Download Figure 1 - Uni Kassel

One centralised question of this work is to explore the performance advantage
of functionally integrated magnetic components in comparison to discrete
components. Many applications allow the introduction of simple magnetic structures
and standard cores or simple modifications of these (flux bypasses) in order to
enable the required component behaviour. The design guidelines introduced in
this work enable the design of functionally integrated magnetic components with
limited effort and, therefore, the application of components which enable superior
performance regarding size and power loss for the applications.
ISBN 978-3-7376-0226-6
9 783737 602266
Thiemo Kleeb
The functional integration of magnetic components is a known technique in
order to enable high power densities for power electronic converters. Magnetic
components are mandatory in many power electronic converters and many
topologies demand more than one magnetic component. Therefore, the functional
integration of magnetic components allows realising several magnetic functions
within one component. This technique promises lower total size, losses and costs
without switching frequency increase. There are several examples in the literature
for coupled inductors, common-differential-mode chokes or transformer-inductor
components.
Investigation on Performance Advantage of Functionally Integrated
Magnetic Components in Decentralised Power Electronic Applications
13
13
Elektrische Energiesysteme
Thiemo Kleeb
Investigation on Per formance Advantage of
Functionally Integrated Magnetic Components in
Decentralised Power Electronic Applications
Elektrische Energiesysteme
Band 13
Herausgegeben vom
Kompetenzzentrum für Dezentrale
Elektrische Energieversorgungstechnik
Investigation on Performance Advantage of
Functionally Integrated Magnetic Components
in Decentralised Power Electronic Applications
Thiemo Kleeb
kassel
university
press
This work has been accepted by the Faculty of Electrical Engineering / Computer Sciences of the
University of Kassel as a thesis for acquiring the academic degree of Doktor der Ingenieurwissenschaften (Dr.-Ing.).
Supervisor:
Prof. Dr.-Ing. habil. Peter Zacharias
Co-Supervisor: Prof. Dr.-Ing. habil. Detlef Schulz
Defense day:
2nd September 2016
Bibliographic information published by Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at http://dnb.dnb.de.
Zugl.: Kassel, Univ., Diss. 2016
ISBN 978-3-7376-0226-6 (print)
ISBN 978-3-7376-0227-3 (online)
DOI: http://dx.medra.org/10.19211/KUP9783737602273
URN: http://nbn-resolving.de/urn:nbn:de:0002-402279
© 2017, kassel university press GmbH, Kassel
www.uni-kassel.de/upress
Printed in Germany
Danksagung
Zunächst möchte ich Prof. Dr.-Ing. habil. Peter Zacharias für die Betreuung meiner
Arbeit danken, der mir stets mit guten Ideen und Rat zur Seite stand und immer für
konstruktive Gespräche offen war. Weiterhin danke ich meinem Zweitgutachter
Prof. Dr.-Ing. habil. Detlef Schulz für das Interesse und die Überarbeitung meiner
Arbeit, sowie Prof. Dr. rer. nat. Ludwig Brabetz und Prof. Dr.-Ing. Mike Meinhardt für
die Teilnahme an der Prüfungskommission.
Außerdem gilt mein Dank meinem Kollegen Dr.-Ing. Samuel Araújo, für seine
Ratschläge bzgl. Halbleitertechnologien und für die vielen fachlichen Gespräche und
Diskussionen. Des Weiteren möchte ich meinen Kollegen Benjamin Dombert und
Dr.-Ing. Christian Nöding danken, die immer bereit waren mir bei der Inbetriebnahme
der Microcontroller und deren Software zu helfen. Meinen Kollegen Dr.-Ing. Mehmet
Kazanbas und Lucas Menezes möchte ich für die Unterstützung und Ratschläge bzgl.
Treiber danken. Bei Florian Fenske bedanke ich mich für die interessanten
Diskussionen über magnetische Bauelemente. Fr. Clark möchte ich für die
Unterstützung bei den vielen organisatorischen Angelegenheiten danken. Natürlich
gilt mein Dank auch den anderen MitarbeiterInnen des KDEE/EVS, die stets mit
gutem Rat und fachlicher Hilfe zur Verfügung standen – nicht zuletzt auch den
technischen Angestellten, die immer gute Ideen bei der praktischen Umsetzung von
Versuchsaufbauten haben.
Zuletzt möchte ich auch meinen Eltern danken, die mich immer, aber vor allem auch
während meines Studiums, unterstützt haben.
5
Vorwort
Die bevorstehenden Herausforderungen in den Anwendungsfeldern der dezentralen
Energieversorgungs-Systeme haben in den letzten Jahren zu einer starken Nachfrage
nach neuen, innovativen leistungselektronischen Wandlern und Komponenten für
diese Wandler geführt. Ein Beispiel hierfür ist die Zunahme von regenerativen
Generatoren in der deutschen Energieversorgung. Vor allem Wind- und SolarKraftwerke spielen eine immer wichtigere Rolle im Energieversorgungskonzept
Deutschlands. Ein anderes Beispiel ist der gerade erst beginnende Wechsel vom
Verbrennungsantrieb zu Hybrid- oder komplett elektrischen Antriebs-Systemen in der
Automobil-Industrie. Die hierfür benötigten grundlegenden leistungselektronischen
Wandler-Konzepte
sind
bereits
bekannt
und
werden
entsprechend
applikationsspezifisch angepasst. Jedoch sind, besonders für Anwendungen welche
der Massenproduktion unterliegen, permanente Optimierungen und Verbesserungen
nötig, um neue Innovationen und Kostensenkungen zu erreichen.
Für leistungselektronische Wandler bedeutet das in der Regel, dass eine Verringerung
von Volumen, Verlustleistung und Kosten eine große Rolle im Entwicklungsprozess
spielen. Speziell Automobil-Anwendungen fordern sehr hohe Leistungsdichten, also
minimales Volumen und Gewicht, bei großen Leistungen, sowie minimale
Verlustleistung und Kosten. Magnetische Bauelemente und Filter nehmen in vielen
Wandlern ein nicht unerhebliches Volumen ein und haben einen entsprechend hohen
Anteil in der Kostenstruktur. Die Verringerung des Filter-Volumens kann daher
signifikant zur Verringerung des Gesamt-Volumens und der Gesamt-Kosten
beitragen. Weiterhin ist zu berücksichtigen, dass ein kleiner Wandler auch ein
kleineres Gehäuse ermöglicht. D.h., durch den Einsatz kleiner magnetischer- und
Filter-Bauelemente werden nicht nur die Kosten dieser Komponenten selbst
verringert, sondern auch die von anderen System-Elementen. Auch die Verringerung
der Verlustleistung der Komponenten kann zu einem ähnlichen Effekt führen: Steigt
der Wirkungsgrad der einzelnen Komponenten, bzw. des Wandlers, kann ggf. der
Aufwand für die Kühlung verringert werden, was ebenfalls zu geringen SystemKosten beitragen kann.
Zum Erreichen hoher Leistungsdichten ist die funktionelle Integration von
magnetischen Bauelementen eine bekannte Technik. Magnetische Bauteile sind in
vielen leistungselektronischen Wandlern obligatorisch, und viele Wandler-Topologien
benötigen mehr als nur ein magnetisches Bauelement. Die funktionelle Integration
6
magnetischer Bauelemente erlaubt die Realisierung mehrerer magnetischer
Funktionen oder Bauelemente in nur einem Bauteil. In der Literatur gibt es mehrere
Beispiele für gekoppelte Drosseln, Gleich-Gegentaktdrosseln und TransformatorSpeicherdrossel-Bauteile. Prinzipiell werden zwei oder mehr magnetische Bauteile
durch eines ersetzt, welches in der Lage ist, alle magnetischen Funktionen zu erfüllen.
Diese Technik verspricht eine Verringerung der Bauteilgröße, Verlustleistung und
Kosten, ohne die Frequenz zu erhöhen.
Obwohl die Grundlagen bereits seit Jahrzehnten bekannt sind, ist die Entwicklung
integrierter magnetischer Bauteile immer noch eine Herausforderung. Für die
Entwicklung funktionell integrierter Bauelemente müssen sowohl das elektrische, als
auch das magnetische Verhalten des Bauelements selbst bekannt sein, sowie die
Funktion der gesamten Schaltung. Funktionell integrierte magnetische Bauelemente
haben normalerweise ein komplett anderes Verhalten als ihre entsprechenden
diskreten Bauelemente. Das ist möglicherweise der Grund dafür, dass funktionell
integrierte magnetische Bauelemente häufig nur in wissenschaftlichen Prototypen zu
finden sind und eher seltener in industriell gefertigten Seriengeräten.
Eine zentrale Fragestellung dieser Arbeit ist es, herauszufinden, wie groß der
Performance-Vorteil von funktionell integrierten Bauelementen gegenüber
konventionellen diskreten Bauelementen ist und ob es überhaupt einen Vorteil gibt.
Die Entwicklung dieser Bauelemente ist sehr applikationsspezifisch und es muss im
Detail untersucht werden, ob und wie Verlustleistung, Größe und Kosten mit dieser
Technik verringert werden können. Nichtsdestotrotz muss geprüft werden, ob der
erhöhte Entwicklungsaufwand den eintretenden Performance-Vorteil rechtfertigt.
Letztendlich lässt sich immer dann ein Vorteil hinsichtlich Größe und Verlustleistung
erreichen, wenn die Bauteilausnutzung (Flussdichte, Verlustleistungsdichte) erhöht
werden kann. Für viele Anwendungen können einfache magnetische Geometrien und
Standard-Kerne verwendet werden, oder entsprechende Modifikationen (FlussNebenschlüsse) an diesen vorgenommen werden, um das geforderte Verhalten des
Bauteils zu ermöglichen. Die in dieser Arbeit eingeführten Entwurfsrichtlinien
ermöglichen den Entwurf von funktionell integrierten magnetischen Bauelementen
mit begrenztem Aufwand und damit den Einsatz von Bauelementen, die zu großen
Vorteilen bzgl. Größe und Verlustleistung in den Anwendungen führen.
7
Preface
The upcoming challenges in the application field of decentralised energy supply
systems have led to a strong demand for novel innovative power electronic
converters and components for these converters in the last years. An example for this
is the growing contingent of renewable generators in the German electrical power
supply. Especially wind and solar power plants are playing a more and more
important role in the electric energy supply concepts of Germany. Another example is
the just started changeover in the automobile industry from conventional
combustion engine drives to hybrid or even totally electric drive systems. Therefore,
conventional power electronic converter concepts are already known and can be
adapted to provide application specific solutions. But especially applications in the
bulk production are strongly forced to claim optimisations regarding performance,
new features and cost reduction.
For power electronic converters this means in general a reduction of size, losses and
costs are in design focus. Especially automotive applications demand very high power
densities, thus minimum weight and size at highest power levels as well as lowest
losses and costs. Magnetic components and filters require a non-negligent size in
power electronic converters and contribute significant to their costs. A decrease of
the filter size can, therefore, enable a significant decrease of the total system size and
costs. Furthermore, it should be taken into account that small converters require only
small housings. This means, the application of small magnetic and filter components
can enable not only a cost reduction for these components themselves, but can
enable a cost reduction of other system parts as well. The decrease of the power loss
of the components can lead to a similar effect: Increasing component and converter
efficiency can enable the reduction of cooling effort along with reduced system
expenditures as well.
In order to enable high power densities the use of integrated magnetic components
is a known technique. Magnetic components are mandatory in many power
electronic converters and many topologies demand more than one magnetic
component. Therefore, the functional integration of magnetic components allows
realising several magnetic functions within one component. There are several
examples in the literature for coupled inductors, common-differential-mode chokes
or transformer-inductor components. In principle, two or more magnetic
components will be replaced by only one component fulfilling all magnetic functions
8
at once. This technique promises lower total size, losses and costs without switching
frequency increase.
But it should be noticed that the development of integrated magnetic components is
still a challenge, even due to the fact that basic approaches are known since decades.
For the development of magnetic components, the magnetic as well as the electrical
behaviour of the component itself and the circuit must be known. But functionally
integrated magnetic components can have a completely different magnetic
behaviour than conventional discrete magnetic components. Maybe this is one of the
reasons why the technique of the functional integration of magnetic components is
mostly used in scientific prototypes and rather less in industrial assembled standard
converters.
One centralised question of this work is to explore the performance advantage of
functionally integrated magnetic components in comparison to discrete components.
The development of these components is very application specific and it must be
examined in detail if and how losses, size and costs can be reduced by using this
technique. Anyway, the effort caused by the more complex development process
must lead to a significant performance advantage, in order to legitimate the
introduction of this method.
Finally, advantages regarding size and power loss can be achieved if the utilisation of
the component (flux density, specific power loss) can be increased. Many applications
allow the introduction of simple magnetic structures and standard cores or simple
modifications of these (flux bypasses) in order to enable the required component
behaviour. The design guidelines introduced in this work enable the design of
functionally integrated magnetic components with limited effort and, therefore, the
application of components which enable superior performance regarding size and
power loss for the applications.
9
Content
1
APPROACH AND OBJECTIVES ............................................................................... 14
1.1
State of the Art ........................................................................................... 16
1.2
Transformers with Integrated Energy Reactor ............................................ 17
1.3
Coupled Inductors ....................................................................................... 22
1.4
Combined Common-Differential Mode Chokes ........................................... 25
2
FUNDAMENTALS OF MAGNETIC COMPONENTS ............................................. 28
2.1
Basic Magnetic Laws ................................................................................... 28
2.2
Magnetisation and Hysteresis ..................................................................... 30
2.3
Magnetic Core Materials ............................................................................. 33
2.4
Magnetic Circuits ........................................................................................ 38
2.5
Definitions of Inductance ............................................................................ 41
2.6
Transformer Principle ................................................................................. 42
2.6.1
Coupling Coefficient ................................................................................. 43
2.6.2
Mutual- and Self-Inductance .................................................................... 45
2.6.3
Leakage Inductance.................................................................................. 45
3
LOSSES IN MAGNETIC COMPONENTS ................................................................. 46
3.1
RMS Power Loss and Temperature Dependency ......................................... 47
3.2
AC Winding Losses ...................................................................................... 49
3.2.1
Skin Effect................................................................................................. 49
3.2.2
Proximity Effect ........................................................................................ 50
3.2.3
High Frequency Losses in Round Solid Wires ........................................... 51
3.2.4
High Frequency Losses in Litz Wires......................................................... 53
3.2.5
Improved Power Loss Calculation for Litz Wires ...................................... 54
3.3
4
Core Losses ................................................................................................. 58
3.3.1
Core Loss Mechanisms ............................................................................. 58
3.3.2
Steinmetz Equation .................................................................................. 61
3.3.3
Modelling Core Losses by Means of Orthogonal Vector Functions ......... 64
COUPLED INDUCTORS ............................................................................................. 70
10
4.1
Direct and Inverse Coupling ......................................................................... 70
4.2
Magnetic Equivalent Circuit ......................................................................... 74
4.2.1
4.3
DC Analysis of Inverse and Direct Coupled Inductors ..............................76
Potential Performance Advantages of Coupled Inductors............................ 77
4.3.1
Equal Phase Current Ripple of Discrete and Coupled Inductor ................78
4.3.2
Equal Converter Output/Input Current Ripple of Discrete and Coupled
Circuit .......................................................................................................80
4.4
Basic Core Geometries of Coupled Inductors ............................................... 82
4.5
Leakage Inductance of Coupled EE Core Inductors ....................................... 83
4.6
Leakage Inductance of Coupled Ring Core Inductors ................................... 85
4.7
Influence of Core Shape and Material on Leakage Inductance ..................... 87
4.7.1
Influence of Permeability and Gap Length on Coupled EE Core Inductors ..
..................................................................................................................87
4.7.2
Leakage of Coupled EE Core Inductors dependent on Core Size..............88
4.7.3
Adjusting the Self-Inductance of Coupled EE Core Inductors...................89
4.7.4
Comparison of Coupled Ring and EE core Inductors ................................90
4.7.5
Measurement of different Materials and Cores.......................................91
4.8
5
Design Methodology ................................................................................... 93
AUTOMOTIVE ON-BOARD POWER SUPPLY WITH MAGNETICALLY
INTEGRATED CURRENT DOUBLER ..................................................................... 97
5.1
Circuit Analysis ............................................................................................ 97
5.1.1
Discrete Current Doubler .........................................................................97
5.1.2
Magnetically Integrated Current Doubler ..............................................101
5.2
Suitable Core Geometries for Magnetically Integrated Current Doublers .. 105
5.2.1
EE Core ...................................................................................................105
5.2.2
EEII or EEUU Core ...................................................................................107
5.2.3
Shell Type Core (5 Leg Core) ...................................................................107
5.2.4
Ring or UU Core with Leakage Segments ...............................................108
5.2.5
Comparison of Integrated and Discrete Current Doubler ......................111
5.3
Design of Integrated Current Doublers ...................................................... 113
5.3.1
Transformer Turn Ratio ..........................................................................113
11
5.3.2
Area Product Approach .......................................................................... 114
5.3.3
Inductance Matrix .................................................................................. 115
5.3.4
Output Current Ripple Calculation ......................................................... 118
5.3.5
Flux Density Swing ................................................................................. 120
5.3.6
DC Pre-Magnetisation ............................................................................ 121
5.3.7
Design Example: Integrated EE Core Current Doubler ........................... 122
5.4
Downsizing Potential of the Integrated Current Doubler .......................... 130
5.5
Analysis of different Current Doubler Technologies .................................. 132
5.5.1
Size and Weight Comparison ................................................................. 133
5.5.2
Power Loss Comparison ......................................................................... 135
5.5.3
Temperature Rise ................................................................................... 138
5.5.4
Summary ................................................................................................ 140
5.6
Experimental Results ................................................................................ 141
5.6.1
12V and 48V Converters with IGBT Bridge operating at 50 kHz ............ 146
5.6.2
48V Converters with SiC MOSFET Bridge operating at 50 kHz ............... 150
5.6.3
48V Converters with GaN Bridge operating at 200 kHz ......................... 152
5.6.4
48V Converters with SiC Bridge operating at 200 kHz ........................... 158
5.7
6
Executive Summary................................................................................... 160
COMBINED COMMON-DIFFERENTIAL MODE CHOKES .............................. 162
6.1
Basics of Common and Differential Mode Noise ....................................... 162
6.2
Parasitic Effects in Filter Chokes ................................................................ 164
6.3
Suitable Core Geometries for Common-Differential Mode Chokes ........... 165
6.3.1
Ring or UU Core with Leakage Segments ............................................... 166
6.3.2
EE Core ................................................................................................... 171
6.3.3
Separated Common- and Differential-Mode Cores ............................... 173
6.3.4
Pot Core with Ferromagnetic Disc or EE Core with Segment ................. 175
6.4
Design of Common-Differential Mode Chokes .......................................... 176
6.4.1
Common Mode Choke ........................................................................... 177
6.4.2
Common-Differential Mode Choke with Additional DM Cores.............. 179
6.4.3
Common-Differential Mode Choke with Inserted Segments ................. 180
12
6.4.4
EE Core Common-Differential Mode Choke ...........................................183
6.4.5
Comparison of Ring and EE Core Common-Differential Mode Chokes ..185
6.4.6
Design Example ......................................................................................187
6.5
6.5.1
Comparison of different Common-Differential Mode Chokes ...............189
6.5.2
Evaluation of Performance Advantage of Integrated Common-Differential
Mode Filters in AC Applications .............................................................200
6.6
7
Experimental Results ................................................................................. 189
Executive Summary ................................................................................... 207
RECAPITULATION AND PERSPECTIVE ............................................................208
APPENDIX ...........................................................................................................................212
I.
EQUIVALENT INDUCTANCE OF COUPLED INDUCTORS .............................212
II.
LEAKAGE CALCULATION FOR COUPLED EE CORE INDUCTORS..............216
III. AIR GAP CALCULATION FOR SEGMENTS OF COMMON-DIFFERENTIAL
MODE CHOKES .........................................................................................................221
IV. FRINGING EFFECT OF AIR GAPS.........................................................................224
V.
LEAKAGE INDUCTANCE OF TRANSFORMERS ...............................................227
VI. THERMAL EQUIVALENT CIRCUITS OF MAGNETIC COMPONENTS.........230
VII. CALCULATION OF AVERAGE MAGNETIC FIELD INTENSITY FOR
TRANSFORMERS .....................................................................................................237
VIII. POWER LOSS EVALUATION IN CIRCUIT SIMULATORS ..............................239
IX. IMPEDANCE AND INDUCTANCE MEASUREMENTS OF COMMONDIFFERENTIAL MODE FILTER CHOKES ...........................................................242
X.
EQUIPMENT AND MEASUREMENT DEVICES .................................................248
A.
LIST OF SYMBOLS ...................................................................................................249
B.
LIST OF TABLES .......................................................................................................252
C.
LIST OF FIGURES .....................................................................................................254
D.
REFERENCES .............................................................................................................264
13
1
Approach and Objectives
A proper design of magnetic components can be very complex, though the basics are
well known. Several appropriate design procedures for discrete inductors and
transformers can be found in literature (e.g. [1], [2] or [3]). The simplification of the
magnetic circuits allows the introduction of simple design rules. But the magnetic
structure of functionally integrated magnetic components can be much more
complex. In many cases the design is very application specific, making the derivation
of general rules difficult.
Therefore, this work will give a brief introduction of the basic magnetic laws and
relationships, required to understand the mode of operation of magnetic
components and their design. To complete the basics a further focus will be the loss
mechanisms occurring in magnetic components. The measurement and calculation of
losses in magnetic components is a very complex and special issue itself. The
different loss mechanisms will be explained and several calculation methods for the
evaluation of winding and core losses will be given.
In order to denote a general context of functionally integrated magnetic components,
the application of coupled inductors will be explained more in detail. The coupled
inductor is a special case of a functionally integrated magnetic component. These
components have at least two windings arranged on a single core. The coupled
inductor can show the two basic principles valid for all functionally integrated
magnetic components:


The component provides an energy reactor enabled by the leakage
inductance of the component.
All windings are magnetically coupled, enabling an energy transfer from one
winding to another one.
The basic operation behaviour of the coupled inductor is used to show possible
advantages of such a component. Furthermore, magnetic leakage effects of common
cores and structures are introduced to enable a proper design of magnetically
coupled components. In this context, questions regarding suitable materials and
geometries for coupled inductors and functionally integrated magnetic components
are treated as well. To complete the required tools and methods for the design of
magnetic components a brief introduction regarding thermal modelling is given in the
appendix.
14
Coupled Inductor
Magnetic
Equivalent Circuit
Functionally
Integrated
Component
Power Loss
Calculation
Thermal Model
Figure 1-1: Required concepts for the design of functionally integrated components
Due to the uncommon application of functionally integrated magnetic components in
industry, it is of interest if functionally integrated magnetic components can offer a
performance advantage compared to their discrete magnetic counterparts. Two
application examples were chosen in order to show possible performance advantages
regarding size, weight and power loss. The focus of this work is the functional
integration of transformers with energy reactors as well as combined commondifferential mode chokes. Both technologies will be explained in detail and
investigated by means of application specific examples in the lower kW power range.
The first application demonstrates how a functionally integrated magnetic
component can be implemented in a push pull converter with current doubler
rectifier. The magnetically integrated current doubler will replace the two discrete
inductors and the transformer, required for the conventional circuit, by only one
component fulfilling the complete functionality required for this circuit.
The second application will be an EMI filter for a photovoltaic inverter topology. The
objective is to design special combined common-differential mode chokes, applicable
to attenuate both common and differential mode noise effectively. In the best case a
common-differential mode choke can replace a conventional differential mode
choke. This objective is in accordance to the current demand for cost reduction in
photovoltaic converters [4], where the EMI filters are responsible for a non-negligent
part of the total system costs.
Several magnetic structures and designs will be developed for both applications,
where the most promising and applicable components will be realised in practice to
demonstrate their performance in comparison to their discrete counterparts. The
design procedure of the functionally integrated components will be explained in
15
detail and includes the calculation of non-negligent leakage inductance effects,
required for a proper design. Calculations and simulations using suitable derived
magnetic models will allow the analysis and comparison of different magnetic
structures and components. Furthermore, applicable operation conditions for the
different components and technologies will be evaluated in order to highlight the
required conditions and specifications for a possible performance advantage.
1.1 State of the Art
The term integrated magnetic components can be distinguished in two concepts: The
structurally integrated magnetic components and the functionally integrated
components. Structural integration of magnetic components implicates the
integration of a discrete magnetic component in a printed copper board (PCB).
Therefore, the windings will be realised with the tracks and several layers of the PCB.
Gaps in the PCB will enable to fit the legs of the core of the magnetic component in
the PCB. The cores will be fixed with glue, clamps or other fixtures.
Integrated Magnetic Components
Structural
Integration /
Discrete
Components
Functional
Integration
Transformer with
Energy Reactor
Coupled Inductror
CommonDifferential Mode
Choke
Transformer
Inductor
Figure 1-2: Functional and structural integration of magnetic components
This work will treat the functional integration of magnetic components, which is
representing a concept to realise different magnetic functions or several discrete
magnetic components within one component. Usually this means, several windings
will use the same core. But there are also other examples, where one winding is
wound on different cores.
However, the target of functionally integrated magnetic components is to fulfil the
application specifications with a reduced effort regarding number of cores and/or
windings. In practice this means that the component parameters (e.g. inductance and
coupling values) must be adjusted in a way that the circuit specifications can be
fulfilled. This is usually done by:
16




Winding arrangement and placement
Adjusting core shape and material
Combination of different core materials
Inserting energy reactors (e.g. air gaps) in the magnetic structure
Subsequent some examples from the literature will be presented in order to show
some applications using integrated magnetic components. The applications are
distinguished in:



Transformers with integrated energy reactor
Coupled inductors
Combined common-differential mode chokes
1.2 Transformers with Integrated Energy Reactor
Subsequent the term transformer with integrated energy reactor describes a
component fulfilling the functionality of galvanic isolation as well as the possibility to
store magnetic energy in the magnetic structure. This energy reactor is used to fulfil
filter functionality (attenuating current ripple) or to release the stored energy at a
specific time instance, e.g. to the output.
Flyback Converter
The flyback converter is a simple converter with galvanic isolation, usually suitable for
the 100 W power range [5]. It requires only one switch, one diode and one
transformer with energy reactor as well as input and output capacitor. Its
transformer is a simple example for a transformer using an energy reactor. The
design of the component is explained e.g. in [2].
The flyback converter uses an inverting transformer to transfer power to the output.
Usually the energy reactor is realised with an air gap in the transformer structure. If
the primary switch is closed, the primary winding of the transformer will store energy
in the energy reactor of the transformer. Due to the inverting transformer structure,
the voltage inducted in the secondary winding will prevent the diode from
conducting. The output is supplied by the output capacitor.
If the primary switch is opened, the secondary winding voltage is positive and the
diode is forward biased. The energy stored in the transformer is released via the
secondary winding, which supplies the output and recharges the output capacitor.
17
L
L
D
D
Cout
Vin
Cout
Vin
S
S
a)
b)
Figure 1-3: Flyback converter: a) switch S closed; b) switch S opened
Resonance Converters
For galvanic isolated resonance converters (e.g. LLC, as example see [6] or [7]) the
leakage inductance of the transformer can be used to replace a necessary resonance
inductor. Therefore, transformers with relatively high leakage inductance are
applicable. Leakage and magnetising inductance must be adjusted dependent on the
converter configuration. The difficulty for the transformer assembly is to realise a
predefined ratio of leakage and magnetising inductance. For some applications this
ratio is that unfavourable that the transformer suffers poor performance (complex
assembly, higher losses). The increase of the leakage inductance by means of the
winding arrangement is a common technique, where the primary and secondary
windings are placed on top of each other (see Figure V-1 b). This winding
arrangement allows implementing very high leakage inductance values because of
the lose coupling between both windings.
S1
S2
Cres
Lres
D1
D2
Cout
Lm
Vin
S3
S4
D3
D4
Figure 1-4: LLC series resonance converter
There are even more possibilities to realise integrated resonance tanks for resonance
converter transformers as can be seen in [8]. Figure 1-5 a) shows a transformer with
increased primary leakage inductance. This leakage inductance is increased by the
displacement of the primary transformer winding away from the secondary winding.
The space between the windings enables a parasitic energy reactor. The
disadvantage of this method is the increasing EMI, caused by the leakage of the
transformer. Therefore, [8] proposes to use an auxiliary core to realise the required
primary resonance inductor. This will decrease leakage as well as EMI effects and
18
enables an easier design of the resonance inductor. As disadvantage, the extra core
causes additional expenditures and component weight. [9] shows some core designs
with integrated leakage segments, e.g. where leakage segments are placed inside the
winding package. These examples show that the core structure and the winding
arrangement have a significant influence on the behaviour of a magnetic component.
But there are also more complex examples: E.g. [10] proposes a multi-resonant
converter with wide voltage range conversion. The use of multi-resonant converter
topologies for photovoltaic applications, in order to enable power point tracking and
galvanic isolation at once is proposed in [11]. A multi-resonant converter with a high
step up capability is presented in [12].
a)
Primary
Secondary
Primary
Secondary
Transformer
Core
Transformer
Core
Auxiliary
Core
b)
Figure 1-5: Transformer with primary resonance tank realised by a) displacement of primary
windings to increase the leakage and b) by expanding the primary winding over an auxiliary
core as proposed by [8] – figure based on representation from [8]
Push-Pull Converter with Magnetically Integrated Current Doubler
The current doubler circuit enables the summation of the transformer and the
inductor current. Due to the 180° phase shift of the two phases the current ripple of
the two inductors cancels out at the output [13]. For the integrated current doubler
several examples designed with EE-cores made of ferrite material can be found in
literature. These cores are easily available on the market and manufacturers offer the
possibility to insert air gaps in the core structure. A low profile integrated current
doubler is proposed by [14], where [15], [16] and [17] are comparing several
transformer structures with integrated current doubler. Figure 1-6 a) shows the
discrete current doubler and the until then two state of the art integrated
alternatives. All integrated alternatives use the EE core structure, where the windings
of the current doubler inductors are placed on the outer legs as well as the air gaps
which enable the required energy reactor. The transformer windings are placed on
19
the ungapped centre leg, where the improved alternative is using the windings from
the current doubler as secondary transformer winding in order to save one winding.
c
a
e
b
c
d
c
a
b
a
e
e
b
d
c
a
d
b
c
a
e
b
d
e
c
d
c
a
b
a
b
e
e
a)
d
b)
d
Figure 1-6: a) Different alternatives of a transformer with current doubler (discrete and
integrated); b) derivation of the today state of the art transformer with integrated current
doubler – figure based on representation from [16]
20
Figure 1-6 b) shows the derivation of the today known state of the art transformer
with integrated current doubler, published by [15]. Therefore, [15] proposes to split
the primary winding and places all windings on the outer legs. The two outer leg air
gaps were replaced by only one centre leg air gap. This changes the magnetic
behaviour of the component and enables a better current ripple cancelation as
demonstrated by [15]. Additionally, the tooling costs for the core can be reduced and
the mechanical stability is improved.
This integrated current doubler is used by [18] to achieve an elevated power density
for a telecom DC-DC converter. This example shows the possible advantage of
functionally integrated magnetic components to enable increasing power densities
with the help of the size reduction of magnetic components.
Push-Pull Forward Converter
[17] and [19] propose a push-pull forward converter with current doubler rectifier
using a full magnetically integrated structure. The discrete circuit requires a
transformer with two primary and one secondary winding, as well as two discrete
inductors. The integrated magnetic component is derived from the push-pull
converter with current doubler, presented in the previous section. In comparison,
this integrated magnetic component requires an additional primary winding. The two
inductors on the secondary were realised by the secondary windings of the
transformer. The energy reactor is realised with air gaps. In [19] all legs of the
component are gapped, where in [17] only one centre leg air gap is required. In
comparison to the discrete circuit, the integrated magnetic component requires only
four windings instead of five and one core instead of three.
c
a
b
a'
b'
a
b
e
b'
d
c
a'
a)
d
b)
e
Figure 1-7: Push-pull forward converter from [19] using magnetically integrated component:
a) circuit; b) integrated magnetic component – figure based on representation from [19]
21
1.3 Coupled Inductors
Coupled inductors are chokes, where at least two or more windings share the same
core. The component provides an energy reactor. Magnetic coupling enables energy
transfer between the windings, which can be advantageous for some topologies.
Thus, a coupled inductor is a kind of a transformer. A brief overview of transformers
and coupled inductors and their design is given in [20]. But in comparison to a
transformer (or transformers with integrated energy reactor) the coupled inductor is
usually not used to enable galvanic isolation.
Interleaved DC-DC Converters
Interleaved converters provide two or more legs which are controlled by a
symmetrical phase shift between each phase. The idea is that the current ripple of
the input or output current will cancel out due to the phase shift - regardless if
coupled inductors are used or not. A reduced current ripple enables the reduction of
capacitive filters and allows filter size and cost reductions [21]. As alternative, the
switching frequency of the converter can be reduced in order to reduce the AC and
switching losses of the converter [21].
A disadvantage of these interleaved converters is that each leg requires one inductor,
even if these can be realised smaller than one larger inductor. This disadvantage can
be cancelled by means of coupled inductors. Two or more windings will share one
core. This technique reduces the number of cores and promises lower total costs.
Furthermore, the coupling can enable a superior current ripple cancelation for each
phase [21] in comparison to the discrete interleaved alternative. E.g. [21] and [22]
propose coupled inductors for buck converters, where [23], [24] and [25] analysed
coupled inductors for boost converters.
[26], [27], [28], [29], and [30] show examples for three-phase DC-DC converters using
coupled inductors. Three discrete inductors are replaced by a coupled counterpart,
where the target is to reduce the total size and power loss of the inductive
components. A multiphase three-level converter using coupled inductors for high
power applications to reduce both differential mode current ripple and common
mode voltage is presented in [31]. The optimisation potential of coupled inductors
for low voltage DC-DC converters is depicted exemplary in [32] by means of a four
phase system, where [33] and [34] show the advantages of different magnetic
structures for coupled inductors. Due to the different operation behaviour of the
22
magnetic coupled inductor [35] investigates special control strategies for interleaved
converters using coupled inductors.
VL1
L1
L2
VL2
Vlow
IL1
S3
IL2
S4
S1
S2
Vhigh
Figure 1-8: Bi-directional interleaved DC-DC converter using coupled inductor
Cúk Converter with Coupled Inductor
The well-known Cúk converter, named after its inventor, is an inverting buck-boost
converter. It can be operated with coupled inductors as well, where the functionality
of the circuit is equivalent - regardless if coupled inductors are used or not [36]. The
capacitor of the circuit is used to store and transfer energy from the input to the
output. If the switch is off, the inductor currents flow through the diode. The
capacitor is charged by the energy from the input supply and the input inductor. The
output is fed by the output inductor. If the switch is on, the inductor currents flow
through the switch (charging the inductors). The capacitor discharges and supplies
the output inductor and the output. A more detailed explanation of the operation
principle is given in [37].
[36] investigates several coupled inductor structures and states that the coupled
inductor allows a downsizing and a loss decrease for the magnetic components of the
Cúk converter. Previously, [38] and [39] investigate the effect that the use of coupled
inductors in a Cúk converter can enable either zero input or output current ripple.
Figure 1-9: Cúk converter with coupled inductor – figure based on representation from [36]
23
Coupled 3-Phase AC Filter Choke
Chokes for sinusoidal three- or multiphase-systems are used e.g. in photovoltaic
inverters to feed the grid or as sinusoidal filters for electric machines to enable
smooth sinusoidal currents. The system can operate with n paralleled half bridges
(multiphase system), but multilevel converters are possible, too (e.g. see [40]). The
filter can be implemented with discrete chokes, where the implementation of
coupled inductors for symmetrically phase-shifted sinusoidal systems is possible as
well. Symmetrically phase shifted sinusoidal systems have the inherent property that
the sum of all currents is zero. Thus, the sum of all magnetic fluxes is zero, too. The n
windings from n discrete chokes can be wound on one core with n legs. An auxiliary
leg is not necessary. This integration technique enables a total size and weight
decrease compared to discrete magnetic designs.
φ1
φ2
Rm1
a)
φ3
Rm2
Rm3
I1
I2
I3
V1 N 1
Vm1 V2 N2
Vm2 V3 N3
Vm3
b)
Figure 1-10: a) Three-Phase coupled inductor; b) symmetrical three-phase current system
DC-AC Converter with Electrical Isolation and Coupled Inductor
A novel DC-AC converter for photovoltaic applications using electrical isolation and a
coupled inductor is presented in [41]. Instead using a transformer, the topology
shown in Figure 1-11 enables the isolation by means of diodes. The topology from
[41] was developed in order to demonstrate that a transformer is not necessary to
enable isolation of photovoltaic converters. Instead, the topology from Figure 1-11
requires a coupled inductor. The coupled inductor enables an energy reactor,
required to release energy to the output stage if all primary switches are opened.
Furthermore, the coupling of the component allows an energy transfer between the
two inductor phases to ensure a continuous current flow at the output. Therefore,
the coupled inductor will fulfil the requirements of an output filter as well, where no
additional output choke is required. The demonstrator from [41] uses only a small
24
680nF capacitor (for a 2.5kW converter) between the diode stage and the H4 bridge
on the secondary.
The primary switches (S1 to S4) can be modulated in a way that the topology can step
the input voltage up or down. The topology is a buck-boost converter requiring no
additional converter stage, which can be advantageous for photovoltaic applications.
The primary switches modulate a 100 Hz unipolar half sinus current, passing the
diodes. To allow the connection to the grid, an unfolding H4 bridge will feed the grid
with correct current polarity, to enable the required 50 Hz full sinus current.
S1
D3
S3
S5
S7
+
C
L2
L1
PV
Grid
D1
D2
D4
S2
S4
S6
DC-DC Stage
S8
Unfolding Bridge
Figure 1-11: PV converter using electrical isolation and coupled inductor – figure taken from
[42] and modified
1.4 Combined Common-Differential Mode Chokes
A common mode choke is a coupled inductor used to attenuate electrical noise in a
converter system. It is not used to enable galvanic isolation or energy transfer
between the windings or to store a significant amount of energy. Instead, it is a filter
required to provide a high impedance path for electrically conducted noise.
Therefore, one winding for each current branch is wound in the same direction e.g.
on a ring core. The currents flowing in the same direction (common mode current)
through each winding will excite fluxes which sum up inside the core. If the currents
flow in the opposite direction, the flux inside the core will cancel out and pass
through external leakage paths.
The ring core structure for common mode chokes is often chosen in order to
minimise these leakage effects and to avoid partial saturation effects inside the core,
which results in permeability decrease, making the filter ineffective. But it is also
possible to increase the leakage effect in order to design a combined common25
differential mode choke. This technique can enable the decrease of differential mode
filters or make them even redundant. The patent [43] gives an example for such an
idea (see Figure 1-12). An additional ferromagnetic material is inserted inside the ring
in order to create a predefined leakage path for the differential mode flux. This will
increase the inductance seen by the differential mode signal and improves the
differential mode attenuation. A similar magnetic circuit can be realised with an EE
core, where the windings are placed on the outer legs (see [44]). The centre leg is
gapped in order to adjust the differential mode inductance.
The same principle of differential mode inductance increase can be realised with pot
cores. The insertion of a ferromagnetic disc into a common mode pot core, to guide
the differential mode flux, is proposed by [45].
Figure 1-12: Common mode choke with predefined leakage path for differential mode signal
(ring core with segment) – figure based on representation from [43]
Figure 1-13 shows an example from [46], where each winding of a common-mode
choke is wound on two cores. The larger core (high permeable) is responsible for the
common mode attenuation, where the two smaller cores (low permeable) implement
two separated differential mode inductors. A similar implementation technique for
combined common-differential mode chokes is presented by [47] – using only one
common mode and one differential model core (see Figure 1-14). Due to the fact that
both windings are wound on both cores, the winding sense of one of the two
windings must be reversed for the differential mode core - because of the different
current flow definitions of differential and common mode signals.
Figure 1-13: Combined common mode/differential mode choke (one winding uses two
different cores) – figure based on representation from [46]
26
Figure 1-14: Combined common mode/differential mode choke with one common mode and
one differential mode core – figure based on representation from [47]
A different strategy for the differential mode inductance increase is proposed by [48]:
Two common mode inductors are connected in series, where the second one will be
flipped over and put on top of the first one, as it is depicted in Figure 1-15. This
technique will increase the differential mode inductance and partially cancel out
external magnetic stray fields [48]. The reduced EMI in the environment of the
component can be a significant advantage for the filter design, because the impact of
parasitic mutual inductances with other filter components disturbing the filter is
reduced. In turn, this technique requires two common mode chokes, resulting in a
heavy component.
Figure 1-15: Two stacked common mode chokes – figure based on representation from [48]
27
2
Fundamentals of Magnetic Components
2.1 Basic Magnetic Laws
Ampere’s Law
A current i, carried by an electrical conductor induces a magnetic field around this
conductor. The magnetic field is characterised by the magnetic field intensity H and
the direction of the field according to the so called right hand thumb rule. The thumb
of the right hand denotes the conductor in the direction of the current flow. The
other fingers indicate the direction of the magnetic field.
Eq. 2-1 expresses ampere’s law in the integral form. The integral of the field intensity
H around a closed loop l is equal to the total current passing through the surface A of
this loop, where J denotes the current density of this loop.
Eq. 2-1
∫ 𝐻 ∙ 𝑑𝑠 = ∫ 𝐽 ∙ 𝑑𝐴
𝑙
𝐴
Assuming a coil with N turns, carrying the current i or n conductors carrying the
currents im, Ampere’s law can be expressed with Eq. 2-2. The sum of all currents is
equivalent to the total magneto motive force (or ampere turns) of a magnetic
component.
𝑛
Eq. 2-2
𝑉𝑚 = 𝑁 ∙ 𝑖 = ∑ 𝑖𝑚 = ∫ 𝐻 ∙ 𝑑𝑠
𝑚=1
𝑙𝑐
Finally, it has to be noticed that Ampere’s law in Eq. 2-1 and Eq. 2-2 is not given
complete. The term ∫ 𝐽𝑑𝐴 does not contain the so called displacement currents. The
complete form of the law, expressed by Maxwell in 1865, including displacement
currents is [49]:
Eq. 2-3
∫ 𝐻 ∙ 𝑑𝑠 = ∫ 𝐽 ∙ 𝑑𝐴 +
𝑙𝑐
𝐴
𝜕
∫ 𝜀𝐸 ∙ 𝑑𝑠
𝜕𝑡
𝐴
28
Magnetic Flux Density
The magnetic field intensity H will lead to a magnetic flux density B. The ratio of flux
density to field intensity gives the product of the permeability of vacuum μ0
-7
(4∙π∙10 V∙s/(A∙m)) and the relative permeability μr of the penetrated material.
Eq. 2-4
𝐵 = 𝜇0 𝜇𝑟 ∙ 𝐻 = 𝜇 ∙ 𝐻
The relative permeability describes the ability of a material to conduct magnetic flux.
For air and electrical conductors as copper or aluminium μr ≈1 can be considered.
Ferrous core materials can offer permeability values of several hundred up to tens of
thousands. The main purpose of magnetic cores is to carry the flux on a predefined
path and concentrate the magnetic field inside the core.
But it has to be taken into account that the B-H characteristic of ferrous materials is
usually non-linear. So the permeability is usually a function of the magnetic field
intensity and describes the slope of the B-H characteristic at a certain operation
point. The deviation of the B-H relationship or the permeability describes the
magnetisation characteristic of a material. If the material is operated close to the
saturation range, a field intensity increase does not lead to a proportional flux density
increase.
Magnetic Flux
The magnetic flux density is the surface density of the magnetic flux passing through
a predefined surface A. The integration of the flux density over the surface area will
give the magnetic flux passing through this area.
Eq. 2-5
𝜑 = ∫ 𝐵 ∙ 𝑑𝐴
𝐴
Faraday’s Law
A time changing magnetic flux passing through an open loop conductor will induce a
time dependent voltage according to Faraday’s law:
Eq. 2-6
𝑣(𝑡) = −𝑁
𝑑𝜑(𝑡)
𝑑Ψ(𝑡)
=−
𝑑𝑡
𝑑𝑡
29
N is the number of turns linked with the magnetic flux φ. The term N∙φ is also
denoted as flux linkage ψ. By expressing the induced voltage with the electric field E
and replacing the magnetic flux with Eq. 2-5, Faraday’s law can be expressed by:
Eq. 2-7
∫ 𝐸 ∙ 𝑑𝑙 = −
𝑙
𝑑
∫ 𝐵 ∙ 𝑑𝐴
𝑑𝑡
𝐴
It should be noted that Eq. 2-6 and Eq. 2-7 are given in the generator convention. To
use the consumer convention the negative algebraic sign must be replaced by
positive algebraic sign.
Gauss’s Law
Gauss’s law states that the total magnetic flux entering a closed surface is equivalent
to the total magnetic flux leaving this surface. This means the total flux penetrating
the surface is zero.
Eq. 2-8
∫ 𝐵 ∙ 𝑑𝐴 = 0
𝐴
2.2 Magnetisation and Hysteresis
All materials can be classified into five magnetic material classes [3], [1]:



Ferromagnetic materials can have a relative permeability much higher than
unity. The magnetisation is not linear and depends on the applied field as
well as on the previous history of the material. If the flux density reaches the
saturation value, the relative permeability value can drop down to unity. The
magnetisation behaviour is described more in detail subsequent in this
chapter.
In antiferromagnetic materials, the net magnetic moment is zero regardless
if a magnetic field is applied or not. However, the relative permeability
increases to values slightly greater than unity if a magnetic field is applied
[1].
Ferrimagnetic materials (e.g. manganese) are listed beside iron in the
periodic table. The spin magnetic moments are very large, unequal and
alternate from atom to atom, resulting in a zero net magnetic moment if no
30


magnetic field is applied. An applied magnetic field results in nonzero net
magnetic moment, though the magnetic moments partially cancel out.
Therefore, the magnetic flux density is lower compared to ferromagnetic
materials, but the relative permeability is much higher than unity as well. If
the material is heated above its Curie temperature, the material becomes
paramagnetic [1].
Diamagnetic materials (e.g. copper, gold) have a relative permeability close
below unity. The relationship between field intensity and flux density is
linear. Diamagnetic materials have the property to create magnetic fields,
which oppose the applied field. A strong field intensity increase causes only
a low flux density increase. These materials exhibit only magnetic properties
if an external field is applied.
Paramagnetic material (e.g. aluminium, platinum, titanium) has a relative
permeability slightly greater than unity. The relationship between field
intensity and flux density is linear and these materials do not retain
magnetisation if no magnetic field is applied. The total magnetisation will
drop to zero if the applied magnetic field is removed.
Figure 2-1 shows a typical hysteresis loop, including the initial magnetisation of a
ferromagnetic material, as well as the magnetisation of para- and diamagnetic
materials. In ferromagnetic materials, the so called Weiss domains or ferromagnetic
domains, containing atoms with a net magnetic moment greater than zero, have a
predefined magnetisation direction. The magnetic moments of adjacent domains are
opposing, resulting in a total magnetic moment equal to zero. The domains are
divided by the domain or Bloch walls, where the direction of the magnetic moment
changes.
If an external magnetic field is applied to a ferromagnetic material, the material starts
to magnetise. This process starts slowly and increases with increasing field intensity.
This means that the domain walls between adjacent domains will be displaced in a
way that the domains with magnetic moments in direction according to the applied
field will increase. For high magnetic field intensity, the displacement of the domain
walls proceeds in jumps (Barkhausen jumps) [3].
If the magnetic field intensity increases further, the material starts to saturate. A
strong field intensity increase will cause only a low flux density increase, because
nearly all atom magnetic moments are aligned in the direction of the applied
31
magnetic field. Practically, this operation point is expressed by the saturation flux
density Bsat. If the magnetic field intensity is reduced to zero, the flux density will not
drop to zero. Instead, a certain flux density will remain – this is the so called
remanence or residual flux density Br. If the flux density becomes zero, the so called
coercive field intensity Hc is applied.
The hysteresis loop is traversed the same number of times per second as the
frequency of the applied current. In many power electronic applications the currents
are not pure sinusoidal. Very often, a DC current or a low frequency AC current is
overlain by a high frequency AC current. In this case, the material is magnetised as
described above, but the high frequency AC component causes additional minor
loops in the hysteresis.
+Bsat
+Br
Initial Magnetisation
(ferromagnetic)
Paramagnetic (μr>1)
Free Space (μr=1)
Diamagnetic (μr<1)
-Hc
+Hc
-Br
-Bsat
Figure 2-1: Hysteresis loop - figure based on representation from [3]
32
2.3 Magnetic Core Materials1
The main purpose of magnetic cores is to guide the flux on a predefined path and
concentrate the magnetic field inside the core. So ferromagnetic materials will
increase the magnetic conductance of a magnetic circuit and have a much higher
relative permeability than air and other non-ferromagnetic materials (μr≈1). This
leads to significant reduced electromagnetic interference (EMI) and size of the
magnetic component. E.g. a certain inductance can be realised with smaller size and
less windings if a ferromagnetic core is used - which can be a superior advantage. This
is the reason why magnetic cores are widely used in many applications, even though
their obvious drawbacks like core loss and saturation effects.
Figure 2-2: Specific core loss vs. saturation flux density for different selected materials
(ferrite: N27, N87; amorphous alloys: Vitrovac 6030, AMCC; nanocrystalline: Vitroperm
500 F, Finemet F3CC; Iron Powder: KoolMμ) [50], [51], [52], [53], [54], [55] – figure taken
from [56]
Today core manufacturers offer many magnetic materials, core shapes and sizes to
cover a wide spectrum of applications. The materials differ in permeability, specific
power loss per weight or volume, saturation flux density, electrical resistivity and
other properties. Table 2-1 shows a comparison of common used materials in power
electronic applications. The materials given in the upper rows (MnZn, NiZn, iron
powder, amorphous and nanocrystalline alloys) are usually used for frequencies
much higher than 10 kHz. Instead, the magnetic steel sheets are usually used below
1
This chapter is partially taken from the ECPE Joint Research Report “Characterization of
Magnetic Materials”.
33
10 kHz. Figure 2-2 shows the plotted specific core loss and saturation flux density of
selected core materials exemplary.
Material
Saturation
flux density
Relative
permeability
Curie
temperature
Resistivity
MnZn
Ferrite
0.3 – 0.5 T
1,000 –
15,000
150 – 220
°C
10 – 10
NiZn
Ferrite
0.4 T
40 – 900
400 °C
10 – 10
Iron
Powder
1 – 1.3 T
10 – 500
700 °C
10
Up to a few
100 kHz
Amorphous
alloys
0.5 – 1.8 T
10,000 –
150,000
350 – 450
°C
1.2 – 2
Below
100 kHz
Nanocrystalline
alloys
1.2 – 1.5 T
15,000 –
150,000
600 °C
0.4 – 1.2
Up to
100 kHz
[μΩm]
Frequency
range
2
4
Up to
several 100
kHz
7
9
Up to
several
MHz
6
Magnetic steel sheets
Iron Silicon,
FeSi (3-6%
Si)
1.9 T
1,000 –
10,000
720 °C
0.4 – 0.7
Up to a few
kHz
Permalloy
(80% Ni)
1T
10,000
500 °C
0.15
Up to a few
kHz
Isoperm
1.6 T
3,000
500 °C
0.35
Up to a few
kHz
0.6 T
2,000
500 °C
0.75
Up to a few
kHz
2.4 T
10,000
450 °C
0.35
Up to 100
kHz
(50% Ni)
Invar
(30-40% Ni)
FeCo
Table 2-1: General properties of different core materials [1], [49]
Ferrite Materials
Many applications of functionally integrated magnetic components are realised with
manganese zinc (MnZn) ferrites. These materials can be manufactured in arbitrary
shapes, giving the designer a high degree of freedom. Their relatively high
34
permeability makes them suitable for transformer applications. Required energy
reactors can be enabled by the insertion of air gaps. Manufacturers offer e.g. tooled E
cores with gaps in the centre leg. For E cores, the gaps will be established by grinding
the cross-sectional area of the core legs. Furthermore, segments as well as custom
shapes can be produced in order to enable the introduction of arbitrary geometries.
Ferrites are ceramic materials. They have a relatively high specific resistance in
comparison to materials with high metal content. Material compositions with very
high specific resistances can prevent the generation of eddy current losses inside the
core. In comparison to other core materials the specific core losses of ferrite
materials are usually very low. This enables the operation at very high switching
frequencies, even in the MHz range (NiZn ferrites).
As drawback, ferrites have only a limited saturation flux density. Functionally
integrated magnetic components are often driven with DC bias flux overlain with an
AC flux component. Materials providing a high saturation flux density can enable a
certain advantage, because they can allow component downsizing.
The operation temperature range of ferrite is limited, too. The materials are usually
very temperature sensitive and show a parabolic temperature behaviour of the core
losses. The material dependent loss minimum is between 80 and 120°C. In case of
elevated operation temperatures, the designer may face the dangerous of thermal
runaways. The operation temperature must be limited due to the relatively low Curie
temperature compared to other materials.
NiZn ferrites are usually manufactured for small core sizes. Manufacturers deliver
only a limited number of core shapes and sizes. In comparison to MnZn ferrites, NiZn
materials provide lower permeability and saturation flux density. Due to their very
high specific resistance, NiZn ferrites are applicable for very high frequencies. They
will be usually manufactured as ring or pot core types in order to minimise EMI
effects at very high operation frequencies (MHz range).
Iron Powder Material
Iron powder cores consist of small iron particles insulated against each other. This
leads to a relatively high saturation flux density, a high electrical resistance and a
distributed air gap inside the core. These materials are usually designed for the
application of differential mode filter inductors. Due to the distributed air gap, the
insertion of air gaps inside the magnetic structure is not necessary, leading to
35
reduced EMI effects. The materials have only a limited permeability due to the
distributed air gap and will be manufactured for several permeability values. The
material is unsuitable for transformer applications, because of the high specific core
losses and the limited permeability. But the application of functionally integrated
magnetic components is possible if low induction values are ensured. An inherent
property of the material is that the permeability depends on the applied ampere
turns, leading to partial saturation and making the design more difficult.
Some iron powder materials tend to age at elevated temperatures. The insulation of
the iron particles will be partially destroyed by excessive heat of the core, leading to
increasing eddy current losses. The eddy current loss increase will lead to additional
heat, causing the damage of even more insulation material. In order to avoid this
aging effect, the cores must operate at limited temperature. To cancel this drawback,
manufacturers have developed materials suitable for elevated operation
temperatures (e.g. Sendust), where these materials are usually much more expensive
than low temperature alternatives. Common core shapes are ring and E cores.
Amorphous and Nanocrystalline Alloys
These metallic alloys will be manufactured as thin ribbons. The ribbons will be
laminated, cut and stacked in order to form the cores. As drawback, only C and U cut
cores as well as ring cores will be manufactured, making the application for
functionally integrated magnetic components difficult. Furthermore, the anisotropy
of these materials can make the design of flux bypasses difficult, because these
materials are usually designed in order to enable a maximum permeability along the
lamination direction.
Due to the very high permeability, these materials suit very well for transformer
applications. Required energy reactors can be established by air gaps. The air gaps
can be enabled by cut cores, where the cutting process will partially destroy the
insulation between the ribbons. The insulation can be re-established by additional
tooling, making the cut cores more expensive. Additionally, the destroyed insulation
is leading to increased eddy current losses at the cutting line.
In general, nanocrystalline and amorphous materials have relatively high saturation
flux densities. Additionally, nanocrystalline materials offer low core losses in general,
where amorphous materials have to suffer higher specific core losses. A high
admissible operation temperature allows the application at elevated ambient
36
temperatures, making these materials e.g. interesting for automotive applications.
The use of these materials for functionally integrated magnetic components may be
difficult, but must be checked application dependent.
Magnetic Steel Sheets2
Iron alloys contain a small amount of silicon, nickel, cobalt or chromium, dependent
on the alloy type. In general, they have a very low resistivity because of the high iron
content of the material. Furthermore, they have very high permeability, saturation
flux density and core losses. Because of the high core losses, they are only suitable
for low frequency applications (in the lower kHz range). At higher frequencies, the
core losses are dominated by eddy current losses - due to the low electrical
resistivity. In order to limit the eddy currents, the cores are made of stacks with thin
laminations, electrically insulated against each other. The single sheets can be
manufactured in arbitrary geometries.
The most common iron alloy is iron-silicon (FeSi). The silicon content is up to about
6 % [1] and leads to a crystalline structure. It will increase the resistance of the
material and, thus, decreases the eddy current losses. Furthermore, the permeability
is increased, which reduces the hysteresis losses. A further advantage is that the
silicon decreases the magnetostriction and, therefore, reduces acoustic noise of the
component [1], [49]. The drawback of the alloyed silicon is a decreasing saturation
flux density and decreasing Curie temperature.
The iron-nickel alloys can be separated in three groups [49]:



High nickel content (80 %) – Permalloy, Mumetal – highest permeability
Medium nickel content (50 %) – Isoperm – highest saturation flux density
(1.6 T)
Low nickel content (30-40 %) – Invar – highest electrical resistivity
(0.7 – 0.8 μΩm)
Some of the iron-nickel alloys can have a relative permeability up to 300,000
(Mumetal). The cores are sensitive to mechanical stress and, therefore, get protected
by plastic or aluminium cases filled with a damping material. The magnetic properties
depend on the temperature and material thickness.
2
This section is taken from the ECPE Joint Research Report “Characterization of Magnetic
Materials” and modified.
37
Iron-Cobalt-Alloys (FeCo) have a cobalt content of approximately 50 % and offer the
highest saturation flux density. The typical material thickness is about 0.05 – 0.1 mm.
These materials are used in special transformer applications at very high flux
densities with limited losses (magnetic amplifiers, space applications) [1].
2.4 Magnetic Circuits
Magnetic components can be described by means of magnetic circuits. They will be
handled similar like electric circuits, with equivalent rules. Kirchhoff’s voltage and
current law e.g. is valid in the magnetic domain as well. A comparison between
electric and magnetic values is given in Table 2-2. There are similarities for voltage,
current, resistance and other values.
φ
I
V
N
Vm
Rm
Figure 2-3: Magnetic circuit
Electric Domain
Magnetic Domain
Voltage: V [V]
Magneto motive force: Vm [A]
Current: I [A]
Flux: φ [Wb]
Resistance: R [Ω]
Reluctance: Rm [A/Wb]
Conductance: G [S]
Permeance: Pm [Wb/A]
Electrical Field Intensity: E [V/m]
Magnetic Field Intensity: H [A/m]
Current Density: J [A/m]
Flux Density: B [T]
Conductivity: σ [S/m]
Permeability: μ [Vs/(Am)]
Electric Charge: Q [As]
Flux Linkage: Ψ [Vs]
Capacitance: C [F]
Inductance: L [H]
Electric Energy: Wel = ½ C v² [J]
Magnetic Energy: Wm = ½ L i² [J]
Kirchhoff’s current law: ∑ ii = 0
Gauss’s law: ∑ φi = 0
Kirchhoff’s voltage law: ∑ vi = 0
∑ Vmi = 0
Table 2-2: Electrical quantities and their magnetic counterparts
38
Figure 2-3 shows a magnetic core carrying a conductor with N turns. The
corresponding ideal magnetic equivalent circuit is drawn in red, where all parasitic
effects in the magnetic as well as in the electric domain will be neglected here. A
conductor with N turns, carrying the current I is generating a magnetic voltage Vm in
the magnetic domain (see also Eq. 2-2):
Eq. 2-9
𝑁 ∙ 𝐼 = 𝑉𝑚 = 𝑅𝑚 ∙ 𝜑 = 1/𝑃𝑚 ∙ 𝜑
Eq. 2-9 is the equivalent to Ohm’s law in the electric domain, where φ is the flux
excited by the ampere turns. Rm is the magnetic resistance called reluctance, where
Pm is the so called permeance.
Reluctance and Permeance
The magnetic resistance is called reluctance and can be defined by the ampere turns
Vm applied to the reluctance Rm and the flux φ passing through this reluctance
according to Eq. 2-9 - or by geometrical and material quantities of the reluctance
itself as depicted in Figure 2-4:
Eq. 2-10
𝑅𝑚 =
1
𝑙𝑒
∙
𝜇0 ∙ 𝜇𝑟 𝐴𝑒
μ0 is the permeability of free space and μr is the relative permeability of the used
material. le is the mean or equivalent magnetic path length. Ae is the equivalent
magnetic cross section, used by the magnetic flux, passing through the reluctance.
Both values can be directly derived by the geometrical quantities, as depicted in
Figure 2-4.
φ
le
µr
Rm
Vm
Ae
Figure 2-4: Reluctance of a ferrous rod core
39
The inverse of the reluctance is the so called permeance, which can be interpreted as
magnetic conductance:
Eq. 2-11
𝑃𝑚 =
1
𝐴𝑒
= 𝜇0 ∙ 𝜇𝑟 ∙
𝑅𝑚
𝑙𝑒
Reluctance and permeance depend only on material and geometry properties. But in
some cases, it is not easy to evaluate these values in practice. The permeability of
ferrous cores can dependent on the operation point of the component - leading to a
non-constant value of μr. Furthermore, the design of magnetic components especially
functionally integrated magnetic components can make the recognition of parasitic
flux paths necessary. Reluctances in magnetic circuits are lumped elements. They are
derived directly from the geometrical structures of a component. But it is not that
easy to define leakage or fringing flux paths of a component, passing through the
environment, by geometrical parameters.
Kirchhoff’s Voltage Law in the Magnetic Domain
The rules for the analysis of electric circuits are valid for magnetic circuits as well,
because of the description with lumped elements. According to potential differences
in magnetic circuits this means: The sum of all magneto motive forces (ampere turns)
and magnetic voltage differences within a loop of a magnetic circuit is zero:
𝑛
Eq. 2-12
𝑛
∑ 𝑉𝑚,𝑖 = ∑ 𝜑𝑖 ∙ 𝑅𝑖 = 0
𝑖=1
𝑖=1
Kirchhoff’s Current Law in the Magnetic Domain (Gauss’s law)
Gauss’s law states the continuity of the magnetic flux. If a lumped circuit model is
assumed, this means:
𝑛
Eq. 2-13
𝑛
∑ 𝜑𝑖 = ∑ 𝐵𝑖 ∙ 𝐴𝑖 = 0
𝑖=1
𝑖=1
So the Gauss’s law for magnetic circuits is the equivalent to Kirchhoff’s current law
for electric circuits. The sum of the total magnetic flux flowing into a node of a
magnetic circuit is zero. In other words: If two boundary surfaces with different
40
surface areas are considered, the ratio of the flux density of each surface is
equivalent to the inverse ratio of the surface areas, because the flux through both
areas must be equal.
2.5 Definitions of Inductance
In principle, an inductance is just wound up wire. A magnetic core is not necessary to
realise an inductance, but applicable in many cases. In literature many definitions of
inductance are given. The most important ones are briefly presented here. More
details can be found in the literature [1], [49].
Small Signal Inductance
The small signal inductance is defined by the change of the flux linkage and the flux
exciting current. It is also known as differential inductance and given by:
Eq. 2-14
𝐿=
𝑑Ψ
𝑑𝑖
The small signal inductance is defined by the ratio of an infinitesimal flux linkage
change to an infinitesimal current change at a given operation point. The inductance
can be interpreted by the slope of the ψ-I curve at a given operation point. For nonlinear materials as ferrous cores this means that the permeability and the inductance
depend on the applied magnetic field.
Magnetic Energy
Inductance can be calculated by the magnetic energy stored in the magnetic
component, too:
Eq. 2-15
𝐿=
2𝑊𝑚
1
1
= 2 ∫ (𝐵 ∙ 𝐻 ∗ )𝑑𝑉 =
∭ 𝐵2 𝑑𝑉
𝑖2
𝑖 𝑉
2𝜇 𝑉
Wm is the energy stored in the magnetic field of the component, excited by the
current with the amplitude i flowing through the excitation winding. This definition is
often used for magnetic components which require an energy reactor (e.g. air gap of
an inductor).
41
Reluctance/Permeance
Figure 2-3 and Eq. 2-9 show that the winding is the interface between the electric and
the magnetic domain. In order to show the duality of both domains Figure 2-3 will be
used as example ( [57] gives a more detailed explanation of the duality of electric and
magnetic circuits). Expanding Eq. 2-9 by the multiplication with the number of turns
N and replacing the factor N∙φ by the flux linkage Ψ leads to:
Eq. 2-16
𝑁 2 ∙ 𝐼 = 𝑅𝑚 ∙ 𝑁𝜑 = 𝑅𝑚 ∙ Ψ
By rearranging Eq. 2-16 according to Eq. 2-14 will give the context of inductance L
and reluctance Rm:
Eq. 2-17
𝐿=
Ψ 𝑁2
=
= 𝑁 2 𝑃𝑚
𝐼
𝑅𝑚
Thus, it is possible to determine the inductance with the reluctance (Rm) or the
permeance (Pm also AL or inductance factor) of a magnetic circuit. Eq. 2-17 is only
dependent on geometry and material parameters. The permeance or inductance
factor (AL), also given by the core manufacturers in their data sheets, represents the
total permeance of a magnetic component - seen by the excitation winding. But it
should be noted that this value can vary material dependent within a range of up to
25%. This shows that the definition of specific values for magnetic materials is not
easy, due to the non-linear behaviour of the core and material tolerances.
2.6 Transformer Principle
Coupled inductors, transformers and functionally integrated magnetic components
have one thing in common: Each of these components offer at least two windings
wound on one core. This means that the windings of these components are coupled
and magnetic energy transfer from one to another winding is possible. This chapter
will explain the common specific values like coupling coefficient, mutual inductance
and leakage inductance.
The magnetic circuit from Figure 2-5 represents a transformer with low leakage
between the windings. This simplified magnetic circuit will lead to the well-known Tequivalent transformer circuit in the electric domain and suits well to explain the
common specific values of coupled inductors.
42
N2 N1
Rm
Vm1
Rσ1
Vm2
Rσ2
Figure 2-5: Transformer with leakage path
2.6.1 Coupling Coefficient
The coupling coefficient can be defined for two windings placed in an arbitrary
magnetic circuit. A primary winding will excite a flux due to current flow through this
winding. A part of this excited flux will pass through the secondary winding, inducing
a voltage drop at this winding - according to Faraday’s law (see Eq. 2-6). The coupling
coefficient is the ratio of the flux passing through the secondary winding and the
excited flux.
In order to get a more manageable definition in the context of magnetic circuits, the
coupling coefficient can be defined by reluctance or permeance values of a magnetic
circuit, similar to the current divider rule for electric circuits. Assuming the example
from Figure 2-5, the coupling coefficient can be defined for the winding N1 as
excitation winding (k12 – see also Figure 2-6 a) as well as for the winding N2 as
excitation winding (k21 – see also Figure 2-6 b):
Eq. 2-18
𝑘12 =
𝜑12
1/𝑅𝑚
𝑅𝜎1
=
=
𝜑1
1/𝑅𝑚 + 1/𝑅𝜎1 𝑅𝜎1 + 𝑅𝑚
Eq. 2-19
𝑘21 =
𝜑21
1/𝑅𝑚
𝑅𝜎2
=
=
𝜑2
1/𝑅𝑚 + 1/𝑅𝜎2 𝑅𝜎2 + 𝑅𝑚
43
N2
N1
φ1σ
Rm
Rm
φ1
Rσ1
Rσ2
N2 N1
φ21
Rσ1
Vm1
φ12
Rσ2
a)
φ2σ
φ2
Vm2
b)
Figure 2-6: Transformer with a) open secondary winding and b) open primary winding
It has to be noticed that the winding which is not excited by a magneto motive force
is left open, leading to a short circuit in the magnetic equivalent circuit, due to the
duality principle (electric open circuit equals magnetic short circuit). For Figure 2-6 a)
the reluctance Rσ2 will be cancelled out due to the short circuit, where Rσ1 will be
cancelled for Figure 2-6 b).
In practice, the coupling coefficient will be measured via the excitation of a sinusoidal
voltage at one winding and the measurement of the induced voltage of the other
winding (open loop). The ratio of induced and excited voltage under the
consideration of the winding turn ratio will give the coupling coefficient, where the
induced open loop voltage is labelled with the dash.
Eq. 2-20
𝑘12 =
𝑁1 𝑉2 ′
∙
𝑁2 𝑉1
Eq. 2-21
𝑘21 =
𝑁2 𝑉1 ′
∙
𝑁1 𝑉2
The resulting total coupling coefficient is defined as:
Eq. 2-22
𝑘 = √𝑘12 ∙ 𝑘21
High coupling coefficients will be characterised as tight coupling, where low coupling
coefficients will be termed loose coupling [5].
44
2.6.2 Mutual- and Self-Inductance
Another definition for the characterisation of coupled inductors is the so called
mutual inductance. It is defined by the self-inductance and the coupling coefficient of
two coupled windings. Definitions of the self-inductance can be found in chapter 2.5.
For the example from Figure 2-5, the self-inductance seen by each winding is given in
Eq. 2-23 and Eq. 2-24, according to the definition of Eq. 2-17.
1
1
+
)
𝑅𝜎2 𝑅𝑚
Eq. 2-23
𝐿1 = 𝑁12 ∙ (
Eq. 2-24
𝐿2 = 𝑁22 ∙ (
1
1
+
)
𝑅𝜎1 𝑅𝑚
The self-inductance is proportional to the flux linkage excited by the winding itself.
The mutual inductance is proportional to the flux linkage, which is linked to the
secondary winding:
Eq. 2-25
𝑀 = 𝑀12 = 𝑀21 = √𝑘12 ∙ 𝑘21 ∙ √𝐿1 ∙ 𝐿2
2.6.3 Leakage Inductance
In comparison to electric circuits, there are no non-conductors in the magnetic
domain. In fact, magnetic flux can pass through non-magnetic materials, too. In
practice, this will lead to effects like leakage inductance or fringing flux.
The leakage or stray inductance is the part of the primary excited flux, which is not
passing through the secondary winding. It is the part of the flux which cannot
contribute to the induction of the secondary voltage and is defined by:
Eq. 2-26
𝐿𝜎1 = (1 − 𝑘) ∙ 𝐿1
Eq. 2-27
𝐿𝜎2 = (1 − 𝑘) ∙ 𝐿2
If power transformer applications are considered, the leakage inductance can be
interpreted as parasitic inductance. But sometimes the leakage will be used as a
design parameter, to improve the functionality of functionally integrated magnetic
components.
45
3
Losses in Magnetic Components3
Due to the fact that the calculation of losses is one of the main issues for the
development of magnetic components for power electronic converters, this chapter
will explain the basic loss mechanisms and treat a few of the current most advanced
loss models. As depicted in Figure 3-1 the losses can be separated in winding and
core losses. The winding losses can be separated further into RMS-, skin effect and
proximity effect related losses.
Three loss mechanism for the core losses are known today: hysteresis, eddy current
and relaxation losses. Application dependent not all loss mechanisms have to occur
[56]. The impact of a loss mechanism depends on the design and operation mode of
the component as well as on the material properties of the used materials [56]. The
most important core loss models, as well as their inherent advantages and
disadvantages, will be briefly discussed in this chapter. [56] gives a more detailed
overview of core loss models and their developments and enhancements.
Power Loss
Core Losses
Winding Losses
Hysteresis Losses
RMS Losses
AC Losses
Eddy Current Losses
Skin Effect
Relaxation Losses
Proximity Effect
Figure 3-1: Losses in magnetic components – figure taken from [56] and modified
3
This chapter is partially taken from the ECPE Joint Research Report “Characterization of
Magnetic Materials”.
46
3.1 RMS Power Loss and Temperature Dependency4
The temperature dependency of a resistance can be expressed by [5]:
Eq. 3-1
𝑅𝑇2 = 𝑅𝑇1 ∙ (1 + 𝛼𝑐𝑢 ∙ (𝑇2 − 𝑇1 ))
α is the temperature coefficient of the given material (e.g. 0.0039/K for copper at
20 °C). RT1 is the reference resistance for the temperature T1. The DC resistance at
20 °C is an important characteristic value for magnetic components. Using Eq. 3-1,
the resulting resistance for an elevated temperature T2 can be calculated in order to
evaluate the temperature dependent winding losses. Eq. 3-1 can be expressed by the
specific resistance ρ or specific conductance σ, too:
Eq. 3-2
𝜌𝑇2 = 𝜌𝑇1 ∙ (1 + 𝛼𝑐𝑢 ∙ (𝑇2 − 𝑇1 ))
Eq. 3-3
𝜎𝑇2 =
𝜎𝑇1
1 + 𝛼𝑐𝑢 ∙ (𝑇2 − 𝑇1 )
The RMS or DC losses of a magnetic component are calculated according to:
Eq. 3-4
2
𝑃𝑟𝑚𝑠 = 𝑅𝐷𝐶 ∙ 𝐼𝑟𝑚𝑠
Taking the temperature dependency into account, Eq. 3-4 changes to:
Eq. 3-5
2
𝑃𝑟𝑚𝑠 = 𝑅𝐷𝐶,20°𝐶 ∙ (1 + 𝛼𝑐𝑢 ∙ (𝑇2 − 20°𝐶)) ∙ 𝐼𝑟𝑚𝑠
Elevated operation temperatures cause higher losses, where higher losses cause
higher temperature rise. By replacing the temperature rise with the thermal
resistance and the power loss (∆T=Rth∙P) Eq. 3-5 can be rewritten [56]:
Eq. 3-6
𝑃𝑟𝑚𝑠 =
2
𝑅𝐷𝐶,20°𝐶 ∙ 𝐼𝑟𝑚𝑠
𝑃𝑟𝑚𝑠,20°𝐶
=
2
1 − 𝛼𝑐𝑢 ∙ 𝑅𝑡ℎ ∙ 𝑅𝐷𝐶,20°𝐶 ∙ 𝐼𝑟𝑚𝑠
1 − 𝛼𝑐𝑢 ∙ 𝑅𝑡ℎ ∙ 𝑃𝑟𝑚𝑠,20°𝐶
The equivalent circuit of Eq. 3-6 is depicted in Figure 3-2, where the thermal
capacitance has no influence on the steady state and is not recognised here.
4
This section is taken from the ECPE Joint Research Report “Characterization of Magnetic
Materials” and modified.
47
Tw
R
I
Rth
W
P
Cth
Ta
Figure 3-2: Thermal (orange) and electric (black) circuit for the calculation of the temperature
dependent winding losses – figure taken from [56]
The term RDC,20°C∙Irms² can be interpreted as power loss at 20 °C without any
temperature dependency. Therefore, Eq. 3-7 describes the RMS power loss increase
dependent on the power loss at 20°C and the thermal resistance.
Eq. 3-7
𝑃𝑟𝑚𝑠
1
=
𝑃𝑟𝑚𝑠,20°𝐶 1 − 𝛼𝑐𝑢 ∙ 𝑅𝑡ℎ ∙ 𝑃𝑟𝑚𝑠,20°𝐶
Figure 3-3 shows an example for a graphical evaluation of Eq. 3-7. For very small
thermal resistances as well as low power loss, the temperature dependency can be
neglected. But the impact of the temperature dependency increases significant with
increasing thermal resistance. A high power loss at 20 °C and high thermal resistances
results in a much higher total power loss than expected for
20 °C. Finally, for heavy windings with high thermal resistance, suffering high current
densities, the context of losses and thermal behaviour cannot be neglected.
Figure 3-3: RMS power loss increase dependent on power loss at 20 °C for different thermal
resistances
48
3.2 AC Winding Losses
3.2.1 Skin Effect5
The skin effect describes the eddy currents induced by a conductor itself. A timevarying current generates a magnetic field, penetrating the conductor and, therefore,
induces eddy currents. The eddy currents themselves generate magnetic fields,
opposing the excitation field according to Lenz’s law.
As a result, the current is displaced to the surface of the conductor, because the eddy
currents will cancel out the excitation current in the centre of the conductor and
enforce the current at the surface. Thus, the current density is decreasing from the
conductor surface to the centre. The current is displaced to the outer surface with
increasing frequency, until the current is flowing only on a narrow skin of the
conductor surface. Therefore, this effect is called skin effect.
The characteristic penetration depth, also skin depth, describes the penetration of a
conductor by the magnetic flux. The skin depth is the distance at which the amplitude
of the electromagnetic wave traveling in a lossy conductor is reduced to the
normalised factor 1/e [1]:
Eq. 3-8
2
𝛿=√
𝜔𝜇0 𝜇𝑟 𝜎
ω is the angular frequency of the applied magnetic field, μr is the relative
permeability and σ is the electrical conductivity of the conductor. Eq. 3-8 can be
derived with the Helmholtz equation - a second order ordinary differential equation
describing a one dimensional field distribution (see [1]).
Due to the described displacement effect, the current distribution inside the
conductor is not uniform. This means that the effective resistance of the conductor is
increased due to a smaller effective conductor cross section. This resistance increase
will also lead to additional losses, which increase frequency dependent.
5
This section is taken from the ECPE Joint Research Report “Characterization of Magnetic
Materials” and modified.
49
Litz Wire
For inductor and transformer designs, the skin effect can be reduced by means of litz
wire. A litz wire consists of a bundle of insulated wires twisted against each other.
The single wires or strands are twisted in a way that each conductor will occupy every
position several times in the total wire cross section along the conductor length. The
nominal diameters of the strands are in a range of 0.04 to 0.5 mm. The idea is to
replace a single solid conductor by several strands using a smaller conductor cross
section - but resulting in an equivalent total cross section according to the single solid
wire. If the strand diameter is chosen in the range of the skin depth, the current in
the conductor cross section will become a more uniform distribution. In other words,
the surface area used by the current will be increased without increasing the
conductor size. Thus, the displacement effect related to the skin effect will have a
lower impact. In practice, it is possible to choose the litz wire in a way to reduce the
skin losses in a component to a negligent minimum - compared to the other loss
components.
Figure 3-4: Skin effect for solid (left) and litz wire (right), where the shaded area indicates the
skin depth – figure based on representation from [1]
3.2.2 Proximity Effect6
The proximity effect occurs if a conductor is penetrated by a time-varying magnetic
field. The time-varying magnetic field will induce circulating currents (eddy currents),
similar to the skin effect. But the magnetic field is caused by currents flowing in
nearby conductors, instead. High frequency currents cause magnetic fields inducing
voltages in adjacent conductors, which in turn cause eddy currents in these adjacent
conductors.
6
This section is taken from the ECPE Joint Research Report “Characterization of Magnetic
Materials” and modified.
50
The skin effect can only occur if the observed conductor is carrying current. In
comparison, the proximity effect occurs whether or not there is a current flow in the
observed conductor, because the proximity effect is caused by high frequency
magnetic fields excited by currents in adjacent conductors. If the observed conductor
carries current, both the skin and the proximity effect will occur. Both effects are
orthogonal and can be considered separately [1]. The impact of the proximity effect
depends on frequency, conductor shape, size and arrangement as well as spacing.
Thus, the mechanism of the proximity effect is quite complex. In [58] two test setups
for the investigation of skin and proximity effects are presented in order to
characterise the impedance of litz wires.
3.2.3
High Frequency Losses in Round Solid Wires
Skin Losses
As mentioned previously, the AC losses can be separated into skin and proximity
related losses. Therefore, the skin losses can be calculated according to the frequency
dependent effective resistance of the wire. The proximity losses, which are caused by
external magnetic fields, can be described by a dissipation factor, as done in [58].
The AC resistance related to the skin effect can be calculated according to [58]:
Eq. 3-9
1
𝐼0 (𝑥𝐿 )
𝑅𝑎𝑐 = 𝑅𝑑𝑐 𝑅𝑒 {𝑥𝐿
}
2
𝐼1 (𝑥𝐿 )
Rdc is the DC resistance of the wire. I0 and I1 are modified Bessel functions of the first
kind of orders 0 and 1 [56]. The parameter xL is the complex skin depth normalised to
the wire diameter dL [58]:
Eq. 3-10
𝑥𝐿 = (1 + 𝑗)√𝜋𝑓𝜇0 𝜎𝐿
𝑑𝐿
2
σL is the specific conductivity of the conductor and f the observed frequency. The
term μr from Eq. 3-8 is replaced by the factor 1, because the most common electrical
conductors (copper and aluminium) are diamagnetic or paramagnetic - having a
relative permeability of about unity (copper: 0.9999936, aluminium: 1.000022).
51
Sometimes it is advantageous to calculate the AC to DC resistance ratio, which
expresses the AC resistance increase compared to the DC resistance of the given
wire.
Eq. 3-11
𝐹𝑎𝑐,𝑑𝑐 =
𝑅𝑎𝑐 1
𝐼0 (𝑥𝐿 )
= 𝑅𝑒 {𝑥𝐿
}
𝑅𝑑𝑐 2
𝐼1 (𝑥𝐿 )
In order to evaluate the power loss, it is necessary to calculate the frequency
spectrum of the given current. The resistance from Eq. 3-9 must be evaluated for
each frequency occurring in the spectrum – which is not feasible. The calculation
must be interrupted at a certain harmonic. For common current wave forms (e.g.
triangular current), occurring in power electronic applications, the magnitudes of the
harmonics are strongly decreasing for higher harmonics. The calculation of many
harmonics is usually not necessary and the skin power losses can be calculated
according to Eq. 3-9.
𝑚
Eq. 3-12
𝑃𝑠𝑘𝑖𝑛
𝐼0 (𝑥𝐿 (𝑖 ∙ 𝑓𝑔 ))
1
= 𝑅𝑑𝑐 ∙ ∑ 𝑅𝑒 {𝑥𝐿 (𝑖 ∙ 𝑓𝑔 )
} ∙ 𝑖𝑖2
2
𝐼1 (𝑥𝐿 (𝑖 ∙ 𝑓𝑔 ))
𝑖
fg is the fundamental frequency of the current and ii is the i-th magnitude in the
current spectrum.
Proximity Losses
The evaluation of the proximity losses is more complex, because the average
magnetic field intensity for each harmonic and for each conductor is required. The
proximity losses caused by external magnetic field intensities can be expressed by
[58]:
Eq. 3-13
𝑃𝑃𝑟𝑜𝑥 =
𝑙
̂2 ∙ 𝐷
∙𝐻
(𝑥 )
𝜎𝐿 𝑒𝑥𝑡 𝑝𝑟𝑜𝑥 𝐿
l is the effective conductor length, σL is the specific conductivity, Hext the average
external field intensity applied to the conductor and Dprox is the proximity dissipation
factor, which is given by [58]:
Eq. 3-14
𝐷𝑝𝑟𝑜𝑥 = 2𝜋 ∙ 𝑅𝑒 {𝑥𝐿
𝐼1 (𝑥𝐿 )
}
𝐼0 (𝑥𝐿 )
52
xL is the complex skin parameter according to Eq. 3-10. By splitting the winding in n
sections and considering m harmonics of the average magnetic field intensity, the
proximity losses can be described by:
Eq. 3-15
3.2.4
𝑃𝑝𝑟𝑜𝑥
𝑛
𝑚
𝑗
𝑖
𝐼1 (𝑥𝐿 (𝑖 ∙ 𝑓𝑔 ))
2𝜋
2
̂𝑒𝑥𝑡,𝑖,𝑗
=
∙ ∑ ∑ 𝑙𝑗 ∙ 𝐻
∙ 𝑅𝑒 {𝑥𝐿 (𝑖 ∙ 𝑓𝑔 )
}
𝜎𝐿
𝐼0 (𝑥𝐿 (𝑖 ∙ 𝑓𝑔 ))
High Frequency Losses in Litz Wires
Skin Losses
The AC resistance of litz wires is [58]:
Eq. 3-16
1
𝐼0 (𝑥𝑠 )
𝑑𝑠2 𝐼1 (𝑥𝑠 )
𝑅𝑎𝑐 = 𝑅𝑑𝑐 𝑅𝑒 {𝑥𝑠 [
+ 𝑁𝑠 (𝑁𝑠 − 1) 2
]}
2
𝐼1 (𝑥𝑠 )
𝑑𝐿 𝐼0 (𝑥𝑠 )
Ns is the number of strands, where ds is the strand diameter and dL is the total wire
diameter. xs is the complex skin parameter similar to Eq. 3-10 [58]:
Eq. 3-17
𝑥𝑠 = (1 + 𝑗)√𝜋𝑓𝜇0 𝜎𝑠
𝑑𝑠
2
σs is the specific conductivity of the wire and f is the frequency, similar as in Eq. 3-10.
Similar to Eq. 3-11, the AC to DC resistance ratio can be expressed for litz wires, too:
Eq. 3-18
𝐹𝑎𝑐,𝑑𝑐 =
𝑅𝑎𝑐 1
𝐼0 (𝑥𝑠 )
𝑑𝑠2 𝐼1 (𝑥𝑠 )
= 𝑅𝑒 {𝑥𝑠 [
+ 𝑁𝑠 (𝑁𝑠 − 1) 2
]}
𝑅𝑑𝑐 2
𝐼1 (𝑥𝑠 )
𝑑𝐿 𝐼0 (𝑥𝑠 )
The calculation of the losses related to the skin effect can be calculated by:
𝑚
𝑃𝑠𝑘𝑖𝑛 = 𝑅𝑑𝑐
Eq. 3-19
1
∙ ∑ 𝑅𝑒 {𝑥𝑠 (𝑖
2
𝑖
∙ 𝑓𝑔 ) [
𝐼0 (𝑥𝑠 (𝑖 ∙ 𝑓𝑔 ))
𝐼1 (𝑥𝑠 (𝑖 ∙ 𝑓𝑔 ))
53
+ 𝑁𝑠 (𝑁𝑠 − 1)
𝑑𝑠2 𝐼1 (𝑥𝑠 (𝑖 ∙ 𝑓𝑔 ))
]} ∙ 𝑖𝑖2
𝑑𝐿2 𝐼0 (𝑥𝑠 (𝑖 ∙ 𝑓𝑔 ))
Proximity Losses
The proximity losses can be calculated according to [58]:
Eq. 3-20
𝑃𝑃𝑟𝑜𝑥 =
𝑙
̂2 ∙ 𝑁 ∙ 𝐷
∙𝐻
(𝑥 )
𝜎𝑠 𝑒𝑥𝑡 𝑠 𝑝𝑟𝑜𝑥 𝑠
The proximity dissipation factor is defined similar to Eq. 3-14 [58]:
Eq. 3-21
𝐷𝑝𝑟𝑜𝑥 = 2𝜋 ∙ 𝑅𝑒 {𝑥𝑠
𝐼1 (𝑥𝑠 )
}
𝐼0 (𝑥𝑠 )
Taking into account that the magnetic field intensity depends on the frequency and
the position, the total proximity losses can be calculated according to:
Eq. 3-22
𝑃𝑝𝑟𝑜𝑥
𝑛
𝑚
𝑗
𝑖
𝐼1 (𝑥𝑠 (𝑖 ∙ 𝑓𝑔 ))
2𝜋
2
̂𝑒𝑥𝑡,𝑖,𝑗
=
∙ ∑ ∑ 𝑙𝑗 ∙ 𝐻
∙ 𝑁𝑠 ∙ 𝑅𝑒 {𝑥𝑠 (𝑖 ∙ 𝑓𝑔 )
}
𝜎𝑠
𝐼0 (𝑥𝑠 (𝑖 ∙ 𝑓𝑔 ))
3.2.5 Improved Power Loss Calculation for Litz Wires
The calculation of high frequency related losses caused by the skin and proximity
effect in litz wires was presented in the previous section. Eq. 3-19 and Eq. 3-20
represent the losses in litz wires under the consideration of ideal AC resistances and
ideal dissipation factors. In practice, the measured AC resistances and dissipation
factors of real litz wires differ from the ideal calculated values. The impedance of a
litz wire is not only dependent on copper cross section, wire length, conductance and
temperature. The strand diameter as well as the number of strands is taken into
account, too. But there are other parameters, like the pitch of layers and the exact
configuration of the bundles, which may influence the impedance behaviour of the
wire. The exact impact of the not recognised parameters is not known yet.
A method to evaluate the impedance of a litz wire more realistic is presented by [58].
Therefore, [58] calculates the resistance of an ideal litz wire and an equivalent solid
wire with the same outer diameter. The AC resistance of the real litz wire was
measured with an impedance analyser and a special measurement setup for the skin
and the proximity effect – explained in [58], too. The measured curve is fitted via a
linear combination of the calculated impedance of the ideal litz and solid wire using a
54
fitting factor. The AC resistance and the proximity dissipation factor can be calculated
according to [58]:
Eq. 3-23
𝑅𝑎𝑐,𝑙𝑖𝑡𝑧 = 𝜆𝑠𝑘𝑖𝑛 𝑅𝑎𝑐,𝑖𝑑𝑒𝑎𝑙 + (1 − 𝜆𝑠𝑘𝑖𝑛 )𝑅𝑎𝑐,𝑠𝑜𝑙𝑖𝑑
Eq. 3-24
𝐷𝑝𝑟𝑜𝑥,𝑙𝑖𝑡𝑧 = 𝜆𝑝𝑟𝑜𝑥 𝐷𝑝𝑟𝑜𝑥,𝑖𝑑𝑒𝑎𝑙 + (1 − 𝜆𝑝𝑟𝑜𝑥 )𝐷𝑝𝑟𝑜𝑥,𝑠𝑜𝑙𝑖𝑑
The ideal AC resistance of the litz wire (Rac,ideal) is calculated according to Eq. 3-16,
where the equivalent solid wire AC resistance (Rac,solid) is calculated with Eq. 3-9. The
dissipation factors are calculated with Eq. 3-14 and Eq. 3-21. Both wires must have
the same diameter as well as the same DC resistance. This is a conflict, because a
solid wire with the same diameter as a litz wire has a much larger conductor cross
section. The litz wire consists of bundled, twisted, insulated strands. The insulation as
well as the twisting of the strands requires space, resulting in the fact that either the
conductor cross section or the conductor diameter can be equivalent. In [58] this
problem is solved by scaling the specific conductance of the solid wire:
Eq. 3-25
𝜎𝐿 = 𝜎𝑐𝑢
𝑑𝑠2 𝑁𝑠
𝑑𝐿2
Replacing the conductance of the solid wire by Eq. 3-25 enables to fulfil both
requirements: Both wires have the same DC resistance as well as the same diameter.
In [58] Eq. 3-26 is proposed to calculate the total wire diameter, where ds’ is the
distance between two adjacent strands (recognising the insulation and the twisting).
Eq. 3-26
2
𝑑𝐿 = 𝑑𝑠′ √ √3𝑁𝑠
𝜋
The factors λskin and λprox can be interpreted as some kind of quality factors,
expressing how ideal the impedance of the litz wire behaves. Figure 3-5 and Figure
3-6 show the AC resistance and the dissipation factor of a litz wire according to the
previously explained procedure evaluated by [59]. It can be seen that both diagrams
show an inflexion point, where resistance and dissipation factor of the solid wire
become lower than for the litz wire. For the AC resistance, this effect can be
explained by the internal proximity effect of the litz wire. The litz wire consists of
several strands, each conducting current. Each strand inside the wire will induce eddy
55
currents in adjacent strands. The skin effect and this internal proximity effect cannot
be measured separately. Therefore, [58] defines the skin losses as the sum of the
losses caused by the skin effect and the internal proximity effect.
For the proximity dissipation factor, the steeper slope of the litz wire can be
explained by the number of strands, too. The twisting of the strands enables that the
magnetic fields inside the conductor can partially cancel out. But the external
proximity effect is becoming more impact with the increase of the number of strands,
because there are more conductors, each enabling the flow of eddy currents caused
by external fields.
However, the mechanisms of AC losses in windings are quite complex, making an
evaluation which wire suits best for a given application difficult. E.g. [60] tried to
calculate the optimum number of strands in litz wires for transformer applications,
where [61] compares the AC to DC resistance ratios of litz and solid wires. But the
interpretation of the AC resistances, dissipation factors or AC to DC resistance ratios
is not that simple. These specific values are used to calculate the AC winding losses of
a component. Finally, only the total loss balance can answer the question which wire
technology suits best. The comparison of AC resistances or dissipation factors is
sometimes not sufficient, because other consideration like the possible copper fill
factors for the different wire technologies have to be taken into account, too. Even if
the resistance and dissipation factors of a litz wire are lower compared to a solid
wire, the solid wire offers the potential to increase the conductor cross-section. The
solid wire enables higher fill factors, resulting in a DC resistance decrease. Even if the
AC losses may increase, this loss increase can be over compensated by a DC loss
decrease. In this case the total losses will decrease by the application of solid wires.
Thus, the correct choice of a suitable wire is not only dependent on the frequency,
but on the AC current and AC magnetic field intensities, too. For low AC currents and
leakage fields, the application of solid wires can be applicable, even for higher
frequencies in the 100 kHz range (e.g. ring core inductors).
56
Figure 3-5: AC resistance for 54x0.2 mm litz wire dependent on frequency (green: ideal litz
wire; red: equivalent solid wire; yellow: measured AC resistance; blue: approximated
measurement curve) – figure taken from [59]
Figure 3-6: Dissipation factor for 54x0.2 mm litz wire dependent on frequency (green: ideal
litz wire; red: equivalent solid wire; yellow: measured dissipation factors; blue:
approximated measurement curve) – figure taken from [59]
57
3.3 Core Losses
3.3.1 Core Loss Mechanisms7
Hysteresis losses
Figure 3-7 shows a typical magnetisation curve of a ferromagnetic core, where B is
the induction and H is the applied magnetic field. If the magnetic field intensity is
increased from zero to a certain value, energy supplied by an external circuit is
transferred to the magnetic core. This energy is equivalent to the area enclosed by
the ordinate and the hysteresis loop, as depicted in Figure 3-7 a). If the field intensity
decreases to zero, the energy is fed back to the electric circuit, as shown in Figure 3-7
b). But the returned energy is lower than the supplied energy. The difference energy
is the so called hysteresis loss. These hysteresis losses are used to align and rotate
magnetic domains inside the core. To overcome the friction to rotate and align the
magnetic domains, this energy is dissipated as heat.
B
B
Energy Out
Energy In
Hysteresis
Losses
H
a)
H
b)
Figure 3-7: Typical hysteresis loop for a ferromagnetic core – figure based on representation
from [1]
In transformers or inductors the hysteresis loop is traversed the same number of
times per second as the frequency of the applied current. The loss energy is
proportional to the area enclosed by the hysteresis loop. Therefore, many cores are
7
This section is taken from the ECPE Joint Research Report “Characterization of Magnetic
Materials” and modified.
58
made of soft magnetic materials, which have a narrow hysteresis, to reduce the
energy loss per cycle caused by the hysteresis. The energy loss density or energy loss
per core volume Vc is given by [1]:
Eq. 3-27
𝑊𝐻
= ∫ 𝐻 ∙ 𝑑𝐵
𝑉𝑐
Eddy Current Losses
All iron-based magnetic materials have not only a high magnetic conductivity, but are
also more or less electrical conductors. Due to Lenz’s law, a time changing magnetic
flux will induce voltages, which drive circulating currents inside the core. The current
flow inside the core will cause the so called eddy current losses. The eddy currents
will flow on circular paths normal to the direction of the flux. The magnetic field
generated by them is opposing the applied excitation field. The field generated by the
eddy currents and the exciting magnetic field will superimpose and the resulting total
field intensity will decrease. This effect will increase with the applied frequency. As a
result, the current will be displaced to the surface of the core. This phenomenon is
equivalent to the skin effect.
If the skin depth is small compared to the cross sectional dimensions of the core, the
applied magnetic field is displaced to the surface of the core. This means, the inside
of the core carries only a small amount of the flux and the AC reluctance of the core
increases. The main function of the core to provide a low reluctance path to guide
the flux is weakened.
The eddy current losses depend significant on the resistance of the core material.
Materials with high electrical resistance tend to have relatively low eddy current
losses, because the resistance will oppose the current flow inside the core.
To prevent the effect of eddy current losses inside magnetic cores, the materials are
designed in a way, that they provide only low electrical conductance. The
manufacturers have the possibility to reduce the conductance for example by adding
non-conductive elements to the material. Other materials based on iron can be
produced as sheets, insulated against each other and assembled to a stack. The
insulation of the sheets avoids the current flow between adjacent sheets [62].
Therefore, only low voltages per sheet can be induced and only small current loops,
limited to a single sheet, are assigned.
59
Relaxation Losses
The publication [62] concludes that the relaxation losses occur during a phase of
constant flux. So the applied winding voltage is zero, according to Eq. 2-6.
The hysteresis and eddy current related losses can only occur if a time changing flux
is applied. But [62] and [63] show that power losses can occur even if the applied
voltage is zero and the resulting flux is kept constant.
Figure 3-8 is showing a rectangular voltage wave form with zero voltage periods
resulting in constant flux within these periods. These voltage waveforms are typical
for transformers in DC-DC converters with galvanic isolation, for example push-pull
converters.
v(t)
φ(t)
t
Figure 3-8: Typical rectangular voltage wave form with zero voltage periods – figure taken
from [56]
The precise interaction mechanism of the relaxation processes is not completely
understood [64]. In [64] relaxation processes are described by the reestablishment of
a thermal equilibrium that was altered by an external force. If the thermal
equilibrium of the system is disturbed, the systems will try to establish a new
equilibrium. These interactions are governed by the response of atomic magnetic
moments and are interactions of these atomic moments with themselves or with the
lattice. The relaxation process can be described by the Landau-Lifshitz equation [64].
The equation expresses the rate of change of an angular momentum, reduced by a
frictional term that opposes the direction of motion. It describes how the system
establishes the new equilibrium that is achieved by the rearrangement of the
magnetic domains to reach a state of lower energy.
The magnetic relaxation changes the magnetisation even if the applied field is kept
constant. As a consequence, residual losses will occur during a period of constant
applied field. The relaxation losses will become important in applications where the
magnetisation changes rapidly such as high frequency applications [62].
60
3.3.2 Steinmetz Equation8
The Steinmetz equation is named for P. Steinmetz, who discovered the effect of
hysteresis losses in magnetic cores in 1890 [65]. Today the so called Steinmetz
equation is described as in Eq. 3-28, where P. Steinmetz did not recognise the
frequency dependency in his publication.
Eq. 3-28
𝑃𝑣 = 𝑘 ∙ 𝑓 𝛼 ∙ 𝐵𝛽
The Steinmetz equation is used to calculate the time average specific core losses per
unit volume (e.g. in mW/cm³ = kW/m³) or per unit weight (W/kg), where k is a
constant factor with the corresponding unit. α and β are unit-less material constants.
The frequency is usually referred to 1 Hz or 1 kHz, where the induction is referred to
1 T. For the fitting of the material dependent parameters k, α and β, it is important to
notice the reference values, because these material constants are only valid in the
context of the given reference values.
Eq. 3-28 is the simplest equation for the description of specific core losses. The
material dependent parameters can be obtained very easy and the calculation is
quite simple. But the Steinmetz equation has some inherent disadvantages:



Eq. 3-28 is used to fit core loss curves evaluated for sinusoidal excitation as
given in the data sheets of the material manufacturers. But in many power
electronic applications the flux waveforms are not sinusoidal.
Only induction and frequency dependency of the material is taken into
account. But several publications show that the specific core losses also
depend on the inductance change (dB/dt) ( [66], [67], [68], [62], [63]) as well
as on the DC pre-magnetization ( [69], [70], [71], [72], [73], [74]) of the
material.
The fitting of the specific core losses with Eq. 3-28 leads to limited accuracy,
even for small data areas.
Figure 3-9 shows the specific core losses of a ferrite material and the corresponding
-6
fitting with Eq. 3-28, where k=1.39∙10 mW/cm³, α=2.08 and β=2.83. The fitting was
performed for 50 mT and the frequency range above 100 kHz. It can be seen that the
measured 50 mT curve is well fitted for frequencies above 100 kHz (error about 7%).
8
This chapter is based on a similar chapter previously published in the ECPE Joint Research
Report “Characterization of Magnetic Materials”.
61
But for other flux densities and other frequency ranges Eq. 3-28 is leading to
significant deviations.
Figure 3-9: Example for core loss fitting according to Steinmetz equation using data from [75]
A significant improvement of the fitting is possible if the measured data is split up in
several small sections [66] and an extra set of Steinmetz parameters is applied to
each section [56]. Therefore, only small sections from the measured curves must be
described by Eq. 3-28. Though this measure leads to higher accuracy for each section,
the effort to describe a complete loss characteristic of a given material increases
significant [56].
Temperature Dependency
The specific core losses of some materials (e.g. ferrites) are temperature dependent.
Therefore, Eq. 3-28 is enhanced by a parabola term as done in Eq. 3-29, because the
temperature dependency is usually of parabolic behaviour. Parabolas of higher order
can be established as well, e.g. in order to improve the accuracy of the
approximation. But in this case it has to be taken into account that the formula is only
valid within the recognised data area. Outside this area the formula is not
representing the correct physical behaviour and estimations are not allowed
anymore.
Eq. 3-29
𝑃𝑣 = 𝑘 ∙ 𝑓 𝛼 ∙ 𝐵𝛽 ∙ (𝑎 ∙ 𝑇 2 + 𝑏 ∙ 𝑇 + 𝑐)
62
Enhancements of the Steinmetz Equation
As mentioned in the previous section, Eq. 3-28 does not take the dB/dt or the DC premagnetisation dependency of the core losses into account. Therefore, several
enhancements of the Steinmetz equation and other loss models were developed over
the years:








[66] presents the so called Modified Steinmetz Equation (MSE), taking the
dB/dt dependency of the core losses into account.
[76] proposes several correction factors for the calculation of core losses,
recognising common flux waveforms in power electronic applications.
[68] denotes several disadvantages of the MSE (e.g. underestimation of
losses for low fundamental frequencies). [68] introduces the so called
Generalized Steinmetz Equation (GSE).
[68] itself recognises a limited accuracy for waveforms containing harmonic
content over a wide frequency range if using the GSE. [77], [78] and [49]
develop the improved Generalized Steinmetz Equation (iGSE) independently
to overcome the drawbacks of the previous Steinmetz loss models. The iGSE
takes the dB/dt dependency of the core losses into account, too. It is the
most advanced Steinmetz based loss model if only loss data from the
manufacturer’s data sheets are available.
[62] introduces the i²GSE in order to enable the calculation of relaxation
losses, where the method from [62] requires additional measurements for
the evaluation of further material dependent parameters.
[79] presents a method to recognise the DC pre-magnetisation dependency
of the core losses, where the compatibility to the i²GSE from [62] is unknown
[79].
Another core loss model for ferrite materials is the Ridley-Nace model [80].
DC pre-magnetisation or dB/dt dependency is not taken into account. [80]
tries to improve the accuracy of the Steinmetz equation by introducing a
frequency dependency for the material parameter k and the flux exponent
th
β. Furthermore, [80] uses a polynomial of 5 order to enhance the accuracy
of the temperature dependency.
[81] publishes the Oliver Model to calculate the core losses of iron powder
materials by two different terms taking the hysteresis and the eddy current
losses into account. Nevertheless, [81] does not recognise any dB/dt or DC
63

pre-magnetisation dependency. The suitability for other materials is not
verified [56].
[82], [56] and [83] present a fitting algorithm based on the Karhunen-Loève
transform in order to model the core losses for inductors, excited with
rectangular voltages and DC pre-magnetisation. Previous, [84] used this
fitting algorithm to model empirical data of electro-physical and chemical
excavation processes.
3.3.3 Modelling Core Losses by Means of Orthogonal Vector Functions
Today many core loss models are based on the Steinmetz equation. The models were
enhanced several times in order to recognise effects like dB/dt or DC bias dependency
of the core losses. At first glance, the mathematical approach of the Steinmetz
equation does not seem to be wrong. But the bottleneck of these models is the
assumption that the material specific parameters are constant values. This might be
the reason why it is almost impossible to calculate the core losses with such a model
over a wide parameter area. Instead, it is probable that the material specific
parameters, used to scale the influencing factors, are not constant, but depend on
these influencing factors themselves.
The use of a mathematical algorithm based on the Karhunen-Loève transform to fit
core loss data is proposed by [82] (see also [85], [86]). This algorithm decomposes a
discrete data set into its eigenvectors. Large data sets can be structured, simplified
and displayed by means of linear combinations of orthogonal functions [83]. The
method requires a full experimental design with two influencing factors. This method
is enhanced by [82] for the analysis of data sets with an arbitrary number of
influencing factors greater than two (see also [83]).
influencing factor 1 →
influencing
factor 2 ↓
x1
x2
x3
…
xn
z1
y11
y12
y13
…
y1n
z2
y21
y22
y23
…
y2n
z3
y31
y32
y33
…
y3n
…
…
…
…
…
…
zm
ym1
ym2
ym3
…
ymn
Table 3-1: Full experimental design – representation taken from [82] and modified
64
For the investigation of core loss data [82] recommends to assign the data to the
logarithmic domain to enable a better approximation.
Eq. 3-30
𝑦 = 𝑙𝑜𝑔 (
𝑃𝑣 (𝐵, 𝑓)
)
𝑘𝑊/𝑚³
The dataset can be centred by subtraction of the arithmetic average in the
logarithmic domain in order to minimise the influence of the error, because the
average of the error will be subtracted, too [83], [82].
Eq. 3-31
1
𝑌 ∗ = 𝑌 − 𝐸[𝑌] = 𝑌 − ∑ 𝑦𝑖
𝑛
𝑖
The mathematical structure is reproduced by a product approach:
Eq. 3-32
𝑌∗ = 𝐴 ∙ 𝐶 𝑇
A and C are sets of eigenvectors of the matrix Y, containing the discrete data set.
A∙CT denotes the dyadic product (also called tensor product) [82]. The optimisation
of this product approach requires the minimisation of the empirical variance [82].
𝑚
𝑛
2
Eq. 3-33
∑ ∑(𝑦𝑖𝑗 − 𝑎𝑖 ∙ 𝑐𝑗 )
𝑖
𝑗
Differentiation of Eq. 3-33 yields the required minimum:
Eq. 3-34
𝑎𝑖 =
∑𝑛𝑗 𝑦𝑖𝑗 ∙ 𝑐𝑗
∑𝑛𝑗 𝑐𝑗2
Eq. 3-35
𝑐𝑗 =
∑𝑚
𝑖 𝑦𝑖𝑗 ∙ 𝑎𝑖
2
∑𝑚
𝑖 𝑎𝑖
Insertion of Eq. 3-34 into Eq. 3-35 and vice versa leads to the following eigenvalue
problem [82]:
Eq. 3-36
𝜆 ∙ 𝐴 = 𝑌∗ ∙ 𝑌∗𝑇 ∙ 𝐴
65
Eq. 3-37
𝜆 ∙ 𝐶 = 𝑌∗ ∙ 𝑌∗𝑇 ∙ 𝐶
λ denotes the eigenvalues and is given by [82]:
𝑚
Eq. 3-38
𝜆=
𝑛
∑ 𝑎𝑖2
∑ 𝑐𝑗2
𝑖
𝑗
The eigenvalue problem has at least r=min(n,m) eigenvalues and eigenvectors.
Finally, the data set can be described by the sum of the dyadic products:
𝑟
Eq. 3-39
∗
𝑌 = ∑ √𝜆𝑣 ∙ 𝐴𝑣 ∙ 𝐶𝑣𝑇
𝑣
It has to be noticed that the eigenvectors Av and Cv in Eq. 3-39 are normalised equal
to one and weighted by their eigenvalues.
Eq. 3-40
√
𝐴𝑣 ∙ 𝐴𝑣
𝐶𝑣 ∙ 𝐶𝑣
=√
=1
𝜆𝑣
𝜆𝑣
The inverse transformation of the data is done by Eq. 3-41, where the previously
subtracted arithmetic average E[Y] must be added again:
Eq. 3-41
𝑟
𝑇
𝑌 ≈ 10𝐸[𝑌]+∑𝑣 √𝜆𝑣∙𝐴𝑣 ∙𝐶𝑣
In order to enable a continuous model, the eigenvectors can be fitted by polynomial
functions:
𝑚
Eq. 3-42
𝑓(𝑥) = ∑ 𝑘𝑛 ∙ 𝑙𝑜𝑔(𝑥)𝑛
𝑛=0
Example
As an example, the core loss data for the ferrite material N87 (compare Figure 3-9)
will be modelled according to the previous described algorithm. Table 3-2 shows a
full experimental design chosen for the model.
66
influencing factor 1 →
influencing
factor 2 ↓
0.025 T
0.05 T
0.1 T
100000 Hz
1.34 mW/cm³
7.41 mW/cm³
50.06 mW/cm³
200000 Hz
5.55 mW/cm³
29.16 mW/cm³
170.45 mW/cm³
300000 Hz
14.97 mW/cm³
71.78 mW/cm³
383.46 mW/cm³
400000 Hz
29.69 mW/cm³
134.86 mW/cm³
682.52 mW/cm³
Table 3-2: Full experimental design for specific core loss data from N87 – data extracted from
[75]
Transformation into the logarithmic domain and subtraction of the arithmetic
average (E[Y] = 1.599) leads to (use Eq. 3-30 and Eq. 3-31):
influencing factor 1 →
influencing
factor 2 ↓
0.025 T
0.05 T
0.1 T
100000 Hz
-1.472
-0.729
0.101
200000 Hz
-0.854
-0.134
0.633
300000 Hz
-0.424
0.257
0.985
400000 Hz
-0.126
0.531
1.235
Table 3-3: Centred data in the logarithmic domain for specific core loss data from N87
The transformed data from Table 3-3 leads to the following eigenvalue problem
according to Eq. 3-36:
Eq. 3-43
3.093
𝜆 ∙ 𝐴 = 𝑌 ∗ ∙ 𝑌 ∗ 𝑇 ∙ 𝐴 = ( 1.012
−1.262
1.012
0.897
0.75
The resulting eigenvalues are:
Eq. 3-44
𝜆1
5.904 ∙ 10−5
𝜆 = (𝜆2 ) = ( 4.283 )
𝜆3
2.612
The corresponding normalised eigenvectors are:
67
−1.262
0.75 ) ∙ 𝐴
2.905
Eq. 3-45
0.798
−1.567
3.048 ∙ 10−3
𝐴1 = (−5.803 ∙ 10−3 ) , 𝐴2 = (−0.172) , 𝐴3 = (0.931)
1.052
1.341
2.826 ∙ 10−3
Eq. 3-46
−1.083
3.133 ∙ 10−3
1.24
−3
−5.648
∙
10
1.068
𝐶1 = (
) , 𝐶2 = (
) , 𝐶3 = (−0.087)
0.938
0.58
−4.711 ∙ 10−4
0.853
1.047
3.026 ∙ 10−3
Figure 3-10 depicts the normalised eigenvectors according to Eq. 3-45 and Eq. 3-46
dependent on the corresponding influencing factors. The eigenvalue λ1 is almost zero,
resulting in the fact that the corresponding eigenvectors A1 and C1 have no significant
influence on the data set. This means, these eigenvectors can be neglected for the
approximation.
Figure 3-10: Normalised eigenvectors for N87 data set
According to Eq. 3-39, the data set can be restored by Eq. 3-47:
3
𝑌 = 𝐸[𝑌] + ∑ √𝜆𝑣 ∙ 𝐴𝑣 ∙ 𝐶𝑣𝑇
Eq. 3-47
𝑣=2
= 1.599 + √4.283 ∙ 𝐴2 ∙ 𝐶2𝑇 + √2.612 ∙ 𝐴3 ∙ 𝐶3𝑇
The eigenvectors A2, A3, C2, and C3 can be fitted by the following polynomial
functions:
Eq. 3-48
𝐴2(𝐵) = 0.31 ∙ 𝑙𝑜𝑔(𝐵)2 + 3.141 ∙ 𝑙𝑜𝑔(𝐵) + 3.479
Eq. 3-49
𝐴3(𝐵) = −0.044 ∙ 𝑙𝑜𝑔(𝐵)2 + 0.146 ∙ 𝑙𝑜𝑔(𝐵) + 0.841
68
Eq. 3-50
𝐶2(𝑓) = −0.107 ∙ 𝑙𝑜𝑔(𝑓)2 + 0.824 ∙ 𝑙𝑜𝑔(𝑓) − 0.848
Eq. 3-51
𝐶3(𝑓) = 0.426 ∙ 𝑙𝑜𝑔(𝑓)2 − 2.343 ∙ 𝑙𝑜𝑔(𝑓) + 0.386
Combining Eq. 3-47 to Eq. 3-51 and performing the inverse transform yields:
𝑃𝑣 (𝐵, 𝑓) =
Eq.
3-52
1.599+√4.283∙(0.31∙𝑙𝑜𝑔(𝐵)2 +3.141∙𝑙𝑜𝑔(𝐵)+3.479)∙(−0.107∙𝑙𝑜𝑔(𝑓)2 +0.824∙𝑙𝑜𝑔(𝑓)−0.848) ...
10
…+√2.612∙(−0.044∙𝑙𝑜𝑔(𝐵)2 +0.146∙𝑙𝑜𝑔(𝐵)+0.841)∙(0.426∙𝑙𝑜𝑔(𝑓)2 −2.343∙𝑙𝑜𝑔(𝑓)+0.386)
Figure 3-11 shows the plotted core loss data according Eq. 3-52 in comparison to the
values from the data sheet. The black rhombus indicates the chosen experimental
design according to Table 3-2. The mean deviation for the values of the complete
experimental design is 0.186 mW/cm³. Thus, the calculated model allows an almost
perfect fit of the specific core losses within the area of the experimental design.
Furthermore, Figure 3-11 shows that the calculated core loss curves allow very
accurate approximations of the measured data even if values outside the
experimental design area are calculated.
Figure 3-11: Example for core loss fitting by means of orthogonal vector functions using data
from [75]
69
4
Coupled Inductors
Coupled inductors are commonly used in interleaved converters, where several
converter legs with one inductor for each leg will add up the current at one node. In
order to avoid n discrete inductors, n windings can share one core to make a
multitude of cores redundant. The coupled inductor must be designed in order to
fulfil the current ripple specification of the converter. Therefore, the self-inductance
and the coupling coefficient must be adjusted in a way that electric (e.g. current
ripple) as well as magnetic (avoid saturation) boundary conditions can be fulfilled.
Usually the self-inductance can be adjusted by the number of turns or the inductance
factor of the core. The adjustment of the coupling coefficient can be performed by
the winding arrangement and the insertion of flux bypasses.
The coupled inductor is the simplest integrated magnetic component, because it will
combine several discrete inductors within one. The filter functionality is the only
required function of the component even if the coupling enables transformer
behaviour. Weather the transformer function is advantageous or not depends on the
application. The theory of the coupled inductor will provide the basic principle of
many functionally integrated magnetic components. Each magnetic component with
two or more windings wound on one core will have at least the behaviour of a
coupled inductor. Therefore, the function of coupled inductors will be introduced
subsequent more in detail.
4.1 Direct and Inverse Coupling
Figure 4-1 shows a two leg interleaved bi-directional DC-DC converter, capable to
transfer energy to high voltage or low voltage load. This is a typical application for
coupled inductors. The coupled inductor can be implemented with inverse (see
Figure 4-1 a) or with direct coupling (see Figure 4-1 b). The joining of two discrete
inductors as depicted in Figure 4-2 leads, dependent on the winding sense, to Figure
4-3 a) (inverse coupling) or to Figure 4-4 a) (direct coupling). For inverse coupling, the
excited flux will add up in the centre leg and cancel in the outer legs. The direct
coupling leads to a vice versa behaviour.
70
VL1
L1
L2
VL2
Vlow
VL1
IL1
S3
IL2
S4
S1
L1
L2
VL2
S2
Vlow
Vhigh
a)
IL1
S3
IL2
S4
S1
S2
Vhigh
b)
Figure 4-1: Bi-directional interleaved DC-DC converter using coupled inductor with a) inverse
coupling and b) direct coupling
The magnetic components in Figure 4-3 a) and Figure 4-4 a) show a low permeable
magnetic path (e.g. due to air gaps) at the outer legs and a high permeable path at
the centre leg. The flux excited by the windings placed on the outer legs is shorted by
the centre leg. Only a small amount of the flux will pass through the opposing leg.
Such magnetic structures enable a loose coupling. If the permeance of the outer legs
is much lower than the permeance of the centre leg, the component behaves more
like two discrete inductors.
If the gaps are shifted from the outer legs to the centre leg, a tight coupling can be
introduced. Now the outer legs enable a high permeable magnetic path, where the
centre leg has a lower permeance than the outer legs. A higher amount of flux
excited by one winding can flow in the opposing outer leg.
The combination of direct and inverse coupling as well as the introduction of loose
and tight coupling allows four basic possibilities to design a coupled inductor. In
practice, the coupling should be designed in a way that the coupled inductor can
improve the performance of the given application. If tight or loose coupling is more
appropriate, depends on the application and the operation conditions. Usually the
coupling has to be adjusted in a certain range to enable a proper working coupled
inductor. Therefore, gapping of the centre or the outer legs is possible, as well as
gapping of all legs, as shown in [33].
I1
V1
φ1
φ2
Rg
Rg
Rm
N
Rm
Vm1
I2
N
V2
Vm2
Figure 4-2: Magnetic equivalent circuit of two discrete inductors
71
φ1
I1
V1
φ1
φ1+φ2
φ2
Rg
I2
Rg
Rm
N
N
Vm1
Rm
Rm
φ1+φ2
I1
V1
V2
φ2
Vm1
N
I2
Vm2
Rg
N
V2
Vm2
a)
b)
Figure 4-3: Magnetic equivalent circuit of inverse coupled inductor: a) loose coupling;
b) tight coupling
φ1
I1
V1
φ1
Rg
Rm
N
Vm1
a)
φ1−φ2 φ2
Rg
Rm
φ1−φ2
I1
I2
N
Rm
V1
V2
N
Vm1
Rg
φ2
I2
Vm2
N
V2
Vm2
b)
Figure 4-4: Magnetic equivalent circuit of direct coupled inductor: a) loose coupling;
b) tight coupling
Figure 4-5 shows the typical voltage, flux and current waveforms for direct and
inverse coupling of a loose coupled inductor operated in a DC-DC converter, where
duty cycles greater and lower than 50 % are distinguished. The corresponding wave
forms for direct and inverse coupling for tight coupled inductors can be found in
Figure 4-6.
For a very tight coupling the current slopes are nearby equal for different phases. If
one winding is tied to a voltage source, the coupling enables the excited flux to pass
through the opposing winding, inducing a voltage drop. Energy is transferred from
one winding to the other one, enabling a current rise or fall in this phase. If the
coupling becomes loose, this effect is weakened.
The out- or input current is always the sum of the phase currents and always shows n
times the converter switching frequency (where n is the number of converter
legs/phases), regardless of any coupling conditions. This is valid for discrete inductors
as well.
72
Inverse, Loose Coupling, D<0.5
Inverse, Loose Coupling, D>0.5
VL1
VL2
VL1
VL2
t1
t2
t3
t4
φ1
φ2
φ3
t2
t3
t4
t1
t2
t3
t4
t1
t2
t3
t4
φ1
φ2
φ3
t1
t2
t4
t3
I1
I2
I3
a)
t1
I1
I2
I3
t1
t2
t4
t3
b)
Direct, Loose Coupling, D<0.5
Direct, Loose Coupling, D>0.5
VL1
VL2
VL1
VL2
t1
t2
t3
t4
φ1
φ2
φ3
t2
t3
t4
t1
t2
t3
t4
t1
t2
t3
t4
φ1
φ2
φ3
t1
t2
t4
t3
I1
I2
I3
c)
t1
I1
I2
I3
t1
t2
t4
t3
d)
Figure 4-5: Schematic current and flux waveforms of loose coupled inductors: a) inverse
coupling and D<0.5; b) inverse coupling and D>0.5; c) direct coupling and D<0.5; d) direct
coupling and D>0.5
73
Inverse, Tight Coupling, D<0.5
Inverse, Tight Coupling, D>0.5
VL1
VL2
VL1
VL2
t1
t2
t3
t4
φ1
φ2
φ3
t2
t3
t4
t1
t2
t3
t4
t1
t2
t3
t4
φ1
φ2
φ3
t1
t2
t4
t3
I1
I2
I3
a)
t1
I1
I2
I3
t1
t2
t4
t3
b)
Direct, Tight Coupling, D<0.5
Direct, Tight Coupling, D>0.5
VL1
VL2
VL1
VL2
t1
t2
t3
t4
φ1
φ2
φ3
t2
t3
t4
t1
t2
t3
t4
t1
t2
t3
t4
φ1
φ2
φ3
t1
t2
t4
t3
I1
I2
I3
c)
t1
I1
I2
I3
t1
t2
t4
t3
d)
Figure 4-6: Schematic current and flux waveforms of tight coupled inductors: a) inverse
coupling and D<0.5; b) inverse coupling and D>0.5; c) direct coupling and D<0.5; d) direct
coupling and D>0.5
4.2 Magnetic Equivalent Circuit
The simplest magnetic circuit of a two leg coupled inductor is described by the circuit
given in Figure 4-7 a). The two windings are separated and wound on different legs of
the core, where at least one leg is required to take and balance the flux from the
exciting windings. Dependent on the core shape, more than one balancing leg is
possible. Additionally, there are several parasitic or leakage paths for the magnetic
flux as well. But for simplification all these balancing legs and paths are paralleled and
74
centralised to one leg in Figure 4-7. In general, the magnetic circuit from Figure 4-7 a)
can be enhanced to an arbitrary number of windings. Finally each coupled inductor
can be described by the circuit given in Figure 4-7b).
Rm1
Rm3 Rm2
Vm1
Rm1
Vm2
a)
Vm1
Rm3 Rm2
Rmx
Vm2
Vmx
b)
Figure 4-7: Basic magnetic circuit of coupled inductors: a) two winding configuration;
b) n winding configuration
The self-inductance of the magnetic circuit from Figure 4-7 a) is given by (Rm1=Rm2,
N1=N2=N):
Eq. 4-1
𝐿=
𝑁2
𝑅 𝑅
𝑅𝑚1 + 𝑚1 𝑚3
𝑅𝑚1 + 𝑅𝑚3
The coupling for the magnetic equivalent circuit given in Figure 4-7 a) is (Rm1=Rm2):
Eq. 4-2
𝑘=
1
𝑅𝑚1
1
1
+
𝑅𝑚1 𝑅𝑚3
=
𝑅𝑚3
1
=
𝑅𝑚1 + 𝑅𝑚3 1 + 𝑅𝑚1
𝑅𝑚3
For Figure 4-7 b), Eq. 4-1 and Eq. 4-2 can be rewritten, where n denotes the number
of windings (Rm1=Rm2=Rmn≠Rm3, N1=N2=Nn=N):
𝐿=
Eq. 4-3
𝑁2
𝑅𝑚1 + 𝑛 − 1
1
𝑅𝑚1
Eq. 4-4
+
1
𝑅𝑚3
1
𝑅𝑚1
𝑘=
𝑛−1
1
+
𝑅𝑚1
𝑅𝑚3
75
4.2.1 DC Analysis of Inverse and Direct Coupled Inductors
The DC analysis for the magnetic circuit from Figure 4-7 a) in case of inverse coupling,
assuming that Rm1=Rm2, φ1=φ2 and Vm1=Vm2 is valid, gives the flux in the exciting
legs as follows:
Eq. 4-5
𝜑1,𝑖𝑛𝑣 =
𝑉𝑚1
𝑅𝑚1 + 2𝑅𝑚3
In case of direct coupling the reluctance Rm3 is not conducting DC flux, because φ1
and φ2 will cancel out if φ1=φ2 is valid. The total DC flux for direct coupling is:
Eq. 4-6
𝜑1,𝑑𝑖𝑟 =
𝑉𝑚1
𝑅𝑚1
The ratio of Eq. 4-5 and Eq. 4-6 is:
Eq. 4-7
𝜑1,𝑖𝑛𝑣
𝑅𝑚1
1
=
=
𝜑1,𝑑𝑖𝑟 𝑅𝑚1 + 2𝑅𝑚3 1 + 2𝑅𝑚3
𝑅𝑚1
Figure 4-8 shows the flux ratio of inverse and direct coupled inductors and the
coupling coefficient dependent on the reluctance ratio Rm1/Rm3. If this reluctance
ratio becomes very small (e.g. because the centre leg is gapped and the outer legs are
ungapped), the coupling is getting tighter. For an increased reluctance ratio (e.g.
because the outer legs are gapped and the centre leg is ungapped) the coupling
decreases until the coupling is that loose that the component behaves like two
discrete inductors.
The curve of the flux ratio in Figure 4-8 shows that the flux of direct coupled
inductors is significant increasing compared to their inverse counterparts if the
coupling becomes tighter. Higher flux requires a larger magnetic cross section,
resulting in a larger component. This means that direct coupled inductors are more
suitable for applications, where loose coupling is required. If tight coupling is
required, the inverse coupling is preferred.
For DC-DC converters the current and the flux have a DC component. For direct
coupled inductors using tight coupling (e.g. see Figure 4-4 b) the outer legs provide a
high permeable path, where add up of the excited fluxes in the outer legs may cause
76
saturation. This effect is more critical for high DC currents, making the application of
direct coupling for DC-DC converters often unsuitable.
Figure 4-8: Flux ratio of direct and inverse coupled inductors dependent on the inductor
reluctance ratio
4.3 Potential Performance Advantages of Coupled Inductors
Figure 4-5 and Figure 4-6 denote that the slope of the current or the flux in coupled
inductors is dependent on inductance, coupling values and on the duty cycle. Due to
the coupling, two current slopes and two equivalent inductances can be defined. In
the first case both winding voltages are equal, giving the first equivalent inductance:
Eq. 4-8
𝐿𝑒𝑞1 = 𝐿 + 𝑀 = 𝐿(1 + 𝑘)
Leq1 can be interpreted as the valid inductor during the transient response [33],
which is only dependent on self- and mutual inductance (or coupling).
The second equivalent inductance can be defined in case of different winding
voltages:
Eq. 4-9
𝐿𝑒𝑞2 =
𝐿2 − 𝑀2
1 − 𝑘2
=𝐿
𝐷
𝐷
𝐿+ 𝑀
1+ 𝑘
𝐷′
𝐷′
It has to be noticed that the mutual inductance (or the coupling) is negative for
inverse coupling and positive for direct coupling. The ratio D/D’ is equivalent to the
voltage ratio of the applied winding voltages and can be expressed by the converter
77
duty cycle (D/(1-D) or (1-D)/D). The complete derivation of Eq. 4-8 and Eq. 4-9 is
given in Appendix I.
In the two following chapters, the performance of the coupled inductor is compared
to its discrete counterpart. Only the inverse coupled inductor is taken into account
for the analysis. Two different comparisons of coupled and discrete inductors are
distinguished.
1.
2.
The phase current ripple of the discrete inductor equals the phase current
ripple of its coupled counterpart: This means that the semiconductors of
the circuit operate under equal conditions. Thus, the semiconductor losses
for the discrete and the coupled case are equal. If coupled inductors are
used, the input or output current ripple will increase. A higher total current
ripple requires an increase of the capacitive filter elements to keep the
voltage ripple at the converter in- or output constant.
The in- or output current ripple of the discrete converter equals the in- or
output current ripple of its coupled counterpart: In this case, the required
capacitive in- or output filter is kept constant if the same voltage ripple is
assumed. But the phase current ripple will decrease if a coupled inductor is
used. Therefore, the converter switches will turn on at higher current and
turn off at lower current (and vice versa for the diodes). This results in
higher turn on losses and lower turn off losses for the switches. If the turn
on losses are dominant (depends on the characteristic of the switch), the
semiconductor losses increase.
Finally, an equivalent input/output current ripple of the discrete and the coupled
circuit can result in higher semiconductor and often (if the semiconductor losses are
dominant) higher converter losses. Instead, equivalent semiconductor power loss
behaviour results in increasing EMI or higher required amount of capacitive filters.
4.3.1 Equal Phase Current Ripple of Discrete and Coupled Inductor
The ratio of the equivalent inductance and self-inductance is evaluated graphically in
Figure 4-9, as it is done in [33]. If the ratio of equivalent inductance (Leq2) to selfinductance is greater one, the coupled inductor enables better current ripple
attenuation [33]. As can be seen, operation at higher duty cycle requires tighter
78
coupling. Furthermore, the capability for better current ripple attenuation increases
in this case. Instead, the operation at low duty cycle requires loose coupling and will
not enable significant performance advantage for the coupled inductor.
Figure 4-9: Equivalent inductance to self-inductance ratio dependent on duty cycle and
coupling – representation taken from [33] and modified
The basic assumption for the derivation of Figure 4-9 is that the self-inductance of
the coupled inductor and its discrete counterpart are equivalent. But a coupled
inductor does not have to be the same self-inductance than its discrete counterpart.
Figure 4-10 gives another interpretation of the self-inductance issue of coupled and
discrete inductors. Figure 4-10 is based on the assumption that the self-inductance of
the discrete inductor is equivalent to the equivalent inductance of the coupled
counterpart, in order to enable equivalent phase current ripple attenuation. In this
case Eq. 4-9 can be rewritten to:
Eq. 4-10
𝐷
𝐿𝑠 1 + 𝐷′ 𝑘
=
𝐿𝑑
1 − 𝑘2
Ls is the self-inductance of the coupled inductor, where Ld is the self-inductance of
the discrete inductor. Figure 4-10 shows the required self-inductance for the coupled
inductor dependent on coupling and duty cycle. The interpretation of Figure 4-10 is
similar to Figure 4-9. The converter operation at increased duty cycles makes the
application of tight coupling values appropriate. If the self-inductance ratio of
coupled to discrete inductor becomes less than unity, the coupling is adjusted
appropriate. This means, the coupled inductor requires less self-inductance
79
compared to its discrete counterpart, to enable the same phase current ripple
attenuation. Lower required inductance values may lead to less required core
material and/or number of turns, enabling the potential for reducing size and weight.
Figure 4-10: Self-inductance ratio of coupled and discrete inductors for constant phase
current ripple
4.3.2
Equal Converter Output/Input Current Ripple of Discrete and
Coupled Circuit
If the equality of the converter output or input current ripple is required, the analysis
of the equivalent inductance is not sufficient to evaluate appropriate values for selfinductance and coupling.
The calculation of the output/input current ripple of a two phase interleaved DC-DC
converter in case of coupled inductors is shown in Appendix I. The output/input
current ripple ratio for the discrete and the coupled case can be given as:
Eq. 4-11
∆𝑖𝑜𝑢𝑡,𝑐 𝐿𝑑 1 + 𝑘
𝐿𝑑
1
=
∙
=
∙
2
∆𝑖𝑜𝑢𝑡,𝑑 𝐿𝑠 1 − 𝑘
𝐿𝑠 1 − 𝑘
Eq. 4-11 shows no duty cycle dependency. The output current ripple ratio depends
only on the self-inductance ratio and the coupling. The graphical evaluation of Eq.
4-11 is depicted in Figure 4-11. For the assumption that the self-inductance of a
coupled inductor is equivalent to the self-inductance of its discrete counterpart, the
converter output/input current ripple is always higher if a coupled inductor is used.
The tighter the coupling of the coupled inductor, the higher is the current ripple in
comparison to the discrete converter. This means that the coupled inductor requires
80
a higher self-inductance than its discrete counterpart, to enable at least the same or
better output/input current ripple attenuation.
Figure 4-11: Output current ripple ratio for coupled and discrete interleaved DC-DC
converters dependent on coupling and self-inductance ratio
If the ratio of the converter output/input current ripple for the discrete and the
coupled inductor is set to unity, Eq. 4-11 can be simplified to:
Eq. 4-12
𝐿𝑑
1
=
𝐿𝑠 1 − 𝑘
Eq. 4-12 is evaluated graphically in Figure 4-12. It is easy to see that tight coupling
requires a high self-inductance for the coupled inductor to enable the same output or
input current ripple attenuation.
Figure 4-12: Self-inductance ratio of coupled and discrete inductors for constant output
current ripple
81
4.4 Basic Core Geometries of Coupled Inductors
Subsequent some assembly possibilities as well as the corresponding advantages and
disadvantages for coupled inductors are discussed. There are even more possibilities,
but the subsequent examples given in Figure 4-13 are restricted to component
structures, which can be assembled with standard cores and materials available on
the market.
a)
b)
c)
Figure 4-13: Assembly possibilities for coupled inductors: a) EE core design; b) UU core;
c) ring core
EE Core
E core configurations suit well for two leg coupled inductors, where the windings are
placed on the outer legs. Additional windings must be added with additional core
segments, where the symmetry of the magnetic structure must be conserved for a
proper design. The self-inductance can be adjusted by the number of turns and the
permeance of the core seen by the winding. The coupling coefficient can be adjusted
with the help of the centre leg air gap. The fringing effect of the air gap can be
reduced by filling the gap with powder core segments, as proposed by [49]. The outer
leakage can be reduced by placing additional I or U core segments beside the
windings.
UU and Ring Core
The adjustment of the coupling is more difficult as for the E core configuration,
because of the missing centre leg. Therefore, it is possible to insert magnetic
conductive segments inside the window, where such segments must be usually
manufactured at special request. Another possibility is the displacement of the
winding away from the core and/or to add additional leakage segments. This leads to
a flux linkage decrease between the coupled windings, resulting in increased leakage
inductance. Due to the fact that leakage paths of the flux are a parasitic effect, it is
difficult to adjust the coupling in this way. In practice, several configurations must be
82
tested until the optimum adjustment is found. However, ring and UU cores can be
manufactured with all materials, giving the designer a certain degree of freedom. The
cores can be cut into two halves, in order to adjust the self-inductance.
In practice it is difficult to fulfil the current ripple specifications with a ring core
coupled inductor, because the very tight coupling can lead to very low equivalent
inductance, which is responsible for the current ripple attenuation. The tight coupling
cannot be compensated by a self-inductance increase. UU cores have a higher
leakage, but their suitability must be checked application dependent. Because of the
low leakage inductance of ring cores [87] suggests to use additional discrete
inductors to increase the equivalent inductance. As a drawback n discrete inductors
will be replaced by n+1 magnetic components. Nevertheless, [87] tries to show that
the total size of the coupled inductor and the discrete inductors is smaller in
comparison to the conventional discrete solution.
4.5 Leakage Inductance of Coupled EE Core Inductors
The magnetic equivalent circuit for the EE core structure can be established according
to Figure 4-14. Several leakage reluctances have been taken into account in order to
model the circuit more realistic.



Leakage path beside the outer legs (Rσ)
Leakage path inside the window (Rw)
Leakage path beside the centre leg and fringing effect of the air gap ( Rf)
Rm3
Rm1
Rσ1
Rw1
Rm2
Rg
Rf
Vm1
Rw2
Rσ2
Vm2
Figure 4-14: Magnetic equivalent circuit for EE core coupled inductor
The reluctance values of the outer legs and the corresponding leakage values are
equal. The self-inductance can be calculated according to:
83
1
𝐿𝑠1 = 𝐿𝑠2 = 𝑁 2 ∙
𝑅𝑚1 +
Eq. 4-13
1
1
1
2
2
𝑅𝑔 ∙ 𝑅𝑓 + 𝑅𝑚2 + 𝑅𝑤 + 𝑅𝜎
𝑅𝑚3 +
𝑅𝑔 + 𝑅𝑓
Eq. 4-13 can be expressed as self-permeance, too:
𝑃𝑠1 = 𝑃𝑠2 =
Eq. 4-14
𝐿𝑠1
= 𝑅𝑚1 +
𝑁2
1
1
1
2
2
𝑅𝑔 ∙ 𝑅𝑓 + 𝑅𝑚2 + 𝑅𝑤 + 𝑅𝜎
𝑅𝑚3 +
𝑅𝑔 + 𝑅𝑓
The coupling coefficient is defined by:
Eq. 4-15
𝜑′2 𝜑′′1
𝑘=
=
=
𝜑′1 𝜑′′2
1
𝑅𝑚2
1
1
2
2
𝑅𝑔 ∙ 𝑅𝑓 + 𝑅𝑚2 + 𝑅𝑤 + 𝑅𝜎
𝑅𝑚3 +
𝑅𝑔 + 𝑅𝑓
The reluctance values of the core (Rm1, Rm2, Rm3) can be calculated with the given
dimensions of the core and the permeability of the material (see Eq. 2-11). The
calculation of the leakage reluctance values is given in Appendix II.
Example Calculation
Figure 4-15 shows the calculated leakage permeance according to the previous
derived model (Eq. 4-14) in comparison to the measurement of EE70 and EE65 ferrite
(μi=2000) cores assembled with different centre leg air gaps. The calculated
permeance values are a bit overestimated, but very close to the measurement. The
characteristic of the calculated model and the measured curves are in well
accordance.
84
Figure 4-15: Comparison of leakage permeance calculation and measurement
4.6 Leakage Inductance of Coupled Ring Core Inductors
The leakage or differential mode inductance of ring cores in order to improve the
design of common mode chokes is evaluated by [88]. The differential mode
inductance depends on core size and permeability. In [88] the differential mode
inductance of ring cores is derived with the help of existing models for rod cores,
assuming that the differential mode flux uses one half of the effective magnetic path
length inside the core. Therefore, [88] defines a unit less geometry factor and depicts
the effective differential mode permeability versus this geometry factor (see Figure
4-16):
Eq. 4-16
Γ=√
𝜋 𝑙𝑒
∙
𝐴𝑒 2
In [88] it is recommended to use the following equation to calculate the effective
differential mode permeability for cores with permeability values about 5,000:
Eq. 4-17
𝜇𝑑𝑚,5000 = 2.5 ∙ Γ1.45
Alternatively or for other permeability values, the effective differential mode
permeability can be read out from Figure 4-16.
The equivalent air inductance is given by [88]:
85
Lair =
Eq. 4-18
𝜇0 ∙ 𝑁 2 ∙ 𝐴𝑒
Θ
sin ( )
√ Θ
2
𝑙𝑒 ∙
+
360°
𝜋
Θ is the angular winding core coverage in degree:
Eq. 4-19
Θ = 180° ∙
N
𝑁𝑚𝑎𝑥
The differential mode inductance is calculated according to [88]:
Eq. 4-20
𝐿𝑑𝑚 = 𝜇𝑑𝑚 ∙ 𝐿𝑎𝑖𝑟
As a result, the differential mode or leakage inductance increases with increasing
core size (larger le and thus higher Γ) and with increasing permeability. But for
permeability values of several thousands the increase of the effective differential
mode permeability is only marginal. A further increase can be observed only for very
large cores.
Figure 4-16: Effective differential mode permeability dependent on Γ factor – representation
taken from [88] and modified
86
4.7 Influence of Core Shape and Material on Leakage
Inductance
4.7.1
Influence of Permeability and Gap Length on Coupled EE Core
Inductors
In order to investigate the influence of the permeability and the gap length on the
current ripple attenuation, the leakage permeance was calculated for an EE65 core
according to the dimensions given in [89]. From the evaluation of Eq. 4-13, it
becomes clear that the gap length has no significant influence on the self-permeance,
because of the high gap reluctance. The self-permeance is only dependent on the
core geometry and the relative permeability. In contrast, permeability and gap length
have an influence on the coupling, as can be seen in Eq. 4-15. It is obvious, that high
permeability as well as increasing gap lengths lead to tight coupling.
The influence on the current ripple attenuation can be investigated with the help of
the leakage permeance (defined according to Eq. 2-26 and Eq. 2-27 or Eq. 4-8), where
high leakage permeance gives the possibility for current ripple attenuation. Figure
4-17 shows that the leakage permeance increases for increasing permeability and
decreasing gap length. For gaps greater than 1 mm, the permeability has only an
influence up to a value about 2,000. This allows the following conclusion:


High permeable materials cannot take advantage of an increasing selfpermeance
Low permeable materials will provide only a poor current ripple attenuation
Thus, materials with a relative permeability of about 2,000 suit very well for the
application of coupled inductors using an EE core structure. This means that ferrite
materials will be applicable in general, because they can be manufactured with
permeability values up to 15,000, where ferrite materials used in power applications
usually provide permeability values about 2,000.
Furthermore, the significant influence of the gap length can be observed in Figure
4-18, where the leakage permeance increases for small gap length and allows better
current ripple attenuation. Nevertheless, it has to be taken into account, that
increasing gap lengths will prevent the saturation of the core. Thus, maybe a lower
leakage permeance must be accepted in order to avoid saturation.
87
Figure 4-17: Coupling coefficients for an EE65 core dependent on permeability and gap length
Figure 4-18: Leakage permeance for an EE65 core dependent on permeability and gap length
4.7.2 Leakage of Coupled EE Core Inductors dependent on Core Size
To show the impact of the core size, the leakage inductance was calculated and
4
plotted versus the area product (window size ∙ magnetic cross-section [m ]) for
several ferrite and iron powder EE cores with the previous derived model from Figure
4-14. Therefore, three materials were taken into account:
 A ferrite with μr=2,000
 Two powder materials with μr=125 and μr=60
Figure 4-19 shows that the ferrite material can provide much higher leakage
permeance, where the higher leakage values can be obtained by small gap lengths.
88
The leakage increases for higher area product, because of the higher self-permeance
values.
For small gap length (< 3 mm), the leakage permeance is higher for the ferrite
material. For increasing gap length the leakage permeance of the iron powder cores
can be higher if the permeability is high enough. It can be seen that the iron powder
cores with medium permeability of 60μ offer only very low leakage permeance.
Therefore, only iron powder materials with higher permeability (about 100μ and
greater) seem to be appropriate in order to achieve acceptable equivalent inductance
values. Nevertheless, it has to be taken into account that the permeability of iron
powder materials will decrease under load conditions. All calculations were
performed under no load conditions. Thus, it can be expected that the current ripple
attenuation in comparison to ferrite will be even lower as depicted in Figure 4-19.
This can be compensated by a number of turns increase, leading to higher winding
losses. Usually iron powder materials suffer higher specific core losses in comparison
to ferrite, which may cancel out the advantage of higher saturation flux densities. The
most suitable material regarding losses, component size and current ripple
attenuation must be evaluated application specific.
Figure 4-19: Leakage permeance of iron powder and ferrite EE cores
4.7.3 Adjusting the Self-Inductance of Coupled EE Core Inductors
The self-inductance is strongly dependent on the permeability of the given core
material. But usually the manufacturers give a large permeability deviation for their
materials (e.g. +- 25 %). This can cause inaccuracies for the calculation of the
component, resulting in the dilemma that the required inductance and coupling
89
values cannot be adjusted. Gapping of the winding carrying legs is one way to
overcome this dilemma. This will lead to a self-inductance and coupling decrease.
Figure 4-20 shows the influence of inserted air gaps on the leakage permeance
dependent on the area product. The leakage permeance decreases with increasing
gap length. The decreasing coupling cannot compensate the self-permeance
decrease. This results in lower equivalent inductance. Increasing the number of turns
will compensate this effect, but results in higher winding losses. If the window is too
small for more turns, the core size must be increased.
The insertion of air gaps should be restricted to very small gaps about 0.1mm. Figure
4-20 shows that large air gaps cause a significant self-permeance decrease, causing
very low equivalent inductance values.
Figure 4-20: Influence of air gaps for self-inductance adjustment
4.7.4 Comparison of Coupled Ring and EE core Inductors
Figure 4-21 shows the calculated leakage permeance of ferrite EE and ring cores,
using Eq. 4-14 and Eq. 4-20, dependent on the area product. The permeability was
set to 2,000 for the calculation. It becomes obvious, that the leakage permeance of
EE cores is usually much higher than the leakage permeance of ring cores even if the
centre legs of the EE cores are realised with large air gaps. The ideal shape of (small)
ring cores provide an even flux density distribution, even at higher operation
frequencies. The ring shape does not have edges or sharp corners, which can tend to
partial saturation and push the flux out of the core.
90
Figure 4-21: Leakage permeance of EE and ring cores (μr=2,000)
4.7.5 Measurement of different Materials and Cores
Figure 4-22 shows the leakage permeance of an EE65 and an EE70 ferrite core for
different centre leg air gap lengths. As evaluated previously, the increase of the gap
length will decrease the leakage permeance. In comparison, Figure 4-22 shows the
leakage permeance of two EE65 iron powder cores without centre leg gap. Though
the remove of the gap leads to the highest leakage permeance, the iron powder
cores have much lower permeance values compared to their ferrite counterparts.
The reason for this effect is the much lower self-inductance of the powder cores,
caused by the low permeability (26 and 60 compared to about 2,000 for the ferrite).
Figure 4-22: Leakage permeance of EE65 and EE70 ferrite cores (μr=2,000) in comparison to
EE65 powder cores (μr=60 and μr=26)
91
Figure 4-23 depicts the leakage permeance of different ring cores dependent on the
area product. In general, it can be seen that larger cores will give higher leakage
permeance values. This is valid for all materials. However, the permeability of the
given material has a certain influence, too. From Figure 4-18 it was derived that
permeability values in the range of several thousand will not cause a significant
leakage permeance change for a given core size. Figure 4-23 gives a hint for that
statement, because some cores of similar size offer similar leakage values, regardless
of their permeability. Nevertheless, the powder cores seem to tend to lower leakage
values as elaborated in the theory.
Figure 4-23: Leakage permeance of ring cores with different core size and material
Figure 4-24 shows a comparison of ring and EE cores of similar size as well as
different materials. It can be seen that EE cores can offer higher leakage values
compared to ring cores. The EE core centre leg acts as leakage path, where the ring
core usually provides very low leakage values. It should be taken into account that
the ring cores were measured without any leakage inductance enhancing segments.
The large ferrite EE cores offer the highest leakage values in this comparison even if
very large centre leg air gaps are established. Their powder material counter parts
have much lower leakage values, where the ferrite ring cores of similar area product
have even lower leakage.
Finally, the use of ferrite E cores can be recommended for coupled inductors,
because they seem to provide the highest leakage values. The leakage of the EE core
can be adjusted by the centre leg air gap. The powder cores have too low
permeability resulting in lower leakage values. The ring cores offer too low leakage,
92
because of their geometry. Furthermore, they have to suffer poor copper fill factors
in comparison to the EE cores.
Figure 4-24: Leakage permeance comparison of ring and EE cores for different materials
4.8 Design Methodology
The design of coupled inductors and integrated magnetic components requires the
adjustment of self- and leakage inductance (or coupling). These component specific
values depend on the reluctance values of the magnetic circuit and the winding
design. Finally, core material, geometry and number of turns (and if applicable turn
ratio) will influence the design. This means, several design parameters can be
changed independent, making a straight forward procedure difficult. Therefore, the
proposed procedure contains the identification of design limitations, the calculation
of technically possible components by variation of the design parameters and the
discard of unsuitable designs.
The proposed design methodology is based on a computer aided design procedure
including iterative steps. The method suits well for an implementation in software’s
like Mathcad or Mathlab. Figure 4-25 shows the flow chart of the design procedure,
where several parts of the design procedure must be adapted application dependent
(e.g. establishment of the magnetic circuit or loss calculation). The method allows the
calculation of all appropriate designs at once. The basic idea is that all technical
possible designs will be calculated iterative according to the design specifications.
Afterwards, the optimum design can be identified with the help of a performance
factor, which considers the importance of the predefined specifications.
93
Magnetic circuit
model
Select core
shapes, materials
and sizes
Select winding
material
(e.g. litz wire, solid
wire, ...)
Calculate possible design range
- Variegate design parameters (e.g.
number of turns, gap length, ...) and
evaluate inductance and coupling
values with the help of the magnetic
circuit
Evaluate electrical
operation points
(e.g. resulting current)
dependent on
calculated inductance
and coupling values
Winding design
- Evaluate maximum number of turns
according to given window area
Evaluate magnetic
operation points
(e.g. flux density swing,
DC bias flux)
dependent on
calculated inductance
and coupling values
Discard all designs not fulfilling all design specifications
(e.g. current ripple too high or core saturation occurs or number of turns do not fit
into the window)
Loss (core and winding losses) and thermal calculation
(Discard all designs which exceed the loss or thermal specifications)
Optimisation
- Arrange remaining designs according to design priority
(e.g. calculate performance factor to find the most appropriate design)
Finished
Figure 4-25: General design procedure
94
If no designs remain, choose other cores / winding material or change design parameters
Set design specifications
(e.g. losses, size, weigth, costs, current ripple, thermal limits,
turn ratios, ...)
The establishment of the magnetic circuit must be done application specific.
Dependent on the core shape, it is necessary to identify the parasitic effects, which
have a non-negligent influence on the component behaviour. This requires a certain
amount of experience as well as a detailed analysis of the proposed component.
Some reluctance values of the magnetic circuit may depend on design parameters
(e.g. air gap length). Usually the number of turns and air gap lengths will be used to
adjust required inductance and coupling values in order to fulfil the design
specifications. In practice, the component specific inductance and coupling values will
be calculated for each possible combination of the design parameters (e.g. number of
turns, air gap length).
The design parameters can be restricted e.g. by geometrical limits. If the window size
as well as the wire type is known, it is possible to calculate the maximum allowed
number of turns per window. If more than one winding is placed in one window, the
turn ratio of each winding pair must be known. Furthermore, other restrictions like
maximum air gap lengths can be calculated in order to avoid the calculation of
designs which cannot be assembled in practice.
If the inductance values for each design are calculated, it is possible to derive the
electric and magnetic operating points of each component. Resulting current ripples
as well as DC bias flux densities and flux density swings in each core segment must be
calculated. This means, constructive parameters will be linked with the resulting
electric and magnetic properties and operating points. Now electric and magnetic
boundary conditions must be recognised, e.g. avoid saturation or too high current
ripples. Designs which do not fulfil the specifications can be discarded.
An important issue is the loss calculation. Many power electronic converters cover a
certain input and/or output voltage range. In this case the losses must be calculated
for the different operating conditions (e.g. maximum input voltage and minimum
input current as well as vice versa), because voltage as well as current have an
influence on the core and winding losses. The different operating conditions of a
converter can change the loss balance (core and winding losses), which must be
taken into account in order to avoid excessive losses in the core or the winding.
As a result, the remaining designs should fulfil all design specifications. If no design is
remaining, the core size, material or winding materials must be changed (e.g. choose
95
a larger core or smaller wire profile). If such measures do not lead to success, the
design specifications may be adapted.
After a successful design procedure, the remaining designs can be rated regarding the
given specifications. It is possible to calculate a performance factor, which is
recognising the predefined importance of the different properties of the calculated
components. Properties like losses, size, weight, costs etc. can be weighted by an
impact factor. Each property can be normalised to a reference value e.g. from a
chosen reference design. This performance factor can be calculated for each design
and allows the comparison of the different design:
𝑛
Eq. 4-21
𝑃𝐹 = 100% −
100%
𝑉𝑎𝑙𝑢𝑒𝑖
∙ ∑ 𝑓𝑎𝑐𝑡𝑜𝑟𝑖 ∙
𝑛
𝑅𝑒𝑓𝑖
𝑖
The reference design is set to 0 %, where all other designs will yield more or less than
0 %. The design with the highest performance factor can be interpreted as the
calculated optimum regarding the given specifications.
96
5
Automotive On-Board Power Supply with Magnetically
Integrated Current Doubler
In order to enable performance comparison of functionally integrated magnetic
components with their discrete counterparts, a transformer with an integrated
current doubler will be presented as an example here. The circuit in this example is a
push-pull converter with current doubler rectifier, where Figure 5-1 is showing the
conventional discrete circuit, known from the literature. The circuit contains a
transformer to provide galvanic isolation as well as two inductors to enable the
output current supply and smoothing. Figure 5-4 depicts an equivalent circuit, where
all magnetic components are realised within only one component [16]. The inductors
are integrated in the structure of the transformer.
The objective of this chapter is to explain the behaviour of the integrated magnetic
component as well as to compare the performance to the discrete magnetic design.
The electric behaviour of both circuits is explained and the differences are pointed
out. A detailed performance comparison was done with a 2.5 kW converter - e.g.
applicable for on-board power supplies of electric and hybrid vehicles, replacing the
conventional alternator. The secondary circuit is capable to handle currents even in
the 100 A range. Therefore, the circuit is favourable to transfer power from a high
voltage (400 V) to a low voltage battery (12 V, 24 V, 48 V).
Different magnetic structures applicable for the integrated magnetic current doubler
are presented, where the most promising designs are analysed regarding power loss,
size and weight. Finally, several magnetic components were assembled and tested
experimentally with appropriate converters.
5.1 Circuit Analysis
5.1.1 Discrete Current Doubler
Figure 5-2 shows the PWM and current waveforms of a push-pull converter with a
discrete current doubler according to the circuit from Figure 5-1. A PWM signal with
180° phase shift between both phases is assumed.
Time Interval t1
If positive input voltage is applied on the transformer, a positive current flow will
start, transferring energy from the transformer primary to the secondary side. The
leakage inductance values from the transformer will be charged. The diode D3 on the
97
secondary will take over the current from diode D4. The inductors L1 and L2 will be
discharged both until the commutation is completed.
Time Interval t2
The transformer primary is still tied to the input voltage, enabling the power transfer
to the secondary. Diode D3 has taken over the full output current. The secondary
circuit is closed via the inductor L2. L2 will be charged, where the output load is
supplied by the current delivered by L1 and the transformer secondary.
Time Interval t3
Both PWM signals are low. Transformer primary and secondary currents will
decrease. The transformer leakage inductances will discharge. The primary
transformer current will commutate from S1 to D2. The diode D4 on the secondary
will start taking over a part of the output current from D3. Due to the fact that no
energy will be transferred from the transformer, the output is supplied by the two
inductors L1 and L2, causing a discharge of both inductors.
Time Interval t4
Both primary switches are still off and no power transfer from primary to secondary
occurs. The output is fed by the energy stored in the inductors L1 and L2. D3 and D4
will share the output current and discharge L1 and L2. The two inductors are
connected in parallel and tied to the output voltage. The currents of both diodes will
add and the current flow from the secondary winding of the transformer is replaced
by the current flow of the inductor L2.
Time Interval t5
This time interval is similar to time interval t1, but negative input voltage is applied to
the transformer. All following time intervals are complementary to the previous
description.
S1
I2
V1
VL2
Vdc/2
CDC2
Iout
Cout
L1
Rout
V2
VL1
I1
IL1
Vout
Vdc/2
D3
D1
CDC1
S2
L2
IL2
D2
D4
Figure 5-1: Push-pull converter with discrete current doubler
98
A further aspect is that the output inductors L1 and L2 operate with the fundamental
switching frequency (see Figure 5-2). But the output current shows twice the
switching frequency. The current ripple at the output cancels due to the 180° phase
shift of both output inductors.
PWM1
PWM2
PWM Signals
t1
t2
I1
I2
t3
t4
t5
t6
t7
t8
t9
t6
t7
t8
t9
t6
t7
t8
t9
t6
t7
t8
t9
t7
t8
t9
Transformer Currents
t1
t2
t3
IL1
IL2
t4
t5
Inductor Currents
t1
t2
t3
ID3
ID4
t4
t5
Rectifier Currents
t1
t2
Iout
t3
t4
t5
Rectifier Output Current
t1
t2
t3
t4
t5
t6
Figure 5-2: Current waveforms of the push-pull converter with discrete current doubler
99
t1
t5
S1
Irect
L1
L2
D2
Irect
Iout
Cout
Rout
L2
Vdc/2
Vdc/2
CDC2 S2
D3
D1
CDC1
Vout
L1
Rout
Vout
Cout
S1
Iout
Vdc/2
Vdc/2
D3
D1
CDC1
D4
t2
CDC2 S2
D2
D4
t6
S1
Irect
L1
L2
D2
Irect
Iout
Cout
Rout
L2
Vdc/2
Vdc/2
CDC2 S2
D3
D1
CDC1
Vout
Rout
Vout
Cout
L1
S1
Iout
Vdc/2
Vdc/2
D3
D1
CDC1
D4
t3
CDC2
S2
D2
D4
t7
Iout
Cout
Rout
L1
Vdc/2
Vdc/2
D2
D4
t4
Iout
Cout
Rout
CDC2 S2
D2
Irect
Iout
Cout
Rout
D4
t8
S1
CDC1
D3
D1
Cout
Rout
S1
CDC1
L1
L2
D2
L2
Vdc/2
Vdc/2
CDC2 S2
D3
D1
D4
CDC2 S2
D2
D4
Figure 5-3: Switching status of the discrete current doubler according to the time intervals
from Figure 5-2
100
Vout
Iout
Vout
L1
Irect
Vdc/2
Vdc/2
Irect
L2
L2
CDC2 S2
D3
D1
CDC1
Vout
L1
S1
Irect
Vout
Vdc/2
D3
D1
Vdc/2
S1
CDC1
5.1.2 Magnetically Integrated Current Doubler
Figure 5-4 shows a half bridge push pull converter with the magnetically integrated
current doubler proposed by [16]. The transformer is split in two transformers placed
on the outer legs. The primary windings are connected in series. They are wound in a
sense that the excited fluxes will add in the outer legs of the core (direct coupling).
Instead, the primary flux cancels out in the centre leg (indicated by the coloured
arrows in Figure 5-4). The secondary windings require a centre tap. They form a
coupled output inductor by means of the gaped centre leg, which is acting as a
predefined leakage path. The secondary fluxes will cancel out each other in the outer
legs, but will add in the centre leg (inverse coupling) to enable the required energy
reactor for the output current smoothing. The outer legs are used for the transformer
function of the component, where the centre leg is used as energy reactor to ensure
a continuous current flow on the secondary if no power is transferred from the
primary.
Figure 5-5 shows the current waveforms of the circuit with the magnetically
integrated current doubler. By comparing Figure 5-5 and Figure 5-2, it can be seen
that the transformer primary current as well as the output current have the same
behaviour as shown in the discrete current doubler circuit. The output current is the
sum of both diode currents. The difference is that the two output inductors (L1 and
L2) from Figure 5-1 are now integrated in the transformer structure. Therefore, the
rectifier diodes D3 and D4 are directly in series to the transformer secondary
windings. The currents in the secondary windings are identical with the
corresponding diode currents. In principle both circuits work similar as briefly
described below.
B1
S1
Vdc/2
CDC1
D1
W1
B2
B3
W2
D3
Irect
W3
D4
D2
Cout
Rout
Figure 5-4: Push-pull converter with magnetically integrated current doubler
101
Vout
Vdc/2
CDC2S2
Iout
W4
Time Interval t1
The transformer primaries are tied to the positive input voltage, causing a positive
current flow. The transformer leakage inductances are charged and the power
transfer to the secondary will start. Diode D3 (or winding W3) will take over the
current from D4 (winding W4).
Time Interval t2
Diode D3 and winding W3 have taken over the full output current. The complete
output power is now supplied via W3 and D3. In contrast to the discrete current
doubler, this means that the secondary windings must carry the full output current,
instead of only one half.
Time Interval t3
All primary switches are off and the primary transformer winding is open. The
transformer back EMF will enable the primary current flow via the freewheeling
diode D2, causing the discharge of the transformer leakage inductances. The rectifier
diode D3 will start commutating the output current to D4. Thus, the secondary
windings will be switched in parallel, tied to the output voltage and act as a coupled
inductor.
Time Interval t4
Both rectifier diodes will share the output current. Each diode is conducting one half
of the output current. No energy is transferred via the transformer from the primary
to the secondary. The output is supplied by the energy reactor of the coupled
inductor via the two secondary windings.
Time Interval t5
This time interval and all the following are equivalent to the previous description,
except that the transformer input is tied to the negative input voltage.
102
PWM1
PWM2
PWM Signals
t1
t2
Iw1=Iw2
t3
t4
t5
t6
t7
t8
t9
t7
t8
t9
t7
t8
t9
t7
t8
t9
t7
t8
t9
Transformer Primary Current
t1
t2
Iw3
Iw4
t3
t4
t5
t6
Transformer Secondary Currents
t1
t2
Iout
t3
t4
t5
t6
Rectifier Output Current
t1
t2
B1
B2
B3
t3
t4
t5
t6
Transformer Flux Densities
t1
t2
t3
t4
t5
t6
Figure 5-5: Current waveforms of the push-pull converter with magnetically integrated
current doubler
103
t1
t5
S1
Vdc/2
D1
W1
Vdc/2
S1
CDC1
W2
CDC1
D1
W1
W2
W3
W4
D3
D3
Irect
Cout
Rout
t2
CDC2S2
D4
D2
Cout
Iout
Rout
Vout
D4
D2
Vout
Vdc/2
CDC2
Irect
W4
S2
Vdc/2
W3
Iout
t6
S1
Vdc/2
D1
W1
Vdc/2
S1
CDC1
W2
CDC1
D1
W1
W2
W3
W4
D3
D3
Irect
Cout
Rout
t3
S2
CDC2
D4
D2
Cout
Iout
Rout
Vout
D4
D2
Vout
Vdc/2
CDC2S2
Irect
W4
Vdc/2
W3
Iout
t7
S1
Vdc/2
D1
W1
Vdc/2
S1
CDC1
W2
CDC1
D1
W1
W2
W3
W4
D3
D3
Irect
Cout
Rout
t4
CDC2S2
D4
D2
Cout
Iout
Rout
Vout
D4
D2
Vout
Vdc/2
CDC2
Irect
W4
S2
Vdc/2
W3
Iout
t8
S1
S1
D1
W1
W2
W3
W4
Vdc/2
Vdc/2
CDC1
CDC1
D1
W1
W2
W3
W4
D3
D3
Irect
Rout
CDC2S2
D2
D4
Cout
Iout
Rout
Vout
Cout
Irect
Vdc/2
D2
D4
Vout
Vdc/2
CDC2S2
Iout
Figure 5-6: Switching status of the magnetically integrated current doubler according to time
intervals from Figure 5-5
104
In principle, the input as well as the output behaviour of the discrete and the
magnetically integrated circuit is equivalent. But the integration of the current
doubler into the transformer will lead to a completely different magnetic behaviour.
Furthermore, the coupling of the different windings of the transformer has a certain
influence on the voltages and currents of the windings. The voltages at the primary
side of the transformer are dependent on the coupling, too. If the primary windings
are tied to the input voltage, the sum of both winding voltages is equivalent to the
input voltage. But the voltage distribution of both windings will dependent on the
coupling and on the electrical conditions of the secondary windings, too.
5.2 Suitable Core Geometries for Magnetically Integrated
Current Doublers
There are different possibilities for the assembly of an integrated current doubler. All
integrated current doublers presented here require a split transformer with series
connected primaries. The secondary is centre taped, forming a coupled output
inductor with energy reactor. In order to improve the adjustment of the energy
reactor, it is recommended to provide a predefined leakage path. The most
transformer structures and types are based on the standard EE core design, because
the EE core provides three legs: Two legs for the two transformers as well as one leg
for the energy reactor, acting as a predefined leakage path.
Nevertheless, the analysis of the disadvantages of the standard EE core structure
leads to several enhanced core geometries and structures with the objective to
cancel these disadvantages. Subsequent a comparison of the different transformer
designs including the most important advantages and disadvantages as well as a
more detailed explanation is given.
5.2.1 EE Core
The EE core structure of the transformer with integrated current doubler is well
known from the literature and can be described as the standard design. The
geometry is the simplest possible. The material and assembly costs are limited. The
centre leg of the E core has twice the magnetic cross section than the outer legs. This
results in the optimal distribution of core material. However, the leakage occurring in
the environment of the core can have a significant influence on the component
behaviour and must be considered, making the design more complex. The leakage is
105
not only important for the inductance calculation of the component but for the
proximity losses, too. Excessive EMI in the environment of the transformer as well as
high proximity losses can occur - especially for applications requiring very high output
currents. High proximity losses make the use of litz wire necessary, where EMI
problems require additional shielding. If the application requires very high power
density, increased EMI in the environment of the component may be unacceptable,
making the application of this current doubler unsuitable.
Furthermore, it should be noticed that the drawback of occurring fringing effects
caused by air gaps can be handled by the insertion of low permeable powder
segments or segmenting the leg by n segments resulting in n+1 gaps. This technique
is possible independent of the used geometry or material and can be used for other
current doubler structures as well, but causes additional material and assembly costs.
Rm1
Rσ1
Vm1
Rm3 Rm2
Rg
Vm3
a)
Vm2
Rσ2
Vm4
b)
Figure 5-7: EE core current doubler: a) component structure; b) magnetic equivalent circuit
Advantages



Disadvantages

Standard core shape
One core replaces three others
Optimised distribution of core
material



Special bobbins for the outer
legs are required
Additional tooling of the core
becomes necessary in order to
adjust the air gap
Possible increased EMI due to
winding placement on the outer
legs
Possible increased proximity
losses due to air gap and
leakage beside outer legs
Table 5-1: Advantages and disadvantages of EE core current doubler
106
5.2.2 EEII or EEUU Core
In order to reduce the leakage effects and the issue of proximity losses of the EE core
design, it is possible to place additional I or U cores beside the outer legs, similar as
proposed by [90]. The assembly requires only standard core shapes. But increased
assembly effort and additional material is necessary for the realisation. This design is
a trade-off between the standard EE core design and the five leg design presented in
the next section. The centre leg of the E core has twice the magnetic cross section
than the outer legs. But the additional I cores lead to a low flux density in the leakage
paths (centre leg and external cores). Thus, this geometry requires more core
material than necessary.
Rg1
Rg3
Rm1
Rm4
Vm1
Rg
Vm3
Rg2
a)
Rm3 Rm2
Vm2
Rm5
Vm4
Rg4
b)
Figure 5-8: EEII core current doubler: a) component structure; b) magnetic equivalent circuit
Advantages



Standard core shapes possible
One component replaces three
others
Decreasing EMI and proximity
losses due to shell type
structure
Disadvantages




Additional material and
assembly costs
Special bobbins for the outer
legs are required
Additional tooling of the core
becomes necessary in order to
adjust the air gap
Distribution of core material not
optimised
Table 5-2: Advantages and disadvantages of EEII core current doubler
5.2.3 Shell Type Core (5 Leg Core)
The five leg configuration represents the best shell type structure. EMI problems and
proximity losses can be reduced to a minimum. The leakage can be adjusted by three
symmetrical air gaps. The structure of the magnetic circuit of this core is equivalent
to the EE and the EEII core design. But the advantage is that the leakage in the
107
environment is very low and can be usually neglected for the design. This makes the
calculation of this structure much easier. In contrast to the EE core design, the five
leg core is no standard core geometry and requires a special tool for the
manufacturing. In a technical sense this design promises the best electric and
magnetic properties, where the use of the special core geometry is only applicable
for large scale productions. In comparison to the EEII structure the optimised
utilisation of the core material is ensured due to the fact that the total magnetic cross
section of the leakage paths (gapped legs) is equivalent to the magnetic cross section
of both winding carrying legs. The legs carrying the windings can be designed in a way
that the use of standard bobbins becomes possible.
Rm4
Rg1
Rm1
Vm1
Rg3
Vm3
a)
Rm3 Rm2
Rm5
Vm2
Rg2
Vm4
b)
Figure 5-9: 5 leg core current doubler: a) component structure; b) magnetic equivalent circuit
Advantages





Disadvantages


One core replaces three others
Very low EMI and proximity
losses due to shell type
structure
Easy design (environment
leakage can be neglected)
Optimised distribution of core
material
Magnetic cross-section can be
scaled to enable the use of
standard bobbins
No standard core shape
Additional tooling of the core
becomes necessary in order to
adjust the air gaps
Table 5-3: Advantages and disadvantages of 5 leg core current doubler
5.2.4 Ring or UU Core with Leakage Segments
Due to the low leakage inductance values of ring cores, the leakage for the coupled
output inductor must be enhanced by the insertion of additional ferromagnetic
material. A high permeable material with air gap (e.g. ferrite) is recommended.
108
Otherwise, the low leakage of this structure will cause only limited current ripple
attenuation. The application of the UU core is possible, too. A further leakage
inductance increase is possible by adding ferromagnetic segments beside the ring or
the UU core.
The ring and UU core structures are available for all core materials, giving the
designer a certain degree of freedom. The windings for the ring core will be wound
directly on the core. No bobbin is required, but the winding assembly itself is more
complex. Furthermore, ring cores have to suffer low copper fill factors. This may
make the use of larger core sizes necessary, in order to increase the required window
size. Cut UU cores can be assembled with bobbins, enabling a higher fill factor.
The ring core can be implemented as cut core, too, where the segment is placed
between the ring halves. This may become applicable if amorphous or
nanocrystalline materials will be used. Otherwise the flux of the secondary windings
will leave the ring core perpendicular to the anisotropic material structure, causing
high eddy current losses in these parts of the core. As a drawback there occur at least
four gaps in the transformer path of the component. The gaps must be kept very
small in order to ensure a high permeable path for the transformer. If the permeance
of the leakage segment must be lowered further to adjust the coupling, additional
gaps can be inserted in the segment itself.
Nanocrystalline materials become interesting because of their higher saturation flux
density compared to ferrite. This may allow the downsizing of an integrated current
doubler by means of flux density increase at lower frequencies. Cut cores can be
avoided if the segments are placed on top and bottom of the core. This avoids a
perpendicular flux leaving for laminated amorphous and nanocrystalline ring or UU
cores. The expensive cutting process for the insertion of a segment is not necessary
anymore. In turn, the segments will increase the component height.
109
a)
b)
Rm1
Rσ1
Rg1
Rm2
Vm1
Rm3
Vm2
Vm3
Rg2
Vm4
Rσ2
c)
Figure 5-10: Current doublers with window segment: a) ring core; b) UU core; c) magnetic
equivalent circuit
a)
b)
Rg3
Rg5
Rm1
Rm4
Rg1 Rm2
Vm1
Rm3
Vm2
Vm3
Rg4
Rg2
Vm4
Rg6
Rm5
c)
Figure 5-11: Current doublers with window and external segment: a) ring core; b) UU core;
c) magnetic equivalent circuit
110
Rg1
Rg2
Rm1
Rσ1
Rm2
Rm3
Vm1
Vm2
Vm3
a)
Vm4
Rg3
b)
Rm1 Rg1
Rσ1
Rσ2
Rg4
Rg3 Rm2
Vm1
Rm3
Rm4
Vm2
Vm3
Rg2
Rg4
Vm4
Rσ2
c)
Figure 5-12: Ring core current doubler with cut core or segment on top and bottom:
a) component structure; b) magnetic equivalent circuit of cut core; c) magnetic equivalent
circuit with top-bottom segment
Advantages




Disadvantages

Standard core shape
All materials possible for the
ring and UU core (except
powder materials)
One component replaces three
others
Limited EMI due to ring or UU
core structure



Additional core segment(s)
necessary
Requires no bobbin but more
complex winding manufacturing
(ring core)
Usually low copper fill factors
(ring core)
Cut core or top-bottom segment
necessary (for amorphous or
nanocrystalline materials)
Table 5-4: Advantages and disadvantages of ring or UU core current doubler
5.2.5 Comparison of Integrated and Discrete Current Doubler
The idea of the magnetically integrated current doubler is, to replace the two
inductors and the transformer of the conventional discrete current doubler by only
one component fulfilling all electric functions. A significant technical advantage is the
fact that only one component must be assembled instead of three for the total
converter assembly. This can be a superior advantage especially for large scale
productions.
111
Table 5-5 shows a comparison of all pervious described integrated current doublers
and the discrete current doubler. The advantage of the discrete current doubler is
that each component can be optimised regarding its functionality. All materials and
core shapes can be used for the design, giving a large degree of freedom for the
design. The only restriction is that the transformer requires a high permeable
material. The transformer does not need a centre tap, which can cause symmetry
problems. Instead, the discrete assembly requires 8 connections (4 for the
transformer and 4 for the two inductors). The tooling of air gaps becomes necessary
if high permeable core material is used for the inductors. If the inductors are realised
with low permeable material, no air gap tooling is necessary.
Current
doubler
Discrete
Integrated
EE core
Integrated
EEII core
Integrated
5 leg core
Ring / UU
core +
segment
Materials
Transformer: all
high
permeable
materials
Ferrite
Ferrite
Ferrite
Ring/UU:
all high
permeable
materials
Segment:
all
materials
Inductor:
all
materials
Cores
3 (rings) - 6
2
4
2
2-5
Bobbins
3
2
2
2
0-2
Windings
4
4
4
4
4
Connectio
ns
8
5
5
5
5
Centre Tap
Not
necessary
For
secondary
For
secondary
For
secondary
For
secondary
Tooled air
gaps
maybe 2
(for
inductors)
1
1
3
-
Table 5-5: Comparison of different current doubler technologies regarding assembly
considerations
112
In comparison, the integrated current doublers require less cores, bobbins and
electrical connections. This can be an advantage regarding the assembly of the
magnetic component itself as well as for the converter assembly. One component
requiring less cores and bobbins can be assembled faster. Furthermore, the assembly
of one component into the converter requires less effort than the assembly of three
components. This effect is cumulative with the number of required electrical
connections. But it has to be taken into account that the advantage regarding the less
assembly effort and the less required number of cores and bobbins does not
automatically lead to an advantage regarding component expenditures.
The integrated current doublers require a centre taped secondary winding, which is
prone to asymmetry problems. The number of required windings is equivalent to the
discrete current doubler. The number of air gaps depends on the assembly. The ring
core current doublers can avoid the tooling of air gaps, where additional segments
must be manufactured, instead. The 5 leg core requires the tooling of three gaps,
where the standard EE core current doubler requires only one gap.
5.3 Design of Integrated Current Doublers
The design procedure of the integrated current doubler will be explained exemplary
with the EE ferrite core current doubler. Nevertheless the EEII, the 5 leg core and the
ring or UU core with segments can be designed similar, because all equivalent circuits
of these core structures can be simplified to a simple three leg magnetic circuit
similar as in Figure 4-7 a).
The requirements of the transformer behaviour and the coupled inductor must be
recognised for the design procedure. Required specifications are:





Nominal power
Input / output voltage as well as currents
Switching frequency
Transformer turn ratio
Output current ripple
5.3.1 Transformer Turn Ratio
The transformer turn ratio must be adjusted according to the input voltage range, the
output voltage and the possible duty cycle. Figure 5-13 shows the required
transformer turn ratio dependent on the maximum duty cycle for different ratios of
113
minimum input voltage to output voltage, valid for half-bridge push-pull converters
using current doubler rectifiers. The turn ratio of both transformers is set to equal
value, according to Eq. 5-1. The transformer turn ratio should be chosen as high as
possible, because the output current ripple cancelation is most effective close to the
50 % duty cycle limit. The transformer secondary voltages are lower at higher turn
ratios, resulting in a lower blocking voltage for the rectifier. The voltage class of the
rectifier semiconductors may be chosen smaller at higher turn ratios, which reduces
the switching losses of the semiconductors as well.
Eq. 5-1
𝑛=
𝑉1 𝑁1 𝑉2 𝑁2
=
=
=
𝑉3 𝑁3 𝑉4 𝑁4
Figure 5-13: Required transformer turn ratio dependent on maximum duty cycle for halfbridge push pull converters with current doubler rectifier (270 V minimum input voltage)
Nevertheless, a certain design limit regarding the duty cycle must be taken into
account, because the leakage inductance of the transformer will limit the power
transfer capability of the converter (e.g. see [91]). Therefore, and to avoid shorts at
the bridge, the duty cycle must kept below 50 % within a certain safety margin.
5.3.2 Area Product Approach
The area product approach is used to estimate an appropriate core size for a
magnetic component, based on the application dependent operation conditions. It is
defined by the multiplication of the window area Aw and the magnetic cross-section
Ae and has the unit m4. The area product is a help for the designer in order to reduce
114
the required iterations to find an appropriate core size. Different expressions for the
area product can be found in literature. E.g. [1] derives the area product for inductors
and transformers based on design considerations, where [2] and [49] propose
approaches based on experience. A general area product method for the design of
integrated magnetic components is expressed by [92]. The modification of the area
product method for inductors given by [1] leads to:
Eq. 5-2
𝐴𝑃 =
2
2 ∙ 𝐿𝑒𝑞 ∙ 𝐼𝑝𝑘
= 𝐴𝑤 ∙ 𝐴𝑒
𝐵𝑝𝑘 ∙ 𝐽𝑟𝑚𝑠 ∙ 𝑘𝑐𝑢
The denominator of Eq. 5-2 contains the allowed peak flux density Bpk, the RMS
current density Jrms and the maximum allowed copper fill factor kcu. The term in the
numerator denotes the magnetic energy stored in the component. This value is
relevant for the current ripple attenuation of the component. The integrated current
doubler requires two secondary windings for the coupled output inductor and two
primaries for the transformer. The multiplication by the factor two indicates that
50 % of the window is utilised by the secondary and the remaining space is left for
the primary, where other spacing is possible as well.
The equivalent inductance Leq can be calculated by rearranging Eq. 5-35, which is
derived in chapter 5.3.4:
Eq. 5-3
𝐿𝑒𝑞 = 𝐿3 (1 − 𝑘34 ) =
2𝑉𝑜𝑢𝑡 1 − 2𝐷
∙
∆𝑖𝑜𝑢𝑡
2𝑓𝑠
5.3.3 Inductance Matrix
The next step is the calculation of the required inductance and coupling values. The
complete inductance matrix of the component can be represented as follows:
Eq. 5-4
𝑉1
𝐿1
𝑉2
−𝑀21
( )=(
𝑉3
𝑀31
𝑉4
−𝑀41
−𝑀12
𝐿2
−𝑀32
𝑀42
𝑀13
−𝑀23
𝐿3
−𝑀43
𝑑𝑖1 /𝑑𝑡
−𝑀14
𝑑𝑖2 /𝑑𝑡
𝑀24
)(
)
−𝑀34
𝑑𝑖3 /𝑑𝑡
𝐿4
𝑑𝑖4 /𝑑𝑡
The mutual inductances of inverse coupled windings are defined with negative sign
according to the magnetic circuit from Figure 5-14. In practice, the leakage
115
reluctances of the windings can be neglected. A simplification leading to the circuit
given in Figure 5-14 b) is possible. If the leakage inductance values are required, the
model from Appendix V can be used.
The previous presented magnetic circuits from the EE, the EEII, the 5 leg shell-type
and the ring or UU core with segment structure can be simplified to the circuit given
in Figure 5-14. Rm3 represents not only the centre leg air gap but other leakage paths
beside the core as well. As a further simplification Rm1=Rm2, Rσ1=Rσ2=Rσ3=Rσ4 as well
as N1=N2 and N3=N4 can be assumed due to symmetry reasons. The self-inductance
values L1, L2, L3 and L4 can be derived as follows:
𝐿1 = 𝐿2 = 𝑁12 ∙
Eq. 5-5
𝐿3 = 𝐿4 = 𝑁32 ∙
Eq. 5-6
1
𝑅𝑚1 𝑅𝑚3
)
𝑅𝑚1 + 𝑅𝑚3
𝑅 𝑅
𝑅𝜎1 + 𝑅𝑚1 + 𝑚1 𝑚3
𝑅𝑚1 + 𝑅𝑚3
𝑅𝜎1 ∙ (𝑅𝑚1 +
1
𝑅𝑚1 𝑅𝑚3
)
𝑅𝑚1 + 𝑅𝑚3
𝑅 𝑅
𝑅𝜎3 + 𝑅𝑚1 + 𝑚1 𝑚3
𝑅𝑚1 + 𝑅𝑚3
𝑅𝜎3 ∙ (𝑅𝑚1 +
The mutual inductance values can be calculated by:
Eq. 5-7
𝑀12 = 𝑀21 = 𝑘12 √𝐿1 𝐿2 = 𝑘12 𝐿1 = 𝑘12 𝐿2
Eq. 5-8
𝑀13 = 𝑀31 = 𝑘13 √𝐿1 𝐿3
Eq. 5-9
𝑀14 = 𝑀41 = 𝑘14 √𝐿1 𝐿4
Eq. 5-10
𝑀23 = 𝑀32 = 𝑘23 √𝐿2 𝐿3
Eq. 5-11
𝑀24 = 𝑀42 = 𝑘24 √𝐿2 𝐿4
Eq. 5-12
𝑀34 = 𝑀43 = 𝑘34 √𝐿3 𝐿4 = 𝑘34 𝐿3 = 𝑘34 𝐿4
116
≈ 𝑁12 ∙
≈ 𝑁32 ∙
1
𝑅 𝑅
𝑅𝑚1 + 𝑚1 𝑚3
𝑅𝑚1 + 𝑅𝑚3
1
𝑅 𝑅
𝑅𝑚1 + 𝑚1 𝑚3
𝑅𝑚1 + 𝑅𝑚3
The coupling coefficients between the different windings can be calculated according
to Eq. 5-13 and Eq. 5-14, where only two different cases must be considered for a
symmetrical circuit: both windings are placed on the same leg (Eq. 5-13) or both
windings are placed on different legs (Eq. 5-14).
Eq. 5-13
𝑅𝑆1
𝑘13 = 𝑘24 =
𝑅𝜎1 + 𝑅𝑚1 +
𝑘12 = 𝑘14 = 𝑘23 = 𝑘34 =
𝑅𝑚1 𝑅𝑚3
𝑅𝑚1 + 𝑅𝑚3
≈1
𝑅𝑚3
∙
𝑅𝑚1 + 𝑅𝑚3 𝑅
𝑅𝑆1
+ 𝑅𝑚1 +
𝜎1
Eq. 5-14
≈
𝑅𝑚1 𝑅𝑚3
𝑅𝑚1 + 𝑅𝑚3
𝑅𝑚3
𝑅𝑚1 + 𝑅𝑚3
By neglecting the winding leakage reluctance Rσ and inserting Eq. 5-5, Eq. 5-6, Eq.
5-13, Eq. 5-14 into Eq. 5-7 to Eq. 5-12 leads to:
Eq. 5-15
Eq. 5-16
Eq. 5-17
Eq. 5-18
𝑀12 = 𝑀21 =
𝑁12
𝑅𝑚1 𝑅𝑚3
𝑚1 + 𝑅
𝑚1 + 𝑅𝑚3
𝑅𝑚3
∙
𝑅𝑚1 + 𝑅𝑚3 𝑅
𝑀13 = 𝑀31 = 𝑀24 = 𝑀42 =
𝑁1 𝑁3
𝑅 𝑅
𝑅𝑚1 + 𝑚1 𝑚3
𝑅𝑚1 + 𝑅𝑚3
𝑀14 = 𝑀41 = 𝑀23 = 𝑀32 =
𝑅𝑚3
∙
𝑅𝑚1 + 𝑅𝑚3 𝑅
𝑀34 = 𝑀43 =
𝑁1 𝑁3
𝑅𝑚1 𝑅𝑚3
𝑚1 + 𝑅
𝑚1 + 𝑅𝑚3
𝑁32
𝑅𝑚1 𝑅𝑚3
𝑚1 + 𝑅
𝑚1 + 𝑅𝑚3
𝑅𝑚3
∙
𝑅𝑚1 + 𝑅𝑚3 𝑅
117
Rm1
Rm3
Rm2
I1 N1
φ1
V1
Rσ1
Rσ2
φ2
I3 N3
V3
Rσ4
Rσ3
a)
φ4
Rm3
Rm2
I1 N1
V2
N4
φ3
Rm1
N 2 I2
V1
φ1
φ2
φ3
φ4
V4
V3
V2
N4
I3 N3
I4
N 2 I2
I4
V4
b)
Figure 5-14: Simplified magnetic equivalent circuits for integrated current doublers
5.3.4 Output Current Ripple Calculation
Two different circuit conditions can be used to calculate the output current ripple:


Power transfer from primary to the secondary (rising current): Only one
rectifier phase is conducting - the other one does not conduct current.
No power transfer from the primary (decreasing current): The output is fed
by the energy stored in the coupled inductor – both secondary windings are
connected in parallel and the current through both windings and rectifier
diodes is equivalent.
Case 1: Power Transfer from Primary to Secondary
The two primary windings are connected in series. The sum of the two primary
voltages is equivalent to the half DC link voltage (for the half-bridge converter),
where the current as well as the derivative of the current from winding 1 and winding
2 are equivalent. The current in one of the two rectifier phases is zero, thus, the
derivative of the current is zero, too.
𝑉𝑖𝑛
2
Eq. 5-19
𝑉1 − 𝑉2 =
Eq. 5-20
𝑑𝑖1
𝑑𝑖2
=−
≠0
𝑑𝑡
𝑑𝑡
Eq. 5-21
𝑑𝑖4
=0
𝑑𝑡
118
Using these assumptions, the first three rows of the matrix from Eq. 5-4 can be
simplified to:
𝑑𝑖1
𝑑𝑖3
+ 𝑀13
𝑑𝑡
𝑑𝑡
Eq. 5-22
𝑉1 = (𝐿1 + 𝑀12 )
Eq. 5-23
−𝑉2 = (𝐿2 + 𝑀21 )
𝑑𝑖1
𝑑𝑖3
+ 𝑀23
𝑑𝑡
𝑑𝑡
Eq. 5-24
𝑉3 = (𝑀13 + 𝑀23 )
𝑑𝑖1
𝑑𝑖3
+ 𝐿3
𝑑𝑡
𝑑𝑡
Add up Eq. 5-22 and Eq. 5-23 under the consideration L1=L2 and M12=M21 yields:
Eq. 5-25
𝑉1 − 𝑉2 = 2(𝐿1 + 𝑀12 )
𝑑𝑖1
𝑑𝑖3
+ (𝑀13 + 𝑀23 )
𝑑𝑡
𝑑𝑡
Rearranging Eq. 5-24 leads to:
Eq. 5-26
𝑑𝑖3
𝑑𝑖1 𝑉3 − 𝐿3 𝑑𝑡
=
𝑑𝑡
𝑀13 + 𝑀23
Inserting Eq. 5-26 into Eq. 5-25 will give the following equation:
Eq. 5-27
𝑑𝑖3 (𝑉1 − 𝑉2 )(𝑀13 + 𝑀23 ) − 2(𝐿1 + 𝑀12 )𝑉3
=
(𝑀13 + 𝑀23 )2 − 2(𝐿1 + 𝑀12 )𝐿3
𝑑𝑡
The output current ripple is equivalent to the phase current ripple in this case. By
inserting Eq. 5-19 as well as V3=Vout and the valid time instance ∆t=D/fs, the current
ripple can be calculated according to:
Eq. 5-28
∆𝑖𝑜𝑢𝑡 = ∆𝑖3 =
𝑉𝑑𝑐 /2(𝑀13 + 𝑀23 ) − 2(𝐿1 + 𝑀12 )𝑉𝑜𝑢𝑡 𝐷
∙
(𝑀13 + 𝑀23 )2 − 2(𝐿1 + 𝑀12 )𝐿3
𝑓𝑠
Case 2: No Power Transfer from the Primary
In this case the primary winding currents are zero and their derivatives are zero as
well. The two secondary windings are in parallel, tied to the output. Therefore, the
derivatives of the two rectifier currents are equivalent, too.
119
Eq. 5-29
𝑑𝑖1
𝑑𝑖2
=−
=0
𝑑𝑡
𝑑𝑡
Eq. 5-30
𝑉3 = 𝑉4 = 𝑉𝑜𝑢𝑡
Eq. 5-31
𝑑𝑖3 𝑑𝑖4
=
≠0
𝑑𝑡
𝑑𝑡
By inserting the conditions from Eq. 5-29 to Eq. 5-31 into the matrix equation from
Eq. 5-4, the third and fourth row yield:
Eq. 5-32
Eq. 5-33
𝑉3 = 𝐿3
𝑑𝑖3
𝑑𝑖4
− 𝑀34
𝑑𝑡
𝑑𝑡
𝑉4 = 𝐿4
𝑑𝑖4
𝑑𝑖3
− 𝑀43
𝑑𝑡
𝑑𝑡
Rearranging, under the consideration of Eq. 5-12 and Eq. 5-30, leads to:
Eq. 5-34
𝑑𝑖3
𝑉3
𝑉𝑜𝑢𝑡
=
=
𝑑𝑡
𝐿3 − 𝑀34 𝐿3 (1 − 𝑘34 )
The insertion of ∆t=(1−2D)/(2fs ) yields the current ripple, where it has to be taken
into account that the output current is the sum of the two equivalent phase currents:
Eq. 5-35
∆𝑖𝑜𝑢𝑡 = ∆𝑖3 + ∆𝑖4 = 2∆𝑖3 =
2𝑉𝑜𝑢𝑡
1 − 2𝐷
∙
𝐿3 (1 − 𝑘34 )
2𝑓𝑠
5.3.5 Flux Density Swing
According to Figure 5-5 it is most appropriate to calculate the flux swing in case of
power transfer to the secondary, because the total flux swing can be calculated at
once using only one time instance. In general, the induction law gives:
Eq. 5-36
𝑑𝜑1 /𝑑𝑡
𝑉1
−𝑁1
𝑑𝜑 /𝑑𝑡
𝑉
−𝑁
( 2) = ( 2) ( 2
)
𝑉3
−𝑁3
𝑑𝜑3 /𝑑𝑡
𝑉4
−𝑁4
𝑑𝜑4 /𝑑𝑡
The considerations from Eq. 5-19 to Eq. 5-21 yield:
120
Eq. 5-37
𝑉1
𝐿1
𝑉2
−𝑀21
( )=(
𝑉3
𝑀31
𝑉4
−𝑀41
−𝑀12
𝐿2
−𝑀32
𝑀42
𝑀13
−𝑀23
𝐿3
−𝑀43
−𝑀14
𝑑𝑖1 /𝑑𝑡
𝑀24
−𝑑𝑖1 /𝑑𝑡
)(
)
−𝑀34
𝑑𝑖3 /𝑑𝑡
𝐿4
0
The calculation of the inductance matrix according to Eq. 5-6, Eq. 5-7 and Eq. 5-15 to
Eq. 5-18 allows the phase current ripple calculation according to Eq. 5-28.
Using Eq. 5-26 will give the primary current slope. This allows the calculation of all
voltages according to Eq. 5-37 and the flux swing calculation under the consideration
of Eq. 5-36 and ∆t=D/fs:
Eq. 5-38
−𝑉1 / 𝑁1
∆𝜑1
∆𝜑2
−𝑉 /𝑁
(
) = ( 2 2 ) (𝐷/𝑓𝑠 )
∆𝜑3
−𝑉3 /𝑁3
∆𝜑4
−𝑉4 /𝑁4
If the leakage reluctances are neglected, the centre leg flux swing is the sum of either
both primary or both secondary windings:
Eq. 5-39
∆𝜑𝑐𝑒𝑛𝑡𝑟𝑒 = ∆𝜑1 + ∆𝜑2 = ∆𝜑3 + ∆𝜑4
It has to be noticed that the frequency in the centre leg is doubled, due to the
interaction of the outer leg fluxes. The corresponding flux densities will be calculated
with the given magnetic cross-sections. For EE cores, the centre leg flux density is
calculated with the equivalent magnetic cross section given in the manufacturer’s
data sheet. For the outer legs, only one half of the equivalent magnetic cross section
must be taken into account.
5.3.6 DC Pre-Magnetisation
The pre-magnetisation of the core is caused by the DC components of the secondary
currents. If a symmetrical balancing is assumed, the pre-magnetisation can be
calculated according to the equivalent circuit given in Figure 5-15. Therefore, only the
parts of the flux flowing through core segments are relevant.
121
Rσ1
Rm1
Rm3
Rm2
φ1dc
φ3dc
φ2dc
N3I3dc
Rσ1/2
Rσ2
Rm3
φ1dc
φ3dc
N3I3dc=N4I4dc
N4I4dc
a)
Rm1/2
b)
Figure 5-15: DC equivalent circuit of the integrated current doubler: a) complete circuit;
b) simplified circuit
The DC current through both secondary windings is equivalent as well as the number
of turns, resulting in an equivalent magneto motive force (ampere turns). The
reluctances of the outer legs as well as the leakage parts beside the core can be
assumed to be symmetrical, too. In this case the equivalent circuit from Figure 5-15
a) can be simplified to Figure 5-15 b). The centre leg flux can be calculated according
to:
Eq. 5-40
𝜑𝑐𝑒𝑛𝑡𝑟𝑒,𝑑𝑐
𝑅𝜎1
𝑁3 𝐼3
2
=
∙
𝑅𝑚3 𝑅𝜎1 𝑅𝜎1
𝑅𝑚1 +
+ 𝑅𝑚3
𝑅𝑚3 + 𝑅𝜎1 2
The resulting peak flux is obtained by combining Eq. 5-39 and Eq. 5-40, under the
consideration that only one half of the flux swing must be added to the mean value:
Eq. 5-41
𝜑𝑐𝑒𝑛𝑡𝑟𝑒,𝑝𝑘 = 𝜑𝑐𝑒𝑛𝑡𝑟𝑒,𝑑𝑐 +
∆𝜑𝑐𝑒𝑛𝑡𝑟𝑒
2
The peak flux in the outer legs is calculated in a similar manner, where the maximum
flux swing from Eq. 5-36 must be evaluated in order to obtain the maximum peak.
5.3.7 Design Example: Integrated EE Core Current Doubler
The design specifications from Table 5-6 will be used to design an EE ferrite core
current doubler according to the previous given theory. Assuming an RMS current
4
density of 4 A/mm², the area product approach from Eq. 5-2 leads to 38.2 cm . The
4
EE70 core provides an area product of 38.9 cm and seems to be applicable.
122
In a first step the inductance matrix according to Eq. 5-1, using the equivalent circuit
from Figure 5-14, must be calculated. To allow a more accurate calculation and a
better adjustment of the inductance values for the assembly, the outer legs will be
gapped 0.1 mm. Figure 5-16 shows the primary self-inductance dependent on the
number of turns and the centre leg air gap. The inductance is increasing with
increasing number of turns. Increasing air gap length reduces the self-inductance,
where the impact of the gap is not as significant as the influence of the number of
turns. In general, small gaps and high number of turns will lead to high inductance
values as in case of discrete inductors.
Output Power
2500 W
Output Voltage
48 V
Input Voltage
270 – 400 V
Switching Frequency
50 kHz
Transformer Turn Ratio
1:1
Max. Flux Density
0.3 T
Copper Fill Factor
22 %
Output Current Ripple
<40 %
Table 5-6: Design specifications
Figure 5-16: Self-inductance values dependent on centre leg gap length and number of turns
123
Figure 5-17 shows the coupling value for the secondary windings dependent on the
centre leg gap length according to Eq. 5-14. For an increasing gap length, the coupling
becomes tighter, because the reluctance in the centre leg increases. A higher amount
of flux tends to flow through the outer legs yielding a better magnetic connection
between the outer legs.
Figure 5-17: Secondary-secondary coupling coefficient dependent on gap length
The current ripple can be calculated by using Eq. 5-35, where the inductance term in
the denominator can be interpreted as a scaling factor for the output current ripple
attenuation. Figure 5-18 shows this factor dependent on the centre leg gap length
and the number of turns, where both parameters have a significant influence. A small
gap in combination with many turns will give very high inductance values and
promises well current ripple attenuation.
Figure 5-18: Effective leakage inductance for output current ripple attenuation
124
Figure 5-19 and Figure 5-20 show the normalised current ripple for the nominal and
minimum input voltage dependent on the gap length and the number of turns. It can
be seen that an appropriate combination of secondary turns and centre leg gap
length must be chosen to ensure specific current ripple attenuation. The current
ripple calculation for nominal input voltage represents the stricter case as can be
seen by comparing Figure 5-19 and Figure 5-20.
Figure 5-19: Normalised current ripple for nominal input voltage
Figure 5-20: Normalised current ripple for minimum input voltage
Figure 5-21 and Figure 5-22 show the calculated peak flux density for the nominal and
the minimum input voltage. The nominal input voltage gives the stricter case for the
125
maximum peak flux density. By comparing Figure 5-19 and Figure 5-21, the possible
combinations of air gap length and number of turns are reduced. E.g. for 15 turns an
arbitrary gap length between 1 and 10 mm is allowed to ensure a current ripple
below 40 %. But in order to limit the peak flux density to 0.3 T, the gap length must
be greater than 5 mm. The comparison of Figure 5-19 and Figure 5-21 yields, that the
gap length must be at least 6 mm. Finally, 13 or more secondary turns are required.
Figure 5-21: Peak flux density for nominal input voltage
Figure 5-22: Peak flux density for minimum input voltage
The allowed combinations of gap length and number of turns are known now. The
next step is the loss calculation according to the theory given in chapter 3.2.4 and
126
3.2.5. Figure 5-23 shows the winding losses dependent on the number of turns,
where RMS, DC as well as skin and proximity losses for the first harmonic are
recognised. For an increase of the number of turns, the winding losses will increase,
because of smaller possible copper cross section and increasing winding length
leading to higher winding resistance. The minimum input voltage causes a higher
power loss, because of the higher current consumption of the circuit.
Figure 5-23: Winding losses for nominal and minimum input voltage
A high required number of turns can cause high winding losses, where, in turn, the
core losses decrease for increasing number of turns. Figure 5-24 shows the core
losses for nominal and minimum input voltage (calculated according to the model
given in chapter 3.3.3). The higher input voltage causes higher induction, resulting in
higher core losses. But in general, the core losses are relatively low because of the
low switching frequency and induction. Additionally, the ferrite material provides low
specific core losses. The influence of the number of turns is dominant, because the
induction is directly dependent on the number of turns. The air gap has only a
negligent influence on the core losses. A small gap causes higher induction on the
outer legs, resulting in higher core losses. The core losses are mainly assigned to the
outer legs. Because of the cancelation effect of the 180° phase shift, the induction in
the centre leg is that low that the core losses in this part of the core can be almost
neglected, though the frequency is doubled. By taking into account that at least 13 or
127
more turns are required, it becomes obvious that the core losses are almost negligent
in comparison to the winding losses.
Figure 5-24: Core losses for nominal and minimum input voltage
Figure 5-25 shows the total power loss of the component for nominal and minimum
input voltage. The minimum power loss occurs at 6 turns. In case of less turns the
core loses will dominate, where for more turns the winding losses will become
dominant. Nevertheless, 6 turns are not applicable in this example, because the core
will saturate. In order to limit the component losses, the number of turns must be
restricted to the possible minimum. Finally, 13 turns per winding are applicable for
this design.
Setting the primary RMS current density to 6.6 A/mm² and the secondary current
density to 5.3 A/mm² and recognising the 22 % copper fill factor, allows the
application of 1840 x 0.05 mm litz wire for the primary and 3060 x 0.05 mm litz wire
for the secondary.
128
a)
b)
Figure 5-25: Loss balance and optimisation
129
5.4 Downsizing Potential of the Integrated Current Doubler
The area product method is used to choose an appropriate core size for the design
procedure in order to reduce required iterative steps. This method can be used to
estimate the resulting size of a component without calculating a complete design,
too. But it must be taken into account, that the area product method is more or less a
rough approach.
Using the area product method from chapter 5.3.2 and the considerations from Table
5-7, allows the calculation of the required area product dependent on the operation
frequency and the output current ripple. To enable a comparison to the discrete
current doubler Eq. 5-42 and Eq. 5-43, both from [1], can be used:
𝐿 ∙ 𝐼𝑟𝑚𝑠 ∙ 𝐼𝑝𝑘
𝐵𝑝𝑘 ∙ 𝐽𝑟𝑚𝑠 ∙ 𝑘𝑐𝑢
Eq. 5-42
𝐴𝑃𝑖𝑛𝑑𝑢𝑐𝑡𝑜𝑟 =
Eq. 5-43
𝐴𝑃𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑟 =
𝑉1,𝑟𝑚𝑠 ∙ 𝐼1,𝑟𝑚𝑠 + 𝑉2,𝑟𝑚𝑠 ∙ 𝐼2,𝑟𝑚𝑠
4.44 ∙ 𝑓 ∙ 𝐵𝑝𝑘 ∙ 𝐽𝑟𝑚𝑠 ∙ 𝑘𝑐𝑢
Output Power
2500 W
Output Voltage
48 V
Input Voltage
400 V
Peak Flux Density
0.3 T
Max. Copper Fill Factor
22 %
RMS Current Density
4 A/mm²
Core Material
Ferrite N87
Core Loss Limit
100 mW/cm³
Table 5-7: Design considerations
Figure 5-26 shows the approximated boxed size for the EE core current doubler from
chapter 5.2.1 and the corresponding approach for a discrete component (transformer
or inductor). It has to be taken into account that the EE core current doubler requires
two winding sets, placed on the outer legs. Therefore, the boxed size is increased due
to the part of the windings surrounding the outer legs. In comparison, the windings of
a discrete component are placed on the centre leg, leading to the minimum boxed
130
size for the EE core geometry. Appropriate fitting formulas describing the context of
area product and boxed size for the two cases are given in Figure 5-26 as well.
Figure 5-27 shows the calculated boxed size for the integrated EE core current
doubler and the discrete counterpart assuming EE cores for the transformer as well
as for the two inductors, according to the fitting formulas given in Figure 5-26.
Frequency as well as output current ripple is varied in order to examine possible
downsizing potentials. The estimated size for the discrete current doublers is smaller
for lower frequencies. For higher frequencies and higher current ripples, the
downsizing potential for the integrated current doubler seems to be higher. The
frequency dependent inflexion point depends on the current ripple condition and can
be in the range of a few 100 kHz if low current ripple conditions (20 %) are assumed.
For moderate current ripple conditions (40 %) the inflexion point is between 100 and
150 kHz.
Figure 5-26: Context of area product and component box size for E cores
Figure 5-27: Estimated size based on area product approach
131
This effect can be explained by the specific core loss limit of the different
components (restricted to 100 mW/cm³). The peak flux density and the induction is
reduced frequency dependent to maintain the 100 mW/cm³ limit. If the power loss
limit is reached, the component size cannot decrease anymore, unless an improved
cooling method is established.
Figure 5-28 shows the calculated specific core losses for the different magnetic
components, calculated with the loss model from chapter 3.3.3 (Figure 3-11). The
transformer is designed loss limited even at 25 kHz. For the inductor and the
magnetically integrated current doubler, the induction must be limited, due to the DC
pre-magnetisation, in order to avoid saturation. The integrated current doubler
operates with lower inductions in comparison to the discrete transformer and the
inductors. Therefore, the integrated current doubler suffers less specific core losses
and allows downsizing even at elevated switching frequencies.
Figure 5-28: Calculated specific core losses dependent on current ripple and frequency for
the discrete and the integrated current doubler components
5.5 Analysis of different Current Doubler Technologies
Three different current doubler technologies were analysed more in detail to
evaluate and compare their possible performance:



The integrated EE core current doubler
The integrated 5 leg core current doubler
The standard discrete current doubler
132
The investigated circuit is a half bridge push pull converter as depicted in Figure 5-1
and Figure 5-4. The general converter and component specifications are given in
Table 5-8. The frequency is varied from 50 to 400 kHz, where the RMS current density
is varied from 4 A/mm² to 9 A/mm², in order to evaluate size, weight and power loss
of the different magnetic technologies. All components are designed with
appropriate litz wires. The number of strands was adapted to adjust the desired
current density. The proposed ferrite core material was N87 – see example chapter
3.3.3. The discrete transformers and inductors were designed with E cores.
Output Power
2500 W
Output Voltage
48 V
Input Voltage
270 – 400 V
Rectifier Current Ripple
< 20 %
Frequency
50 – 400 kHz
RMS Current Density
4, 6, 9 A/mm²
Transformer Turn Ratio
1:1
Litz Wire (strand diameter)
0.05 mm
Copper Fill Factor
< 30 %
Peak Flux Density
< 300 mT
Core Material
N87
Table 5-8: Converter and component specifications
The integrated current doublers are designed according to the design procedure from
Figure 4-25. The losses are simulated with a circuit simulator, according to the
calculation methods given in chapter 3.2.5 and 3.3.3. RMS, skin, proximity and core
losses are taken into account. The implementation of the power loss models is given
in Appendix VIII. All losses are evaluated for 25 °C ambient temperature as well as
nominal power and voltage. The magnetic circuits used for the simulation are
equivalent to Figure 5-7 b) and Figure 5-9 b).
5.5.1 Size and Weight Comparison
Figure 5-29 and Figure 5-30 show the weight and size of the different current doubler
technologies. In general, size and weight decrease for increasing frequency and
133
current density. A frequency increase enables an induction decrease and allows the
application of a smaller core with smaller magnetic cross section. The increase of the
current density, by means of a conductor cross section decrease, enables more turns
inside a smaller window to achieve required inductance values. In practice, both
effects are used to downsize magnetic components. However, the downsizing of a
component can lead to a power loss density increase. Even if the losses for a smaller
component are kept constant, the smaller component must dissipate the power loss
through a smaller surface. This leads to higher hot spot temperatures of the
component and may require the application of improved cooling methods (see
chapter 5.5.3).
For lower frequencies (up to 100 kHz), the discrete current doubler seems to have
the potential to be smaller and lighter than the integrated ones. But it has to be
taken into account, that only the boxed size was considered in this analysis.
Transformer and inductor boxed size were added, without consideration of any
spacing. But in practice some space between the components must be considered.
For 200 and 400 kHz, the integrated current doubler technologies can be smaller and
lighter than their discrete counterparts, especially for higher RMS current densities.
Figure 5-29: Weight for different current doubler technologies – data partially
published in [93]
134
Figure 5-30: Size for different current doubler technologies
5.5.2 Power Loss Comparison
Figure 5-31 shows the simulated power loss for nominal output power. In many
calculations the integrated current doublers have to take higher losses than their
discrete counterparts. This effect can be explained by the higher harmonic content of
the secondary currents of the integrated current doublers - causing higher skin and
proximity losses in the windings [94]. In comparison, the transformer of the discrete
current doubler is stressed with trapezoidal currents, where the inductors are
stressed with DC current overlain with triangular AC current. Both, trapezoidal and
triangular current shapes have relatively low harmonic content. For a similar RMS
current density, the discrete current doublers have the potential to take less AC
losses than their integrated counterparts. For 400 kHz and for high current densities,
the losses for the integrated current doubler can become excessive, making the
application of this technology difficult and perhaps unsuitable in this operating range.
For frequencies below 400 kHz and for low current densities the losses of the
integrated current doublers are controllable and are in a similar range compared to
the discrete current doublers. For 4 A/mm² the 5 leg core current doubler even
provides the lowest power loss compared to the other technologies.
135
Figure 5-31: Losses for different current doubler technologies – data partially
published in [93]
A more detailed analysis of the losses for the three current doubler technologies is
depicted in Figure 5-32, showing the loss balance for different frequencies and RMS
current densities. The diagrams show that the power loss of the integrated current
doublers is dominated by the skin and proximity losses. These losses tend to increase
for higher frequencies as well as for higher RMS current densities. This effect is more
critical for the integrated EE core current doubler, because of the leakage fields
beside the outer legs. Furthermore, the EE core current doubler usually requires
more turns, causing higher winding resistance, to ensure certain current ripple
attenuation.
The RMS or DC losses can have a significant influence, too. Especially for higher
current densities the DC resistance of the windings increases, causing higher DC
current losses. For higher frequencies the components become smaller and require
less turns with smaller mean length per turn, resulting in lower DC winding
resistances. Therefore, the DC losses can have less significance at higher frequencies.
The core losses for the integrated current doublers are almost negligent for lower
frequencies (100 kHz and below) and increase with higher frequency, because of
increasing specific core losses.
136
a)
b)
c)
Figure 5-32: Loss balance of different current doublers for a) 4A/mm², b) 6A/mm² and
c) 9 A/mm² RMS current density
137
For the discrete current doubler, the skin and proximity losses can have a significant
influence, too, where these losses increase with increasing frequency and are mainly
assigned to the transformer. The DC losses behave vice versa and occur only in the
inductors. They decrease with increasing frequency, because higher switching
frequencies lead to lower required inductance values, resulting in smaller inductors
with lower DC resistance. The DC losses become dominant for higher current
densities. The core losses increase with higher frequency and are mainly assigned to
the transformer. The winding loss components become dominant for higher current
densities.
Each diagram shows a loss minimum for a fixed current density. For lower
frequencies the DC losses become more significant. But for higher frequencies the AC
losses in the windings and the core increase and compensate the decreasing DC
power loss. Above the optimum frequency the decreasing DC losses cannot
compensate the increasing AC losses anymore, leading to a total loss increase. For
low current densities the loss minimum occurs at lower frequencies, because the
influence of the DC losses is limited. For higher current densities this loss minimum is
shifted to higher frequencies.
5.5.3 Temperature Rise
In order to investigate the cooling effort and the feasibility of the previous calculated
components, the hot spot temperature is calculated according to the thermal models
given in Appendix VI. A forced air cooling of 235 m³/h and an ambient temperature of
25°C was assumed exemplarily.
Figure 5-33 shows the calculated hot spot temperatures for the 5 leg and EE core
integrated current doublers as well as for the discrete transformers and inductors
dependent on the RMS current density and the operation frequency. Due to the fact
that a major part of the losses is generated in the windings, the hot spot temperature
will rise with the RMS current density.
The high operation frequencies can cause high AC winding losses for the integrated
current doublers, leading to higher hot spot temperatures. This effect can get more
significant if the component size and the cooling surface reduces. The downsizing can
also lead to lower DC resistance, leading to lower RMS losses counteracting this
effect. In general, the hot spot temperatures for the 5 leg current doublers are higher
compared to the EE core current doublers. This can be explained by the higher
138
thermal resistance of the 5 leg shell-type current doublers. The core of the shell-type
structure partially encloses the winding, which contains the dominant heat source.
Due to the limited heat conductivity of ferrite, the total thermal resistance of the
component will increase and cause higher hot spot temperatures. In contrast, the
winding structure of the EE core current doublers allow a superior cooling especially
with forced cooling methods, because a major part of the winding is easy accessible
by coolant flow or passive heat conductors.
a)
b)
Figure 5-33: Calculated hot spot temperature for a) integrated current doublers and
b) discrete components
Figure 5-33 shows that a few designs of the 5 leg shell-type current doublers can
cause excessive hot spot temperatures, especially for very high RMS current densities
and operation frequencies. Some designs exceed the thermal class B limit and even
the thermal class H limit, though 25°C ambient temperature was assumed. This
139
makes the application of these designs difficult or almost impossible in practice,
because the application ambient temperature can be much higher than 25 °C. In
comparison, the EE core current doublers and the discrete transformers exceed the
thermal class H limit only for 400 kHz and 9 A/mm². All other designs have the
potential to be applicable even if some of them may require superior cooling.
Most of the discrete transformers and all inductors are thermally uncritical if the
previously mentioned cooling effort is assumed. All inductor designs are far below
the thermal class B limit. Their losses occur mainly in the windings and the core losses
are quite low. The low total power loss of the three discrete components along with
the higher total surface and a better heat spread explains the better thermal
performance. The discrete components have the highest potential for low cooling
effort. This means, the application of passive or even pure convectional cooling may
become possible.
5.5.4 Summary
The previous analysis shows that the integrated current doublers can be smaller and
lighter than their discrete counter parts, where the higher harmonic content of the
current can cause higher AC winding losses. Nevertheless, the performance
advantage must be checked application and operation point dependent.
Furthermore, other basic conditions like costs and assembly must be taken into
account as well.
Figure 5-34 shows the power loss of all calculated current doublers dependent on the
component weight. It can be seen that some designs have a similar performance
regarding size and power loss. Other designs seem to be unsuitable due to too high
power loss or weight. The lightest weight at lowest power loss is calculated for the
200 kHz designs, regardless of the current doubler technology. The discrete and the
5 leg shell-type current doublers seem to provide the best performance, where the
EE core current doublers are a bit heavier and cause a bit more losses.
Nevertheless, the EE core current doublers can be cooled very well, due to the
accessible winding structure, allowing a thermal stable operation even for higher
power loss. The thermal behaviour of the discrete components requires the lowest
cooling effort. For the 5 leg shell-type structure, the cooling effort may be a bit
higher, where the losses are usually lower compared to the EE core current doubler.
140
Figure 5-34: Weight and power loss for different current doubler technologies
5.6 Experimental Results
Different integrated current doublers for 50 kHz and 200 kHz were assembled and
benchmarked experimentally in comparison to discrete components. Two half bridge
push pull converters (one for 50 kHz and one for 200 kHz), according to the
specifications given in Table 5-8, were used to test the magnetic components. Table
5-9 shows the specific values of assembled magnetic components for 50 kHz and
48 V output voltage, where Table 5-10 presents the data of the 200 kHz components.
Size and weight of the different components and current doubler technologies is
depicted graphically in Figure 5-36 a) and b). By comparing the figures the size
decrease of the magnetically integrated components becomes obvious. The
integrated EE core current doubler for 50 kHz is about 13 % smaller and 16 % lighter
than its discrete counterpart. The nanocrystalline ring core current doubler from
Table 5-9 shows that a size decrease by means of flux density increase is possible as
well (43 % smaller and lighter), where this component suffered excessive power loss
and tended to be thermally unstable.
The integrated ferrite current doublers for 200 kHz enable an even more significant
size decrease compared to their discrete counterpart. The EE55 current doubler is
141
about 37 % smaller and 9 % lighter, where the 5 leg core is 47 % smaller and 17 %
lighter than the 200 kHz discrete current doubler. Thus, the weight decrease is in a
similar range as for the 50 kHz components, where the downsizing is more significant.
The comparison of the 50 kHz and the 200 kHz components shows that the size of the
discrete current doubler was reduced about 42 %, where the weight decreased 44 %.
The downsizing of the integrated EE70 to the EE55 current doubler enabled a size
decrease of even 56 % and 45 % lower weight.
Table 5-11 shows the data of an integrated EE core current doubler suitable for 12 V
output voltage and 50 kHz. Size and weight is similar to the corresponding 48 V EE70
core current doubler from Table 5-9. The core size is equivalent and the total copper
fill factor is similar, though the secondary wire cross sections and the transformer
turn ratio are different.
a)
b)
c)
Figure 5-35: Magnetically integrated current doublers for 48 V converter output voltage:
a) EE55 (200kHz); b) 84/21/20 5 leg (200kHz); c) nanocrystalline ring core with ferrite
segment (50kHz) – figures taken from [93] and modified
142
a)
b)
Figure 5-36: a) Size and b) weight of different assembled current doublers
143
Discrete
Transformer
Discrete
Inductors 1/2
Integrated EE
Core
Current
Doubler
Integrated Ring
Core
Current
Doubler
Core Size
ETD59
ETD54
E70
54/38/26
Material
N87
3C90
Mf102
Vitroperm 500F,
N97 segment
Prim. Litz Wire
180x0.2mm
180 x 0.2mm
1980x0.05mm
480x0.1mm
Sec. Litz Wire
180x0.2mm
-
3300x0.05mm
720x0.1mm
Number of
Turns
Npri = Nsec = 9
N = 20
Npri = Nsec = 13
Npri = Nsec = 15
Sec.-Prim. Turn
Ratio
Ns/Np = 0.999
-
Ns/Np = 0.983
Ns/Np = 0.993
Ns/Np = 0.989
Ns/Np = 0.961
Prim. DC
Resistance
[mΩ]
3.7
6.5 / 6.8
8.9 / 9.2
9.2 / 8
Sec. DC
Resistance
[mΩ]
3.6
-
6.2 / 5.2
5.8 / 5.4
Self-/ Magn.
Inductance
[μH]
555
51.7 / 51.4
146.1 / 149.6
79.9 / 69
-
-
0.851 / 0.827
0.137 / 0.159
0.183
0.144
0.41
0.27
929
590
24
20.8
Sec.-Sec.
Coupling
Boxed Size
[dm³]
Total Size
[dm³]
Component
Weight [g]
Total Weight
[g]
Current Ripple
@ PN [A]
0.47
452
328 / 324
1104
15 (inductor), 7.2 (rectifier
output)
Table 5-9: Data of discrete and integrated current doublers for 48 V, 50 kHz – data partially
previous published in [93]
144
Discrete
Transformer
Discrete
Inductors 1 / 2
Integrated EE
Core Current
Doubler
Integrated
5
Leg
Current
Doubler
Core Size
ETD49
ETD39
E55
84/21/20
Material
N87
N97
Fi325
Fi325
Prim. Litz
Wire
1980x0.05mm
2640x0.05mm
1980x0.05mm
1980x0.05mm
Sec. Litz Wire
1980x0.05mm
-
2640x0.05mm
2640x0.05mm
Number of
Turns
Npri = Nsec = 9
N = 12
Npri = Nsec = 9
Npri = Nsec = 7
Sec.-Prim.
Turn Ratio
Ns/Np = 0.999
-
Ns/Np = 0.982
Ns/Np = 0.99
Ns/Np = 0.993
Ns/Np = 0.986
Prim. DC
Resistance
[mΩ]
5.2
3.4 / 3
4.6 / 5.2
5/5
Sec. DC
Resistance
[mΩ]
4
-
3.4 / 3.6
3.4 / 3.5
435
11.8 / 11.6
73.1 / 73.5
38.2 / 38.3
-
-
0.787 / 0.783
0.603 / 0.604
0.106
0.091
0.18
0.153
510
468
Self-/ Magn.
Inductance
[μH]
Sec.-Sec.
Coupling
Boxed Size
[dm³]
Total Size
[dm³]
Component
Weight [g]
Total Weight
[g]
0.288
251
155 / 151
557
Table 5-10: Data of discrete and integrated current doublers for 48 V, 200 kHz – data partially
previous published in [93]
145
Integrated EE Core Current Doubler
Core Size
E70
Material
Mf102
Prim. Litz Wire
1035 x 0.071mm
Sec. Litz Wire
4140 x 0.071mm
Number of Turns
Npri = 12, Nsec = 4
Sec.-Prim. Turn
Ratio
Ns/Np = 0.332
Prim. DC Resistance
[mΩ]
14.1
Sec. DC Resistance
[mΩ]
0.8
Pri. Magnetising
Inductance [μH]
439.5 / 441.1
Sec. Magnetising
Inductance [μH]
47.2 / 45.6
Sec.-Sec. Coupling
0.919
Boxed Size [dm³]
0.41
Component Weight
[g]
988
Current Ripple @
PN [A]
38.27
Table 5-11: Data of integrated current doubler for 12 V, 50 kHz – data partially previous
published in [94]
5.6.1 12V and 48V Converters with IGBT Bridge operating at 50 kHz
Table 5-12 shows the converter specifications for the 50 kHz push pull converters.
The 12 V system was investigated with the integrated EE core current doubler, where
the 48 V system was tested with the discrete current doubler as well. Figure 5-38
shows the measured efficiency values for the three 50 kHz systems. By comparing the
efficiency curves, it can be seen that the maximum efficiency values for all converters
are higher at lower input voltage. For the 48 V converters, this is valid for nominal
power, too. But the 12 V converter shows higher nominal power efficiency for
nominal input voltage.
146
Figure 5-37: Measured secondary currents of EE core current doubler (magenta/green:
rectifier diode currents; red: rectifier output current)
Converter
1
2
Output Power
Output Voltage
Input Voltage
3
2500 W
12 V
48 V
48 V
260 V – 400 V
270 V – 400 V
270 V – 400 V
Operating
Frequency
50kHz
Magnetic
Components
EE70 Integrated
Current Doubler
EE70 Integrated
Current Doubler
ETD59
Transformer /
2 x ETD 54 Choke
Primary
Semiconductors
650 V, 40 A, F5
IGBT
650 V, 50 A, F5
IGBT
650 V, 40 A, F5
IGBT
(IKW40N65F5)
(IKW50N65F5)
(IKW40N65F5)
170 V, 400 A
Schottky Diodes (2
x STPS 200170TV1)
400 V, 60 A Si
Diodes
400 V, 60 A Si
Diodes
(VS-60CPU04-F3)
(VS-60CPU04-F3)
none
3.3 nF, 12 Ω
3.3 nF, 24 Ω
Nominal Efficiency
(270V / 400V)
87 % / 89 %
93.6 % / 93.5 %
92.4 % / 91.8 %
Maximum
Efficiency (270V /
400V)
92.6 % / 92.3 %
95 % / 94.1 %
93.1 % / 92.2 %
Rectifier
Turn off Snubber
(Rectifier)
Table 5-12: Data of investigated Si based push pull converters (50 kHz) [93]
147
For lower output power, the switching and AC losses will dominate, where for higher
output power, the conduction and RMS power loss will be dominant. This can be
explained by the high output currents required for higher output power (>1000 W),
causing dominant conduction losses in the rectifier - especially for 12V output
voltage. For an output power of higher than 750 W (400 V) or 500 W (260 V), the
efficiency values are strongly decreasing, because of the increasing input and output
current. The conduction losses of the semiconductors and the RMS losses of the
transformer increase. The efficiency values at nominal power are much lower (3.3 %
for 400V and even 5.6 % for 260 V) than the maximum partial load efficiency.
In comparison, the 48 V converters show very flat efficiency curves. For 400 V input
voltage the maximum efficiency is only slightly higher than the efficiency at nominal
power. By comparing the 270 V efficiency curves, the maximum partial load efficiency
values are a bit higher than the nominal efficiency, because of the previous
mentioned higher current consumption at higher output power. For the 48 V systems
with integrated current doubler, the nominal efficiency is only 0.1 % higher at 270 V
in comparison to 400 V input voltage. For the converter with discrete current
doubler, the efficiency is 0.6 % higher. For the 12V converter, the efficiency
difference at nominal power for 400 V and 260 V is 2 %. This shows that the
conduction losses for the 12 V converter are dominant, where for the 48 V converters
the switching losses have significant influence, too.
Finally, the 48 V systems have a significant higher efficiency than the 12 V system
(6.6% for 260 / 270 V and 4.5 % for 400 V). The high output current of the 12 V
converter is causing excessive conduction and RMS losses, requiring the application
of a large amount of copper and silicon. Though the switching losses of the 48 V
converters have a significant influence on the efficiency, the low RMS losses enable
high efficiency values even at nominal power. As a result, the total efficiency for 48 V
is superior higher in comparison to the 12 V system. The maximum partial load
efficiency values for the 12 V systems are not significant lower. The publication [95]
investigated several semiconductor technologies in order to increase the efficiency of
a 12 V system.
148
Figure 5-38: Efficiency measurement of Si based push-pull converters operating at 50 kHz
Beside the higher efficiency, the 48 V converters should be cheaper, because of the
smaller required chip size of the rectifier as well as the reduced cooling effort. In
order to feed the remaining 12 V consumers in the vehicle, a small 12 V grid, maybe
with less power, is remaining. Thus, an additional 48 V to 12 V DC-DC converter
without galvanic isolation is required. This converter can be much smaller and
cheaper than the 48 V converter. The 12 V converter provides 89 % efficiency at
nominal power (400 V), where the 48 V converter with integrated current doubler
enables 93.5 % efficiency. Thus, an additional 48-12 V DC-DC converter must have at
least 95.2 % efficiency, in order to obtain an efficiency advantage. By comparing the
total power loss, the 12 V converter has 309 W losses, where the 48V converter
dissipates only 174 W. Thus, the 48-12 V DC-DC converter requires less than 135 W
losses, in order to obtain a better total performance regarding power loss.
Figure 5-39: Efficiency values for 50 kHz IGBT converters
149
Figure 5-40: 50 kHz push pull half bridge converter (equipped with 12V rectifier)
5.6.2 48V Converters with SiC MOSFET Bridge operating at 50 kHz
The bridge of the 50 kHz push pull converter was also equipped with SiC normally off
MOSFETs. Figure 5-41 shows the efficiency measurement of the converter
configurations given in Table 5-13. The 48 V converter given in Table 5-12 (converter
3) was equipped with a SiC bridge, in order to show the possible performance
advantage of SiC technology. The voltage class of the rectifier diodes was reduced to
300 V. A further nanocrystalline ring core integrated current doubler was tested as
well.
Converter
4
Output Power
Output Voltage
Input Voltage
5
2500 W
48 V
48 V
270 V – 400 V
270 V – 400 V
Operating Frequency
Magnetic Components
50 kHz
Nanocrystalline
Integrated Ring Core
Current Doubler
ETD59 Transformer /
2 x ETD 54 Choke
1200 V, 40 A,
SiC MOSFET
1200 V, 40 A,
SiC MOSFET
300 V, 120 A Si Diodes
300 V, 120 A Si Diodes
(DPG120C300QB)
(DPG120C300QB)
Nominal Efficiency
(270V / 400V)
93.6%* / 91.9 %
93.4 % / 93.3 %
Maximum Efficiency
(270V / 400V)
95.5 % / 94.6 %
94.6 % / 94.2 %
Primary Semiconductors
Rectifier
Table 5-13: Data of investigated SiC based push pull converters (50 kHz, * evaluated at 2kW)
150
The comparison of the efficiency curves in Figure 5-41 shows that the SiC equipped
push pull converter achieved a nominal power efficiency of 93.3 % at 400 V. This is
1.5 % higher than for the Si based variant. The maximum efficiency at 400 V is even
2% higher. For 270 V, the efficiency increase is 1 % for nominal power and 1.5 % for
the maximum efficiency. It has to be noticed that the voltage class of the rectifier
diodes was changed to 300 V. Thus, the switching losses on the secondary side of the
converter are reduced as well. Nevertheless, a major part of the superior efficiency
advantage can be explained by low switching losses of the primary SiC bridge. As
depicted in Figure 5-41, this efficiency increase can be observed over the complete
power range.
Figure 5-41: Efficiency measurement of SiC based push-pull converters operating at 400 V
input voltage, 50 kHz – figure taken from [93] and modified
The circuit was also equipped with a nanocrystalline ring core current doubler. The
efficiency measurement for 400 V is depicted in Figure 5-41 as well. The maximum
efficiency is about 0.4 % to 0.9 % higher compared to the discrete current doubler,
where the nominal power efficiency is 1.4 % lower. The efficiency measurement at
270 V and 2500 W was not possible, because it was not possible to transfer the full
power for this operation point - the transformer leakage inductance of the integrated
current doubler was too high.
151
Figure 5-42: Efficiency values for 50 kHz SiC converters
The lower nominal power efficiency of the ring core current doubler can be explained
by high AC and DC winding losses, which become excessive at nominal power. A
thermal stable operation was only possible with forced air cooling at 25 °C ambient
temperature. Especially the edges, where the ferrite segment and the nanocrystalline
rings come together, have to accept excessive magnetic AC fields, causing high
proximity losses. This results in decreasing nominal power efficiency.
5.6.3 48V Converters with GaN Bridge operating at 200 kHz
Table 5-14 shows the specifications of different 200 kHz push-pull half bridge
converters with current doubler rectifier. The bridge was assembled with 650 V GaN
semiconductors, where the rectifier was realised with SiC diodes. The application of
these innovative semiconductor technologies enable the converter to operate at
elevated switching frequencies. The objective is to enable high power densities at
acceptable converter efficiency values.
Three different concepts for the magnetic components were benchmarked in this
experiment. The conventional discrete current doubler using one transformer and
two chokes was compared to the standard EE core integrated current doubler and
the proposed 5 leg integrated alternative – all components were realised with ferrite
core material. The operation frequency of 200 kHz was chosen because of the result
from chapter 5.5, where the magnetically integrated current doublers seem to
provide the best trade-off between limited power loss and high power density at
200 kHz.
152
Figure 5-43: 200 kHz push pull converter with magnetically integrated 5 leg current doubler
Figure 5-44 shows the measured efficiency values for 300 V and 400 V input voltage
for all converters. The efficiency values for 300 V input voltage are a bit less than
94 % in the kW range and higher than the 400 V efficiency values. The
semiconductors operate in a better operating point (higher current but less voltage)
with higher duty cycles, causing less power loss. The efficiency curves for all
converter configurations are similar and quite flat if the output power is higher than
700 W.
For 400 V, the efficiency of the converter using the discrete current doubler is still
about 93 %, even at 500 W. The efficiency for the converters with magnetically
integrated current doublers is a bit lower and reaches 92 % at about 1 kW. The
simulation and calculation results from chapter 5.5 predicted that the integrated
current doublers suffer higher losses. Instead, they have the potential to be smaller
and lighter than their discrete counterparts. In fact, the integrated current doublers
have about half of the size and weight, compared to the discrete magnetic
components (see Table 5-10). Finally, the magnetically integrated current doublers
were not able to improve the efficiency of the converter in this frequency range. But
the required magnetic material and the converter size was reduced, leading to a
power density increase.
153
Converter
6
7
Output Power
2500 W
Output Voltage
48 V
Input Voltage
300 V – 400 V
Operating
Frequency
Magnetic
Components
Primary Switches
8
200kHz
ETD49 Transformer
/ 2 x ETD 39 Choke
EE55 Integrated
Current Doubler
5 Leg (84/21/20)
Integrated Current
Doubler
650 V, 30 A, GaN (RFJS3006F)
Freewheeling
Diodes
600V, 20A, SiC Schottky Diodes (C3D20060D)
Rectifier
650 V, 50 A SiC Schottky Diodes (C5D50065D)
Turn off Snubber
(Rectifier)
none
Efficiency @ 0.5PN
(300V / 400V)
93.4 % / 93.1 %
93.7 % / 92.0 %
93.8 % / 92.3 %
Maximum
Efficiency
(300V / 400V)
94.3 % / 93.2 %
93.7 % / 92.0 %
95.2 % / 93.2 %
Table 5-14: Data of investigated GaN based push pull converters (200 kHz)
Figure 5-44: Efficiency measurement for 200 kHz converters with GaN and SiC
semiconductors
154
Figure 5-45 shows the thermal behaviour of all magnetic components, where the
winding losses are dominant according to the temperature rise. The temperature of
the integrated 5 leg current doubler is the highest in this experiment, because of the
shell type core structure. The integrated EE55 current doubler can dissipate the heat
more efficient, because the winding surface to the ambient is larger, compared to the
5 leg configuration. The discrete transformer suffers a similar temperature rise than
the integrated EE55 current doubler. Instead, the temperature of the discrete chokes
is a bit lower because of lower AC losses.
a)
b)
c)
d)
Figure 5-45: Thermal measurement of a) integrated 5 leg current doubler, b) integrated EE55
current doubler, c) discrete transformer and d) discrete inductor (all transformers cooled
with 18 m³/h forced air cooling; inductors are natural convection cooled)
155
However, the efficiency values at lower input voltage are higher for all converter
configurations. This is a hind that the switching losses of the semiconductors are
dominant compared to the conduction losses. The switches operate with very low
dV/dt values (about 10 V/ns). Higher converter efficiency is possible if the switches
will operate with higher dV/dt values. But in order to improve the EMI behaviour of
the circuit, a low switching speed was chosen. Furthermore, different RCD snubber
configurations for the rectifier were tested in order to limit the voltage oscillations at
the diodes. But the effect of the snubbers on the EMI behaviour was limited and they
decreased the efficiency of the converter. Nevertheless, the 650 V reverse blocking
voltage capability of the SiC diode technology is capable to handle even excessive
voltage overshoots in this application.
Despite different trials to minimize EMI effects with snubbers, the gates of the
semiconductors and the currents on the primary as well as on the secondary suffered
considerable EMI and oscillations. This is the reason why the efficiency measurement
was only performed up to 1250W. For higher power, the EMI at the gate of the
transistors tended to destroy the gate or cause parasitic turn on. The non-linear
capacitances of the switch can lead to oscillating behaviour, which is partially
influenced by the commutation path.
The transformer leakage inductance is forming parasitic resonance circuits with the
capacitances of the semiconductors. Especially the Schottky technology of the diodes
provides high component capacitances, resulting in the fact that high transformer
leakage inductance values and high component capacitances cause low parasitic
resonance frequencies. If the converter is operated at elevated switching
frequencies, the probability for parasitic oscillations increases. This means, the
transformers must be connected with very short cables in order to restrict the
parasitic leakage inductance to a minimum. Improved assembly and connection
techniques of the components are required, in order to get proper operating
converters - capable of transferring power in the kW range with several 100 kHz.
Otherwise, the superior performance advantages of GaN and SiC semiconductor
technologies cannot be fully utilised.
156
Figure 5-46: Measured voltage and current waveforms of the integrated EE55 current
doubler - green/purple: gate source voltages; dark/light blue: secondary transformer
currents; red: rectifier output current
Figure 5-47: Measured voltage and current waveforms of the integrated EE55 current
doubler - green/purple: gate source voltages; dark blue: rectifier diode current; light blue:
rectifier diode voltage
Figure 5-48: Efficiency values for 200 kHz GaN converters
157
5.6.4 48V Converters with SiC Bridge operating at 200 kHz
The GaN switches were replaced by SiC semiconductors, because nominal power was
not reached with the GaN semiconductors. The components of the converters can be
found in Table 5-15. The replacement of the semiconductors enabled a more robust
converter behaviour regarding EMI. Finally, the converter operation at nominal
power was possible.
Converter
9
10
11
Output Power
2500 W
Output Voltage
48 V
Input Voltage
300 V – 400 V
Operating
Frequency
Magnetic
Components
200 kHz
ETD49 Transformer
/ 2 x ETD 39 Choke
Primary Switches
EE55 Integrated
Current Doubler
5 Leg (84/21/20)
Integrated Current
Doubler
1200 V, 20 A, SiC (SCT2080KE)
Freewheeling
Diodes
600V, 20A, SiC Schottky Diodes (C3D20060D)
Rectifier
650 V, 50 A SiC Schottky Diodes (C5D50065D)
Turn off Snubber
(Rectifier)
none
Efficiency @ PN
(300V / 400V)
89.2 % / 88.5 %
89.6 % / 89.4 %
89.6 % / 89.3 %
Maximum
Efficiency
(300V / 400V)
93.3 % / 92.6 %
93.8 % / 92.1 %
93.9 % / 93 %
Table 5-15: Data of investigated SiC based push pull converters (200 kHz)
Figure 5-49 shows the efficiency measurements of the converters for 300 V and
400 V. For 300 V, the efficiency is higher, especially in the partial load range. But at
nominal power the efficiency difference is almost negligent. For 400 V, the efficiency
values for the integrated EE55 current doubler are similar to its discrete counterpart.
For 300 V, the integrated EE55 current doubler provides a bit higher efficiency in the
partial load range. But for nominal power the converters with integrated current
doublers have no significant advantage. The efficiency values for the integrated 5 leg
158
current doubler converter are a bit higher over the complete power range, for 300
and 400 V as well. But finally, as already mentioned in the previous chapter, there is
no significant efficiency increase for the magnetically integrated current doublers.
The efficiency values for the converters drop down to about 89 % at nominal power.
The higher current consumption of the circuit at higher power increases the
conduction and RMS losses of the converter components. Especially the
semiconductor chip sizes for the bridge and the rectifier are relatively small. A chip
size increase or paralleling of chips can reduce the conduction losses in the higher
power range and enable a flatter efficiency curve with the result of higher nominal
power efficiency values.
Figure 5-49: Efficiency measurement for 200 kHz converters with SiC semiconductors
Figure 5-50: Measured voltage and current waveforms of the integrated current doubler green/purple: gate source voltages; dark blue: rectifier diode current; light blue: rectifier
diode voltage
159
Figure 5-51: Efficiency values for 200 kHz SiC converters
5.7 Executive Summary
It was shown that the integrated current doublers can be smaller and lighter than
their discrete counterparts. If the discrete transformers operate at power loss limit, a
further downsizing is not possible. The integrated components can take advantage by
operating under additional DC bias condition. Therefore, the core is better utilised
and the discrete chokes become obsolete. For the ferrite components, a weight
decrease of the integrated current doublers compared to their discrete counterparts
of about 10 to 15 % is possible in practice – even at low operation frequencies. The
size decrease is even more significant: about 15 % for 50 kHz and 35 – 45 % for
200 kHz.
The components can be optimised to increase converter efficiency and/or power
density. The frequency increase from 50 kHz to 200 kHz enabled a size and weight
decrease in a range of about 45 % for the discrete current doubler. The downsizing
advantage of the integrated current doubler technologies can be even higher at
elevated frequencies. The magnetically integrated current doublers allowed
downsizing of about 55 % for the frequency increase from 50 to 200 kHz.
The power loss of the integrated current doublers depends on the winding structure
and the used litz wire. The use of litz wire with small strand diameter is absolutely
recommended, in order to prevent high AC winding losses. Especially the proximity
losses can become excessive for some core geometries if inappropriate wire is used.
Nevertheless, the assembly of low-loss integrated current doublers is possible, where
160
the ferrite designs seems to be most appropriate, because the core losses do not
contribute that much to the total power loss.
The nanocrystalline design does not seem to lead to a better performance, because
of the high manufacturing and assembly costs. Though, the power density can be
improved significant, it is hard to decrease power loss with this technology.
Furthermore, the application is restricted much below 100 kHz for this material.
The experiments demonstrated that SiC and GaN technology have the potential to
allow a superior efficiency increase for hard switched push-pull converters with
current doubler rectifier. These innovative semiconductor technologies allow high
efficiency values, even at elevated switching frequencies. Though the efficiency
values for all converters were at least acceptable, [96] and [97] showed that
resonance converters can enable superior efficiency values (> 94 %). Nevertheless,
the operation in the 100 kHz range allows reducing size, weight and costs of the
magnetic components, where the higher semiconductor expenditures must be
accepted. Alternatively, low-priced IGBT are applicable hard switched even at 50 kHz.
Thus, higher costs for the magnetic components are acceptable if the semiconductor
costs can be reduced disproportionately high.
Though the operation at 200 kHz was possible, the operation of the converters in the
100 kHz range requires a low inductive assembly and connection technique for the
transformers. Low leakage inductances of the transformer itself as well as very short
connection wires are absolutely recommended. Finally, the operation at 400 kHz is
critical because of the leakage inductance and the high specific winding losses,
requiring superior cooling.
161
6
Combined Common-Differential Mode Chokes
Many power electronic applications have to face the issue that the converters have
to become smaller, lighter and low-priced. The filters contribute a non-negligent part
to the total system size and expenditures. In general, the filter size is decreased by
increasing the switching frequency of the system. Small required inductance and
capacitance values reduce the expenditures and the required space. This effect is
cumulative with the fact that small systems require only small and cost-efficient
housings as well. Today innovative semiconductor technologies like SiC or GaN
enable power electronic converters in the lower kW range to operate at frequencies
in the range of several hundred kHz, even for hard switched applications. In the last
years, the semiconductor manufacturers enhanced Si technologies (e.g. CoolMOS) as
well. Increasing the switching frequency is still the common method of choice to
downsize the filters.
However, another method to decrease the effort regarding filters is the assembly of
combined common-differential mode chokes. Conventional EMI filters are assembled
with discrete common and differential mode chokes as well as X and Y capacitors.
This work will only treat the magnetic components of the EMI filters. In theory, the
replacement of two chokes, fulfilling two different functions, by only one choke
fulfilling both functions with a similar performance will enable a superior advantage
regarding the assembly effort.
Therefore, the following questions are in focus of this chapter:



Can combined common-differential mode chokes reduce the size and weight
of EMI filters?
Is it possible to reduce the power loss of a filter by using combined commondifferential mode chokes?
Is there a performance advantage of common-differential mode chokes
regarding insertion loss?
6.1 Basics of Common and Differential Mode Noise
This work will only treat conductive emissions, where noise can be separated into
differential and common mode noise signals. The differential mode noise uses the
line conductor to flow to the sink and will return via the neutral conductor to the
source. In comparison, the common mode signal will be distributed via the line and
the neutral conductor in the same direction, where the ground is the return path. The
162
common mode noise is distributed via capacitive coupling effects to the ground. The
filters of a system are used to change the impedance behaviour in a way that the
noise does not disturb the sink.
IL
IDM
Zi
IL
UL
Zi
ZL
UDM
ZL
ICM
IN
PE
ICM/2
Zi
UNutz
IN
Zi
IDM
UN
ZN
UL
UNutz
UDM,0
a)
ICM/2
UN
ZN
ICM
b)
ZCM
PE
UCM,0
Figure 6-1: Propagation of a) differential mode and b) common mode noise – figure taken
from [98]
The objective of the chokes is to provide high impedance paths for the noise, to
reflect or absorb the noise. A part of the noise is attenuated by the impedance of the
choke. In a physical sense this means that a part of the energy of the electrical noise
is converted into heat and dissipated through the component surface. Dependent on
the complete filter and system structure/impedance, a part of the noise is blocked by
the choke and flowing through other parasitic paths. In contrast to the chokes, the
idea of a capacitor is to short the noise source and provide a low impedance path for
the noise back to the source. In general, there are three basic principles how to face
EMI noise:



Try to avoid noise if possible (e.g. try to enable low dV/dt values and
appropriate PWM)
Block noise by high impedance in the noise path
Short noise to provide a low impedance path back to the source
In practice, the generation and distribution of noise is so complex that EMI problems
will be faced by combining all the three previous mentioned methods. A good
introduction regarding EMI basics and the explanation of generation and
transmission theory of noise is given by [99].
163
6.2 Parasitic Effects in Filter Chokes
In principle the main issues for the design of filter chokes are similar to the
development of other magnetic components. The components should provide small
size and weight, low power loss and expenditures and they have to be thermally
stable. A further aspect for filter components is the issue of parasitic effects. Beside
the inductive behaviour, each choke has a capacitive behaviour caused by winding
(turn to turn and intra winding capacitance) and core-winding capacitances. Filter
chokes must provide inductive behaviour even at elevated frequencies, to attenuate
the dominant parts of the noise.
R
C
L
Figure 6-2: Simplified RLC circuit for chokes
A simplified electrical model of a choke is the RLC parallel circuit (see Figure 6-2 or
[100]). The capacitance C represents the total capacitive behaviour of the
component. The resistance can be interpreted as a lumped equivalent loss resistance.
The component behaves inductive until the resonance frequency occurs – the
impedance increases linear with increasing frequency. If the resonance is exceeded,
the capacitive behaviour will become dominant and the impedance is decreasing with
increasing frequency. More detailed models and equivalent circuits especially for
common mode chokes can be found in [101].
The lowest winding capacitance can be achieved by single layer windings (see Figure
6-3 a), because the potential difference between two adjacent windings is the
minimum possible [45]. Increasing the distance between the different turns leads to a
further decrease of the winding capacitance. This is a technique leading to good
electrical behaviour of the choke, though only very low copper fill factors can be
achieved. Two layers (see Figure 6-3 b) provide the worst case, enabling the highest
possible winding capacitance [102] – the first and the last turn are adjacent to each
other, yielding the highest possible potential difference. However, better fill factors
and higher inductance values can be achieved by using multilayer techniques.
A trade-off between the previous described winding techniques can be found in
Figure 6-3 c) and d), called bank and progressive winding [45]. The bank winding
164
allows the implementation of multi-layer windings with limited potential difference
between adjacent turns. This technique can enable a capacitance decrease of about
25% compared to conventional multilayer windings [103]. The progressive winding
allows even better fill factors, but provides higher potential differences between
adjacent turns. Another important design issue, beside the winding capacitance, is
the manufacturability [45]. Complex winding structures may cause higher effort and
expenditures regarding assembly of the choke.
11
a)
1
2
3
8
5
c)
1
5
6
9
6
2
4
17
7
3
b)
14
4
10
18
15
11
1
13
d)
1
9
3
8
9
7
16
12
10
2
6
2
8
4
6
14
5
3
7
5
13
4
12
10
11
Figure 6-3: Different winding structures: a) single layer; b) double layer; c) bank;
d) progressive – figure based on representation from [45]
6.3 Suitable Core Geometries for Common-Differential Mode
Chokes
The conventional common-mode choke, realised with a high permeable core, is the
basis for many common-differential mode chokes. A conventional common mode
choke is wound in a way that the common mode signals will add up inside the core
(direct coupling, see chapter 4.1). For the conventional common mode choke, the
differential mode signal will cancel out and does not contribute significant to the flux
inside the core, instead, it is restricted to flow on external paths outside the core
(inverse coupling, see chapter 4.1). Common-differential mode chokes can increase
the magnetising inside the core to enable a better utilisation. However, this effect
must be recognised for a proper design in order to avoid saturation effects.
Furthermore, the leakage flux of common mode chokes can induce parasitic currents
in adjacent filter components (e.g. capacitors). If heavy leakage effects occur, the
adjacent filter components must be displaced away from the choke. E.g. [104] shows
possibilities to model and to avoid these effects. Instead, some common-differential
mode chokes are improved by means of shielding measures, weakening EMI issues in
the environment.
165
6.3.1 Ring or UU Core with Leakage Segments
The ring core geometry is used in many filter applications, because it provides the
best common mode properties. Uncut UU cores can behave quite similar but provide
a higher magnetic path length, resulting in a lower inductance factor (for the
equivalent magnetic cross-section). Therefore, the application of UU cores for
common mode chokes is uncommon. Dependent on the shape of the edges, the UU
cores provide a higher basic leakage inductance compared to ring cores. The
proposed methods for the enhancement of the differential mode inductance might
be more efficient.
Window Segment
The simplest way to improve the differential mode behaviour of a common-mode
choke is to insert a leakage segment inside the window of the common mode core.
For ring cores this technique is proposed by [43]. The ring or UU core is responsible
for the common mode attenuation, where the segment will increase the differential
mode inductance. The segment can be high permeable, where small gaps between
the common mode core and the segment must be inserted to adjust the reluctance
of the leakage path. Another possibility is to use a low permeable material (iron
powder) and to melt or glue the segment directly on the common mode core. The
windings are wound only on the common mode core and, therefore, provide the
minimum mean length per turn and winding length.
φcm
φcm
φdm
φdm
a)
b)
Rg1
Rm1
Rσ1
Rm3
Vm1
Rg2
Rm2
Rσ2
Vm2
c)
Figure 6-4: Common mode choke with window segment: a) ring core; b) UU core; c) magnetic
equivalent circuit
166
Advantages


Disadvantages

Simple component assembly –
only one segment must be
added, standard chokes can be
modified (ring core)
Minimum mean length per turn
Limited increase of the
differential mode inductance
Table 6-1: Advantages and disadvantages of common mode choke with window segment
Top-Bottom Segments
Another possibility is the insertion of two segments - one on top and one on bottom.
The leakage inductance depends on the segment geometry and the gap length
between the common mode core and the segment. Rectangular block segments are
possible as well as disc or disk like segments covering the complete ring or UU core. If
the segment covers the complete ring or UU core, the minimum gap length between
common mode core and segment is restricted by the wire diameter and the winding
structure. The insertion of very small gaps is not possible in this case. The
combination with other segments inside or outside the common mode core is
possible as well.
The top-bottom placement of segments is advantageous for amorphous or
nanocrystalline common mode cores, because the leakage flux tends to leave the
core in direction of the segments on top and bottom and only a limited amount of
flux leaves the core perpendicular to the lamination.
φcm
φcm
φdm
φdm
a)
b)
Rg1
Rg5
Rm3
Rm4
Rg2
Rg6
Rm1
Rσ1
Vm1
Rm2
Rσ2
Vm2
c)
Figure 6-5: Shielded common-differential-mode choke with top-bottom segment: a) ring
core; b) UU core; c) magnetic equivalent circuit
167
Advantages





Disadvantages
Standard chokes can be
modified (ring cores)
Insertion of segment and/or
discs is possible (more freedom
for segment design)
Minimum mean length per turn
Better flux guidance for
amorphous and nanocrystalline
cores
Reduced leakage flux on top and
bottom


Requires at least two segments
for symmetry reasons
Minimum gap length between
common mode core and
segment is limited by wire
diameter and winding structure
if segment overlaps the winding
Table 6-2: Advantages and disadvantages of common mode choke with top-bottom
segments
External Segments
An alternative method is the placement of external segments parallel to the common
mode core, beside the windings. The differential mode flux tends to leave the
common mode core at the front edges. Instead of providing a high permeable path
inside the window of the common mode core, a high permeable path surrounding
the common mode core is provided. Figure 6-6 a) shows a modified ring core
common mode choke, where the external segments are realised with cut ring cores.
Figure 6-6 b) shows a special external segment, which allows a more precise
adjustment of the air gap between ring core and segment. Figure 6-6 c) depicts an
equivalent assembly for UU cores, where U cores were used as external segments as
well (I cores are possible, too).
However, the use of external segments for the differential mode flux guidance can be
combined with the technique of using a segment inside or on top and bottom of the
common mode core. The use of additional external segments is more expensive,
because more material is necessary to enhance the differential mode flux path.
Nevertheless, all techniques can be combined to enhance a standard common mode
choke.
An advantage of the external segment technique is that the component is well
shielded. A certain amount of leakage flux can leave the component only on top and
bottom. The component will cause lower EMI, making the placement of other filter
components beside the choke possible without taking critical EMI issues.
168
φcm
φcm
φdm
φdm
a)
b)
φcm
Rg3
Rg1
Rm1
φdm
c)
Rg5
Rm2
Rm4
Rm3
Vm1
Rg4
Rg2
Rm5
Vm2
Rg6
d)
Figure 6-6: Shielded common-differential-mode choke with external and window segment:
a) ring core with cut ring segment; b) ring core with adjusted segment; c) UU core with U
segments; d) magnetic equivalent circuit
Advantages




Disadvantages

Standard ring core segments
possible (Figure 6-6 a)
Insertion of other segment is
possible, too
Lower leakage and EMI beside
the choke
Minimum mean length per turn
Heavy external segments
makes the component heavier
and more expensive
Table 6-3: Advantages and disadvantages of common mode choke with external segments
Window Inlay
To address the different behaviour of the common and differential mode, the
component is assembled with two separated cores. One high permeable ring core is
used for the common mode attenuation. The differential mode inductance is
improved by a second ring core with a centre leg. This special core is placed inside the
169
common mode ring core as an inlay (decreases window size) or can be placed on top
of the common mode ring (increases component height). Each winding is wound on
both cores in the same direction. The differential mode flux will add up in the centre
leg of the inlay, where the common mode flux will add up in both ring cores.
Therefore, the inner ring core will contribute to the common mode attenuation as
well even if its permeability might be much lower. The increased magnetic cross
section of both cores reduces the risk of differential mode caused core saturation.
This technique is applicable for UU cores, too. The disadvantage is the requirement of
a special differential mode core. An applicable material for this inlay is iron powder or
a gapped ferrite, where the tooling of a gap causes extra effort.
φcm
Rm1
φdm
Rm3
Rσ2
Vm1
Rg1
Rm5
Rm4
Rm2
Rσ2
Vm2
Rg2
a)
b)
Figure 6-7: Common-mode choke with differential mode inlay: a) component structure;
b) magnetic equivalent circuit
Advantages



Disadvantages
Simple design – scaling of
differential mode inductance by
the geometry of the inlay
Simple component and winding
assembly
Second ring (or UU) core will
contribute to the common mode
inductance



A special inlay (ring or UU with
segment) is required
Either window size is decreased
(inlay technique) or component
height is increased (both cores
assembled on top of each
other)
Mean length per turn increases
Table 6-4: Advantages and disadvantages of common mode choke with inlay
170
6.3.2 EE Core
The E core is a standard core shape. The leakage segment must not be inserted
additionally. The core must be manufactured with very high permeable ferrite
material. The standard power materials are not recommended, because of the
limited permeability (usually up to about 3,000). Another advantage of the
component is the assembly of the windings. The windings can be wound on bobbins,
though no standard bobbins can be used because the windings will be placed on the
outer legs of the core. However, the use of bobbins enables an automatic winding
assembly, even for larger copper cross sections.
The windings are placed on the outer legs in the same direction. Therefore, the
common-mode flux will circulate in the outer legs of the core. The differential mode
flux will cancel out in the outer legs and add up in the centre leg, which acts as a
predefined leakage path. In theory, this magnetic structure fulfils the requirement to
provide a significant common mode and differential mode inductance. But the
component provides non-negligent leakage paths beside the core, contributing to the
differential mode inductance. Additional EMI in the environment of the component
must be accepted, unless improvements like shell-type structures (see Figure 6-10) or
shielding measures (see Figure 6-9) are introduced. Such shielded structures allow a
better adjustment of the differential mode inductance, too.
A more important issue is the common mode path of the flux. In comparison to a ring
core, the flux must pass four edges. The differential mode flux can saturate the inner
edges. This will decrease the effective permeability of the common-mode inductance
and limit the common-mode attenuation.
A more suitable possibility in a technical sense is the use of nanocrystalline or
amorphous E cores. Due to the very high permeability of the material and the
rounded edges, only a very small amount of the flux may contribute to unwanted
component leakage or partial saturation. As a drawback, the centre leg must be
tooled in order to adjust the leakage (differential mode path). In comparison to
ferrite materials this is more expensive for amorphous materials, because a
reestablishment of the insulation at the cutting surfaces is necessary. Another
material independent drawback is the lowered permeability due to the small gaps,
occurring at the legs where the core halves are put together.
171
φcm
φdm
Rm1
Rm3 Rm2
Rσ1
Rσ2
Vm1
a)
Rg
Vm2
b)
Figure 6-8: EE core common mode choke: a) component structure; b) magnetic equivalent
circuit
Rg1
φcm
Rg3
Rm1
φdm
Rm4
Vm1
Rm3 Rm2
Rg
Rg2
a)
Vm2
Rm5
Rg4
b)
Figure 6-9: EE core common mode choke with external U segments: a) component structure;
b) magnetic equivalent circuit
φcm
φdm
Rm4
Rg1
a)
Rm1
Vm1
Rm3 Rm2
Rg3
Rm5
Vm2
Rg2
b)
Figure 6-10: 5 leg common mode choke: a) component structure; b) magnetic equivalent
circuit
172
Advantages



Disadvantages

Standard core shapes (except
structure from Figure 6-10)
Simple winding assembly
Scaling of differential mode
inductance by air gaps




Leakage can cause additional
EMI in environment (Figure 6-8)
Decrease of common mode
permeability due to edges in the
magnetic structure
EMI application suitable
material is usually not
manufactured in E core shape
Expensive tooling of the centre
leg for amorphous materials
Gaps between the core halves
reduce the common mode
inductance
Table 6-5: Advantages and disadvantages of EE core common-differential mode chokes
6.3.3 Separated Common- and Differential-Mode Cores
[46] and [47] present two possibilities to enhance a common mode choke with one or
two additional low permeable ring cores for the differential mode attenuation.
Therefore, each winding of the common mode choke is wound additionally on a
second low permeable (iron powder or gapped ferrite) ring core. This technique is
applicable to U cores, too. Differential and common mode attenuation will be
performed by at least two different cores. This allows an easy and more or less
independent design of the common and differential mode inductance. The different
materials can enable high performance for both common and differential mode
filtering. Such a component is more a structurally integrated component. However,
the advantage of replacing two or three discrete counterparts is still valid.
If only one low permeable core is used, the second winding must be wound in the
opposite direction on the differential mode core, in order to avoid the cancelation of
the differential mode flux inside the low permeable core. This requires a complex
winding arrangement. The second alternative is, to place only one of the two
windings on the low permeable differential mode core, leading to an unsymmetrical
differential mode inductance of the component. If two identical components are put
in series, this asymmetry can be balanced. The low permeable differential mode core
can be put inside or outside the common mode core or on top and/or bottom.
173
If two differential mode cores are provided, the winding sense for these cores is not
in conflict with the common winding sense for the common mode core. The winding
assembly is a bit easier compared to the method using only one differential mode
core wound with two windings. A further advantage is the symmetry of the
inductances. For a high required differential mode inductance, it is necessary to
choose large low permeable cores, placed on top and bottom of the common mode
ring core. For a small differential mode inductance, the use of small low permeable
ring cores, placed inside a larger common mode ring core is possible. In this case the
component is smaller in height, but requires a larger outer diameter. However, the
complex winding arrangements requires the assembly by hand and increases the
mean length per turn.
The insertion of other differential mode core shapes (rods, C or U cores) is possible,
too. But the ring core enables a low leakage and the best adjustment of the
differential mode inductance.
φcm
φdm1
φdm2
Rm1
Rσ1
Rm3
Rσ3
Rm4
Vm1
a)
Rm2
Rσ2
Vm2
b)
Figure 6-11: Separated common-differential-mode choke with three rings: a) component
structure; b) magnetic equivalent circuit
φcm
Rm1
φdm
Rσ1
Vm12
Vm11
a)
Rm3
Rσ3
Rm2
Rm4
Vm22
Rσ4
Rσ2
Vm21
b)
Figure 6-12: Separated common-differential-mode choke with two rings: a) component
structure; b) magnetic equivalent circuit
174
Advantages



Disadvantages

Standard ring cores (ideal for
EMI applications)
Easy design
High differential mode
inductance by scaling of low
permeable core(s) possible


Complex winding assembly if
two differential mode windings
are required – or asymmetrical
differential mode inductance if
only one differential mode
winding is used
At least two or three cores
necessary
Mean length per turn increases
Table 6-6: Advantages and disadvantages of common-differential-mode choke with different
cores
6.3.4 Pot Core with Ferromagnetic Disc or EE Core with Segment
The winding of pot cores is almost completely shielded by the core. This avoids the
effect of unwanted induced leakage fields in adjacent components. The windings are
placed according to the top-bottom structure as depicted in Figure 6-13 (or Figure
V-1 b). A bifilar winding sense enables add up of the common mode flux inside the
core. The differential mode flux will flow in opposing direction inside the core and
leave the core between the windings. In fact, the component offers exactly the same
behaviour as a transformer with high leakage.
The leakage or differential mode inductance can be increased further by placing a
ferromagnetic disc with centre hole between the windings, as suggested by [45]. The
disc has the same effect as the block segment used for the ring core – it simply
increases the permeability for the differential mode flux path. If high permeable
material is used for the disc, the permeability must be adjusted by the air gap
between the disc and the pot core, by scaling the inner and outer diameter of the
disc. The application of the E core structure is possible, too, where the ferromagnetic
disc must be replaced by two rectangular segments.
However, pot cores are quite expensive and the use of additional ferromagnetic discs
manufactured on special request makes the component even more expensive.
Instead, the ferromagnetic disc can be replaced by a non-magnetic disc. The
displacement of both windings by a non-magnetic disc, as it is done for transformers,
will increase the leakage inductance, too. The increase of the differential mode
175
inductance should be less effective, but the material costs decrease. However, the
insertion of the disc will decrease the available window space, regardless of the used
disc material. This means the component suffers low copper fill factors and high
differential mode inductance values may require large pot cores.
φcm
Rm1
Rg1 Rm3 Rg2 Vm1
φdm
Rg3 Rm4 Rg4
Vm2
Rm2
a)
b)
Figure 6-13: Pot or EE core common mode choke with ferromagnetic disc or segment:
a) component structure; b) magnetic equivalent circuit
Advantages



Disadvantages

Simple design – scaling of
differential mode inductance by
the geometry of the
ferromagnetic disc / segment
Simple component and winding
assembly
Low to very low EMI in
environment


Expensive core geometry (pot
core)
Special ferromagnetic disc may
increase the costs significant
Low copper fill factors for large
discs / segments
Table 6-7: Advantages and disadvantages of common-differential-mode pot core choke with
ferrous disc and E core choke with segment
6.4 Design of Common-Differential Mode Chokes
Several design guidelines for common mode chokes can be found in literature (see
[45], [88], [105]). Due to the fact that each common mode choke provides a certain
amount of leakage inductance, it is convenient to start the design of a combined
common-differential mode choke according to the common mode design guidelines.
As explained in the previous section, common and differential mode signals use
different magnetic paths in the choke, due to their different characteristic. This
allows designing the common mode part partially separated from the differential
mode part of the component. However, it must be taken into account that
176
considerations defined for the common mode design will affect the differential mode
design as well and vice versa. Dependent on the chosen differential mode technique,
some designs require the definition of a priority. Is it more important to meet the
common mode or the differential mode requirements? The subsequent explained
design procedures will prioritise the common mode design. This approach suits
better if it is more important to meet the common mode specifications first. But a
vice versa approach is possible as well.
6.4.1 Common Mode Choke
A good approach for the design of common mode chokes (ring cores) is given in [88],
because the calculation of leakage inductance is considered in the design procedure,
too. The subsequent explained procedure is similar to the guideline from [88]:
1.
Eq. 6-1
2.
3.
Eq. 6-2
4.
Eq. 6-3
Select an appropriate wire according to DC/50/60Hz RMS current
specification. For convectional cooling, current densities from 4 to 8 A/mm²
are usually possible.
𝐴𝑐𝑢 =
𝐼𝑟𝑚𝑠
𝐽𝑟𝑚𝑠
Chose an appropriate core material and core size for the common mode
inductor according to the required common mode inductance (see Table
6-8).
Calculate the required number of turns with the given inductance factor (AL
value).
𝐿𝑐𝑚
𝑁=√
𝐴𝐿
Determine the resulting copper fill factor or the number of layers/core
coverage factor (see chapter 4.6). If the winding does not fit on the core,
chose a material with higher permeability or increase the core size. Both
measures increase the AL value and reduce the required number of turns.
𝑘𝑐𝑢 =
𝑁 ∙ 𝐴𝑐𝑢
𝐴𝑤
177
5.
Eq. 6-4
6.
Eq. 6-5
Calculate the leakage inductance (see chapter 4.6).
𝐿𝑑𝑚 = 𝜇𝑑𝑚 ∙
𝐿𝑐𝑚
𝜇𝑐𝑚
Check the current handling limit with the predicted leakage inductance. If
the current handling limit exceeds the nominal current, larger core size
(increases magnetic cross section Ae), material with higher permeability
(decreases number of turns) or higher saturation flux density (increased Bmax
possible) can help.
𝐼𝑚𝑎𝑥 =
𝐵𝑚𝑎𝑥 ∙ 𝑁 ∙ 𝐴𝑒
𝐿𝑑𝑚
Material
Permeability
Material Properties
Ferrite
(broadband)
Up to 5,000
Stable permeability over a wide frequency
range (up to several 100 kHz) allows
moderate attenuation even at elevated
frequencies, but only limited attenuation in
lower frequency range.
Ferrite
(high
permeability)
Up to 15,000
High permeability allows high attenuation
at lower frequencies. The material
bandwidth is limited (constant permeability
up to about 100 kHz), leading to limited
attenuation at very high frequencies.
Amorphous or
nanocrystalline
Up to
150,000
Ultra-high permeability enables superior
attenuation over a wide frequency range,
though the permeability decrease occurs in
a range about 10 kHz. Furthermore, the
high permeability allows realising very high
inductance values with less number of
turns. The drawback of these materials is
the high manufacturing costs compared to
ferrite, which is even more critical for the
nanocrystalline materials.
Table 6-8: Core materials for common mode chokes
178
6.4.2 Common-Differential Mode Choke with Additional DM Cores
The differential mode inductance will be enhanced according to the method given in
chapter 6.3.3. The low permeable differential mode core(s) can be designed
independent from the common mode inductance.
The required differential mode core size can be calculated by means of the magnetic
energy. Rearranging Eq. 2-15 gives:
Eq. 6-6
𝑊=
1 ′
∙ 𝐿 ∙ 𝐼2
2 𝑑𝑚 𝑑𝑚
Usually, manufacturers offer charts, where the core size (and initial permeability) is
plotted vs. the energy or the Li² product.
The no load inductance for powder or gapped ferrite cores is defined by:
Eq. 6-7
𝐿′𝑑𝑚,0 = 𝐴𝐿 ∙ 𝑁 2
The number of turns can be set at will, where equal number of turns according to the
common mode design is possible as well. If core size and number of turns are known,
the ampere turns and the magnetic field intensity can be calculated.
Eq. 6-8
𝑁 ∙ 𝐼𝑑𝑚 = 𝐻𝑑𝑚 ∙ 𝑙𝑒
To calculate the inductance under load conditions, the manufacturers provide
permeability vs. ampere turns curves for their different materials. These plots enable
to read out the permeability decrease under load conditions.
𝜇𝑑𝑚,𝐿 ′
𝐿′𝑑𝑚,𝐿 =
∙𝐿
Eq. 6-9
𝜇𝑑𝑚,0 𝑑𝑚,0
The total differential mode inductance of the common-differential mode component
is usually dominated by the low permeable differential mode cores. The total
differential mode inductance under load conditions can be approximated as:
Eq. 6-10
𝐿∗𝑑𝑚,𝐿 = 𝐿𝑑𝑚 + 𝑛 ∙ 𝐿′𝑑𝑚,𝐿
Ldm is the leakage inductance of the common mode core according to Eq. 6-4 and n
denotes the number of low permeable differential mode cores (e.g. n=1 for
assemblies according to Figure 6-12 and n=2 for assemblies according to Figure 6-11).
179
6.4.3 Common-Differential Mode Choke with Inserted Segments
The calculation of the common mode inductance can be performed according to
chapter 6.4.1. The maximum possible differential mode inductance factor can be
calculated according to:
Eq. 6-11
𝐴𝐿,𝑑𝑚,𝑚𝑎𝑥 =
𝐵𝑚𝑎𝑥 ∙ 𝐴𝑒
𝑁 ∙ 𝑖̂
Bmax is the maximum allowed flux density and Ae is the magnetic cross section of the
common mode core. Eq. 6-11 assumes that the saturation of the common mode core
limits the differential mode inductance and the insertion of applicable segments is
possible to enable this maximum inductance. The differential mode inductance can
be calculated according to:
Eq. 6-12
𝐿𝑑𝑚 = 𝑁 2 ∙ 𝐴𝐿,𝑑𝑚
If two dominant leakage paths are considered, the differential mode inductance
factor can be assumed as:
Eq. 6-13
𝐴𝐿,𝑑𝑚 =
1
1
+
𝑅𝑑𝑚1 𝑅𝑑𝑚2
Rdm1 and Rdm2 represent the reluctance values for segments placed inside the window
and/or joined external on the common mode core. Three different cases for the
evaluation of these reluctances will be considered here:
1. Segment inserted into the window of the common mode core (see Figure 6-14 a):
The differential mode inductance is dominated by the leakage path inside the
window. The insertion of the segment provides two gapes in the magnetic structure
(Rg1, Rg2). The gap length can be calculated according to the model given in Appendix
III (all segment surfaces facing the ring core must be considered). If the permeability
of the segment is very high, the reluctance of the segment itself can be neglected
(Rm1):
Eq. 6-14
𝑅𝑑𝑚1 = 𝑅𝑔1 + 𝑅𝑚1 + 𝑅𝑔2
180
The leakage inductance outside the common mode ring usually contributes about
95 % to the total leakage inductance of a conventional common mode choke (see
Ldm,ext Table 6-11, use Eq. 4-20):
Eq. 6-15
𝑅𝑑𝑚2 =
𝑅𝜎1 ∙ 𝑅𝜎2
1
≈
𝑅𝜎1 + 𝑅𝜎2 0.95 ∙ 𝜇𝑑𝑚 ∙ 𝐿𝑎𝑖𝑟
The calculation of the inlay technology from Figure 6-7 can be calculated in a similar
manner.
2.
External joined segment (see Figure 6-14 b):
The differential mode inductance is dominated by the leakage path provided by the
external segments. The magnetic structure provides four gaps (Rg3, Rg4, Rg3, Rg4). The
gaps can be calculated according to the model given in Appendix III. If the
permeability of the segments is very high, the reluctance of the segments themselves
can be neglected (Rm2, Rm3):
Eq. 6-16
𝑅𝑑𝑚1 =
(𝑅𝑔3 + 𝑅𝑚2 + 𝑅𝑔4 ) ∙ (𝑅𝑔5 + 𝑅𝑚3 + 𝑅𝑔6 )
𝑅𝑔3 + 𝑅𝑚2 + 𝑅𝑔4 + 𝑅𝑔5 + 𝑅𝑚3 + 𝑅𝑔6
The leakage inductance inside the common mode ring is almost negligent if no
segment is inserted (compare/use Eq. 4-20):
Eq. 6-17
3.
𝑅𝑑𝑚2 = 𝑅𝜎3 ≈
1
0.05 ∙ 𝜇𝑑𝑚 ∙ 𝐿𝑎𝑖𝑟
Segments inserted inside the window and external joined segments (see Figure
6-14 c):
Rdm1 and Rdm2 are defined by:
Eq. 6-18
𝑅𝑑𝑚1 = 𝑅𝑔1 + 𝑅𝑚1 + 𝑅𝑔2
Eq. 6-19
𝑅𝑑𝑚2 =
(𝑅𝑔3 + 𝑅𝑚2 + 𝑅𝑔4 ) ∙ (𝑅𝑔5 + 𝑅𝑚3 + 𝑅𝑔6 )
𝑅𝑔3 + 𝑅𝑚2 + 𝑅𝑔4 + 𝑅𝑔5 + 𝑅𝑚3 + 𝑅𝑔6
181
Rcm1
Rσ1
Rg1
Rm1
+
Vm1
Rcm2
+
Vm2
Rσ2
Rg2
a)
Rg3
Rm2
Rcm1
Rσ3
+
Vm1
Rcm2
Rg5
+
Vm2
Rm3
Rg4
Rg6
b)
Rg3
Rm2
Rg4
Rcm1
+
Vm1
Rg1
Rm1
Rg2
Rcm2
Rg5
+
Vm2
Rm3
Rg6
c)
Figure 6-14: Magnetic equivalent circuits for common mode ring core chokes with enhanced
leakage inductance: a) segment inserted inside the window; b) external segment; c) window
segment and external segments
The gap reluctance values for the models from Figure 6-14 a), b) and c) can be
calculated according to:
Eq. 6-20
𝑅𝑔𝑥 =
1
𝛿̅
∙
𝜇0 𝐴𝑑𝑚𝑥
The mean gap length δ can be calculated according to the rough approaches given in
Appendix III.
The resulting maximum flux density for each segment can be calculated according to:
182
Eq. 6-21
Eq. 6-22
𝐵𝑑𝑚1 =
𝑅𝑑𝑚2
𝑁 ∙ 𝑖̂
1
∙
∙
𝑅𝑑𝑚1 + 𝑅𝑑𝑚2 𝑅𝑐𝑚 + 𝑅𝑑𝑚1 𝑅𝑑𝑚2 𝐴𝑑𝑚1
2
𝑅𝑑𝑚1 + 𝑅𝑑𝑚2
𝐵𝑑𝑚2 =
𝑅𝑑𝑚1
𝑁 ∙ 𝑖̂
1
∙
∙
𝑅
𝑅
𝑅
𝑅𝑑𝑚1 + 𝑅𝑑𝑚2 𝑐𝑚 + 𝑑𝑚1 𝑑𝑚2 𝐴𝑑𝑚2
2
𝑅𝑑𝑚1 + 𝑅𝑑𝑚2
The resulting flux density inside the common mode ring is given by:
Eq. 6-23
𝐵𝑐𝑚 =
𝑁 ∙ 𝑖̂
1
∙
𝑅𝑐𝑚
𝑅𝑑𝑚1 𝑅𝑑𝑚2 𝐴𝑐𝑚
+
2
𝑅𝑑𝑚1 + 𝑅𝑑𝑚2
6.4.4 EE Core Common-Differential Mode Choke
The common mode inductance is defined by the reluctance of the outer legs.
Assuming the model from Figure 4-14 leads to:
Eq. 6-24
𝐿𝑐𝑚 =
𝑁2
= 𝑁 2 ∙ 𝐴𝐿,𝑐𝑚
2𝑅𝑚1
The required number of turns can be calculated according to Eq. 6-2.
The calculation of the differential mode inductance can be performed according to
the model given in chapter 4.5. The mmf sources from Figure 4-14 can be assumed to
be in parallel, where the reluctances Rm1 and Rm2 are in parallel, too. The magnetic
equivalent circuit for the differential mode inductance is depicted in Figure 6-15. The
inductance can be calculated according to:
𝐿𝑑𝑚 =
Eq. 6-25
𝑁2
𝑅𝑚1
1
+ 2
2
1
2
+
+
𝑅𝑓 𝑅𝑔
𝑅𝑤 𝑅𝜎
𝑅𝑚3 +
𝑅𝑓 + 𝑅𝑔
= 𝑁 2 ∙ 𝐴𝐿,𝑑𝑚
The maximum possible inductance factor can be calculated by means of the
maximum allowed flux density and the amplitude of the current – assuming that the
number of turns is defined by the required common mode inductance:
183
Eq. 6-26
𝐴𝐿,𝑑𝑚,𝑚𝑎𝑥 =
𝐴𝑒
2
𝑁 ∙ 𝑖̂
𝐵𝑚𝑎𝑥 ∙
It must be taken into account that the windings are placed on the outer legs, where
the magnetic cross section is only one half of the total cross section of the centre leg.
The required air gap length can be evaluated by plotting the inductance factor
calculated with the help of Eq. 6-25, as it is done in Figure 6-16. If the maximum
allowed inductance factor from Eq. 6-26 is lower than the required inductance factor
from Eq. 6-25, the core size must be increased in order to avoid saturation.
Rm3
Rm2
Rm1
Rw1
Rw2 Rg
Rf Rσ1
Rσ2
Vm1
Figure 6-15: Differential mode magnetic equivalent circuit for common-differential mode EE
core choke
Figure 6-16: Differential mode inductance factor for different EE cores
184
6.4.5 Comparison of Ring and EE Core Common-Differential Mode Chokes
Figure 6-17 shows the maximum possible differential mode inductance factor for
4
selected ring and E cores with similar area products (in a range from 0.7 cm to
4
13 cm ). The windings of the E cores will be placed on the outer legs. The calculation
is performed according to Eq. 6-11 and Eq. 6-26. The relative permeability is set to
5,000 and the maximum flux density is restricted to 0.2 T. The differential mode
inductance factor decreases for increasing ampere turns. The ring cores seem to
provide the potential for higher inductance factors compared to the E cores. If the
insertion of ferromagnetic segments into the dominant leakage paths of the ring
cores is possible, the ring core can enable high differential mode inductance values.
This effect can be explained by the larger magnetic cross section of the ring cores.
But it must be taken into account that the insertion of heavy segments may be
necessary, to enable the calculated inductance factors. As indicated in Eq. 6-26, the
EE cores can use only one half of the total magnetic cross section per winding for the
differential mode inductance. In turn, the EE core can take advantage of its
rectangular window shape which allows higher copper fill factors if multilayer
windings are allowed for the design.
Figure 6-17: Maximum possible differential mode inductance factor for ring and EE cores
Figure 6-18 shows the common mode inductance factors for the selected cores. The
common mode inductance factors of the EE cores are 40 to 65 % lower compared to
the ring cores. This effect can be explained by the smaller magnetic cross sections
and the increased magnetic path length of the EE cores. Thus, the EE core geometry
requires more turns to achieve a specific differential and common mode inductance.
185
Figure 6-18: Common mode inductance factor for ring and EE cores
In turn, the mean length per turn for the EE cores is smaller in comparison to ring
cores. If the specific conductance and the wire cross section are kept constant, the
comparison of the wire length is equivalent to the comparison of the expected DC
resistance values. Figure 6-19 shows the resistance ratio of ring and EE cores with the
same area product. If the common mode inductance for both geometries is
equivalent, the EE cores will have about 10 to 15 % higher resistance values than the
ring cores. If the same analysis is performed for a specific differential mode
inductance, the ring cores have 5 to 15 % higher resistance values. Finally, ring cores
can enable high inductance values with low DC resistance, where EE cores suit better
if high differential mode inductance value are required.
Figure 6-19: Ring core to EE core DC Resistance ratio for constant common mode and
differential mode inductance
186
6.4.6 Design Example
The design of a common mode choke using the L659-X830 ring core from Table 6-10
resulted in a choke according to Table 6-9. The nominal 50 Hz sinus RMS current is
10 A. The differential mode inductance will be enhanced by inserting a segment
inside the window.

The segment length is calculated dependent on the segment width b
according to Eq. III-5: 𝑙 = 2√𝑟𝑖2 −

𝑏2
4
The segment height is set to 16 mm (height of the ring core without
coating).
2
The mean gap length is calculated according to Eq. III-3: 𝛿 ̅ = ℎ𝑚𝑎𝑥
3
hmin is set to zero because the maximum possible segment length according
to Eq. III-5 is assumed. hmax is calculated according to Eq. III-8: ℎ𝑚𝑎𝑥 =




2𝑟−𝑙
2
The reluctance values for the differential mode inductance are calculated
according to Eq. 6-14 and Eq. 6-15: 𝑅𝑑𝑚1 (𝛿̅) ≈ 𝑅𝑔1 (𝛿̅) + 𝑅𝑔2 (𝛿̅),
1
𝑅𝑑𝑚2 ≈
= 8.98 ∙ 106 𝐴/𝑊𝑏
0.95 ∙ 𝜇𝑑𝑚 ∙ 𝐿𝑎𝑖𝑟
The total differential mode inductance factor is calculated according to Eq.
6-13:
1
1
𝐴𝐿,𝑑𝑚 (𝛿̅) =
+
̅
𝑅𝑑𝑚1 (𝛿 ) 𝑅𝑑𝑚2
Finally, Eq. 6-12 will give the inductance: 𝐿𝑑𝑚 (𝛿̅) = 𝑁 2 ∙ 𝐴𝐿,𝑑𝑚 (𝛿̅)
The maximum flux density in the segment is evaluated with Eq. 6-21:
𝑅𝑑𝑚2
𝑁 ∙ 𝑖̂
1
𝐵𝑑𝑚1 =
∙
∙
𝑅𝑑𝑚1 (𝛿̅) + 𝑅𝑑𝑚2 𝑅𝑐𝑚
𝑅𝑑𝑚1 (𝛿̅)𝑅𝑑𝑚2 𝐴𝑑𝑚1 (𝑏)
+
2
𝑅𝑑𝑚1 (𝛿̅) + 𝑅𝑑𝑚2
Eq. 6-23 is used to check the flux density inside the ring core:
𝑁 ∙ 𝑖̂
1
𝐵𝑐𝑚 =
∙
̅
𝐴
𝑅
(𝛿 )𝑅𝑑𝑚2
𝑐𝑚
𝑅𝑐𝑚
+ 𝑑𝑚1 ̅
2
𝑅𝑑𝑚1 (𝛿 ) + 𝑅𝑑𝑚2
187
CM Core
Number of
Turns
L659-X830
9
LCM @ 100
kHz [μH]
LDM @ 100
kHz [μH]
Rdc @ 25°C
[mΩ]
Weight [g]
537
7.6
4.9
75
(567)
(9.5)
(5)
(77.1)
Table 6-9: Data of common mode choke without segment (calculated values in brackets)
Figure 6-20 shows the calculated inductance and flux density values dependent on
the segment width. The inductance increases with increasing segment width, because
of the increasing magnetic cross section of the segment. The insertion of a 3 mm
segment enabled a measured inductance of 12.5 μH (increase about 64 %), where a
segment with 8 mm width enabled 14.2 μH differential mode inductance (increase
about 87 %). Both values are in well accordance to the calculated values (compare
Figure 6-20).
The flux density inside the segment decreases with increasing magnetic cross section.
In turn, the flux density inside the ring increases with increasing segment width. The
magnetic cross section of the ring is kept constant and the increasing total
differential mode inductance causes a higher total flux inside the ring. This means,
the core is better utilised. But it must be taken into account that the total flux density
(common and differential mode) is kept below the saturation limit.
Figure 6-20: Differential mode inductance and peak flux density dependent on segment
width (N=9, N∙î=127.3A)
188
6.5 Experimental Results
The previous presented methods for the improvement of differential mode
attenuation in common mode chokes will be investigated experimentally in this
chapter. Five conventional ring core common mode chokes were designed as
reference. The common- and differential mode inductance of these five chokes will
be investigated to evaluate the most promising methods for a differential mode
attenuation improvement. The main focus of the following analysis of the chokes is:




What are the advantages and disadvantages of the different methods for the
differential mode attenuation improvement?
How is inductance and weight of the chokes influenced?
Is it possible to decrease DC resistance and RMS losses of the filter?
Is the common mode attenuation influenced by the improvements of the
differential mode?
6.5.1 Comparison of different Common-Differential Mode Chokes
Five different common mode chokes were designed, in order to evaluate the
performance advantage of the different differential mode inductance enhancement
techniques. The technical data of these chokes is given in Table 6-11, where the data
of the chosen ring cores is given in Table 6-10.
The differential mode inductance of the designed common mode chokes was
enhanced by the insertion of cores and segments given in Table 6-12. To relieve the
insertion of window segments, two chokes were assembled with a two layer bank
winding (L618-X38 N11Z and L659-X830 N19Z). The bank winding enables a more
efficient use of the window area. In turn, the winding length increases, resulting in
higher DC resistance. The leakage inductance increases, too, because the second
layer of the winding is displaced away from the core. The value Ldm,ext in Table 6-11
indicates the relative part of the leakage inductance which is assigned to the external
environment of the choke. This value was measured by means of shielding the
window of the ring with a circular copper strap.
The diagrams of the impedance (vs. frequency) and inductance (vs. DC bias)
measurement of the chokes are depicted in Appendix IX. Subsequent the
measurement results of the chokes are summarised. The specific values (differential
mode inductance, differential mode resonance, saturation current, DC resistance and
weight) of the chokes are summarised in tables. The normalised differential mode
189
inductance and weight is given graphically. A summary and analysis is given in the last
section of this chapter.
Core
Material
Dout
[mm]
Din
[mm]
H
[mm]
μi
AL
[nH]
L618-X38
T38
26.6
13.5
11
10,000
10,700
L659-X830
N30
41.8
22.5
17.2
4,300
7,000
Table 6-10: Ring cores for common mode filtering – material see [106] [107];
cores see [108], [109]
CM
Choke
Number
of Turns
Lcm @
100 kHz
[μH]
Ldm @
100 kHz
[μH]
Ldm,ext
[%]
Rdc @
25°C
[mΩ]
Weight
[g]
ΔT @
10Arms
[K]
L618X38 N7
7
541
2.4
95.4
2.6
24.5
14.2
L618X38
N11
11
1246
6.3
95.2
4.1
30
27.7
L618X38
N11Z
11
1217
7.9
95.8
4.7
31
26.2
L659X830
N9
9
537
7.6
91.0
4.9
75
16.5
L659X830
N19
19
2305
22.6
93.2
10.1
91
33.3
L659X830
N19Z
19
2408
34.1
91.5
10.9
93
33.5
Table 6-11: Characteristic values of common mode chokes
190
Segment
Shape
Material
μi
Dimensions
(L x W x H) or
(Dout x Din x H) [mm]
13.3x10x
3
block
Ferrite Mf196
2,000
13.3 x 10 x 3
22x16x3
block
Ferrite Mf196
2,000
22 x 16 x 3
20.5x16x
8
block
Ferrite Mf196
2,000
20.5 x 16 x 8
18.5x13.
6x4
block
Ferrite Mf196
2,000
18.5 x 13.6 x 4
R22
ring
Ferrite Mf196
2,000
22 x 14.5 x 18.5
RK40
cut ring
Ferrite Mf102
2,000
40 x 24 x 16
RK63
cut ring
Ferrite Mf196
2,000
63 x 38 x 25
Disc
40x4.5
disc
Ferrite Mf196
2,000
40 x 0 x 4.5
T106-26
ring
Iron powder Amidon 26
75
26.9 x 14.5 x 11.1
Table 6-12: Segments for differential mode improvement of common mode chokes
a)
c)
b)
d)
e)
Figure 6-21: Different common-differential mode chokes: a) L618-X38 N11 and L618-X38 N11
+ T106-24 chokes; b) L659-X830 N19Z + 22x16x3 + RK60 choke; c) L659-X830 N19Z + 22x16x3
choke; d) L659-X830 N7 + Disc40x4.5 choke; e) assembly of L659-X830 + R22 + 18.5x13.6x4
191
L616-X38 N7
Choke
Ldm [μH]
fres [MHz]
Isat [A]
Rdc [mΩ]
Weight [g]
L618-X38 N7
2.4
> 40
> 30
2.6
24.5
L618-X38 N7 +
13.3x10x3
4.9
> 40
> 30
2.6
25
L618-X38 N7 +
RK40
6.3
40
> 30
2.6
81
L618-X38 N7 +
13.3x10x3 +
RK40
8.3
36
27.5
2.6
89
L618-X38 N7 +
Disc 40x4.5
4.8
> 40
> 30
2.6
77
L618-X38 N7 +
Disc 40x4.5 +
13.3x10x3
6.6
37
> 30
2.6
79
L618-X38 N7 +
T106-26
12.7 (no
load)
> 40
Partial
saturating
4.3
87.5
Table 6-13: Specific values for L618-X38 N7 chokes
Figure 6-22: Normalised differential mode inductance and weight for L618-X38 N7 chokes
192
The insertion of the small block segment (13.3x10x3) inside the window does not
increase the component weight significant, where the differential mode inductance is
doubled. But it must be taken into account, that the basic leakage inductance without
segments is very low. External segments like the RK40 and the disc on top and
bottom in combination with the window segment can improve the differential mode
inductance even more significant. But the weight increases significant, too. The
insertion of the T106-26 powder ring gives the highest differential mode inductance
increase, where DC resistance, size and weight of the component increase significant.
L618-X38 N11
Choke
Ldm [μH]
fres [MHz]
Isat [A]
Rdc [mΩ]
Weight [g]
L618-X38 N11
4.7
39
> 30
4.1
30
L618-X38 N11Z
7.3
26.5
> 30
4.6
31
L618-X38 N11Z
+ 13.3x10x3
12.2
20
22.5
4.6
32.5
L618-X38 N11Z
+ RK40
16.6
16.7
22
4.6
87
L618-X38 N11Z
+ 13.3x10x3 +
RK40
21.5
14.5
17
4.6
89
L618-X38 N11Z
+ Disc 40x4.5
10.3
23.5
> 30
4.6
84.5
L618-X38 N11Z
+ Disc 40x4.5 +
13.3x10x3
15.6
18.1
23
4.6
86.5
L618-X38 N11
+ T106-26
32.1
(no load)
25.5
Partial
saturating
8.7
96
Table 6-14: Specific values for L618-X38 N11 chokes
193
Figure 6-23: Normalised differential mode inductance and weight for L618-X38 N11Z chokes
The relative differential mode inductance increase for the block segment (13.3x10x3)
is lower compared to the L618-X38 N7 choke. But the basic leakage inductance is
higher because of the higher number of turns, resulting in a higher absolute
inductance value. The external ring and disc segments increase the weight significant.
The relative differential mode inductance increase is a bit lower compared to the
L618-X38 N7 choke, where the absolute inductance is of course much higher.
194
L659-X830 N9
Choke
Ldm [μH]
fres [MHz]
Isat [A]
Rdc [mΩ]
Weight [g]
L659-X830 N9
7.8
31
> 30
4.9
75
L659-X830 N9 +
22x16x3
12.5
24.8
30
4.9
81
L659-X830 N9 +
20.5x16x8
14.2
22.5
> 30
4.9
88
L659-X830 N9 +
RK63
24.9
15.8
28
4.9
299
L659-X830 N9 +
22x16x3 + RK63
29.1
14.6
25
4.9
304
L659-X830 N9 +
20.5x16x8 +
RK63
30.9
13.9
22
4.9
311
L659-X830 N9 +
Disc 40x4.5
14.2
19.1
> 30
4.9
130
L659-X830 N9 +
Disc 40x4.5 +
22x16x3
18.2
16.3
> 30
4.9
135
L659-X830 N9 +
Disc 40x4.5 +
20.5x16x8
19.7
15.4
> 30
4.9
142
L659-X830 N9 +
T106-26
24.1
(no load)
25.5
Partial
saturating
8.9
142
L659-X830 N9 +
R22
9.4
24
> 30
10.8
89.5
L659-X830 N9 +
R22 +
18.5x13.6x4
25.3
14
17
10.8
94.5
L659-X830 N9 +
R22 + RK63
27.3
12.8
30
10.8
322.5
L659-X830 N9 +
R22+18.5x13.6x4
+ RK63
43
10.6
15
10.8
327.5
Table 6-15: Specific values for L659-X830 N9 chokes
195
Figure 6-24: Normalised differential mode inductance and weight for L659-X830 N9 chokes
Similar as for the previous chokes, the weight increase for the insertion of block
segments (22x16x3 and 20.5x16x8) is limited and the inductance increase is
moderate. The insertion of the external ring segment seems to increase the
differential mode inductance and the weight even more significant as it is the case for
the smaller common mode chokes. Furthermore, it can be recognised that the
increase of the block segment can increase the inductance, too. The disc segments
enable a moderate inductance and weight increase. The T106-26 powder ring core
does not provide the highest inductance increase anymore.
Figure 6-25: Normalised differential mode inductance and weight for L659-X830 N9 chokes
with inlay segment
196
The insertion of the ring inlay (R22) does not give a significant advantage regarding
the differential mode inductance. But the combination of ring inlay and block
segment (R22 and 18.5x13.6x4) can enable a significant inductance increase at
limited weight increase. The additional insertion of an external ring segment (RK63)
can enable superior inductance increase, but causes a heavy weight.
L659-X830 N19
Choke
Ldm [μH]
fres [MHz]
Isat [A]
Rdc [mΩ]
Weight [g]
L659-X830 N19
20.7
12
> 30
10.1
91
L659-X830 N19Z
34.1
9.4
> 30
10.9
93
L659-X830 N19Z
+ 22x16x3
56.1
7.1
14
10.9
98.5
L659-X830 N19Z
+ 20.5x16x8
64
6.7
23
10.9
106
L659-X830 N19Z
+ RK63
110.6
4.9
12.5
10.9
326
L659-X830 N19Z
+ 22x16x3 +
RK63
132
4.5
11
10.9
331.5
L659-X830 N19Z
+ 20.5x16x8 +
RK63
138.6
4.3
10
10.9
339
L659-X830 N19Z
+ Disc 40x4.5
51.4
6.9
27
10.9
147
L659-X830 N19Z
+ Disc 40x4.5 +
22x16x3
72.9
5.8
18
10.9
153
L659-X830 N19Z
+ Disc 40x4.5 +
20.5x16x8
78.5
5.5
18
10.9
160
Table 6-16: Specific values for L659-X830 N19 chokes
197
Figure 6-26: Normalised differential mode inductance and weight for L659-X830 N19 / N19Z
chokes
The insertion of the top-bottom discs provides similar inductance increase as the
small block segment (22x16x3), but causes a weight increase of about 50 %. The
combination of top-bottom discs and window segments allow even higher differential
mode inductance increase. The highest inductance along with highest weight is
achieved with the external ring segments (RK63).
Summary
The increase of the differential mode inductance causes a decrease of the resonance
frequencies of the differential mode impedance. But in comparison to discrete
differential mode chokes, the resonance frequencies of the common-differential
mode chokes are still relatively high. The saturation current decreases with increasing
inductance, too. High inductance values are possible, but lower ampere turns must
be accepted (see Appendix IX), in order to avoid saturation. E.g. Figure IX-7 shows
how the differential mode inductance can be adjusted by means of the gap length of
external ring segments.
Figure 6-27 shows the relative inductance increase vs. the required material increase
for the different chokes. A high inductance increase requires the insertion of a high
amount of material. The insertion of segments inside the common mode core
window requires only a low amount of material, but leads only to a limited
differential mode inductance increase. The insertion of external segments (as the cut
ring cores) enables a significant differential mode inductance increase. But the
198
material expenditures can increase significant, too. The insertion of discs on top and
bottom enable a trade-off. This effect shows that the leakage inductance of a ring
core common mode choke is dominated by the leakage paths outside of the ring. The
leakage caused by the winding displacement and the leakage inside the ring are not
negligent, but less significant.
Figure 6-27: Differential mode inductance increase vs. weight increase for different chokes
The insertion of low permeable ring cores (one winding uses two cores) seems to be
appropriate if small common mode cores are used. In turn, this technique increases
the DC resistance of the component, because the winding must be wound on two
cores. The inductance increase by means of inserting segments into the given
structure is limited for small cores – but will not increase the DC resistance. If the
winding is wound on an additional core, the inductance can be increased more
efficient.
For larger common mode cores, the insertion of segments is more appropriate,
because the chokes can have high basic leakage inductance values, caused by their
large geometry and higher number of turns. The insertion of additional powder ring
cores seems to be less effective and causes a high effort.
The characteristic of the common mode choke does not change by the increase of
the differential mode inductance, regardless which technique is applied. The only
exception is the insertion of an additional ring inside the window (inlay technology)
of the common mode core. The winding is wound on both cores and the inductance
factor of the additional ring must be added to the inductance factor of the common
mode ring (see Figure IX-8).
199
6.5.2
Evaluation of Performance Advantage of Integrated CommonDifferential Mode Filters in AC Applications
The possible performance advantages of the previous presented common-differential
mode chokes will be investigated by means of an AC filter for a solar inverter. Figure
6-28 shows the investigated topology. A buck converter performs a half-sinus PWM
(100 Hz), where the unfolding bridge turns the current into a full wave sinus (50 Hz),
in order to enable the supply of the AC grid. For the converter specifications see
Table 6-17.
Output Power
2300 W
Output Voltage
230 Vrms
Output Current
10 Arms
Line Frequency
50 Hz
Switching Frequency
100 – 300 kHz
Table 6-17: Inverter specifications
The objective of this chapter is to show if combined common-differential mode
chokes can reduce or even replace other differential mode chokes. The differential
mode inductor of the buck converter is not investigated here. This inductor can be
designed with very low inductance value (e.g. 50 μH), taking a relatively high current
ripple. This enables a small and cost-efficient inductor. The following capacitor can be
a small rated X-capacitor (in the nF range), because no large DC link is necessary for
unfolding bridges [110].
Buck Converter
(Half-Sinus Modulated)
Unfolding Bridge
S01
EMI Filter
VL
Vin
Cin
IL
S1
S3
Iout
Ldm
S02
Lcm
CX
CX
S4
CY
CY
S2
Figure 6-28: Solar inverter topology using buck converter and unfolding bridge
200
230V
50Hz
Due to the small rated filter of the buck converter, the topology requires additional
EMI filter components, in order to ensure sufficient attenuation to fulfil the EMI
standards. A two-stage or double L differential mode filter topology has been chosen
for the EMI filter. In comparison to a single L topology, the double L requires twice
the amount of components. But the inductors as well as the capacitors can be
designed much smaller [111]. The double L topology suits well for high switching
frequencies and power applications [111]. The insertion loss measurements of the
filters were performed in the 50 Ω system according to Figure 6-29.
50Ω
EMI Filter
Vin
50Ω
Vout
Figure 6-29: Insertion loss measurement
Figure 6-31 shows the different AC EMI filters for the converter. Filter A provides two
equal differential mode chokes with 50 μH. Filter B is assembled with 20 μH
differential mode chokes and improved common-differential mode chokes. Finally,
for Filter C the differential mode inductance of the common mode chokes is
increased to replace the discrete differential mode inductors. The comparison of
Filter D and E presents another example, where the differential mode chokes were
replaced.
Figure 6-30: EMI Filter A
201
Table 6-18 shows the data of the filter components used for the different filter
configurations, where Figure 6-32 and Figure 6-33 show the filter weight and DC
resistance graphically. A comparison of the filter configurations A, B and C shows that
Filter A must accept the highest weight and DC resistance. Filter B enables about 30%
less weight and DC resistance. Filter C reduces the DC resistance about 80%. Instead,
the additional segments required for the common mode chokes cancel the advantage
regarding filter weight. By comparing Filter D and E, it becomes obvious that the
replacement of the differential mode choke enables weight and DC resistance
decrease of about 50%.
Differential Mode Choke
Common Mode Choke
Filter
Core
L
[μH]
Rdc
[mΩ]
Weight
[g]
A
ETD29
50
16.8
62
B
ETD24
20
8.5
C
none
-
D
ETD24
E
none
Core
Lcm
[mH]
Ldm
[μH]
Rdc
[mΩ]
Weight
[g]
L618-X38
N11
1.25
4.7
4.1
30
34
L618-X38
N11Z +
13.3x10x3
1.25
12.2
4.6
32.5
-
-
L618-X38
N11Z+
13.3x10x3
+ RK40
1.25
21.5
4.6
89
20
8.5
34
L618-X38
N11
1.25
4.7
4.1
30
-
-
-
L618-X38
N11Z +
13.3x10x3
1.25
12.2
4.6
32.5
Table 6-18: Filter components for the different filter configurations
202
Filter A
Lcm=1.25mH
Ldm=6.3µH
50μH
Lcm=1.25mH
Ldm=6.3µH
1μF
0.68μF
50μH
22nF
1nF
a)
Filter B
Lcm=1.25mH
Ldm=13µH
20μH
Lcm=1.25mH
Ldm=13µH
0.68μF
0.68μF
20μH
22nF
1nF
b)
Filter C
Lcm=1.25mH
Ldm=22.8µH
Lcm=1.25mH
Ldm=22.8µH
0.68μF
0.68μF
22nF
1nF
c)
Filter D
20μH
Lcm=1.25mH
Ldm=6.3µH
Lcm=1.25mH
Ldm=6.3µH
0.68μF
0.68μF
20μH
22nF
1nF
d)
Filter E
Lcm=1.25mH
Ldm=13µH
Lcm=1.25mH
Ldm=13µH
0.68μF
22nF
0.68μF
1nF
e)
Figure 6-31: EMI filter configurations
203
Figure 6-32: Weight for the different filter configurations
Figure 6-33: Total DC resistance for the different filter configurations
Though the weight decrease depends on the assembly of the common-differential
mode chokes, it becomes obvious that the decrease of the differential mode chokes
enabled a significant resistance and, therefore, power loss decrease for the filter. E.g.
a solar converter including boost and inverter stage with 2 kW nominal power and
98 % efficiency will cause at least 40 W losses. RMS filter losses of about 5 W
contribute 12.5 % to the total converter power loss. The reduction of 1 W RMS filter
losses enables an efficiency increase of 0.2 %, leading to a converter efficiency of
98.2%.
Figure 6-34 shows the measured insertion loss of Filter A, B and C. These filters are
applicable for a converter switching frequency of 144 kHz. The differential mode
nd
insertion loss at 288 kHz (2 harmonic) is about 70 dB (see Figure 6-34 a). The cut-off
204
frequency of Filter A occurs at 90 kHz, where for Filter B and C the cut-off frequency is
shifted down to about 60 kHz. For the frequency range from 150 kHz to 1.5 MHz, the
three filters provide a quite similar behaviour. Only Filter B shows a small resonance
at about 700 kHz. In comparison to the conventional discrete filter, the filters using
the enhanced differential-common mode chokes show very similar attenuation
behaviour. For frequencies higher than 2 MHz, the filters show different attenuation
behaviour. Filter A shows a resonance at 2 MHz, caused by the differential mode
chokes. Filter B shows a similar resonance at 3.5 MHz. The resonance is shifted into a
higher frequency range, because of the lower inductance value of the differential
mode chokes. Filter C does not show this resonance, because Filter C is not
assembled with any differential mode chokes. Instead, Filter C provides a bit lower
attenuation up to about 10 MHz.
a)
b)
Figure 6-34: Insertion loss for the filters A, B, and C: a) differential mode; b) common mode
205
Up to 1 MHz, the common mode attenuation for Filter A, B and C are identical. The
frequency range from 1 to 10 MHz shows several component resonances. Filter A
provides attenuation about 100 dB in this frequency range, where the attenuation of
Filter B drops down below 80 dB at about 2.5 MHz. At 6 MHz, Filter C provides 10 dB
less attenuation than Filter A, where the attenuation curve is similar.
a)
b)
Figure 6-35: Insertion loss for the filters D and E: a) differential mode; b) common mode
Figure 6-35 shows the insertion loss measurement for Filter D and E. The differential
mode cut-off frequency for Filter D occurs at 100 kHz, where Filter E provides a cutoff frequency at about 80 kHz (see Figure 6-35 a). The attenuation up to 250 kHz is
slightly higher for Filter E, where Filter D provides a bit higher attenuation over
250 kHz. But in general, the attenuation is quite similar up to 2 MHz. Filter D provides
206
a resonance frequency at about 3.5 MHz, limiting the attenuation from 1 to 6 MHz.
Filter E does not have this resonance, because of the removed differential mode
chokes, but accepts lower attenuation above 4 MHz.
The common mode attenuation of both filters is quite similar up to 1 MHz (see Figure
6-35 b). From 1 to 10 MHz, Filter E provides an even better attenuation. Instead, the
attenuation of Filter D is higher for frequencies above 10 MHz.
6.6 Executive Summary
Different techniques for the implementation of enhanced common-differential mode
chokes were presented and evaluated by practical assemblies. The inductance
evaluation of the different chokes shows that high required differential mode
inductance values require the insertion of a certain amount of material. But for the
filter design, a limited inductance increase may enable sufficient attenuation to fulfil
the specifications. If the differential mode chokes provide a limited inductance, it is
possible to downsize these chokes or to remove them. The experimental results
show, that a high amount of additional core material is necessary to compensate
large differential mode chokes. Thus, the advantage regarding component costs must
be evaluated application dependent. It must be taken into account that assemblies
on special request cause extra expenditures. Finally, material, assembly and
manufacturability have a non-negligent influence on the total component costs.
However, the improved common-differential mode chokes fulfil the filter
specifications and allow decreasing the DC resistance of the filter. The examples
showed that a significant resistance decrease, resulting in less filter RMS power loss,
is possible. Avoiding several Watts power loss can even improve the total converter
efficiency.
207
7
Recapitulation and Perspective
The application of functionally integrated and coupled magnetic components
requires the analysis of the circuit and its operating range. Possible advantages
regarding size and power loss may appear only in a limited operation range. The
application of magnetically coupled components, replacing discrete counterparts,
requires the adjustment of coupling and self-inductance, dependent on converter
specifications.
For the design of coupled and integrated components, the choice of core shape and
material is of importance. In general, ferrite materials suit well for coupled and
integrated magnetic components. Due to low specific core losses, ferrite materials
can operate well even at several 100 kHz. Instead, ferrite materials suffer limited
saturation flux densities. Amorphous or nanocrystalline materials can be an
alternative if the application frequency is below 50 kHz. These materials suffer high
specific core losses. Instead, they can enable a significant downsizing, due to their
high saturation flux density level.
Common core shapes are E, ring or U cores. E cores can enable high leakage
inductance values, where ring cores become applicable if low leakage is required.
Nevertheless, the application of the E core is preferred in many applications, because
the coupling can be adjusted by dimensioning the centre leg air gap. Ring and U cores
require the insertion of segments for a proper adjustment of the leakage, making the
application of these shapes often extensive. The influence of the material
permeability on the leakage is restricted to permeability values below 2,000 or to
very large cores. Very high permeable materials cannot increase the leakage
inductance much more.
Beside the desired inductance values of the component, the leakage inductance is of
importance regarding proximity losses and EMI. The proximity effect related power
loss can be reduced by the application of litz wires and/or by measures enabling welldefined flux guidance. The second option enables the limitation of EMI related issues
as well. Possible measures are the application of shell-type core structures or the
insertion of segments and other shielding measures. Nevertheless, the application of
custom core shapes and the insertion of segments for improved flux guidance will
increase the component costs.
208
The cost reduction of magnetic components is done by means of downsizing. The
reduction of material is enabled either by increasing the frequency or by increasing
the flux density. Both effects were demonstrated with the magnetically integrated
current doubler. The integrated ferrite current doublers have the highest downsizing
potential at frequencies above 100 kHz, because the transformer of the discrete
alternative must be operated at the specific core loss limit in this frequency range.
The integrated component is usually operating at lower induction values, enabling
further frequency increase and downsizing potential of the component. The
bottleneck at elevated frequencies is the specific power loss of the component.
Higher specific losses require enhanced cooling methods. For discrete magnetic
components this issue can be handled easier, because the total power loss can be
dissipated through larger surfaces.
Magnetic functions like power transfer or storing magnetic energy as required for the
magnetically integrated current doubler can be enabled by standard core shapes if
higher EMI in the environment is acceptable. For EMI filter chokes the EMI issue is
more important. An effective improvement of EMI filters by means of combined
common-differential mode chokes requires the use of custom core shapes and
segments. The insertion of core segments into the dominant leakage paths of a
common mode choke can improve the differential mode inductance significant. But
the reduced saturation current limit caused by the inductance increase must be
handled with care. A saturated filter choke will not work proper and can cause
exceeding of EMI specification limits under load conditions. Nevertheless, even a
limited improvement of the differential mode inductance can enable superior filter
performance and even the replacement of complete differential mode chokes. In
addition, the reduction of filter components is not only contributing to the total filter
size and cost reduction, but will also help to avoid parasitic effects in the filter and
reduces the power loss.
In general, the implemented examples show that reduced power loss and downsizing
is possible by means of functionally integrated magnetic components, due to better
utilisation of the component.
Recommendations for Future Work
Two basic investigations regarding functionally integrated magnetic components
maybe interesting in future:
209


Theoretically, functional and structural integration of magnetic components
can be implemented at once - e.g. by the realisation of windings within the
PCB. The use of both techniques may lead to superior power densities and
enables a higher degree of automation for the converter manufacturing. But
the structural integration may accept drawbacks, like limited applicable core
shapes. The possible implementation of both techniques should be
investigated in future scientific works.
A detailed investigation of U cores may become interesting if medium
leakage inductance values are required. The insertion of segments is
possible and many materials can be manufactured in U core shape, giving
the designer a certain degree of freedom. Several applications like coupled
inductors or common-differential mode chokes may take advantage of a
better understanding and analysis of leakage effects of U cores.
The following aspects for the magnetically integrated current doubler technologies
can be investigated more in detail:



Due to the DC current component of the magnetically integrated current
doubler, it might be possible to downsize the component by inserting
permanent magnets into the magnetic structure. In [112] this technique is
investigated for discrete inductors.
Innovative semiconductor components like SiC and GaN can enable
operation frequencies of several 100 kHz in the kW range. The analysis of
the magnetically integrated current doublers showed that these
technologies can take advantage regarding downsizing especially in the
100 kHz range. But the magnetically integrated current doublers suffer high
AC winding losses in this frequency range, too. For a better utilisation of this
technology and to enable the operation at high ambient temperatures, the
investigation of advanced cooling methods, especially for windings, can be
advantageous.
The designed nanocrystalline ring core current doubler showed a significant
downsizing potential. But the power loss and the temperature rise of the
component in the presented application was too high. Further
improvements in the magnetic structure (arrangement of leakage segment,
uncut ring core) and the use of litz wire with smaller strand diameter can
improve the design.
210
More detailed investigations on common-differential mode chokes can be performed
regarding the following issues:



Further investigations on leakage inductance of different core shapes and
winding arrangements are interesting regarding common-differential mode
chokes. In this work the ring core was investigated in detail, because it
enables the best common mode behaviour. A more detailed investigation on
E, pot and U cores regarding the differential mode inductance increase as
well as common mode inductance, size and weight will help to select the
best core geometry for a given application. Alternative design strategies
regarding differential mode inductance design are possible, too.
The design of special segments for common mode ring cores, especially for
external segments may improve the performance of these components. The
external segments enabled a significant differential mode inductance
increase. But the weight of the components increased significant, too.
Further optimisations regarding segment weight may make this technique
more applicable.
A further not investigated aspect was the influence of the EMI behaviour of
the combined common-differential mode components on the environment.
Some combined common-differential mode chokes can enable better EMI
behaviour, where others may lead to critical EMI in the environment.
211
Appendix
I. Equivalent Inductance of Coupled Inductors
Figure I-1 shows the two phase interleaved boost converter with coupled inductors.
The circuit operates with 180° phase shifted PWM. If one of the switches is on, the
inductor voltage is tied to the input voltage. Otherwise the inductor voltage is the
difference between input and output voltage.
L1
VL1
D1
IL1
Iin
IL2
L2
S1
S2
D2
Cout
Rout
Vout
Vin
Cin
VL2
Figure I-1: Interleaved boost converter
Eq. I-1 and Eq. I-2 describe the inductor voltage equations. The winding resistance is
neglected.
Eq. I-1
𝑉𝐿1 = 𝐿1 ∙
𝑑𝑖𝐿1
𝑑𝑖𝐿2
+𝑀∙
𝑑𝑡
𝑑𝑡
Eq. I-2
𝑉𝐿2 = 𝐿2 ∙
𝑑𝑖𝐿2
𝑑𝑖𝐿1
+𝑀∙
𝑑𝑡
𝑑𝑡
Subsequent the self-inductances L1 and L2 are equivalent (L1 = L2 = L).
212
Case 1: D < 0.5 – VL1 = Vin, VL2 = Vin - Vout
The insertion of VL1 = Vin and VL2 = Vin - Vout yields:
Eq. I-3
𝑑𝑖𝐿1
𝑑𝑖𝐿2 𝑉𝑖𝑛 − 𝐿 ∙ 𝑑𝑡
=
𝑑𝑡
𝑀
Eq. I-4
𝑑𝑖𝐿1
𝑑𝑖𝐿2 𝑉𝑖𝑛 − 𝑉𝑜𝑢𝑡 − 𝑀 ∙ 𝑑𝑡
=
𝑑𝑡
𝐿
Equalising Eq. I-3 and Eq. I-4 allows cancelling the term diL2/dt – further
rearrangements lead to:
Eq. I-5
(𝐿2 − 𝑀2 ) ∙
𝑑𝑖𝐿1
= 𝐿 ∙ 𝑉𝑖𝑛 − 𝑀 ∙ 𝑉𝑖𝑛 + 𝑀 ∙ 𝑉𝑜𝑢𝑡
𝑑𝑡
The output voltage can be replaced by the voltage transfer ratio of the boost
converter (Vout = Vin/(1-D)). Replacing the differential term diL1/dt by the difference
values gives the current ripple:
Eq. I-6
∆𝑖𝐿1 = 𝑉𝑖𝑛 ∙ ∆𝑡 ∙
𝐷
𝐷
∙ 𝑀)
(1 +
∙ 𝑘)
1−𝐷
1
−
𝐷
=
𝑉
∙
∆𝑡
∙
𝑖𝑛
(𝐿2 − 𝑀2 )
𝐿(1 − 𝑘 2 )
(𝐿 +
The current ripple of the second winding can be calculated by inserting Eq. I-6 into
Eq. I-2 or Eq. I-4. Several rearrangements lead to:
Eq. I-7
∆𝑖𝐿2 = 𝑉𝑖𝑛 ∙ ∆𝑡 ∙
𝐷
𝐷
∙ 𝐿)
(𝑘 +
)
1−𝐷
1
−
𝐷
=
𝑉
∙
∆𝑡
∙
𝑖𝑛
(𝐿2 − 𝑀2 )
𝐿(1 − 𝑘 2 )
(𝑀 +
Replacing the input voltage by the output voltages gives (VL2 = Vin - Vout = -D Vout):
∆𝑖𝐿2 = 𝑉𝑜𝑢𝑡 ∙ (−𝐷) ∙ ∆𝑡 ∙
1−𝐷
𝑀)
𝐷
2
2
(𝐿 − 𝑀 )
(𝐿 +
Eq. I-8
= 𝑉𝑜𝑢𝑡 ∙ (−𝐷) ∙ ∆𝑡 ∙
213
1−𝐷
𝑘)
𝐷
2
𝐿(1 − 𝑘 )
(1 +
The equivalent inductance values according to Eq. I-6 and Eq. I-8 are:
Eq. I-9
Eq. I-10
𝐿𝑒𝑞 =
(𝐿2 − 𝑀2 )
𝐿(1 − 𝑘 2 )
=
𝐷
𝐷
(𝐿 +
∙ 𝑀) (1 +
∙ 𝑘)
1−𝐷
1−𝐷
𝐿′𝑒𝑞 =
(𝐿2 − 𝑀2 )
𝐿(1 − 𝑘 2 )
=
1−𝐷
1−𝐷
(𝐿 +
𝑀) (1 +
𝑘)
𝐷
𝐷
Case 2: D < 0.5 – VL1 = VL2 = Vin - Vout , ΔiL1 = ΔiL2
The voltages across both windings as well as the current ripples are equivalent. Thus
Eq. I-1 or Eq. I-2 leads to:
Eq. I-11
𝑉𝑖𝑛 − 𝑉𝑜𝑢𝑡 = 𝐿 ∙
𝑑𝑖𝐿1
𝑑𝑖𝐿1
+𝑀∙
𝑑𝑡
𝑑𝑡
Insertion of the voltage transfer ratio (Vout = Vin/(1-D)) yields:
Eq. I-12
∆𝑖𝐿1 = ∆𝑖𝐿2
−𝐷
−𝐷
)
(
)
1
−
𝐷
1
−𝐷
= 𝑉𝑖𝑛 ∙ ∆𝑡 ∙
= 𝑉𝑖𝑛 ∙ ∆𝑡 ∙
(𝐿 + 𝑀)
𝐿(1 + 𝑘)
(
Case 3: D > 0.5 – VL1 = VL2 = Vin , ΔiL1 = ΔiL2
Voltages and current ripples for both windings are equivalent, leading to:
Eq. I-13
∆𝑖𝐿1 = ∆𝑖𝐿2 =
𝑉𝑖𝑛 ∙ ∆𝑡
𝑉𝑖𝑛 ∙ ∆𝑡
=
(𝐿 + 𝑀) 𝐿(1 + 𝑘)
Eq. I-11 and Eq. I-13 results in the following equivalent inductance:
Eq. I-14
𝐿𝑒𝑞 = (𝐿 + 𝑀) = 𝐿(1 + 𝑘)
The current ripple equations are (the input current ripple is the sum of both phase
current ripples):
214
Case
Conditions
D < 0.5,
VL1 = Vin
1
VL2 = Vin - Vout
Current Ripple
D < 0.5,
2
VL1 = Vin - Vout
VL2 = Vin - Vout
∆𝑖𝐿1 = ∆𝑖𝐿2
−𝐷
)
= 𝑉𝑖𝑛 ∙ ∆𝑡 ∙ 1 − 𝐷
𝐿(1 − 𝑘)
(
D > 0.5,
3
VL1 = Vin
Time Period
𝐷
(1 −
∙ 𝑘)
1
−
𝐷
∆𝑖𝐿1 = 𝑉𝑖𝑛 ∙ ∆𝑡 ∙
2
𝐿(1 − 𝑘 )
𝐷
(𝑘 −
)
1
−
𝐷
∆𝑖𝐿2 = 𝑉𝑖𝑛 ∙ ∆𝑡 ∙
2
𝐿(1 − 𝑘 )
∆𝑖𝐿1 = ∆𝑖𝐿2
VL2 = Vin
𝑉𝑖𝑛 ∙ ∆𝑡
=
𝐿(1 − 𝑘)
VL1 = Vin
Δt = (0.5 – D) ∙ Ts
Δt = (D – 0.5) ∙ Ts
Δt = (1 – D) ∙ Ts
D > 0.5,
4
Δt = D ∙ Ts
Equivalent to case 1
VL2 = Vin - Vout
Table I-1: Current ripple equations for interleaved boost converter
The equivalent inductance values are:
Conditions
VL1 = Vin
VL2 = Vin - Vout
Inductance
𝐿𝑒𝑞 =
(𝐿2 − 𝑀2 )
𝐿(1 − 𝑘 2 )
=
𝐷
𝐷
(𝐿 +
∙ 𝑀) (1 +
∙ 𝑘)
1−𝐷
1−𝐷
𝐿′𝑒𝑞 =
VL1 = Vin
VL2 = Vin
(𝐿2 − 𝑀2 )
𝐿(1 − 𝑘 2 )
=
1−𝐷
1−𝐷
(𝐿 +
𝑀) (1 +
𝑘)
𝐷
𝐷
𝐿𝑒𝑞 = (𝐿 + 𝑀) = 𝐿(1 + 𝑘)
Table I-2: Equivalent inductance values for interleaved boost converter
The values from Table I-1 and Table I-2 are valid for direct coupling. Inverse coupling
can be obtained by negative mutual inductance values. Setting the mutual inductance
to zero gives the valid equations for the discrete interleaved converter. The
evaluation of the inductor equations for the buck converter will give the equivalent
results.
215
II. Leakage Calculation for Coupled EE Core Inductors
Subsequent the calculation of the magnetic leakage reluctance values from Figure
4-14 is performed. The geometric dimensions of the core are named according to
Figure II-1.
The air gap reluctance is given by the ideal geometric values of the gap, where δ is
the gap length. The insertion of a fringing effect correction factor is possible, too (see
Appendix IV):
𝑅𝑔 =
Eq. II-1
1
𝛿
∙
∙ 𝑘 (𝛿)
𝜇0 𝐹 ∙ 𝐶 𝑓𝑓
A
B
D
A
δ
D
B
a)
L
M
C
F
M
L
L
b)
M
F
M
L
Figure II-1: Geometric dimensions for E core: a) front; b) top
Leakage beside the Centre Leg
According to [113] the flux beside the centre leg can be separated into two
components: The flux passing through the surface area of the centre leg and the flux
passing through the edges of the centre leg and flowing through the spherical shells.
As simplification, [113] assumes the maximum magneto motive force applied to the
leakage reluctance / permeance. Therefore, the permeance of the leakage flux
passing through the surfaces of the centre leg can be calculated according to the
surface integrals [113]:
𝐷
𝐷
𝛿/2
𝛿/2
𝑑𝐴1
𝑑𝐴2
𝐹 ∙ 𝑑𝑟
𝐶 ∙ 𝑑𝑟
𝑃𝑓1 = 2𝜇0 ∙ (∫
+∫
) = 2𝜇0 ∙ ( ∫
+ ∫
)
𝑠
𝑠
𝜋∙𝑟
𝜋∙𝑟
Eq. II-2
=
2𝜇0
2𝐷
∙ (𝐹 + 𝐶) ∙ 𝑙𝑛 ( )
𝜋
𝛿
216
The permeance of the leakage flux passing through the four quadrants of the
spherical shells can be estimated according to [113]:
Eq. II-3
𝜋
𝛿
𝛿
∙ (𝐷 − ) ∙ (𝐷 + )
8
2
2 = 𝜇 ∙ (𝐷 − 𝛿/2)
𝑃𝑓2 = 4 ∙ 𝜇0 ∙
0
𝜋
𝛿
∙ (𝐷 + )
2
2
Finally, the complete leakage permeance beside the centre leg can be calculated by:
Eq. II-4
𝑃𝑓 = 𝑃𝑓1 + 𝑃𝑓2
C
A2
F
r
D−δ/2
A1
δ
A1
D−δ/2
dr
A2
Figure II-2: Schematic leakage flux beside centre leg
Leakage Effect inside the Windows
The leakage flux crossing the windows is considered under the restriction by
neglecting the field deformation caused by the air gap fringing [113]. The mean path
length of the flux is estimated by the window height. The surface area used by the
leakage flux is estimated by the window width and depth resulting in:
Eq. II-5
𝑃𝑤 = 𝑃𝑤1 + 𝑃𝑤2 =
𝑅𝑤1 ∙ 𝑅𝑤2
𝐶∙𝑀
𝐶∙𝑀
= 2 ∙ 𝜇0 ∙
= 𝜇0 ∙
𝑅𝑤1 + 𝑅𝑤2
2𝐷
𝐷
D
D
C
M
Figure II-3: Calculation of leakage flux inside the windows
217
Leakage effect beside the outer legs
For the estimation of the leakage flux beside the outer legs, it has to be taken into
account that the magnetic potential inside the leg itself is not constant, because the
windings are placed on these legs. As it is done in [113] a linear increase of the
magnetic potential is considered. The magnetic flux leaving the outer legs at the front
side and the outer side has an elliptical shape. Nevertheless, the estimation with half
rings is possible [113], because the increasing magnetic path length is compensated
by an increasing magnetic surface area. The integration along the surface borders
leads to the corresponding part of the leakage permeance for both outer legs [113]:
𝐷
Eq. II-6
𝑃𝑠1 = 2 ∙ 𝜇0 ∫
0
𝑧 (𝐶 + 2𝐿)𝑑𝑧
𝜇0
= 2 ∙ ∙ (𝐶 + 2𝐿)
𝐷
𝑧∙𝜋
𝜋
Therefore, the term z/D describes the linear increase of the magnetic voltage across
the leg.
Furthermore, the leakage flux at the edges must be recognised [113]:
Eq. II-7
1 𝜋 2
∙ ∙𝐷
𝜇0
𝑃𝑠2 = 2 ∙ 𝜇0 ∙ 2 4
=
∙𝐷
𝐷
2
𝜋∙
2
dz
D
z
D
C
L
Figure II-4: Calculation of leakage flux beside the outer legs
Leakage Effect of the Top and Bottom Core Segments
The leakage flux leaving these core segments can be separated into three parts:
1.
Flux part leaving the core at the front ends (all four sides).
This permeance can be estimated to [113]:
218
Eq. II-8
(𝐵 − 𝐷) ∙ (𝐴 + 𝐶) 4
(𝐵 − 𝐷) ∙ (𝐴 + 𝐶)
𝑃𝑡1 = 2 ∙ 𝜇0 ∙ 𝜋
= ∙ 𝜇0 ∙
𝐵+𝐷
∙ (𝐵 − 𝐷 + 2𝐷) 𝜋
2
Therefore, the mean magnetic path lengths are assumed as half rings.
2.
Eq. II-9
Leakage flux passing through the quadrants of the spherical shells bordering
the front ends.
The mean magnetic path length is estimated with half rings – similar as in
Eq. II-8. The mean magnetic cross section is estimated by one half of the
section of a circular ring. The resulting permeance for these four quadrants
is given by [113]:
1 𝜋
𝜋
[ ∙ (2𝐵)2 − ∙ (2𝐷)2 ]
𝐵2 − 𝐷2
4
4
4
𝑃𝑡2 = 4 ∙ 𝜇0 ∙
=
2
∙
𝜇
∙
0
𝜋
𝐵+𝐷
∙ (𝐵 − 𝐷 + 2𝐷)
2
A
B
B
D
C
D
Figure II-5: Calculation of leakage flux at the front ends
3.
Elliptical leakage flux path between the outer top and bottom surfaces.
These parts of the leakage flux leave the top side of the upper core segment,
flowing along elliptical arcs and re-enter the core at the corresponding core
surface at the opposed side. The mean magnetic path length can be
estimated by a large half ring and two smaller quarter rings [113], resulting
in the corresponding permeance:
219
Eq. II-10
𝐴
∙𝐶
2
𝑃𝑡3 = 2 ∙ 𝜇0 ∙
2∙
𝜋
𝐴 𝐴
∙ (𝐵 + + )
2
4 4
=
2
𝐴∙𝐶
∙ 𝜇0 ∙
𝜋
2𝐵 + 𝐴
The total leakage flux of the core (including the windows and without the leakage
effect of the centre leg) is calculated according to:
Eq. II-11
𝑃𝜎 = 𝑅𝜎1 ∥ 𝑅𝜎2 ∥ 𝑅𝑤1 ∥ 𝑅𝑤2 = 𝑃𝑠1 + 𝑃𝑠2 + 𝑃𝑡1 + 𝑃𝑡2 + 𝑃𝑡3 + 𝑃𝑤
A/4
C
A
B
B+A/4
B
Figure II-6: Calculation of leakage flux between the outer top and bottom surfaces
220
III. Air Gap Calculation for Segments of Common-Differential
Mode Chokes
Block Segments inserted into the Window of Ring Cores
Figure III-1 shows the schematic of a cuboid segment inside the window of a ring
core. Fringing and leakage effects of the flux are neglected. It is assumed that the flux
is entering the cores perpendicular. The border of the inner ring can be described as
parabola:
𝑦 = 𝑎 ∙ 𝑥2 + 𝑏 ∙ 𝑥 + 𝑐
Eq. III-1
The constants a, b and c can be evaluated with the boundary conditions c=hmax,
y=hmin for x=b/2 and y=hmin for x=-b/2 (according to Figure III-1 a):
𝑦 = 𝑎 ∙ 𝑥2 + 𝑐 =
Eq. III-2
4
(ℎ
− ℎ𝑚𝑎𝑥 )𝑥 2 + ℎ𝑚𝑎𝑥
𝑏 2 𝑚𝑖𝑛
The mean air gap length is given by the integration of y along the x-axis:
𝑏/2
𝛿̅ =
Eq. III-3
1
2
1
∫ (𝑎 ∙ 𝑥 2 + 𝑐)𝑑𝑥 = ℎ𝑚𝑎𝑥 + ℎ𝑚𝑖𝑛
𝑏
3
3
−𝑏/2
y
hmax h
min
hmin
b
x
b
h'
l
r
a)
l/2
r
b)
Figure III-1: Schematic of ring core with block segment (cuboid)
The relationship of the segment distances hmax and hmin can be calculated according to
Figure III-1 b) by means of:
221
Eq. III-4
ℎ𝑚𝑎𝑥 = ℎ𝑚𝑖𝑛 + ℎ′
r is the inner radius of the ring core. The half segment length l/2 is given by:
Eq. III-5
𝑙
𝑏2
= √𝑟 2 −
2
4
h’ can be evaluated by:
Eq. III-6
ℎ′ = 𝑟 −
𝑙
𝑏2
= 𝑟 − √𝑟 2 −
2
4
Inserting Eq. III-6 into Eq. III-4 gives:
Eq. III-7
ℎ𝑚𝑖𝑛 = ℎ𝑚𝑎𝑥 − 𝑟 + √𝑟 2 −
𝑏2
4
The value hmax can be expressed by the segment length l and the inner ring radius r:
Eq. III-8
ℎ𝑚𝑎𝑥 =
2𝑟 − 𝑙
2
Inserting Eq. III-8 into Eq. III-7 and Eq. III-4 yields:
𝛿̅ =
2 (2𝑟 − 𝑙) 1 (2𝑟 − 𝑙)
𝑏2
+ (
− 𝑟 + √𝑟 2 − )
3
2
3
2
4
Eq. III-9
=
(2𝑟 − 𝑙) 1
𝑏2
− (𝑟 − √𝑟 2 − )
2
3
4
Eq. III-9 gives the single sided gap length of a segment inserted into a ring core
dependent on the inner ring radius r as well as the segment length l and width b.
222
Cut Ring Segments joined on a Ring Core
Figure III-2 shows the schematic of a cut half-ring joined on a ring core. The course of
the leakage flux path is assumed as a segment of a circle. It is assumed that the flux is
entering the outer radius of the ring core perpendicular. Fringing and other leakage
flux effects are neglected. The length of the flux path can be described by a segment
of a circle:
𝛼
Eq. III-10 𝑓(𝑏′) = 𝜋 ∙ 180° ∙ 𝑏′
Integration over the total length of the cross section gives the average gap length:
𝑏+𝑐
𝛿̅ =
Eq. III-11
1
𝛼
𝛼
𝜋
∫ 𝜋∙
∙ 𝑏′ ∙ 𝑑𝑏 ′ =
∙
∙ ((𝑏 + 𝑐)2 − 𝑐 2 )
𝑏
180°
180° 2𝑏
𝑐
=
𝛼 𝜋
∙ ∙ (𝑏 + 2𝑐)
180° 2
f
cut half ring
segment
b
outer radius
of ring core
b
α
c
Figure III-2: Schematic for cut ring segment joined on a ring core
223
IV. Fringing Effect of Air Gaps
All air gaps in magnetic circuits show the so called fringing effect. It is caused by stray
flux using paths beside the ideal air gap geometry (see Figure IV-1). As a result, noise
and EMI can be coupled to circuit components beside the air gap. Furthermore, there
will be a certain amount of magnetic energy stored inside the fringing field, outside
the ideal air gap geometry. This energy will contribute to the energy reactor of the
component. Thus, specific values like inductance or coupling coefficient of a
component can be estimated inaccurate if the fringing effect is not taken into
account.
a)
b)
Figure IV-1: FEM simulation of gapped EE core: a) field intensity plot; b) flux density plot
Figure IV-2 shows the magnetic equivalent circuit for the inductor from Figure IV-1.
The core reluctance was taken into account via Rc, where Rg represent the reluctance
from the geometrical air gap. The fringing field was taken into account via the parallel
reluctance Rf, as suggested by [1]. The inductance of the inductor from Figure IV-2
can be modelled by [1]:
Eq. IV-1
𝐿=
𝑁2
𝑅𝑔 ∙ 𝑅𝑓
𝑅𝑐 +
𝑅𝑔 + 𝑅𝑓
If the relative permeability of the core is much higher than unity, Eq. IV-1 can be
simplified to [1]:
Eq. IV-2
𝐿=
𝑁2
𝜇0 𝐴𝑒 𝜇0 𝐴𝑓
= 𝑁 2 ∙ (𝑃𝑔 + 𝑃𝑓 ) = 𝑁 2 ∙ (
+
)
𝑅𝑔 ∙ 𝑅𝑓
𝛿
𝑙𝑓
𝑅𝑔 + 𝑅𝑓
224
Eq. IV-2 represents the parallel circuit of the fringing and the air gap permeance. Af is
the equivalent cross section used by the fringing flux beside the air gap and lf the
corresponding magnetic path length. In practice, it is not easy to assign the fringing
flux cross section and the path length by geometrical quantities. In literature, it is
suggested to use an air gap correction factor based on the geometry of the air gap.
Eq. IV-2 can be rearranged in order to introduce a correction factor by replacing Af
and lf:
Eq. IV-3
𝐴𝑓 = 𝐴𝑒 (1 + 𝐾𝐴 )
Eq. IV-4
𝑙𝑓 = 𝛿(1 + 𝐾𝛿 )
By inserting and combining Eq. IV-3 and Eq. IV-4 into Eq. IV-2, the permeance and the
inductance can be expressed by a single correction factor:
Eq. IV-5
(𝑃𝑔 + 𝑃𝑓 ) =
Eq. IV-6
𝐿 = 𝑁2 ∙
(1 + 𝐾𝐴 )
𝜇0 𝐴𝑒
𝜇0 𝐴𝑒
∙ (1 +
)=
∙ 𝐾𝑓𝑓 (𝛿, 𝐴𝑒 )
(1 + 𝐾𝛿 )
𝛿
𝛿
𝜇0 𝐴𝑒
∙ 𝐾𝑓𝑓 (𝛿, 𝐴𝑒 )
𝛿
Kff can be interpreted as fringing field or air gap correction factor, which can be
evaluated via empirical measurements or FEM simulations dependent on the gap
length δ and/or the gap cross section Ae. Eq. IV-5 can be used as general expression in
order to model the fringing effect of air gaps via two parallel permeances. For the
inductor from Figure IV-1, it has to be taken into account that the air gap and the
fringing field are enclosed by the winding. For air gaps which are not enclosed by a
winding, the fringing field will spread out even more. As a result, the air gap
correction factor will increase dependent on the enclosure of a winding and
dependent on the gap length and cross section.
For rectangular legs, [2] proposes (C∙F is the ideal magnetic cross section):
Eq. IV-7
𝐾𝑓𝑓 =
(𝐹 + 𝛿) ∙ (𝐶 + 𝛿)
𝐹∙𝐶
For round legs with the diameter Ds, the gap correction factor proposed by [2] is:
225
Eq. IV-8
𝐾𝑓𝑓 =
(𝐷𝑠 + 𝛿)2
𝐷𝑠 2
Rc
N
Rg
Rf
Vm1
Figure IV-2: Equivalent magnetic circuit for an inductor with air gap and fringing effect
226
V. Leakage Inductance of Transformers
The evaluation of leakage inductance values of transformer engages magnetic
designers since decades and is still in the focus of research today. It limits the power
transfer capability of hard switched converters [114], [91] or is used as design
parameter, e.g. for resonance converters. The leakage inductance of a transformer is
mainly dependent on the window size (height and width) as well as on the winding
arrangement. There are three general winding arrangements for primary and
secondary windings of a transformer, also mentioned in [115], [116] and depicted in
Figure V-1:



Side by Side
Top-Bottom
Interleaved
It is also possible to combine the side by side structure with the interleaved structure
as depicted in Figure V-1 d). Generally, interleaved structures offer the lowest
leakage inductance values. In turn, the effort for the assembly can increase significant
(e.g. if conductor cross sections of primary and secondary are extremely different).
Therefore, the side by side interleaved technique is a compromise (Figure V-1 d) –
providing very low leakage, too. In general, the side by side structure will provide also
low leakage values, where the leakage can be adjusted by the displacement of the
primary and secondary winding or by the insertion of ferromagnetic material
between them as showed in [8] (see also Figure 1-5). The top-bottom structure is
used to generate very high leakage inductance – e.g. to replace an external series
inductor of a resonance converter.
There are three general methods for the evaluation of leakage inductances values:



Measurement
Analytical estimation
FEM simulation
The evaluation via measurements is more a try and error method becoming
applicable if other estimation or evaluation methods fail or are too inaccurate. Due to
the fact that processing power is not a problem for today’s personal computers, the
simulation of FEM models becomes applicable. Nevertheless, the effort for
implementation of the models is increasing with the complexity of the model and
some simplifications must be accepted even for complex models. For a quick
227
evaluation or for optimisation processes the use of FEM methods is often
inappropriate. Therefore, [117] and [118] use the support of FEM tools to develop
analytical models in order to obtain both the advantages of analytical descriptions as
well as increased accuracy of FEM.
N2
N1
N1
N2
a)
b)
c)
d)
N2
N2
N1
N1
Figure V-1: Winding arrangements for transformers: a) side by side; b) top-bottom;
c) interleaved; d) side by side interleaved
But there are many publications proposing analytical evaluation methods for some
common winding and core arrangements. E.g. [119], [120] and [119] investigate the
leakage of transformers and inductors analytically. There are much more publications
regarding analytical leakage inductance estimation, but [121] states that many of
them show formulas which are modifications of the method established by Rogowski
[122]:
Eq. V-1
𝐿𝜎 = 𝜇0 ∙ 𝑁 2 ∙ 𝑙𝑚𝑒𝑎𝑛 ∙ 𝜆 ∙ 𝑘𝜎
228
N is the number of turns, lmean is the mean length per turn, λ is a leakage conductive
parameter depending on the geometry and interleaving of the windings and kσ is the
so called Rogowski factor also depending on geometrical parameters [121]. In order
to provide a manageable formula, [121] describes the leakage inductance as follows:
𝐿𝜎 = 𝜇0 ∙ 𝑁 2 ∙
Eq. V-2
2
𝑛𝑖𝑓
∑ 𝑋𝑝𝑒𝑟𝑝−𝑙𝑓
𝑙𝑚𝑒𝑎𝑛
∙(
+ ∑ 𝛿)
3
∙ 𝑋𝑝𝑎𝑟−𝑙𝑓
The parameters of Eq. V-2 are given as follows [121]:


N1
Xpar-lf
N2
N1
Xpar-lf
N2
N2
Xperp-lf

𝑙𝑚𝑒𝑎𝑛 : mean length per turn
∑ 𝑋𝑝𝑒𝑟𝑝−𝑙𝑓 : geometrical sum of all sub-windings oriented perpendicular to
the leakage flux
𝑋𝑝𝑎𝑟−𝑙𝑓 : geometrical sum of sub-windings oriented in parallel to the leakage
flux
∑ 𝛿: geometrical sum of all interspaces between the sub-windings
𝑛𝑖𝑓 : number of interspaces between the sub-windings
N2
N1
Xperp-lf


N1
δ
Xperp-lf
Xperp-lf
δ
Figure V-2: Winding arrangement and parameters for the calculation of leakage inductance –
figure based on representation from [121]
229
VI. Thermal Equivalent Circuits of Magnetic Components
Thermal models for magnetic components enable a recursive temperature
dependent loss calculation. Losses will dissipate as heat - and heat will change the
thermal operation point of the component and, therefore, the power loss of the
component.
The challenge for the implementation of thermal models is:





Heat will propagate in all directions. Thus, a three dimensional problem will
be simplified by means of a one dimensional thermal equivalent circuit.
There are different types of heat transfer: conductance, convection and
radiation.
Windings usually do not consist of pure copper. Dependent on the wire type
there is a certain amount of insulation material inside the winding package.
The same problem occurs for laminated core materials.
Core and winding will also heat each other.
The thermal properties of some materials like core or insulation materials
are unknown, because they are not given in the data sheets.
From a thermal point of view, magnetic components are very complex and the
evaluation of lumped thermal elements in a model is very difficult. Even simplified
models can cause a high calculation or simulation effort, though the accuracy is
limited. A very complex model for the thermal resistance of windings is developed by
[123]. But [123] itself states a limited accuracy of the model. The literature offers also
some formulas for rough approaches enabling estimation by hand [124], [1], [2].
These formulas calculate the temperature rise by treating the complete component
as a lumped uniform mass and are based on rough experimental approaches for
standard components/cores operating under pure convectional cooling.
A more detailed but still simple model suitable for ring cores is given by [1], where
the winding enables full core coverage. The core will transfer its heat completely to
the winding, where the winding will transfer its own heat and the heat of the core to
the ambient. But this is a special case. Usually core and winding can transfer heat to
the ambient at multiple surfaces. Therefore, [49] presents a more detailed thermal
model. Winding and core are modelled separately, where their interaction is taken
into account, too.
In general, [49] considers four heat paths for each component (also see Figure VI-1):
230




Internal winding resistance - required to model the winding hot spot
(conductance)
Thermal resistance for heat transfer between winding and core (convection
and radiation)
Thermal resistance from winding surface to ambient (convection and
radiation)
Thermal resistance from core to ambient (convection and radiation)
Ths
Rhs
Rwa,c
Tw
Rcw,c
Pw
Rcw,r
Rwa,r
Rca,c
Tc
Rca,r
Pc
Ta
Figure VI-1: General thermal equivalent circuit for magnetic components – figure based on
representation from [49]
In general, the thermal resistance for the convection can be described as [49]:
Eq. VI-1
𝑅𝑡ℎ,𝑐𝑜𝑛 =
1
ℎ𝑐𝑜𝑛 ∙ 𝐴𝑐
hcon is the convectional heat transfer coefficient in W/(m² K) and depends on the
cooling method, where Ac is the surface dissipating the heat. In [49] the following
approximation for hcon is proposed:
Eq. VI-2
ℎ𝑐𝑜𝑛 = 1.42 (
∆𝑇 0.25
)
ℎ
∆T is the temperature rise and h is the height of the component. For forced air
convection, [49] proposes to use:
Eq. VI-3
ℎ𝑐𝑜𝑛 = (3.33 + 4.8𝑣 0.8 )ℎ−0.288
231
v is the velocity of the flow of coolant and h is the distance of the boundary layer of
the component. More complex formulas for the heat transfer coefficients e.g. taking
the pressure and more fitting coefficients into account are given in [49], too. But in
practice, values like the exact pressure are usually not known, making the exact
evaluation difficult and limiting the accuracy of the approach.
Assuming heat conductance, the heat transfer coefficient from Eq. VI-1 can be
replaced by length of the heat path l and the thermal conductivity kcon in W/(m K),
resulting in Eq. VI-4. Table VI-1 gives some examples for the thermal conductivity
[49], [125], [126].
Eq. VI-4
𝑅𝑡ℎ,𝑐𝑜𝑛 =
𝑙
𝑘𝑐𝑜𝑛 ∙ 𝐴𝑐
Material
Thermal
conductivity
[W/(m K)]
Pure copper
379
Aluminium (Al)
206
Pure iron (Fe)
67
Ferrite (MnZn, NiZn)
3.8
Epoxy resin
1.1
Polyethylene
0.33
Polypropylene
0.16
Transformer oil
0.12
Air @ 30°C
0.026
Air @ 70°C
0.03
Table VI-1: Thermal conductivity of selected materials at 100°C (unless specified otherwise)
[49], [125], [126]
The thermal resistance representing the radiation can be described as [49]:
Eq. VI-5
𝑅𝑡ℎ,𝑟𝑎𝑑 =
𝑇2 − 𝑇1
𝜀 ∙ 𝜎 ∙ (𝑇24 − 𝑇14 ) ∙ 𝐴𝑐
232
ε is the emissivity of the surface, σ is the Stefan-Boltzmann constant
-8
4
(5,67∙10 W/(m²K )) and Ac is the radiation surface. T2 is the surface temperature of
the radiating body and T1 is the ambient temperature. This thermal resistance is
temperature dependent. It increases if the ambient temperature increases and if the
temperature difference from ambient to surface increases. This means that the
thermal radiation of a body is increasing at higher temperatures.
The most part of the heat of a magnetic component is dissipated by convection and
radiation, where the convection is usually dominant. The conduction can be usually
neglected [49] – unless heat pipes, heat straps or heat sinks are directly connected to
the component.
Evaluation of Thermal Equivalent Circuit
Subsequent the evaluation of the thermal resistances from the thermal equivalent
circuit depicted in Figure VI-1 will be given according to [49].
Rhs is the internal winding resistance, describing the temperature difference inside
the winding package from the hot spot to the coil surface [49]:
Eq. VI-6
𝑅ℎ𝑠 =
𝑙ℎ𝑠
𝑘 ∙ (𝐴𝑐𝑤 + 𝐴𝑤𝑎 )
The parameters from Eq. VI-6 are [49]:




𝑙ℎ𝑠 represents the parasitic gaps inside the winding package and between
the winding and the coil former.
𝑘 is the conductivity of air (0.026 W/(m K) @ 30°C to 0.031W/(m K) @ 100°C
[49])
𝐴𝑐𝑤 is the surface of the winding facing the core (winding to core area)
𝐴𝑤𝑎 is the surface part of the winding transferring heat direct to the
ambient
The thermal resistance between winding and core can be expressed by [49]:
Eq. VI-7
𝑅𝑤𝑐,𝑐𝑜𝑛 =
𝑙𝑤𝑐
𝑘 ∙ 𝐴𝑤𝑐
𝑙𝑤𝑐 is the air gap distance between winding and core. The radiation can be taken into
account by [49]:
233
Eq. VI-8


𝑅𝑤𝑐,𝑟𝑎𝑑 =
𝑇𝑤 − 𝑇𝑐
𝜀 ∙ 𝜎 ∙ (𝑇𝑤4 − 𝑇𝑐4 ) ∙ 𝐴𝑐𝑤
𝜀 is the emissivity of the coil (e.g. 0.8 for enamelled copper [49])
-8
4
𝜎 is the Stefan-Boltzmann constant (5,67∙10 W/(m²K ))
For the part of the winding which will dissipate heat to the ambient, the convection
can be described by [49]:
Eq. VI-9

𝑅𝑤𝑎,𝑐𝑜𝑛 =
1
ℎ𝑐𝑜𝑛 ∙ 𝐴𝑤𝑎
ℎ𝑐𝑜𝑛 can be evaluated by Eq. VI-2 or Eq. VI-3
The radiation is represented by:
Eq. VI-10
𝑅𝑤𝑎,𝑟𝑎𝑑 =
𝑇𝑤 − 𝑇𝑎
𝜀 ∙ 𝜎 ∙ (𝑇𝑤4 − 𝑇𝑎4 ) ∙ 𝐴𝑤𝑎
The thermal resistances between the core and the ambient are calculated equivalent
to Eq. VI-9 and Eq. VI-10 [49]:
Eq. VI-11
𝑅𝑐𝑎,𝑐𝑜𝑛 =
1
ℎ𝑐𝑜𝑛 ∙ 𝐴𝑐𝑎
Eq. VI-12
𝑅𝑐𝑎,𝑟𝑎𝑑 =
𝑇𝑐 − 𝑇𝑎
𝜀 ∙ 𝜎 ∙ (𝑇𝑐4 − 𝑇𝑎4 ) ∙ 𝐴𝑐𝑎


𝐴𝑐𝑎 is the surface of the core transferring heat direct to the ambient
The emissivity 𝜀 for ferrite is about 0.9 – 0.95 [49]
Evaluation of Temperature Rise
The temperature rise of a component can be evaluated by solving the thermal circuit
from Figure VI-1. The temperature rise is calculated by means of the superposition
law [49]. Figure VI-2 shows the required equivalent circuits, where the convection
and radiation thermal resistances have been combined to the equivalent parallel
circuits.
234
Ths
Pw
Rhs
Tw
Rwa
Tw
Rcw
Rwa
Rcw
Rca
Rca
Tc
Tc
Pc
Ta
a)
Ta
b)
Figure VI-2: Thermal equivalent circuit with a) winding loss source and b) core loss source –
figure based on representation from [49]
Considering the equivalent circuit from Figure VI-2 a), the power loss generated by
the windings will pass through the thermal resistance Rhs as well as the parallel circuit
of Rwa||(Rcw+Rca). The core losses are set to zero and the following power
relationships are valid:
Eq. VI-13
′
′
𝑃𝑐𝑤
𝑃𝑐𝑎
𝑅𝑤𝑎
=
=
𝑃𝑤
𝑃𝑤 𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎
Eq. VI-14
′
𝑃𝑤𝑎
𝑅𝑐𝑤 + 𝑅𝑐𝑎
=
𝑃𝑤
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎
The resulting temperature rises for the nodes w, c and a are as follows:
Eq. VI-15
′
′
∆𝑇𝑤𝑎
= 𝑅𝑤𝑎 ∙ 𝑃𝑤𝑎
= 𝑅𝑤𝑎 ∙
Eq. VI-16
′
′
∆𝑇𝑐𝑤
= 𝑅𝑐𝑤 ∙ 𝑃𝑐𝑤
= 𝑅𝑐𝑤 ∙
Eq. VI-17
′
′
∆𝑇𝑐𝑎
= 𝑅𝑐𝑎 ∙ 𝑃𝑐𝑎
= 𝑅𝑐𝑎 ∙
𝑅𝑐𝑤 + 𝑅𝑐𝑎
∙𝑃
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎 𝑤
𝑅𝑤𝑎
∙𝑃
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎 𝑤
𝑅𝑤𝑎
∙𝑃
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎 𝑤
By considering only the core losses, the thermal equivalent circuit from Figure VI-2 b)
is valid. The equations for the power flow are:
Eq. VI-18
′′
′′
𝑃𝑐𝑤
𝑃𝑤𝑎
∓𝑅𝑐𝑎
=
=
𝑃𝑐
𝑃𝑐
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎
235
Eq. VI-19
′′
𝑃𝑐𝑎
𝑅𝑐𝑤 + 𝑅𝑤𝑎
=
𝑃𝑐
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎
It has to be noticed that the power flow through the resistance Rcw is reversed in
comparison to Figure VI-2, resulting in a negative algebraic sign in Eq. VI-18.
Therefore, the following equations for the temperature rise can be obtained:
𝑅𝑐𝑎
∙𝑃
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎 𝑐
Eq. VI-20
′′
′′
∆𝑇𝑤𝑎
= 𝑅𝑤𝑎 ∙ 𝑃𝑤𝑎
= 𝑅𝑤𝑎 ∙
Eq. VI-21
′′
′′
∆𝑇𝑐𝑤
= 𝑅𝑐𝑤 ∙ 𝑃𝑐𝑤
= −𝑅𝑐𝑤 ∙
Eq. VI-22
′′
′′
∆𝑇𝑐𝑎
= 𝑅𝑐𝑎 ∙ 𝑃𝑐𝑎
= 𝑅𝑐𝑎 ∙
𝑅𝑐𝑎
∙𝑃
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎 𝑐
𝑅𝑐𝑤 + 𝑅𝑤𝑎
∙𝑃
𝑅𝑤𝑎 + 𝑅𝑐𝑤 + 𝑅𝑐𝑎 𝑐
Finally, the resulting temperature rises can be calculated by the superposition:
Eq. VI-23
′
′′
∆𝑇𝑤𝑎 = ∆𝑇𝑤𝑎
+ ∆𝑇𝑤𝑎
Eq. VI-24
′
′′
∆𝑇𝑐𝑤 = ∆𝑇𝑐𝑤
− ∆𝑇𝑐𝑤
Eq. VI-25
′
′′
∆𝑇𝑐𝑎 = ∆𝑇𝑐𝑎
+ ∆𝑇𝑐𝑎
The temperature rise ∆𝑇𝑐𝑤 is only an auxiliary quantity giving no additional
information - the calculation of this value is not necessary. Moreover, it should be
noticed that ∆𝑇𝑐𝑤 can be negative, too.
More interesting for the calculation of magnetic components is the calculation of the
resulting absolute temperatures at the different surfaces (core and winding – Tc and
Tw) and hotspots (inside winding package Ths) of the component:
Eq. VI-26
𝑇𝑤 = 𝑇𝑎 + ∆𝑇𝑤𝑎
Eq. VI-27
𝑇𝑐 = 𝑇𝑎 + ∆𝑇𝑐𝑎
Eq. VI-28
𝑇ℎ𝑠 = 𝑅ℎ𝑠 ∙ 𝑃𝑤 + 𝑇𝑤
236
VII. Calculation of Average Magnetic Field Intensity for
Transformers
The calculation of the proximity losses according to Eq. 3-13 or Eq. 3-20 requires the
evaluation of the average quadratic magnetic field intensity. As example, a side by
side winding transformer with separated but very tight coupled windings is assumed.
Figure VII-1 shows the idealised magnetic field distribution of the transformer for the
case of infinite and finite core permeability. The primary winding N1 will increase the
ampere turns and the field intensity, where the secondary N2 will decrease the field
intensity by means of negative ampere turns. For infinite core permeability, the field
intensity between section x0 and x1 can be described according to:
Eq. VII-1
̂01 (𝑥) =
𝐻
𝑁1 ∙ 𝑖̂1 (𝑥 − 𝑥0 )
∙
(𝑥1 − 𝑥0 )
𝑙
The quadratic average is calculated according to:
𝑥1
̅̅̅̅̅
2
̂01
𝐻
=
1
̂01 (𝑥)|2 𝑑𝑥
∙ ∫ |𝐻
(𝑥1 − 𝑥0 )
𝑥0
Eq. VII-2
𝑥2
1
𝑁1 ∙ 𝑖̂1 (𝑥 − 𝑥0 ) 2
=
∙ ∫|
∙
| 𝑑𝑥
(𝑥1 − 𝑥0 )
(𝑥1 − 𝑥0 )
𝑙
𝑥1
Solving the integral leads to:
Eq. VII-3
1
1 𝑁 ∙ 𝑖̂ 2
̅̅̅̅̅
2
̂01
̂1 )2 = ∙ ( 1 1 )
𝐻
= ∙ (𝐻
3
3
𝑙
The calculation of the average quadratic field intensity of section x1 to x2 leads to a
similar solution, where the ampere turns must be adapted according to amperes law:
Eq. VII-4
2
1
1 𝑁 ∙ 𝑖̂
𝑁 ∙ 𝑖̂
̅̅̅̅̅
2
̂12
̂1 + 𝐻
̂2 )2 = ∙ ( 1 1 + 2 2 ∙ cos(𝜑1 − 𝜑2 ))
𝐻
= ∙ (𝐻
3
3
𝑙
𝑙
The second term in the bracket denotes the ampere turns generated by the
secondary winding, recognising the phase shift of the primary and secondary
currents.
237
Ĥ
Ĥ
Ĥ1
N1
Ĥ1
N2
N1
N2
l
Ĥ0
a)
x0
Ĥ0
Ĥ2
x1
x2
x
b)
x0
Ĥ2
x1
x2
x
Figure VII-1: Side by side winding transformer with ideal magnetic field distribution for
a) infinite and b) finite core permeability
In case of finite core permeability (see Figure VII-1 b), the quadratic average field
intensity can be calculated by the following integral of the section x1 to x2:
𝑥2
1
̅̅̅̅̅
2
̂12
̂12 (𝑥)|2 𝑑𝑥
𝐻
=
∙ ∫ |𝐻
(𝑥2 − 𝑥1 )
𝑥1
Eq. VII-5
𝑥2
1
(𝑥 − 𝑥1 ) 2
̂
̂
=
∙ ∫ |(𝐻2 − 𝐻1 ) ∙
| 𝑑𝑥
(𝑥2 − 𝑥1 )
(𝑥2 − 𝑥1 )
𝑥1
The solution of the integral is:
1
̅̅̅̅̅
2
̂12
̂12 + 𝐻
̂1 ∙ 𝐻
̂2 + 𝐻
̂22 )
𝐻
= ∙ (𝐻
3
Eq. VII-6
1 𝑁1 ∙ 𝑖̂1 2 𝑁1 ∙ 𝑖̂1 𝑁2 ∙ 𝑖̂2
= ((
) +
∙
3
𝑙
𝑙
𝑙
∙ cos(𝜑1 − 𝜑2 ) + (
238
𝑁2 ∙ 𝑖̂2 2
) ∙ cos 2(𝜑1 − 𝜑2 ))
𝑙
VIII. Power Loss Evaluation in Circuit Simulators
State of the art circuit simulator tools offer the opportunity to enable power loss
calculations of the components by means of deposited data, e.g. in the form of look
up tables. This enables the loss calculation via the simulation of the circuit on the fly.
But usually the user does not know how the given loss model works and if the loss
model is appropriate for the given component. This is a problem especially for
magnetic components, because of the many different loss mechanisms and due to
the fact that not all of them occur in the application. The losses are often dependent
on winding design and arrangement as well as cooling. Therefore, the loss models
must be adapted application dependent. Nevertheless, it is possible to implement a
set of standard functions in order to model the basic loss mechanisms of magnetic
components. Subsequent the implementation of the pervious derived loss models
from chapter 3 are presented. All models were implemented in the circuit simulator
PLECS.
DC Losses
Figure VIII-1 shows the implementation of Eq. 3-5. The moving average block will
gather the DC component of the current. The block Temperature Dependent
Resistance includes the DC resistance for 20°C as well as the formula for the
resistance increase. Input of the block is the average winding temperature.
Figure VIII-1: Model for DC power loss calculation
Skin Losses
The skin effect related winding losses will be calculated according to Eq. 3-12 or Eq.
3-19. In practice it is more suitable to use the AC to DC resistance ratio to describe
the loss function:
239
𝑚
Eq. VIII-1
𝑃𝑠𝑘𝑖𝑛 = 𝑅𝐷𝐶,20°𝐶 ∙ (1 + 𝛼𝑐𝑢 ∙ (𝑇2 − 20°𝐶)) ∙
1
∙ ∑ 𝐹𝑎𝑐,𝑑𝑐,𝑖 ∙ 𝑖𝑖2
2
𝑖
At first, the discrete Fourier transform (DFT) of the current must be calculated. The
number of required harmonics will be set in the Discrete Fourier block. It has to be
noticed that the output of this block is a vector. The output of the function
(AC_Losses) is a vector, too. The sum of the skin related losses must be calculated by
splitting the output signal and using a summation block. Triangular current wave
forms of constant frequency have not that much harmonic content. The calculation
of the first ten harmonics or even less is usually sufficient. The block Resistance
Factors includes the AC to DC resistance ratio for each harmonic. These factors must
be calculated according to the given wire technology, e.g. by Eq. 3-11 or Eq. 3-18. The
temperature dependency is taken into account similar as done in the DC loss model
from Figure VIII-1.
Figure VIII-2: Model for skin losses
Proximity Losses
Figure VIII-3 shows the PLECS function for the evaluation of proximity related power
loss of a transformer according to Eq. 3-20 combined with Eq. VII-3 and Eq. VII-6.
Magnitude and phase of the primary and secondary current is generated by the DFT
function of the simulation. The winding specific data (e.g. number of turns, number
of strands, mean length per turn, window height and proximity dissipation factor)
must be supplied by the user. The temperature dependency is considered, too.
240
Figure VIII-3: Model for proximity losses
Core Losses
The core loss calculation was performed by means of the Karhunen-Loève transform
as described in chapter 3.3.3. Figure VIII-4 shows the implemented loss model. The
experience showed that two sets of eigenvectors are usually sufficient to describe a
complete core loss data set. The input of the function requires the parameters of the
eigenvector functions (e.g. Eq. 3-48 to Eq. 3-51), the eigenvalues (e.g. Eq. 3-44), the
mean value of the data set and the core volume. The parameters for a parabolic
temperature dependency, e.g. like in Eq. 3-29 can be inserted, too. Because of the
required logarithmic functions of the model, zero induction is not allowed as input.
Thus a small default value must be set for the induction, for the case that the
magnetic part of the simulation is calculation zero induction values.
Figure VIII-4: Core loss model according to Karhunen-Loève transform
241
IX.
Impedance and Inductance Measurements of CommonDifferential Mode Filter Chokes
The impedance of the chokes was measured with the Bode 100 network analyser
according to Figure 1-1, where the inductance curves were measured with the Wayne
Kerr magnetics analyser (see Appendix X).
A
AC
A
V
AC
a)
V
b)
Figure IX-1: Impedance measurement: a) differential mode; b) common mode
L616-X38 N7
a)
b)
Figure IX-2: Differential mode impedance and inductance for L618-X38 N7 choke
242
L618-X38 N11
a)
b)
Figure IX-3: Differential mode impedance and inductance for L618-X38 N11 choke
243
L659-X830 N9
a)
b)
Figure IX-4: Differential mode impedance and inductance for L659-X830 N9 choke
244
a)
b)
Figure IX-5: Differential mode impedance for L659-X830 N9 choke with inlay segment
245
L659-X830 N19
a)
b)
Figure IX-6: Differential mode impedance and inductance for L659-X830 N19 choke
246
Figure IX-7: Differential mode inductance vs. DC bias for L659-38 N19 choke assembled with
external segment
Common Mode Impedance
Figure IX-8: Common mode impedance for different filter chokes
247
X.
Equipment and Measurement Devices
Insertion Loss, Impedance, Inductance and DC Resistance Measurement
Both inductance and DC resistance measurement was performed by means of four
wire measurement method. Current and voltage are measured independent to
enable improved accuracy. The Precision Magnetics Analyser allows small signal
measurement of the impedance by means of sinusoidal (inductance measurement)
or DC excitation (DC resistance measurement). The DC bias unit is used to measure
the inductance dependent on DC current loading in order to observe saturation
effects of the components/materials. Frequency dependent impedance and insertion
loss measurements were performed with the Bode 100 vector network analyser.



Wayne Kerr Precision Magnetics Analyser 3260B
Wayne Kerr 25A DC Bias Unit 3265B
Omicron Bode 100 Vector Network Analyser
Current and Voltage Probes





Tektronix DPO 2024 Oscilloscope, 4 channels, 200 MHz, 1GS/s
Active High Voltage Differential Probe P5205, 1.3 kV, 100 MHz
AC/DC Current Probe TCP0030, 30 A, 120 MHz
AC/DC Current Probe TCP0150, 150 A, 20 MHz
AC/DC Current Probe TCP305 for TCPA300 Current Probe Amplifier, 50 A, 50
MHz
Power Analyser


ZES Zimmer LMG-500 Power Analyser
ZES Zimmer PSU 200HF-L50 measuring current transformer
Temperature Measurement

Flir i40 infrared camera
Microcontroller

Texas Instruments TI XDS100 C2000 TMS 320F28335
Voltage Sources and Sinks



Delta Elektronika SM 600-10 Voltage Source, 600 V, 10 A
GW Instek GPS 4303 Voltage Source, 2 x 30 V/ 3 A, 3-6 V/1 A, 8-15 V/1 A
EA Electronic Load EA-EL 9160-300HP, 0-160 V, 0-300 A
248
A. List of Symbols
A
Cross-section [m²]
AL
Inductance factor [H/Turn²]
AP
Area Product [m ]
4
B
Magnetic flux density [T]
b
Width [m]
C
Capacitance [F]
D
Dissipation factor [1] or duty cycle [1]
d
Diameter [m]
E
Electric field intensity [V/m]
Fac,dc
AC to DC resistance ratio [1]
f
Frequency [Hz]
G
Electrical conductance [S]
H
Magnetic field intensity [A/m]
h
Heat transfer coefficient [W/(m² K)] or height [m]
I, i
Current [A]
J
Current density [A/m²]
Kff
Air gap correction factor [1]
k
Coupling coefficient [1], thermal conductivity [W/(m K)] or
Steinmetz constant [mW/cm³ or W/kg]
kcu
Copper fill factor [1]
L
Self- or leakage inductance [H]
l
Length [m]
M
Mutual inductance [H]
N
Number of turns [1]
Ns
Number of strands [1]
n
Turn ratio [1]
P
Permeance [Wb/A] or Power [W]
Q
Electric charge [As]
r
Radius [m]
R
Electric [Ω], magnetic (reluctance) [A/Wb] or thermal [K/W]
resistance
249
T
Period time [s] or Temperature [°C, K]
t
Time [s]
V
Electric voltage [V], magnetic voltage (mmf) [A] or Volume [m³]
v
Velocity [m/s]
W
Energy [J]
x
Complex skin parameter [1]
Z
Impedance [Ω]
Greek Symbols
-1
α
Temperature coefficient [K ] or Steinmetz frequency parameter [1] or
angle [°]
β
Steinmetz flux density parameter [1]
Γ
Geometry factor for ring cores [1]
δ
Skin depth, gap length or spacing [m]
ε
Permittivity [As/(Vm)] or emissivity [1]
θ
Angle [°]
λ
Quality factor for litz wires [1] or eigenvalue
μ
Permeability [Vs/(Am)]
ρ
Specific electrical resistance [Ωm]
σ
Conductivity [S/m]
φ
Magnetic flux [Wb]
Ψ
Magnetic flux linkage [Vs]
ω
Angular frequency [s ]
-1
Abbreviations
a
AC, ac
Ambient
Alternating current
c
Core or coupled
ca
Core-ambient
CD
Current Doubler
cu
Copper
cw
Core-winding
cm
Common mode
con
Convection or conduction
250
d
DC, dc
Discrete
Direct current
dir
Direct
dm
Differential mode
e, eq
Equivalent
el
Electric
ext
External
f, ff
Fringing effect
g
Gap
H
Hysteresis
hs
Hot spot
i
Initial
in
Input or inner
inv
Inverse
L
Inductance
m
Magnetic
max
Maximum
min
Minimum
mmf
Magneto Motive Force
pk
prox
r
Peak
Proximity
Relative
rad
Heat radiation
res
Resonance
RMS
Root mean square
th
Thermal
s
Self or switching
sat
Saturation
out
Output
w
Winding or window
wa
Winding-ambient
σ
Leakage
251
B. List of Tables
Table 2-1: General properties of different core materials [1], [49] ............................. 34
Table 2-2: Electrical quantities and their magnetic counterparts ................................ 38
Table 3-1: Full experimental design – representation taken from [82] and modified . 64
Table 3-2: Full experimental design for specific core loss data from N87 – data
extracted from [75] ...................................................................................................... 67
Table 3-3: Centred data in the logarithmic domain for specific core loss data from N87
..................................................................................................................................... 67
Table 5-1: Advantages and disadvantages of EE core current doubler ..................... 106
Table 5-2: Advantages and disadvantages of EEII core current doubler ................... 107
Table 5-3: Advantages and disadvantages of 5 leg core current doubler .................. 108
Table 5-4: Advantages and disadvantages of ring or UU core current doubler ......... 111
Table 5-5: Comparison of different current doubler technologies regarding assembly
considerations ............................................................................................................ 112
Table 5-6: Design specifications ................................................................................. 123
Table 5-7: Design considerations ............................................................................... 130
Table 5-8: Converter and component specifications ................................................. 133
Table 5-9: Data of discrete and integrated current doublers for 48 V, 50 kHz – data
partially previous published in [93] ........................................................................... 144
Table 5-10: Data of discrete and integrated current doublers for 48 V, 200 kHz – data
partially previous published in [93] ........................................................................... 145
Table 5-11: Data of integrated current doubler for 12 V, 50 kHz – data partially
previous published in [94] ......................................................................................... 146
Table 5-12: Data of investigated Si based push pull converters (50 kHz) [93] ........... 147
Table 5-13: Data of investigated SiC based push pull converters (50 kHz, * evaluated
at 2kW)....................................................................................................................... 150
Table 5-14: Data of investigated GaN based push pull converters (200 kHz) ............ 154
Table 5-15: Data of investigated SiC based push pull converters (200 kHz) .............. 158
Table 6-1: Advantages and disadvantages of common mode choke with window
segment ..................................................................................................................... 167
252
Table 6-2: Advantages and disadvantages of common mode choke with top-bottom
segments ....................................................................................................................168
Table 6-3: Advantages and disadvantages of common mode choke with external
segments ....................................................................................................................169
Table 6-4: Advantages and disadvantages of common mode choke with inlay ........170
Table 6-5: Advantages and disadvantages of EE core common-differential mode
chokes ........................................................................................................................173
Table 6-6: Advantages and disadvantages of common-differential-mode choke with
different cores ............................................................................................................175
Table 6-7: Advantages and disadvantages of common-differential-mode pot core
choke with ferrous disc and E core choke with segment ...........................................176
Table 6-8: Core materials for common mode chokes ................................................178
Table 6-9: Data of common mode choke without segment (calculated values in
brackets) .....................................................................................................................188
Table 6-10: Ring cores for common mode filtering – material see [106] [107]; cores
see [108], [109] ..........................................................................................................190
Table 6-11: Characteristic values of common mode chokes ......................................190
Table 6-12: Segments for differential mode improvement of common mode chokes
....................................................................................................................................191
Table 6-13: Specific values for L618-X38 N7 chokes ..................................................192
Table 6-14: Specific values for L618-X38 N11 chokes ................................................193
Table 6-15: Specific values for L659-X830 N9 chokes ................................................195
Table 6-16: Specific values for L659-X830 N19 chokes ..............................................197
Table 6-17: Inverter specifications .............................................................................200
Table 6-18: Filter components for the different filter configurations ........................202
Table I-1: Current ripple equations for interleaved boost converter .........................215
Table I-2: Equivalent inductance values for interleaved boost converter .................215
Table VI-1: Thermal conductivity of selected materials at 100°C (unless specified
otherwise) [49], [125], [126] ......................................................................................232
253
C. List of Figures
Figure 1-1: Required concepts for the design of functionally integrated components15
Figure 1-2: Functional and structural integration of magnetic components ............... 16
Figure 1-3: Flyback converter: a) switch S closed; b) switch S opened ........................ 18
Figure 1-4: LLC series resonance converter ................................................................. 18
Figure 1-5: Transformer with primary resonance tank realised by a) displacement of
primary windings to increase the leakage and b) by expanding the primary winding
over an auxiliary core as proposed by [8] – figure based on representation from [8] 19
Figure 1-6: a) Different alternatives of a transformer with current doubler (discrete
and integrated); b) derivation of the today state of the art transformer with
integrated current doubler – figure based on representation from [16] .................... 20
Figure 1-7: Push-pull forward converter from [19] using magnetically integrated
component: a) circuit; b) integrated magnetic component – figure based on
representation from [19] ............................................................................................. 21
Figure 1-8: Bi-directional interleaved DC-DC converter using coupled inductor ......... 23
Figure 1-9: Cúk converter with coupled inductor – figure based on representation
from [36] ...................................................................................................................... 23
Figure 1-10: a) Three-Phase coupled inductor; b) symmetrical three-phase current
system .......................................................................................................................... 24
Figure 1-11: PV converter using electrical isolation and coupled inductor – figure
taken from [42] and modified ...................................................................................... 25
Figure 1-12: Common mode choke with predefined leakage path for differential
mode signal (ring core with segment) – figure based on representation from [43].... 26
Figure 1-13: Combined common mode/differential mode choke (one winding uses
two different cores) – figure based on representation from [46] ............................... 26
Figure 1-14: Combined common mode/differential mode choke with one common
mode and one differential mode core – figure based on representation from [47] ... 27
Figure 1-15: Two stacked common mode chokes – figure based on representation
from [48] ...................................................................................................................... 27
Figure 2-1: Hysteresis loop - figure based on representation from [3]........................ 32
254
Figure 2-2: Specific core loss vs. saturation flux density for different selected
materials (ferrite: N27, N87; amorphous alloys: Vitrovac 6030, AMCC;
nanocrystalline: Vitroperm 500 F, Finemet F3CC; Iron Powder: KoolMμ) [50], [51],
[52], [53], [54], [55] – figure taken from [56] ...............................................................33
Figure 2-3: Magnetic circuit .........................................................................................38
Figure 2-4: Reluctance of a ferrous rod core ................................................................39
Figure 2-5: Transformer with leakage path ..................................................................43
Figure 2-6: Transformer with a) open secondary winding and b) open primary winding
......................................................................................................................................44
Figure 3-1: Losses in magnetic components – figure taken from [56] and modified ...46
Figure 3-2: Thermal (orange) and electric (black) circuit for the calculation of the
temperature dependent winding losses – figure taken from [56] ...............................48
Figure 3-3: RMS power loss increase dependent on power loss at 20 °C for different
thermal resistances ......................................................................................................48
Figure 3-4: Skin effect for solid (left) and litz wire (right), where the shaded area
indicates the skin depth – figure based on representation from [1] ...........................50
Figure 3-5: AC resistance for 54x0.2 mm litz wire dependent on frequency (green:
ideal litz wire; red: equivalent solid wire; yellow: measured AC resistance; blue:
approximated measurement curve) – figure taken from [59] .....................................57
Figure 3-6: Dissipation factor for 54x0.2 mm litz wire dependent on frequency (green:
ideal litz wire; red: equivalent solid wire; yellow: measured dissipation factors; blue:
approximated measurement curve) – figure taken from [59] .....................................57
Figure 3-7: Typical hysteresis loop for a ferromagnetic core – figure based on
representation from [1] ...............................................................................................58
Figure 3-8: Typical rectangular voltage wave form with zero voltage periods – figure
taken from [56] ............................................................................................................60
Figure 3-9: Example for core loss fitting according to Steinmetz equation using data
from [75] ......................................................................................................................62
Figure 3-10: Normalised eigenvectors for N87 data set...............................................68
Figure 3-11: Example for core loss fitting by means of orthogonal vector functions
using data from [75] .....................................................................................................69
255
Figure 4-1: Bi-directional interleaved DC-DC converter using coupled inductor with a)
inverse coupling and b) direct coupling ....................................................................... 71
Figure 4-2: Magnetic equivalent circuit of two discrete inductors .............................. 71
Figure 4-3: Magnetic equivalent circuit of inverse coupled inductor: a) loose coupling;
b) tight coupling ........................................................................................................... 72
Figure 4-4: Magnetic equivalent circuit of direct coupled inductor: a) loose coupling;
b) tight coupling ........................................................................................................... 72
Figure 4-5: Schematic current and flux waveforms of loose coupled inductors: a)
inverse coupling and D<0.5; b) inverse coupling and D>0.5; c) direct coupling and
D<0.5; d) direct coupling and D>0.5 ............................................................................ 73
Figure 4-6: Schematic current and flux waveforms of tight coupled inductors: a)
inverse coupling and D<0.5; b) inverse coupling and D>0.5; c) direct coupling and
D<0.5; d) direct coupling and D>0.5 ............................................................................ 74
Figure 4-7: Basic magnetic circuit of coupled inductors: a) two winding configuration;
b) n winding configuration ........................................................................................... 75
Figure 4-8: Flux ratio of direct and inverse coupled inductors dependent on the
inductor reluctance ratio ............................................................................................. 77
Figure 4-9: Equivalent inductance to self-inductance ratio dependent on duty cycle
and coupling – representation taken from [33] and modified .................................... 79
Figure 4-10: Self-inductance ratio of coupled and discrete inductors for constant
phase current ripple ..................................................................................................... 80
Figure 4-11: Output current ripple ratio for coupled and discrete interleaved DC-DC
converters dependent on coupling and self-inductance ratio ..................................... 81
Figure 4-12: Self-inductance ratio of coupled and discrete inductors for constant
output current ripple ................................................................................................... 81
Figure 4-13: Assembly possibilities for coupled inductors: a) EE core design; b) UU
core; c) ring core ......................................................................................................... 82
Figure 4-14: Magnetic equivalent circuit for EE core coupled inductor ...................... 83
Figure 4-15: Comparison of leakage permeance calculation and measurement ........ 85
Figure 4-16: Effective differential mode permeability dependent on Γ factor –
representation taken from [88] and modified ............................................................. 86
256
Figure 4-17: Coupling coefficients for an EE65 core dependent on permeability and
gap length .....................................................................................................................88
Figure 4-18: Leakage permeance for an EE65 core dependent on permeability and gap
length ...........................................................................................................................88
Figure 4-19: Leakage permeance of iron powder and ferrite EE cores ........................89
Figure 4-20: Influence of air gaps for self-inductance adjustment ..............................90
Figure 4-21: Leakage permeance of EE and ring cores (μr=2,000) ...............................91
Figure 4-22: Leakage permeance of EE65 and EE70 ferrite cores (μ r=2,000) in
comparison to EE65 powder cores (μr=60 and μr=26) .................................................91
Figure 4-23: Leakage permeance of ring cores with different core size and material .92
Figure 4-24: Leakage permeance comparison of ring and EE cores for different
materials.......................................................................................................................93
Figure 4-25: General design procedure ........................................................................94
Figure 5-1: Push-pull converter with discrete current doubler ....................................98
Figure 5-2: Current waveforms of the push-pull converter with discrete current
doubler .........................................................................................................................99
Figure 5-3: Switching status of the discrete current doubler according to the time
intervals from Figure 5-2 ............................................................................................100
Figure 5-4: Push-pull converter with magnetically integrated current doubler ........101
Figure 5-5: Current waveforms of the push-pull converter with magnetically
integrated current doubler ........................................................................................103
Figure 5-6: Switching status of the magnetically integrated current doubler according
to time intervals from Figure 5-5 ...............................................................................104
Figure 5-7: EE core current doubler: a) component structure; b) magnetic equivalent
circuit ..........................................................................................................................106
Figure 5-8: EEII core current doubler: a) component structure; b) magnetic equivalent
circuit ..........................................................................................................................107
Figure 5-9: 5 leg core current doubler: a) component structure; b) magnetic
equivalent circuit ........................................................................................................108
Figure 5-10: Current doublers with window segment: a) ring core; b) UU core; c)
magnetic equivalent circuit ........................................................................................110
257
Figure 5-11: Current doublers with window and external segment: a) ring core; b) UU
core; c) magnetic equivalent circuit .......................................................................... 110
Figure 5-12: Ring core current doubler with cut core or segment on top and bottom:
a) component structure; b) magnetic equivalent circuit of cut core; c) magnetic
equivalent circuit with top-bottom segment ............................................................. 111
Figure 5-13: Required transformer turn ratio dependent on maximum duty cycle for
half-bridge push pull converters with current doubler rectifier (270 V minimum input
voltage) ...................................................................................................................... 114
Figure 5-14: Simplified magnetic equivalent circuits for integrated current doublers
................................................................................................................................... 118
Figure 5-15: DC equivalent circuit of the integrated current doubler: a) complete
circuit; b) simplified circuit ........................................................................................ 122
Figure 5-16: Self-inductance values dependent on centre leg gap length and number
of turns ....................................................................................................................... 123
Figure 5-17: Secondary-secondary coupling coefficient dependent on gap length .. 124
Figure 5-18: Effective leakage inductance for output current ripple attenuation ..... 124
Figure 5-19: Normalised current ripple for nominal input voltage ............................ 125
Figure 5-20: Normalised current ripple for minimum input voltage ......................... 125
Figure 5-21: Peak flux density for nominal input voltage .......................................... 126
Figure 5-22: Peak flux density for minimum input voltage ........................................ 126
Figure 5-23: Winding losses for nominal and minimum input voltage ...................... 127
Figure 5-24: Core losses for nominal and minimum input voltage ............................ 128
Figure 5-25: Loss balance and optimisation ............................................................... 129
Figure 5-26: Context of area product and component box size for E cores .............. 131
Figure 5-27: Estimated size based on area product approach ................................... 131
Figure 5-28: Calculated specific core losses dependent on current ripple and
frequency for the discrete and the integrated current doubler components ........... 132
Figure 5-29: Weight for different current doubler technologies – data partially
published in [93] ........................................................................................................ 134
Figure 5-30: Size for different current doubler technologies .................................... 135
258
Figure 5-31: Losses for different current doubler technologies – data partially
published in [93].........................................................................................................136
Figure 5-32: Loss balance of different current doublers for a) 4A/mm², b) 6A/mm² and
c) 9 A/mm² RMS current density ................................................................................137
Figure 5-33: Calculated hot spot temperature for a) integrated current doublers and
b) discrete components..............................................................................................139
Figure 5-34: Weight and power loss for different current doubler technologies ......141
Figure 5-35: Magnetically integrated current doublers for 48 V converter output
voltage: a) EE55 (200kHz); b) 84/21/20 5 leg (200kHz); c) nanocrystalline ring core
with ferrite segment (50kHz) – figures taken from [93]and modified .......................142
Figure 5-36: a) Size and b) weight of different assembled current doublers .............143
Figure 5-37: Measured secondary currents of EE core current doubler
(magenta/green: rectifier diode currents; red: rectifier output current) ..................147
Figure 5-38: Efficiency measurement of Si based push-pull converters operating at 50
kHz ..............................................................................................................................149
Figure 5-39: Efficiency values for 50 kHz IGBT converters .........................................149
Figure 5-40: 50 kHz push pull half bridge converter (equipped with 12V rectifier) ...150
Figure 5-41: Efficiency measurement of SiC based push-pull converters operating at
400 V input voltage, 50 kHz – figure taken from [93] and modified ..........................151
Figure 5-42: Efficiency values for 50 kHz SiC converters ............................................152
Figure 5-43: 200 kHz push pull converter with magnetically integrated 5 leg current
doubler .......................................................................................................................153
Figure 5-44: Efficiency measurement for 200 kHz converters with GaN and SiC
semiconductors ..........................................................................................................154
Figure 5-45: Thermal measurement of a) integrated 5 leg current doubler, b)
integrated EE55 current doubler, c) discrete transformer and d) discrete inductor (all
transformers cooled with 18 m³/h forced air cooling; inductors are natural convection
cooled)........................................................................................................................155
Figure 5-46: Measured voltage and current waveforms of the integrated EE55 current
doubler - green/purple: gate source voltages; dark/light blue: secondary transformer
currents; red: rectifier output current .......................................................................157
259
Figure 5-47: Measured voltage and current waveforms of the integrated EE55 current
doubler - green/purple: gate source voltages; dark blue: rectifier diode current; light
blue: rectifier diode voltage ....................................................................................... 157
Figure 5-48: Efficiency values for 200 kHz GaN converters ....................................... 157
Figure 5-49: Efficiency measurement for 200 kHz converters with SiC semiconductors
................................................................................................................................... 159
Figure 5-50: Measured voltage and current waveforms of the integrated current
doubler - green/purple: gate source voltages; dark blue: rectifier diode current; light
blue: rectifier diode voltage ....................................................................................... 159
Figure 5-51: Efficiency values for 200 kHz SiC converters ......................................... 160
Figure 6-1: Propagation of a) differential mode and b) common mode noise – figure
taken from [98] .......................................................................................................... 163
Figure 6-2: Simplified RLC circuit for chokes .............................................................. 164
Figure 6-3: Different winding structures: a) single layer; b) double layer; c) bank; d)
progressive – figure based on representation from [45] ........................................... 165
Figure 6-4: Common mode choke with window segment: a) ring core; b) UU core; c)
magnetic equivalent circuit........................................................................................ 166
Figure 6-5: Shielded common-differential-mode choke with top-bottom segment: a)
ring core; b) UU core; c) magnetic equivalent circuit ................................................ 167
Figure 6-6: Shielded common-differential-mode choke with external and window
segment: a) ring core with cut ring segment; b) ring core with adjusted segment; c)
UU core with U segments; d) magnetic equivalent circuit ........................................ 169
Figure 6-7: Common-mode choke with differential mode inlay: a) component
structure; b) magnetic equivalent circuit .................................................................. 170
Figure 6-8: EE core common mode choke: a) component structure; b) magnetic
equivalent circuit ....................................................................................................... 172
Figure 6-9: EE core common mode choke with external U segments: a) component
structure; b) magnetic equivalent circuit ................................................................... 172
Figure 6-10: 5 leg common mode choke: a) component structure; b) magnetic
equivalent circuit ....................................................................................................... 172
260
Figure 6-11: Separated common-differential-mode choke with three rings: a)
component structure; b) magnetic equivalent circuit ...............................................174
Figure 6-12: Separated common-differential-mode choke with two rings: a)
component structure; b) magnetic equivalent circuit ...............................................174
Figure 6-13: Pot or EE core common mode choke with ferromagnetic disc or segment:
a) component structure; b) magnetic equivalent circuit ...........................................176
Figure 6-14: Magnetic equivalent circuits for common mode ring core chokes with
enhanced leakage inductance: a) segment inserted inside the window; b) external
segment; c) window segment and external segments ...............................................182
Figure 6-15: Differential mode magnetic equivalent circuit for common-differential
mode EE core choke ...................................................................................................184
Figure 6-16: Differential mode inductance factor for different EE cores ...................184
Figure 6-17: Maximum possible differential mode inductance factor for ring and EE
cores ...........................................................................................................................185
Figure 6-18: Common mode inductance factor for ring and EE cores .......................186
Figure 6-19: Ring core to EE core DC Resistance ratio for constant common mode and
differential mode inductance .....................................................................................186
Figure 6-20: Differential mode inductance and peak flux density dependent on
segment width (N=9, N∙î=127.3A) ..............................................................................188
Figure 6-21: Different common-differential mode chokes: a) L618-X38 N11 and L618X38 N11 + T106-24 chokes; b) L659-X830 N19Z + 22x16x3 + RK60 choke; c) L659-X830
N19Z + 22x16x3 choke; d) L659-X830 N7 + Disc40x4.5 choke; e) assembly of L659X830 + R22 + 18.5x13.6x4 ..........................................................................................191
Figure 6-22: Normalised differential mode inductance and weight for L618-X38 N7
chokes ........................................................................................................................192
Figure 6-23: Normalised differential mode inductance and weight for L618-X38 N11Z
chokes ........................................................................................................................194
Figure 6-24: Normalised differential mode inductance and weight for L659-X830 N9
chokes ........................................................................................................................196
Figure 6-25: Normalised differential mode inductance and weight for L659-X830 N9
chokes with inlay segment .........................................................................................196
261
Figure 6-26: Normalised differential mode inductance and weight for L659-X830 N19
/ N19Z chokes ............................................................................................................ 198
Figure 6-27: Differential mode inductance increase vs. weight increase for different
chokes ........................................................................................................................ 199
Figure 6-28: Solar inverter topology using buck converter and unfolding bridge ..... 200
Figure 6-29: Insertion loss measurement .................................................................. 201
Figure 6-30: EMI Filter A ............................................................................................ 201
Figure 6-31: EMI filter configurations ........................................................................ 203
Figure 6-32: Weight for the different filter configurations ........................................ 204
Figure 6-33: Total DC resistance for the different filter configurations ..................... 204
Figure 6-34: Insertion loss for the filters A, B, and C: a) differential mode; b) common
mode .......................................................................................................................... 205
Figure 6-35: Insertion loss for the filters D and E: a) differential mode; b) common
mode .......................................................................................................................... 206
Figure I-1: Interleaved boost converter ..................................................................... 212
Figure II-1: Geometric dimensions for E core: a) front; b) top ................................... 216
Figure II-2: Schematic leakage flux beside centre leg ................................................ 217
Figure II-3: Calculation of leakage flux inside the windows ....................................... 217
Figure II-4: Calculation of leakage flux beside the outer legs .................................... 218
Figure II-5: Calculation of leakage flux at the front ends ........................................... 219
Figure II-6: Calculation of leakage flux between the outer top and bottom surfaces220
Figure III-1: Schematic of ring core with block segment (cuboid) .............................. 221
Figure III-2: Schematic for cut ring segment joined on a ring core ............................ 223
Figure IV-1: FEM simulation of gapped EE core: a) field intensity plot; b) flux density
plot ............................................................................................................................. 224
Figure IV-2: Equivalent magnetic circuit for an inductor with air gap and fringing
effect .......................................................................................................................... 226
Figure V-1: Winding arrangements for transformers: a) side by side; b) top-bottom; c)
interleaved; d) side by side interleaved ..................................................................... 228
262
Figure V-2: Winding arrangement and parameters for the calculation of leakage
inductance – figure based on representation from [121] ..........................................229
Figure VI-1: General thermal equivalent circuit for magnetic components – figure
based on representation from [49] ............................................................................231
Figure VI-2: Thermal equivalent circuit with a) winding loss source and b) core loss
source – figure based on representation from [49] ...................................................235
Figure VII-1: Side by side winding transformer with ideal magnetic field distribution
for a) infinite and b) finite core permeability ............................................................238
Figure VIII-1: Model for DC power loss calculation ....................................................239
Figure VIII-2: Model for skin losses ............................................................................240
Figure VIII-3: Model for proximity losses ...................................................................241
Figure VIII-4: Core loss model according to Karhunen-Loève transform ...................241
Figure IX-1: Impedance measurement: a) differential mode; b) common mode ......242
Figure IX-2: Differential mode impedance and inductance for L618-X38 N7 choke ..242
Figure IX-3: Differential mode impedance and inductance for L618-X38 N11 choke 243
Figure IX-4: Differential mode impedance and inductance for L659-X830 N9 choke 244
Figure IX-5: Differential mode impedance for L659-X830 N9 choke with inlay segment
....................................................................................................................................245
Figure IX-6: Differential mode impedance and inductance for L659-X830 N19 choke
....................................................................................................................................246
Figure IX-7: Differential mode inductance vs. DC bias for L659-38 N19 choke
assembled with external segment .............................................................................247
Figure IX-8: Common mode impedance for different filter chokes ...........................247
263
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One centralised question of this work is to explore the performance advantage
of functionally integrated magnetic components in comparison to discrete
components. Many applications allow the introduction of simple magnetic structures
and standard cores or simple modifications of these (flux bypasses) in order to
enable the required component behaviour. The design guidelines introduced in
this work enable the design of functionally integrated magnetic components with
limited effort and, therefore, the application of components which enable superior
performance regarding size and power loss for the applications.
ISBN 978-3-7376-0226-6
9 783737 602266
Thiemo Kleeb
The functional integration of magnetic components is a known technique in
order to enable high power densities for power electronic converters. Magnetic
components are mandatory in many power electronic converters and many
topologies demand more than one magnetic component. Therefore, the functional
integration of magnetic components allows realising several magnetic functions
within one component. This technique promises lower total size, losses and costs
without switching frequency increase. There are several examples in the literature
for coupled inductors, common-differential-mode chokes or transformer-inductor
components.
Investigation on Performance Advantage of Functionally Integrated
Magnetic Components in Decentralised Power Electronic Applications
13
13
Elektrische Energiesysteme
Thiemo Kleeb
Investigation on Per formance Advantage of
Functionally Integrated Magnetic Components in
Decentralised Power Electronic Applications