 is varied. Therefore, different control variables are found forU1/U2 < 1 and U1/U2 > 1 for the same power.


d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(a) U1/U2 = 1/2 and L1 = L2 = ∞.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(b) U1/U2 = 3/4 and L1 = L2 = ∞.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(c) U1/U2 = 1/2 and L1 = L2 = 10LA.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(d) U1/U2 = 3/4 and L1 = L2 = 10LA.

Figure 3.4: Plots of φ(d2) for the three-level topology with a star-star connected transformer,considering the expressions of iA(A1) = 0 (red line), ia(a2) = 0 (green line), iA(A1) = ia(a2)(blue line), and power levels of 0.1 p.u. and 1 p.u. (solid and dashed black lines), for differentvoltage ratios and magnetizing inductances L1 and L2.

Analysis for U1/U2 = 1

For the voltages U1 and U2 being equal, symmetric two-level operation (d1 = d2 =1/2) results in the lowest rms current in the transfer-inductor, which is also visiblein figure 3.1. The required phase-shift can be directly calculated with the powerequations corresponding to switching modes 32 and 34 or using (2.11).


d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(a) U1/U2 = 5/4 and L1 = L2 = ∞.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(b) U1/U2 = 2 and L1 = L2 = ∞.

Figure 3.5: Plots of φ(d1) for the three-level topology with a star-star connected transformer,considering the expressions of iA(A1) = 0 (red line), ia(a2) = 0 (green line), iA(A1) = ia(a2)(blue line), and power levels of 0.1 p.u. and 1 p.u. (solid and dashed black lines), for differentvoltage ratios.

d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

(a)

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

(b)

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2

(c)

Figure 3.6: The control variables d1 (a), d2 (b), and φ (c), calculated with the proposed an-alytical modulation scheme for the three-level three-phase DAB converter with a star-starconnected transformer.

Resulting modulation scheme

The derived modulation scheme requires six switching modes (21, 22, 23, 31, 32,and 33) for the whole voltage and power range given in section 3.1. The corre-sponding control variables, derived with the proposed modulation scheme, areshown in figure 3.6. The result is very similar to the optimised control variables, us-ing two degrees of freedom, as shown in figure 3.1b. The corresponding equationsfor the duty-cycle, phase-shift and switching-mode boundaries for each switchingmode are given in appendix B.1.


U1/U2[-]

P [p.u.]

8% 4% 2% 1% 1%

64% 32% 16% 8% 4% 2% 1%

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

Figure 3.7: Contour plot of ∆Imod,opt3, defined by (3.6), representing the difference of rmscurrent through the transfer-inductor, in percent, for the symmetrically operated three-leveltopology with a star-star connected transformer using the analytical modulation scheme andoptimal modulation scheme with three degrees of freedom.

Comparison of the analytical and optimal modulation scheme

The control variables, derived with the proposed modulation scheme, are not op-timal since the result from the numerical optimiser is approximated and only twoinstead three degrees of freedom are used. Therefore, the rms current through thetransfer-inductor corresponding to the analytical modulation scheme is comparedto the current that results from optimiser with three degrees of freedom. The dif-ference in current is normalised to the current that results from the optimiser withthree degrees of freedom, as given by

∆Imod,opt3 =Imod − Iopt3

Iopt3× 100 %. (3.6)

The outcome is presented by a contour plot given in figure 3.7. It can be seen thatthe analytical modulation scheme generates higher circulating currents comparedto the optimal modulation scheme. However, the increase remains within 8 % forpower levels above 0.5 p.u. and, is therefore considered useful for the targetedapplications.


Simulation results

The proposed analytical modulation scheme is implemented using Matlab®

Simulink and the circuit is modelled with the PLECS® blockset. The simulation iscarried out using ideal switches and circuit parameters as given in table 3.1.

Table 3.1: Simulation parameters for three-level three-phase DAB converter with a star-starconnected transformer.

Parameter Value

DC-link voltage, U1 375-1500 VDC-link voltage, U2 750 VTransfer-inductors, LA, LB, and LC 11.7 µHCommutation inductors, L1 and L2 10LA = 117 µHMaximum power level 100 kWSwitching frequency, fsw 20 kHzTransformer turns ratio, N 1:1Dead-time, tdt 500 ns

Using the proposed modulation scheme, the control variables are simulated for areference power of 10 kW and 100 kW (0.1 p.u. and 1 p.u.), while the voltage U1 isvaried from 375 V to 1500 V with U2 kept constant, considering steady-state condi-tions. The resulting power and the corresponding control variables are shown infigure 3.8. From this, it can be concluded that the actual power closely follows thereference power for a wide voltage-range. The control variables show continuousbehaviour for both power levels, which is highly desirable from a control perspec-tive.

Circuit simulations are performed to verify ZVS operation of the three-level topol-ogy with a star-star connected transformer, using the proposed analytical mod-ulation scheme, for a wide range of voltage and power levels. Simulations of thephase-leg voltages and currents of both inverter bridges, considering different volt-age and power levels, are presented in figure 3.9 and figure 3.10. This shows thatthe converter is operating as expected and achieves ZVS for the investigated rangeof voltage and power levels.


P=0.1 p.u.

P

2φ/π

d2

d1

P=1 p.u.

P

2φ/π

d2

d1

U1 [V]

375 500 625 750 875 1000 1125 1250 1375 15000

0.2

0.4

0.6

0.8

1

Figure 3.8: Steady-state simulation of the control variables and the power, utilising theproposed modulation scheme, for the three-level topology with a star-star connected trans-former considering a reference power of 10 kW and 100 kW (0.1 p.u. and 1 p.u.) while thevoltage U1 is varied from 375 V to 1500 V with U2 = 750 V.


i a[A]

uan[V]

t [µs]

i A[A]

φ =0.095d2=0.247d1=0.500

uAN[V]

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

100

−500

−250

0

250

500

−100

−50

0

50

100

−500

−250

0

250

500

uan

ia

iA

uAN

(a)

i a[A]

uan[V]

t [µs]

i A[A]

φ =0.044d2=0.500d1=0.282

uAN[V]

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

100

−500

−250

0

250

500

−100

−50

0

50

100

−500

−250

0

250

500

uan

ia

iA uAN

(b)

Figure 3.9: Simulated waveforms of the symmetrically operated three-level topology with astar-star connected transformer for U1 = 500 V (a) and U1 = 1000 V (b), with U2 = 750 Vand P = 10 kW (0.1 p.u.), using the analytical modulation scheme.


i a[A

]

uan[V

]

t [µs]

i A[A

]

φ =0.845d2=0.325d1=0.500

uAN[V

]

0 10 20 30 40 50 60 70 80 90 100

−250

−125

0

125

250

−500

−250

0

250

500

−250

−125

0

125

250

−500

−250

0

250

500

uAN

iA

iauan

(a)

i a[A]

uan[V]

t [µs]

i A[V]

φ =0.403d2=0.500d1=0.320

uAN[V]

0 10 20 30 40 50 60 70 80 90 100

−250

−125

0

125

250

−500

−250

0

250

500

−250

−125

0

125

250

−500

−250

0

250

500uAN

iA

ia

uan

(b)

Figure 3.10: Simulated waveforms of the symmetrically operated three-level topology witha star-star connected transformer for U1 = 500 V (a) and U1 = 1000 V (b), with U2 = 750 Vand P = 100 kW (1 p.u.), using the analytical modulation scheme.


3.2.3 Optimal modulation for a star-delta connected transformer

In this section an optimal modulation scheme is derived for the three-level three-phase DAB converter with a star-delta connected transformer, following the firststep of the procedure described in section 3.1. The three optimised control variablesd1, d2, and φ are shown in figure 3.11a.

Similar to the three-level topology with a star-star connected transformer, the resultfrom the optimal modulation scheme with three control variables is approximatedby d1 = 1/2 for U1/U2 ≤ 1 and d2 = 1/2 for U1/U2 ≥ 1. This results in two degreesof freedom and creates two optimisation problems, respectively, for voltage ratiosbelow and above one. The control variables that result from optimising with twodegrees of freedom are shown in figure 3.11b.


The rms current through the transfer-inductor utilising an optimal modulationscheme with two and three degrees of freedom is compared using (3.2). Theresult is given as a contour plot which is presented in figure 3.12. Similar to thethree-level topology with a star-star connected transformer, there is no significantdifference in the cost function for powers above 0.5 p.u.. The reduced degrees offreedom, therefore, only affect the efficiency at low power levels (P < 0.5 p.u.).Furthermore, using two degrees of freedom reduces the number of switchingmodes from 108 to 24 for 0 ≤ φ ≤ π/2. The remaining modes are found on the(φ, d1) (d2 = 1/2) and (φ, d2) (d1 = 1/2) planes visible in figure 2.19.

3.2.4 Analytical modulation for a star-delta connected transformer

Similar to the three-level topology with star-star connected transformer, it is shownthat two, instead of three, control variables still preserve low circulating currentsfor high power levels and reduce the number of switching modes significantly. Us-ing the same approach as presented in section 3.2.2, the derivation of an analyti-cal modulation scheme for the symmetrically operated three-level topology with astar-delta connected transformer is given in the following.


d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2


d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2


Figure 3.11: Control variables, obtained with a numerical optimiser, of the three-level DABconverter with a star-delta connected transformer for three (a) and for two degrees of free-dom (b). The control variables d1, d2, and φ are displayed from top to bottom, respectively.

Analysis for U1/U2 < 1

As shown in section 3.2.3, the duty-cycle d1 can be fixed to 1/2 for voltage ratiosbelow one while a close-to-minimal rms current in the transfer-inductor is still pos-sible for P > 0.5 p.u. This results in twelve switching modes for 0 ≤ φ ≤ π/2, for


U1/U2[-]

P [p.u.]

1%1%

1%

16%

8%

4%

2%

1%1%

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

32%

64%

Figure 3.12: Contour plot of ∆Iopt2,opt3, defined by (3.2), representing the difference of rmscurrent through the transfer-inductor, in percent, for the symmetrically operated three-leveltopology with a star-delta connected transformer using an optimal modulation scheme withtwo and three degrees of freedom.

φ

d2

34

33

32

31

24

23

22

21

14

13

12

11

0 1/6 1/3 1/2

-π/6

0

π/6

2π/6

3π/6

4π/6

Figure 3.13: Switching modes for the three-level three-phase DAB converter with a star-deltaconnected transformer with d1 = 1/2.

which the switching mode boundaries are shown in figure 3.13.

As a consequence of the duty-cycle d1 being substituted by 1/2, only one ZVS in-equality constraint for inverter bridge 1 is required since the current is odd sym-metric (iA(0) = −iA(π)). Furthermore, it is found from the optimisation that onlyone inequality constraint of inverter bridge 2, i.e. ia(a2) ≤ 0, is an active constraint.


Therefore, one ZVS inequality constraint of each inverter bridge will be investi-gated, utilising the analytical model, for the operating range given in section 3.1.To simplify the analysis, the magnetizing currents (iL1 , iL2 ) and the minimum ZVScommutation currents (IZVS1 , IZVS2 ) are considered zero. As a consequence, the twoZVS inequality constraints are given by

iA(A1) = iLA(A1) ≤ 0 (3.7)

ia(a2) = iLA(a2)− iLC(a2) ≤ 0, (3.8)

where ’A1’ and ’a2’ refer to the rising and falling voltage transition of, respectively,phase-leg ’A’ of inverter bridge 1 and 2, as shown in figure 2.18.

The boundaries of the ZVS inequality constraints, given above, are solved using thePWL model for a given voltage ratio below one, as shown in figure 3.14. The ZVSboundaries, i.e. iA(A1) = 0 (red line) and ia(a2) = 0 (green line) enclose an areawhich contains control variables that result in ZVS. By introducing the expressioniL(A1) = iL(a2), the corresponding solution curve (blue line) is located betweenthe ZVS boundaries (green and red lines) and intersects the curves for a powerlevel of 0.1 and 1 p.u. (solid and dashed black lines). Thus, the control variablescan be found by solving the power equality and including the expression iL(A1) =iL(a2) for the appropriate switching modes, utilising the PWL model presented inchapter 2.

However, when the boundaries (green and red lines) intersect or do not enclose asingle area, as shown in figure 3.14a and figure 3.14b, no control variables can befound for a certain power range which lead to ZVS. This can be solved by includ-ing L1 and L2, representing the transformer magnetizing inductance or externalcommutation inductors. As an example, a magnetizing current is included withL1 = L2 = 8LA. This is shown in figure 3.14c and figure 3.14d. As a consequence,the ZVS boundaries (green and red lines) are shifted and the enclosed area has in-creased. The solutions for iA(A1) = ia(a2) (blue line) remain unchanged and arenow resulting in ZVS for the complete power range.

Analysis for U1/U2 > 1

Due to the symmetry in the power flow, given by (3.5), the switching modes forvoltage ratios below one can also used for voltage ratios above one. For this,only the voltages (U1 and U2) and the duty-cycles (d1 and d2) need to be inter-changed. As a consequence, similar or equivalent curves for the current expres-sions are found, as shown in figure 3.15. The power curves, however, are not the


d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(a) U1/U2 = 1/2 and L1 = L2 = ∞.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(b) U1/U2 = 3/4 and L1 = L2 = ∞.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(c) U1/U2 = 1/2 and L1 = L2 = 8LA.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(d) U1/U2 = 3/4 and L1 = L2 = 8LA.

Figure 3.14: Plots of φ(d2) for the three-level topology with a star-delta connected trans-former, considering the expressions of iA(A1) = 0 (red line), ia(a2) = 0 (green line),iA(A1) = ia(a2) (blue line), and power levels of 0.1 p.u. and 1 p.u. (solid and dashed blacklines), for different voltage ratios and magnetizing inductances L1 and L2.

same since the input voltage U1 is varied. Therefore, different control variables arefound for U1/U2 < 1 and U1/U2 > 1 considering the same power.

Analysis for U1/U2 = 1

For a unity voltage ratio, symmetric two-level operation (d1 = d2 = 1/2) is ap-plied. The required phase-shift can be directly calculated with the power equationscorresponding to switching modes 31 and 33 or using (2.16).


d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(a) U1/U2 = 5/4 and L1 = L2 = ∞.

d2

342414

332313

322212

312111

0 1/6 1/3 1/20

(b) U1/U2 = 2 and L1 = L2 = ∞.

Figure 3.15: Plots of φ(d1) for the three-level topology with a star-delta connected trans-former, considering the expressions of iA(A1) = 0 (red line), ia(a2) = 0 (green line),iA(A1) = ia(a2) (blue line), and power levels of 0.1 p.u. and 1 p.u. (solid and dashed blacklines).

d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

(a)

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

(b)

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2

(c)

Figure 3.16: The control variables d1 (a), d2 (b), and φ (c), calculated with the proposed mod-ulation scheme for the three-level three-phase DAB converter with a star-delta connectedtransformer.

Resulting modulation scheme

The derived modulation scheme requires six switching modes (21, 22, 23, 24, 31,and 32) for the investigated voltage and power range given in section 3.1. Thecorresponding control variables, derived with the proposed modulation scheme,are shown in figure 3.16.


U1/U2[-]

P [p.u.]

64%

32% 16% 8% 4% 2% 1%

1%

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

Figure 3.17: Contour plot of ∆Imod,opt3, defined by (3.6), representing the difference of rmscurrent through the transfer-inductor, in percent, for the symmetrically operated three-leveltopology with a star-delta connected transformer using the analytical modulation schemeand optimal modulation scheme with three degrees of freedom.


The control variables, derived with the proposed modulation scheme, are not op-timal since the result from the numerical optimiser is approximated and only twoinstead three degrees of freedom are used. Therefore, the rms currents throughthe transfer-inductor using the analytical and the optimal modulation scheme withthree degrees of freedom are compared. The current corresponding to the analyticmodulation scheme is normalised to the current resulting from the optimal mod-ulation scheme with three degrees of freedom, as given in (3.6). The outcome ispresented by a contour plot in figure 3.17.

It can be seen that the analytical modulation scheme generates higher rms circulat-ing currents compared to the optimal modulation scheme. However, the increaseremains within 10 % for power levels above 0.5 p.u., which is considered useful forthe targeted applications.

Simulation results

The proposed modulation scheme is implemented using Matlab® Simulink and thecircuit is modelled with the PLECS® blockset. The simulation is carried out usingideal switches and circuit parameters as given in table 3.2.


Table 3.2: Simulation parameters for three-level three-phase DAB converter with a star-deltaconnected transformer.

Parameter Value

DC-link voltage, U1 375-1500 VDC-link voltage, U2 750 VTransfer-inductors, LA, LB, and LC 11.5 µHCommutation inductors, L1 and L2 8LA = 92 µHMaximum power level 100 kWSwitching frequency, fsw 20 kHzTransformer turns ratio, N 1:

√3

Dead-time, tdt 500 ns

Using the proposed modulation scheme, the control variables are simulated for areference power of 10 kW and 100 kW (0.1 p.u. and 1 p.u.), while the voltage U1 isvaried from 375 V to 1500 V with U2 kept constant, considering steady-state condi-tions. The resulting power and the corresponding control variables are shown infigure 3.18. From this, it can be concluded that the actual power closely follows thereference power for a wide voltage-range. In addition, continuous behaviour of thecontrol variables can be seen for both power levels, which is highly desirable froma control perspective.

Simulations of the three-level topology with a star-delta connected transformer, us-ing the proposed modulation scheme, are carried out to verify ZVS operation. Fig-ure 3.19 and figure 3.20 show the simulated phase-leg voltage and current for bothinverter bridges considering different voltage and power levels. This confirms ZVSoperation for the investigated voltage and power range.


P=0.1 p.u.

P

2φ/π

d2

d1

P=1 p.u.

P

2φ/π

d2

d1

U1 [V]

375 500 625 750 875 1000 1125 1250 1375 15000

0.2

0.4

0.6

0.8

1

Figure 3.18: Steady-state simulation of the control variables and the power, using the pro-posed modulation scheme, for the three-level topology with a star-delta connected trans-former considering a reference power of 10 kW and 100 kW (0.1 p.u. and 1 p.u.) while thevoltage U1 is varied from 375 V to 1500 V with U2 = 750 V.


i a[A

]

uan[V

]

t [µs]

i A[A

]

φ =0.088d2=0.254d1=0.500

uAN[V

]

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

100

−500

−250

0

250

500

−100

−50

0

50

100

−500

−250

0

250

500

iA

uAN

iauan

(a)

i a[A

]

uan[V

]

t [µs]

i A[A

]

φ =0.039d2=0.500d1=0.285

uAN[V

]

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

100

−500

−250

0

250

500

−100

−50

0

50

100

−500

−250

0

250

500uAN

uan

ia

i A

(b)

Figure 3.19: Simulated waveforms of the symmetrically operated three-level topology witha star-delta connected transformer for U1 = 500 V (a) and U1 = 1000 V (b), with U2 = 750 Vand P = 10 kW (0.1 p.u.), using the analytical modulation scheme.


i a[A

]

u an[V

]

t [µs ]

i A[A

]

φ =0.848d2=0.327d1=0.500

u AN[V

]

0 10 20 30 40 50 60 70 80 90 100

−300

−150

0

150

300

−500

−250

0

250

500

−300

−150

0

150

300

−500

−250

0

250

500

uAN

iA

iauan

(a)

i a[A

]

uan[V

]

t [µs]

i A[A

]

φ =0.375d2=0.500d1=0.320

uAN[V

]

0 10 20 30 40 50 60 70 80 90 100

−300

−150

0

150

300

−500

−250

0

250

500

−300

−150

0

150

300

−500

−250

0

250

500

iA

ia

uAN

uan

(b)

Figure 3.20: Simulated waveforms of the symmetrically operated three-level topology witha star-delta connected transformer for U1 = 500 V (a) and U1 = 1000 V (b), with U2 = 750 Vand P = 100 kW (1 p.u.), using the analytical modulation scheme.


3.3 Asymmetrical operation of two-level three-phaseDAB converters

The two-level three-phase DAB converter can be operated asymmetrically by con-trolling the duty-cycle of both inverter bridges and the phase-shift between them,resulting in three control variables as described in section 2.3. Similar to the sym-metrically operated three-level topologies, a numerical optimiser is used to find thecontrol variables that result in the lowest rms current through the transfer-inductorwhile ZVS is achieved for a wide voltage and power range. The correspondingoptimisation problem and the investigated voltage and power range is given insection 3.1.

3.3.1 Optimal modulation schemes

The optimised control variables, derived with a numerical optimiser, for three de-grees of freedom are shown in figure 3.21a. It can be observed that the duty-cyclesof the inverter bridges do not exceed 1/2, even though this is allowed for asymmet-ric operation.

The result from the optimal modulation scheme with three control variables is ap-proximated by d1 = 1/2 for U1/U′2 ≤ 1 and d2 = 1/2 for U1/U′2 ≥ 1, similar asdone for the symmetrically operated three-level topologies. This results in two de-grees of freedom and creates two optimisation problems, respectively, for voltageratios below and above one. The control variables that result from optimising twodegrees of freedom are shown in figure 3.21b.


The rms current through the transfer-inductor is compared for operating with theoptimal modulation scheme with two and three degrees of freedom. The differenceis normalised to the result for three degrees of freedom using (3.2). The result isgiven as a contour plot in figure 3.22. It can be observed that using only two degreesof freedom, with d1 = 1/2 for U1/U′2 ≤ 1 and d2 = 1/2 for U1/U′2 ≥ 1, results in asignificant increase of the rms current through the transfer-inductor. This impliesthat all three degrees of freedom must be utilised to achieve minimal or close-to-minimal circulating currents. Therefore, as a third step, the switch turn-off currentsfor optimal operation will be analysed below.


d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2


d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2


Figure 3.21: Control variables, obtained with a numerical optimiser, for asymmetric opera-tion of the two-level DAB converter, considering three (a) and for two degrees of freedom(b). The control variables d1, d2, and φ are displayed from top to bottom, respectively.

Analysis of the switch turn-off currents

Since two degrees of freedom do not allow close-to-optimal operation, three de-grees of freedom need to be considered for asymmetric operation of the two-level


U1/U2[-]

P [p.u.]

8%128% 64% 32% 16%

256% 128% 64% 32%

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

Figure 3.22: Contour plot of ∆Iopt2,opt3, defined by (3.2), representing the difference of rmscurrent through the transfer-inductor, in percent, for the asymmetrically operated two-leveltopology with a star-star connected transformer using an optimal modulation scheme withtwo and three degrees of freedom.

three-phase DAB converter. In order to solve the control variables (d1,d2, and φ),two equations, in addition to the power equation, are required. Therefore, theswitch turn-off currents are analysed for similarities or patterns. These are shownshown in figure 3.23 for the given operating range. From this, the following char-acteristics can be seen:

1. The switch turn-off current iA(A1) (figure 3.23a), belonging to inverter bridge1, is zero for approximately the same operating range as the switch turn-offcurrent ia(a2) (figure 3.23d) that belongs to inverter bridge 2.

2. For voltage ratios below one, the switch turn-off current iA(A1) (figure 3.23a)is zero for approximately the same operating range as the switch turn-off cur-rent iA(A2) (figure 3.23b), with both currents belonging to inverter bridge 1.

3. For voltage ratios above one, the switch turn-off current ia(a1) (figure 3.23c)is zero for approximately the same operating range as the switch turn-off cur-rent ia(a2) (figure 3.23d), with both currents belonging to inverter bridge 2.

These findings are used to approximate the optimal modulation with analyticalequations, as will be brought forward in the following.


U1/U2

P [p.u.]

0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.5

1

1.5

2

(a) iA(A1)U2P [p.u.].

U1/U2

P [p.u.]

0 0.5 10

0.5

1

1.5

2

0.5

1

1.5

2

(b) iA(A2)U2P [p.u.].

U1/U2

P [p.u.]

0 0.5 10

0.5

1

1.5

2

0.5

1

1.5

2

(c) ia(a1)U2P [p.u.].

U1/U2

P [p.u.]

0 0.5 1

−1.5

−1

−0.5

0

0.5

1

1.5

2

(d) ia(a2)U2P [p.u.].

Figure 3.23: Switch turn-off currents of phase ’A’ for both inverter bridges considering asym-metric operation of the two-level three-phase DAB converter using an optimal modulationscheme, as shown in figure 3.21a.

3.3.2 Analytical modulation scheme

Based on the similarities in the switch turn-off currents using an optimal modula-tion scheme, given in section 3.3.1, a number of equations are introduced to derivethe control variables analytically. The first characteristic, which is valid for the com-plete voltage range, resulted in the following equation

iA(A1)U2 − ia(a2)U1 = 0. (3.9)

Another equation, found from the second characteristic, is true for voltage ratiosbelow one, and is given by

iA(A1) + iA(A2) = 0. (3.10)


d1

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

0.1

0.2

0.3

0.4

0.5

(a)

d2

U1/U2 P [p.u.]0

0.5

1

0.51

1.520

0.1

0.2

0.3

0.4

0.5

(b)

φ

U1/U2P [p.u.]0

0.5

1 0.51

1.52

0

π/6

π/3

π/2

(c)

Figure 3.24: The control variables d1 (a), d2 (b), and φ (c), calculated with the proposedmodulation scheme for asymmetric operation of the two-level three-phase DAB converter.

Consequently, the control variables are found, for voltage ratios lower than one,by combining (3.9), (3.10), and the power equality utilising the PWL model. Thisrequires only three switching modes, and the corresponding equations for theduty-cycles, phase-shift and switching-mode boundaries are given in appendix B.2.Then, similar to the three-level topologies, the equations are reused for voltageratios higher than one by interchanging the voltages (U1,U2) and duty-cycles(d1,d2). As a result, the control variables for the complete operating range can becalculated with the analytic modulation scheme, this is shown in figure 3.24. It canbe seen that the control variables obtained with the analytic modulation schemematches the optimal control variables presented in figure 3.21a.


Since the result of the optimal modulation scheme is approximated by the expres-sions (3.9) and (3.10), it is expected that the analytical modulation scheme is notoptimal. Therefore, the rms current through the transfer-inductor is compared foroperating with the analytical and optimal modulation scheme. For this, the differ-ence is normalised to the result from the optimal modulation scheme, accordingto (3.6). The resulting contour plot, shown in figure 3.25, reveals that the analyt-ical modulation scheme is close-to-optimal for a large operating range, whereasthe largest difference is less than 10 % for the lowest voltage ratio and maximumpower level. Therefore, the proposed analytical modulation is considered usefulfor the targeted applications.


U1/U2[-]

P [p.u.]

1%

3%2%1%

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

Figure 3.25: Contour plot of ∆Imod,opt3, defined by (3.6), representing the difference of rmscurrent through the transfer-inductor, in percent, for the asymmetrically operated two-leveltopology with a star-star connected transformer using the analytical modulation scheme andoptimal modulation scheme with three degrees of freedom.

Simulation results

The proposed modulation scheme is implemented using Matlab® Simulink and thecircuit is modelled with the PLECS® blockset. The simulation is carried out usingideal switches and circuit parameters as given in table 3.1.

Using the proposed modulation scheme, the control variables are simulated for areference power of 10 kW and 100 kW (0.1 p.u. and 1 p.u.), while the voltage U1 isvaried from 375 V to 1500 V with U2 kept constant, considering steady-state condi-tions. The resulting power and the corresponding control variables are shown infigure 3.26. From this, it can be concluded that the actual power closely follows thereference power and the control variables, as function of the input voltage, showcontinuous behaviour for both power levels.

Circuit simulations of the asymmetrically operated two-level topology, using theproposed analytical modulation scheme, are performed to verify ZVS operationfor a wide voltage and power range. The phase-leg voltages and currents of bothinverter bridges are simulated for different voltage and power levels, as shown infigure 3.27 and figure 3.28. The simulations confirm ZVS operation and presentasymmetric voltage and current waveforms, as expected.


P=0.1 p.u.

P

2φ/π

d2

d1

P=1 p.u.

P

2φ/π

d2

d1

U1 [V]

375 500 625 750 875 1000 1125 1250 1375 15000

0.2

0.4

0.6

0.8

1

Figure 3.26: Steady-state simulation of the control variables and the power, utilising the pro-posed modulation scheme, for the asymmetrically operated two-level topology consideringa reference power of 10 kW and 100 kW (0.1 p.u. and 1 p.u.) while the voltage U1 is variedfrom 375 V to 1500 V with U2 = 750 V.


i a[A

]

uan[V

]

t [µs]

i A[A

]

φ =0.176d2=0.112d1=0.168

uAN[V

]

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

100

−500

−250

0

250

500

−100

−50

0

50

100

−500

−250

0

250

500

uAN

uan

ia

iA

(a)

i a[A]

uan[V]

t [µs]

i A[A]

φ =0.101d2=0.129d1=0.097

uAN[V]

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

100

−500

−250

0

250

500

−100

−50

0

50

100

−500

−250

0

250

500

uan

ia

uAN

iA

(b)

Figure 3.27: Simulated waveforms of the asymmetrically operated two-level topology forU1 = 500 V (a) and U1 = 1000 V (b), with U2 = 750 V and P = 10 kW (0.1 p.u.), using theanalytical modulation scheme.


i a[A]

uan[V]

t [µs]

i A[A]

φ =0.861d2=0.296d1=0.412

uAN[V]

0 10 20 30 40 50 60 70 80 90 100

−350

−175

0

175

350

−500

−250

0

250

500

−350

−175

0

175

350

−500

−250

0

250

500

iA

uAN

ia

uan

(a)

i a[A]

uan[V]

t [µs]

i A[V]

φ =0.382d2=0.351d1=0.264

uAN[V]

0 10 20 30 40 50 60 70 80 90 100

−350

−175

0

175

350

−500

−250

0

250

500

−350

−175

0

175

350

−500

−250

0

250

500

iA

uAN

ia

uan

(b)

Figure 3.28: Simulated waveforms of the asymmetrically operated two-level topology forU1 = 500 V (a) and U1 = 1000 V (b), with U2 = 750 V and P = 100 kW (1 p.u.), using theanalytical modulation scheme.


3.4 Summary

This chapter presents modulation schemes for symmetric and asymmetric opera-tion of, respectively, three- and two-level three-phase DAB converters. As a firststep, optimal control variables were derived using a numerical optimiser that re-sult in ZVS operation and minimum rms current through the transfer-inductor fora wide voltage and power range.

In a second step, it is shown that the symmetrically operated three-level topolo-gies require two, instead of three, degrees of freedom to achieve soft-switching andminimal circulating currents for high power levels. This reduces the total num-ber of switching modes significantly and enables graphical analysis of the ZVSboundaries and power equations. Consequently, analytical modulation schemesare derived for the symmetrically operated three-level topologies that result in soft-switching and close-to-minimal circulating currents for high power levels.

Two degrees of freedom did not result in close-to-minimal circulating currents forthe asymmetrically operated two-level topology. Therefore, three degrees of free-dom are required for this converter. In order to derive an analytical modulationscheme, a third step was taken that involved analysis of the switch turn-off cur-rents corresponding to optimal operation. This resulted in two equations whichapproximate the optimal switch turn-off currents. Together with the power equa-tion, the three control variables of the asymmetrically operated two-level topologycan be solved analytically and result in soft-switching and close-to-minimal circu-lating currents.

The proposed modulation schemes for the different topologies and operating meth-ods are verified in simulation for several voltage and power levels. Furthermore,the equations of the modulation schemes and the corresponding switching modesare given in appendix B for the symmetrically and asymmetrically operated topolo-gies with star-star connected transformer.

The performance of both asymmetric and symmetric operation of, respectively,two-level and three-level three-phase DAB converters will be evaluated in chap-ter 5, using the modulation schemes presented in this chapter. The next chapter,first, investigates the impact of different transformer winding configurations onthe two-level topology, considering only symmetric operation as modelled in sec-tion 2.1.

Part II

Evaluation and ExperimentalResults

93

Chapter 4

Impact of transformerconnections on the two-level

topology

T HE ac-link of the three-phase DAB converter is realised with a three-phasetransformer. The transformer can be realised with one, preferably symmetri-cal, three-phase transformer or with three single-phase transformers which

are connected in a three-phase configuration. This provides the same function-ality since the inter-phase coupling does not affect the power [27]. Single-phasetransformers can be constructed more conveniently with commercially availablemagnetic cores, in contrast to a symmetric three-phase transformer. However, thesingle-phase transformer configurations require more magnetic core material andthereby have more hysteresis losses due to the absence of the flux cancelling effectin three-phase transformers [83, 104].


• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Impact of different transformer-winding configurations on the performance of a three-phase dual active bridge dc-dc converter’,in Proceedings of the IEEE Energy Conversion Congress and Exposition, pp. 637–644, Sep. 2015.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Performance evaluation of a three-phase dual active bridge dc-dc converter with different transformer winding configurations’,IEEE Transactions on Power Electronics, vol. 31, no. 10, pp. 6814–6823, Oct. 2016.

95

96 Chapter 4 Impact of transformer connections on the two-level topology

The transformer windings can be connected in different ways, regardless of therealizations discussed above. The primary and secondary windings are often con-nected in a star configuration [20, 27, 31, 52, 53, 80, 89, 96, 97, 100, 101]. This isfurther referred to as star-star winding configuration or in short Y-Y. Another com-mon transformer connection is the star-delta configuration, in short Y-∆, where oneside of the transformer is connected in star and the other side in delta. In [60], theimpact of these configurations on a three-phase single active bridge (SAB) is investi-gated. It is reported that the selected configurations shape the current through thetransfer-inductor differently and, consequently, influence the switch turn-off cur-rents and the current through the dc-link capacitors. This resulted in the highestefficiency and power density for the star-delta connection. Furthermore, a three-phase DAB converter that comprises a transformer with a double secondary wind-ing, connected in star-star-delta, is reported in [15, 16]. However, the impact ofthe winding configuration on the converter performance was not investigated. Be-sides the star-star and star-delta winding configuration, it is also common to exploita delta-delta connected transformer. However, the currents towards the inverterbridges are equal to those resulting from a star-star winding configuration [65].

The impact of different winding configurations on the performance of the symmet-rically operated two-level three-phase DAB dc-dc converter will be presented inthe following. For this, the investigation is restricted to the star-star and star-deltatransformer winding configuration.

4.1 Performance evaluation

To investigate the impact of the selected winding configurations on the perfor-mance of the dc-dc converter, the stress on the three main components is compared.These are the switches, the transformer, and the capacitors in the dc-link.

The comparison is carried out under the condition that the converter reaches thesame maximum power (φ = π/2) for each configuration. Also, the transformerturns-ratio is chosen such that voltage sources in the equivalent circuits have equalamplitudes for the first harmonic for each winding configuration. This results inN = 1 for the star-star configuration and N = 1/

√3 for the star-delta configuration.

4.1.1 Switches

The stress on the switches is determined by the switching and conduction losses.The switching loss can be divided in turn-on and turn-off losses, where the turn-on

4.1 Performance evaluation 97

losses are usually higher than the turn-off losses. Also, the switch turn-on actioncauses reverse recovery of the complementary diode which lead to additional lossesand deteriorate EMI performance. Therefore, zero voltage turn-on is of importanceto reach a high converter efficiency and power density. For this reason, the ZVSregion of the two-level three-phase DAB converter with a star-star and star-deltaconnected transformer is investigated further on. Thereafter, the switch turn-offcurrent is presented to compare the turn-off loss. Furthermore, the average andrms values of the switch current are presented to asses the impact of the windingconfigurations on the conduction losses.

ZVS region

The ZVS region is derived for each winding configuration considering idealswitches (IZVS1 = IZVS2 = 0) and no magnetizing or commutation current(L1 = L2 = ∞). Consequently, the ZVS conditions are given by

iLA(A1) ≤ 0 (4.1)

iLA(a1) ≥ 0, (4.2)

with ’A1’ and ’a1’ corresponding to the rising transition of the phase-leg voltagesof, respectively, inverter bridge 1 and 2.

Star-star winding configuration: The condition given by (4.1) must be met toachieve ZVS operation for inverter bridge 1. Then, the voltage ratio for ZVSoperation is found by rewriting (2.4). This is given by

U1

U2>

2π−3φ

2π N ∀ 0 ≤ φ ≤ π3

3π−6φ2π N ∀ π

3 ≤ φ ≤ 2π3

. (4.3)

ZVS operation for inverter bridge 2 is given by (4.2). The corresponding voltage ra-tio is derived by combining (2.4) with the equations given in table 2.1 and table 2.2.This is found to be

U1

U2<

2π

2π−3φ N ∀ 0 ≤ φ ≤ π3

2π3π−6φ N ∀ π

3 ≤ φ ≤ 2π3

. (4.4)

Star-delta winding configuration: Similarly as for the star-star winding configura-tion, the voltage ratio for ZVS operation of inverter bridge 1 is found by rewriting


U1U2

Power [%]

0 20 40 60 80 1000

Inverter bridge 2 hard-switching

Inverter bridge 1hard-switching

ZVS region

Figure 4.1: The ZVS region for the two-level three-phase DAB converter with star-star (Y-Y)or star-delta (Y-∆) connected transformer. The voltage ratio is given for U2 being constantand the power is normalised to the maximum power P(U1, φ = π/2).

(2.13). This results in

U1

U2>

32 N ∀ 0 ≤ φ ≤ π

69π−18φ

4π N ∀ π6 ≤ φ ≤ π

2

. (4.5)

Due to the delta connection, the phase-legs of inverter bridge 2 carry currents oftwo phases. Considering a balanced three-phase system and a symmetric current,the phase-leg current ia of inverter bridge 2 is given by

ia(θ) = iLA(θ)− iLC(θ) = iLA(θ) + iLA(θ + π/3). (4.6)

The voltage ratio for ZVS operation of inverter bridge 2 is found by combining(2.13) with the equations given in table 2.3 and table 2.4. This is found to be

U1

U2<

2N ∀ 0 ≤ φ ≤ π

64π

3π−6φ N ∀ π6 ≤ φ ≤ π

2. (4.7)

Finally, the boundaries of the voltage ratios given above, for the star-star and star-delta winding configuration, are presented in figure 4.1. The area enclosed by theboundaries is called the ZVS region, where the switches of both inverter bridgesbenefit from ZVS turn-on. The ZVS boundaries corresponding to the star-deltaconfiguration are independent of the power for phase-shifts up to π/6, as shown by(4.5) and (4.7), which relates to a power level of approximately 50 %. As a result,


iIdc

Power [%]

0 20 40 60 80 100−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

Figure 4.2: Normalised switch turn-off currents iA(A1) and ia(a1) of, respectively, inverterbridge 1 and 2 for U1/U2 = 1 and L1 = L2 = ∞.

the star-delta winding configurations favours a wider ZVS region, compared to astar-star connected transformer, for power levels up to 30 %. Whereas, a smallerZVS range can be seen for higher power levels.

Switch turn-off current

Considering ZVS operation, the turn-on losses of a switch are negligible since theanti-parallel diode is conducting when the switch closes while the voltage acrossthe switch is almost zero. Opening a switch, however, causes turn-off losses be-cause, both, the voltage and current are changing due to the finite time of theswitching process. Therefore, the switch turn-off current for both inverter bridgesis evaluated to investigate the impact of the transformer winding configurations onthe switching losses. This is shown in figure 4.2 for a unity voltage ratio, wherethe current is normalised to the dc-link current Idc that corresponds to the maxi-mum power level. This reveals that the star-delta configuration exhibits a higherand constant switch turn-off current at no or light load conditions which, in con-trast to the star-star configuration, ensures ZVS at no or light load conditions. Thereason for this constant turn-off current, even with a zero phase-shift (φ = 0), isthe disparity in the voltage waveforms due to the difference in the primary andsecondary winding connection. For power levels between 30 % and 80 %, a lowerswitch turn-off current can be seen for the star-delta configuration.

For voltage ratios other than unity, the switch turn-off currents loose the symmetryaround the zero current axis, as shown in figure 4.3. The switch turn-off currents


iIdc

Power [%]

0 20 40 60 80 100

iIdc

Figure 4.3: Normalised switch turn-off currents iA(A1) and ia(a1) of, respectively, inverterbridge 1 and 2 for voltage ratios U1/U2 = 1.25 and U1/U2 = 0.75 and L1 = L2 = ∞.

become more positive for voltage ratios lower than one and are conflicting with theZVS condition for inverter bridge 1. The opposite happens for voltage ratios higherthan one, where the switch turn-off currents become more negative and opposes theZVS condition for inverter bridge 2.

Switch current

The rms and average value of the switch current is included to compare the relatedconduction losses. For this comparison, ideal switches are considered, therefore,the absolute value of the switch current is used to calculate the average value. Asshown in figure 4.4, both the rms and average switch current for inverter bridge 1show small differences between the two winding configurations. A small offsetis visible for the star-delta connected transformer at low-load conditions due tothe disparity in the voltage waveforms, which is also found for the switch turn-offcurrent in figure 4.2. However, the star-delta configuration generates slightly lowerrms and average switch currents for power levels between 40 % and 90 %. Thiswill result in a small reduction of conduction losses compared to a converter witha star-star connected transformer. The same result is found for the switch currentcorresponding to the output bridge and considering different voltage ratios.


Y-∆

Y-Y

iIdc

Power [%]0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

rms

avg.

Figure 4.4: Normalised average and rms switch current for inverter bridge 1, with U1/U2 =1 and L1 = L2 = ∞.

4.1.2 Transformer

To compare the impact of the selected winding configurations on the transformerlosses, it is considered that the transformers are designed such that the peak fluxdensity and magnetic core volume, thus the core losses, are kept equal for the se-lected winding configurations. Under this condition, the current through the trans-former windings is compared below.

Transformer winding current

The rms current through the primary winding, considering zero transformer mag-netizing current (L1 = L2 = ∞), is shown in figure 4.5 for the investigated trans-former connections. Similar to the switch current, small differences are presentbetween the winding currents for a star-star and star-delta connected transformer.For power levels up to 50 % the star-star configuration generates lower rms cur-rents in the transformer, whereas the star-delta configuration result in marginallylower winding currents for power levels between 50 % and 90 %. The same result isfound for other voltage ratios. Note that the rms current of the secondary windingis equal to the rms current of the primary winding multiplied by the turns ratio N.This means that the rms current in the secondary winding of the star-delta config-uration is a factor of

√3 smaller. However, it should be noted that the lower rms

current caused by a delta connection does not lead to lower ohmic losses. This canbe explained by considering an equal peak flux density in the magnetic core and


Y-∆

Y-Y

iIdc

Power [%]0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

Figure 4.5: Normalised rms current through the primary-winding of the selected trans-former connections, with U1/U2 = 1 and L1 = L2 = ∞.

a fully utilised winding area for both connections. This results in more turns forthe delta connection, compared to a star connection, which increases the windinglength and reduces the cross section of the conductor. Both effects cause a quadraticincrease in the resistance of the winding, while the rms current is reduced inverselyproportional by N and, thus, the ohmic losses remain constant.

Power flow

The power as function of the phase-shift φ is evaluated using (2.11) and (2.16), andis presented in figure 4.6. From this, it is visible that phase-shifts smaller than π/6

lead to slightly lower power levels for the star-delta configuration compared to thestar-star configuration. For phase-shifts bigger than π/6 the opposite is true. Thismeans that for phase-shifts larger than π/6 the star-delta configuration requires asmaller phase-shift for the same power level, which leads to less reactive powerflow in the ac-link. This is also visible in rms current through the primary windingas shown in figure 4.5.

Transformer utilization factor

To get more insight in the effective usage of the transformer, the transformer uti-lization factor (TUF) is calculated for the selected winding configurations. The TUFcan be found by dividing the active power by the average apparent power of the


Y-∆

Y-Y

Pow

er[%

]

Phase shift [rad]0 π/6 π/3 π/2

0

20

40

60

80

100

Figure 4.6: Power versus phase-shift φ for U1/U2 = 1.

ac-terminals for both inverter bridges. This is given by

TUF =2P

S1 + S2=

2P3(UANIA + UanIa)

, (4.8)

where S1 and S2 denote the apparent power of, respectively, inverter bridge 1 and2. The resulting TUF for three voltage ratios is presented in figure 4.7. This confirmsthat the star-delta configuration has a better transformer utilization for power levelsbetween 50 % and 90 %, whereas the star-star configurations performs better forpower levels below 50 %.

The TUF reduces to zero in all configurations when the power is going to zero fornon-unity voltage ratios. This is caused by the circulating current that is generateddue to the unequal voltage sources. The same is true for the star-delta configura-tion, even with a unity voltage ratio, because of the disparity in the voltage wave-forms, as shown in figure 2.10. Depending on the application, the TUF for nominalpower is of main interest since the transformer size is related to this operating point.

4.1.3 Capacitors

The rms currents through the dc-link capacitors, considering different voltage ra-tios, are investigated to compare the impact of the selected transformer windingconfigurations. The capacitor currents corresponding to inverter bridge 1 and 2 arepresented in figure 4.8 and figure 4.9, respectively. This reveals that the star-deltaconfiguration features lower capacitor currents for power levels between 30 % and


TUF

P S

Power [%]

0 20 40 60 80 1000

0

Figure 4.7: Transformer utilization factor (TUF) for different voltage ratios.

iIdc

Power [%]

0 20 40 60 80 1000

Figure 4.8: Normalised rms current through the dc-link capacitors of inverter bridge 1 fordifferent voltage ratios.

80 %, whereas the star-star configuration result in a lower current between powerlevels of 80 % and 100 %.

For power levels below 50 %, a constant capacitor current can be seen for the star-delta configuration. This is caused by a constant switch turn-off current, presentedin figure 4.2 and figure 4.3, resulting in a constant peak current through the capac-itor. Similarly, the rms capacitor current increases significantly for increasing thepower to 100 % for all winding configurations, which can also be seen in the switchturn-off current.


iIdc

Power [%]

0 20 40 60 80 1000

Figure 4.9: Normalised rms current through the dc-link capacitors of inverter bridge 2 fordifferent voltage ratios.

4.1.4 Conclusions

The stress on the switches, transformer, and dc-link capacitors was investigatedconsidering a star-star and star-delta connected transformer. This revealed that thestar-delta configuration features a wider and phase-shift independent ZVS regionfor power levels below 30 %. For higher power levels, the star-star configurationhas a similar or wider ZVS region. A constant and phase-shift independent switchturn-off current is found for the star-delta configuration for power levels below50 %, ensuring ZVS even with zero phase-shift. In addition, the switch turn-offcurrent for the star-delta configuration is lower for powers between 30 % and 80 %.Furthermore, the star-delta configuration features lower rms current in the wind-ings, resulting in a better transformer utilization, and lower rms capacitors currentsfor power levels between 50 % and 80 %.

Note that the phase-shift for a typical DAB converter design does not exceed π/3,which results in a power flow of approximately 87 %, to limit the reactive powerin the ac-link. Therefore, the star-delta winding configuration is preferred for thesymmetrically operated two-level three-phase DAB converter.


Figure 4.10: Prototype of the symmetrically operated two-level three-phase DAB dc-dc con-verter, using star-star and star-delta transformer winding configurations.

4.2 Measurements

Measurements are conducted, using a prototype dc-dc converter, to verify the theo-retical model and evaluate the performance. The prototype converter, shown in fig-ure 4.10, contains two three-phase inverters, three single-phase transformers, andthree inductors which represent the transfer-inductor L. The dc output of the con-verter is connected to the dc input, while a dc power-supply only provides the totalpower loss. Parameters of the experimental setup are given in table 4.1.

Table 4.1: Parameters of the prototype two-level three-phase DAB dc-dc converter.

ParameterWinding configuration:

Y-Y Y-∆

dc-link voltages, U1, U2 375 V 375 Vdc-link currents, Idc 90.1 A 89.7 AMaximum power level 33.8 kW 33.6 kWTransformer turns ratio, N(N1 : N2) 11 : 11 7 : 12Switching frequency, fs 16 kHz 16 kHzTransfer-inductance, L 25 µH 25 µH

4.2 Measurements 107

4.2.1 Waveforms

The transfer-inductor current iLA and the phase-to-neutral voltages uAN and uanof, respectively, inverter bridge 1 and 2 are captured using an oscilloscope. Thewaveforms for both winding configurations are shown in figure 4.11. The shape ofthe current waveforms matches the idealised waveforms presented in section 2.1.

The phase-leg voltage uAN of the star-delta configuration, presented in figure 4.11b,shows small spikes after a rising or falling edge is completed. This is due to thefixed dead time (4 µs) of the inverter bridge used in the experimental setup. Reduc-ing the dead time or increasing the phase-leg current by the magnetizing induc-tance is necessary to prevent these unwanted voltage commutations.


iLA

uAN

uan

(a) Star-star winding configuration with 500 V/div, 50 A/div, and 20 µs/div.

iLA

uAN

uan

(b) Star-delta winding configuration with 500 V/div, 20 A/div, and 20 µs/div.

Figure 4.11: Measured converter waveforms for the symmetrically operated two-level topol-ogy using star-star (a) and star-delta (b) winding configurations with U1/U2 = 1. Themeasured signals are: the phase-to-neutral voltage uAN (top) corresponding to phase A ofinverter bridge 1, the phase-to-neutral voltage uan (middle) corresponding to phase a of in-verter bridge 2, and the transfer-inductor current iLA (bottom).

4.2 Measurements 109

ia model

ia measured

iA model

iA measured

iIdc

Power [%]0 20 40 60 80 100

−1.5

−1

−0.5

0

0.5

1

1.5

(a) Star-star winding configuration.

ia model

ia measured

iA model

iA measured

iIdc

Power [%]0 20 40 60 80 100

−1.5

−1

−0.5

0

0.5

1

1.5

(b) Star-delta winding configuration.

Figure 4.12: Measured and calculated turn-off currents for a converter with a star-star andstar-delta connected transformer with, respectively, a transformer turns ratio N of 1:1 and7:12. The current is normalised to Idc and the power is normalised to the maximum power,which are given in table 4.1.

4.2.2 Turn-off current

To verify the theoretical model, the switch turn-off currents are measured for thetwo winding configurations. This measurement is conducted by capturing thephase currents iA and ia together with the phase-to-neutral voltages uAN and uan,by means of an oscilloscope, for phase-shift between zero and π/2. From this data,the switch turn-off current for the two inverter bridges is derived by reading thevalue of the phase current during the rising edge of the corresponding phase-to-neutral voltage. The resulting measurement points of the switch turn-off current,together with the results obtained from the theoretical model, are shown in fig-ure 4.12. The measured switch turn-off current matches with the model, confirmingthe theoretical analysis.

4.2.3 Efficiency

The impact of the selected winding configurations on the converter components,presented in section 4.1, influences the overall efficiency of the converter. Therefore,the efficiency of the dc-dc converter is measured for both transformer connectionsin the full power range using a power analyser. Both configurations have an equaltransfer-inductance L and approximately the same maximum power, as shown in


Y-∆

Y-Y

Efficien

cy[%

]

Power [%]0 20 40 60 80 100

88

89

90

91

92

93

94

95

Figure 4.13: The measured efficiency versus the normalised power for U1/U2 = 1.

Table 4.1. The measured efficiencies are shown in figure 4.13. This reveals that thestar-delta configuration features the highest efficiency of 94.6 % which is approx-imately 0.5 % higher than the peak efficiency of the star-star configuration. Simi-larly, a maximum loss reduction of 15 % is measured for the star-delta connectedtransformer. The power range for which this converter generates lower losses isapproximately from 20 % to 80 %. This matches with the results of the performanceevaluation presented in section 4.1.

4.3 Summary

Different transformer winding configurations may be employed in the two-levelthree-phase DAB dc-dc converter. The impact of two common transformer con-nections on the performance of converter was investigated, considering the star-star and star-delta transformer connection. For this, the stress on the switches,transformer, and the capacitors in the dc-link is compared for equal conditions.Overall, the star-delta configuration performs best, compared to the star-star con-figuration, for power levels between 50 % and 80 %, regarding the current stresson the switches, the transformer, and the capacitors in the dc-link. Measurementsof waveforms, switch turn-off currents, and the efficiency, obtained from a high-power experimental setup, support the theoretical analysis and confirm the out-come of the comparison. In the next chapter, a comparative evaluation is madebetween two- and three-level three-phase DAB converter with, respectively, asym-metric and symmetric modulation.

Chapter 5

Comparative evaluation ofthree-phase DAB converters

T HROUGHOUT this thesis, five different three-phase DAB converters havecome to attention. Whereas converter models have been presented inchapter 2 and in chapter 3, modulation schemes have been derived for

the converters with more than one degree of freedom. This gives rise to thequestion which topology or operating method leads to the lowest power losses orthe smallest converter, given a certain application. In the previous chapter, thisquestion is answered to some extent for specifically the symmetrically operatedtwo-level topology, where the impact of three-phase transformer connections onthe converter performance has been investigated. However, only voltage ratiosclose to unity were considered and the converters which have more than onedegree of freedom were not included. Therefore, to assess all five converters forwide voltage-range high-power applications, a comparative evaluation is required.This chapter presents a procedure for such a comparative evaluation, and itsresults, for the selected three-phase DAB dc-dc converters.


• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Evaluation of a high-power three-phase dual active bridge dc-dc converter with three-level phase-legs’, in Proceedings of the IEEEEuropean Conference on Power Electronics and Applications, pp. 1–10, Sep. 2016.

111

112 Chapter 5 Comparative evaluation of three-phase DAB converters

Table 5.1: Selected three-phase DAB dc-dc converters for the comparative evaluation.

Operating Three-phase DAB topology Transformer Abbreviationmethod connection

Symmetrical Two-level topology star-star S2L-YYSymmetrical Two-level topology star-delta S2L-Y∆Symmetrical Three-level topology star-star S3L-YYSymmetrical Three-level topology star-delta S3L-Y∆

Asymmetrical Two-level topology star-star A2L-YY

5.1 Method

The comparative evaluation is carried out in two parts. In the first part the currentstress of the switches, transformer, and the dc-link capacitors is evaluated for theselected three-phase DAB converters given in table 5.1. The parameters and operat-ing range used for the comparison is given in table 5.2. The transfer-inductances forthe symmetrically operated two-level topologies are derived by limiting the phase-shift to π/3, while the maximum power level must be reached at the lowest voltageU1. These inductance values are also used for the remaining topologies with thesame transformer connections to have a fair comparison. The commutation induc-tances L1 and L2 for the converters with more than one degree of freedom (S3L-YY,S3L-Y∆, and A2L-YY) are chosen such that ZVS turn-on occurs for the whole op-erating range, considering a dead-time of 500 ns. No commutation inductances areconsidered for the symmetrically operated two-level topologies (S2L-YY and S2L-Y∆), since very low values of the commutation inductances are required to ensuresoft-switching [66]. Therefore, these converters are hard-switching for a large partof the operating range. The result of the component current stress comparison forthe selected converters is given in section 5.2.

In the second part, the minimum amount of semiconductor chip area is derived foreach converter topology, such that an equal chip junction temperature is reachedin every switch. This semiconductor chip area comparison, including the semi-conductor loss modelling, is given in section 5.3. This is useful to compare thesemiconductor utilisation and it indicates the related costs. In addition, the totalsemiconductor losses are derived, while the total chip area is kept constant for allconverters to have a fair comparison. Furthermore, the required dc-link capaci-tance of both inverter bridges is compared, based on a voltage-ripple specificationand the maximum permissible rms current of an existing film capacitor.

5.2 Component current stress comparison 113

Table 5.2: Converter parameters and operating conditions for the comparative evaluation.

Parameter Three-phase DAB dc-dc converters

Converter S2L-YY S2L-Y∆ S3L-YY S3L-Y∆ A2L-YYTransformer turns ratio, N 1 : 1 1 :

√3 1 : 1 1 :

√3 1 : 1

Transfer-inductance, L 11.7 µH 11.8 µH 11.7 µH 11.8 µH 11.7 µHInductance, L1 = L2 ∞ ∞ 10L 6L 8LVoltage, U1 375-1500 VVoltage, U2 750 VMaximum power level 100 kWSwitching frequency, fsw 20 kHzDead-time, tdt 500 ns

5.2 Component current stress comparison

In this section, the currents through the switches, transformer, and the capacitorsin the dc-link are evaluated for the three-phase DAB dc-dc converters given in ta-ble 5.1. For this evaluation, the three-level T-type topology [88] is utilised for sim-ulating two- and three-level operation considering ideal components as shown infigure 5.1. The switches (S3A,B,C and S3a,b,c) connected to the neutral voltages(’N’ and ’n’) are disabled for two-level operation. Three-level operation can also berealised by other multi-level topologies, such as the neutral-point-clamped (NPC)and the flying capacitor (FC) topology [17, 77]. However, the T-type is selected forits straightforward structure and operation. The transformer is connected in eitherstar-star or star-delta and the magnetizing or commutation inductances L1 and L2are connected in star, their values are given in table 5.2.

5.2.1 Switches

The stress of the switches is caused by conducting the phase current for a certaintime interval and turning the switch on and off, leading to power loss. The con-duction related stress is compared for the selected three-phase DAB converters byevaluating the rms current through the switches for the given operating range. It isobserved that the highest current is found at the maximum power level. Therefore,to compare the maximum stress due to conduction, the rms value of the currentthrough the switches is compared with a power level of 100 kW. This is shown infigure 5.2.


U12

U12

U22

U22

A

B

C

a

b

c

N n

1A

2A

1B 1C

2B 2C

1a 1b 1c

2a 2b 2c

iA ia

L1A,B,C L2a,b,c

I1 I2

3A

3B

3C

3a

3b

3c

C11

C12

C21

C22

Three-phase transformer

Figure 5.1: Circuit diagram with ideal components used for evaluation of the componentstress of the selected three-phase DAB converters. Component values and transformer wind-ing configurations are given in table 5.2 for the corresponding converters.

The comparison reveals that the top and bottom switches of both inverter bridges(S1A,S2A and S1a,S2a) have comparable rms currents for U1 < 900 V. For highervoltages the rms current of top and bottom switches in symmetrically operatedtwo-level topologies (S2L-YY and S2L-Y∆) increases significantly. This is due to theinability of these converters to compensate for the voltage difference between U1and U2 since they are controlled with only one degree of freedom. The symmetri-cally operated three-level topologies (S3L-YY and S3L-Y∆) are controlled with twodegrees of freedom. As a result, the duty-cycle of one of the inverter bridges iscontrolled for non-unity voltage ratios (three-level operation) which reduces therms current in the top and bottom switches, while an additional current flows inthe switch connected to the third voltage level (S3A,S3a). Three degrees of freedomare used for asymmetric operation of the two-level topology (A2L-YY), reducingthe rms current in the top and bottom switches compared to symmetric opera-tion (S2L-YY), without the need for a third voltage level and the correspondingswitches. However, the A2L-YY converter generates slightly higher rms currentsin most of the top and bottom switches compared to the three-level topologies. Fur-thermore, unequal current stress is observed for the top and bottom switches in thesame inverter bridge.


A2L-YY

S3L-Y∆

S3L-YY

S2L-Y∆

S2L-YY

I S1A[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(a) Rms current through S1A (top sw.).

A2L-YY

S3L-Y∆

S3L-YY

S2L-Y∆

S2L-YY

I S1a[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(b) Rms current through S1a (top sw.).

A2L-YY

S3L-Y∆

S3L-YY

S2L-Y∆

S2L-YY

I S2A[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(c) Rms current through S2A (bot. sw.).

A2L-YY

S3L-Y∆

S3L-YY

S2L-Y∆

S2L-YY

I S2a[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(d) Rms current through S2a (bot. sw.).

S3L-Y∆

S3L-YY

I S3A[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(e) Rms current through S3A (mid. sw.).

S3L-Y∆

S3L-YY

I S3a[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(f) Rms current through S3a (mid. sw.).

Figure 5.2: Evaluation of the rms currents trough the switches of inverter bridge 1 (left) and2 (right) for the selected three-phase DAB converters, with U2 = 750 V and P = 100 kW.


The stress related to switching losses is compared by evaluating the instantaneousphase-current at the switching instants. For the symmetrically operated two-leveltopologies, the current at two switching instants is included. Those are ’A1’ and’a1’ which correspond to, respectively, the rising voltage transition of phase ’A’ atinverter bridge 1 and 2. The resulting currents during the switching transition areshown in figure 5.3 for power levels between 10 kW and 100 kW.

Similarly, the two- and three-level topologies which are, respectively, operatedasymmetrically and symmetrically, require two additional switching instants toevaluate the corresponding currents. Those are ’A2’ and ’a2’ which correspond to,respectively, the falling voltage transition of phase ’A’ at inverter bridge 1 and 2.The resulting currents during the switching transition are shown in figure 5.4 forpower levels between 10 kW and 100 kW.

The selected three-phase DAB converters can be dived in two groups. The firstgroup is distinguished by the converters which have one degree of freedom (S2L-YY and S2L-Y∆) and the second group by those which have two or three degrees offreedom (S3L-YY, S3L-Y∆, and A2L-YY). The switch current during the switchinginstant for the first group, presented in figure 5.3, does not result in soft-switchingsince the ZVS conditions (section 2.4) are not met in the complete operating range.This can lead to higher switching losses due to the switch turn-on and diode reverserecovery losses, and an increased electromagnetic interference (EMI) generation[43, 82]. Furthermore, large switching currents are observed for high values of U1,even for low power levels.

In the second group, the converters operate with ZVS in the whole operating rangeand generate significantly lower currents during the switching instants for highvoltage ratios, as shown in figure 5.4. The three-level topologies (S3L-YY, S3L-Y∆)have comparable turn-off currents, whereas, the A2L-YY converter generateshigher turn-of currents, especially for low and high voltage ratios. Therefore,higher turn-of losses are expected for the A2L-YY converter compared to thethree-level topologies.


10 kW

100 kW

i A(A

1)[A

]

U1 [V]375 750 1125 1500

−400

−300

−200

−100

0

100

200

300

400

10 kW

100 kW

i a(a

1)[A

]

U1 [V]375 750 1125 1500

−400

−300

−200

−100

0

100

200

300

400

(a) S2L-YY converter.

10 kW

100 kW

i A(A

1)[A

]

U1 [V]375 750 1125 1500

−400

−300

−200

−100

0

100

200

300

400

10 kW

100 kW

i a(a

1)[A

]

U1 [V]375 750 1125 1500

−400

−300

−200

−100

0

100

200

300

400

(b) S2L-Y∆ converter.

Figure 5.3: Comparison of the phase current in both inverter bridges at, respectively, switch-ing instants ’A1’ and ’a1’, corresponding to the rising voltage transitions of phase ’A’ atinverter bridge 1 and 2, respectively.


10 kW

100 kW

i A(A

2)[A

]i A

(A1)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

10 kW

100 kW

i a(a

1)[A

]i a(a

2)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0


10 kW

100 kW

i A(A

2)[A

]i A

(A1)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

10 kW

100 kW

i a(a

1)[A

]i a(a

2)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0


10 kW

100 kW

i A(A

2)[A

]i A

(A1)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

10 kW

100 kW

i a(a

1)[A

]i a(a

2)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

(c) A2L-YY converter.

Figure 5.4: Comparison of the phase current in both inverter bridges at, respectively, switch-ing instants ’A1’, ’A2’, ’a1’, and ’a2’, corresponding to the rising and falling voltage transi-tions of phase ’A’ at inverter bridge 1 and 2, respectively.


A2L-YY

S3L-Y∆

S3L-YY

S2L-Y∆

S2L-YY

I LA[A

]

U1 [V]375 750 1125 1500

50

100

150

200

250

Figure 5.5: Transfer-inductor rms current ILA , evaluated for the selected three-phase DABconverters to compare the related transformer winding losses, with U2 = 750 V and P =100 kW.

5.2.2 Transformer

The stress of the transformer is considered to be dominated by the currents throughthe windings. For this reason, the rms current through the primary winding is com-pared for the selected three-phase DAB converters. The transformer magnetizingcurrent is neglected and the commutation inductances (L1 and L2) are consideredto be connected externally. Consequently, the transformer’s primary winding cur-rent equals the current through the transfer-inductor iLA . This is presented in fig-ure 5.5 for the selected three-phase DAB converters with a power level of 100 kW.From this it can be observed that, the converters which have two or three degrees offreedom (S3L-YY, S3L-Y∆, and A2L-YY) create significantly lower transfer-inductorrms currents for high voltage ratios. Consequently, the transformer winding lossesare lower for this operating range. For low voltage ratios, the rms currents increasefor all converters since the power is kept constant.

5.2.3 Capacitors

The capacitors, present in the dc-link of both inverter bridges, provide energy stor-age to compensate for the high-frequency power fluctuation of the inverter bridges.Consequently, the capacitors carry an ac-current which causes a voltage ripple onthe dc-link voltages due to the charge displacement and the voltage drop over the


equivalent series resistance (ESR). Furthermore, losses in the capacitors are gener-ated due to the ESR. This, together with the voltage-ripple specification, determinesthe type, size, and losses of the capacitors. For this reason, the rms current throughthe dc-link capacitors of both inverter bridges are compared in figure 5.6 and fig-ure 5.7 for, respectively, the symmetrically operated two-level topologies and two-and three-level converters with more than one degree of freedom.

Considering the symmetrically operated two-level topologies in figure 5.6, the ca-pacitor rms currents of both inverter bridges have comparable characteristics. Thecurrent increases for non-unity voltage ratios, even for zero power. Consideringthe three-level converters (S3L-YY,S3L-Y∆), the capacitors of inverter bridge 1 ex-perience only about two-third of the maximum capacitor rms current, compared tothe two-level topologies. However, the capacitors in inverter bridge 2 experienceabout a factor of two higher maximum rms currents for low voltage ratios. TheA2L-YY converter generates the highest capacitor currents for inverter bridge 2,whereas the rms current of the capacitors of inverter 1 is comparable with that ofthe two-level topologies.


10 kW

100 kW

I C1x[A

]

U1 [V]375 750 1125 15000

50

100

150

200

10 kW

100 kW

I C2x[A

]

U1 [V]375 750 1125 15000

50

100

150

200


10 kW

100 kW

I C1x[A

]

U1 [V]375 750 1125 15000

50

100

150

200

10 kW

100 kW

I C2x[A

]

U1 [V]375 750 1125 15000

50

100

150

200


Figure 5.6: Capacitor rms current IC1x and IC2x of, respectively, inverter bridge 1 (left) and 2(right) for the symmetrically operated two-level converters S2L-YY (a) and S2L-Y∆ (b), withU2 = 750 V. The result for two- and three-level converters with more than one degree offreedom is given in figure 5.7.


10 kW

100 kW

I C1x[A

]

U1 [V]375 750 1125 15000

50

100

150

200

10 kW

100 kW

I C2x[A

]

U1 [V]375 750 1125 15000

50

100

150

200


10 kW

100 kW

I C1x[A

]

U1 [V]375 750 1125 15000

50

100

150

200

10 kW

100 kW

I C2x[A

]

U1 [V]375 750 1125 15000

50

100

150

200


10 kW

100 kW

I C1x[A

]

U1 [V]375 750 1125 15000

50

100

150

200

10 kW

100 kW

I C2x[A

]

U1 [V]375 750 1125 15000

50

100

150

200

(c) A2L-YY converter.

Figure 5.7: Capacitor rms current IC1x and IC2x of, respectively, inverter bridge 1 (left) and 2(right) for the symmetrically operated three-level converters S3L-YY (a) and S3L-Y∆ (b), andthe asymmetrically operated two-level converter A2L-YY (c), with U2 = 750 V. The resultfor symmetrically operated two-level converters is given in figure 5.6.


Table 5.3: Overview of the maximum values of the current stress concerning the switches,transformer, and the dc-link capacitors.

Component S2L-YY S2L-Y∆ S3-YY S3L-Y∆ A2L-YY

Maximum rms switch current with P = 100 kW

IS1A 172 172 162 171 172IS2A 172 172 162 171 142IS3A 0 0 96 117 0IS1a 172 172 121 127 150IS2a 172 172 121 127 188IS3a 0 0 170 188 0

Minimum/maximum phase current at switching instants

iA(A1) -395/167* -407/133* -180/-16 -215/-13 -150/2*iA(A2) -167*/395 -133*/407 17/192 13/231 12/352ia(a1) -357*/270 -264*/277 36/371 25/386 15/459ia(a2) -270/357* -277/264* -190/-11 -223/-14 -306/-13

Maximum transfer-inductor rms current with P = 100 kW

iLA 243 244 224 234 218

Maximum rms current of dc-link capacitors

IC1x 109 117 67 72 108IC2x 104 80 162 176 224

*no ZVS turn-on

5.2.4 Overview of the component stress comparison

In the process of designing a converter, the components are selected to withstandthe maximum stress. Therefore, an overview of the maximum values of the currentstress concerning the switches, transformer, and the dc-link capacitors are collectedand given in table 5.3. In the case of the current during the switching instants, themaximum and the minimum current is provided since it can cause both turn-onand turn-off switching losses, depending on the sign of the current.


U1

U12

U12

C11

C12

S1A

S2A

AN

(a)

S1A

S2A

S4AS3A

U1

U12

U12

C11

C12

AN

(b)

Figure 5.8: Circuit diagram of a single two-level (a) and three-level (b) phase-leg, employedwith MOSFETs, for phase ’A’ of inverter bridge 1. Each inverter bridge contains three corre-sponding phase-legs.

5.3 Converter comparison

This section will further elaborate on the required semiconductor chip area, semi-conductor losses, and required dc-link capacitance, based on the numbers givenin table 5.3. Power electronic converters can be compared in many ways. Oneway to compare different converter topologies or operating methods is to derivethe required semiconductor chip area for each switch, such that the chip junctionreaches a certain temperature [37, 44, 91]. This is carried out for the selected three-phase DAB converters using two- and three-level phase-legs utilising metal-oxide-semiconductor field-effect transistors (MOSFET), as shown in figure 5.8. Anotherinteresting performance indicator to compare is the converter efficiency. However,the converter losses in comparable systems are dominated by semiconductor losses[1, 70, 99]. Therefore, this section includes a loss comparison that is limited to thesemiconductor losses for an equal total chip area. Furthermore, the required dc-linkcapacitance of both inverter bridges is derived for each converter. For this, both avoltage ripple specification and the rms current rating of a commercially availablefilm capacitor are respected.

5.3.1 Semiconductor chip area comparison

A semiconductor chip area based converter comparison is carried out for the se-lected three-phase DAB converters by deriving the required semiconductor chiparea to limit the junction temperature to 100 C for the complete operating range.The chip area and semiconductor losses are based on a commercially available

5.3 Converter comparison 125

Ron[m

Ω]

ids [A]0 100 200 300 400

0

2

4

6

8

10

12

(a)

Ron[p.u.]

TJ [C]25 50 75 100 125 1500

0.5

1

1.5

2

2.5

(b)

Figure 5.9: MOSFET on-state resistance of the CAS300M17BM2 half-bridge module as func-tion of the drain-source current ids (a), extrapolated from the datasheet values (’+’ markers),and the normalised resistance as function of the junction temperature TJ (b) with a linearcurve fit (1 + (TJ − 25)/150) on the data-sheet values (’+’ markers) [25].

MOSFET, packaged in the CAS300M17BM2 half-bridge module [25]. This is a sili-con carbide (SiC) MOSFET with a breakdown voltage UBV of 1700 V, of which thecharacteristic parameters will be scaled with the required blocking voltage for theconsidered converters.

Semiconductor losses

The switch conduction characteristic is modelled by an on-state resistance whichis dependent on the drain-source current ids and the junction temperature TJ. Thedependency on ids is linearly extrapolated from the on-resistance Ron at 25 C andthe temperature dependency is modelled by a linear curve fit, as shown in fig-ure 5.9. Furthermore, the on-resistance is modelled to be inversely proportional tothe semiconductor chip area [37, 91]. For the switches in the three-level topologywhich require only half the blocking voltage, a factor kBV is included to scale theon-resistance according to the silicon limit [63]. This factor is given by

kBV =

(U∗BVUBV

)α

, (5.1)

with α = 2.5. Thus kBV = 1 for all top and bottom switches (U∗BV = UBV) andkBV ≈ 0.18 for all switches for which U∗BV = UBV/2. The resulting on-resistance isgiven by

R∗on(ids, TJ) = kBVRon(ids)

[1 +

TJ − 25150

]Asc

A∗sc, (5.2)


Eoff

Eon

E[m

J]

ids [A]−400 −200 0 200 400

0

5

10

15

20

25

30

Figure 5.10: Switching losses as function of the drain-source current ids, interpolated fromthe CAS300M17BM2 data-sheet values (’+’ markers) for a drain-to-source voltage of 1200 V[25].

where Asc is the semiconductor chip area of the CAS300M17BM2 and A∗sc is therequired area to limit the junction temperature to 100 C. Consequently, the con-duction losses are calculated by

Pcond = I2dsR∗on(ids, TJ). (5.3)

The switching losses are found in the device data-sheet [25] and are linearly ex-trapolated for negative drain-source currents, as shown in figure 5.10. For positivedrain-source currents, turn-on loss is considered zero. Similarly, the turn-off en-ergy loss is considered zero for negative switch currents. The switching losses arelinearly scaled by the drain-source voltage specified in the data-sheet (1200 V), thisis given by

Psw = [Eon(ids) + Eoff(ids)]uds

1200 Vfsw. (5.4)

Thermal model

The thermal resistance from the chip junction to the case RthJ−Cis scaled, like the

on-resistance, inversely proportional to the chip area. This results in

R∗thJ−C=

RthJ−CAsc

A∗sc, (5.5)

where RthJ−Cis the specified thermal resistance for the MOSFET chip junction to

case, for which the typical value is 0.067 C/W. By considering a heat sink whichhas a constant temperature of 50 C and neglecting the thermal interface material,


Table 5.4: Required semiconductor chip area, relative to the MOSFET chip area of theCAS300M17BM2 half-bridge module, to limit the junction temperature to 100 C over thewhole operating range.

Converter Semiconductor chip area A∗sc/Asc [p.u.]

S1A S2A S3A/S4a S1a S2a S3a/S4a Total (converter)

S2L-YY 1.08 1.08 0 0.87 0.87 0 11.72S2L-YD 1.10 1.10 0 0.86 0.86 0 11.72S3L-YY 0.66 0.66 0.16 0.53 0.53 0.37 10.27S3L-YD 0.70 0.70 0.19 0.57 0.57 0.40 11.21A2L-YY 0.69 0.57 0 0.72 0.96 0 8.82

the required maximum junction temperature is reached by a temperature increaseof 50 C due to the thermal resistance R∗thJ−C

. This is given by

∆TJ = (Pcond + Psw)R∗thJ−C= 50 C. (5.6)

Required semiconductor chip area

Finally, the required semiconductor chip area, relative to the MOSFET chip areaof the CAS300M17BM2 half-bridge module, to limit the junction temperature TJto 100 C in all operating points is derived. This is given in table 5.4. The totalconverter chip area is calculated by

Aconverter = 3(AS1A + AS2A + AS3A + AS4A) + 3(AS1a + AS2a + AS3a + AS4a) (5.7)

From this, it is visible that the conventional symmetrically operated two-leveltopologies (S2L-YY and S2L-Y∆) require the largest chip area. In comparison, thesymmetrically operated three-level topologies (S3L-YY and S3L-Y∆) require about10 % less chip area while these topologies have the double amount of switches. Thisis explained by the lower rms current in the transfer-inductor, especially for highinput voltages, which result in lower switching and conduction losses. In addition,the switches in the three-level topology that require half the blocking voltage have aconsiderably lower on-resistance and switching loss. The asymmetrically operatedtwo-level topology (A2L-YY) requires the smallest chip area which is about 25 %lower than the symmetrically operated two-level topology (S2L-YY). Similar to thethree-level topologies, this result is due to the reduced rms current in the transfer-inductor, especially for high input voltages, and the absence of the switches for the


A2L-YY

S3L-Y∆

S3L-YY

S2L-Y∆

S2L-YY

P[kW

]

U1 [V]375 562.5 750 937.5 1125 1312.5 15000

1

2

3

4

5

6

7

8

9

Figure 5.11: Semiconductor losses for the selected three-phase DAB converters for a equaltotal chip area of 12 p.u., considering a power level of 100 kW and U2 = 750 V.

third voltage level. However, the required area for the top and bottom switches forboth inverter bridges is unequal, which requires custom designed power modules.

5.3.2 Semiconductor loss comparison

In order to compare the total conduction and switching losses in a fair way, the totalsemiconductor area must be equal for all three-phase DAB converters. Therefore,a total semiconductor area of 12 p.u. is chosen while the distribution among theswitches is based on the finding presented in table 5.4. Consequently, total semi-conductor losses are calculated for the complete voltage range considering a powerlevel of 100 kW. The result is shown in figure 5.11.

Comparing these results revealed that the symmetrically operated two-leveltopologies generate the lowest amount of semiconductor losses for voltage ratiosbelow one and close to one (U1=U2). This is expected since DAB converters ingeneral operate most efficiently for voltage ratios close to one. In addition, allsemiconductor chip area per inverter bridge is distributed equally between thetop and bottom switches which is favourable for symmetrically operation. Forhigher voltage ratios, however, the semiconductor losses rise sharply, whereas theconverters with more than one degree of freedom (S3L-YY, S3L-Y∆, and A2L-YY)


generate significantly lower losses. Among these converters, the symmetricallyoperated three-level topology with a star-star connected transformer generates thelowest average semiconductor losses. This is explained by the lower circulatingcurrents and switch turn-off currents which, especially at high voltage ratios, leadto lower conduction and switching losses.

5.3.3 Capacitance comparison

A certain amount of capacitance must be present in the dc-link of both inverterbridges to limit the voltage ripple and reduce the differential mode (DM) conductedelectromagnetic interference (EMI) [75, 114]. To compare the required capacitance,a maximum peak-to-peak voltage ripple of 1 % is allowed considering the nominalvoltage of 750 V for both inverter bridges. This results in ∆u1 ≤ 7.5 V and ∆u2 ≤7.5 V for the complete operating range.

By increasing the capacitance until the voltage ripple remains within the specifica-tion given above, the minimum capacitance is found. This is carried out using thecapacitance value and equivalent series resistance (ESR) of a commercially avail-able film capacitor with type number B32776E8306 [34]. This a 800 V rated capaci-tor with a capacitance of Cref = 30 µF and an ESR of RESR = 6 mΩ at a frequencyof 100 kHz. Consequently, by paralleling this capacitor n times, the dc-link voltagefor inverter bridge x is given by

ux = Ux +qCx1 + qCx2

nCref+ (iCx1 + iCx2)

RESR

n, (5.8)

where qCx1 and qCx2 represent the instantaneous charge of, respectively the capacitorcurrents iCx1 and iCx2 for inverter bridge x. Finally, the peak-to-peak voltage ripple∆ux for inverter bridge x is defined as

∆ux = max(ux)−min(ux). (5.9)

By evaluating the equations given above, the minimum capacitance to meet thevoltage-ripple specification for the complete operating range is found. This is givenin table 5.5. In addition to the voltage ripple specification, the maximum permis-sible rms current of the film capacitor must be respected. This is specified as 18 A,at a temperature of 70 C [34]. An overview of the required number of capacitorsin parallel, to meet both the voltage ripple and the capacitor rms current specifi-cation, is given in table 5.5. The highest number is shown in bold. From this, itis clear that the capacitor size, considering film capacitors, is mainly determinedby the maximum capacitor rms current, as given in table 5.3. The asymmetricallyoperated two-level topology(A2L-YY) is an exception to this statement, since the


Table 5.5: Required number of capacitors in parallel for both inverter bridges to limit thevoltage ripple to 1 % of the nominal voltage and respect the maximum permissible capacitorrms current [34].

Converter To limit voltage ripple To limit capacitor rms currentC11,C12 [p.u.] C21,C22 [p.u.] C11,C12 [p.u.] C21,C22 [p.u.]

S2L-YY 3.82 3.71 6.07 5.80S2L-Y∆ 4.15 2.87 6.51 4.43S3L-YY 2.00 2.76 3.74 9.05S3L-Y∆ 2.33 2.93 3.98 9.75A2L-YY 7.39 14.99 5.97 12.44

iC22

iC21

i[A

]

t [µS]0 10 20 30 40 50

−800

−600

−400

−200

0

200

400

600

800

(a)

iC22

iC21

i[A

]

t [µS]0 10 20 30 40 50

−800

−600

−400

−200

0

200

400

600

800

(b)

Figure 5.12: Capacitor currents iC21 and iC22 of inverter bridge 2 for the symmetrically op-erated three-level topology (S3L-YY) (a) and the asymmetrically operated two-level topol-ogy (A2L-YY) (b), both with a star-star connected transformer, considering a power level of100 kW and U1 = 375 V.

voltage ripple specification demands the largest number of capacitors. Comparedto the three-level topologies, the asymmetrically operated two-level topology re-quires more capacitors in parallel by a factor of 2 and 1.5 for, respectively, inverterbridge 1 and 2. This difference can be explained by analysing the current throughthe capacitors of an inverter bridge. For instance, the currents through the capaci-tors of inverter bridge 2 are shown in figure 5.12 for both converters (A2L-YY andS3L-YY), considering the maximum power level and U1 = 375 V. This revealsthat the two capacitor currents for the three-level topology are shifted by Tsw/6,whereas the same currents for the asymmetrically operated two-level topology arein phase. Consequently, the voltage ripple of the two capacitors in series is signifi-cantly lower for the three-level topology considering (5.8).

5.4 Summary 131

Table 5.6: Overview of the total semiconductor chip area, capacitance, average values ofthe semiconductor losses, and average transfer-inductor rms current, considering a uniformmission profile with a voltage range between 375 V and 1500 V and a power level of 100 kW.

ParameterThree-phase DAB dc-dc converters

S2L-YY S2L-Y∆ S3L-YY S3L-Y∆ A2L-YY

Required chip area [p.u.] 11.72 11.71 10.27 111.21 8.82Required capacitance [p.u.] 11.86 10.94 12.76 13.73 18.71Avg. Semicon. losses [ W] 3123 3140 2299 2750 2566Avg. inductor current IL [ A] 152 153 118 119 124

5.4 Summary

The different topologies, operating methods, and modulation schemes, which havebeen introduced in chapter 2 and chapter 3, have led to five different three-phaseDAB dc-dc converters. This gives rise to the question which converter generatesthe lowest power losses or can be built most compactly for the given operatingrange. To answer this question, a comparative evaluation is carried out. In the firstpart, the current stress of the switches, transformer, and the capacitors in the dc-link is evaluated for the selected three-phase DAB converters. In the second part,the required semiconductor area, the total semiconductor losses, and the requiredcapacitance are compared. An overview of this is given in table 5.6.

Based on the derived component stress, a design is derived for each converter thatuses a minimum amount of semiconductor chip area, while the maximum chipjunction temperature is kept equal. This semiconductor chip area comparison re-vealed that the asymmetric two-level topology (A2L-YY) has the best utilisation ofthe total chip area, however, it requires custom designed power modules. Conse-quently, the semiconductor chip area is kept constant for all converter designs inorder to calculate and compare the semiconductor losses. From this it is shownthat the conventional symmetrically operated two-level topologies have the lowestsemiconductor losses for voltage ratios below and close to unity. However, for widevoltage-range application, as considered in this work, the converter with two ormore degrees of freedom provide lower semiconductor losses. Among these con-verters, the symmetrically operated three-level topology with star-star connectedtransformer (S3L-YY) performs best. Furthermore, the required dc-link capaci-tance of both inverter bridges is derived for the complete operating range to meetthe voltage-ripple specification and respect the permissible capacitor rms current.


This revealed that the asymmetric two-level topology (A2L-YY) requires signifi-cant more capacitance compared to the symmetrically operated two- and three-level topologies.

Considering wide voltage-range applications, the symmetrically operated three-level topology with star-star connected transformer (S3L-YY) performs best interms of average total semiconductor losses and low rms transformer and capac-itor currents. The asymmetrically operated two-level topology (A2L-YY) has aslightly lower performance, however, it is more cost effective due to the reducedcomplexity of the phase-leg and busbar structure. Therefore, a prototype converteris constructed, which allow operation of both converters, to experimentally verifythe modelling and modulation schemes, as will be brought forward in the nextchapter.

Chapter 6

Experimental verification

T HE three-phase DAB converters with more than one degree of freedom havethe ability to minimise the circulating current for a wide voltage-range. Thisreduces the transformer ohmic losses and the total semiconductor losses,

as shown in the previous chapter. Among these converters, the symmetricallyoperated three-level topology with a star-star connected transformer seems mostpromising in view of the average total semiconductor losses and required dc-linkcapacitance. To verify the modelling and modulation scheme derived for this topol-ogy, a prototype converter is developed. In order to have an experimental verifica-tion which is representative for a high-power application, the prototype is designedto achieve a power level of 100 kW with a nominal input and output voltage of750 V. Furthermore, the prototype is designed such that two-level operation, con-sidering symmetric and asymmetric modulation, can also be experimentally veri-fied.

In the following, the design and realisation of the prototype dc-dc converter isdescribed. This includes the three-level inverter bridges, dc-link busbar, magneticcomponents, and the control system. Thereafter, measurements of the two- andthree-level topologies, considering symmetric and asymmetric modulation, areprovided to verify the modelling and modulation schemes.

6.1 High-power dc-dc converter prototype design

The prototype dc-dc converter is designed according to the specifications given intable 6.1. As shown in the previous chapter, the current stress for low voltage ratios

133

134 Chapter 6 Experimental verification

Table 6.1: Converter design specifications of the prototype three-phase DAB dc-dc converter.

Parameter Values

Transformer turns ratio, N 1 : 1Transfer-inductance, L 11.7 µHInductance, L1, L2 10LVoltage, U1 375-1500 VVoltage, U2 750 VMaximum power level 100 kWMaximum current, I1 200 ASwitching frequency, f s 20 kHz

U1/U2 and high power levels is relatively large. To limit the size and losses of theprototype, the current I1 is restricted to a maximum of 200 A. This result in a lineardrop of power for voltages below 500 V.

6.1.1 Three-level inverter bridge realisation

The relatively large voltage and current levels hinders a converter realisation withdiscrete transistors on a printed circuit board (PCB). For this reason, the inverterbridges are constructed with power modules placed on heat sinks and with allcomponents connected through busbars or cables, supported by a mechanical con-struction.

The phase-legs of the inverter bridges use the T-type topology to generate threevoltage levels. The T-type circuit is basically a half-bridge topology with a bidirec-tional switch from the output to the mid-point of the dc-link capacitors. Two-leveloperation is possible, simply, by controlling only the half-bridge switches. In re-cent years, SiC MOSFETs are competing or exceeding the performance of Si IGBTsfor voltage ratings of 1700 V [49, 85]. However, at the time of designing the pro-totype, no power modules were available to realise a T-type phase-leg or inverterbridge with SiC MOSFETs, considering the voltage and current levels. Therefore,the three-level phase-legs are realised with CAS300M17BM2 half-bridge SiC MOS-FET power modules [25] and SKM300GM12T4 bidirectional-switch Si IGBT powermodules [90]. This is shown in figure 6.1, were the grey boxes indicate the powermodules.

The selected power modules for a single phase-leg are placed on a heat sink withfans for forced air cooling. The location of the modules is such that the area of the

6.1 High-power dc-dc converter prototype design 135

S1A

S2A

S1B

S2B

S1C

S2C

U1

U12

U12

NS4AS3A

S4BS3B

S4CS3C

C

B

A

C11

C12

Figure 6.1: Circuit diagram of the prototype three-level inverter bridge, employingCAS300M17BM2 half-bridge SiC MOSFET power modules [25] and SKM300GM12T4bidirectional-switch Si IGBT power modules [90]. The grey boxes indicate the power mod-ules.

commutation loops is as small as possible while the required clearance distance be-tween the gate drivers is respected. Also, it is in favour of the cooling performanceif the power modules are distributed on the heat sink. The resulting layout for asingle phase-leg is shown in figure 6.2a. Consequently, a prototype three-level in-verter bridge is realised by mounting three heat sinks with the presented modulelayout. This is shown in figure 6.2b.

For driving the power modules, the Prodrive PT62SCMD17 [84] was selected forthe half-bridge SiC MOSFET power modules, while the Concept 2SC0435TC0-17[24] was selected for the bidirectional-switch Si IGBT power modules.


200 mm

180

mm

+

− n.c.

N

Bidi

rect

iona

lsw

Heat sink

dc

Hal

f-br

idge

Phase-leg output

Gate driver Gate driver

(a) (b)

Figure 6.2: Power module layout on a heat sink for a single phase-leg, indicating the differentpower modules and the corresponding connections (a). Realised construction of a prototypethree-level inverter bridge (b), containing three heat sinks with the power module layoutfrom (a) including fans for forced air cooling.

6.1.2 Dc-link capacitors and busbar design

The dc-link capacitors of both inverter bridges can be realised by paralleling EPCOSB32776E8306 film capacitors [34]. From the comparative evaluation presented inthe previous chapter, the required amount of capacitors in parallel for the three-level topology, using the B32776E8306 film capacitor, was 4 and 10 capacitors forinverter bridge 1 and 2, respectively. However, since the current I2 for the prototypeconverter is limited to 200 A (75 kW for U1 = 375 V), the capacitor rms currentsremain within 100 A, as shown in section 5.2.3. Therefore, the dc-link capacitanceis realised by paralleling the B32776E8306 film capacitor five times for the top andbottom capacitor (Cx1 and Cx2). This results in ten film capacitors for each inverterbridge.

All capacitors are mounted on a busbar which connects the power modules andcapacitors to the corresponding voltage levels. The busbar is designed to be placedcentrally in each inverter bridge to provide a low inductive path for each phase-leg to all capacitors. This is realised with copper sheet material to create the threevoltage rails on which the capacitors are soldered. The resulting design is shownin figure 6.3a. The voltage rails are separated by a kapton layer with a thickness


(a) (b)

Figure 6.3: Busbar design (a) using copper sheets for the positive (red), neutral (beige), andnegative (blue) voltages rails. The sheets are separated by a kapton layer with a thickness0.075 mm and a dielectric strength of 177 kV/mm. The selected film capacitors (2× 5) aredirectly soldered to the realised busbar (b).

of 0.075 mm and a dielectric strength of 177 kV/mm to isolate the different voltagelevels. Special care in designing the busbar is taken such that the minimum clear-ance and creepage distances, specified in [59] for pollution degree 2 and materialgroup II, are respected. The realisation of the busbar including the film capacitorsis shown in figure 6.3b. It was successfully tested up to a voltage of 1500 V.


Figure 6.4: Prototype three-phase transformer, realised by three single-phase transformers.

6.1.3 Three-phase transformer

The three-phase transformer is realised by three single-phase transformers, asshown in figure 6.4. The transformers have turns ratio of one and are designedto withstand rms currents up to 150 A and a maximum magnetic flux-density of0.2 T for two-level operation with a voltage of 1000 V. Each transformer contains4 UU magnetic core sets of type B67345B0001X087 [33]. To support the maximumrms current, four Litz wires are used in parallel with each wire containing 250strands and each has a diameter of 0.2 mm (δ20 kHz ≈ 0.46 mm). The primary andsecondary winding have 8 turns and are interleaved to reduce proximity losses.This results in 4 layers which are isolated with kapton tape between each layer.

6.1.4 Prototype isolated dc-dc converter

The final assembly of the inverter bridges, together with the three-phase trans-former and the inductors for realising the desired transfer-inductance L and thecommutation inductances L1 and L2 for each phase, is shown in figure 6.5. To verifyoperation at different voltage ratios without requiring a resistive load, two powersupplies are connected to the dc-link terminals of both inverter bridges. Both powersupplies, a switched-mode power supply and a motor-generator set, can source andsink a power of 100 kW. However, the maximum voltage is limited to 1250 V.


Inverter bridge 1 Inverter bridge 2

U1 U2Transformer

L

L1 L2

Figure 6.5: Final assembly of the prototype isolated dc-dc converter with both inverterbridges together with the three-phase transformer and the inductors for realising the desiredtransfer-inductance L and the commutation inductances L1 and L2 for each phase.

The inverter bridges are controlled by a dSPACE MicroLabBox, containing a digi-tal signal processor (DSP) and a field programmable gate array (FPGA). The FPGAis responsible for all time-critical tasks, such as pulse width modulation (PWM),dead-time generation, and hardware protections. Real-time execution of MatlabSimulink models, containing the modulation schemes, and communication to ahost computer is carried out by the DSP.

Measurement setup

Operation of the selected converters is verified by measuring the phase-leg voltageand current of both inverter bridges. This is carried out using an oscilloscope withtwo voltage and current probes, as given in table 6.2. In order to measure the con-version efficiency, the input and output power of the converter is measured usinga power analyser with external current sensors. Measurement, or burden, resis-tors are used in combination with the selected power analyser and external currentsensors, as given in table 6.2, in order to obtain a high measurement accuracy. Theresulting accuracy of power measurements is±0.0684 % of the measured value and±0.1 % of the power analyser’s range. The measurement errors in are included aserror bars in the efficiency measurements presented in the following sections.


Table 6.2: Overview of the measurement equipment.

Equipment Type

Oscilloscope Teledyne LeCroy HDO6024- Voltage probes Tektronix P5200A High voltage differential probe- Current probes Tektronix TCP 404XL probe + TCPA400 amplifier

PEM CWT MiniHF 1 rogowski coilPower analyser Yokogawa WT3000E (4-channel 1000 V/30 A)

DC accuracy: ±0.05 % of reading ±0.1 % of range- Current sensors LEM IT 200-S ±0.0084 % accuracy- Measurement resistor Vishay VPR221 10 Ω ±0.01 %, ±5 ppm/ C

6.2 Measurement results

Measurements are carried out to verify the simulations of the two- and three-leveltopology, considering the proposed symmetric and asymmetric modulationschemes. Among the different operating methods, the symmetrically operatedtwo-level topology is included in the experimental verification since it representsthe conventional three-phase DAB converter. The other two converters representthe three- and two-level topologies with, respectively, symmetric and asymmetricmodulation schemes. The measurements for each converter include voltage andcurrent waveforms, rms transfer-inductor current, switch turn-off currents, andefficiencies for various voltage ratios and power levels.

6.2.1 Symmetrical operation of the two-level topology

Waveforms

Voltage and current waveforms of the symmetrically operated two-level topologyare captured using an oscilloscope and probes as given in table 6.2. The measure-ment includes the phase-to-neutral voltages uAN and uan, together with the phase-leg currents iA and ia, of inverter bridge 1 and 2, respectively. The waveforms arecaptured for U1 = 500 V and U1 = 1000 V, considering a power level of 10 kW and100 kW.

The measurements for the voltages mentioned above with a power level of 10 kWare shown in figure 6.6. As expected, the non-unity voltage ratios result in large

6.2 Measurement results 141

reactive currents in the ac-link and, depending on the voltage ratio, cause hard-switching in inverter bridge 1 or 2. This creates high dV/dt in the phase-leg volt-ages which is reflected in noise on the voltage and current measurements.

Increasing the power level to 100 kW for the same voltage ratios still result in hard-switching. The measured waveforms for these operating points are shown in fig-ure 6.7. However, the ratio between reactive and active power flow in the ac-link isbetter for high power levels.


iA

uan

uAN

300 V/div100 A/div10 µs/div

ia

(a) U1 = 500 V.

iA

uan

uAN


ia

(b) U1 = 1000 V.

Figure 6.6: Measured phase-leg voltages and currents for the symmetrically operated two-level topology with a star-star connected transformer. Plots show operation with U1 = 500 V(a) and with U1 = 1000 V (b), both for U2 = 750 V and a power level of 10 kW.


iA

uan

uAN


ia

(a) U1 = 500 V.

iA

uan

uAN


ia

(b) U1 = 1000 V.

Figure 6.7: Measured phase-leg voltages and currents for the symmetrically operated two-level topology with a star-star connected transformer. Plots show operation with U1 = 500 V(a) and with U1 = 1000 V (b), both for U2 = 750 V and a power level of 100 kW.

Converter currents

The rms current through the transfer-inductor LA is measured for voltages U1 be-tween 500 V and 1000 V, considering power levels of 50 kW and 100 kW. The mea-sured values matches the simulations, as shown in figure 6.8. Furthermore, theswitch turn-off currents are measured and compared with the simulated values for


Simulated P = 50 kW

Measured P = 50 kW

Simulated P = 100 kW

Measured P = 100 kW

I LA[A

]

U1 [V]375 750 1125 15000

50

100

150

200

250

Figure 6.8: Measured and simulated rms transfer-inductor current of the symmetrically op-erated two-level topology with a star-star connected transformer, considering a power levelof 50 kW and 100 kW.

Simulated

Measured

i A(A

1)[A

]

U1 [V]375 750 1125 1500

−400

−200

0

200

400

(a)

Simulated

Measured

i a(a

1)[A

]

U1 [V]375 750 1125 1500

−400

−200

0

200

400

(b)

Figure 6.9: Measured and simulated switch turn-off currents for inverter bridge 1 (a) and 2(b) of the symmetrically operated two-level topology with a star-star connected transformer,considering a power level of 100 kW and with U2 = 750 V.

different voltage ratios and a power level of 100 kW. The measurements are inagreement with the simulations, as shown in figure 6.9.


U1=1000V

U1=750V

U1=500V

Efficien

cy[%

]

Power [kW]

0 10 20 30 40 50 60 70 80 90 10090

91

92

93

94

95

96

97

98

99

100

Figure 6.10: Measured converter efficiency for the symmetrically operated two-level topol-ogy with a star-star connected transformer, considering three different voltages U1 =500 V , 750 V , 1000 V and with U2 = 750 V.

Efficiency

Measurements of the converter efficiency are carried out using a YokogawaWT3000E power analyser with external LEM IT 200-S high-precision currentsensors using Vishay VPR221 measuring resistors. The measured efficiency forseveral voltage ratios is shown in figure 6.10. High efficiencies are measured fora voltage ratio of one (U1 = U2 = 750 V), which reaches 98.4 % for a power levelof 100 kW. For non-unity voltage ratios, the efficiency is reduced, especially atlower power levels. This is due to the high reactive current flow in the ac-link andhard-switching of one inverter bridge, as shown earlier.

6.2.2 Symmetrical operation of the three-level topology

Waveforms

Waveforms of the symmetrically operated three-level topology are captured to ver-ify the simulations performed in section 3.2.2. For this reason, the phase-to-neutralvoltages uAN and uan, together with the phase-leg currents iA and ia, of inverterbridge 1 and 2, respectively, are captured using an oscilloscope. This is shown infigure 6.11 for U1 = 500 V and U1 = 1000 V, considering a power level of 10 kW.It can be seen that the three-level topology is soft-switching and generates lowphase-leg rms currents for non-unity voltage ratios. This would not be the case


with the symmetrically operated two-level three-phase DAB converter, as shownin the previous section. Measured waveforms for the same voltage ratios with apower level of 100 kW are shown in figure 6.12. These operating points also resultin soft-switching and show near sinusoidally shaped phase-leg currents. Further-more, the measured waveforms for voltage and power levels presented here are ingood agreement with the simulated waveforms, shown in figures 3.9 and 3.10. Thevoltage oscillations that are visible for three-level operation are caused by the par-asitic inductance due to the interconnection of half-bridge and bidirectional-switchpower module. It is expected that a three-level phase-leg integrated in one powermodule would reduce these oscillations.


iA

uan

uAN


ia

(a) U1 = 500 V.

iA

uan

uAN


ia

(b) U1 = 1000 V.

Figure 6.11: Measured phase-leg voltages and currents for the symmetrically operated three-level topology with a star-star connected transformer. Plots show operation with U1 = 500 V(a) and with U1 = 1000 V (b), both for U2 = 750 V and a power level of 10 kW.


iA

uan

uAN


ia

(a) U1 = 500 V.

iA

uan

uAN


ia

(b) U1 = 1000 V.

Figure 6.12: Measured phase-leg voltages and currents for the symmetrically operated three-level topology with a star-star connected transformer. Plots show operation with U1 = 500 V(a) and with U1 = 1000 V (b), both for U2 = 750 V and a power level of 100 kW.

Converter currents

The rms value of the current through the transfer-inductor LA is measured for anumber voltages U1 between 500 V and 1000 V, considering power levels of 50 kWand 100 kW. The measured values match the simulations, as shown in figure 6.13.Furthermore, the switch turn-off currents are derived from captured waveforms


Simulated P = 50 kW

Measured P = 50 kW


Measured P = 100 kW

I LA[A

]

U1 [V]375 750 1125 15000

50

100

150

200

250

Figure 6.13: Measured and simulated rms value of the current through the transfer-inductorcurrent of the symmetrically operated three-level topology with a star-star connected trans-former, considering a power level of 50 kW and 100 kW.

i A(A

2)[A

]

Simulated

Measured

i A(A

1)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

(a)

i a(a

1)[A

]

Simulated

Measured

i a(a

2)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

(b)

Figure 6.14: Measured and simulated switch turn-off currents for inverter bridge 1 (a) and2 (b) of the symmetrically operated three-level topology with a star-star connected trans-former, considering a power level of 100 kW and with U2 = 750 V.

and compared with the simulated values for different voltage ratios and a powerlevel of 100 kW. The measurements are in good agreement with the simulations, asshown in figure 6.14.


U1=1000V

U1=750V

U1=500V

Efficien

cy[%

]

Power [kW]

0 10 20 30 40 50 60 70 80 90 10090

91

92

93

94

95

96

97

98

99

100

Figure 6.15: Measured converter efficiency for the symmetrically operated three-level topol-ogy with a star-star connected transformer, considering three different voltages U1 =500 V , 750 V , 1000 V and with U2 = 750 V.

Efficiency

Measurements of the converter efficiency are carried out using a power analyserwith high-precision current sensors. The measured efficiency for several voltageratios is shown in figure 6.15. High efficiencies are measured for a voltage ratio ofone (U1 = U2 = 750 V), reaching 98.4 % for a power level of 100 kW. This voltageratio results in two-level operation whereby only the SiC half-bridge modules areoperated, which explaines the high efficiency. For U1 = 500 V and a power levelof 100 kW, an efficiency of 95.1 % is measured. This operating point suffers fromhigh values of the rms current through the transfer-inductor and the bidirectional-switch in inverter bridge 2, as shown in the previous chapter. The Si IGBTs in thebidirectional-switch power modules have relative high conduction and switchinglosses, which causes a lower efficiency. This is of less influence on the efficiency forvoltage ratios above one, as shown in figure 6.15 for U1 = 1000 V, since the rmsvalue of the current through the transfer-inductance is lower.


6.2.3 Asymmetrical operation of the two-level topology

Waveforms

Asymmetrical operation of the two-level topology is verified by comparing mea-sured waveforms of the phase-to-neutral voltages uAN and uan, together with thephase-leg currents iA and ia, with the simulations in section 3.3.2. The capturedwaveforms are shown in figure 6.16 for two voltage ratios and a power level of10 kW. Similar as with the three-level topology, it can be seen that the two-leveltopology is soft-switching and generates low phase-leg rms currents for non-unityvoltage ratios. This would not be the case with the symmetrically operated two-level three-phase DAB converter, as shown in the previous chapter. Measuredwaveforms for the same voltage ratios with a power level of 100 kW are shownin figure 6.17. These operating points also result in soft-switching, however, dueto the asymmetric currents, the peak currents are higher compared to those of thesymmetrically operated three-level topology. Furthermore, the measured wave-forms for voltage and power levels presented here are in good agreement with thesimulated waveforms, shown in figure 3.27 and figure 3.28.


iA

uan

uAN


ia

(a) U1 = 500 V.

iA

uan

uAN


ia

(b) U1 = 1000 V.

Figure 6.16: Measured phase-leg voltages and currents for the asymmetrically operated two-level topology with a star-star connected transformer. Plots show operation with U1 = 500 V(a) and with U1 = 1000 V (b), both for U2 = 750 V and a power level of 10 kW.


iA

uan

uAN


ia

(a) U1 = 500 V.

iA

uan

uAN


ia

(b) U1 = 1000 V.

Figure 6.17: Measured phase-leg voltages and currents for the asymmetrically operated two-level topology with a star-star connected transformer. Plots show operation with U1 = 500 V(a) and with U1 = 1000 V (b), both for U2 = 750 V and a power level of 100 kW.

Converter currents

The rms value of the current through the transfer-inductor LA is measured for anumber voltages U1 between 500 V and 1000 V, considering power levels of 50 kWand 100 kW. The measured values match the simulations, as shown in figure 6.18.Switch turn-off currents are derived from captured waveforms and compared with


Simulated P = 50 kW

Measured P = 50 kW


Measured P = 100 kW

I LA[A

]

U1 [V]375 750 1125 15000

50

100

150

200

250

Figure 6.18: Measured and simulated rms transfer-inductor current of the asymmetricallyoperated two-level topology with a star-star connected transformer, considering a powerlevel of 50 kW and 100 kW.

i A(A

2)[A

]

Simulated

Measured

i A(A

1)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

(a)

i a(a

1)[A

]

Simulated

Measured

i a(a

2)[A

]

U1 [V]375 750 1125 1500

100

200

300

400

−400

−300

−200

−100

0

(b)

Figure 6.19: Measured and simulated switch turn-off currents for inverter bridge 1 (a) and2 (b) of the asymmetrically operated two-level topology with a star-star connected trans-former, considering a power level of 100 kW and with U2 = 750 V.

the simulated values for different voltage ratios and power of 100 kW. The mea-sured results are in good agreement with the simulation, as shown in figure 6.19.


U1=1000V

U1=750V

U1=500V

Efficien

cy[%

]

Power [kW]

0 10 20 30 40 50 60 70 80 90 10090

91

92

93

94

95

96

97

98

99

100

Figure 6.20: Measured converter efficiency of the asymmetrically operated two-level topol-ogy for three different voltages U1 = 500 V , 750 V , 1000 V and with U2 = 750 V.

Efficiency

The measured efficiency of the asymmetrically operated two-level topology for sev-eral voltage ratios is shown in figure 6.20. Compared to the symmetrically operatedthree-level topology, higher efficiencies are measured for non-unity voltages ratios.An efficiency of 96.5 % and 98.0 % is measured for, respectively, U1 = 500 V andU1 = 1000 V, considering a power level of 100 kW. For comparison, the efficiencyof the symmetrically operated three-level topology for the same operating pointsis measured to be 95.1 % and 97.7 %, respectively. This is explained by the fact thatthe Si IGBTs, which have higher losses compared to the SiC MOSFETs, are not usedin the asymmetrically operated two-level topology. For a unity voltage ratio and apower level of 100 kW, an efficiency of 98.3 % is measured for asymmetric opera-tion, whereas, an efficiency of 98.4 % was found for symmetric modulation.


A2L-YY

S3L-YY

S2L-YY

Efficien

cy[%

]

U1 [V]

500 625 750 875 1000 1125 125094

95

96

97

98

99

100

Figure 6.21: Measured efficiency versus U1 for the symmetrically operated two- and three-level topology (S2L-YY and S3L-YY) and the asymmetrically operated two-level topology(A2L-YY), with a power level of 100 kW and U2 = 750 V.

6.3 Discussion

In this chapter, different three-phase DAB topologies and operating methods areexperimentally verified. From the comparative evaluation in chapter 5, the three-level topology with a symmetric modulation scheme (S3L-YY) and the two-leveltopology with an asymmetric modulation sheme (A2L-YY), both with a star-starconnected transformer, were selected for their efficient operation in a wide voltage-range. These converters, together with the conventional symmetrically operatedtwo-level topology (S2L-YY), are included in the experimental verification.

A high-power prototype dc-dc converter was developed which permits operationof the two- and three-level topologies, using the proposed modulation schemes.For each converter, the waveforms, rms values of the current through the transfer-inductor, switch turn-off currents, and efficiencies are measured for various volt-ages ratios and power levels up to 100 kW. The measurements are in good agree-ment with the simulations, thereby supporting the theoretical analysis of the pre-ceding chapters.

Efficiencies are measured for each converter, considering a wide range of voltageand power levels. The highest efficiencies are found for voltage ratios close to unity(U1 = U2), reaching efficiencies of well over 98 %. An overview of the measuredefficiency as function of the voltage U1 for each converter, considering a powerlevel of 100 kW, is given in figure 6.21. Because of the absence of the commuta-

6.3 Discussion 157

tion inductances (L1, L2) and utilising half-bridge modules with SiC MOSFETs, theconventional symmetrically operated two-level topology (S2L-YY) has the high-est efficiency up to a voltage U1 of approximately 900 V. However, for highervalues of U1 the current through the transfer-inductor increases significantly, asshown in figure 6.8, which reduces the efficiency. Additionally, soft-switching islost for non-unity voltage ratios. Since the SiC MOSFETs have negligible diode re-verse recovery losses, this does not affect the total losses considerably. However,hard-switching causes high dV/dt values of the phase-leg voltages. Therefore, itis expected that EMI emissions are higher compared to soft-switching converters.Furthermore, hard-switching causes a delayed commutation, which introduces adifference between the actual phase-shift and the applied phase-shift. As a result,large deviations of the actual and required power exist when the converter is hard-switching.

The two- and three-level topology with, respectively, asymmetric and symmet-ric modulation schemes (A2L-YY and S3L-YY) have proven to be able to reducethe current through the transfer-inductor for non-unity voltage ratios. Therefore,higher efficiencies are achieved for light loads and for maximum power with highvoltage ratios. For low values of U1, when the current in the ac-link is high, thesymmetrically operated three-level topology (S3L-YY) demonstrates a relativelylow efficiency compared to the other converters. This is caused by the relative highconduction and switching losses of Si IGBTs of the bidirectional-switch module.Still, for higher voltage ratios the efficiency is relatively constant and surpasses theother converters. For U1 = 1250 V the measured efficiency is 97.5 %, as shown infigure 6.21. The symmetrically operated three-level topology (S3L-YY) has poten-tially the highest average efficiency if the Si IGBTs are replaced with SiC MOSFETs,as shown in the comparative evaluation in chapter 5. The asymmetrically operatedtwo-level topology (A2L-YY), on the other hand, also demonstrates high efficien-cies for a wide voltage-range while it has the benefit of a simple phase-leg andbusbar structure. However, the losses are not equally distributed among the topand bottom switches and the maximum switch turn-off currents are significantlylarger for low and high voltage ratios compared to the symmetrically operatedthree-level topology. In addition, the converter requires more capacitance in bothinverter bridges due to the higher switch turn-off currents and the asymmetric cur-rent waveform, as shown in the comparative evaluation in chapter 5. Nevertheless,it is expected that the asymmetrically operated two-level converter can be realisedat a lower cost due to the reduced complexity of the phase-leg and busbar structure.


6.4 Summary

The two- and three-level topologies with a star-star connected transformer andtheir associated modulation schemes have been experimentally verified using ahigh-power prototype. The obtained measurement results are in good agreementwith the simulations, thereby supporting the theoretical analysis of the preced-ing chapters. The two- and three-level topology with, respectively, asymmetricand symmetric modulation schemes demonstrate low circulating currents, soft-switching, and high efficiencies for a wide voltage and power range. On the con-trary, the conventional symmetrically operated two-level topology is only soft-switching for voltage ratios close to unity and is exposed to large circulating cur-rents for non-unity voltage ratios. The symmetrically operated three-level topologyhas potentially the highest average efficiency, whereas the asymmetrically oper-ated two-level topology is more cost effective due to the reduced complexity of thephase-leg and busbar structure.

Part III

Closing

159

Chapter 7

Conclusions andrecommendations

T HIS research has focussed on isolated dc-dc converters, based on the three-phase DAB topology, to convert electric power while providing galvanic iso-lation from input to output. The main goal was to derive converter topolo-

gies and operating methods that provide efficient power conversion for applica-tions that exhibit a wide voltage and power range. This thesis provides modelling,modulation schemes, and comparison of existing and proposed three-phase DABconverter topologies.

The most important conclusions that have been presented throughout this thesisare summarised in this chapter. This is followed by the main contributions of theresearch, together with a list of publications. Furthermore, several aspects of three-phase DAB converters are still unexplored, therefore, recommendations are givenfor future research.

7.1 Conclusions

The goal of this research has been to derive isolated dc-dc converter topologies thathave a high efficiency and power density for a wide range of voltage and power lev-els. To accomplish this goal, a multi-level topology is adopted in the three-phaseDAB converter. This has led to the three-level three-phase DAB topology with astar-star or star-delta connected transformer. Considering a symmetric operating

161

162 Chapter 7 Conclusions and recommendations

method, three degrees of freedom (control variables) are available for fixed fre-quency operation. This allows for regulation of the power and minimisation of thecirculating current while soft-switching boundaries can be included as constraints.Similar to the three-level topology, the conventional two-level three-phase DABconverter can also be operated with three degrees of freedom by controlling theduty-cycles of both inverter bridges. This results in asymmetric voltage and cur-rent waveforms and, therefore, is designated as ”asymmetric operating method”.By combining different operating methods, topologies, and transformer windingconfigurations, a family of three-phase DAB converters is created.

In order to provide analytical equations for the power transfer and verify the soft-switching behaviour, the converters are modelled by piecewise-linear equations.As a result, a large number of switching modes are found for the converters withthree degrees of freedom. Therefore, a systematic approach is given to identifythe switching modes and their boundaries. Consequently, the derivation of theanalytical equations, describing the converter currents and the power, is presentedfor any switching mode.

Using a numerical optimiser, control variables have been derived for a wide rangeof voltage and power levels that result in soft-switching and minimal circulatingcurrents. Based on this optimal modulation scheme, it is found that the symmet-rically operated three-level topologies can be operated using only two degrees offreedom, while ZVS and minimum circulating currents are maintained for powerlevels above 50 %. In a second step, using the piecewise-linear model, closed-formanalytical solutions of two control variables are derived that result in soft-switchingand low circulating currents. A similar approach is used for the asymmetrically op-erated two-level topology, however, two degrees of freedom did not result in lowcirculating currents. Therefore, as a third step, equations have been derived, whichclosely match with the optimal switch turn-off currents, that resulted in an analyti-cal modulation scheme with three degrees of freedom.

The impact of a star-star and star-delta connected transformer on the performanceof the symmetrically operated two-level topology has been investigated for appli-cations with a limited voltage range. The comparison revealed that the two-leveltopology with a star-delta connected transformer features lower component stressfor medium to high power levels. In addition, a wider soft-switching range, in-dependent of the phase-shift, is found for phase-shifts below π/6. This is experi-mentally verified with a high-power prototype. The measurements supported thetheoretical analysis and demonstrated that a converter with a star-delta configura-tion achieves a higher efficiency due to a loss reduction up to 15 %.

Several topologies, operating methods, and modulation schemes have come to at-tention throughout this thesis. This has led to the composition of five different

7.2 Contributions 163

three-phase DAB dc-dc converters. These are compared over a wide range of volt-age and power levels. It has been demonstrated that, for equal conditions, the con-verters which have multiple degrees of freedom can achieve the lowest circulatingcurrents, whereas the conventional symmetrically operated two-level topologiesgenerate more than twice the amount of circulating current for high voltage ratios.This result also reflects in the current stress of the components and the requiredtotal semiconductor chip area. The latter revealed that the asymmetrically oper-ated two-level topology requires approximately 25 % less chip area compared to theconventional symmetrically operated two-level topologies. For an equal total chiparea, the symmetrically operated three-level topology with a star-star connectedtransformer has the lowest average total semiconductor losses. This is explainedby the low circulating currents and switch turn-off currents which, especially athigh voltage ratios, lead to lower conduction and switching losses.

A high-power prototype converter has been designed and built to experimentallyvalidate two- and three-level topologies with a star-star connected transformer andtheir associated modulation schemes. The experimental results show good agree-ment with the predictions and, thereby, support the theoretical analysis. For aunity voltage ratio, all three converters have ZVS, low circulating currents, andefficiencies exceeding 98 % for power levels up to 100 kW. For non-unity volt-age ratios, however, the conventional symmetrically operated two-level topologyis hard-switching and suffers from large circulating currents. On the contrary, thetwo- and three-level topology with, respectively, asymmetric and symmetric mod-ulation schemes demonstrate low circulating currents, soft-switching, and high ef-ficiencies, especially for high voltage ratios.

Overall, it has been demonstrated that the two- and three-level topology with, re-spectively, asymmetric and symmetric modulation schemes are promising convert-ers for high-power applications with a wide voltage-range. The symmetrically op-erated three-level topology can potentially achieve the highest average efficiency,whereas the asymmetrically operated two-level topology is more cost effective dueto the reduced complexity of the phase-leg and busbar structure.

7.2 Contributions

• Introduction of a three-level topology of the three-phase DAB converter.By adopting a three-level inverter topology in the conventional three-phaseDAB converter, additional degrees of freedom arise. A three-level topologyis introduced which can achieve soft-switching and low internal circulating


currents for a wide voltage-range, considering symmetric operation and dif-ferent transformer winding configurations.

• Development of a piecewise-linear modelling method for three-phaseDAB converters. To accurately model three-phase DAB converters andobtain closed-form analytical solutions of the control variables, a piecewise-linear model is required. However, a large number of switching modesare involved for the converters with multiple degrees of freedom. For thisreason, a systematic and generic modelling method is given to identifythe switching modes and their boundaries. Using this modelling method,analytic equations are given for several three-phase DAB converters.

• Modulation schemes for three-phase DAB converters. In order to operatethree-phase DAB converters as efficiently as possible, analytical modulationschemes are derived, based on the results obtained from numerical optimi-sation algorithms. By using the piecewise-linear models, equations are givenfor, respectively, the symmetric and asymmetric operation of the three- andtwo-level topologies that result in soft-switching and low circulating currents.

• Investigation of the impact of different transformer connections on the per-formance of the symmetrically operated two-level topology. A comparisonis presented in which the impact of different transformer connections on thestress of the switches, transformer, and capacitors is investigated. The com-parison is carried out for a star-star and star-delta connected transformer, con-sidering an input-to-output voltage ratio close to one. Experimental resultsare included which support the theoretical analysis.

• Comparative evaluation of three-phase DAB converters. Several topologies,operating methods, and modulation schemes have come to attention through-out this thesis. This has led to the composition of five different three-phaseDAB dc-dc converters. To assess the appliance for wide voltage-range high-power applications, a comparative evaluation is carried out. For this purpose,the current stress of the components, the required semiconductor chip area,and the semiconductor losses are compared.

• Experimental verification of the selected topologies and modulationschemes using a high-power prototype converter. A three-phase DABconverter is designed and constructed to verify symmetric operation ofthe two- and three-level topology, as well as asymmetric operation of thetwo-level topology. The measurements support the theoretical models andshow efficient power conversion for a wide voltage-range and powers up to100 kW.

7.2 Contributions 165

7.2.1 Journal publications

• N. H. Baars, J. Everts, H. Huisman, J. L. Duarte, and E. A. Lomonova, ‘A 80-kW isolated dc-dc converter for railway applications’, IEEE Transactions onPower Electronics, vol. 30, no. 12, pp. 6639–6647, Dec. 2015.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Performanceevaluation of a three-phase dual active bridge dc-dc converter with differenttransformer winding configurations’, IEEE Transactions on Power Electronics,vol. 31, no. 10, pp. 6814–6823, Oct. 2016.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Modelling andmodulation of three-level three-phase dual active bridge dc-dc converters’,IEEE Transactions on Power Electronics, under review, 2017.

7.2.2 Conference publications

• N. H. Baars, ‘80 kW auxiliary converter for railway applications’, in Proceedingof the IEEE Young Researchers Symposium in Electrical Engineering, pp. 1–5, Apr.2014.

• N. H. Baars, H. Huisman, J. L. Duarte, and J. Verschoor, ‘A 80 kW isolateddc-dc converter for railway applications’, in Proceedings of the IEEE EuropeanConference on Power Electronics and Applications, pp. 1–10, Aug. 2014.

• N. H. Baars, C. G. E. Wijnands, and J. L. Duarte, ‘Reduction of thermal cyclingto increase the lifetime of MOSFET motor drives’, in Proceedings of the IEEEAnnual Conference of the Industrial Electronics Society, pp. 1740–1746, Oct. 2014.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Impact ofdifferent transformer-winding configurations on the performance of a three-phase dual active bridge dc-dc converter’, in Proceedings of the IEEE EnergyConversion Congress and Exposition, pp. 637–644, Sep. 2015.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Evaluation ofa high-power three-phase dual active bridge dc-dc converter with three-levelphase-legs’, in Proceedings of the IEEE European Conference on Power Electronicsand Applications, pp. 1–10, Sep. 2016.

• N. H. Baars, C. G. E. Wijnands, and J. Everts, ‘ZVS modulation strategy fora three-phase dual active bridge dc-dc converter with three-level phase-legs’,in Proceedings of the IEEE European Conference on Power Electronics and Applica-tions, pp. 1–10, Sep. 2016.


• N. H. Baars, C. G. E. Wijnands, and J. Everts, ‘A three-level three-phase dualactive bridge dc-dc converter with a star-delta connected transformer’, in Pro-ceedings of the IEEE Vehicle Power and Propulsion Conference, pp. 1–6, Oct. 2016.

• N. H. Baars, J. Everts, C. G. E. Wijnands, and E. A. Lomonova, ‘Modulationstrategy for wide-range ZVS operation of a three-level three-phase dual activebridge dc-dc converter’, in Proceedings of the IEEE Applied Power ElectronicsConference and Exposition, pp. 3357–3364, Mar. 2017.

• H. Huisman, F. Baskurt, A. Bouloukos, N. H. Baars, and E. A. Lomonova,‘Optimal trajectory control of a series-resonant inverter with a non-linear res-onant inductor’, in Proceedings of the IEEE International Symposium on PredictiveControl of Electrical Drives and Power Electronics, pp. 1–6, Sep. 2017.

7.3 Recommendations for future work

In this thesis, several three-phase DAB converter topologies and operation methodshave been presented, for which models, modulation schemes, and comparisons arederived. However, some areas require further research, as will be brought forwardin the following.


• Both inverter bridges of the three-phase DAB converters in this work areconsidered to be operated both with symmetric or asymmetric modulation.However, a hybrid solution which utilises both operating methods is notincluded. This could potentially be interesting for high-to-low voltage con-version, where the high-voltage side is equipped with a three-level inverterbridge and the low-voltage side with a two-level inverter bridge. In this way,three degrees of freedom are available for hybrid symmetrical-asymmetricaloperation while the high-voltage side has reduced voltage stress from thethree-level circuit and the low-voltage side has a lower component count andreduced complexity.

• The family of three-phase DAB converters is much more extensive than theconverters presented in this thesis. For example, asymmetric operation couldbe considered for the three-level (or multi-level) topology. This would, how-ever, lead to increased complexity of modelling and modulating the convert-ers. Therefore, further research on how to deal with the increased complexityof three-phase converters with a high number of control variables is required.

7.3 Recommendations for future work 167

• The presented modelling assumes steady-state operation, this may not giveadequate results for highly dynamical operation. Additional research is re-quired if fast converter response and ZVS during dynamical operation is re-quired.

• In this thesis, a fixed switching frequency is considered for operation of three-phase DAB converters. However, a variable switching frequency could re-duce the required current from the commutation inductors to achieve ZVSover a wide operating range. This could be beneficial for applications whichexperience a very wide voltage range, such as single-stage ac-dc converters.

• The proposed analytical modulation schemes provide close-to-minimal rmscirculating currents for high loads. However, additional research is requiredto decrease the circulating currents also for light load conditions. Especiallythe symmetrically operated three-level topologies, which exploit only two de-grees of freedom, should be investigated on this matter.

• Currently, the modelling and numerical optimiser, used to obtain optimalcontrol variables, does not include the required charge to achieve ZVS nordoes it take the resonant transition into account. This has to be implementedto further optimise the modulation scheme and minimise the commutationinductances.


• In addition to the star-star and star-delta transformer connections, many morewinding configurations exist. For instance, the extended delta, zig-zag, andpolygon winding configurations could be considered. Further research is re-quired to find the impact of these transformer connections on two-level ormulti-level three-phase DAB converters.

• The comparative evaluation carried out in this work includes only the T-typecircuit for the three-level topologies. Additional multi-level circuits, such asthe neutral point clamped (NPC) and flying capacitor (FC) topology, whichallow operation using devices with lower blocking voltages, should be in-cluded to find the most efficient or cost effective solution.

• The prototype converter presented in this thesis is built to experimentallyverify operation of the topological concepts and the associated modulationschemes. While the prototype converter meets the design specifications, thetrade-off between overall efficiency, cost, and volume can be optimised byperforming a pareto-front analysis.


• Currently, the three-phase transformer is realised by three single-phase trans-formers while external inductors are used to achieve the required transfer-inductance and commutation inductances for practical convenience. How-ever, to reduce the volume and losses of the magnetic components, an opti-mal transformer design, preferably a symmetric three-phase structure, withintegrated inductances is recommended.

Appendix A

Nomenclature

A.1 Notation

Notation Description

x, X Small or capital letters in non-italic font are used for circuit,waveform, and converter designations

x Instantaneous value of variable x

X Mean or rms value of x

x′ Variable x referred to the opposed transformer winding side

∆x Peak-to-peak value of x, given by max(x)−min(x)

x Vector with m × 1 elements, typically describing variable xfor each interval angle

X Matrix with m× n elements, typically describing variable xfor each interval angle (rows) and phases of each inverterbridge (columns)

xᵀ Transpose of x

x′ Manipulation of x, typically a selection or circular shift of x

〈x〉 Average value of x between two adjacent interval angles

169

170 Appendix A Nomenclature

A.2 Circuit, waveform, and converter designations

Designation Description

A,B,C Phases of inverter bridge 1

a,b,c, Phases of inverter bridge 2

N Neutral-point of inverter bridge 1

n Neutral-point of inverter bridge 2

S Transformer star-point on inverter bridge 1 side

s Transformer star-point on inverter bridge 2 side

X1, X2, X3, X4 Designation of rising and falling transitions of the phase-to-neutral voltage, with X corresponding to phases A,B,C anda,b,c, starting with the rising transition of the positive volt-age level

Y-Y Star-star connected transformer

Y-∆ Star-delta connected transformer

S2L-YY Symmetrically operated two-level topology with a star-starconnected transformer

S2L-Y∆ Symmetrically operated two-level topology with a star-deltaconnected transformer

S3L-YY Symmetrically operated three-level topology with a star-starconnected transformer

S3L-Y∆ Symmetrically operated three-level topology with a star-delta connected transformer

A2L-YY Asymmetrically operated two-level topology with a star-starconnected transformer

A.3 Scalar symbols

Scalar Unit Description

Asc p.u. Normalised semiconductor chip area

c - integration constant

C11, C12 F DC-link capacitors of inverter bridge 1

A.3 Scalar symbols 171


C21, C22 F DC-link capacitors of inverter bridge 2

Cs F Switch output capacitance

d1 - Duty-cycle of inverter bridge 1

d2 - Duty-cycle of inverter bridge 2

Eoff J Turn-off energy of a transistor

Eon J Turn-on energy of a transistor

fsw Hz Switching frequency

I1 A DC-link current of inverter bridge 1

I2 A DC-link current of inverter bridge 2

iA, iB, iC A Phase-leg currents of inverter bridge 1

ia, ib, ic A Phase-leg currents of inverter bridge 2

ids A Drain-source current of a MOSFET

IC11 , IC12 A RMS value of the current through, respectively, C11and C12

IC21 , IC22 A RMS value of the current through, respectively, C21and C22

iL A Current through transfer-inductor L for a given phase

IL A RMS value of iLiLA , iLB , iLC A Current through transfer-inductor L for each phase

iL1A , iL1B , iL1C A Current through commutation inductor L1 for eachphase of inverter bridge 1

iL2a , iL2b , iL2c A Current through commutation inductor L2 for eachphase of inverter bridge 2

IS1A, IS2A, IS3A A RMS value of the current through, respectively, thetop, bottom, and middle switches of phase-leg ’A’

IS1a, IS2a, IS3a A RMS value of the current through, respectively, thetop, bottom, and middle switches of phase-leg ’a’

IZVS A Minimum switch turn-off current to achieve ZVS

kBV - Scaling factor of the switch on-resistance as functionof the blocking voltage

L H Transfer-inductance

LA, LB, LC H Transfer-inductance for each phase

L1 H Commutation inductance for inverter bridge 1



L1A, L1B, L1C H Commutation inductors for each phase in inverterbridge 1

L2 H Commutation inductance for inverter bridge 2

L2a, L2b, L2c H Commutation inductors for each phase in inverterbridge 2

m - Switching mode number

N - Transformer winding turns ratio

P W Active power

Pcond W Semiconductor conduction losses

Psw W Semiconductor switching losses

QA C Charge

RESR Ω Equivalent series resistance

Ron Ω On-resistance of a transistor

RthJ−CC/ W Thermal junction-to-case resistance of a transistor

S VA Apparent power

t s Time

tdt s Dead-time

TJC Semiconductor junction temperature

TUF - Transformer utilisation factor

U1 V DC-link voltage of inverter bridge 1

U2 V DC-link voltage of inverter bridge 2

uAN, uBN, uCN V Phase-to-neutral voltages of inverter bridge 1

uan, ubn, ucn V Phase-to-neutral voltages of inverter bridge 2

uAS, uBS, uCS V Phase-to-star-point voltages of inverter bridge 1

uas, ubs, ucs V Phase-to-star-point voltages of inverter bridge 2

uab, ubc, uca V Phase-to-phase voltages of inverter bridge 2

uds V Drain-source voltage of a MOSFET

UBV V Rated blocking voltage of a transistor

uL V Voltage across transfer-inductor L for a given phase

φ rad Phase-shift between the first harmonic of the phase-to-star voltages of the associated phases in both in-verter bridges

A.5 Abbreviations 173


θ rad Angle, given by θ = ωt

ω rad/s Angular frequency

A.4 Vector and matrix symbols

Vector/Matrix Unit Description

Am - Linear coefficient of the switching mode boundaries

bm - Constant coefficient of the switching mode bound-aries

x - Control variable vector, given by x = [d1 d2 φ]ᵀ

θm rad Angle vector, containing the interval angles of a halfor full switching cycle for an arbitrary switching mode

iLm A Vector of the transfer-inductor current for an arbi-trary switching mode, containing current values cor-responding to the interval angles of θm

uA, uB, uC - state vector of the phase-legs of inverter bridge 1

ua, ub, uc - state vector of the phase-legs of inverter bridge 2

uL V Transfer-inductor voltage for a half or full switchingcycle corresponding to an arbitrary switching mode

Um - Converter state matrix, containing the states of the in-verter bridges for a half or full switching cycle corre-sponding to an arbitrary switching mode

A.5 Abbreviations

Abbreviation Description

ac Alternating current

APU Auxiliary power unit

BEV Battery electric vehicle

DAB Dual active bridge


Abbreviation Description

dc Direct current

DM Differential mode

DSP Digital signal processor

EMI Electro magnetic interference

FC Flying capacitor

ESR Equivalent series resistance

FPGA Field-programmable gate array

HESS Hybrid energy storage system

IGBT Insulated gate bipolar transistor

LF Low frequency

MOSFET Metal-oxide-semiconductor field-effect transistor

NPC Neutral point clamped

PCB Printed circuit board

PFC Power factor correction

PHEV Battery hybrid electric vehicle

PWM Pulse-width modulation

PWL Piecewise linear

RMS Root mean square

SAB Single active bridge

SPS Single phase-shift

Si Silicon

SiC Silicon carbide

SST Solid-state transformer

ZVS Zero voltage switching

Appendix B

Converter models andanalytical modulation schemes

B.1 Symmetrically operated three-level topology witha star-star connected transformer

As proposed in chapter 3, the symmetrically operated three-level topologies re-quire only two degrees of freedom to achieve soft-switching and close-to-optimalcirculating currents. Therefore, the switching modes, PWL equations, and the ana-lytical modulation schemes are derived for voltage ratios less than or equal to one(U1/U2 ≤ 1) with d1 = 1/2. The same switching modes, equations, and modulationschemes can also be used for voltage ratios larger than or equal to one (U1/U2 ≥ 1)by interchanging the duty-cycles (d1,d2) and dc-link voltages (U1,U2).

B.1.1 Switching modes

The voltage transitions sequences of the three-level three-phase DAB converterwith a star-star connected transformer for d1 = 1/2 and 0 ≤ φ ≤ π/2 define theswitching modes. This is shown in table B.1. These are only valid for certain con-trol variables, as shown in figure B.1. This can be verified for each switching modem byAmx+ bm ≤ 0, for which matricesAm and bm are given below. (x = [d2 φ]ᵀ)

175

176 Appendix B Converter models and analytical modulation schemes

Table B.1: Sequences of the voltage transition for half a switching cycle, describing theswitching modes of the symmetrically operated three-level topology with a star-star con-nected transformer for positive power levels (0 ≤ φ ≤ π/2) with d1 = 1/2.

φ, d2 0 ≤ d2 ≤ 1/6 1/6 ≤ d2 ≤ 1/3 1/3 ≤ d2 ≤ 1/2

−π/6 ≤φ≤ π/6 A1b3b4C2a1a2B1c3c4 A1c2a1C2b4c3B1a2b1 A1a1c2C2c3b4B1b1a2

0 ≤φ≤ 2π/6 A1c2b3C2b4a1B1a2c3 A1b3c2C2a1b4B1c3a2 A1a4a1C2c2c3B1b4b1π/6 ≤φ≤ 3π/6 A1c1c2C2b3b4B1a1a2 A1a4b3C2c2a1B1b4c3 A1b3a4C2a1c2B1c3b4

2π/6 ≤φ≤ 4π/6 A1a4c1C2c2b3B1b4a1 A1c1a4C2b3c2B1a1b4 A1b2b3C2a4a1B1c2c3

φ

d2

34

33

32

31

24

23

22

21

14

13

12

11

0 1/6 1/3 1/2

-π/6

0

π/6

2π/6

3π/6

4π/6

Figure B.1: Switching modes for the three-level three-phase DAB converter with a star-starconnected transformer with d1 = 1/2.

A11 =

π −1

π 1

−1 0

A21 =

−π −1

−π 1

1 0

A31 =

π −1

π 1

−1 0

A12 =

−π −1

−π 1

1 0

A22 =

π −1

π 1

−1 0

A32 =

−π −1

−π 1

1 0

A13 =

π −1

π 1

−1 0

A23 =

−π −1

−π 1

1 0

A33 =

π −1

π 1

−1 0

A14 =

−π −1

−π 1

1 0

A24 =

π −1

π 1

−1 0

A34 =

−π −1

−π 1

1 0

B.1 Symmetrically operated three-level topology with a star-star connected transformer 177

b11 =

− π/6

− π/6

0

b21 =

π/6

π/6

− 1/3

b31 =

− 3π/6

− 3π/6

1/3

b12 =

π/6

π/6

− 1/6

b22 =

− π/6

− 3π/6

1/6

b32 =

3π/6

π/6

− 1/2

b13 =

π/6

− 3π/6

0

b23 =

3π/6

− π/6

− 1/3

b33 =

− π/6

− 5π/6

1/3

b14 =

3π/6

− 3π/6

1/6

b24 =

π/6

− 5π/6

1/6

b34 =

5π/6

− π/6

− 1/2


B.1.2 PWL equations

Power flow

P11 =8 U1 U2 d2 φ

L w

P12 =U1 U2

(−36 π2 d2

2 + 216 π d2 φ + 12 π2 d2 − 36 φ2 + 12 π φ− π2)

36 L π w

P13 =2 U1 U2 d2 (π + 6 φ)

3 L w

P14 =−U1 U2

(12 π2 d2

2 − 16 π2 d2 + 12 φ2 − 12 π φ + 3 π2)

6 L π w

P21 =2 U1 U2 φ (6 d2 + 1)

3 L w

P22 =U1 U2

(−36 π2 d2

2 + 216 π d2 φ + 12 π2 d2 − 36 φ2 + 12 π φ− π2)

36 L π w

P23 =U1 U2

(−108 π2 d2

2 + 72 π d2 φ + 84 π2 d2 − 108 φ2 + 84 π φ− 19 π2)

36 L π w

P24 =−U1 U2

(12 π2 d2

2 − 16 π2 d2 + 12 φ2 − 12 π φ + 3 π2)

6 L π w

P31 =2 U1 U2 φ (6 d2 + 1)

3 L w

P32 =−U1 U2

(12 π2 d2

2 − 12 π2 d2 + 12 φ2 − 16 π φ + 3 π2)

6 L π w

P33 =U1 U2

(−108 π2 d2

2 + 72 π d2 φ + 84 π2 d2 − 108 φ2 + 84 π φ− 19 π2)

36 L π w

P34 =−U1 U2

(36 π2 d2

2 − 36 π2 d2 + 36 φ2 − 36 π φ + 11 π2)

9 L π w


T ran

sfer

-ind

ucto

rcu

rren

t

i L11=

−π

U1−

2U

1φ+

2π

U1

d 2−

4π

U2

d 23

Lw

2U

1φ−

πU

1+

2π

U1

d 2+

2π

U2

d 23

Lw

−2

π(U

1−

3U

2d 2)

9L

w2(2

U1

φ−

2π

U1

d 2+

πU

2d 2)

3L

w2(2

U1

φ+

2π

U1

d 2−

πU

2d 2)

3L

w2

π(U

1−

3U

2d 2)

9L

wπ

U1+

2U

1φ−

2π

U1

d 2−

2π

U2

d 23

Lw

πU

1+

2U

1φ+

2π

U1

d 2−

4π

U2

d 23

Lw

4π(U

1−

3U

2d 2)

9L

w

i L12=

6U

1φ−

5π

U1+

6π

U1

d 2+

12π

U2

d 29

Lw

−π

U1−

2U

1φ+

2π

U1

d 2−

4π

U2

d 23

Lw

−4

πU

1+

πU

2−

6U

2φ−

18π

U2

d 218

Lw

2(6

U1

φ−

2π

U1+

6π

U1

d 2+

3π

U2

d 2)

9L

w2(2

U1

φ−

2π

U1

d 2+

πU

2d 2)

3L

w2

πU

1−

πU

2+

6U

2φ

9L

wπ

U1+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

πU

1+

2U

1φ−

2π

U1

d 2−

2π

U2

d 23

Lw

8π

U1−

πU

2+

6U

2φ−

18π

U2

d 218

Lw

i L13=

−5

πU

1−

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

6U

1φ−

5π

U1+

6π

U1

d 2+

12π

U2

d 29

Lw

−2

π(U

1−

6U

2d 2)

9L

w

−4(π

U1−

3U

1φ+

3π

U1

d 2−

3π

U2

d 2)

9L

w2(6

U1

φ−

2π

U1+

6π

U1

d 2+

3π

U2

d 2)

9L

w2

π(U

1+

3U

2d 2)

9L

wπ

U1+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

πU

1+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

2π(2

U1−

3U

2d 2)

9L

w

i L14=

6U

1φ−

7π

U1+

6π

U1

d 2+

6π

U2

d 29

Lw

−5

πU

1−

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

−4

πU

1−

3π

U2+

6U

2φ−

18π

U2

d 218

Lw

4(3

U1

φ−

2π

U1+

3π

U1

d 2+

3π

U2

d 2)

9L

w

−4(π

U1−

3U

1φ+

3π

U1

d 2−

3π

U2

d 2)

9L

w4

πU

1−

3π

U2+

6U

2φ+

18π

U2

d 218

Lw

6U

1φ−

πU

1+

6π

U1

d 2+

6π

U2

d 29

Lw

πU

1+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

4π

U1−

3π

U2+

6U

2φ

9L

w

i L21=

πU

2−

5π

U1+

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

−π

U1+

πU

2−

6U

1φ+

6π

U1

d 2−

12π

U2

d 29

Lw

πU

2−

4π

U1+

18U

2φ+

6π

U2

d 218

Lw

2(π

U2−

2π

U1+

6U

1φ+

6π

U1

d 2−

3π

U2

d 2)

9L

w2(2

πU

1−

πU

2+

6U

1φ−

6π

U1

d 2+

3π

U2

d 2)

9L

w4

πU

1−

πU

2+

18U

2φ−

6π

U2

d 218

Lw

πU

1+

πU

2+

6U

1φ+

6π

U1

d 2−

12π

U2

d 29

Lw

−π

U2−

5π

U1−

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

−π(U

2−

4U

1+

6U

2d 2)

9L

w

i L22=

πU

2−

3π

U1+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

πU

2−

5π

U1+

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

−4

πU

1+

πU

2−

6U

2φ−

18π

U2

d 218

Lw

−π

U2−

12U

1φ+

12π

U1

d 2−

12π

U2

d 29

Lw

2(π

U2−

2π

U1+

6U

1φ+

6π

U1

d 2−

3π

U2

d 2)

9L

w2

πU

1−

πU

2+

6U

2φ

9L

w3

πU

1−

2π

U2+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

πU

1+

πU

2+

6U

1φ+

6π

U1

d 2−

12π

U2

d 29

Lw

8π

U1−

πU

2+

6U

2φ−

18π

U2

d 218

Lw


i L23=

6U

1φ−

πU

2−

7π

U1+

6π

U1

d 2+

12π

U2

d 29

Lw

πU

2−

3π

U1+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

π(U

2−

2U

1+

6U

2d 2)

9L

wπ

U2−

8π

U1+

12U

1φ+

12π

U1

d 2+

6π

U2

d 29

Lw

−π

U2−

12U

1φ+

12π

U1

d 2−

12π

U2

d 29

Lw

4π

U1−

5π

U2+

18U

2φ+

6π

U2

d 218

Lw

2π

U2−

πU

1+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

3π

U1−

2π

U2+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

8π

U1−

7π

U2+

18U

2φ−

6π

U2

d 218

Lw

i L24=

−5

πU

1−

2π

U2−

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

6U

1φ−

πU

2−

7π

U1+

6π

U1

d 2+

12π

U2

d 29

Lw

−4

πU

1−

3π

U2+

6U

2φ−

18π

U2

d 218

Lw

πU

2−

4π

U1+

12U

1φ−

12π

U1

d 2+

6π

U2

d 29

Lw

πU

2−

8π

U1+

12U

1φ+

12π

U1

d 2+

6π

U2

d 29

Lw

4π

U1−

3π

U2+

6U

2φ+

18π

U2

d 218

Lw

πU

1−

πU

2+

6U

1φ−

6π

U1

d 2+

12π

U2

d 29

Lw

2π

U2−

πU

1+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

4π

U1−

3π

U2+

6U

2φ

9L

w

i L31=

πU

2−

πU

1+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

5π

U2−

5π

U1+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

πU

2−

4π

U1+

18U

2φ+

6π

U2

d 218

Lw

4(π

U1−

πU

2+

3U

1φ−

3π

U1

d 2+

3π

U2

d 2)

9L

w4(π

U2−

πU

1+

3U

1φ+

3π

U1

d 2−

3π

U2

d 2)

9L

w4

πU

1−

πU

2+

18U

2φ−

6π

U2

d 218

Lw

5π

U1−

5π

U2+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

πU

1−

πU

2+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

−π(U

2−

4U

1+

6U

2d 2)

9L

w

i L32=

πU

2−

7π

U1+

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

πU

2−

πU

1+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

2(π

U2−

πU

1+

3U

2φ)

9L

w5

πU

2−

8π

U1+

12U

1φ+

12π

U1

d 2−

6π

U2

d 29

Lw

4(π

U1−

πU

2+

3U

1φ−

3π

U1

d 2+

3π

U2

d 2)

9L

w2(π

U1−

πU

2+

6U

2φ)

9L

w4

πU

2−

πU

1+

6U

1φ+

6π

U1

d 2−

12π

U2

d 29

Lw

5π

U1−

5π

U2+

6U

1φ−

6π

U1

d 2+

6π

U2

d 29

Lw

2(2

πU

1−

2π

U2+

3U

2φ)

9L

w

i L33=

−3

πU

1−

5π

U2−

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

πU

2−

7π

U1+

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

π(U

2−

2U

1+

6U

2d 2)

9L

wπ

U2+

12U

1φ−

12π

U1

d 2+

6π

U2

d 29

Lw

5π

U2−

8π

U1+

12U

1φ+

12π

U1

d 2−

6π

U2

d 29

Lw

4π

U1−

5π

U2+

18U

2φ+

6π

U2

d 218

Lw

3π

U1−

4π

U2+

6U

1φ−

6π

U1

d 2+

12π

U2

d 29

Lw

4π

U2−

πU

1+

6U

1φ+

6π

U1

d 2−

12π

U2

d 29

Lw

8π

U1−

7π

U2+

18U

2φ−

6π

U2

d 218

Lw

i L34=

6U

1φ−

4π

U2−

9π

U1+

6π

U1

d 2+

12π

U2

d 29

Lw

−3

πU

1−

5π

U2−

6U

1φ+

6π

U1

d 2+

6π

U2

d 29

Lw

−2(π

U1−

3π

U2+

3U

2φ)

9L

wπ

U2−

12π

U1+

12U

1φ+

12π

U1

d 2+

6π

U2

d 29

Lw

πU

2+

12U

1φ−

12π

U1

d 2+

6π

U2

d 29

Lw

2(π

U1+

3U

2φ)

9L

w5

πU

2−

3π

U1+

6U

1φ+

6π

U1

d 2−

6π

U2

d 29

Lw

3π

U1−

4π

U2+

6U

1φ−

6π

U1

d 2+

12π

U2

d 29

Lw

2(2

πU

1−

3π

U2+

6U

2φ)

9L

w


B.1.3 Analytical modulation scheme

As presented in chapter 3, the analytical modulation scheme can be found by solv-ing the following two equations:

• P− Pref = 0

• iL(A1)− iL(a2) = 0

The resulting solutions of the phase-shift and duty-cycle are given below. For theinvestigated voltage and power range, only 6 switching modes are required. Theseare 21, 22, 23, 31, 32, and 33.

Phase-shift equations

φ11 = −π (2 U1 + 6 U1 d11 − 9 U2 d11)

6 U1

φ12 = −π (10 U1 −U2 + 12 U1 d12 − 30 U2 d12)

12 U1 + 6 U2

φ13 = −π (5 U1 + 6 U1 d13 − 12 U2 d13)

6 U1

φ14 = −π (3 U2 − 11 U1 + 6 U1 d14 + 6 U2 d14)

6 (U1 −U2)

φ21 = −π (5 U1 + 6 U1 d21 − 18 U2 d21)

6 U1

φ22 = −π (10 U1 + U2 + 12 U1 d22 − 42 U2 d22)

12 U1 + 6 U2

φ23 = −π (5 U2 − 22 U1 + 12 U1 d23 + 30 U2 d23)

12 U1 − 18 U2

φ24 = −π (2 U2 − 11 U1 + 6 U1 d24 + 12 U2 d24)

6 (U1 −U2)

φ31 = −π (5 U1 − 2 U2 + 6 U1 d31 − 12 U2 d31)

6 U1

φ32 = −π (5 U2 − 11 U1 + 6 U1 d32 + 6 U2 d32)

6 (U1 −U2)

φ33 = −π (9 U2 − 22 U1 + 12 U1 d33 + 18 U2 d33)

12 U1 − 18 U2

φ34 = −π (7 U2 − 16 U1 + 12 U1 d34 + 6 U2 d34)

12 (U1 −U2)

See below for solutions of d11, d12, ..., d34.

Duty-cycle equations


d 11=−

√ U2( 9L

Pw(3

U2−

2U1)

+4π

U2 1U

2) +2√

πU

1U2

6√π

U2(

2U1−

3U2)

d 12=

√π

U1U

2( −20

U2 1+

10U

1U2+

U2 2) −(

2U1+

U2)√ U

1U2( 9L

Pw( −8U

2 1+

16U

1U2+

U2 2) +π

U1U

2(2U

1+

U2)(1

4U1+

U2))

6√π

U1U

2( 8U

2 1−

16U

1U2−

U2 2)

d 13=−

√ U2( 4π

U2 1U

2−

9LP

w(U

1−

2U2)) +2

√π

U1U

2

6√π

U2(

U1−

2U2)

d 14=

(U1−

U2)√ U

1U2( 4π

U1U

2(U

1+

U2)

2−

9LP

w( U

2 1+

U2 2)) +

2√π

U1U

2( 3U

2 1+

U2 2)

6√π

U1U

2( U

2 1+

U2 2)

d 21=

3√π

U2(

U2−

2U1)−√ U

2(π

U2(

4U1+

3U2)

2−

36LP

w(U

1−

3U2))

12√

πU

2(U

1−

3U2)

d 22=

√π

U1U

2( −20

U2 1+

12U

1U2+

3U2 2) −(

2U1+

U2)√ U

1U2( π

U1U

2( 28

U2 1+

44U

1U2+

7U2 2) −1

8LP

w( 4U

2 1−

12U

1U2+

U2 2))

12√

πU

1U2( 4U

2 1−

12U

1U2+

U2 2)

d 23=

(2U

1−

3U2)√ U

1U2( π

U1U

2( 28

U2 1+

156U

1U2+

63U

2 2) −1

8LP

w( 4U

2 1+

4U1U

2+

9U2 2)) +

√π

U1U

2( 44

U2 1+

28U

1U2+

27U

2 2)

12√

πU

1U2( 4U

2 1+

4U1U

2+

9U2 2)

d 24=

(U1−

U2)√ U

1U2( π

U1U

2( 16

U2 1+

88U

1U2+

31U

2 2) −1

8LP

w( 2U

2 1+

2U1U

2+

5U2 2)) +

3√π

U1U

2( 4U

2 1+

3U1U

2+

2U2 2)

6√π

U1U

2( 2U

2 1+

2U1U

2+

5U2 2)

d 31=−

√ U2( 4π

U2 1U

2−

9LP

w(U

1−

2U2)) +3

√π

U1U

2−

2√π

U2 2

6√π

U2(

U1−

2U2)

d 32=

(U1−

U2)√ U

1U2( 4π

U1U

2(U

1+

U2)

2−

9LP

w( U

2 1+

U2 2)) +

√π

U1U

2( 5U

2 1+

U2 2)

6√π

U1U

2( U

2 1+

U2 2)

d 33=

(2U

1−

3U2)√ U

1U2( π

U1U

2( 28

U2 1+

72U

1U2+

27U

2 2) −9

LPw( 8U

2 1+

9U2 2)) +

2√π

U1U

2( 22

U2 1−

3U1U

2+

9U2 2)

6√π( 8U

3 1U2+

9U1U

3 2)

d 34=

2(U

1−

U2)√ U

1U2( π

U1U

2( 40

U2 1+

4U1U

2+

19U

2 2) −9

LPw( 8U

2 1−

4U1U

2+

5U2 2)) +

√π

U1U

2( 32

U2 1−

16U

1U2+

11U

2 2)

6√π

U1U

2( 8U

2 1−

4U1U

2+

5U2 2)

B.2 Asymmetrically operated two-level topology 183

B.2 Asymmetrically operated two-level topology

B.2.1 Switching modes

The asymmetrically operated two-level topology requires three degrees of freedomto achieve low circulating currents and soft-switching. For positive power, thiswould result in 72 switching modes. However, since the duty-cycles do not exceed0.5 for optimal operation, 32 switching modes are sufficient to model the converter.The sequences of the voltage transitions given in table B.2 and the correspondingboundaries are shown in figure B.2.

Table B.2: Sequences of the voltage transitions for half a switching cycle, describing theswitching modes of the asymmetrically operated two-level topology with a star-star con-nected transformer for positive power levels (0 ≤ φ ≤ π/2).

0 ≤ d2 ≤ 1/3 1/3 ≤ d2 ≤ 2/3

0 ≤ d1 ≤ 1/3

Switching modes 11 to 18:A1a1a2A2B1b1b2B2C1c1c2C2

A1A2a2b1B1B2b2c1C1C2c2a1

A1a1A2a2B1b1B2b2C1c1C2c2

A1A2a1a2B1B2b1b2C1C2c1c2

A1c2a1A2B1a2b1B2C1b2c1C2

A1c2A2a1B1a2B2b1C1b2C2c1

A1A2c2a1B1B2a2b1C1C2b2c1

A1c1c2A2B1a1a2B2C1b1b2C2

Switching modes 21 to 28:A1A2b1a2B1B2c1b2C1C2a1c2

A1c2b1A2B1a2c1B2C1b2a1C2

A1c2A2b1B1a2B2c1C1b2C2a1

A1a1c2A2B1b1a2B2C1c1b2C2

A1A2c2b1B1B2a2c1C1C2b2a1

A1a1A2c2B1b1B2a2C1c1C2b2

A1A2a1c2B1B2b1a2C1C2c1b2

A1b2a1A2B1c2b1B2C1a2c1C2

1/3 ≤ d1 ≤ 2/3

Switching modes 31 to 38:A1C2a1a2B1A2b1b2C1B2c1c2

A1c2a1C2B1a2b1A2C1b2c1B2

A1c2C2a1B1a2A2b1C1b2B2c1

A1C2c2a1B1A2a2b1C1B2b2c1

A1c1c2C2B1a1a2A2C1b1b2B2

A1c1C2c2B1a1A2a2C1b1B2b2

A1C2c1c2B1A2a1a2C1B2b1b2

A1b2c1C2B1c2a1A2C1a2b1B2

Switching modes 41 to 48:A1a1c2C2B1b1a2A2C1c1b2B2

A1C2c2b1B1A2a2c1C1B2b2a1

A1a1C2c2B1b1A2a2C1c1B2b2

A1C2a1c2B1A2b1a2C1B2c1b2

A1b2a1C2B1c2b1A2C1a2c1B2

A1b2C2a1B1c2A2b1C1a2B2c1

A1C2b2a1B1A2c2b1C1B2a2c1

A1c1b2C2B1a1c2A2C1b1a2B′2


φ[rad]

d2 [-]d1 [-]

0

1/3

2/3 0

1/3

2/3−π/3

0

π/3

2π/3

π

Figure B.2: Boundaries of 32 switching modes for the asymmetrically operated two-leveltopology with a star-star connected transformer for 0 ≤ φ ≤ π/2.

The proposed modulation scheme requires only three switching modes, the cor-responding boundaries are visualised in figure B.3. Consequently, the switchingmodes numbers are 11, 34, and 44, and the boundaries can be found byAmx+bm ≤0, for which matricesAm and bm are given below. (x = [d1 d2 φ]ᵀ)

A11 =

−π π −1

0 −2π 0

−π π 1

2π 0 0

A34 =

−2π 0 0

π −π −1

0 2π 0

π −π 1

A44 =

−2π 0 0

π π −1

0 −2π 0

π π 1

b11 =

0

0

0− 2π/3

b34 =

2π/3

0− 2π/3

− 2π/3

b44 =

2π/3

− 2π/3

2π/3

− 4π/3


φ[rad]

d2 [-]d1 [-]

0

1/3

2/3 0

1/3

2/3−π/3

0

π/3

2π/3

(a) 3D view

0 1/3 2/30

1/3

2/3

d1 [-]

d2 [-]

(b) Top view

Figure B.3: Boundaries of switching modes 11, 34, and 44, which are used by the analyti-cal modulation scheme of the asymmetrically operated two-level topology with a star-starconnected transformer considering positive power levels (0 ≤ φ ≤ π/2).

B.2.2 PWL equations

Below, equations for the power flow and the transfer-inductor current are given forthe three switching modes corresponding to the analytical modulation scheme.

Power flow

P11 =8 U1 U2 d2 φ

L wP34 = P44 =

U1U2(π2 (−27d2

1 + 6d1(3d2 + 2) + 3d2(4− 9d2)− 4)+ 6φπ(3d1 + 3d2 + 2)− 27φ2)

9Lπw



iL11 =

4 (U1 φ−π U1 d2+π U2 d2)3 L w

4 (U1 φ+π U1 d2−π U2 d2)3 L w

4 π (U1 d1−U2 d2)3 L w

4 π (U1 d1−U2 d2)3 L w

− 2 (U1 φ−π U1 d1−π U1 d2+2 π U2 d2)3 L w

− 2 U1 (φ−π d1+π d2)3 L w

0

0

− 2 U1 (φ+π d1−π d2)3 L w

− 2 (U1 φ+π U1 d1+π U1 d2−2 π U2 d2)3 L w

− 4 π (U1 d1−U2 d2)3 L w

− 4 π (U1 d1−U2 d2)3 L w

iL34 =

2 (6 π U1 d1−3 U2 φ−4 π U1+3 π U2 d1+3 π U2 d2)9 L w

4 (3 U1 φ−2 π U1+3 π U1 d2+3 π U2 d2)9 L w

4 (U1 φ−π U1 d2+π U2 d2)3 L w

4 (2 π U1−2 π U2+3 U2 φ−3 π U1 d1+3 π U2 d1)9 L w

4 (π U1+3 U2 φ−3 π U2 d1)9 L w

− 2 (3 U1 φ−2 π U1−3 π U1 d1+3 π U1 d2+6 π U2 d2)9 L w

− 2 (U1 φ−π U1 d1−π U1 d2+2 π U2 d2)3 L w

− 2 (2 π U1−2 π U2+3 U2 φ−6 π U1 d1+3 π U2 d1+3 π U2 d2)9 L w

− 2 (3 U2 φ−2 π U1+6 π U1 d1−3 π U2 d1+3 π U2 d2)9 L w

− 2 U1 (3 φ−2 π+3 π d1+3 π d2)9 L w

− 2 U1 (φ+π d1−π d2)3 L w

− 4 π U1−4 π U2+6 U2 φ+6 π U2 d1−6 π U2 d29 L w


iL44 =


4 (π U2+3 U1 φ−3 π U1 d2)9 L w

4 (2 π U2−2 π U1+3 U1 φ+3 π U1 d2−3 π U2 d2)9 L w

4 (2 π U1−2 π U2+3 U2 φ−3 π U1 d1+3 π U2 d1)9 L w

4 (π U1+3 U2 φ−3 π U2 d1)9 L w


− 2 (2 π U2−2 π U1+3 U1 φ−3 π U1 d1+3 π U1 d2)9 L w

− 2 (2 π U1−2 π U2+3 U2 φ−6 π U1 d1+3 π U2 d1+3 π U2 d2)9 L w



− 2 (2 π U2−2 π U1+3 U1 φ+3 π U1 d1+3 π U1 d2−6 π U2 d2)9 L w

− 4 π U1−4 π U2+6 U2 φ+6 π U2 d1−6 π U2 d29 L w

B.2.3 Analytical modulation scheme

As presented in chapter 3, the analytical modulations scheme can be found by solv-ing the following three equations:

• P− Pref = 0

• iL(A1)U2 − iL(a2)U1 = 0

• iL(A1) + iL(A2) = 0

The resulting solutions for the phase-shift and the duty-cycles are given below forthe switching modes 11, 34, and 44.


Swit

chin

gm

ode

11:

d 1=

LPw

2√2√

πU

1√ L Pw(U

2−

U1)

U2

d 2=−

√ L Pw(U

2−

U1)

U2

2√2√

π(U

1−

U2)

φ=

√π√ LP

w(U

2−

U1)

U2

2√2U

1

Swit

chin

gm

ode

34:

d 1=

2√π( 4U

4 1+

10U

3 1U2+

5U2 1U

2 2+

U1U

3 2+

4U4 2) −U

2(U

1+

U2)

√ 4π( 4U

4 1+

12U

3 1U2+

5U2 1U

2 2+

4U4 2) −18

LPw(U

4 1+

3U3 1

U2+

2U2 1U

2 2+

U4 2)

U1U

2

12√

π( U

4 1+

3U3 1U

2+

2U2 1U

2 2+

U4 2)

d 2=

( U2 1−

2U2 2)√ 4π

( 4U4 1+

12U

3 1U2+

5U2 1U

2 2+

4U4 2) −18

LPw(U

4 1+

3U3 1

U2+

2U2 1

U2 2+

U4 2)

U1U

2+

2√π( 4U

4 1+

9U3 1U

2+

2U2 1U

2 2+

4U4 2)

12√

π( U

4 1+

3U3 1U

2+

2U2 1U

2 2+

U4 2)

φ=

2π( 4U

4 1+

9U3 1U

2+

5U2 1U

2 2+

3U1U

3 2+

4U4 2) −√

2√π( U

2 1+

3U1U

2+

U2 2)√ 2π

( 4U4 1+

12U

3 1U2+

5U2 1U

2 2+

4U4 2) −9L

Pw(U

4 1+

3U3 1

U2+

2U2 1

U2 2+

U4 2)

U1U

2

12( U

4 1+

3U3 1U

2+

2U2 1U

2 2+

U4 2)

Swit

chin

gm

ode

44:

d 1=

8√π−

U2 2

√ 16π

U1U

2−

18LP

wU

5 1U

2−

U3 1U

3 2+

U1U

5 2

12√

π

d 2=

( U2 1−

2U2 2)√

16π

U1U

2−

18LP

wU

5 1U

2−

U3 1U

3 2+

U1U

5 2+

8√π

12√

π

φ=

1 12√

π

( 8√π−( U

2 1+

U2 2

)√16

πU

1U2−

18LP

wU

5 1U2−

U3 1U

3 2+

U1U

5 2

)

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Acknowledgement

N A een periode van hard werken, kan ik nu mijn promotieonderzoek metvoldoening afsluiten. Gedurende deze periode hebben vele mensen, iederop hun eigen manier, bijgedragen aan dit proefschrift. Bij deze neem ik de

gelegenheid om hen te bedanken.

In het bijzonder wil ik graag Elena Lomonova, Korneel Wijnands, Jordi Everts enJorge Duarte bedanken voor hun begeleiding tijdens de afgelopen jaren. Elena, be-dankt voor het geven van deze unieke kans en het vertrouwen in mij om dezeuitdaging aan te gaan. Jouw adviezen en kritische blik zijn erg waardevol enleerzaam voor mij geweest. Daarnaast hebben de, veelal niet-technische, discussiestijdens vergaderingen en conferenties mijn kijk op de wereld veranderd en mijngeschiedeniskennis verbeterd. Korneel en Jordi, bedankt voor jullie enorme inzetals begeleiders in de afgelopen jaren. Door jullie expertise en bevlogenheid ver-liep het promotieonderzoek voorspoedig en heb ik meer uit mezelf kunnen halen.Ondanks dat het roer na een jaar compleet om ging, hebben jullie mij gesteund engeholpen richting te geven aan het onderzoek. De samenwerking en ongedwongensfeer tijdens besprekingen heb ik erg gewaardeerd. Verder wil ik jullie bedankenvoor het lezen en verbeteren van de teksten die ik aanleverde voor artikelen en ditproefschrift. Zeker omdat deze soms pas vlak voor deadlines bij jullie aankwamen.Jorge, je was al tijdens mijn afstuderen betrokken als begeleider en tot mijn genoe-gen vervulde je deze rol ook in het begin van mijn promotieonderzoek. Zelfs nadatje de rol als begeleider had doorgegeven, kon ik altijd bij jou terecht voor zoweltechnische als niet-technische vragen. Hiervoor wil ik je graag bedanken. Verderwil ik Henk Huisman en Bas Vermulst bedanken voor de waardevolle correctiesvan dit proefschrift.

Dit promotieonderzoek is mogelijk gemaakt door het ARMEVA project dat bestaatuit een consortium van zes partners, te weten vier bedrijven en twee universiteiten

201

202 Acknowledgement

waaronder de TU/e. Hierbij wil ik de partners bedanken voor de fijne samenwerk-ing. In het bijzonder Elias van Wijk en Saphir Faid van Punch Powertrain N.V. enPieter Janssen en Thomas Thiemann van Prodrive Technologies B.V..

Hereby, I would like to express my gratitude to the committee members, prof. DeDoncker and prof. Kjær for their participation in my promotion committee. Thankyou for your time and effort to read my thesis and for travelling to EindhovenUniversity of Technology for the defense.

Een proefopstelling en metingen zijn essentiele onderdelen van wetenschappelijkonderzoek. Voor de faciliteiten en de realisatie van een proefopstelling, heb ik ookhulp van buiten de EPE groep gekregen. Allereerst wil ik Jeroen van de Keybus enPiet van Assche van Triphase N.V. in Leuven-Heverlee Belgie hartelijk bedankenvoor het beschikbaar stellen van ruimte, hardware en expertise. Hierdoor heb ikmetingen (hoofdstuk 4) kunnen doen tijdens de verhuizing van het EPE laborato-rium en vertraging van publicaties kunnen voorkomen. Verder wil ik Hans Bolten Arno Moorthamer van BM Constructies B.V. in Andelst bedanken voor onderandere het frezen van de heat sinks en het watersnijden van de koperen busbarsten behoeve van het prototype.

Bij deze wil ik graag alle (oud) collega’s van de EPE groep bedanken voor de goedesfeer tijdens kantoortijden, maar ook vooral daarbuiten tijdens borrels en uitjes. Inhet speciaal gaat mijn dank uit naar Marijn, Rutger en Wim voor de hulp met hetrealiseren van de prototypes en de opstelling daar omheen. Verder ben ik Nancydankbaar voor alle administratieve zaken afgelopen jaren. Jouw hulp was vangroot belang voor de goede afwikkeling van mijn promotie. Tevens wil ik ookmijn (oud) kantoor genoten van Impuls 1.11 en Flux 2.112 (Bas, Mark, Erik, Sjef,Ya, Jing, Mert, Georgios, Baris, Nikola en Lie) bedanken voor de gezelligheid opkantoor maar ook vooral daarbuiten met de onvergetelijke bierproeverijen, (pub)quizzen en chinees nieuwjaar. Erik bedankt voor al het gevraagde en ongevraagdeadvies, het bundelen van loshangende kabels zowel op kantoor als bij mijn proto-type en het ’mooi’ maken van de transformatoren. Het ziet er inderdaad een stukbeter uit zo! Verder wil ik Maurice en Dave bedanken dat ik altijd kon binnenvallenmet vragen over LATEX, MatlabFrag en andere thesis gerelateerde vragen. Tom, wilik bedanken voor het carpoolen in de afgelopen jaren. De velen discussies en dediverse muziekstijlen brachten de nodige ontspanning voor en na het werk.

Naast werk zijn voor mij de sociale- en sportactiviteiten van groot belang geweestom te ontspannen en nieuwe energie te op te doen. Daarom wil ik de (oud) ledenvan Spielen Und Erleben bedanken voor de leuke zaalvoetbalwedstrijden tijdensde pauzes. Met dank aan Bart voor de organisatie hiervan de laatste jaren. Ook degrote hoogtes en frisse berglucht tijdens het fietsen in Winterberg met Jordi, Marie,Bas en Paola hebben bijgedragen aan de nodige ontspanning en een betere conditie.

Ook wil ik de (oud) spelers, (oud) trainers en coaches van mijn volleybalteam be-danken voor alle gezelligheid en de sportieve prestaties, maar ook voor het begripvan mijn afwezigheid door conferenties. Daarnaast wil ik Mark bedanken voorhet opnemen van de taken als secretaris tot mijn verdediging. Bovenal wil ik Bart,Koen, Roy en Sven bedanken voor alle gezellige middagen en avonden. Dit zorgdevoor ontspanning en inspiratie maar ook voor zware ochtenden!

Tot slot ben ik mijn ouders, Marcel en Tineke, en mijn zussen, Annelies en Martine,heel erg dankbaar voor alle steun en liefde die jullie altijd hebben gegeven. Hoewelik de afgelopen jaren niet overal bij kon zijn hadden jullie begrip hiervoor en blevenmij steunen. Ik kan geen betere familie wensen!

Nico BaarsZetten, September 2017

About the author

N. H. (Nico) Baars was born in Kamerik, The Netherlands,on the 2nd of June 1988. He received the BEng degree inelectrical engineering in 2010 from HAN University of Ap-plied Sciences, Arnhem, The Netherlands. In 2013 he re-ceived the MSc degree in electrical engineering from Eind-hoven University of Technology, Eindhoven, The Nether-lands. His master graduation assignment was carried outat Strukton Rolling Stock in Alblasserdam, the Nether-lands, where he investigated and designed an isolated dc-dc converter for railway applications.

In October 2013 he started as a PhD researcher in the Electromechanics and PowerElectronics (EPE) group of the Electrical Engineering department at EindhovenUniversity of Technology. His research focussed on wide voltage-range isolateddc-dc converters, based on the three-phase dual active bridge topology, for high-power applications, of which the results are presented in this dissertation.

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