Contactless Energy Transmission System with Rotatable

Contactless Energy Transmission System with Rotatable Transformer – Modeling, Analyze and Design

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Instytut Elektrotechniki E l e c t r o t e c h n i c a l I n s t i t u t e

Ph.D. Thesis

M. Sc. Artur J. Moradewicz

Contactless Energy Transmission

System with Rotatable Transformer

- Modeling, Analyze and Design

Energoelektroniczny System Zasilania Bezstykowego

z Transformatorem Obrotowym -

- Modelowanie, Analiza i Projektowanie

Thesis supervisor

Prof. Dr Sc. Marian P. Kazmierkowski

Warsaw, Poland 2008


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Preface

The work presented in this thesis was carried out during my Ph.D. work at the

Electrotechnical Institute – Department of Electric Drives.

I would like to express my gratitude for my supervisor Professor Marian P.

Kazmierkowski, whose guidance, encouragement and continuous support made this

thesis possible. His excellent communication skill and insight made each discussion

become a valuable chance for me to learn about my work and scientific inspiration.

I am also grateful to Prof. Jan Iwaszkiewicz from the Electrotechnical Institute,

Gdansk Branch and Prof. Jerzy T. Matysik from the Institute of Control and Industrial

Electronics, Warsaw University of Technology, for their interest in this work and

holding the post of referee.

Furthermore, I thank all Colleagues from the Department of Electric Drives for

their support, assistance and friendly atmosphere.

Finally, I am very grateful for my wife Dagmara’s and son Adam’s love,

patience and faith. I would also like to thank my whole family, particularly my parents

for their care over the years.


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Table of Contents

1. Introduction ……………………………………………………………… 5

2. Review of Contactless Inductive Coupled Energy Transmission (CET) Systems ……………………………………………………………. 82.1 Basic Principles of Operation……………………………………………. 8

2.2 CET Systems with Multiple Secondary Winding………………………... 9

2.3 CET Systems with Cascaded Transformers……………………………… 10

2.4 CET Systems with Sliding Transformers………………………………... 11

2.5 CET Systems with Multiple Primary Winding…………………………... 13

2.6 Summary and Conclusion………………………………………………... 14

3. Transformer Model in CET System…………………………………….. 16

3.1 Introduction………………………………………………………………. 16

3.2 Two winding transformers……………………………………………….. 16

3.2.1 Ideal Transformer……………………………………………….. 17

3.2.2 Π – model as a Practical Transformer………………………….. 18

3.2.3 Conversion of Transformer Π-model to Coupled Inductor Model …………………………………….. 21

3.3 Magnetic and Electrical Model Analogy………………………………… 21

3.4 Examples of the Rotating Transformer Construction……………………. 23

3.5 Transformation of Magnetic to Electric Model………………………….. 25

3.6 Calculation of Section Reluctance in the Pot Core Rotatable Transformer. ………………………………………………….. 303.7 Measurement of Transformer Parameters………………………………... 32

3.7.1 No Load Test……………………………………………………. 32

3.7.2 Transformer Short Circuit Test…………………………………. 33

3.7.3 Leakage and Mutual Inductances Measurements………………. 33

3.8 Transformer Copper and Core Losses Ratio…………………………….. 36

3.9 Conclusion……………………………………………………………….. 37

4. Power Converters Used in Contactless Energy Transmission Systems 39

4.1 Introduction………………………………………………………………. 39


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4.2 Basic Principles of Resonant Converters………………………………… 40

4.2.1 Resonant Converters Topologies……………………………….. 40

4.2.2 The Leakage Inductances Compensation in CET System……… 44

4.2.3 Analysis of Series-Series Resonant Converter…………………. 46

4.2.4 Analysis of Series-Parallel Resonant Converter………………... 53

4.2.5 Summary and Conclusions……………………………………... 57

4.3. CET System with Series Topology Compensation……………………… 61

4.3.1. CET System with Secondary Compensation…………………... 62

4.3.2 CET System with Primary Compensation ……………………... 65

4.3.3. CET System with Compensation Capacitors on Both Side of the Transformer……………………………………………… 68

4.3.4. Voltage Gain Behavior of the SS-compensated Circuit……….. 70

4.4 Conclusions……………………………………………………………… 76

5. Control and Protection System…………………………………………. 77

5.1 Introduction……………………………………………………………… 77

5.2 Control System Operation and Behavior………………………………… 77

6. Design and Description of Laboratory Prototype …………………….. 85

6.1 Introduction…………………………………………...………………….. 85

6.2 Design Procedure of CET System……………………………………….. 85

6.3 Description of the Laboratory Prototype………………………………… 86

7. Simulation and Experimental Results………………………………….. 91

7.1 Introduction………………………………………………………………. 91

7.2 Performance Characterization of SS and SP Compensation Circuits……. 917.3 Investigation Results of Developed CET System ……………………….. 93

8. Summary and Closing Conclusions……………………………………... 101

References…………………………………………………………………… 103

Appendixes…………………………………………………………………... 108


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1. Introduction

Recently, the contactless energy transmission (CET) systems are developed and

investigated widely [see list of References]. This innovative technology creates new

possibilities to supply mobile devices with electrical energy because elimination of cables,

connectors and/or slip-rings increase reliability and maintenance-free operation of such a

critical systems as in aerospace, biomedical and robotics applications. Figure 1.1 shows

classification of the CET systems. As “medium” for contactless energy transfer could be used

electromagnetic waves including light, acoustic waves (sound) as well as electric field. In the

most popular applications, the core of CET system is inductive or capacitive coupling

between power source and load, and high switching frequency converter.

Fig. 1.1. Classification of Contactless/Wireless Energy Transmission Systems

The capacitive coupling (Fig. 1.2b) is used in low power range (e.g. supply systems for

sensors) whereas inductive coupling (Fig. 1.2a) allows transferring power from a few mW up

to hundred kW [26]. It should be noted that there is no commonly accepted nomenclature in

CET systems. Some authors use term “wireless” [6, 18-20, 27, 48, 51] instead of

“contactless” energy transmission or power supply. In author opinion the term “wireless”

energy transmission (or power supply) should be used only to describe systems where energy

is transmitted on longer distance (several meter), like for cellular phone or wireless-sensor

technology [6, 27].

In this work only inductive coupled CET systems are considered. The potential applications

for such a technology are practically endless and can range from the transfer of energy

between low power home and office devices to high powered industrial applications. Medical,


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marine, and other applications where physical electrical contact might be dangerous (battery

chargers), impossible or very problematic, are all prospective candidates for the use of

contactless energy transfer CET systems.

Air gap

b)a)

Ferromagneticmaterial

Fig. 1.2. Inductive (a) and capacitive (b) coupling used in CET systems.

Because of many parameters used in specification of a CET system, it has to be designed and

adapted to individual conditions and there is no one universal solution. This thesis is limited

to investigation of inductive coupled CET system with rotatable transformer which is used in

industrial robots and manipulators, however, the results are valid for wide range other

applications. In spite of many papers presenting individual solution of inductive coupled CET

systems (5), [4, 8-15, 18, 23-25, 33-35, 37, 45-46, 56] there is no commonly accepted control

and design methodology. Because of high switching frequency (fsw ≥ 20 kHz) used in CET

converters, most of the reported systems has been build in hardware technology (5), (8), [30-

35, 45, 46, 55] and implemented control and protection methods were characteristic for

hardware based approach. However, there is a need to develop more sophisticated method

which could easy be implemented in digital signal processors (DSP) or programmable logical

controllers like FPGA circuits. Therefore, author of this work has formulated the following

thesis:

“Use of an extreme regulator which controls amplitude of primary current or the phase

angle between primary side voltage and current in inductive coupled Contactless Energy

Transmission (CET) systems provides high efficiency energy transfer for wide range of

inductive coupling factor values”.

In order to prove the above thesis, the author used an analytical and simulation based

approach, as well as experimental verification on the laboratory setup with a 3 kW CET

system with rotatable transformer. Thus, the new CET system should guarantees:

- High total efficiency thanks to optimization of all system-components,


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- Optimal operation of high-switching frequency power transistor based resonant converter,

- Low cost single board FPGA based controller,

- Reliable and fast operation,

- Flexibility to parameter changes of the main circuit

The thesis consists of seven chapters. Chapter 1 is an introduction. In Chapter 2 a review

of basic concepts and solutions of inductive coupled contactless energy transmission systems

are presented. Chapter 3 describes in a systematical way modeling and parameter optimization

of transformer used in CET systems. Also, the measurement technique of practical

transformer is presented and results are compared with analytical calculations. Very important

is Chapter 4 in which resonant power converter is analyzed and several variants of leakage

inductance compensation method as well as operation characteristic are discussed. Chapter 5

describes the control, monitoring and protection system implemented in programmable logic

circuit FPGA (Stratix II EP2S60F1020C3ES – device). The 3 kW series-series (SS) resonant

converter laboratory prototype with rotatable transformer is described in Chapter 6. In

Chapter 7 simulation study and experimental verification of the developed CET system are

presented. Finally, Chapter 8 includes summary and general conclusions. The thesis is

supplemented by 5 Appendices.

In the author’s opinion the following parts of the thesis represent his original

contribution:

• Development of simulation models for the contactless energy transmission system

including:

a) Rotatable transformer using QuickField and Matlab packages (Chapter 3)

b) Analyze of resonant circuits using Matlab packages (Section 4.3)

c) Series–series high-frequency resonant converter using OrCAD-PSpice and

SABER packages (Chapters 4 and 5)

d) Control and Protection block using OrCAD-PSpice and SABER packages

(Chapters 6 and 7).

• Design, construction and verification of the experimental setup with rotatable

transformer and 3 kVA resonant converter.

• Elaboration and implementation of FPGA (Stratix II EP2S60F1020C3ES – device)

based control and protection system for resonant converter (Chapter 6).

• Elaboration of design methodology for contactless energy transmission system

verified by simulation and laboratory investigations.


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2. Review of Contactless Inductive Coupled Energy

Transmission (CET) Systems

2.1 Basic Principles of Operation

Figure 2.1 shows the block diagram of typical inductive coupled CET systems. It

consists of primary side DC/AC resonant converter which converts DC into high frequency

AC energy. Next the AC energy via transformer with inductive coupling factor k is

transmitted to the secondary side. The secondary side is not connected electrically with

primary and, therefore, can be movable (linearly or/and rotating) giving flexibility, mobility

and safeness for supplied loads. In the secondary side the high frequency AC energy is

converted safety by AC/DC converter to meet requirements specified by the load parameters.

In most cases as the AC/DC converter simply diode rectifier with capacitive filter is used.

DC

SO

UR

CE

LOA

D

Fig. 2.1. Block scheme of CET system

However, in some applications an active rectifier or inverter (for stabilized DC or AC loads)

is required [13, 14, 37]. Hence, the inductive coupled CET system consists mainly of resonant

converter and large air gap transformer. Depending on the power range and air gap length

different transformer cores can be used (see Chapter 3).


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Fig. 2.2. Power range of Contactless Energy Transmission (CET) systems

based on inductive coupling versus air gap wide.

A general overview representing construction of inductive coupling used in CET systems is

shown in Fig. 2.2. It can be seen that for high power and low air gap, transformers with

magnetic cores in primary and secondary side are applied. Contrary, for large air gap and low

power air transformers (coreless) are preferred. A special case is a sliding transformer which

can have construction for linear or circular movement [2, 30, 36]. The final configuration of

CET systems depends also strongly on number of loads to be supplied. In such cases

transformer with multiwinding secondary or primary side are used. In the next subsection

some selected examples of inductive coupled CET systems will be presented.

2.2 CET Systems with Multiple Secondary Winding

The CET system of Fig. 2.1 can be equipped with multiple secondary winding as

shown in Fig. 2.3. This is very flexible solution in which several isolated and/or moving loads

can be supplied. In situations when stabilized AC or DC loads are required, an additional

active DC/AC or DC/DC converter has to be added (Fig. 2.3). Of course, it results in

additional losses and efficiency reduction. Based on this idea, in [13, 14], a CET system has

been proposed which can be compared to a plug-and-socket extension cable. Instead of


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inserting a plug into a socket, a connection between supply line (cable) and loads (clamps) is

established using CET. Also, ABB Corporate Research, Ladenburg, Germany has developed

a factory communication and wireless power supply system for sensors and actuators called

WISA [49, 50, 51]. In this solution a coreless single winding primary side (constructed in

form of a frame) is coupled with distributed multiple secondary windings to supply sensors

and actuators with 10 mW output power each.

DC

DC

DC

AC

DC

AC

C1

C1

L Lk

DC

DCDC

AC

SECONDARY SIDEAC/DC - DC/DC

CONVERSIONRESONANT CIRCUIT

C1

DC

ACC1

SECONDARY SIDEAC/DC

CONVERSIONRESONANT CIRCUIT

SALVE INDUCTANCE

COUPLING FACTOR

PRIMARY SIDESUPPLY CONVERTER

RESONANT CIRCUIT

SECONDARY SIDEAC/DC - DC/DC

CONVERSION

RESONANT CIRCUIT

PRIMARY WINDINGS

SECONDARYWINDINGS

Fig. 2.3. Contactless energy transmission (CET) system with multiple secondary winding

The transformers used in the system of Fig. 2.3 can have different construction:

stationary, rotating, rotatable, with magnetic core or coreless. As an example a rotating

transformer with double parallel connected secondary windings is used in contactless energy

transmission (CET) system for power supply of airborne radar systems [44, 45].

2.3 CET Systems with Cascaded Transformers

In Fig. 2.4 a CET system used in power supply for robots and manipulators [11] is

shown. The indirect DC link AC/DC/AC power converter generates a square wave voltage of

200V– 600V and 20–60 kHz frequency. This voltage is fed to the primary winding of first

rotatable transformer located on the first axis of the robot. The transformer secondary side is

connected to the next DC link AC/DC/AC power converter, which using pulse width


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modulation (PWM) technique, generates variable frequency AC voltage to supply first three

phase motor. The transformer secondary is also connected to the primary of the next rotatable

transformer which is located on the second joint of the robot. The transformer feeds the

second axis drive in similar way as described above for the first machine. More transformers

may be added to create arrangement of an AC bus throughout the robot. Similar system is

applied for multi-layer optical disc used in data storage systems [15]. However, the output

power in optical disc is in the range of 20–30 mW, whereas in robots supply 10-20 kW.

Fig. 2.4. Contactless energy transmission (CET) system with cascaded transformers.

2.4 CET Systems with Sliding Transformers

Contactless electrical energy delivery systems used in long distance are based on

sliding transformers with long primary windings [2, 30, 36]. Basically, two configurations are

applied: primary winding forming elongated loop as long as range of receiver movement is

required (Fig.2.6a) or circular form for circular movement (Fig. 2.6b). The output converter(s)

and load(s) are directly connected to secondary winding placed on movable magnetic core.

PRIMARY WINDINGS

SECONDARY WINDINGS

FERRITE CORE

1U

Fig. 2.5. Example of sliding transformer construction for linearly moving secondary


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a)

b)

Fig. 2.6. Basic configuration of CET system with sliding transformer; (a) for linear movement, (b) for circular movement

The magnetic core constructions enable for free movement of secondary winding along of the

primary winding loop (Fig. 2.5). The sliding transformer gives possibility to construct long

contactless, electrical energy delivery systems for mobile receivers. These transformer cores

are composed of many strips of magnetic materials. Regarding magnetic and mechanical

properties, the amorphous or nanocristalic magnetic materials are preferable.


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However, when high dynamic properties of mobile receiver are required, some problem may

appear because of core inertia. Heavy magnetic core is fixed with the energy receiver (Fig.

2.5), therefore it increases mass of on the secondary side. The length of primary winding is in

the range of 1–70m and output power 1–200kW [36].

2.5 CET Systems with Multiple Primary Winding

Contactless Energy Transfer (CET) system with multiple primary winding without

magnetic core is presented in Fig. 2.7. In these solutions electrical energy is transferred

between primary and secondary coils through inductive coupling across an air gap. The

multiple primary winding consist of a matrix of small (20-40 mm) hexagon or circular shape

spiral coils embedded into desktop and creating a power supply array to transfer energy for

consumer electronic devices placed on the table. An electronic device like: laptop, portable

music-player, cellular phone, etc., should have the secondary coils mounted in their casing.

When such a devices are placed on the desktop array, the primary and secondary coils will

automatically coupled enabling contactless energy transfer. To reduce magnetic field stray

and improve total efficiency, only limited number of primary coils, located closest to the

device, are excited (darkened coils in the Fig. 2.7).

Fig. 2.7. Contactless Energy Transfer (CET) system with multiple primary winding creating a power

supply array for consumer devices; only darkened coils placed closest to the device are excited.


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When design such a CET system care has to be taken on optimized primary and secondary

coils to get magnetic coupling that is as constant as possible for a sufficiently large area. Also

special problem is separate power supply for every primary winding and its control [52].

The same approach is used, as competitive solution to sliding transformer, for moving

actuators and so called “flying robots” [4].

2.6 Summary and Conclusion

A brief review of basic contactless energy transfer (CET) systems, with special focus

on inductively coupled solution, is given in this Chapter. Several group of application with

typical specification are summarized in the Table 2.1. Key conclusions include the following:

● The CET systems are used in power range from mW (biomedicine, sensors,

actuators, etc.) till several hundred kW (cranes, fast battery charging);

● The final efficiency achieved by inductively coupled CET systems is in the range of

60-90% for low and high power applications, respectively;

Table 2.1. Overview of Inductive Coupled CET Systems

Transformer construction

DC/ AC converter O

utpu

t Po

wer

Out

put

Vol

tage

Air

Gap

le

ngth

Max

. ef

ficie

ncy

Application

/ Primary side Secondary side Topology Freq.

[kHz] [W] [V] [mm] [%] -

1 Single

winding ferrite core

Single winding ferrite core

Full bridge MOSFET/

IGBT

20 - 100

1 - 150 kW

15 - 350 0.2 – 1 1- 300

≥ 90 ≥ 80

Battery chargers [12, 17, 23-26, 28-29, 33-35,40, 56]

2 Single Coreless

Triply ferrite core moving

Flyback MOSFET 125 0.1 3.0 DC - Biomedical

[40, 43, 48,]

3 Single

winding ferrite

Double ferrite rotating

Full bridge MOSFET 100 1000 54 DC 0.25 - 2 ≥ 90 Biomedical

[40, 43, 48,]

4 Single

winding coreless

Multiwinding ferrite core movable

Full bridge MOSFET 80 2 x

240 240 AC 50 Hz 2 - 5 ≈ 90

Multiple users Mobile devices

[6, 13, 14]

5 Single

winding coreless

Multiwinding coreless movable

Full/Half Bridge

MOSFET 120

Each load 0.01

5 - 15 1000 - 7000 -

Industrial sensors and actuators, ABB [1,18-20, 41-42, 49-

51]

6 Single

winding ferrite core

Single winding ferrite core

rotatable/linear

Full bridge IGBT 20-40

10 – 60 kW

3 x 230V AC

0.2 - 2 ≥ 92 Robots and manipulators [11,15]

7

Multiple winding coreless

(Desktop)

Single winding coreless movable

Half bridge MOSFET

100 - 400

30 - 300 12 2 - 5 ≈ 90

Stationary (laptops, phone) or mobile

(actuators) [4, 6, 8-10, 27, 52, ]


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● In high power (>1kW) transformers with core winding are applied;

● In low power (<100 mW) air gap coupling and very high transmission frequency

(100 – 1000 kHz) is preferred;

● For long distance mobile loads CET systems with sliding transformers are used;

● There is no one standard solution of CET system, every design has to take into

account several specific parameters and user conditions.


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3. Transformer Model in CET Systems 3.1 Introduction Magnetic components are a vital part of most power electronic equipment, and the models

used in a simulation must faithfully reproduce or predict the behavior of the circuit. Most of

the other electronic components in these circuits have predetermined models that have been

derived from standardized components. Magnetic components, however, are rarely

standardized and are generally designed for specific applications. In most cases the model, or

at least the component values within the model, must be altered for each new circuit

simulation.

L1 L3

L2

M12 M23

M13

Fig. 3.1. The transformer model as coupled inductances

Figure 3.1. show the model under assuming that a transformer can be represented by an

inductor for each winding (L1, L2, … Ln) and a series of mutual inductances between the

windings (M12, M2, … Ln).

3.2 Two Winding Transformers

The usual method of a transformer simulation is via the specification of the open-

circuit inductance that is seen at each winding, and then the addition of the coupling

coefficients to a pair of coupled inductors.

1υ 2υ

1i 2iN1 N2

Fig. 3.2. A two-winding transformer.


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This approach tends to lose the physical meaning associated with leakage and magnetizing

inductance and does not allow the insertion of a nonlinear core. However, it provides a

transformer model that is simple to create and simulate efficiently. The CET transformer, its

related equations, and its relationship to an ideal transformer with added leakage and

magnetizing inductance are discussed in this Section.

Algebraically, the voltages for two winding transformer equations, using the self and mutual

inductances can be express as:

dtdiL

dtdiM

dtdiM

dtdiL

22

1122

212

111

+=

+=

υ

υ (3.1)

And for the multi-winding transformer (Fig. 3.1) equation (3.1) can be rewritten in the matrix

form as:

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

dtdi

dtdi

LM

LMML

nnnn

nij

n

.

.

.

.................................................................................

.....................................

.

.

.

1

1

22

1111

υ

υ

(3.2)

The mutual inductance is related to the magnetic coupling factor k and self winding

inductances of primary and secondary side:

21LL

Mk = (3.3)

3.2.1 Ideal Transformer

Considering the two winding transformer, and applying the Ampere’s law, yields

2211 ININMMF += (3.4)

Substituting CRMMF ⋅Φ= we obtain

2211 ININRC +=⋅Φ (3.5)

However, in the ideal transformer the resistances of the winding are neglected and assuming

the core reluctance RC = 0. Thus, above Eq. (3.5) becomes:


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22110 ININ += (3.6)

and applying the Faraday’s law to an ideal transformer are obtains:

dtdN

dtdN CC Φ

=Φ

= 2211 υυ (3.7)

For the ideal transformer Eq. (3.6 - 7) can be rewritten as:

1

2

2

1

2

1

2

1

NN

ii

NN

−=

=υυ

(3.8)

N1 N2

+ +

- -

1υ 2υ

1i 2i

Fig. 3.3 Equivalent scheme of ideal transformer.

3.2.2 Π – Model as a Practical Transformer

The transformer model shown in Fig. 3.3 is an abstract model. The reluctance RC in practical

transformer is nonzero. Substituting the expression for Φ Eq. (3.7) into (3.5), we obtain:

dtNNiid

RN

C

⎟⎟⎠

⎞⎜⎜⎝

⎛+

= 1

2212

11υ (3.9)

The Eq. (3.9) consists of two main terms. The first term: CR

NL2

112 = – magnetizing

inductance, referred to the primary transformer side. Second term, 1

221 N

Niiim += -

magnetizing current referred to the primary transformer side. To more deeply analysis we

need, however, a circuit model that includes leakage and magnetizing inductance and a turn’s

ratio. An example of this type of model is shown in Fig. 3.4.


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L12

L22L11

N1 N2

1υ 2υ

1i 2i

Fig. 3.4. Structure of the Π model. Two-winding transformer model including magnetizing inductance L12, primary and secondary leakage inductances L11, L22 and number of turns N1, N2.

The leakage inductances of transformer winding are mainly determinate by leakage fluxes

Φ11, Φ22, which are linked only to one winding primary or secondary, respectively.

2

22222

1

11111 i

NLi

NL Φ=

Φ= (3.10)

The leakage inductances are in series with the windings, so the transformer self inductances

can be written as:

1

2122221222

2

1121112111

NNMLLLL

NNMLLLL

+=+=

+=+= (3.11)

where

L1 – primary self inductance,

L2 – secondary self inductance,

L12 – magnetizing inductance referred to the primary side,

L21 – magnetizing inductance referred to the secondary side,

M12 – mutual inductance between primary and secondary winding.

The magnetizing and mutual inductance are expressed as:

2

2

12112

2

121

1

21212

⎟⎟⎠

⎞⎜⎜⎝

⎛=

==

NNLL

NNL

NNLM

(3.12)

If the transformer turns ratio n is define as

1

2

NNn = (3.13)

the relationship between the two winding transformer inductances in model of Fig. 3.1 and

Fig. 3.4 yields:


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12

122

222

12111

nLMLnLL

LLL

=+=

+=

(3.14)

and coupling factor

( )( )122

221211

12

LnLLLnLk

++= (3.15)

So, the equation for circuit shown in Fig. 3.4. can be expressed as:

( )

( )dtdiLnL

dtdinL

dtdinL

dtdiLL

212

222

1122

212

112111

++=

++=

υ

υ (3.16)

If we assume the same numbers of primary and secondary winding N1 = N2, the inductances

in Eq. (3.14) can be described as follows:

12

12222

12111

LMLLLLLL

=+=+=

(3.17)

Additionally, if LLL == 21 (Fig. 3.2) the leakage inductances and coupling factor will

reduce to:

121222111 M

kkMLLL −

=−== (3.18)

LMk 12= (3.19)

Figure 3.5 show another equivalent transformer scheme. The following equation for this

model can be written:

2221

2112

2112

NPL

NPL

NNPM

m

m

m

=

=

=

(3.20)

Pm

L22L11

1υ 2υ

1i 2iN1 : 1 1 : N2

Fig. 3.5. Transformer model with magnetizing permeance.


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The presented above inductance value L1, L2, L12, L21 and the permeance Pm are always

positive. The turn’s ratio n can be positive or negative depending on the transformer winding

direction. The mutual inductance M12 can be also negative or positive in that aspect (7). In

conventional transformer, coupling factor is normally very close to 1 and the leakage

inductances are close to zero. The range of k ∈(-1, 1) represents the degree of magnetic

coupling between the primary and secondary transformer side. The coupling factor k

decreases when the core saturates.

3.2.3 Conversion of Transformer Π - model to Coupled Inductor Model

To use the coupled inductor model (Fig. 3.1), it is necessary to determine the values in

the Π model and then convert them to the values for the coupled inductor model.

The main problem in the analysis of coupled inductor transformer model is precision of

determination circuit parameters. In a typical transformer, the magnetizing inductance (L12)

has a value of few mH. The leakage inductances, however, are below of μH level. The value

of coupling coefficient k must be specified with enough accuracy to recreate this difference

accurately. This problem is good illustrated by inversion of Eq. (3.14).

2112

21222

21111

LLnk

L

LLnkLL

LLnkLL

=

−=

−=

(3.21)

Example value:

For L12 = 5 mH, leakage inductances = 0.5 μH, n = 1, k12 = 0.99990 for the preceding

values. It could appear some differences between determined inductances values. Therefore,

the coupling factor should be computed, in general, to four decimal places.

3.3 Magnetic and Electrical Model Analogy

The basic problem in the simulation model building with magnetic components is

conversion physical structure of the device into equivalent electrical circuit. The reluctance

based transformer model showed in Fig. 3.11 is then converted into an electric model based

on the duality properties (Tab. 3.1) between the magnetic and electric fields, provides a means

to accomplish this task. Reluctance modeling creates a magnetic circuit model that can then


- 22 -

be converted into an electric circuit model. Table 3.1 shows analogous quantities between

electric and magnetic circuits.

Tab. 3.1. Magnetic and Electric Circuit Analogous Quantities

Magnetic Electric NIF ≡ magnetic circuit voltage

(Magnetomotive force) V - electric circuit voltage

(Electromotive force) H - magnetic field intensity E - electric field intensity

∫ =⋅= mm HlldHF ∫ =⋅−= cc ElldEV

mm lNI

lFH ==

clVE =

B - magnetic flux density J - current density HB μ= EJ σ=

μ - permeability σ - conductivity

mH7

0 104 −⋅= πμ

φ - magnetic flux I - electric current

∫ =⋅= mBAsdBφ csJAsdJI =⋅−= ∫

'R - reluctance R - resistance

LN

AlFR

m

m2

' ==Φ

=μ

c

c

Al

IVR

σ==

'1

RP = - permeance RG 1= - conductance

By comparing the form of the equations in each column, the following analogous between

magnetic and electrical quantities can be identified:

• MMF (F) and EMF (V)

• Magnetic field (H) and Electric field (E) intensities

• Flux density (B) and current density (J)

• Flux (φ ) current (I)

• Reluctance (R’) and resistance (R)

• Permeability (μ) and conductivity (σ)

However, the analogy quantities are of course, not complete. There are some differences:

• The relationship between B and H in soft magnetic materials, is usually non-linear,

• The leakage flux in the magnetic structure with an air gap, change the total reluctance

of the magnetic circuit. In the electrical circuit there is no such effect. The electrical

isolation conductivity is on the order 1020 times lower than the conductivity of circuit


- 23 -

wires. Contrarily, the permeance of the air μ0 is only 103 times less than the

permeance of magnetic material.

• M – mutual inductance and k – coupling factor also do not have an analogue in

electrical circuit,

• The power losses in the wires carrying current are RI ⋅2 , however there is no loses in

magnetic circuits describes as '2 R⋅φ .

Reluctance is computed in the same manner as resistance, that is, from the dimensions of the

magnetic path and the magnetic conductivity (μ). For a constant cross-sectional area (Am) and

the magnetic path (lm), the magnetic reluctance is given by:

m

m

AlRμ

=' (3.22)

where: rμμμ 0= , μr - relative permeability.

The inductance of a magnetic circuit is directly related to reluctance R and the number

of winding turns N by:

PNRNL 2

'

2== (3.23a)

and

PNNRNNM 21

2112 == (3.23b)

3.4 Examples of the Rotating Transformer Construction

In this section several transformer constructions used in the industry devices based on

the pot cores are presented. The tentative selection of a magnetic core and number of winding

turns is the first step in designing the rotating transformer. Equation (3.24) is directly derived

from Faraday’s law and gives the required number of primary winding turns N for optimum

utilization (ΔBmax) of a magnetic core with an effective area Ae, when a voltage pulse U1 of

duration dmax/fs is applied across the winding.

es ABfdUN

⋅Δ⋅⋅

=max

max11 2

(3.24)

Equation (3.24) can be iteratively evaluated for various sizes of a pot-core family. Each core

size requires a specific number of turns for optimum utilization. In turn, N1 defines the

magnetizing current and, hence, the wire thickness that is required in each case. The


- 24 -

appropriate magnetic core is the smallest one that can accommodate the required copper in its

window area. This is only a tentative selection, as in a rotating transformer, there are some

additional issues (such as the fringing-field effect) that need to be considered before the final

decision is made.

Fig. 3.6. Pot core rotatable transformer with separating cores and adjacent windings.

Secondary winding

Air gap

Ferrite

Primary winding

Fig. 3.7. Pot core rotatable transformer with separating cores and overlapping coaxial windings.


- 25 -

Fig. 3.8. Rotatable transformer with overlapping coaxial windings on the common part of the core.

3.5 Transformation of Magnetic to Electric Model.

This subsection describes and shows transformation process of reluctance based

transformer model to inductances based electrical model. Figure 3.9 presents a cut section of

the rotatable pot core transformer and magnetic paths in the transformer structure. The

transformer core is divided into the single sections. Each section is represented by a magnetic

reluctance. According to Eq. (3.22) the magnetic reluctance Ri for a magnetic flux in the i

section is defined as:

ir

ii A

lRi

μμ0

= ⎥⎦⎤

⎢⎣⎡

H1 (3.25)

li – length of the magnetic path in the section,

Ai – active core section area,

μri – relative permeability of the section.


- 26 -

Fig.3.9. The pot core transformer with separating cores and adjacent windings.

.

Fig. 3.10. The cross section area of the pot core transformer with depicted the main flux path flow

(on the left part) and reluctance in the single section of the both cores.


- 27 -

The building process of the transformer model can by divided into following steps:

- divide the core (magnetic flux path), including the air gap, into the single sections

(Fig. 3.10),

- assign and compute the reluctance values for each sections,

- assign to the primary and secondary windings magnetic voltage sources,

- draw the reluctance based transformer model (as shown in Fig. 3.10),

- convert reluctance model into the permeance model, (Fig. 3.13),

- scale the permeance model to the transformer winding turns,

- replace the scaled permeances by inductors,

- in order to provide the correct voltages, for multiple transformer windings, use ideal

transformer model.

The reluctance based model of the CET transformer, which is show in Fig. 3.11 includes:

- voltage source for primary and secondary winding N1 and N2 respectively,

- cores section reluctances and air gap reluctance for the common flux path,

- reluctances for the leakage flux associated with primary and secondary windings.

Rc34

N1*I1

Rc23

Rc12

Ra23

Ra34

Ra12

Ra23

Rc34

Rc12

N2*I2

Rc23

Fig. 3.11. Reluctance based transformer model.

Similarly as resistances in the electric circuit, the series connected reluctances in magnetic

circuit can be summed. For the convenience of analysis, we assumed that the primary and

secondary transformer cores have the same shapes and dimensions. In such case the

reluctance of both cores have the same value Rpc =Rsc

342312 cccscpc RRRRR ++== (3.26)

The reluctance signified as Ra12 and Ra34 on the Fig. 3.11, are on the path of the main

magnetic flux, and currying the flux in the same direction. So, the resultant reluctance in the

air for the main magnetic flux yields:


- 28 -

3412 aaa RRR += (3.27)

Based on Eq. (3.20-21), the transformer reluctance model can be modified as shown Fig. 3.12

N1*I1

RpcRa23

Ra

Ra23

N2*I2

Rsc

Fig. 3.12. Reluctance based model for two winding CET transformer.

The reluctance model is transformed into a permeance model in the next, as shows Fig. 3.13.

The reluctances have become permeances, the magnetic current (φ ) has become a magnetic

voltage, the magnetic voltage source has become a magnetic current source, and the series

branches has become parallel branches.

1/Ra23 1/Ra23

1/Rpc 1/Ra 1/Rpc

N1I1 N2I2

Fig. 3.13. Permeance based model for two winding CET transformer. Transformer M – model.

The transformer model presented in the Fig. 3.13 is also called M – model. The saturable parts

of the magnetic path are well coupled with the windings, whereas the air gap exhibits neither

saturation nor core losses. In case of the air gap, the small central magnetizing inductance is

present. This is the two windings transformer model build on the pot cores using in non-

contact rotational axis power systems.

To simplified the analysis and obtain parameters of transformer Π - model, the parallel

connected permeances 1/Rpc, 1/Rsc, 1/Ra can be added:

pcapc RRRP 111

12 ++= (3.28)

Thus, the model from the Figure 3.13 can be transformed to Fig. 3.14.


- 29 -

The leakage permeances of primary and secondary winding respectively, are obtain by

2223

111 P

RP

a

== (3.29)

N1P11 N1P22

N1P12

I1 N2/N1 I2

Fig. 3.14. Permeance based T – model of transformer.

This model is scaled, multiplied by primary turns N1 as the reference winding, in order to

remove N from the current source, thereby leaving only the winding current I. To keep

constant magnetic fluxφ , the current source is multiplied by 1/N1 and each of the permeances

is multiplied by N1. Based on Faraday’s low, the winding voltage is given by:

φ⋅= NV (3.30)

In the next step each element of Fig. 3.14 is multiplied by N1 turns. The resulting network in

terms of the winding voltage and the permeances multiplied by N12 is now. Then by using Eq.

(3.23a), this permeances model can be replaced by the inductances based model including

ideal transformer model as it is shown in Fig. 3.15.

1υ 2υ

Fig. 3.15. Inductance model for two winding CET transformer.

The transformer turns ratio is maintained via the use of an ideal transformer. The leakage

inductance L22 can by moved to the secondary side dividing by the square turns ratio n2. As a

result a transformer Π - model of Fig.3.16 is achieved.


- 30 -

L12

L22L11

N1 N2

1υ 2υ

1i 2i

Fig. 3.16. Structure of the Π model. Two-winding transformer model including magnetizing inductance L12, primary and secondary leakage inductances L11, L22 and number of turns N1, N2.

3.6 Calculation of Section Reluctance in the Pot Core

Rotatable Transformer.

In this subsection the expression for reluctance of the CET transformer shown in the

Fig. 3.10 are delivered.

)( 21

22

12 rrhR

rc −⋅⋅

=πμ

(3.31)

)( 23

24

34 rrhR

rc −⋅⋅

=πμ

(3.32)

In order to determine the reluctance denoted as Rc23, the active cross section area for the

magnetic flux flow, has to be calculated:

)(2

2323

23

3

2 rrarr

rdraA

r

r −⋅⋅=−

⋅⋅⋅=∫

ππ

(3.33)

)( 23

2323 rra

rrRr

c +⋅⋅⋅−

=πμ

(3.34)

The sum of the core sections reluctances on the main magnetic flux path is:

233412 cccc RRRR ++= (3.35)

Based on the Eq. (3. 32- 33) and the scheme from Fig. 3.10 the primary and secondary sides

reluctances express:

ca

cacscp RR

RRRR+⋅

==23

23 (3.36)

Similarly as reluctance Rc23, the reluctance in the air section (r3-r2) Ra23 can be expressed as:

)()( 230

2323 rraha

rrRa +⋅−⋅⋅⋅−

=πμ

(3.37)


- 31 -

The sum of air gap reluctances between the cores on the main magnetic path is:

Ra = Ra12 + Ra34 (3.38)

where:

)( 21

220

12 rrl

R ga −⋅⋅

=πμ

(3.39a)

)( 23

240

34 rrl

R ga −⋅⋅

=πμ

(3.39b)

As a results of above analysis, the magnetizing inductance expression, for the selected pot

core transformer and air gap path (Rcp + Rcs + Ra), can be written as:

acscp RRRN

RNNL

++==

∑2

2112 (3.40)

and the coupling factor based on Fig. 3.12 can be approximated by following expression:

aacp

a

RRRR

k++

=23

23 (3.41)

When designing a rotatable transformer, based on the pot cores with an air gap in the

centimeter range, the design priority is different from the conventional transformer. In a

power supply with a conventional isolating transformer, the magnetic-core size and,

ultimately, the transformer size are determined by losses. The core losses limit can be set in

absolute terms (in watts) or in terms of temperature rise (12). A temperature rise of 20 ◦C−40

◦C is usually acceptable. Consequently, the appropriate core size is the minimum size that can

handle the required amount of power by satisfying the losses requirement. In the CET

transformers, however, the core size is selected based on the magnetic coupling and the

window area.

Fig. 3.17. Calculated magnetizing inductance


- 32 -

Fig. 3.18. Calculated magnetic coupling factor

As shown in Fig. 3.17-18, the magnetizing inductance (also leakage inductance) and magnetic

coupling factor, primarily depends on the air-gap area (which, for a short air gap, is equivalent

with the magnetic-core area) and its length. So, increasing the core area and keeping the air-

gap length results in coupling factor increase. Furthermore, a larger core window is usually

required to fit the additional copper that handles the excess magnetizing current. As a result,

the core size of a rotating transformer is greater than that of an equally rated ungapped

transformer. Consequently, the design is typically winding-losses limited rather than core-

losses limited [44].

3.7 Measurement of Transformer Parameters

3.7.1 No Load Test

This measurements test assumed that the copper losses are negligible. As result of test

the transformer core losses can be measured (see Fig. 3.19). Additionally also the primary and

secondary inductances can also be measure (see Fig. 3.20 a, b).

1υ

fePP ≅

Fig. 3.19. Transformer no loads test. Measurement of the core losses.


- 33 -

a)

1L

b)

2L

Fig. 3.20. No load test of transformer. Measurement primary (a) and secondary (b) self inductances.

3.7.2 Transformer Short Circuit Test

This measurements test assumes that the all transformer losses are caused by the

ohmic resistance of the windings, and the magnetizing inductance is very high. The measured

resistance of the circuit is the sum of primary winding resistance and reflected to the primary

side, secondary winding resistance

2

12121 N

NRRRRR r +=+= (3.42)

1υ

cuPP ≅

Fig. 3.21. Measurement transformer copper losses and winding resistance.

3.7.3 Leakage and Mutual Inductances Measurements In this measurement method the primary and secondary winding of the transformer are

connected in series and inverse series, according to two possible ways of coupling (7) (Fig.

3.22). Based on two measurements inductances, defined as La and Lb in the test:

1221 2MLLLa ++= (3.43a)

1221 2MLLLb −+= (3.43b)


- 34 -

a)

aL

b)

bL

Fig. 3.22. Measurement inductances of the transformer. a) La, inverse series connection, b) Lb, series connection.

and two previously measured self inductances in the no load test (Fig. 3.20), then the mutual

and leakage inductances of the transformer can be figured out.

Appointed results La and Lb can be check by the equation

221ba LLLL +

=+ (3.44)

If the condition given by Eq. (3.44) is met, then the measured values are correct. The absolute

value of difference in that condition, for measured laboratory transformer is shown in the

Fig.6. The mutual inductance can be obtained from Eq. (3.45):

412ba LLM −

= (3.45)

Next the primary and secondary coupling factor can be calculated from:

21

12

LLMk = , (3.46)

and the leakage inductance factor

21

121LL

M−=σ . (3.47)

The factors from Eq. (3.46-47) are obtained only by measurements and no windings turns

ratio n is introduced to calculations. So, the transformer windings coupling factor is

independent of the actual turns ratio. Measured coupling factor of the pot core transformer in

the laboratory model obtained by the above presented method is shown in Fig. 3.23.


- 35 -

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 2 4 6 8 10 12 14 16 18 20 22 24 26

air gap [mm]

k – coupling factor

Fig. 3.23. Magnetic coupling factor of the pot core transformer

used in the laboratory model, measured (blue line) and calculated (red line)

0 5 10 15 20 25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x 10-3

Air gap [mm]

Mag

netiz

ing

indu

ctan

ce [H

]

Fig. 3.24. Magnetizing inductance of the pot core transformer

used in the laboratory model, measured (dashed red line), calculated (blue line)


- 36 -

3.8 Transformer Copper and Core Losses Ratio

The total of transformer power losses consist of copper and core losses. Assuming a

level for copper and core losses a condition at which that level is optimal can be derived.

feCutot PPP += (3.48)

One of the ways core losses modeling is assuming that the losses are proportional to square of

magnetic induction level –B. 2~ BPfe (3.49)

To simplify further analysis, the magnetizing current is neglected. So, the primary and

secondary winding currents are proportional to each other. If the relative number of the turns

ε increases, then the copper losses also increase.

NNN Δ+

=ε (3.50)

where: ε - relative number of the turns, N – turns number, ΔN – change of turns number.

Additionally, if we assume the constant copper volume:

wCuCu lAV = (3.51)

where: ACu – active copper cross section, lw – winding wire length.

So, by increasing relative number of the turns - ε, the copper cross section area is reduced and

the winding resistance increases

[ ]Ω=Cu

wcu

Al

Rρ

(3.52)

where: ρCu – copper conductance.

As result the copper losses increases by ε2. Magnetic induction, however, decreases when the

factor ε increases. According to the Eq. (3.53), the core losses are assumed to decrease in a

quadratic factor ε2. The equation for the total losses can be write as

22

εε fe

Cutot

PPP += (3.53)

Figure 3.25 shows the dependence of Eq. (3.53). The minimal total transformer losses are

achieved when ε = 1. It means that the optimal turns numbers are well designed and have not

to be changed if:

feCu PP = (3.54)

In practice, if the design is not saturation limited, the optimal efficiency and minimum power

losses are obtain when copper losses are equal to the core losses.


- 37 -

ε

totP

Fig. 3.25. Transformer per unit total losses as a function of relative turns number.

3.9. Conclusion

In this Chapter an analysis of the transformer modeling and designing was presented.

The presented pot cores magnetic transformer model can be described by equivalent electric

circuit (Fig. 3.16). The magnetizing inductance and the leakage inductances of contactless

power supply system mainly depends on the dimensions of the primary and secondary system

parts, the applying magnetic cores on primary and secondary side and the air gap length. CET

systems are characterized by a large leakage inductances and small magnetizing inductance.

The windings effective resistance is an important parameter that determines the transformer

efficiency and depends on several factors. The length of the windings is a definitive

parameter; however, in a high-frequency converter, phenomena such as the skin and

proximity effects are equally important. Due to the high magnetizing current of a rotating

transformer, the effective resistance becomes critical. Incorporating more winding turns in the

transformer reduces the magnetizing current but increases the windings resistance and vice

versa; if the number of turns are reduced, the magnetizing current increases and the winding

resistance decreases. So, there are an optimum number of turns that results in minimum losses

in the transformer windings.

Summing up it can be said summarized those values of mutual and self inductances of the

CET transformer depends on following parameters:

air gap length between the cores / windings,


- 38 -

used core sizes and material permeability,

turns number of primary and secondary windings,

the active cross section area for the main magnetic flux flow,

length of the magnetic flux path.

The optimal transformer efficiency and minimum power losses are obtained when copper

losses are equal to the core losses (see Fig. 3.25).

Fig. 3.26. The magnetic flux flow for a two values of adjusted air gap length in pot core rotatable transformer with separating cores and adjacent windings.

Fig. 3.27. The magnetic flux flow for a two values of adjusted air gap length in pot core rotatable transformer with separating cores and overlapping coaxial windings.


- 39 -

4. Power Converters Used in Contactless Energy Transmission (CET) Systems 4.1 Introduction

Recently, various kinds of soft-switching techniques for switching power converters

have been proposed in order to satisfy the ever-increasing requirements for smaller size,

lighter weight, and higher efficiency [35]. Soft-switching techniques reduce the switching

losses, enabling high-frequency operation and, consequently, reduce the overall system size

(inverter sink, transformer cores). Generally, the soft-switching techniques can be classified

into two groups: zero voltage switching (ZVS) and zero current switching (ZCS). In the

metal-oxide-semiconductor-field-effect-transistors (MOSFET’s) large turn-on losses are

caused by the large output capacitance. For these devices ZVS technique is desirable. In the

IGBT’s transistors the main part of switching loses occurs by turn-off due to the current tail

characteristics and, consequently, the ZCS approaches are desirable for these devices.

Comparing the power transistors IGBT to MOSFET they have higher voltage rating, higher

power density a and lower production cost. Nowadays, the MOSSFET power transistors are

replacing by IGBT for high power application.

In conventional applications transformer is used for galvanic isolation between source

and load, and its operation is based on high magnetic coupling coefficient between primary

and secondary windings. Assumed that two half cores are used, the CET transformers operate

under much lower magnetic coupling factor.

Fig. 4.1. Rotatable transformer with adjustable air gap (lg).

As a result the main inductance L12 is very small whereas leakage inductances (L11, L22) are

large, comparing them with conventional transformers. Consequently, the increasing

magnetization current causes higher conduction losses. Also, winding losses increase because


- 40 -

of large leakage inductances. Another disadvantage of transformers with relatively large gap

is EMC problem (strong radiation). To minimize the above disadvantages of CET

transformers several power converter topologies have been proposed which can be classified

in following categories: the flyback, resonant, quasi-resonant and self-resonant [9]. The

common for all these topologies is that they all utilize the energy stored in the transformer. In

this work resonant soft switching technique has been used. From Fig.4.2 is clearly to see that

inductive coupled CET system is based on resonant converters and large air gap transformer.

Energy is transmitted without galvanic contact via inductive coupling between windings

placed on separated rotatable parts of a core of a single phase transformer.

Switch mode DC/DC converters with galvanic insulation are widespread used for

power supplies in different applications. The various kinds of soft-switching techniques for

switching power DC/DC converters have been developed in order to satisfy the ever-

increasing requirements for reduction size, volume and weight compared to systems with a

low frequency transformer. Additionally they offer much more flexibility by applying of such

a critical system as electric vehicles battery chargers, in aerospace, biomedical and robotics

applications etc. [11, 12, 15, 17, 23-26, 28-29, 33-35, 40, 43, 48, 56].

Fig.4.2. Block diagram of contactless energy transmission system

Hence, the next Section is devoted to resonant conversion technique.

4.2 Basic Principles of Resonant Converters

4.2.1 Resonant Converters Topologies

Resonant power converters contain resonant L–C networks, called also resonant

circuit – RC or resonant tank network, whose voltage and current waveforms vary

sinusoidally during one or more subintervals of each switching period (1). These converters


- 41 -

contain low total harmonic distortion because switching frequency is equal to first harmonic

frequency.

Resonantcircuit

Cs

Cp

L

L

)(2 tu

)(2 ti)(1 ti

)(1 tu

DCsource

Ez

SNRC

S1

S2Re

Switchnetwork

Fig.4.3 a) The basic structure of resonant inverter SN - switch network and RC - resonant circuit.

zE−

zEπ4

)(1 tu

Fundamentalcomponent

zE

1ϕ

tsω)(1 ti

)()1(1 tu

Fig.4.3 b) Basic waveforms in resonant inverter of Fig.4.4c,

u1(t) - square wave output voltage of switch network and it’s fundamental component i1(t) - fundamental primary current.

A switching network (Fig.4.3a) produces a square wave voltage u1(t) (Fig.4.3b), which

spectrum contains fundamental plus odd harmonics and feed the resonant circuit RC.

Depending on the used converter topology (Fig. 4.4) square wave voltage u1(t) can be

expressed in Furrier series as:

-for Fig. 4.4 a) and b) topologies:

( )tnn

Etu sn

z ωπ

sin12

4)(...5,3,1

1 ∑=

⋅= (4.1)

-for Fig. 4.4 c) topology:

( )tnn

Etu sn

z ωπ

sin14)(...5,3,1

1 ∑=

⋅= (4.2)


- 42 -

a) )(1 tu)(1 ti

)(2 tu

)(2 ti)(1 ti

tsω

1U

ST

zE

b)

DCsource

Ez

)(1 tu)(1 ti

Resonant circuit

Cs

Cp

L

L

)(2 tu

)(2 ti)(1 tiRC

Re

SN

Switch network

T1

T2

2zE

tsω

1U

ST

2zE

c) )(1 tu)(1 ti

)(2 tu

)(2 ti)(1 ti

tsω

1U

ST

zE

Fig.4.4 Basic topologies of series resonant converter and resonant circuit voltage u1(t) waveforms.

a) half-bridge uni-polar converter, b) half-bridge bi-polar converter, c) full-bridge converter.

The full bridge (Fig. 4.4c) inverter composed by four switches and the resonant circuit are

commonly used in high power application. The half bridge inverter (Fig. 4.4a) has only two

switches and two others can be replaced by capacitors (Fig. 4.4b).

The RC resonant frequency f0 is tuned to the fundamental component of u1(t) and is equal to

the inverter switching frequency fs. Therefore, it can be defined as follows:

πω2

ssf = (4.3)


- 43 -

rr CLf

⋅==

ππω

21

20

0 (4.4)

where Lr and Cr - resonance inductance and capacitance of the RC circuit.

The fundamental component of the primary voltage u1(t) is expressed as:

( ) ( )tUtEtu ssz ωω

πsinsin4)( )1(1)1(1 =

⋅= (4.5)

If the switch network SN operates with the resonant frequency f0, the primary current i1(t) is

well approximated by a sinusoidal waveform of amplitude I1 and phase ϕ1. Value of the

primary side RC current i1(t) is equal to the source input dc current Is.

By changing the switching frequency fs in respect to the resonant frequency f0, the magnitude

of voltage u0(t), currents i1(t) and i0(t) can be controlled continuously. Such a phase shift

control of resonant circuit RC can also be used for control of voltage and current magnitude.

The DC/DC resonant converter shown in Fig.4.4 consists of three main components: switch

network (SN), resonant circuit (RC) and rectifier network (RN) with low-pass filter (FN).

RN

Rectifiernetwork

Low-passfilter network

FN

Ro)(0 tu

+

-

)(0 ti

Rectifier withfiltering part

Resonant circuitand

magnetic system

Cs

Cp

L

L

)(2 tu

)(2 ti)(1 ti

)(1 tu

DCsource

Ez

SN

RC

S1

S2

Switchnetwork

)(tiR

)(tiR

Co

Co

Lf

Fig.4.5 The DC/DC full bridge resonant converter circuit model build on fundamental components.

The sinusoidal tank output current iR(t) is rectified by a diode bridge rectifier, and next

is filtered by a low-pass filter with large capacitor C0. Filtered dc voltage and current fed dc

load R0. Harmonics of the switching frequency are neglected, and the thank waveforms are

assumed to be purely sinusoidal. This allows simple equivalent circuits to be derived for the

bridge inverter, thank, rectifier, and output filter parts of the converter, whose operation can

be understood and solved using standard linear AC analysis. This intuitive approach is quite

accurate for operation in continuous conduction mode with high Q-factor response, but


- 44 -

becomes less accurate when the resonant circuit RC is operated with a low Q-factor or for

operation of DC/DC resonant converters in or near discontinuous conduction mode (1).

The main advantage of resonant technique is reduction of switching losses, via

mechanism known as Zero Current Switching (ZCS), and Zero Voltage Switching (ZVS) (1,

2). The switch-on and/or switch-off converter semiconductor components can occur at zero

crossing of the resonant quasi-sinusoidal waveforms. This eliminates some of the switching

loss mechanism. Hence, switching losses are reduced, and resonant converters can operate at

switching frequencies that are considerably higher than in comparable PWM hard switching

converters. ZVS can also eliminate or reduce some of the electromagnetic emission sources

called also as Electromagnetic Interference (EMI) (1). Another advantage is that both ZVS

and ZCS converters can utilize transformer leakage inductance and diode junction capacitors

as well as the output parasitic capacitor of the power switch (2).

However, resonant converters exhibit several disadvantages. Although, the

components of resonant circuit RC can be chosen such that good performance with high

efficiency is obtained at a single operating point, typically it is difficult to optimize the

resonant components in such way that good performance is obtained over a wide range of

load currents and input voltages variations. Significant currents may circulate through the tank

components, even when the load is removed, leading to poor efficiency at light loads. Also, a

quasi-sinusoidal waveforms of resonant converters exhibit greater peak values than in the

rectangular waveforms of PWM converters, under assumption that the PWM current spikes

due to diode stored charge are ignored. For these reasons, resonant converters exhibit

increased conduction losses, which can offset their reduced switching losses. Moreover,

the ZVS and ZCS techniques require variable frequency control to regulate the output power.

This is undesirable since it complicates the control circuit and generates EMI harmonics,

especially under large load variations.

4.2.2 The Compensation Leakage Inductances in CET System

The CET system shown in Fig. 4.2 assures electrical energy transmission from a

power supply via an air gap towards a load. The primary transformer side is fed by a high

frequency inverter while a secondary feeds the DC load via a rectifier. By the use of the air

gap leads to a safe electrical energy transfer without any electrical or physical contact

between the power supply and the load. Because of the separation transformer cores, the


- 45 -

leakage inductances increase. Additionally, a large air gap causes a low magnetizing

inductance. In consequence there are significant winding losses due to strong magnetizing

current. The most suitable converter for this type of contactless energy transmission are

resonant converters. In the resonant converter the, RC drives a resistive load as in Fig. 4.3a.

The reactive component of the load impedance, if any, can be effectively incorporated into the

resonant circuit RC. Figure 4.5 shows resonant dc-dc converter. The resonant circuit is

connected to an uncontrolled rectifier network RN, filter network FN and load R0. The

resonant capacitance Cr (in the Fig. 4.5 shows as Cs, Cp) inserted to the system circuit causes,

those leakage inductances of the CET transformer winding becomes resonant inductances Lr.

To form resonant circuits, two methods of leakage inductances compensation can be

used: S-series or P-parallel giving four basic topologies [56]: SS, SP, PS, and PP (first letter

denotes primary and second a secondary compensation respectively).

Series – Series (SS)

Series – Parallel (SP)

Parallel – Series (PS)

Parallel – Parallel (PP)

L12

L22L11

N2

Cr1 Cr2N1

k

Fig. 4.6. SS – compensation topologies.

L12

L22L11

N2

Cr1

Cr2

N1

k

Fig. 4.7. SP – compensation topologies.

L12

L22L11

N2

N1

kCr1

Cr2

Fig. 4.8. PS – compensation topologies.


- 46 -

L12

L22L11

N2

Cr2

N1

kCr1

Fig. 4.9. PP– compensation topologies.

The parallel compensated primary transformer winding is required to generate large primary

current. This causes that PS and PP topologies require an additional series inductor to regulate

the inverter current flowing into the parallel resonant circuit. This additional inductor increase

EMC distortion and total cost of CET system. Therefore, only SS and SP topology has been

considered in this work.

)(1 ti )(2 ti

αZ

)(1 tu

)(sH

)(tiR

Fig. 4.10. Resonant converter circuit model build on fundamental components.

For transfer electric energy through the air gap transformer in CET system with high

efficiency, a high voltage gain with small variation and small circulating current through

magnetizing inductance is preferred. The important variable for the circuit analysis are:

total impedance of considered circuit Zα ( Fig. 4.10),

the normalized angular frequency ω, describing the ratio between operating frequency

ωs and the circuit resonance frequency ω0; ω = ωs / ω0,

the circuit quality factor Q.

In order to study analytically the contactless energy transmission system presented in this

work, the first harmonic method is used.

4.2.3 Analysis of Series-Series Resonant Converter

The series resonant DC/DC converter with series – series SS compensated resonant

circuit is presented in Fig.4.11. The sinusoidal tank output current feeds the rectifier network

RN. Next the current iR(t) is filtered by a large capacitor C0.


- 47 -

Filtered dc current i0(t) and voltage u0(t), which contains negligible numbers

harmonics of switching frequency, fed connected dc load (R0). Therefore, by approximation

we can write u0(t) ≈ U0 and i0(t) = I0. Figure 4.12 shown the behavior of voltage u2(t) and

current i2(t) of the secondary side transformer.

)(0 tu

)(0 ti

)(2 tu

)(1 ti

)(1 tu

)(tiR

)(tiR

)(2 ti

αZ βZ γZ

Fig. 4.11. Equivalent circuit of SS resonant converter shown in Fig.4.4

0U

0U−

)(2 tu

)(2 ti2ϕ

tsω

04 Uπ

Fig. 4.12. Waveforms in circuit of Fig.4.11.

Square wave voltage u2(t) and current i2(t) of secondary transformer side.

The secondary side voltage u2(t) change the sign when the secondary side current i2(t) passes

through zero. Hence, the rectifier input voltage is a square wave, equal to +U0 for positive

and –U0 for negative current i2(t) appropriately.

If the current i2(t) is sinusoidal with peak amplitude and phase shift ϕ2, then the fundamental

component is expressed as:

( )2)1(2)1(2 sin)( ϕω −= tIti s (4.6)

Expending the secondary side voltage u2(t) in Fourier series yields:


- 48 -

( )2...5,3,1

02 sin14)( ϕω

π−

⋅= ∑

=tn

nUtu s

n (4.7)

The fundamental secondary voltage component u2(1)(t) could by written as:

( ) ( )2)1(220

)1(2 sinsin4)( ϕωϕωπ

−=−⋅

= tnUtnUtu ss (4.8)

The sinusoidal tank output current i2(t) is rectified by a diode bridge rectifier, and next is

filtered by a large capacitor C0. Hence, the dc component of |i2(t)| is equal to load current I0.

( ) )1(222

0 )1(22sin2 IdttI

TI s

T

so

s

πϕω =−= ∫ (4.9)

Therefore, the load current I0 and the secondary side current i2(t) in steady state conditions are

directly related. Substitution of (4.8) into (4.7) gives expression for the effective load

resistance yields as:

02

0

)1(2

)1(2 8)()(

IU

titu

Res π== (4.10)

oI)(tiR

)(2 tuRI

π2

Fig. 4.13. An effective load resistance of SS compensated converter.

A simplified equivalent circuit for the SS compensated resonant converter given in Fig.4.14 is

similar to the model for SS compensation topology presented in Fig.4.11. However, in the

equivalent model the rectifier network with low pass filter is replaced by an effective

resistance Res equal to 81% of the actual load resistance R0 = U0 / I0.


- 49 -

Impedances of SS – Compensated Resonant Converter

αZ βZ γZ

)(1 tu

)(1 ti

)(2 ti

)(tiR

)(

2t

u

Fig. 4.14. Simplified equivalent circuit for SS – compensated resonant converter.

The circuit impedances shows on Fig.4.14 can be written as fallowing equations if

transformer inductances can be obtain from Eq. (3.18):

-impedance of secondary side in case of chosen series compensation

( )⎩⎨⎧

+⋅

+⋅−⋅= esr

RCj

LkjZ2

211

ωω (4.11)

-reflected secondary impedance seen from primary side can be fund by dividing reflected

voltage by primary current:

2

2

1

2'2 Z

ZI

IZZ MM =

⋅= (4.12)

with LkjZM ⋅⋅= ω (4.13)

tsω

tsω

tsω

tsω

Fig. 4.15. Simulation results: primary voltage source (u1), secondary voltage (u2), primary current (i1)

and secondary current (i2) for zero primary current switching mode and two load resistance values R1, R2.

Thus the reflected secondary side impedance for SS compensation can be expressed as:

⎪⎪⎩

⎪⎪⎨

⎧

⋅⋅+−−⋅⋅⋅−−⋅⋅⋅⋅⋅⋅⋅

−

⋅⋅+−−⋅⋅⋅⋅⋅⋅⋅

=

222

222

22

2222

3

222

222

2

2222

4

'2

)1)1(()1)1((

)1)1((

esrr

rr

esrr

esr

RCkLCkLCLkCj

RCkLCRLkC

Z

ωωωω

ωωω

(4.14)


- 50 -

In the Fig. 4.15 are shown the simulation result waveforms of primary source voltage (u1),

primary current (i1) and secondary current (i2) for zero primary current switching mode. The

oscilograms were simulated for the same resonant circuit parameters and different loads

resistances R02 > R01. The loading effect of the secondary on primary circuit is corresponding

to reflected impedance '2Z .

a)

b)

Fig. 4.16.1. Real and imaginary component of reflected secondary impedance '

2Z for SS -compensation mode, as a function of operating frequency and load resistance. k = 0.247

a) Reflected resistance )Re( '2Z , b) Reflected reactance )Im( '

2Z

The power transfer from primary side trough the air gap to secondary side can be expressed

as a function of reflected resistance multiplied by the square of primary current: 2

1'2 )Re( IZP ⋅= (4.15)

Therefore, even with low coupling coefficient – k of CET transformer, the high electric power

can be transferred, if the circuit is working with the secondary current resonant frequency and

primary winding resonant frequency is equal to secondary. In order to ensure that condition

primary resonant capacitor should be chosen depending on the compensation topology.


- 51 -

a)

b)

Fig. 4.16.2. Real and imaginary component of reflected secondary impedance '

2Z for SS - compensation mode, as a function of operating frequency and load resistance. k = 0.526


2Z

The required resonant capacitors values, for desired resonant frequency, can be expressed as

follows:

- for series secondary compensation - SS

22112211

221 LLrrr CC

LLC === (4.16)

- for parallel secondary compensation - SP

( ) 222211222112211

2222

1 11

rLLr

r CkLLkLL

CLC−

==⋅⋅−⋅

⋅= = (4.17)

The above equations are illustrated in Fig. 4.17. The normalized value of the primary capacitance should to be unity:

11

222

11

LLC

CCr

rnr ⋅= (4.18)

Therefore, by combining the primary and secondary tank networks, the total impedance seen

by the power supply for a series compensated primary system can be written as:


- 52 -

( )⎩⎨⎧

+⋅

+⋅−⋅= '2

1

11 ZCj

LkjZrω

ωα (4.19)

123456789

10

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Magnetic coupling coefficient - k

SP

SS

21 rr CresultC ⋅=

21 2 rr CC ≈example: for k = 0.7

Fig. 4.17. Primary capacitance for SS and SP compensation versus coupling coefficient.

The phase angle of the total impedance seen by the power supply for the SS - compensation

topology versus normalized frequency fs/fo, for a different values of load resistance R0 are

shown in Fig. 4.18. a)

b)

Fig. 4.18. Phase angle of the total impedance seen by the power supply for a SS - compensation versus

normalized frequency fs/fo, for a different values of load resistance R0. Computing circuit parameters:

a) k = 0.247, Cr1 = Cr2=63 nF, L=296 uH b) k = 0.52, Cr1 = Cr2=63 nF, L=372 uH


- 53 -

Summing up, operation at the secondary resonant frequency depends on selected

compensation topology, the coupling coefficient between primary and secondary, and the

quality factor of secondary circuit.

4.2.4 Analysis of Series-Parallel Resonant Converter

The series - parallel (SP) compensated resonant circuit is shown in Figure 4.19. The

main deferens between SP and SS compensation topology is that resonant capacitor Cr2 is

parallel connected to the output rectifier. Moreover, the output low-pass filter consists of

additional inductor Lf. In SP resonant converter the output rectifier is supplied by nearly

sinusoidal voltage resonant capacitor uCr2(t). The diode rectifier is switching (Fig.4.20) when

the voltage uCr2(t) passes through zero.

)(0 tu

)(0 ti

)(2 tuCr

)(1 ti

)(1 tu

)(tiR

)(tiR

)(2 ti

αZ βZ γZ

Fig. 4.19. An equivalent circuit of SP resonant converter

In steady-state conditions the filter inductor current |iR(t)| is equal to the dc load current I0.

The input current of the rectifier iR(t) is therefore a square wave of amplitude I0 (Fig.4.19) and

is in phase with the resonant capacitor voltage uCr2(t). The iR(t) current can be expressed in

Furrier series as:

( )Rsn

R tnn

Iti ϕωπ

−⋅

= ∑=

sin14)(...5,3,1

0 (4.20)

where ϕR is the phase shift of voltage uCr2(t) (Fig.4.19).

The fundamental component of rectifier input current is:

( )RsR tIti ϕωπ

−⋅

= sin4)( 0 (4.21)

The effective load resistance representing the output rectifier yields as:


- 54 -

)1(

)1(2

)1(

)1(2

4)()(

R

Cr

R

Crep I

Utitu

Rπ

== (4.22)

The AC components of the rectifier resonant capacitor voltage |uCr2(t)| are removed by the

output low pass filter. In steady state, the output voltage Uo is equal to the DC component of

voltage |uCr2(t)|:

( ) )1(2

2

0)1(2

2sin2CrRs

T

Crs

o UdttUT

Us

πϕω =−= ∫ (4.23)

Fundamentalcomponent

tsω

0I

0I−

04 Iπ

)(2 tuCr

)(tiRRϕ

Fig. 4.20. Waveforms in SP resonant converter of Fig. 4.19. Fundamental components voltage uCr2(t) and current iR(t).

oI)(tiR

)(2 tuCr

22

CrUπ

Fig. 4.21. Equivalent circuit of the output rectifier of SP compensated converter.

Therefore, the load voltage Uo and the resonant capacitor voltage uCr2(t) in steady state

conditions are directly related.

Substitution of (4.23) into (4.22) yields expression for the effective load resistance:

ooep RRR ⋅== 23,18

2π (4.24)

An equivalent circuit for the SP compensated resonant converter shown in Fig.4.22 is similar

to the model for SS compensation topology presented in Fig.4.14. However, the roles of the


- 55 -

rectifier input voltage and current are interchanged and therefore the effective resistances Res

and Rep values are different.

Impedances of SP – Compensated Resonant Converter

ReL12

L22L11

N2

Cr1N1

k

jX1 jX2

jXm

αZ βZ γZ

Cr2

δZ

)(tiR)(2 ti)(1 ti

)(1 tu

Fig. 4.22. Simplified equivalent circuit of SP resonant converter from Fig.4.19.

From Fig. 4.20 the circuit impedances can be written as follows:

-impedance of secondary side in case of chosen series compensation

( )⎪⎩

⎪⎨

⎧

+⋅+⋅−⋅=

epr RCj

LkjZ 111

22 ω

ω (4.25)

tsω

tsω

tsω

tsω

Fig. 4.23. Simulation results: primary voltage source (u1), secondary voltage (u2), primary current (i1)

and secondary current (i2) for zero primary current switching mode and two load resistance values.

-reflected secondary impedance seen from primary side can be fund by dividing reflected

voltage by primary current:

2

2

1

2'2 Z

ZI

IZZ MM =

⋅= (4.26)

with LkjLjZM ⋅⋅=⋅= ωω 12 (4.27)

Thus, the reflected secondary side impedance for parallel compensation can be stated as:


- 56 -

⎪⎪

⎩

⎪⎪

⎨

⎧

−⋅⋅+−−⋅⋅⋅⋅−⋅+−−⋅⋅⋅⋅⋅⋅⋅⋅

−

−⋅⋅+−−⋅⋅⋅⋅⋅⋅⋅

=

22222

222

222

223

22222

22

222

'2

)1()1)1(())1()1)1(((

)1()1)1((

kLkLCRkLkLCRCLk

j

kLkLCRRLk

Z

rep

repr

rep

ep

ωωωω

ωωω

(4.28)

a) b)

Fig. 4.24. Real and imaginary component of reflected secondary impedance '

2Z for SP compensation mode, as a function of operating frequency and load resistance. k = 0.247


2Z

The total impedance Zα seen from the power supply is close to zero when the circuit is in

resonance and voltage gain is almost constant and independent of the load variety.

( )⎩⎨⎧

+⋅

+⋅−⋅= '2

1

11 ZCj

LkjZrω

ωα (4.29)


- 57 -

a) b)

Fig. 4.25. Real and imaginary component of reflected secondary impedance '

2Z for SP compensation mode, as a function of operating frequency and load resistance. k = 0.52


2Z

4.2.5. Summary and Conclusions In this subsection basic topologies of CET transformer leakage inductances compensation

methods have been analyzed. The contactless power supply system essentially is comprised of

two magnetically coupled electrical systems, as shown in Fig. 4.4, driven by a high-frequency

switching power supply. A significant problem of CET system is the large secondary leakage

inductance, which causes a voltage drop and limitation of the transfer power range.

Consequently, the secondary winding (rotatable - movable pickup) compensation is required

to enhance the power transfer capability. The leakage inductance of primary winding is

normally compensated in order to minimize the VA rating of the supply. The compensation

can be realized either by series resonance capacitor or by parallel resonance capacitor

connected at the secondary transformer winding (see basic compensation circuits at Fig. 4.6-

9). Depending on the selected compensation topology the impedance of secondary network is

changing (Table 4.1). The performed analysis shows the main differences between the two

commonly used resonant topologies.


- 58 -

a)

b)

Fig. 4.26. Phase angle of the total impedance seen by the power supply for SP compensation versus normalized frequency fs/fo for a different values of load resistance R0.

Computing circuit parameters: a) k = 0.247, Cr1 = Cr2=63 nF, L=296 uH

b) k = 0.52, L=372 uH

Resonant frequency is increasing at parallel compensation and decreasing at series

compensation when the load resistance increases. Changes of the load resistance have an

influence on resonant frequency primary current. Moreover, the values of reflected

impedances are strongly depended on coupling coefficient for both compensation topologies,

and decreasing when the coupling coefficient is lower. By comparing the phase shift

characteristic of the total impedance for the both compensation topologies, we can see that

the SS compensation circuit is more sensitive to the load resistance changes at the same

circumstances. While the SS compensation is very sensitive to the coupling factor changes

and need a different resonance capacitances by the coupling changing. In both presented

topologies with increasing load resistance the phase shift between primary and secondary

currents increase as well as output voltage. It can be summarized as fallows: series – parallel

(SP) compensated converter exhibits some advantages to the series – series (SS) resonant

converter only for application when coupling factor is constant.


- 59 -

Table 4.1 Secondary Circuit Properties as a Function of Compensation Topology.

Compensation topology Series Parallel

Secondary impedance

( ) esr

RCj

Lkj +⋅

+⋅−⋅2

11ω

ω ( )ep

r RCjLkj 1

112 +⋅

+⋅−⋅ω

ω

Reflected resistance 22

222

22

2222

4

)1)1(( esrr

esr

RCkLCRLkC

⋅⋅+−−⋅⋅⋅⋅⋅⋅⋅ωω

ω 22222

22

222

)1()1)1(( kLkLCRRLk

rep

ep

−⋅⋅+−−⋅⋅⋅⋅

⋅⋅⋅

ωωω

Reflected resistance at the secondary

resonant frequency esRLk 222 ⋅⋅ω

222

22

)1( kLRLk ep

−⋅⋅

⋅⋅

ω

Reflected reactance 22

222

22

2222

23

)1)1(()1)1((

esrr

rr

RCkLCkLCLkCj⋅⋅+−−⋅⋅⋅−−⋅⋅⋅⋅⋅⋅⋅

−ωω

ωω2222

222

222

2223

)1()1)1(())1()1)1(((

kLkLCRkLkLCRCLk

jrep

repr

−⋅⋅+−−⋅⋅⋅⋅

−⋅+−−⋅⋅⋅⋅⋅⋅⋅⋅−

ωωωω

Reflected reactance at the secondary

resonant frequency 0

)1(

22

kLLk

−⋅⋅⋅

−ω

Secondary circuit quality factor

esRLk ⋅−⋅ )1(ω

LkRep

⋅−⋅ )1(ω

Load resistance oes RR 28π

= oep RR8

2π=

The advantages of a series compensated secondary is that there is no reflected reactance

when the secondary operates at the resonant frequency, while the parallel compensated

secondary reflects a capacitive reactance at the secondary resonant frequency. The favorably

characteristic of parallel compensated secondary is mainly because it is independent of the

connected load. Parallel secondary compensation, which gives a current source output, is

well suited for the applications such as a battery charging (electric vehicles). However, series

secondary compensation topology, is better for systems with intermediate DC voltage bus,

such as PWM-controlled converter feds variable speed AC drives (robotics). In this case

compensation topology should be selected suitable according to outputs requirements.

Theoretically, SS is the best topology, as the primary capacitance is then independent of either

the magnetic coupling or the converter load. The other three topologies (Table 4.2) are all

dependent on the magnetic coupling. In this work, however, we assume that magnetic

coupling factor between CET transformer windings will be changed in case of operation with

different air gap length. Therefore, in the next Subsection 4.4 only variants of SS

compensation circuits will be considered.


- 60 -

Table 4.2. Variations of Circuits Leakage Inductance Compensation

/ Simplified circuits Abbreviation Comments Sensitivity for coupling and load changes

1 L12

L22L11

N2

Cr1 Cr2N1

k

Series - Series SS

systems with intermediate

DC voltage bus

sensitive for load changes

2 L12

L22L11

N2

Cr1

Cr2

N1

k

Series - Parallel SP

(current source output) battery

charging

sensitive for coupling changes

3 L12

L22L11

N2

N1

kCr1

Cr2

Parallel - Series PS

systems with intermediate

DC voltage bus


4 L12

L22L11

N2

Cr2

N1

kCr1 Parallel - Parallel PP

(current source output) battery

charging



- 61 -

4.3. CET System with Series Topology Compensation

In previous chapter the analytical analysis of series-series (SS) and series-parallel (SP)

compensated resonant converter used in CET systems have been presented. It was proven

that for contactless power supply system with variable coupling factor series compensation

topology is preferable.

The series compensation topology of the CET system can by divided on three types

(Table 4.3.):

the series resonant circuit made by connection resonant capacitor to the primary

transformer winding,

the series resonant circuit made by connection resonant capacitor to the secondary

transformer winding,

the series resonant circuit made by connection resonant capacitors to the primary

and secondary transformer windings.

The SS compensation circuit can be divided in three variants (Tab. 4.1.)

Table 4.3. Variations of Series Circuits Leakage Inductance Compensation

/ Simplified circuits Abbreviation

1

Series - Series S1

2

Series - Series S2

3

Series - Series SS


- 62 -

4.3.1. CET System with Secondary Compensation

The concept of leakage inductances compensation of the CET transformer by adding

series resonant capacitor on secondary side is discussed in following literature, e.g. [2, 3, 4].

)(0 tu

)(0 ti

)(2 tu

)(1 ti

)(1 tu

)(tiR

)(tiR

)(2 ti

Fig. 4.27. CET system with secondary series compensation S2

The resonant circuit RC with S2 compensation topology of contactless energy transmission

system shown in Fig.4.27 is feed by U1 square voltage source. The primary winding of

transformer is directly connected to the output of converter. In this case the transformer is not

protected by the DC-component of primary current flow. The secondary winding is connected

with series resonant capacitor and rectifying circuit RN with capacitance filter C0 and dc load

R0. The operating frequency sf of the resonance converter should be equal or close to

resonance frequency of secondary circuit. However, to increase system efficiency, operating

frequency of supply converter is adjusted to ensure zero current switching. The phase shift

(phase error) between square wave voltage u1(t) and primary current i1(t), is a parameter

which can be detected by a control circuit on primary side.

Assuming the same numbers of primary and secondary winding N1 = N2, the leakage

inductances can be described as:

LkLL )1(2211 −== (4.30)

Resonance frequency of circuit shown in Fig.4.27 including rectifying circuit with

capacitance filter can be express as:

( ) ( ) 020 12

112

121

LCkLCkf

r −+

−=

π (4.31)

However 20 rCC >> , and the resonance frequency can be written as:


- 63 -

( ) 20 12

121

rLCkf

−≈

π (4.32)

The frequency of supply voltage u1(t) can be lower, equal or higher then resonant frequency

fo. The values of resonance converter output voltage and rectifier input voltage can be

calculated from following equations:

( )

( )⎪⎩

⎪⎨

⎧

−−−=−

+−=

22

22

11

121

21

IC

jIkLjILkjU

IkLjILkjU

rm

m

ωωω

ωω (4.33)

The magnetizing inductance current is equal to:

21 IIIm −= (4.34)

From the equations (4.33) and (4.34) it follows that, value rectifier input voltage can be

written as:

( ) ( ) mrr

IC

LkjIC

LkjUU ⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−=

ωω

ωω 1211212 112 (4.35)

The waveform of magnetizing current mi has triangular shape and is phase shifted to the

primary voltage u1(t) by an angle πϕ21

−= . The magnetizing current frequency is equal to

operating frequency of resonant converter. The value of the voltage drop on the reactance XLC

as a result of first harmonic magnetizing current flow 2)1()1(

πj

mm eII−

= are defined as:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−−==

−−

rm

j

LC

j

mI CLkIeXeIU

m ωω

ππ 121)1(22

)1()1( (4.36)

Depending on the operating frequency changes above or below resonance frequency, the

series reactance XS2 of tank network character is volatile.

( )r

S CLkX

ωω 12122 −−=

⎪⎩

⎪⎨

⎧

>=<

0

0

0

ffffff

S

S

S

- capacitive character

0 (4.37)

- inductive character


- 64 -

a) 0ffs =

tsω

tsω

b) 0ffs >

tsω

tsω

c) 0ffs <

tsω

tsω

Fig. 4.28. Simulated basic waveforms of currents and voltages in contactless energy transmission system S2 - compensation.

The values of the voltage drop on the reactance XS2 as a result of flow first harmonic primary

current are defined as:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

=

=

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −−

22)1(1

22)1(1

22)1(1

22)1(1

)1(1 0π

ϕπϕ

πϕπ

ϕ

j

S

j

Sj

j

S

j

Sj

I

eXIeXeI

eXIeXeI

U

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

=

<

0

0

0

ff

ff

ff

S

S

S

(4.38)

From (4.34) and (4.38) it follows that, value of secondary side voltage can be approximated

by the following expression:

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−+−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−++

=

rm

r

rm

rm

r

CLktI

CLktIu

CLktIu

CLktI

CLktIu

u

111)1(

111)1(11

111)1(1

11)1(1

11)1(11

2

121sin1212cos

121sin

121sin1212cos

ωωω

ωωϕω

ωωω

ωωω

ωωϕω

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

=

<

0

0

0

ff

ff

ff

S

S

S

(4.39)


- 65 -

From presented simulation results Fig.4.28 and equation (4.39) follows that independent of

the operating converter frequency, the value of the output voltage is reduced by the voltage

drop caused by magnetizing current. If we assume the same numbers of primary and

secondary winding coils, the value of rectifying output voltage U0 is lower than feeding

voltage Ez for any values of load resistance.

4.3.2 CET System with Primary Compensation

The concepts of leakage inductances compensation of the CET transformer by adding series

resonant capacitor on primary side are possible to find in literature e.g. [25, 57, 58].

)(0 tu

)(0 ti

)(2 tu

)(1 ti

)(1 tu

)(tiR

)(tiR

)(2 ti

Fig. 4.29. CET system with primary series compensation S1

The resonant circuit RC with S1 compensation topology of contactless energy transmission

system shown in Fig.4.29 is feed by U1 square voltage source. The primary winding of

transformer is connected in series with resonant capacitor to the output of switch network SN.

Series connected capacitor Cr1, protects the transformer winding against the DC-component of

primary current. To the secondary winding is connected to rectifying network (RN) with

capacitance filter C0 and dc load R0. The operating frequency sf of the SN should be adjusted

to ensure zero current switching. The phase shift (phase error) between square wave voltage

u1(t) and primary current i1(t), is a parameter which can be detected by a control circuit on

primary side. Assuming the same numbers of primary and secondary winding N1 = N2, the

leakage inductances can be written as:

LkLL )1(2211 −== (4.40)

Resonance frequency of circuit shown in Fig.4.29 can be express as:


- 66 -

( ) 10 12

121

rLCkf

−≈

π (4.41)

The values of resonance converter output voltage and rectifier input voltage can be calculated

from following equations:

( )

( )⎪⎩

⎪⎨

⎧

−−=−

−+−=

m

rm

IkLjILkjU

IC

jIkLjILkjU

ωωω

ωω

22

11

11

21

121 (4.42)

From the equations (4.34) and (4.42) it follows that, value of rectifier input voltage can be

written as:

( ) ( ) mr

ILkjIC

LkjUU 211212 112 −+⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= ωω

ω (4.43)

a) 0ffs =

tsω

tsω

b) 0ffs >

tsω

tsω

c) 0ffs <

tsω

tsω

Fig. 4.30. Simulated basic waveforms of currents and voltages in contactless energy transmission system S1 - compensation.

The waveform of magnetizing inductance current mi , if the influence of voltage drop on the

leakage inductance is negligible, has triangular shape. The angle of phase shift mi to the

primary voltage u1(t) is equal πϕ21


operating frequency of resonant converter.


- 67 -

2)1()1(

πj

mm eII−

= (4.44)

From the equation (4.45) follows that the voltage drop in case of magnetizing current flow is

combined with its amplitude and leakage inductance of the CET transformer. The sign of this

voltage drop is independent of the operating frequency.

( ) ( )LkILkjjIU mmIm2121 )1()1()1(

−=−⋅−= ωω (4.45)

Depending on the operating frequency changes above and below resonance frequency the

reactance of tank network character is volatile.

( )r

S CLkX

ωω 12121 −−=

⎪⎩

⎪⎨

⎧

>=<

0

0

0

ffffff

S

S

S - capacitive character

0 (4.46)

- inductive character

The value of the voltage drop on the reactance XS1 as a result of first harmonic primary

current flow are defined as:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

=

=

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −−

21)1(1

21)1(1

21)1(1

21)1(1

)1(1 0π

ϕπϕ

πϕπ

ϕ

j

S

j

Sj

j

S

j

Sj

I

eXIeXeI

eXIeXeI

U

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

=

<

0

0

0

ff

ff

ff

S

S

S

(4.47)

By using equation (4.42 – 47) the value rectifier input voltage can be written as:

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−−+−

−+

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−−++

=

LktIC

LktIu

LktIu

LktIC

LktIu

u

mr

m

mr

21sin1212cos

21sin

21sin1212cos

11)1(1

11)1(11

11)1(1

11)1(1

11)1(11

2

ωωω

ωϕω

ωω

ωωω

ωϕω

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

=

<

0

0

0

ff

ff

ff

S

S

S

(4.48)

From the presented simulation results and equation (4.48) follows that independent of the

converter operating frequency, the value of the output voltage increases because of the

voltage drop on leakage reactance caused by magnetizing current. As is shown in Fig. 4.30 for

all operating frequencies the amplitude of output voltage u2 can be higher then amplitude of

the source voltage u1. Therefore, there is no possible to maintain a fixed level of output

voltage, by adjustment of voltage source frequency only. Voltage gain may exceed the

nominal value GV > 1 what is very danger for the load (over voltage).


- 68 -

4.3.3. CET System with Compensation Capacitors

on Both Side of the Transformer

The CET system with resonant circuit created due to connection in series capacitor to

primary or secondary winding of the transformer (Fig.4.27, 4.29) gives very poor possibility

of keeping voltage gain on the constant level during CET operation, with variable coupling

factor and load resistance. In this Subsection CET system solution with compensation

voltages dropping on leakage inductances by adding series resonant capacitors to primary and

secondary windings of transformer is presented. This concept is described literature e.g. [8,

10, 34, 35, 46, 59-66].

)(0 tu

)(0 ti

)(2 tu

)(1 ti

)(1 tu

)(tiR

)(tiR

)(2 ti

Fig. 4.31. CET system with primary and secondary series compensation mode SS.

Resonance frequency which ensures zero current switching conditions for switch network of

the circuit shown in Fig. 4.31 is:

( ) ( ) 00 12

11

121

LCkLCkf

r −+

−=

π (4.49)

If we assume that rCC >>0 , where Cr = (Cr1 + Cr2) is resonant capacitance, the resonance

frequency of the circuit shown in Fig. 4.31 is follows:

( ) rLCkf

−≈

11

21

0 π (4.50)

The output voltage of resonant converter and rectifier input voltage can be express by

following equations:


- 69 -

( )

( )⎪⎪⎩

⎪⎪⎨

⎧

−−−=−

−+−=

21

22

11

11

121

121

IC

jIkLjILkjU

IC

jIkLjILkjU

rm

rm

ωωω

ωωω

(4.51)

Finally the secondary side voltage equation by using Eq.(4.51) can be rewritten as:

( ) ( ) mrr

IC

LkjIC

LkjUU ⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−=

ωω

ωω 1211212 112 (4.52)

0ff s =

tsω

tsω

0ff s >

Ts <To

u1 u2

i1

i2

im

Gv < 1tsω

tsω

0ffs <

tsω

tsω

Fig. 4.32. Simulated basic waveforms of currents and voltages

in CET system with SS - compensation.

If the influence of voltage on the leakage inductance is negligible, the waveform of

magnetizing inductance current mi has triangular shape. The angle of phase shifted mi to the

primary voltage u1(t) is equal πϕ21


operating frequency of resonant converter.

2)1()1(

πj

mm eII−

= (4.53)

The value of series reactance for primary current 1I is a function of operating frequency and

is two times larger than reactance seen by the magnetizing current mI .


- 70 -

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

rSS C

LkXω

ω 1212 ⎪⎩

⎪⎨

⎧

>=<

0

0

0

ffffff

S

S

S -capacitive character

0 (4.54)

-inductive character

The value of the voltage drop on the reactance XSS caused by first harmonic primary current is

defined as:

⎪⎪

⎩

⎪⎪

⎨

⎧

=

=

=

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −−

2)1(1

2)1(1

2)1(1

2)1(1

)1(1 0π

ϕπϕ

πϕπ

ϕ

j

SS

j

SSj

j

SS

j

SSj

I

eXIeXeI

eXIeXeI

U

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

=

<

0

0

0

ff

ff

ff

S

S

S

(4.55)

I this case using equation (4.51 – 4.55) the value of the input rectifier voltage can be written

as:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−+−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−++

=

rm

r

rm

r

CLktI

CLktIu

uC

LktIC

LktIu

u

111)1(

111)1(11

1

11)1(1

11)1(11

2

121sin1212cos

121sin1212cos

ωωω

ωωϕω

ωωω

ωωϕω

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

=

<

0

0

0

ff

ff

ff

S

S

S

(4.56)

From the equation (4.56) and simulated results from Fig.4.32 follows that, when primary

voltage source operating frequency 0ff S = , the secondary side output voltage has the same

value and the circuit voltage gain is unity GV = 1. The compensation of the voltage drop on

the leakage reactances is possible only for operation with resonant frequency 0ff S = . The

output voltage can be controlled without measurement and adjustment of supply voltage

amplitude Ez, just by keeping converter voltage in resonance. It is a very favorable case

because control may be very precise and input inverter configuration may be simple (full

bridge or half bridge inverter).

4.3.4. Voltage Gain Behavior of the SS – Compensated Circuit

It was proven that for SS-compensated resonant circuit in CET system, the voltage gain Gv

is unity. It is necessary to explore closer SS compensation topologies by means of voltages

gain expression for variable CET system parameters.


- 71 -

To calculate the voltage gain Gv of the SS-compensated circuit, the value of impedance and

reactance (shown at various points in Fig. 4.14) should be derived. For effective transfer

electric energy through the CET transformer, a high-voltage gain with small variation as well

as small circulating current through the magnetizing inductance is important. To achieve these

requirements, the SS-compensated circuit is applied.

The higher turn ratio requires more windings of the secondary side for a given converter

operating frequency, and the lower turn ratio requires high voltage of the input side.

Therefore, the turn ratio of the transformer is considered to unity.

The SS - compensated circuit inductances are expressed as (see Fig. 4.14):

-the inductance seen at the secondary side

esRjXZ +=

2γ (4.57)

above equation can be rewritten as follows:

( )⎪⎩

⎪⎨⎧

+⋅

+⋅−⋅= esRrCj

LkjZS

S

2

11ω

ωγ (4.58)

γ

γβ ZmjX

ZmjXZ

+

⋅= (4.59)

-the inductance seen at the primary side:

βZjXαZ += 1 (4.60)

The reactances X1, X2, Xm are equal to:

1

1111

rCsLsX

ωω −= (4.61)

2

1222

rC

sL

sX

ωω −= (4.62)

MsmX ω= (4.63)

ωs = 2πfs – inverter operation frequency.

The transfer voltage gain of CET system with SS compensation topology from Fig.4.14 is:

γα

βZesR

Z

ZVG = (4.64)

From equations (4.57) to (4.64), the resulting equation for the voltage gain GV can be

expressed as:


- 72 -

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ ⋅++

+⎟⎟⎠

⎞⎜⎜⎝

⎛+=−

es

m

mV R

XXXXX

jXXG

2121

11 1 (4.65)

21

221

212

11

−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ ⋅++

+⎟⎟⎠

⎞⎜⎜⎝

⎛+=

es

m

mV R

XXXXX

XXG (4.66)

It follows from equation (4.66) that GV is unity at resonant frequency, even though the

leakage inductances of the CET transformer are very large. In order to analyze the voltage

gain for frequency variations, Eq. (4.66) should be expressed as a function of frequency. The

compensation frequency ω0 = 2πf0 – (resonance frequency) is calculated for condition X1 =

X2 = 0 as a fallow:

222/1111/1/1 rCLrCLrCrLo ===ω (4.67)

Based on equations (4.61 -67) the expressions for X1 and X2 can be rewritten as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

11111ω

ω LsX (4.68)

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

11222ω

ω LsX (4.69)

Where normalized frequency ω is defined as:

os ωωω /= (4.70)

As result of used two halves cores and large air gap, the CET transformer operates under

lower and variable magnetic coupling factor k. It is desirable to express the voltage gain GV in

terms of coupling factor. For simplification we assuming that the configurations of primary

and secondary transformer cores are the same, and Eq. (3.18) is fulfill. Then, by comparing

equations (4.68-69) to (4.66), the voltage gain can be written as function of operating

frequency:

( )22

22211222112

211

/1111111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −+=

eV R

MLLLL

MLG ω

ωω

ωω

ω (4.71)

By the use Eq. (4.68-69), the voltage gain (4.71) can be rewritten as function of operating


- 73 -

frequency and coupling coefficient:

2

2

2

211

21111111 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

−+⎟

⎠⎞

⎜⎝⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

−+=

ωωω

ω kkQ

kkG acV (4.72)

Where the circuit quality factor for SS compensation topology is:

( )es

r

esac R

LR

LLQ ωω=

+= 2211 (4.73)

The transfer gain of voltage CET system for SP compensation topology is:

γ

δ

α

βZZ

ZZ

VG = (4.74)

where, the equations for component impedance above equation (shown at various points in

Fig. 4.22) can be written as:

esr RCj

Z 11

2 +⋅=

ωδ (4.75)

( ) 21 ZZLkjZ =+⋅−⋅= δωγ (4.76)

The equation (4.74) can by used to compare voltage gain SS and SP compensated CET

system for the same values of the circuit parameters.

The choice of quality factor is an important design parameter. Larger secondary

quality factor increased power transfer capability. The value of primary side quality factor

depends on the geometry primary transformer winding, and the required primary current.

The voltage gain Gv characteristics calculated from equation (4.72) are shown in Fig.4.33

These analytical results are presented for the varying frequency ω, various acQ , and two

different cases of the coupling coefficient. Results show that the curves of voltage gain GV for

the two cases are almost identical except for a small deviation in low circuit quality factor

acQ . Thus, the analytic curves can be used for design and control of the system. From these

curves, the points at unity of the normalized frequency keep a unity gain because the

impedance of the leakage inductances is canceled by the additional capacitors at this

frequency. For this operating frequency the current and the voltage of the resonant converter

are in phase, consequently, reactive power is minimize. As long as the converter operates at

this frequency, the voltage gain GV keeps unity gain, and circulating current through the

magnetizing inductance is suppressed. Furthermore, these characteristics do not depend on

load as well as coupling coefficient.


- 74 -

a)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.5

1

0→acQ

32=acQ16=acQ8=acQ4=acQ2=acQ1=acQ

ω

vG k=0.2A B C

b)

k=0.6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.5

1

0→acQ

32=acQ16=acQ8=acQ4=acQ2=acQ1=acQ

ω

vG

A B C

Fig. 4.33. The dc voltage gain of inductive CET system versus frequency for SS topology compensation.

a) k = 0.2, b) k = 0.6

This frequency, however, varies as the coupling coefficient varies. For the output-voltage

regulation, the feedback control of output voltage should be applied by selecting the desirable

region from three different regions of the voltage gain GV in Fig. 4.33. Range A is the lowest

frequency region. The voltage gain GV increases up to the unity as the switching (operating)

frequency increases. The unity gain frequency ω depends on coupling coefficient k. Range B

is the middle frequency region. The gain largely depends on load variations and switching


- 75 -

frequency in this region. Range C is the highest frequency range. The voltage gain GV

decreases from the unity as the switching frequency of the converter increases. Range B, also

called the double-turned circuit, provides the maximum transfer gain of the voltage. However,

the GV in Range B is very sensitive to load changes as well as coupling coefficient k.

Furthermore, it has nonlinear characteristics versus switching frequency. Thus, it is difficult to

control the output voltage. Regions A or C are able to control the output voltage because the

gain is a monotonic function of switching frequency. Range C is more desirable because the

gain voltage for each load conditions is much less sensitive than in the Range A. In this work,

Range C is suggested as a reliable region to control the output voltage in CET system with

variable air gap length and the load (Fig. 4.35). Moreover, when increasing switching

frequency, the changes in output voltage are very fast. This is important in control system, for

example when danger state is detecting (fs < f0).

0.5

1

0.2 0.4 0.6 0.8 1

14=ρ

2.22=ρ

4.44=ρ

6.66=ρ111=ρ

5.155=ρ

200=ρ

max00 / II

ω0ω

30ω

Fig. 4.34. The output current characteristics in CET system with SS compensation topology versus

operating frequency, for a different values wave impedance by R0 = const.


- 76 -

sω

Fig. 4.35. The basic output power characteristics of CET system versus operating frequency.

4.4 Conclusions

In this Chapter the series resonant converter used in Contactless Energy Transmission

(CET) system operated at resonance has been described. This means, that primary and

secondary leakage inductances of the CET transformer are compensated by series resonance

capacitors. By the use of these capacitors, the connected load to the CET system is seen as an

ohmic load by power supply converter. Therefore, switching instance could occur at zero

current of power transistors. The full bridge inverter topology composed by four switches is

preferred because of better utilization of voltage supply. With a unity transformer turns ratio,

the dc output voltage is ideally equal to the dc input voltage only when the transistor

switching frequency is equal to the tank resonance frequency. The output voltage is reduced if

the switching frequency increase or decrease away from the resonance frequency. Thanks to

secondary circuit compensation the power transfer capability from primary to secondary

circuit though the air gap significantly increase. The series resonant converters are the most

commonly used because of simplicity in realization and are preferable for systems in which

the parameters like coupling factor and circuit quality factor change in wide range. Therefore,

in this work the SS compensation circuit will be used.


- 77 -

5. Control and Protection System

5.1 Introduction

Control and protection of resonant converter is the critical part of CET system and should

meet following requirements:

very fast operation (switching frequency fs ≥ 20 kHz),

fully digital implementation,

no need to parameter adjustment,

stable operation,

flexibility to main circuits parameter changes,

reduced number of sensors (sensorless operation),

low cost,

high reliability.

Several control strategies has been proposed in the literature. Most of the methods are based

on a Phase Lock Loop (PLL) approach, see works e.g. (5), (8) and [37]. However, in this

Thesis another strategy based on extreme regulator has been proposed. This strategy is very

flexible and meets all above formulated requirements. The control system with extreme

regulator adjusts operating frequency and guaranties zero current switching of converter’s

IGBT power transistors.

5.2 Control System Operation and Behavior

The block diagram of CET Main Control and Protection System (MCPS) implemented in

FPGA Stratix II is shown in Fig.5.1. The significant assumption which determines the

behavior and control strategy of the CET system, is that the control circuit does not use

measurement signals from the secondary side of the CET transformer. Note that the secondary

transformer side and load another Secondary Side Protection Circuit (SSPC) is implemented.

The FPGA’s clock frequency is 100 MHz and the resonant switching frequency is 60 kHz.

The primary capacitor voltage Ucr1 and inverter output current i1 are sensed and sent to A/D

converters via operation amplifiers. A 12-b A/D converter, (AD9433) is used in the designed

system. The typical analog input signal is 5V (VDD), hence maximum of amplitude

corresponds to 2.5V. The FPGA device is Stratix II EP2S60F1020C3ES Fig. 5.2.


- 78 -

Fig.5.1 Block diagram of the CET control and protection system.

Fig. 5.2. Stratix II EP2S60 DSP Development Board Components & Interfaces


- 79 -

a)

b)

Execution of

the algorithm

Execution of

the algorithm

Execution of

the algorithm

Execution of

the algorithm

Fig. 5.3. Control algorithm of extreme primary current regulator a) block diagram, b) basic waveforms primary voltage u1 (red line) and current i1 (green line).


- 80 -

To attenuate noise in input signals (i1’, uc

’) a digital recursive filtering algorithm has been

applied. These filtering signals i1f and ucf are used in extreme regulator (ER) based on

reversed counter which determinates converter switching frequency fref. Signal fref is delivered

to signal generator (SG) which generates gate pulses for MOSFET power transistors T1…T4.

Also, death time compensation is implemented in this block (SG).

To guarantee stabile operations, the regulator ER in every N-period sequence searches for

highest current amplitude i1fm and, after comparison with previous data, adjusts reference

value of switching frequency fref. Additionally, the filtered input signals i1f and ucf are

delivered to protection block (PR) which continuously watch up and in case when limit values

i1f(lim) and ucf(lim) are achieved, blocks gate pulses T1 …T4. To limit current during the

converter start-up, the regulator (ER) sets the switching frequency higher as resonance

frequency fref > fo and then in every N-period sequence reduces fref to fallow the maximum of

current amplitude. If the amplitude of current or voltage reaches nominal value i1f(N) < i1f(lim),

ucf(N) < ucf(lim), the regulator (ER) increases converter switching frequency. The period N of

regulator operation has been selected experimentally N=7. Control algorithm and registered

signals of extreme primary current regulator are presented on Fig. 5.3-4. Simulation results of

this control mode are shown in Fig. 5.5.

i1f

ucf

sg1 sg2

per_ch

per_ref

measure.

Over_voltage ucf

Over_current i1f

Fig. 5.4. Measured and control signals in FPGA device, observed by Altera Quartus software.

Another control algorithm realized by the control block of Fig. 5.1, is shown in Fig. 5.6. In

this case the phase angle between primary voltage and primary current is controlled. The

control algorithm behaves very similar to the previous mode.


- 81 -

a)

b)

c)

Fig. 5.5. Simulated results of control signals behaviors when control algorithm work as extreme

regulator of primary current. a) output voltage, maximum pick value of primary current, reference period (operating frequency) ,

b) period sign and primary current, c) Steady-state waveforms of the voltages u1, u2, and currents i1, i2, iin


- 82 -

However, in this case the extreme regulator follows the minimum difference of actual phase

angle between primary voltage and current and its referenced values. By the change of sign

and value of the reference phase angle, the reference operating frequency is also changed.

Finally, the system can operate below, above or with resonant circuit frequency. The

significant advantage of this control mode is stabile operation in safe working area (see Fig.

4.35) close to resonant frequency. This feature is very important when circuit quality factor is

low. Simulated results are shown in Fig. 5.7-8.

MEASUREMENTAND SIGNALINTERFACE

RECURSIVEFILTER

IS PRESENT VALUE OF PHASE ERROR

GRATER THENPREVIOUS

Did frequencyincrease in previous

algorithm cycle

YES NO

YES YESNO NO

Increaseoperating frequency

Decreaseoperatingfrequency

Increaseoperating frequency

Changeoperating frequency

Data save

IMPULS SIGNALS GENERATOR

NO CHANGES

Did frequencyincrease in previous

algorithm cycle

PHASE ERRORDETECTION

REFERENCEVALUE OF

PHASE ERROR

Fig. 5.6. Extreme regulator algorithm for phase angle ϕ - mode.


- 83 -

a)

b)

Fig. 5.7. Simulated result of control signals behaviors when control algorithm operates in the ϕ - mode for two values of load resistance (6, 16 Ohm).

a) reference period (operating frequency) and output voltage b) phase angle between primary voltage and primary current as a clk_impulse value.

Note, that the DC output voltage changes depending on the value of load resistance (Fig.

5.7a bottom). The regulator adjusts switching frequency for new load conditions (Fig. 5.7a –

upper). This is typical behavior of series compensated resonant converter.


- 84 -

a)

b)

Fig. 5.8. Steady-state waveforms of the voltages u1, u2, and currents i1, i2, iin, io a) Ro = 6Ω, a) Ro = 16Ω


- 85 -

6. Design and Description of Laboratory Prototype

6.1 Introduction

Based on the discussion presented in previous Chapters 3-5, the methodology of CET

system design will be given in this Chapter. Further, the description of laboratory prototype

constructed in the Department of Electrical Drives, Electrotechnical Institute is provided.

6.2 Design Procedure of CET System

The block diagram design procedure of a CET system is presented in Fig. 6.1.

Fig. 6.1. Block diagram of design procedure of a CET system


- 86 -

A short description of the blocks included in design procedure is as follows:

1. The specification of CET system mainly depending on the three cases:

desired shape, power range and application destination.

2. The power transfer capability and efficiency is directly dependent upon the coil design.

So, the transformer design plays the most important role in the development of the CET

system.

3. Power transfer simulations and calculations enables to calculate theoretically whether

CET system with will be viable and able to reach the specification.

4. Advanced issues such as communications, shielding, heating problems and stability also

needs to be taken into consideration. Important issues such as human safety and regulation

issues regarding magnetic fields.

5. Optimization of the coil geometries and the circuit analyze allows to develop the

most efficient solution for the CET system.

6. Choice of primary and secondary leakage inductances of the CET transformer

compensation by resonance capacitors. By the use of these capacitors, the connected load

to the CET system is seen by power supply converter as an ohmic load. Selected

compensation topology determines system behavior and sensitivity for variable load and

magnetic coupling factor.

7. Finally the development, building and testing of the CET system prototype

6.3 Description of the Laboratory Prototype

In this Subsection a 3kW laboratory set-up of CET system is presented. The view of

the laboratory setup is depicted in Fig. 6.3-4. The primary winding of CET system is fed

through a high frequency full (Fig. 6.2) (or half) bridge converter while secondary winding

placed on different part of core is an integral part of the load. The resonant converter consists

of a frequency controlled insulated gate bipolar transistor (IGBT) bridge and converts the

rectified DC link voltage into an AC high resonant frequency voltage.


- 87 -

230

V 5

0 H

z )(1 ti

)(2 ti)(1 tu

0U

)(2 tu

0I

)(2 tizE

)(tim

Fig. 6.2. Configuration of contactless energy transmission (CET) system

To compensate high leakage inductance of the rotatable transformer with large air gap and

reduce converter switching losses, the series resonant capacitive circuit SS has been used. The

resonance frequency is adjusted by extreme regulator which follows the instantaneous value

of primary peak current and guaranties zero current switching of converter’s IGBT power

transistors.

Resonant converter is build on high power switching Mitsubishi IGBT modules CM200DU-

24NFH. The CET rotatable transformer is build on two PM 114/93 – B65733 Epcos ferrite

core (Fig.6.5-6). Coupling between the transformer windings changes in a wide range and it is

not known when the system start-up. Hence, leakage inductances of the windings may change

in a wide range as it was shown in Fig. 3.23.

Fig. 6.3. View of laboratory setup


- 88 -

Fig. 6.4. View of 3kW contactless power supply system with rotatable transformer and series resonant inverter

Fig. 6.5. Ferrite pot cores dimensions used in laboratory setup CET system.

(PM 114/93 – B65733)


- 89 -

a)

b)

Fig. 6.6. Ferrite pot cores,

a) before put into resin, b) in resin prepared to use in CET system prototype.

The basic work parameters of the CET system, operating with about 60 kHz resonance

frequency are given in Table 6.1. Measured self inductances of primary and secondary

rotating transformer winding in laboratory model are shown in Fig. 6.7. The measured

coupling factor and magnetizing inductance are shown in Fig. 3.23 and Fig. 3.24 respectively

and measured leakage factor σ is shown in Fig. 6.8.

Table 6.1 Basic Parameters of Rotatable Transformer and Resonant Circuit

Parameter Value Unit N1 32 coils N2 32 coils CR1 63 nF CR2 63 nF C1 1000 μF C0 100 μF

Adjustable air gap 0 - 30 mm


- 90 -

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

0 2 4 6 8 10 12 14 16 18 20 22 24

Air gap [mm]

Self inductance[mH] primary

winding

secondarywinding

Fig. 6.7. Measured self inductance primary and secondary winding of rotating transformer in laboratory model.

0,000

0,100

0,200

0,300

0,400

0,500

0,600

0,700

0,800

0 5 10 15 20 25 30

Fig. 6.8. Measured leakage factor σ of the pot core transformer in the laboratory model.


- 91 -

7. Simulation and Experimental Results 7.1 Introduction

In this Chapter simulation and experimental results of developed CET system are

presented. The research has been carried out in two main directions:

Performance characterization of different compensation circuits (SS, SP) used in

CET systems,

Explaining and presenting the performance of the developed CET system with

proposed extreme regulator and protection circuit.

7.2 Performance Characterization of SS and SP Compensation Circuits

Simulation model of control and power structures of CET system were implemented in

SABER and OrCAD-PSpice packages (see Appendix 3). These packages provide analysis and

behavior of the complete analog and mixed-signal systems including electrical subsystem.

Simulated output voltage characteristics of CET system for SS and SP compensation

topologies are shown in Fig.7.1-4, respectively.

0

20

40

60

80

100

120

140

160

180

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

fr/fs

Uo

5101520253035404550

0

20

40

60

80

100

120

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

fs/fr

Uo

5101520253035404550

Fig. 7.1. Simulated output voltage characteristics for SS circuit compensation Cr1 = Cr2,versus different load resistances and converter operating frequency.

Simulated circuit parameters: a) L1 = L2 = 296 μH, Cr1 = Cr2 = 63 nF, k = 0,247, b) L1 = L2 = 775 μH, Cr1 = Cr2 = 63 nF, k = 0,8


- 92 -

020406080

100120140160180200

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

fr/fs

Uo

5101520253035404550

020406080

100120140160180200

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

fr/fs

Uo

5101520253035404550

Fig. 7.2. Fig. Simulated output voltage characteristics for SP circuit compensation Cr1 > Cr2,versus different load resistances and converter operating frequency.

Simulated circuit parameters: a)L1 = L2 = 296 μH, Cr1 = 67 nF, Cr2 = 63 nF, k = 0,247

b) L1 = L2 = 296 μH, Cr1 = 63 nF, Cr2 = 59.2 nF, k = 0.247

0

20

40

60

80

100

120

140

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2fr/fs

Uo

5101520253035404550

0

20

40

60

80

100

120

140

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

Uo

5101520253035404550

Fig. 7.3. Simulated output voltage characteristics for SP circuit compensation Cr1 > Cr2,versus different load resistances and converter operating frequency.

Simulated circuit parameters: a) L1 = L2 = 775 μH, Cr1 = 175 nF, Cr2 = 63 nF, k = 0,8

b) L1 = L2 = 775 μH, Cr1 = 63 nF, Cr2 = 22,7 nF, k = 0,8

020406080

100120140160180200

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

fr/fs

Uo

5101520253035404550

0

20

40

60

80

100

120

140

160

180

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2fs/fr

Uo

5101520253035404550

Fig. 7.4. Simulated output voltage characteristics for SP circuit compensation Cr1 = Cr2,versus different load resistances and converter operating frequency.

Simulated circuit parameters: a) L1 = L2 = 296 μH, Cr1 = Cr2 = 63 nF, k = 0,247 b) L1 = L2 = 775 μH, Cr1 = Cr2 = 63 nF, k = 0,8


- 93 -

The simulated results are in good agreement with analytically calculated

characteristics presented in Subsection 4.3. Also, it can be seen that, SP compensation is more

sensitive to operating frequency and circuit parameter changes. Therefore, in the CET systems

where the coupling factor changes in wide range, the SS compensation is recommended.

In Fig. 7.5 the per unit output power versus operating (switching) frequency are

shown. As parameter the value of circuit wave impedance ρ is given. It can be seen that,

independent of wave impedance value, the highest transferred power is achieved when

converter switch with resonance frequency fs/f0 =1.

maxo

o

PP

o

s

ff

1ρ

12 4ρρ =

51

0ρρ =

Ω== 47/1 rr CLρ

Fig. 7.5. CET system Per unit output power versus operating frequency

for a different values of circuit wave impedance ρ.

7.3 Investigation Results of Developed CET System

Two variants of converter topologies have been investigated: full bridge and half

bridge. Also in experimental setup two types of transistor gate drivers were used. The first one

was Semikron-SKHI26W and the other has been designed and constructed by author. Because

of 3μs minimal dead time in the Semikron drivers, only 25 kHz switching frequency could be

achieved. The simulated and experimental results for this case are given in Appendix 5.

The steady-state and transient waveforms of the voltage and current in CET system with

rotating transformer and full bridge resonant converter operating at 60 kHz switching

frequency and 1μs dead time are shown in Figures 7.6-8


- 94 -

Simulated

Experimental

a)

CH2 – primary current - i1CH1 – primary voltage - u1

CH4 – secondary voltage - u2CH3 – secondary current - i2

u1

i2 i1u2

P1

3mm - air gap, 10 – resistance load

b)



u1

i2

i1

u2

P1


c)



u1

i2

i1

u2

P1


Fig. 7.6. Steady-state waveforms of the voltages u1, u2, currents i1, i2 and primary side power P1. Air gap length 3mm, load resistance 10Ω.

a) Converter operation with resonance frequency , b) Converter operation below resonance frequency, c) Converter operation above resonance frequency.


- 95 -

Experimental results (full bridge)

a)



u1

i1i2u2

P1


b)



u1

i1

i2

u2

P1


c)



u1 i1

i2u2

P1


Fig. 7.7. Steady-state waveforms of the voltages u1, u2, currents i1, i2 and primary side power P1. Air gap length 10mm, load resistance 10Ω.



- 96 -

Experimental results (full bridge)

Air gap length 25.5mm, load resistance 10Ω. Air gap length 25.5mm, load resistance 20Ω. a)

2.52kW

1kVA10.0A

10.0A



u1

i1i2

u2

P1

25.5 mm - air gap, 10 – resistance load



u1i1

i2

u2

P1


b)



u1i1

i2

u2

P125.5 mm - air gap,

10 – resistance load



u1

i1

i2u2

P1


c)



u1

i1i2

u2

P125.5 mm - air gap,

10 – resistance load



u1i1

i2

u2

P1


Fig. 7.8. Steady-state waveforms of the voltages u1, u2, currents i1, i2 and primary side power P1.



- 97 -

Table 7.1 Basic parameters of rotating transformer and resonant circuit for results present on Fig.7.5-10

Parameter Value Unit

N1 32 coils N2 32 coils L11 166,5 μH L12 203,5 μH L22 166,5 μH M 203,5 μH k See Fig. 3.23. -

CR1 63 nF CR2 63 nF

Adjustable air gap 0 - 30 mm

Please note that very good agreement between simulated and experimental results are

achieved (Fig. 7.6). Further, experimental results for different air gap length and load

resistance are shown (Fig. 7.7-8). It can be seen that operation of the converter is stable for all

presented values of air gap length and load conditions. As expected the highest transferred

power occurs for operation with resonant frequency (see waveforms in Fig. 7.6a, 7.7a, 7.8a).

Experimental results (half bridge)

Fig. 7.9. Experimental waveforms of the voltages u1, u2, currents i1, i2, and primary side power P1 at start-up CET system.

The steady-state and transient waveforms of the voltage and current in CET system with

rotating transformer and half bridge (see Fig. 4.4a) resonant converter operating at 60 kHz

switching frequency and 1μs dead time are shown in Figures 7.9-11. Also in this case CET

system with extreme regulator operates stable and adjusts the converter switching frequency

for different air gap length and load conditions. However, to achieve in the half bridge


- 98 -

converter the same output power as in full bridge topology the supply DC voltage Ez has to be

doubled.


a)

b)

c)

Fig. 7.10. Steady-state waveforms of the voltages u1, u2, uCr1, uL1

currents i1, i2 and primary side power P1. Air gap length 3mm, load resistance 20Ω. a) Converter operation with resonance frequency , b) Converter operation below resonance frequency,

c) Converter operation above resonance frequency.


- 99 -


Air gap length 9mm, load resistance 20Ω. Air gap length 22.5mm, load resistance 20Ω. a)

b)

u1

u2

i1i2



P1

22.5mm - air gap, 20 – resistance load

c)

u1

u2

i1

i2




Fig. 7.11. Steady-state waveforms of the voltages u1, u2, currents i1, i2 and primary side power P1. a) Converter operation with resonance frequency , b) Converter operation below resonance frequency,

c) Converter operation above resonance frequency.


- 100 -

Very important is safe start-up of the CET system. Normally, the control circuit has no

information about the load value. Therefore, the regulator starts with frequency above

resonance (with high impedance Zα of the resonant tank) to protect converter against

overload. As it is shown in Fig. 7.9 after several periods (about 125 ms) the switching

frequency is adjusted to resonance frequency. This guarantees safe start-up even when the air

gap length changes in the range limited by max and min operation frequency set in the

extreme regulator.

Fig. 7.12. Total efficiency versus air gap width and load resistance for SS compensation method,

(simulated results- dashed/dotted line, experimental results – continuous line).

Fig. 7.12 shows the total DC input-output efficiency of the CET system versus transformer air

gap length and the load (for laboratory and simulation model). The efficiency curves show

(Fig. 7.12) a slight difference between simulation and experimental values. This is result of

different transistor models (in simulation were used available in PSpice library International-

Rectifier transistors model - IRGPC50S), whereas in the laboratory prototype Mitsubishi

IGBT power transistors CM200DU-24NFH) have been mounted). Note that for higher load

resistances higher efficiency can be achieved, but the circuit becomes more sensitive for

magnetic coupling coefficient changes. The total efficiency achieves 0,93 – 0,85 for rotating

transformer air gap 0.1 up to 3 cm, respectively.


- 101 -

8. Summary and Closing Conclusions

The thesis has been devoted to modeling, analyze and design of modern inductive

coupled Contactless Energy Transmission (CET) system with rotatable transformer.

Various problems were addressed and discussed as follows:

• Rotatable transformer model developed in step-by-step manner: starting from

constructions, geometrical and material data through reluctance model till ∏ -

equivalent circuits with inductances as parameters (Chapter 3);

• Analyze and discussion of resonant circuits characteristic used for transformer leakage

inductance compensation SS: series-series compensation and SP: series-parallel

compensation (Chapter 4);

• Analyze and modeling of resonant converter operation modes: below, at resonance

and above resonance frequency;

• Development of control and protection methods: with detection of phase angle

between primary current and voltage (primary power factor), or extreme primary

current regulator (Chapter 5);

• Power circuit design and construction (Chapter 6).

To analyze and comparative study several simulation and calculation CET models were

developed using SABER, Matlab and OrCad- PSpice software packages.

For experimental validation a laboratory set-up based on 3 kW H-Bridge converter with

rotatable coupling transformer (ferrite core) and FPGA based extreme regulator has been

constructed.

Among important results of the thesis are:

• Application of high-switching frequency IGBT converter operating at resonance

frequency provides more efficient CET system with reduced transformer size and

weight;

• Theoretically, there is no power transfer limit, even with low coupling coefficient, if

the circuit operates with the resonance frequency of secondary current, in condition

that the primary winding is compensated and resonant frequency of primary current is

equal to secondary fp = fs;


- 102 -

• Application of control method which guarantees Zero Current Switching (ZCS)

conditions in the resonant converter allows to reduce the switching losses, hence

improves total efficiency;

• As shown, the use of FPGA-based digital controller allows implementing very fast

and flexible extreme regulator and protection algorithm.

• The developed CET system has following features and advantages: high total

efficiency up to 93% (depending on air gap length), operating frequency 60 kHz, the

extreme regulator system is self adjusted for actual resonance frequency and variable

load as well as variable magnetic coupling factor.

In the author opinion the results of this thesis can be used in design and development of

modern inductive coupled CET systems with rotatable transformers for robots and

manipulators. The presented methodology can also be easy expanded for design power supply

systems for other mobile or special loads (battery chargers, transport systems, etc.).


- 103 -

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Appendix 1

Remarks

The references in the text are given as follows:

books and PhD Theses are cited in round parenthesis for example (1), (2,3),

journal papers and conferences contributions are cited in square parenthesis for

example [1], [2,3].

List of basic symbols

Lr– leakage inductance

Cr– resonant capacitor

C – Capacitance

I – root mean square value of current

L – Inductance

R – Resistance or reluctance

T – time period

P – active power or permeance

Q – quality circuit factor

Z – Impedance

Z ‘– Reflected Impedance

X– reactance

Ez– Input voltage source

Uo– Output voltage

U1– Converter output voltage

U2– Rectifier input voltage

I1, I2 - Primary and secondary current

ϕ - phase shift angle between primary voltage and primary current

f – frequency

ω - Normalized operating angular frequency


- 109 -

k– coupling factor of the transformer

σ - leakage inductance factor

li– length of the magnetic path in the section

Ai– active core section area

μri– relative permeability of the section


- 110 -

Appendix 2

Basic Definitions

Wave impedance

r

r

CL

=ρ

Self resonant angular frequency of RLC circuit

( )21 αω −=rrCLr ;

r

r

LR

2=α - damper factor

Rr – summarized circuit resistance

rrCL1

0 =ω -resonant angular frequency when assuming, resistance of RLC circuit Rr =0

Quality circuit factor

rRrCrL

rRrRrCrRrL

Q ====ρ

ω

ω

0

10

Harmonic Distortion

%1001MnM

HD =

M1 – RMS value of first harmonic of voltage or current

Mn – RMS value of n harmonic of voltage or current

Total Harmonic Distortion

%1001

21

M

nMnTHD >Σ

=

M1 – RMS value of first harmonic of voltage or current

Mn – RMS value of n harmonic of voltage or current


- 111 -

Power Factor

1 cosIPFI

ϕ=

Partial Weighted Harmonic Distortion

2

14

1

100%h

h

hIPWHD

I

∞

==∑

Harmonic Constant

2 2

2

1

100%h

h

h IHC

I

∞

==∑


- 112 -

Appendix 3

Simulation models

SABER model scheme is presented in Fig. A.3.1. OrCAD-PSpice scheme is presented

in Fig. A.3.2.

a)

b)

c)

Fig. A.3.1. Scheme of SABER model, a) CET system structure, b) h-bridge inverter, c) output diode rectifier


- 113 -

+-

+

-Sbreak

S1

d4

d4

Lf1 2

D4D4t

z

D3t

ip

Dd4

D3

0

D1

AMP1

0

Ui200

d1GEN

L2

1

2

0

D1t

0

C2

63n

K K1

+-

+

-Sbreak

S4

+-

H1

H

d1

D2

0

D2t

zas2

U4

SYG_MINUS

0 1XG1 YG1

d3

Dd3

Dd2

ip

C1

63n

Rg

d2

AMP2

Ro

0

U1

SYG_PLUS

0 1XG1 YG1

+-

+

-Sbreak

S2

U2

SYG_MINUS

0 1XG1 YG1

SRC

d3

0

DT

U5

VPULSE_DT

01

231

DTOUTAMP

0

L1

1

2

zas

zas2

control circuit

BFUN2

0

1

2

3SDC

SRC

SV

AMP

Co

d2

+-

+

-Sbreak

S3

0

Dd1

U3

SYG_PLUS

0 1XG1 YG1

Fig. A.3.2. Scheme of OrCad - PSpice model.


- 114 -

Appendix 4

Switching Losses

Most converter switches need to turn-on or turn-off the full load current at a high voltage,

resulting in what is known as hard switching. Figure. A.4.1. shows typical hard-switching

during power transistors commutation in full bridge converter operating with below and

above resonant frequency. In small time scale the switching loci for a hard-switching

converter are depicted in Fig. A.4.2. As the frequency of operation increases, the switching

losses also increase.

a) fs < fo

b) fs > fo

Fig. A.4.1. Time waveforms of power transistors and antiparallel diodes in H-bridge converter.

Operating frequency a) below resonance, b) above resonance.


- 115 -

There are two types of switching losses:

• Turn-on, when the power transformer leakage inductance produces high di/dt, which

results in a high voltage spike across it.

• Turn-off, when the switching loss in mainly caused by the dissipation of energy stored

in the output parasitic capacitor of the power switch.

In a soft-switching circuit converter topology, an LC resonant network is added to shape the

switching device’s voltage or current waveform into a quasi-sinewave in such a way that a

zero voltage (ZVS) or current (ZCS) condition is created. This eliminates the turn-on or turn-

off loss associated with charging or discharging of the energy stored in the transistor’s

parasitic junction capacitors.

Fig. A.4.2. Typical switching loci for a hard-switching converter.

Simulation in OrCAD-Pspice used IRGPC50S – spice transistor model.


- 116 -

Appendix 5

Investigation Results of Developed CET System Operating at 25 kHz

In Figures A.5.1-2 steady-state waveforms of the voltage and current of CET system with

rotating transformer and full bridge resonant converter operating at 25 kHz switching

frequency and 3μs dead time are shown.

Simulation

Experimental

a)

b)

c)

Fig. A.5.1. Steady-state waveforms of the voltages u1, u2, uCR1 and primary current i1 ,

system works as a battery charger. a) converter operation with resonance frequency, b) converter operation above resonance frequency,

c) converter operation below resonance frequency. SS - method of leakage inductances compensation applied.


- 117 -

Simulation Experimental a)

b)

c)

d)

e)

Fig. A.5.2. Steady-state waveforms of the voltages u1, u2, uCR1 and primary current i1 (at 0,95 rated load -

18Ω, dead time of power transistors gate drivers -3μs), a) converter operation with resonance frequency, b) converter operation above resonance frequency,c) converter operation below resonance frequency,

d,e) start-up laboratory model with 0.95 rated load of the converter .


- 118 -

Basic Parameters of Rotatable Transformer and Resonant Circuit for results presented

in Fig. A.5.1-2.

Parameter Value

Unit

N1 32 coils N2 32 coils L11 166,5 μH L12 203,5 μH L22 166,5 μH M 203,5 μH k 0,55 -

CR1 250 nF CR2 250 nF

Adjustable air gap 10,5 mm

Documents

Contactless Energy Transmission System with Rotatable