POLITECNICO DI MILANO · 2019. 4. 18. · POLITECNICO DI MILANO School of Industrial and Information Engineering Master of Science in Energy Engineering - Renewables and Environmental

POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science in Energy Engineering - Renewables and Environmental

Sustainability

Wireless Power Transfer System Design and Analysis for

E-bikes Application

Advisor: Prof. Michela Longo

Co-Advisor: Prof. Alicia Triviño Cabrera

M. Sc. Dissertation of:

Federico GENCO

884135

Academic Year: 2018-2019

“We continue to believe that

the future is wireless”

Dan Riccio

v

Sommario

Il presente lavoro tratta una analisi del comportamento dei sistemi di ricarica wireless applicati alle

biciclette elettriche. In particolare, lo studio si è concentrato sul design e sulla stabilità delle bobine

mutualmente accoppiate di tale sistema.

Nella prima parte del lavoro sono stati affrontati gli aspetti teorici dei sistemi ICPT (Inductive coupled

power transfer). Qui sono state espresse le principali leggi e relazioni che regolano il funzionamento del

circuito equivalente, definendo le tensioni e le correnti presenti nel sistema passando poi alla potenza

trasferita e alla frequenza di lavoro. Particolare attenzione è stata posta sulle topologie di compensazione

e sulla loro stabilità, in modo da permettere al sistema di lavorare in condizioni di risonanza. In seguito,

è stata affrontata una classificazione dei sistemi di ricarica wireless e delle loro applicazioni.

Nella seconda parte del lavoro sono state analizzate le caratteristiche induttive degli avvolgimenti

accoppiati. Basandosi su simulazioni effettuate in precedenti studi di ricerca, sono state ipotizzate diverse

configurazioni per le bobine, ottenendo i valori di auto induttanza e mutua induttanza. Questo processo

è stato effettuato attraverso il software Ansys Maxwell, eseguendo simulazioni in 3D. In tutti i casi, le

simulazioni sono state effettuate applicando le stesse caratteristiche del filo e gli stessi parametri di

soluzione, questo in modo da ottenere risultati comparabili tra di loro. Infine, è stata scelta la

configurazione con le migliori performance per studiarne la stabilità in funzione della frequenza.

La configurazione presa in considerazione per lo studio della stabilità, suggerita dall’autore, richiede un

maggiore accoppiamento meccanico; tuttavia, questa garantisce il miglior fattore di accoppiamento

magnetico con un air-gap ridotto. La stabilità è stata analizzata variando solo il numero di avvolgimenti

delle bobine, fino a raggiungere un comportamento stabile del sistema.

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Abstract

The present work reports an analysis of Inductive Coupled Power Transfer Systems behaviour applied

to electrical bikes. In particular, the study has focused on the design and stability of the coupled coils of

this system.

In the first part of the work, the theoretical aspects of the wireless power system are handled. Here, the

principal laws and relations that control the equivalent circuit are expressed. The voltages and the

currents of the system are defined and the power transfer and the working frequency are analysed. In

particular, the attention has focalized on the compensation topologies and on their stability in order to

allow to the system to work in resonance condition. Lastly, this section copes with the ICPT systems

classification and their applications.

In the second part of the work, the inductive features of the coupled windings are analysed. Basing on

simulations performed in previous research studies, some configurations have been supposed, obtaining

their auto inductance and mutual inductance values. In order to perform this process, the Ansys Maxwell

software has been used, performing 3D simulations. In all the cases, the simulations have been performed

keeping constant the feature of the cables and of the simulation process. This in order to obtain

comparable results. In the end, the configuration that presents the best performance has been chosen and

its stability as a function of the frequency is studied.

The configuration considered for the stability analysis, suggested by the author, presents a high

mechanical coupling degree; however, it guarantees the best coupling factor with a small air-gap. The

stability is analysed managing the number of turns of the coils till the stable behaviour is achieved.

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ix

Contents Capitolo 1 : ICPT SYSTEM ................................................................................................................ 1

1.1 INTRODUCTION ............................................................................................................... 1

1.2 EQUVALENT CIRCUIT AND TRANSFERRED POWER ................................................ 2

1.3 WORKING FREQUENCY ................................................................................................. 4

1.4 COMPENSATION SYSTEMS ............................................................................................ 6

1.4.1 SS Compensation ......................................................................................................... 7

1.4.2 SP Compensation ......................................................................................................... 9

1.4.3 PS Compensation ....................................................................................................... 12

1.4.4 PP Compensation ....................................................................................................... 14

1.4.5 Nominal Parameters ................................................................................................... 17

1.5 STABILITY ...................................................................................................................... 18

1.6 CLASSIFICATION OF ICPT SYSTEM ............................................................................ 21

1.6.1 Dynamic charge systems ............................................................................................ 21

1.6.2 Fixed charge systems ................................................................................................. 22

1.6.3 Strongly coupled systems ........................................................................................... 23

1.6.4 Chained ring systems ................................................................................................. 23

1.6.5 Loosely coupled systems ............................................................................................ 24

1.7 ICPT OVERALL STRUCTURE........................................................................................ 26

1.8 GEOMETRICAL DESIGN ............................................................................................... 27

Capitolo 2 : ICPT system applications ............................................................................................... 31

2.1 INTRODUCTION ............................................................................................................. 31

2.2 HOUSEHOLD APPLIANCES AND MOBILE DEVICES................................................. 32

2.3 ELECTRIC VEHICLES APPLICATION .......................................................................... 34

2.3.1 E-BIKE application .................................................................................................... 35

2.3.2 Automotive application .............................................................................................. 37

Capitolo 3 : COIL GEOMETRY DESIGN ........................................................................................ 41

3.1 INTRODUCTION ............................................................................................................. 41

3.2 ANSYS MAXWELL ......................................................................................................... 42

3.3 SIMULATIONS ................................................................................................................ 43

3.3.1 Two Circular Coils with 9 turns ................................................................................. 43

3.3.2 Two Circular Coils with 7 turns ................................................................................. 57

Federico Genco

Federico Genco

Federico Genco

Federico Genco

Federico Genco

Chapter

Federico Genco

Chapter

Federico Genco

Federico Genco

Chapter

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3.3.3 Two Circular Coils with 12 turns................................................................................ 58

3.3.4 Two Circular Coils with 9 and 7 turns ........................................................................ 59

3.3.5 Two Vertical Coils ..................................................................................................... 61

3.4 CONCLUSIONS ............................................................................................................... 65

Capitolo 4 : STABILITY ANALYSIS .............................................................................................. 67

4.1 INTRODUCTION ............................................................................................................. 67

4.2 TWO VERTCAL COILS N1=N2=8 ................................................................................... 68

4.3 TWO VERTICAL COILS N1=16 AND N2=6 .................................................................... 72



4.6 CONCLUSIONS ............................................................................................................... 87

Conclusion ........................................................................................................................................ 89

Bibliography ..................................................................................................................................... 90

Federico Genco

Federico Genco

Federico Genco

Chapter

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List of Figures

Figure 1.1: ICPT equivalent circuit without compensation[12] ............................................................ 2

Figure 1.2: Equivalent circuit seen from the primary side[12] .............................................................. 4

Figure 1.3: Overall structure of ICPT system[4] .................................................................................. 6

Figure 1.4: Compensation system topologies ....................................................................................... 7

Figure 1.5: Equivalent circuit of ICPT system in SS configuration[12] ................................................ 7

Figure 1.6: Equivalent circuit of ICPT system in SP configuration[12] ................................................ 9

Figure 1.7: Equivalent circuit of ICPT system in PS configuration[12] .............................................. 12

Figure 1.8: Equivalent circuit of the PS topology seen from the primary circuit[12] ........................... 13

Figure 1.9: Equivalent circuit of ICPT system in PP configuration[12] .............................................. 14

Figure 1.10: Response of the electrical parameters with frequency variation in SS topology[12] ........ 20

Figure 1.11: Scheme of dynamic distributed ICPT system[4]............................................................. 22

Figure 1.12: Scheme of dynamic concentrated ICPT system[4] ......................................................... 22

Figure 1.13: Example of strongly coupled ICPT system .................................................................... 23

Figure 1.14: Chained ring structure: difference between shorttrack and longtrack .............................. 24

Figure 1.15: Normal and Tangential component capture system[12] .................................................. 25

Figure 1.16: Overall structure of ICPT system ................................................................................... 26

Figure 1.17: Geometries analysed[12] ............................................................................................... 28

Figure 1.18: Different types of cable disposition................................................................................ 28

Figure 1.19: Coupling coefficient of circular and horizontal disposition............................................. 29

Figure 1.20: Ratio M2^2/L2^2 of circular and horizontal disposition[12] .......................................... 29

Figure 2.1: ICPT system in an electric toothbrush …………………………………………………………………………….33

Figure 2.2: ICPT in a drill[32]…………………………………………………………………………………………………………..33

Figure 2.3: Multiple charging station ................................................................................................. 33

Figure 2.4: ICPT system in mouse pad [33] ....................................................................................... 34

Figure 2.5: Additional component for ICPT system ........................................................................... 34

Figure 2.6: Charging system proposed in [53] ................................................................................... 35

Figure 2.7: Configuration proposed in [54] ........................................................................................ 36

Figure 2.8: Configuration proposed in [55] ........................................................................................ 36

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Figure 2.9: Configuration proposed in [3] .......................................................................................... 37

Figure 2.10: Common configuration of ICPT system in car applications ............................................ 37

Figure 2.11: BMW transmitting case ................................................................................................. 38

Figure 2.12: Camera system .............................................................................................................. 38

Figure 2.13: HALOIPT charging system ........................................................................................... 39

Figure 2.14: Dynamic wireless charging ............................................................................................ 39

Figure 2.15: Concept of dynamic wireless charging in highways [25] ................................................ 40

Figure 3.1: Solution process of EDDY CURRENT solver ................................................................. 43

Figure 3.2: Command for selecting the desired project ...................................................................... 44

Figure 3.3: Procedure for selecting the solver .................................................................................... 44

Figure 3.4: Procedure for setting the solver type ................................................................................ 45

Figure 3.5: Procedure for defining the geometry ................................................................................ 45

Figure 3.6: Definition of geometry properties .................................................................................... 46

Figure 3.7: Material definition ........................................................................................................... 47

Figure 3.8: Commands for creating the region ................................................................................... 47

Figure 3.9: Physical configuration of coils......................................................................................... 48

Figure 3.10: Physical configuration of the coils top view (left) and top view (right) ........................... 48

Figure 3.11: Definition of the current parameters .............................................................................. 49

Figure 3.12: Procedure for defining the excitation ............................................................................. 49

Figure 3.13: Excitations of the model ................................................................................................ 50

Figure 3.14: Procedure for assigning the matrix ................................................................................. 50

Figure 3.15: Procedure for adding the solution step ........................................................................... 51

Figure 3.16: Mesh settings procedure ................................................................................................ 52

Figure 3.17: Mesh ............................................................................................................................. 52

Figure 3.18: Frequency setting .......................................................................................................... 53

Figure 3.19: Bfield circular configuration .......................................................................................... 54

Figure 3.20: Trend of the resistance with respect to the frequency ..................................................... 55

Figure 3.21: Trend of k factor with respect to the misalignment ......................................................... 56

Figure 3.22: Maximum misalignment configuration side view and top view ...................................... 57

Figure 3.23: Physical configuration of the coils ................................................................................. 59

Figure 3.24: Bfield of the configuration ............................................................................................. 60

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Figure 3.25: Trend of the k factor with respect to the misalignment ................................................... 61

Figure 3.26: Physical configuration of "two vertical coils” ................................................................ 62

Figure 3.27: Charging tower station and bicycle with integrated ICPT system ................................... 63

Figure 3.28: 2D Bfield ...................................................................................................................... 64

Figure 3.29: 3D Bfield ...................................................................................................................... 64

Figure 3.30: Comparison between k factors behaviour with respect to the misalignment .................... 66

Figure 4.1: SS compensation topology .............................................................................................. 67

Figure 4.2: Primary current of "two vertical coils N1=N2=8" model .................................................. 69

Figure 4.3: Voltage on primary capacitor of "two vertical coils N1=N2=8" model ............................. 69

Figure 4.4: Secondary current of "two vertical coils N1=N2=8" model .............................................. 70

Figure 4.5: Voltage on secondary capacitor of "two vertical coils N1=N2=8" model.......................... 70

Figure 4.6: Load voltage of "two vertical coils N1=N2=8" model ...................................................... 71

Figure 4.7: Load power of "two vertical coils N1=N2=8" model ....................................................... 71

Figure 4.8: two vertical coils n1=16 and n2=6 ................................................................................... 72

Figure 4.9: 3D Bfield in logarithmic scale of "two vertical coils N1=16 and N2=6" model ................ 73

Figure 4.10: Primary current of "two vertical coils N1=16 and N2=6" model ..................................... 74

Figure 4.11: Voltage on primary capacitor of "two vertical coils N1=16 and N2=6" model ................ 74

Figure 4.12: secondary current of "two vertical coils N1=16 and N2=6" model ................................. 75

Figure 4.13: Voltage on the secondary capacitor of "two vertical coils N1=16 and N2=6" model ....... 75

Figure 4.14: Load voltage "two vertical coils N1=16 and N2=6" model ............................................. 76

Figure 4.15: Load power "two vertical coils N1=16 and N2=6" model .............................................. 76

Figure 4.16: Two vertical coils N1=25 and N2=5 .............................................................................. 77

Figure 4.17: Bfield in logarithmic scale of "two vertical coils N1=25 and N2=5" model .................... 78

Figure 4.18: Primary current of "two vertical coils N1=25 and N2=5" model ..................................... 79

Figure 4.19: Voltage of primary capacitor of "two vertical coils N1=25 and N2=5" model ................ 80

Figure 4.20: Secondary current of "two vertical coils N1=25 and N2=5" model ................................. 80

Figure 4.21: Voltage of secondary capacitor of "two vertical coils N1=25 and N2=5" model ............. 81

Figure 4.22: Load voltage of "two vertical coils N1=25 and N2=5" model ......................................... 81

Figure 4.23: Load power of "two vertical coils N1=25 and N2=5" model .......................................... 82

Figure 4.24: two vertical coils N1=30 and N2=4 ............................................................................... 82

Figure 4.25: 3D Bfield in logarithmic scale of “two vertical coils N1=30 and N2=4” model .............. 83

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Figure 4.26: Primary current of “two vertical coils N1=30 and N2=4” model .................................... 84

Figure 4.27: Voltage on primary capacitor of “two vertical coils N1=30 and N2=4” model................ 84

Figure 4.28: Secondary current of “two vertical coils N1=30 and N2=4” model ................................ 85

Figure 4.29: Voltage on secondary capacitor of “two vertical coils N1=30 and N2=4” model ............ 85

Figure 4.30: Load voltage of “two vertical coils N1=30 and N2=4” model ........................................ 86

Figure 4.31: Load power of “two vertical coils N1=30 and N2=4” model .......................................... 86

Figure 4.32: Comparison of Pl in the different configurations ............................................................ 88

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List of Tables

Table 1.1: Nominal parameters of the equivalent circuit in the different topologies[12] ..................... 18

Table 1.2: Stability conditions for the four topologies[12] ................................................................. 20

Table 3.1: Results of "2 circular coils with 9 turns" model ................................................................. 53

Table 3.2: results of the reference case .............................................................................................. 54

Table 3.3: Results of "2 circular coils with 9 turns" model with airgap=2 cm ..................................... 54

Table 3.4: Results of "2 circular coils with 9 turns" model with frequency sweep .............................. 55

Table 3.5: results of "2 circular coils with 9 turns" model with misalignment..................................... 56

Table 3.6: Per cent decrease of the coupling factor ............................................................................ 57

Table 3.7: Results of “2 circular coils with 7 turns” model ................................................................ 58

Table 3.8: results of "2 circular coils with 12 turns" model ................................................................ 58

Table 3.9: Results of “two circular coils with 9 and 7 turns” model with misalignment ...................... 60

Table 3.10: Per cent decrease of the coupling factor .......................................................................... 61

Table 3.11: Results of “2 vertical coils” model .................................................................................. 63

Table 3.12: Comparison between circular coils with different number of turns .................................. 65

Table 3.13: Results best configuration ............................................................................................... 65

Table 4.1: Results "two vertical coils N1=N2=8" model .................................................................... 68

Table 4.2: Stability factors of "two vertical coils N1=N2=8" ............................................................. 68

Table 4.3: Results of "two vertical coils N1=16 and N2=6" model ..................................................... 73

Table 4.4: Stability factors "two vertical coils N1=16 and N2=6" model ............................................ 73

Table 4.5: Results of "two vertical coils N1=25 and N2=5" model ..................................................... 78

Table 4.6: stability factors of "two vertical coils N1=25 and N2=5" model ........................................ 78

Table 4.7: Simulation results of “two vertical coils N1=30 and N2=4” model .................................... 83

Table 4.8: stability factors of “two vertical coils N1=30 and N2=4” model ........................................ 83

Table 4.9: Comparison of electrical values of the different geometries .............................................. 87

Table 4.10: Comparison of stability parameters of the different geometries ....................................... 88

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Nomenclature

𝐼1 Complex current of primary circuit

𝐼2 Complex current of secondary circuit

𝐼𝐶1 Complex current through the primary capacitor

𝐼𝐶2 Complex current through the secondary capacitor

𝐼𝑆𝐶 Complex short circuit current

𝐼𝑝 Complex current through the primary coil

𝐼𝑠 Complex current through the secondary coil

𝑆2max Complex maximum transferred power

𝑆2𝑀𝐴𝑋 Complex maximum transferred power with compensation

�⃗⃗�1 Complex input voltage of primary circuit

�⃗⃗�2 Complex output voltage of secondary circuit

�⃗⃗�𝑂𝐶 Complex open circuit voltage

�⃗�1 Impedance of primary circuit

�⃗�1∗ Conjugate impedance of primary circuit

�⃗�2 Impedance of secondary circuit

�⃗�𝑇 Total impedance of the system

�⃗�𝑇_𝑃𝑃 Total impedance of the system in PP topology

�⃗�𝑇_𝑃𝑆 Total impedance of the system in PS topology

�⃗�𝑇_𝑆𝑃 Total impedance of the system in SP topology

�⃗�𝑇_𝑆𝑆 Total impedance of the system in SS topology

�⃗�𝑟 Impedance of secondary circuit reflected to the primary

𝐶1 Capacitance of primary capacitor

𝐶2 Capacitance of secondary capacitor

𝐿1 Self-inductance of primary coil

𝐿2 Self-inductance of secondary coil

𝑃2𝑀𝐴𝑋 Maximum transferred active power

𝑅1 Resistance of primary circuit

xviii

𝑅2 Resistance of secondary circuit

𝑅𝐵𝐴𝑇𝑇 Resistance of the battery

𝑅𝐿 Resistance of the load

𝑞𝑃 Quality factor of primary circuit

𝑞𝑠 Quality factor of secondary circuit

𝜑2 Load factor of secondary circuit

𝜔 Pulsation

𝜔𝑀𝐴𝑋 Maximum pulsation

𝜔𝑅 Resonance pulsation

𝜔𝑑 Design pulsation

H Henry

Δy Misalignment along y axis

𝐼𝑚( ) Imaginary part

𝐾 Coupling factor

𝑀 Mutual inductance

𝑅𝑒( ) Real part

𝑗 Imaginary part suffix

𝜂 Efficiency

____________________________________________________________ Chapter 1: ICPT SYSTEM

__________________________________________________________________________________ 1

Capitolo 1 : ICPT SYSTEM

1.1 INTRODUCTION

The Inductive Coupled Power Transfer (ICPT) systems are systems based on the phenomenon of the

electromagnetic induction between mutually coupled circuits. This phenomenon has been deeply studied

by the physicist Faraday who stated a law that is able to describe it. Nowadays, several devices that base

their functioning on electromagnetic induction exist, such as electrical motors, electrical generators,

speakers, etc. In these devices, two or more coupled coils are always present and they can present a

ferromagnetic core or an “air-core”. Depending on the type of core used, the electromagnetic induction

presents different features: the ferromagnetic core improves it, however, the system can be subjected to

higher losses.

In this thesis, we are going to focus on the magnetic coupling between two coils with an air gap of

dimensions comparable to the dimensions of the coils. Moreover, we are going to principally focus on

systems with an air-core.

Depending on the intensity of the linked fluxes, therefore on the coupling coefficient, the coupled

systems can be divided in Strongly coupled or Loosely coupled. In the former, it is possible to work with

low frequency values (50-60 Hz), since the flux leakages are low and the coupling coefficient is high.

As a consequence of this, the power transfer presents a high efficiency. On the other hand, in loosely

coupled systems, due to the higher flux leakages and to the lower coupling coefficient, it is needed to

work with higher frequencies (tens of kHz) in order to obtain acceptable efficiencies for the power

transfer.

Inductive coupled power transfer systems usually work with a distance between the two coils relatively

high with respect to the dimensions of the them, leading to a low coupling coefficient. Therefore, we can

define this type of ICPT system as loosely coupled system. Because of this, as said above, the system

needs to work at high frequency. Working in this condition, the choice of the type of core has to be

carefully managed due to the possible high losses in it (hysteresis losses and Foucault currents). For this

reason, ICPT systems typically work with an air-core or ferrite core. Moreover, to avoid high losses in

the cables, due to high frequencies, particular types of cables are used, such as Litz wire. With this type

of conductor, the skin effect, that is proportional to the frequency, is really reduced with respect to a

normal copper wire.

Federico Genco

Federico Genco

Federico Genco

Chapter


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1.2 EQUVALENT CIRCUIT AND TRANSFERRED POWER

The ICPT system can be easily modelled as a system formed by two electrically insulated coils,

magnetically coupled through the air, able to transfer power from the transmitting circuit to the receiver.

The equivalent circuit of the system is showed in the Figure 1.1.

FIGURE 1.1: ICPT EQUIVALENT CIRCUIT WITHOUT COMPENSATION [12]

The electrical representation of the system is made by a Primary circuit, it is fed with a sinusoidal tension

of effective value V1 and pulsation ω, and a Secondary circuit, it feeds a purely resistive load RL. In this

equivalent circuit we are not taking into account a compensation system, it is used for increasing the

performance of the ICPT device, allowing it to work in resonant conditions using capacitors.

The Kirchhoff voltage equations for the primary and secondary circuits are the following:

�⃗⃗�1 = (𝑅1 + 𝑗𝜔𝐿1)𝐼𝑝 − 𝑗𝜔𝑀𝐼𝑠 = �⃗�1 𝐼𝑝 − 𝑗𝜔𝑀𝐼𝑠 (1.1)

0 = 𝑅𝐿𝐼2 + (𝑅2 + 𝑗𝜔𝐿2)𝐼𝑠 − 𝑗𝜔𝑀𝐼𝑝 (1.2)

Neglecting R2 with respect to RL, the voltage V2 results:

�⃗⃗�2 = 𝑅𝐿𝐼2 = 𝑗𝜔𝑀𝐼𝑝 − 𝑗𝜔𝐿2𝐼𝑠 (1.3)

In case of open secondary circuit (I2=Is=0), the voltage VOC in it results:

�⃗⃗�𝑂𝐶 = 𝑗𝜔𝑀𝐼𝑝 (1.4)

While, in case of short-circuit in the secondary circuit (RL=0), the current ISC in it results:


__________________________________________________________________________________ 3

𝐼𝑆𝐶 =

𝑗𝜔𝑀

𝑗𝜔𝐿2𝐼𝑝 =

𝑀

𝐿2𝐼𝑝

(1.5)

The condition of maximum transferred power from the primary circuit to the secondary is obtained

multiplying the short circuit current for the open circuit voltage:

𝑆2max = �⃗⃗�𝑂𝐶 ∙ 𝐼𝑆𝐶 = 𝑗𝜔𝑀𝐼𝑝 ∙

𝑀

𝐿2𝐼𝑝 = 𝑗 (

𝜔𝑀2𝐼𝑝2

𝐿2)

(1.6)

Supposing to work in resonant conditions in the secondary circuit, thanks to the introduction of a

compensation system, the maximum transferred power results:

𝑆2𝑀𝐴𝑋 = 𝑆2max∙ 𝑞𝑠 = 𝑗 (

𝜔𝑀2𝐼𝑝2

𝐿2) 𝑞𝑠

(1.7)

The factor qs is the quality factor of the secondary circuit, it is a term that increases the performances of

the device. Moreover, it is a term that has to be carefully managed in order to ensure the stability of the

system.

To conclude, the maximum active power that can be transferred from the primary to the secondary circuit

is:

𝑃2𝑀𝐴𝑋 = 𝑆2𝑀𝐴𝑋 ∙ 𝑐𝑜𝑠𝜑2 (1.8)

In order to obtain the highest value of active power it is needed to work with the load factor equal to 1

(resonance condition). Thus, the transferred power to the secondary circuit in resonance condition

results:

𝑃2𝑀𝐴𝑋 = 𝜔

𝑀2

𝐿2𝐼𝑝

2 𝑞𝑠 (1.9)

From the last equation we can deduce that for increasing P2MAX we need a high mutual inductance value,

a low self-inductance of the secondary circuit and a high inductor quality factor. Moreover, the

performances of the system depend also on the geometry used for the coils. Lastly, with a high frequency,

the primary circuit current (Ip) needed is going to be lower, as a consequence, the amount of material

used for the wire is lower.


__________________________________________________________________________________ 4

1.3 WORKING FREQUENCY

The working frequency is the principal feature to consider for the design of a Wireless Power Transfer

(WPT) system. Basing on it, the compensation capacitors are designed.

Initially, supposing to work at the frequency that guarantees the maximum power transfer condition

(P2MAX), the voltage’s Kirchhoff laws in the primary and secondary circuit result:

�⃗⃗�1 = �⃗�1𝐼𝑝 − 𝑗𝜔𝑀𝐼𝑠 (1.10)

0 = �⃗�2𝐼𝑠 − 𝑗𝜔𝑀𝐼𝑝 (1.11)

Expressing the current in the secondary circuit as a function of the primary current:

𝐼𝑠 =

𝑗𝜔𝑀

�⃗�2

𝐼𝑝 (1.12)

and replacing it in the equation (1.10), this result:

�⃗⃗�1 = �⃗�1𝐼𝑝 −

𝜔2𝑀2

�⃗�2

𝐼𝑝 = (�⃗�1 +𝜔2𝑀2

�⃗�2

) 𝐼𝑝 = (�⃗�1 + �⃗�𝑟)𝐼𝑝 (1.13)

where �⃗�𝑟 is the impedance of the secondary circuit reflected to the primary.

The Figure 1.2 shows the circuit represented by the equation (1.13).

FIGURE 1.2: EQUIVALENT CIRCUIT SEEN FROM THE PRIMARY SIDE [12]

The maximum active power transfer condition is obtained when the reflected impedance of the secondary

circuit equals the conjugate of the primary circuit:

�⃗�𝑟 = �⃗�1∗ (1.14)

Knowing that �⃗�2 is equal to:


__________________________________________________________________________________ 5

�⃗�2 = (𝑅2 + 𝑅𝐿) + 𝑗𝜔𝐿2 (1.15)

Replacing the equation (1.15) in the following equation:

�⃗�𝑟 =

𝜔2𝑀2

�⃗�2

(1.16)

We obtain:

�⃗�𝑟 =

𝜔2𝑀2(𝑅2 + 𝑅𝐿)

(𝑅2 + 𝑅𝐿)2 + (𝜔𝐿2)2− 𝑗

𝜔2𝑀2𝜔𝐿2

(𝑅2 + 𝑅𝐿)2 + (𝜔𝐿2)2

(1.17)

To conclude, in order to respect the maximum power transfer condition, equation (1.14), the following

relations have to be respected:

�⃗�1 = 𝑅1 + 𝑗𝜔𝐿1 (1.18)

𝑅1 =

𝜔2𝑀2(𝑅2 + 𝑅𝐿)

(𝑅2 + 𝑅𝐿)2 + (𝜔𝐿2)2

(1.19)

𝜔𝐿1 =

𝜔2𝑀2𝜔𝐿2

(𝑅2 + 𝑅𝐿)2 + (𝜔𝐿2)2

(1.20)

For easily guaranteeing the equation (1.20) it is possible to introduce tuning capacitors in the system; as

a consequence, the imaginary part of both the impedances is null and the maximum power transfer

condition depends only on R1. From the equation (1.18) it is possible to obtain the resonance frequency:

𝜔𝑀𝐴𝑋 = √𝑅1(𝑅2 + 𝑅𝐿)

𝑀

(1.21)

Working with this frequency the power transfer is going to be maximum. However, high values of power

are suitable for lower values of frequencies; consequently, in this type of systems is more important

focusing on the frequency that increases their efficiency rather that that one that increases the power.


__________________________________________________________________________________ 6

1.4 COMPENSATION SYSTEMS

From the previous paragraph it is possible to deduce the principal two conditions for increases as much

as possible the performance of an ICPT system:

• Working at resonance frequency in the secondary circuit in order to have maximum active power

supplied to the load, as shown in the equation (1.8)

• The total impedance seen from the primary circuit has to be purely resistive in order to decrease

the current absorbed by the converter and, as a consequence, increase the efficiency

For respecting these conditions is needed to work in both circuits in resonant condition. This is possible

applying capacitors in the primary and secondary circuit (compensation system).

The overall ICPT system (Figure 1.3) is formed by:

• Primary system (Transmitter): made by a coil with N1 turns and section S1, a tuning capacitor

C1 that can be connected in series or in parallel with the inductor, and a high frequency

alimentation system

• Secondary system (Receiver or Pick-up): made by a coil with N2 turns and section S2, a tuning

capacitor C2 that can be connected in series or in parallel with the inductor, and an AC/DC

converter for feeding the load.

Depending on the ways the capacitors in the primary and secondary circuit are connected, it is possible

to have four different compensation topologies:

• Series-Series (SS)

• Series-Parallel (SP)

• Parallel-Series (PS)

FIGURE 1.3: OVERALL STRUCTURE OF ICPT SYSTEM [4]


__________________________________________________________________________________ 7

• Parallel-Parallel (PP)

The previous topologies are graphically represented in the Figure 1.4.

Each topology presents different behaviours depending on the parameters of the circuit. The choice of

the type of compensation is based on several factors, such as the type and dimension of load, fixed or

moving supply, etc. However, this decision is not so easy because each topology presents its advantages

and disadvantages, it does not exist a univocal solution, it is strictly connected to the single case.

1.4.1 SS Compensation

In this topology, the capacitors are placed in series with the inductances of the primary and of the

secondary circuit. The equivalent circuit is shown in the Figure 1.5.

FIGURE 1.4: COMPENSATION SYSTEM TOPOLOGIES

FIGURE 1.5: EQUIVALENT CIRCUIT OF ICPT SYSTEM IN SS CONFIGURATION [12]


__________________________________________________________________________________ 8

The equations of the circuit are:

�⃗⃗�1 = [𝑅1 + 𝑗 (𝜔𝐿1 −

1

𝜔𝐶1)] 𝐼1 − 𝑗𝜔𝑀𝐼2 = �⃗�1𝐼1 − 𝑗𝜔𝑀𝐼2

(1.22)

0 = [(𝑅2 + 𝑅𝐿) + 𝑗 (𝜔𝐿2 −

1

𝜔𝐶2)] 𝐼2 − 𝑗𝜔𝑀𝐼1 = �⃗�2𝐼2 − 𝑗𝜔𝑀𝐼1

(1.23)

The impedance of the secondary circuit reflected to the primary results:

�⃗�𝑟 =𝜔2𝑀2

�⃗�2

=𝜔2𝑀2(𝑅𝐿 + 𝑅2)

(𝑅𝐿 + 𝑅2)2 + (𝜔𝐿2 −1

𝜔𝐶2)

2 − 𝑗𝜔2𝑀2 (𝜔𝐿2 −

1𝜔𝐶2

)

(𝑅𝐿 + 𝑅2)2 + (𝜔𝐿2 −1

𝜔𝐶2)

2

(1.24)

Appling the condition of maximum power transfer stated in the equation (1.14):

𝑅1 = 𝑅𝑒(�⃗�𝑟) =

𝜔2𝑀2(𝑅𝐿 + 𝑅2)

(𝑅𝐿 + 𝑅2)2 + (𝜔𝐿2 −1

𝜔𝐶2)

2 (1.25)

(𝜔𝐿1 −1

𝜔𝐶1) = −𝐼𝑚(�⃗�𝑟) = 𝑗

𝜔2𝑀2 (𝜔𝐿2 −1

𝜔𝐶2)

(𝑅𝐿 + 𝑅2)2 + (𝜔𝐿2 −1

𝜔𝐶2)

2

(1.26)

Thus, the relation between the parameters of the circuit and the resonance frequency results:

𝜔𝑑 =

1

√𝐿1𝐶1

=1

√𝐿2𝐶2

(1.27)

On the other hand, fixing the frequency and then finding the values of the capacitances that allow to the

system to work in maximum power transfer condition, these results:

𝐶1 =

1

𝜔𝑑2 𝐿1

(1.28)

𝐶2 =

1

𝜔𝑑2 𝐿2

(1.29)

The main point of the design of the ICPT system is working in resonant condition in the secondary

circuit. It means that the equation (1.29) is valid for all the topologies; while, the value of the capacitor

in the primary circuit is evaluated finding the value of C1 that respect the following equation:


__________________________________________________________________________________ 9

𝐼𝑚(�⃗�𝑇𝑂𝑇) = 𝐼𝑚(�⃗�1 + �⃗�𝑟) = 0 (1.30)

For the SS topology, since the design frequency allows resonance in both the circuits the values of the

capacitances are evaluated in the same way.

The power transfer efficiency results:

𝜂 =

𝑅𝐿𝐼22

𝑅𝐿𝐼22 + 𝑅2𝐼2

2 + 𝑅1𝐼12 =

𝑅𝐿

𝑅𝐿 + 𝑅2 + 𝑅1 (𝐼1𝐼2

)2

(1.31)

Considering that:

𝐼1

𝐼2=

(𝑅2 + 𝑅𝐿)

𝜔𝑅𝑀

(1.32)

Replacing the equation (29) in the (28), the efficiency results:

𝜂 =

𝑅𝐿

(𝑅𝐿 + 𝑅2) (1 + 𝑅1(𝑅2 + 𝑅𝐿)

𝜔𝑅2 𝑀2 )

(1.33)

1.4.2 SP Compensation

In this topology, the capacitor in the primary circuit is in series with the inductor, while the capacitor in

the secondary circuit is in parallel with the inductor. The equivalent circuit is shown the Figure 1.6.

The equations of the system are:

FIGURE 1.6: EQUIVALENT CIRCUIT OF ICPT SYSTEM IN SP CONFIGURATION [12]


__________________________________________________________________________________ 10

�⃗⃗�1 = [𝑅1 + 𝑗 (𝜔𝐿1 −

1

𝜔𝐶1)] 𝐼1 − 𝑗𝜔𝑀𝐼𝑠 = �⃗�1𝐼1 − 𝑗𝜔𝑀𝐼𝑠

(1.34)

0 = (𝑅2 + 𝑗𝜔𝐿2)𝐼𝑠 + 𝑅𝐿𝐼2 − 𝑗𝜔𝑀𝐼1 (1.35)

𝐼𝑠 = 𝐼𝐶2 + 𝐼2 (1.36)

Expressing the current in the secondary circuit as a function of 𝐼1:

𝐼2 =

𝑗𝜔𝑀

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝐶2𝑅𝐿)𝐼1

(1.37)

The equation (1.34) becomes:

�⃗⃗�1 = [�⃗�1 +

𝜔2𝑀2(1 + 𝑗𝜔𝐶2𝑅𝐿)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝐶2𝑅𝐿)] 𝐼1 = [�⃗�1 + �⃗�𝑟]𝐼1 = �⃗�𝑇𝐼1

(1.38)

Being:

�⃗�𝑟 =

𝜔2𝑀2(1 + 𝑗𝜔𝐶2𝑅𝐿)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝐶2𝑅𝐿)

(1.39)


𝑅1 = 𝑅𝑒(�⃗�𝑟) =

𝜔2𝑀2(𝑅𝐿 + 𝑅2) + 𝜔4𝑀2𝐶22𝑅𝐿

2𝑅2

(𝑅2 + 𝑅𝐿 − 𝜔2𝐿2𝐶2𝑅𝐿)2 + 𝜔2(𝐿2 + 𝐶2𝑅2𝑅𝐿)2

(1.40)

(𝜔𝐿1 −

1

𝜔𝐶1) = −𝐼𝑚(�⃗�𝑟) =

𝜔3𝑀2[𝐶2𝑅𝐿2(𝜔2𝐿2𝐶2 − 1) + 𝐿2]

(𝑅2 + 𝑅𝐿 − 𝜔2𝐿2𝐶2𝑅𝐿)2 + 𝜔2(𝐿2 + 𝐶2𝑅2𝑅𝐿)2

(1.41)

Working in resonant condition in the secondary circuit:

𝜔𝑑 =

1

√𝐿2𝐶2

(1.42)

In this case, the design pulsation does not guarantee resonant condition in the primary circuit. As a

consequence, it is needed a value of C1 that compensates the imaginary part of the total impedance.

Neglecting R2 (R2<<RL), for making the calculations easier, the reflected impedance results:

𝑅𝑒(�⃗�𝑟) ≈

𝑀2𝑅𝐿

𝐿2

(1.43)


__________________________________________________________________________________ 11

𝐼𝑚(�⃗�𝑟) ≈

𝜔𝑑𝑀2

𝐿2

(1.44)

From:

(𝜔𝑑𝐿1 −

1

𝜔𝑑𝐶1) = (

𝐿1

√𝐿2𝐶2

−√𝐿2𝐶2

𝐶1) =

𝑀2

√𝐿2𝐶2 𝐿2

(1.45)

The value of C1 results:

𝐶1 =

𝐿22 𝐶2

𝐿1𝐿2 − 𝑀2

(1.46)

While, as said above, the value of C2 is:

𝐶2 =

1

𝜔𝑑2 𝐿2

(1.47)

Analysing the efficiency of the system:

𝜂 =

𝑅𝐿𝐼22


2 + 𝑅1𝐼12

(1.48)

knowing that:

𝐼𝑠 = 𝐼2(1 + 𝑗𝜔𝑑𝐶2𝑅𝐿) (1.49)

𝐼𝑠 = 𝐼2√1 + 𝑅𝐿

2𝐶22𝜔𝑑

2 (1.50)

𝐼1

𝐼2=

√𝑅22 + (𝐿2𝜔𝑑 + 𝑅2𝑅𝐿𝐶2𝜔𝑑)2

𝜔𝑑𝑀

(1.51)

Moreover, expressing C2 with the equation (1.47) and considering the ratio R2/ωd≈0, the efficiency result

𝜂 =

𝑅𝐿

𝑅𝐿 + 𝑅2 +𝑅1𝐿2

𝑀2 +𝑅2𝑅𝐿

2

𝜔𝑑2 𝐿2

2 +𝑅1𝑅2

2

𝜔𝑑2𝑀2

(1.52)


__________________________________________________________________________________ 12

1.4.3 PS Compensation

In this topology, the capacitor in the primary circuit is in parallel with the inductor, while the capacitor

in the secondary circuit is in series with the inductor. The equivalent circuit is shown the Figure 1.7.

The equations of the circuit are:

�⃗⃗�1 = (𝑅1 + 𝑗𝜔𝐿1) − 𝑗𝜔𝑀𝐼2 (1.53)

0 = [𝑅2 + 𝑅𝐿 + 𝑗 (𝜔𝐿2 −

1

𝜔𝐶2)] 𝐼2 − 𝑗𝜔𝑀𝐼𝑝

(1.54)

𝐼1 = 𝐼𝐶1 + 𝐼𝑝 (1.55)

Expressing the current of the secondary circuit as a function of 𝐼𝑝:

𝐼2 =

𝑗𝜔𝑀

(𝑅2 + 𝑅𝐿) + 𝑗 (𝜔𝐿2 −1

𝜔𝐶2)

𝐼𝑝 (1.56)

The equation (1.53) becomes:

�⃗⃗�1 = [(𝑅1 + 𝑗𝜔𝐿1) + (𝜔2𝑀2

(𝑅2 + 𝑅𝐿) + 𝑗 (𝜔𝐿2 −1

𝜔𝐶2)

)] 𝐼𝑝 = [�⃗�1 + �⃗�𝑟]𝐼𝑝

(1.57)

�⃗⃗�1 = [(𝑅1 + 𝑗𝜔𝐿1) + (𝜔2𝑀2

(𝑅2 + 𝑅𝐿) + 𝑗 (𝜔𝐿2 −1

𝜔𝐶2)

)] (𝐼1 − 𝑗𝜔𝐶1 �⃗⃗�1)

(1.58)

FIGURE 1.7: EQUIVALENT CIRCUIT OF ICPT SYSTEM IN PS CONFIGURATION [12]


__________________________________________________________________________________ 13

�⃗�𝑇 =�⃗⃗�1

𝐼1

=(𝑅1 +

𝜔𝑑2𝑀2

𝑅2 + 𝑅𝐿) + 𝑗𝜔𝑑𝐿1

1 + 𝑗𝜔𝑑𝐶1 [𝑅1 +𝜔𝑑

2 𝑀2

𝑅2 + 𝑅𝐿]

(1.59)

The equation (1.57) reflects the circuit in the Figure 1.8.

The receiving circuit has always to work in resonant condition, as a consequence, the designed frequency

results:

𝜔𝑑 =

1

√𝐿2𝐶2

(1.60)

However, the imaginary part of total impedance seen from the voltage generator in the primary circuit is

not null working at ωd. Thus, it is needed a value of C1 that compensates the imaginary part of the total

impedance.

𝐼𝑚(�⃗�𝑇) =𝐿1𝜔𝑑(1 − 𝐿1𝐶1𝜔𝑑

2) − 𝜔𝑑𝐶1 (𝑅1 +𝜔𝑑

2𝑀2

𝑅2 + 𝑅𝐿)

2

(1 − 𝐿1𝐶1𝜔𝑑2) + 𝜔𝑑

2𝐶12 (𝑅1 +

𝜔𝑑2 𝑀2

𝑅2 + 𝑅𝐿)

2 = 0

(1.61)

The equation (1.61) is verified for:

𝐶1 =

𝐿2𝐶2

𝐿1 +𝑀4

𝐿1𝐿2𝐶2𝑅𝐿2

(1.62)

While, C2 results:

𝐶2 =

1

𝜔𝑑2 𝐿2

(1.63)

Focusing on the efficiency of the system, it results:

FIGURE 1.8: EQUIVALENT CIRCUIT OF THE PS TOPOLOGY SEEN FROM THE PRIMARY CIRCUIT


__________________________________________________________________________________ 14

𝜂 =

𝑅𝐿𝐼22


2 + 𝑅1𝐼𝑝2 =

𝑅𝐿

𝑅𝐿 + 𝑅2 + 𝑅1

𝐼𝑝2

𝐼22

(1.64)

Knowing that:

𝐼𝑝

𝐼2=

√(𝑅2 + 𝑅𝐿)2 + (𝜔𝑑𝐿2 −1

𝐶2𝜔𝑑)

2

𝜔𝑑𝑀=

𝑅2 + 𝑅𝐿

𝜔𝑑𝑀

(1.65)

Remembering that we are working in resonant conditions, the efficiency results:

𝜂 =

𝑅𝐿

(𝑅𝐿 + 𝑅2) (1 + 𝑅1(𝑅2 + 𝑅𝐿)

𝜔𝑑2 𝑀2 )

(1.66)

1.4.4 PP Compensation

In this topology, the capacitors are placed in parallel with the inductance of the primary and of the

secondary circuit. The equivalent circuit is shown in the Figure 1.9.

The equations of the systems are:

�⃗⃗�1 = (𝑅1 + 𝑗𝜔𝐿1)𝐼𝑝 − 𝑗𝜔𝑀𝐼𝑠 (1.66)

0 = (𝑅2 + 𝑗𝜔𝐿2)𝐼𝑠 + 𝑅𝐿𝐼2 − 𝑗𝜔𝑀𝐼𝑝 (1.67)

𝐼𝑠 = 𝐼2 + 𝐼𝐶2 (1.68)

𝐼1 = 𝐼𝐶1 + 𝐼𝑝 (1.69)

FIGURE 1.9: EQUIVALENT CIRCUIT OF ICPT SYSTEM IN PP CONFIGURATION [12]


__________________________________________________________________________________ 15

The relation between the currents 𝐼𝑠 and 𝐼𝑝 results:

𝐼𝑠 =

𝑗𝜔𝑀(1 + 𝑗𝜔𝑅𝐿𝐶2)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝑅𝐿𝐶2)𝐼𝑝

(1.70)

Replacing the equation (1.70) in the (1.66) we obtain:

�⃗⃗�1 = (𝑅1 + 𝑗𝜔𝐿1)𝐼𝑝 +

𝜔2𝑀2(1 + 𝑗𝜔𝑅𝐿𝐶2)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝑅𝐿𝐶2)𝐼𝑝 = (�⃗�1 + �⃗�𝑟)𝐼𝑝

(1.71)

being:

�⃗�𝑟 =

𝜔2𝑀2(1 + 𝑗𝜔𝑅𝐿𝐶2)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝑅𝐿𝐶2)

(1.72)


𝑅1 = 𝑅𝑒(�⃗�𝑟) =

𝑀2(𝑅𝐿 + 𝑅2) + 𝜔2𝑀2𝐶22𝑅𝐿

2𝑅2

(𝐿2 + 𝐶2𝑅2𝑅𝐿)2

(1.73)

(𝜔𝐿1 −

1

𝜔𝐶1) = −𝐼𝑚(�⃗�𝑟) =

𝜔3𝑀2[𝐶2𝑅𝐿2(𝜔2𝐿2𝐶2 − 1) + 𝐿2]

(𝑅2 + 𝑅𝐿 − 𝜔2𝐿2𝐶2𝑅𝐿)2 + 𝜔2(𝐿2 + 𝐶2𝑅2𝑅𝐿)2

(1.74)

Considering to work in resonant conditions at the secondary circuit:

𝜔𝑑 =

1

√𝐿2𝐶2

(1.75)

Working with this pulsation, the imaginary part of �⃗�𝑇, the total impedance seen from the primary circuit,

is different from zero. It means that the capacitor C1 has to compensate it.

Expressing �⃗⃗�1 as a function of 𝐼1:

𝐼𝑝 = 𝐼1 − �⃗⃗�1𝑖𝜔𝐶1 (1.76)

�⃗⃗�1 = (�⃗�1 + �⃗�𝑟)𝐼𝑝 = (�⃗�1 + �⃗�𝑟)(𝐼1 − �⃗⃗�1𝜔𝐶1) (1.77)

�⃗⃗�1 (1 + 𝑗𝜔𝐶1(�⃗�1 + �⃗�𝑟)) = (�⃗�1 + �⃗�𝑟)𝐼1 (1.78)

It is possible to obtain the total impedance as:


__________________________________________________________________________________ 16

�⃗�𝑇 =

�⃗⃗�1

𝐼1

=(�⃗�1 + �⃗�𝑟)

(1 + 𝑗𝜔𝐶1(�⃗�1 + �⃗�𝑟))=

=

(𝑅1 + 𝑗𝜔𝑑𝐿1) +𝜔2𝑀2(1 + 𝑗𝜔𝑑𝐶2𝑅𝐿)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝑑𝐿2)(1 + 𝑗𝜔𝑑𝑅𝐿𝐶2)

1 + 𝑗𝜔𝑑𝐶1 [(𝑅1 + 𝑗𝜔𝑑𝐿1) +𝜔2𝑀2(1 + 𝑗𝜔𝑑𝐶2𝑅𝐿)

𝑅𝐿 + (𝑅2 + 𝑗𝜔𝑑𝐿2)(1 + 𝑗𝜔𝑑𝐶2𝑅𝐿)]

(1.79)

Solving the following equation:

𝐼𝑚(�⃗�𝑇) = 0 (1.80)

The value of the capacitance C1 results:

𝐶1 =

(𝐿1𝐿2 − 𝑀2)𝐿22 𝐶2

𝑀4𝑅𝐿2𝐶2

𝐿2+ (𝐿1𝐿2 − 𝑀2)2

(1.81)

While, the value of the capacitance in the secondary circuit results:

𝐶2 =

1

𝜔𝑑2 𝐿2

(1.82)

For what concern the efficiency, it results:

𝜂 =

𝑅𝐿𝐼22

𝑅𝐿𝐼22 + 𝑅1𝐼𝑝

2 + 𝑅2𝐼𝑠2 =

𝑅𝐿

𝑅𝐿 + 𝑅1

𝐼𝑝2

𝐼22 + 𝑅2

𝐼𝑠2

𝐼22

(1.83)

Knowing that:

𝐼𝑠

𝐼2= √1 + 𝑅𝐿

2𝐶22𝜔𝑑

2 (1.84)

𝐼𝑝

𝐼2=

√𝑅22 + (𝐿2𝜔𝑑 + 𝑅𝐿𝑅2𝐶2𝜔𝑑)2

𝜔𝑑𝑀

(1.85)

Moreover, replacing the equation (1.82) and considering the ratio R2/ωd ≈ 0, the equation (1.83) results:

𝜂 =

𝑅𝐿

𝑅𝐿 + 𝑅2 +𝑅1𝐿2

𝑀2 +𝑅2𝑅𝐿

2

𝜔𝑑2𝐿2

2 +𝑅1𝑅2

2

𝜔𝑑2 𝑀2

(1.86)


__________________________________________________________________________________ 17

1.4.5 Nominal Parameters

Once the design frequency is defined and the values of the capacitors are evaluated, the next step is

calculating the parameters of the system. They depend on the type of topology applied.

To sum up, the total impedance seen from the primary circuit in the different topologies result:

�⃗�𝑇_𝑆𝑆 = (𝑅1 + 𝑗 (𝜔𝐿1 −

1

𝜔𝐶1)) +

𝜔2𝑀2

(𝑅2 + 𝑅𝐿 + 𝑗 (𝜔𝐿2 −1

𝜔𝐶2))

(1.87)

�⃗�𝑇_𝑆𝑃 = (𝑅1 + 𝑗 (𝜔𝐿1 −

1

𝜔𝐶1)) +

𝜔2𝑀2

(𝑅2 + 𝑗𝜔𝐿2 +𝑅𝐿

1 + 𝑗𝜔𝐶2𝑅𝐿)

(1.88)

�⃗�𝑇_𝑃𝑆 =

1

(𝑅1 + 𝑗𝜔𝐿1) +𝜔2𝑀2

(𝑅2 + 𝑅𝐿 + 𝑗 (𝜔𝐿2 −1

𝜔𝐶2))

+ 𝑗𝜔𝐶2

(1.89)

�⃗�𝑇_𝑃𝑃 =

1

1

(𝑅1 + 𝑗𝜔𝐿1) +𝜔2𝑀2(1 + 𝑗𝜔𝐶2𝑅𝐿)

(𝑅𝐿 + (𝑅2 + 𝑗𝜔𝐿2)(1 + 𝑗𝜔𝐶2𝑅𝐿))

+ 𝑗𝜔𝐶1

(1.90)

Considering these values for the total impedances, the nominal values of the circuit parameters are shown

in the table 1.1.


__________________________________________________________________________________ 18

TABLE 1.1: NOMINAL PARAMETERS OF THE EQUIVALENT CIRCUIT IN THE DIFFERENT TOPOLOGIES

1.5 STABILITY

The design of the ICPT system is made considering resonant conditions thanks to the support of the

tuning capacitors. In this way, the system is seen from the grid as a purely ohmic device that works at

the maximum power transfer condition. The nominal parameters, shown in the table 1.1, are defined

considering the design pulsation ωd. However, the pulsation can vary.

The stability analysis studies the response of the system in case of frequency variations. The response

principally involves the electrical parameters: they can either remain equal or lower than the nominal


__________________________________________________________________________________ 19

value (stable condition) or they can assume higher values (unstable condition). The latter case is not

wanted, it can cause damages of the electrical components and dangerous working conditions.

The system’s stability depends on the quality factor of the primary and secondary circuit and,

consequently, on the type of topology applied. The q factor is defined as the ratio between the power

stored in a circuit and the power lost from it. In this work, it is evaluated considering the resistance of

the load.

For the SS and PS compensation, where there is not imaginary part in the reflected impedance, the quality

factors of the primary and of the secondary circuit, neglecting R1 and R2 with respect to RL, result:

𝑞𝑃 =

𝜔𝑑𝐿1𝐼12

𝜔𝑑2 𝑀2

𝑅𝐿𝐼1

2

=𝑅𝐿𝐿1

𝜔𝑑𝑀2

(1.91)

𝑞𝑆 =

𝜔𝑑𝐿2𝐼22

𝑅𝐿𝐼22 =

𝜔𝑑𝐿2

𝑅𝐿=

1

𝜔𝑑𝐶2𝑅𝐿

(1.92)

While, the quality factors for SP and PP topology result:

𝑞𝑃 =

𝜔𝑑𝐿1𝐼12

𝑀2𝑅𝐿

𝐿22 𝐼1

2=

𝐿22 𝐿1𝜔𝑑

𝑀2𝑅𝐿

(1.93)

𝑞𝑆 =

𝐿2𝜔𝑑𝐼𝑠2

𝑅𝐿𝐼22 =

𝐿2𝜔𝑑(1 + 𝑅𝐿2𝐶2

2𝜔𝑑2)

𝑅𝐿≈

𝐿2𝜔𝑑3𝑅𝐿

2𝐶22

𝑅𝐿=

𝑅𝐿

𝐿2𝜔𝑑

(1.94)

The stability analysis is performed focusing on the imaginary part of the total impedance. For a proper

working and for guaranteeing the stability of the system, the equation (1.95) has to present only one real

solution that coincide with the design frequency.

𝐼𝑚 (�⃗�𝑇(𝜔)) = 0 (1.95)

However, sometimes the imaginary part of the total impedance can present more solutions; this

phenomenon is called Bifurcation [36], this make the system unstable.

In order to guarantee the system stability, avoiding the Bifurcation, we should analytically solve the

equation (1.95) and analyse for which condition it present only one resonance frequency. It depends on

the relation between the quality factor of the primary and secondary circuit. Considering the four

topologies, the relations are expressed in the table 1.2.


__________________________________________________________________________________ 20

SS SP PS PP

𝑞𝑝 >4 𝑞𝑠

3

4 𝑞𝑠2 − 1

𝑞𝑝 > 𝑞𝑠 +1

𝑞𝑠 𝑞𝑝 > 𝑞𝑠 𝑞𝑝 > 𝑞𝑠 +

1

𝑞𝑠

Respecting these relations, considering the specific type of compensation, the system is stable. In general,

it is possible to state that for the stability we need that qp >> qs; this condition allows the proper working

of the device even though it is not the best option from an economical point of view.

In the Figure 1.10, it is possible to see an example that show the responses of the electrical parameter as

a function of the frequency variation in stable and unstable conditions.

TABLE 1.1.2: STABILITY CONDITIONS FOR THE FOUR TOPOLOGIES

FIGURE 1.10: RESPONSE OF THE ELECTRICAL PARAMETERS WITH FREQUENCY VARIATION IN SS

TOPOLOGY [12]


__________________________________________________________________________________ 21

The system analysed present the same SS topology and the same design frequency ω0. It is easy to see

the Bifurcation phenomenon in the case of unstable condition, where the parameters assume higher value

than the nominal one.

1.6 CLASSIFICATION OF ICPT SYSTEM

WPT systems can be classified depending on the type of supply system and on the coupling between

transmitting and receiving coil.

Depending on the supply system they can be classified in:

• Dynamic charge

• Fixed charge

While, depending on the coupling they can be classified in:

• Strongly coupled systems

• Chained ring systems

• Loosely coupled systems:

o Tangential flux

o Normal flux

The second type of classification is related to the coupling factor K; it is defined as:

𝐾 =

𝑀

√𝐿1𝐿2

(1.96)

The greater is K the greater is the coupling and so the performances of the system.

1.6.1 Dynamic charge systems

This type of ICPT system is made by a moving secondary coil (pick-up) that captures the magnetic flow

from a primary coil (track). This technology is used for charging batteries in moving applications

(automotive) or for directly providing electrical supply (i.e. trains).

Two different types of dynamic charge exist:


__________________________________________________________________________________ 22

• Distributed, it is present only one primary coil that presents dimensions greater than the

secondary (Figure 1.11)

• Concentrated, more coils are placed in the path of the secondary coil with comparable dimension

(Figure 1.12)

FIGURE 1.12: SCHEME OF DYNAMIC CONCENTRATED ICPT SYSTEM [4]

1.6.2 Fixed charge systems

This type of system presents two coupled coils that when they interact are fixed in the space. The

dimensions of the coils are always comparable, and they can manage a higher air gap with respect to the

dynamic charging. They are principally used for batteries charging in several type of devices.

FIGURE 1.11: SCHEME OF DYNAMIC DISTRIBUTED ICPT SYSTEM [4]


__________________________________________________________________________________ 23

1.6.3 Strongly coupled systems

In the strongly coupled ICPT systems (Figure 1.13) the two coils, separated by an almost negligible air

gap, present a ferromagnetic core performing a high coupling coefficient. Thanks to the working

conditions, the receiver is able to capture almost the flux generated from the primary coil.

FIGURE 1.13: EXAMPLE OF STRONGLY COUPLED ICPT SYSTEM

Even though this technology presents a high coupling coefficient, in addition, it requires mechanical

coupling that in some cases it could make senseless the use the ICPT system.

1.6.4 Chained ring systems

The “chained ring” is a type of ICPT system where the coupled coils are separated with a small airgap.

Usually, the secondary coil presents a ferromagnetic core, while, the primary either presents an air-core

(short-track) or it is formed by only one long turn (long-track); the difference between these two

configurations is showed in Figure 1.14. The principal limit of this type of system is the low capacity of

lateral movement.


__________________________________________________________________________________ 24

FIGURE 1.14: CHAINED RING STRUCTURE: DIFFERENCE BETWEEN SHORTTRACK AND LONGTRACK

This configuration in principally used in application of moving elements on tracks, such as local public

transports, trains, Magnetic Levitation [53,54].

1.6.5 Loosely coupled systems

These systems are usually formed by two coupled coils with comparable dimensions. Even though a

loosely coupled system presents a low coupling factor, they present the advantage of having more

freedom in terms of misalignment and distance between the coils.

The magnetic field created by the primary circuit is circular and depending on the component that links

the secondary circuit, it is possible to define two further type of ICPT systems:

• Tangential component capture system (Figure 1.15)

• Normal component capture system (Figure 1.15)


__________________________________________________________________________________ 25

FIGURE 1.15: NORMAL AND TANGENTIAL COMPONENT CAPTURE SYSTEM [12]

The tangential component capture system presents a low mutual inductance value (M), as a consequence,

it is usually used a ferromagnetic core in order to increase it. Moreover, these systems are not suitable in

case of misalignment due to a sharp decrease of performances. The highest performances are achieved

when the receiving coil is placed exactly above one conductor of the transmitter coil.

On the other hand, the normal component capture systems present higher values of M. Moreover, they

present a better response in case of misalignment, for this reason they are the most applied in automotive

applications.


__________________________________________________________________________________ 26

1.7 ICPT OVERALL STRUCTURE

Considering an ICPT system used for batteries charging, the AC nature of the inductive coupling goes

against the DC working of the battery. Because of this, the system has to be managed applying inverters

and converters. The overall system is shown in the Figure 1.16.

FIGURE 1.16: OVERALL STRUCTURE OF ICPT SYSTEM

On the transmitter side, the most common topology presents a cascade of AC-DC rectifier and a DC-AC

inverter. In the middle of these, it is present a boost converter in order to manage the voltage. This step

allows to pass from the grid voltage and frequency (50 Hz) to the required voltage and the design

frequency.

The secondary circuit needs to rectify the AC current in order to supply the battery. Schematically, the

rectifier and the battery are included in the load. Consequently, the value of the load is not only the

resistance of the battery, but it results:

𝑅𝐿 =

8

𝜋2𝑅𝐵𝐴𝑇𝑇

(1.97)

Depending on the case and on the technology used, further types of alimentation system can be applied.

They depend on the inverters and rectifiers used, on the topology, etc.


__________________________________________________________________________________ 27

1.8 GEOMETRICAL DESIGN

For what concern the physical design of the ICPT system, the first parameter to take into account is the

geometry of the coupled coils.

The transferred power is expressed as:

𝑃2𝑀𝐴𝑋 = 𝜔𝑑

𝑀2

𝐿2𝐼𝑝

2𝑞𝑠 (1.98)

It is possible to observe that the maximum transferred power depends on the mutual inductance

coefficient and it is inversely proportional to the mutual inductance of the receiving coil. This relation

between the two coefficients can be seen in the coupling factor:

𝐾 =

𝑀

√𝐿1𝐿2

(1.99)

This factor, that is proportional to the performance of the power transfer, increase for high values of M

and for low values of the self-inductances of the two circuits.

The self and mutual inductances are strictly related to the physical features of the primary and secondary

coils, such as geometry, cable disposition, length, number of turns, type of core and air gap.

The analytical evaluation of the self and mutual inductances could be performed applying already known

formulas. However, when the geometry becomes more complex, it is needed the use of a proper software,

such as FEMM (Finite Element Method Magnetics) or Ansys Maxwell. These evaluate the required

parameters of the ICPT system applying a FEM (finite element method) analysis.

Several researches have analysed several types of geometries for the ICPT system [7,10,11,12] finding

out which are the best conditions for achieving high performances.

The previous researches have focused basically on three types of geometries: rectangular, squared and

circular (see Figure 1.17).


__________________________________________________________________________________ 28

FIGURE 1.17: GEOMETRIES ANALYSED [12]

Moreover, it has been analysed the geometries considering different types of dispositions of the cables:

• Horizontal (Figure 1.18(a))

• In row (Figure 1.18(b))

• Vertical (Figure 1.18(c))

• Circular (Figure 1.18(d))

FIGURE 1.18: DIFFERENT TYPES OF CABLE DISPOSITION

From the analysis and the simulations realized, considering the same cable features and air gap for the

different geometries, it has found out that the horizontal disposition presents the highest mutual

inductance coefficient and the lowest for the self-inductance. This means that it presents the highest

coupling coefficient. Moreover, the circular geometry presents the highest mutual induction coefficient,

presenting a 20% more than the squared geometry and a 50% more than the rectangular. As a

consequence, this presents also the best coupling coefficient. Therefore, the best configuration analysed

is the circular geometry with horizontal disposition.

Further studies have been carried out considering the misalignment. The analysis of the system

behaviour, as a function of this parameter, is important due to the fact that main advantage of the ICPT

technology is the freedom in the placement of the device with respect to the charging station. The


__________________________________________________________________________________ 29

simulations have shown that the circular geometry with horizontal disposition is the best also in this case.

This configuration guarantees better mutual inductance values (see Figure 1.19 and Figure 1.20).

FIGURE 1.19: COUPLING COEFFICIENT OF CIRCULAR AND HORIZONTAL DISPOSITION

FIGURE 1.20: RATIO M2^2/L2^2 OF CIRCULAR AND HORIZONTAL DISPOSITION

__________________________________________________________________________________ 30

_____________________________________________ Chapter 2: ICPT SYSTEM APPLICATIONS

__________________________________________________________________________________ 31

Capitolo 2 : ICPT SYSTEM APPLICATIONS

2.1 INTRODUCTION

Nowadays, the Inductive Coupling Power Transfer is a technology that is going to be introduced in

several fields. It has innovated the classic idea of charging and energy supply. In addition, due to the

always increasing number of loads it can provide support to the electric grid. However, this technology

is still not widely used due to economical disadvantages.

The first attempt of applying inductive power transfer has been performed in 1894 by M. Huntin and M.

Le-Blanc. These proposed this system in order to supply an electric vehicle; however, internal

combustion engines proved more popular. After that, Professor Don Otto, in 1972, proposed a vehicle

powered by induction using transmitters in the road and the receiver on the vehicle. The first applications

of the inductive charging happened in the United States, in the last years of the 70s, applying it to cars

and buses.

The wireless charging can be used in low power applications, it concerns small devices such as handheld

devices, and in high-power applications, usually used for inductive charging of batteries at power level

above 1 kW.

Comparing the ICPT system with a classic wire-equipped charging system, the main advantages are:

• Protected connections, the conductive wires are enclosed, this avoids the corrosion of the cables

and of the terminals.

• Less risk of fault, with this technology the risk of fault due to insulation failure is smaller

• Durability, without the need of constantly plug in and out, the mechanical stress really decreases

• Increase the aesthetic quality

• It can operate automatically without dependence on people to plug and unplug, this improve the

reliability

On the other hand, the disadvantages of the technology are:

• Lower efficiency, the absence of physical connection decreases the efficiency of the system

• Slower charging, due to the lower efficiency, on average it takes the 15% more of time

• Cost, due to higher complexity of system, the design and manufacturing costs increase

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__________________________________________________________________________________ 32

• Inconvenience for mobile devices, usually the ICPT system doesn’t allow movement of the

mobile device during the charging period

• Compatible standards, not all devices are compatible with different inductive chargers; however,

some devices have started to support multiple standards

Even though the disadvantages, it is continuously under study in order to solve these problems and make

it competitive in the market. Moreover, bidirectional ICPT systems are studied in order to provide a

further support to the grid. Thanks to its advantages, this technology can be perfectly integrated in a

Smart Grid.

2.2 HOUSEHOLD APPLIANCES AND MOBILE DEVICES

The application of ICPT system for household appliances and mobile devices is part of the low power

applications. It can be applied in devices with low power batteries, such as mobile phones, electric

toothbrushes, laptops, tablet, etc. The power range for these devices varies from some Watts to tens of

Watts.

The charging station usually utilized for these applications is a platform where the device is left on, this

automatically start charging without any further action of the user; however, this concept decreases the

mobility of the device during the charging period.

The principal difficulty in the diffusion of this WPT systems was the low compatibility between brands

of charging stations producers and devices producers. This was the first step to overcome for the

introduction into the market. For what concerns the mobile phones, in order to solve the problem, the

main companies of the sector (ConvenientPower, Fulton Innovation, Logitech, Motorola, National

Semiconductor Corporation, Royal Philips, Sanyo and SangFei Consumer Communications) recognized

that a standard wireless charging interface was needed. Hence, they founded the Wireless Power

Consortium (WPC) in November 2008. This published the Qi interface specification in August 2010 and

in the first Qi-compliant product was certified in September 2010. Nowadays, more than 200 companies

have joined the WPC.

Different type of configurations of charging stations have been made and they have been applied to any

kind of device. In the Figure 2.1 and Figure 2.2, it is showed the wireless power system applied to a

toothbrush and a drill.


__________________________________________________________________________________ 33

FIGURE 2.1: ICPT SYSTEM IN AN ELECTRIC TOOTHBRUSH FIGURE 2.2: ICPT IN A DRILL[32]

In addition, platforms able to manage more devices at the same time are present (Figure 2.3)

FIGURE 2.3: MULTIPLE CHARGING STATION

Moreover, an interesting application for a wireless mouse is showed in the Figure 2.4. In this case the

wireless charging station has been integrated in the mouse pad.


__________________________________________________________________________________ 34

FIGURE 2.4: ICPT SYSTEM IN MOUSE PAD [33]

Due to the recent development of the wireless power transfer, old devices that are still present in the

market could not be compatible with this type of technology due to the lack of the needed components.

In order to solve this problem, additional components are present in the market; these can make the

device able to be coupled with a wireless charger. An example is shown in the Figure 2.5.

FIGURE 2.5: ADDITIONAL COMPONENT FOR ICPT SYSTEM

2.3 ELECTRIC VEHICLES APPLICATION

Electric vehicles (EV) charging occurs at significantly higher power levels, it ranges from hundreds of

W, such as for E-bike applications, to tens of kW for electric buses. Unfortunately, the Wireless Electric


__________________________________________________________________________________ 35

Vehicle Charging (WEVC) is not completely competitive in the market, as a consequence, it is still far

from the full commercialization and standardization.

For vehicle applications it is possible to recognize three big categories:

• Static wireless charging, EV charging is performed with not moving vehicles and no one inside

• Quasi-dynamic wireless charging, EV charging is performed with not moving vehicles but they

can host persons

• Dynamic wireless charging, EV charging is performed in moving vehicles

The principal vehicles the ICPT system is applied are cars, bicycles and buses.

2.3.1 E-BIKE Applications

Electric bicycles are bikes in which the propulsion of an electric motor is added to the human propulsion.

These light and compact vehicles represent a good solution for the green and sustainable mobility in

cities. However, as well as all the electric means of transport, they have to cope with the charging. The

ICPT system could be considered a solution for the charging system.

Several configurations of charging station and device have been studied. Firstly, the Figure 2.6 shows

the configuration proposed in [55]. In this, the receiving coils is present on the frontal part of the basket

of the bike, while the transmitter is fixed to the wall at the same height of the receiver. This type of

charging system is easy to install, and it could be also introduced as additional component of the E-bike.

FIGURE 2.6: CHARGING SYSTEM PROPOSED IN [53]


__________________________________________________________________________________ 36

Another type of configuration analysed is shown in Figure 2.7.

FIGURE 2.7: CONFIGURATION PROPOSED IN [54]

This charging system, studied in [54], present the secondary coil under the saddle of the bike. Due to the

position of the receiver, the mechanical coupling between the coils results a bit more complex but it can

be handled with a movable transmitter placed in the charging tower station.

A further category of charging system analysed is shown Figure 2.8. This presents the receiver in the

kickstand of the bicycle and the transmitter buried in the soil.


Lastly, as shown in Figure 2.9, it has been analysed a configuration where the receiver is placed in the

fork of the bike and the transmitter in the charging tower [3].


__________________________________________________________________________________ 37


In this thesis, the configuration proposed is shown in Figure 3.26. The reason of the choice is explained

in the paragraph 3.3.5.

2.3.2 Automotive Applications

With the increasing electric cars production, due to the need of more sustainable transport systems, the

manufacturers are working in more innovative charging system, such as ICPT. This allows the consumer

to charge the car only parking it, avoiding risks of faults during the charge and need of physical

connection. The principal configuration used is shown in Figure 2.10. It presents the receiving coil placed

in the chassis of the car and the transmitter in the road, this could be buried or placed in a case. The

coupling system in a car is more difficult to manage due to bigger dimensions and to the lower movement

flexibility; the main consequence of this is a greater air gap. The position of the receiving guarantees the

lowest air gap possible in this type of application.

FIGURE 2.10: COMMON CONFIGURATION OF ICPT SYSTEM IN CAR APPLICATIONS


__________________________________________________________________________________ 38

The main companies that are interested wireless charging systems are Toyota, Nissan, General Motors,

Ford and BMW. In particular, BMW launched a mobile charging station [21] that can be easily connected

to the grid and placed everywhere (Figure 2.11); it is a case containing the primary coil. Moreover, the

car presents a camera system in order to permit the best alignment between the receiver and the

transmitter (Figure 2.12).

FIGURE 2.11: BMW TRANSMITTING CASE

FIGURE 2.12: CAMERA SYSTEM

A further example is the charging system proposed by HaloIPT, it is one of the leader company in ICPT

system for EVs (Figure 2.13).


__________________________________________________________________________________ 39

FIGURE 2.13: HALOIPT CHARGING SYSTEM

Dynamic wireless charging is the most interesting technology for automotive application. This is formed

by a series of transmitters buried in the road that induct a current in the receiver when it is close to them

(Figure 2.14). Thus, the electric car is continuously fed and, consequently, it can be adopted a smaller

battery in the car, decreasing the costs and the space occupied by it. More advanced systems are able to

switch on the transmitter only when the receiver is close to it.

FIGURE 2.14: DYNAMIC WIRELESS CHARGING

This system could solve the charging problems of the electric cars, avoiding long charging periods.

However, it is more compatible with vehicles that move on a well-defined path, for example buses for

public transport. In this field, the Korea Advanced Institute of Science and Technology (KAIST)


__________________________________________________________________________________ 40

proposed the OLEV (Online Electric Vehicle) project [27]. It is an electric bus inductively fed by

transmitters built-in the road.

A further application of dynamic charge can be in highways (Figure 2.15). This can solve the

disadvantages of the electric car in the long distance movements, allowing them long journey without

the need of any charging stop.

FIGURE 2.15: CONCEPT OF DYNAMIC WIRELESS CHARGING IN HIGHWAYS [25]

_________________________________________________ Chapter 3: COIL GEOMETRY DESIGN

__________________________________________________________________________________ 41

Capitolo 3 : COIL GEOMETRY DESIGN

3.1 INTRODUCTION

The aim of this thesis is designing an ICPT system for E-bike applications, in particular, the work has

focalised on the coil’s design. As said in the previous chapter, this step presents high relevance because

on it, all the coupling parameters depend. Basing on the previous researches, it has been decided to

principally handle circular coils. They present the highest coupling coefficient in normal condition and

in case of misalignment.

The design part has been carried out considering all the constrains that can be related to E-bike

applications. This has been made in terms of dimensions and air gap. In fact, the biggest geometry

analysed is a circular geometry with an outer diameter of 19 cm that can be considered a suitable

dimension for a bike. For what concerns the airgap, the interval considered is included between 1 cm

and 3 cm. A bicycle, with respect to the other types of vehicles, has the possibility to be mechanically

managed in an easier way, this allows to analyse smaller airgaps.

The software used for modelling the coils is Ansys Maxwell. The results of the simulations give us the

resistances of the circuits, the self and mutual inductances and the coupling coefficient K. In particular,

the configurations analysed are:

• 2 circular coils with 9 turns



• 2 circular coils one with 9 turns and 7 turns

• 2 vertical coils

These configurations have been analysed considering variations in frequency, airgap and misalignment

in order to study the response of the system.

The aim of the simulations is finding out which is the best configuration to apply in E-bike field in terms

of power transfer performances.

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Chapter


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3.2 ANSYS MAXWELL

The evaluation of the inductances of a coupled system is a complex matter to cope with. The analytic

evaluation is possible; however, it requests time and complex calculations. In order to solve this problem,

proper software is used. The software that has been used in this work is Ansys Maxwell. It is one of the

best simulation software in this field offering an intuitive and easy interface. One great advantage of this

software is the capability of creating a proper mesh on its own, without any need of the user.

Ansys Maxwell is the industry-leading electromagnetic field simulation software for the design and

analysis of electric motors, actuators, sensors, transformers and other electromagnetic and

electromechanical devises. It uses finite element analysis (FEA) to solve three-dimensional (3D) electric,

magnetostatic, eddy current and transient problems.

The Eddy current solver is that one used in the simulations. It computes steady-state, time-varying (AC)

magnetic fields at a given frequency. This is a frequency domain solution and assumes frequency of

pulsating fields to be same throughout the domain. Moreover, it utilizes adaptive mesh refinement

technique to achieve best mesh required to meet defined accuracy level. The solution process of the

solver is automated as shown in the Figure 3.1.


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FIGURE 3.1: SOLUTION PROCESS OF EDDY CURRENT SOLVER

3.3 SIMULATIONS

3.3.1 Two Circular Coils with 9 turns

The first case analysed is a circular geometry configuration with 9 turns. This choice has been made in

order to state the reliability of the results. It can be considered a test. The model presents the same

specifics of a model previously analysed in another work [11]; then the results have been compared.

The specifics of the configuration used are:

• Inner radius: 4.4 cm

• Wire radius: 1.5 mm

• Number of turns of coils: 9

• Distance center-center: 4.5 mm

• Material: copper

• Frequency: 85 kHz


__________________________________________________________________________________ 44

In order to run this model, the first step is defining, in Ansys Electronic Desktop, the type of project that

it is going to be used. The type chosen is Maxwell 3D Design (see Figure 3.2)

FIGURE 3.2: COMMAND FOR SELECTING THE DESIRED PROJECT

Once the project is defined, the next step is setup the type of solver. As we said, the type of solver used

is the Eddy current. For selecting it, the procedure is shown in the Figure 3.3 and Figure 3.4.

FIGURE 3.3: PROCEDURE FOR SELECTING THE SOLVER


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FIGURE 3.4: PROCEDURE FOR SETTING THE SOLVER TYPE

Now the project and the solver are done, the next step consists in creating the coil geometry. The software

already presents some default geometries, the circular geometry is present in this library. It is selected as

shown in the Figure 3.5.

FIGURE 3.5: PROCEDURE FOR DEFINING THE GEOMETRY


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After that, we need only to insert the properties of the coils (Figure 3.6).

FIGURE 3.6: DEFINITION OF GEOMETRY PROPERTIES

For creating the second coil, the procedure is the same. However, they are created exactly in the same

position, in order to move one of them in the space the command “move” is used. Using it, it is possible

to define the airgap and the misalignment. To conclude the physical coil design part, the material is

specified. The material used in these simulations is always copper (Figure 3.7).


__________________________________________________________________________________ 47

FIGURE 3.7: MATERIAL DEFINITION

The software creates exclusively the coils, as a consequence of this, it is needed to manually create the

terminals for the supply.

The next step consists in creating the region where the coils operate. This is done by drawing a “box”

and then selecting the command “create region” (Figure 3.8). It automatically creates the region on the

box previously created. It is convenient to design a region with dimensions bigger enough, with respect

to the coils, for guaranteeing a proper simulation process.

FIGURE 3.8: COMMANDS FOR CREATING THE REGION


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In order to run the model, it is important that the ends of the terminals coincide wuth the surface of the

region; whether it is not respected, it is not possible to launch the simulation.

At this point, the geometry setup is concluded, the final result is shown in the Figure 3.9 and Figure 3.10.

All the following steps are related to the alimentation of the system and to the setup of the analysis.

FIGURE 3.9: PHYSICAL CONFIGURATION OF COILS

FIGURE 3.10: PHYSICAL CONFIGURATION OF THE COILS TOP VIEW (LEFT) AND TOP VIEW (RIGHT)


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The excitation is assigned using the right click on the section of the terminals (Figure 3.11 and Figure

3.12). The same circuit always need an input current and an outlet current (Figure 3.13).

FIGURE 3.12: PROCEDURE FOR DEFINING

THE EXCITATION

FIGURE 3.11: DEFINITION OF THE CURRENT

PARAMETERS


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FIGURE 3.13: EXCITATIONS OF THE MODEL

Now all the physical and electrical features are defined. Before passing to the analysis we have to assign

the matrix to the parameters, where the sources are the input currents (Figure 3.14).

FIGURE 3.14: PROCEDURE FOR ASSIGNING THE MATRIX

Lastly, the solution step has to be defined (Figure 3.15). In this, all the features for the analysis are stated.


__________________________________________________________________________________ 51

FIGURE 3.15: PROCEDURE FOR ADDING THE SOLUTION STEP

Usually, the defaults properties are suitable for running the model. However, depending on the features

of the calculator they may need some changes. Whether the computer is not enough powerful, the

simulation could take too much time (several hours) or it could crash. In the present case, the features of

the analysis have been changed. The new values adopted are:

• General -> Adaptive Pass -> Maximum number of passes: 7

• Convergence -> Standard -> Refinement per Pass: 15%

Moreover, in order to make the simulation lighter, the properties of the mesh have been changed; the

procedure is showed in Figure 3.16.


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FIGURE 3.16: MESH SETTINGS PROCEDURE

FIGURE 3.17: MESH

The Figure 3.17 shows the final mesh on the coil.


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The curved surface meshing resolution has been set at 3 over 10, considering the default value equal to

5. All these devices allow a faster and lighter simulation, in terms of memory and CPU, still maintaining

a good level of accuracy of the results. The same specifics are applied in all the further simulations.

To conclude, the last step before running the model consists in setting the working frequency at 85 kHz

(Figure 3.18).

FIGURE 3.18: FREQUENCY SETTING

Now the model is ready to be launched and after tens of minutes the results are carried out (table 3.1).

The values extracted from the software are the self-inductances of the primary and secondary circuit (L1,

L2), the mutual inductance (M) and the resistances of the circuits (R1, R2). Since the two coils present

the same physical features, it is possible to suppose equal values of the two resistances and considering

only the parameter R (R=R1=R2).

TABLE 3.1: RESULTS OF "2 CIRCULAR COILS WITH 9 TURNS" MODEL

Airgap [cm] L1 [µH] L2 [µH] M [µH] K R [mΩ]

3 13.174 13.1844 5.291 0.391 47.327


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Comparing these results with the results showed in the table 3.2 [11], it is possible to confirm the

reliability of the simulation.

TABLE 3.2: RESULTS OF THE REFERENCE CASE

The following figure (Figure 3.19) shows the distribution of the magnetic field on a plane between the

two coils.

FIGURE 3.19: BFIELD CIRCULAR CONFIGURATION

Once the results of the simulation have been confirmed, it has been possible carrying out other

simulations considering the variations of some parameters.

Firstly, the attention has focused on the case of a thinner airgap. The airgap considered is 2 cm. The

results are shown in the table 3.3.

TABLE 3.3: RESULTS OF "2 CIRCULAR COILS WITH 9 TURNS" MODEL WITH AIRGAP=2 CM

It is easy to see that in this case it is present an increase of the mutual inductance value, it is due to the

thinner distance between the coils. On the other hand, the values of the self-inductances remain almost

the same, the variations can be related to the simulation process. As a consequence, remembering the

Airgap [cm] L1 [µH] L2 [µH] M [µH] K

3 13 13 4.9 0.377

Airgap [cm] L1 [µH] L2 [µH] M [µH] K

2 13.153 13.150 6.954 0.529


__________________________________________________________________________________ 55

equation (1.99), this configuration guarantees a higher coupling coefficient, increasing the performances

of the system.

The next simulation performed concerns the frequency variation. It has been performed considering the

previous circular geometry with a constant airgap of 3 cm. The frequency interval taken into account is

from 60 kHz to 90 kHz. In order to make this simulation easier and faster to perform, it was used the

optimetrics solution analysis where the software automatically changes the required variable.

TABLE 3.4: RESULTS OF "2 CIRCULAR COILS WITH 9 TURNS" MODEL WITH FREQUENCY SWEEP

The table 3.4 shows the results of this configuration. The parameters related to the inductive coupling

show really small variation, while the resistance presents an increase coherently to the frequency

increase. This trend (Figure 3.20) is due to the fact that the frequency increase causes the increase of the

skin effect and consequently of the coil’s resistances.

FIGURE 3.20: TREND OF THE RESISTANCE WITH RESPECT TO THE FREQUENCY

F [kHz] L1 [µH] L2 [µH] M [µH] K R [mΩ]

60 13.193 13.199 5.296 0.4014 43.235

70 13.186 13.192 5.294 0.4014 45.211

80 13.180 13.186 5.293 0.4014 46.849

85 13.178 13.184 5.292 0.4014 47.565

90 13.176 13.181 5.292 0.4014 48.222


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To conclude, the last simulation with this type of configuration has done dealing with the misalignment.

The variable considered is the position of the centre of the secondary coil along the y axes. Using the

parametric analysis, the obtained results are showed in the table 3.5.

TABLE 3.5: RESULTS OF "2 CIRCULAR COILS WITH 9 TURNS" MODEL WITH MISALIGNMENT

It is possible to see that increasing the misalignment between the two coils causes the decrease of the

coupling coefficient achieving pretty low values. The trend of the K factor is showed in the plot of the

figure 3.21.

FIGURE 3.21: TREND OF K FACTOR WITH RESPECT TO THE MISALIGNMENT

The condition of maximum misalignment is showed in the Figure 3.22.

Δy [cm] L1 [µH] L2 [µH] M [µH] K

0 13.209 13.194 5.297 0.4012

2 13.213 13.196 4.866 0.3685

4 13.219 13.198 3.800 0.2877

6 13.227 13.205 2.520 0.1907

8 13.221 13.197 1.317 0.0997


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TABLE 3.6: PER CENT DECREASE OF THE COUPLING FACTOR

FIGURE 3.22: MAXIMUM MISALIGNMENT CONFIGURATION SIDE VIEW AND TOP VIEW


The second type of configuration analysed presents the following specifics:







Displacement [cm] K decrease [%]

0-2 8.15

2-4 21.93

4-6 33.74

6-8 47.72

0-8 75.56


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The only difference with respect to the previous case is the number of turns. The aim of this simulation

is analysing the behaviour of the system as a function of the number of turns. The results are shown in

the table 3.7.

TABLE 3.7: RESULTS OF “2 CIRCULAR COILS WITH 7 TURNS” MODEL

Comparing this case to the previous, due to the smaller dimension of the circuits, the resistance results

smaller. Furthermore, the value of the self and mutual inductances result smaller. This condition leads

to a lower coupling coefficient.


This configuration is defined by the following specifics:







Considering the previous simulations, the increase of the number of turns should increase the coupling

coefficient of the system. This is confirmed by the obtained results.

TABLE 3.8: RESULTS OF "2 CIRCULAR COILS WITH 12 TURNS" MODEL

The increase of the self inductance values could lead to a decrease of the coupling coefficient (equation

(1.99)). However, the increase of the mutual inductance is more incisive.


3 8.174 8.177 2.905 0.3553 32.533


3 23.276 23.294 10.570 0.4539 78.716


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3.3.4 Two Circular Coils with 9 and 7 turns

The principal aim of this configuration (Figure 3.23) is analysing its behaviour in case of misalignment.

Theoretically, it should guarantee a better response of the system.

The specifics of the configuration are:



• Number of turns coil 1: 9

• Number of turns coil 2: 7




• Airgap: 3 cm

The coil with 7 turns is designed for staying in the bicycle, while the biggest coil is designed for being

connected to the charging station.

FIGURE 3.23: PHYSICAL CONFIGURATION OF THE COILS

The Figure 3.24 shows the distribution of the magnetic field between the two coils.


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FIGURE 3.24: BFIELD OF THE CONFIGURATION

The results of this simulation are shown in the table 3.9.

TABLE 3.9: RESULTS OF “TWO CIRCULAR COILS WITH 9 AND 7 TURNS” MODEL WITH

MISALIGNMENT

The maximum value of the coupling coefficient is in case of zero misalignment. It presents a smaller

value with respect to the model with two coils with 9 turns. Increasing the misalignment, the K factor

decreases. This trend is showed in the Figure 3.25.

Δy [cm] L1 [µH] L2 [µH] M [µH] K

0 13.196 8.120 3.589 0.3467

2 13.199 8.116 3.263 0.3153

4 13.209 8.116 2.479 0.2395

6 13.221 8.119 1.553 0.1498

8 13.232 8.121 0.705 0.0679


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FIGURE 3.25: TREND OF THE K FACTOR WITH RESPECT TO THE MISALIGNMENT

The per cent decrease of the coupling coefficient due to the misalignment is shown in the table 3.10.

TABLE 3.10: PER CENT DECREASE OF THE COUPLING FACTOR

Displacement [cm] K decrease [%]

0-2 9.08

2-4 24.04

4-6 37.45

6-8 54.62

0-8 80.39

The table 3.10 shows an overall coupling factor decrease of the 80.39% with respect to the initial

condition (misalignment=0).

3.3.5 Two Vertical Coils

The last configuration analysed (Figure 3.26) presents a different geometry with respect to the previous

ones. In this case, an inner and outer coil that evolve in the vertical direction are present. This geometry

is showed in the Figure 3.25.


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FIGURE 3.26: PHYSICAL CONFIGURATION OF "TWO VERTICAL COILS”

The specifics of the configuration are:

• Inner coil radius: 83 mm

• Outer coil radius: 90 mm

• Turns inner coil: 8

• Turns outer coil: 8


• Pitch: 8.5 mm

• Airgap: 7 mm



Differently to the circular geometry, the two coils do not present the same dimensions. Moreover, one

coil is designed for being connected to the charger and the other to the bicycle, these roles cannot be

swapped. The inner coil is the transmitter (coil1), while the external coil is the receiver (coil2). The

configuration supposed is showed in the following Figure 3.27.


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FIGURE 3.27: CHARGING TOWER STATION AND BICYCLE WITH INTEGRATED ICPT SYSTEM

In the system supposed, the transmitter is placed in the part B of the charging station (Figure 3.26) while

the receiver is placed in the part A of the bicycle (Figure 3.26); the presence of the coils is represented

by the light blue circles. The position of the secondary coil has two aims. Firstly, it allows the connection

to the charging system supposed: the part B, that is vertically movable, should be inserted in the hole

(part A) of the bicycle. Secondly, it is closer to the battery of the E-bike; this avoids long connections

between the storage system and the ICPT system. Consequently, it decreases the possibility of damages.

The main disadvantage of this system is the high mechanical coupling required. However, a bike is a

light vehicle, this allows manual movements that can be easily performed by a person. The high coupling

coefficient between the two coils makes this charging system interesting.

TABLE 3.11: RESULTS OF “2 VERTICAL COILS” MODEL

The results show a high coupling coefficient, it is the highest found out in all the simulations. The strong

interaction between the coils is promoted by the geometry and by the close mechanical coupling.

The magnetic fields in 2D and 3D are showed in the Figure 3.28 and Figure 3.29.

L1 [µH] L2 [µH] M [µH] K R1 [mΩ] R2 [mΩ]

13.086 14.397 11.314 0.8243 34.619 47.614


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FIGURE 3.28: 2D BFIELD

FIGURE 3.29: 3D BFIELD


__________________________________________________________________________________ 65

3.4 DISCUSSION OF RESULTS

The conclusions that we can deduce from the simulations are:

• Considering the circular geometry, the coupling factor is strictly related to the number of turns;

it increases increasing them. From the results of the simulation the best K factor occurs for 12

turns, while the worst for 7 turns.

TABLE 3.12: COMPARISON BETWEEN CIRCULAR COILS WITH DIFFERENT NUMBER OF TURNS

In the table 3.12 it is possible to see how the parameters change with respect to the number of

turns. Even though the K factor increases with the number of turns, also the resistances of the

coils increase. This could lead to higher losses and managing problems of the system.

• The best response in case of misalignment happens when the two coils present different

dimension.

N1-N2 Airgap [cm] L1 [µH] L2 [µH] M [µH] K R [mΩ]

7-7 3 8.174 8.177 2.905 0.3553 32.533

9-9 3 13.174 13.184 5.291 0.4014 47.327

12-12 3 23.276 23.294 10.571 0.4539 78.716

9-7 3 13.196 8.1203 3.589 0.3467


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FIGURE 3.30: COMPARISON BETWEEN K FACTORS BEHAVIOUR WITH RESPECT TO THE

MISALIGNMENT

The Figure 3.30 shows the response of the coupling factor in case of misalignment. The blue line

represents the case with two identical circular coils with 9 turns; the orange line represents the

configuration with one coil with 7 turns and the other with 9 turns. Plotting the linear trend lines

and analysing their equations, it results that the trend line of the configuration with N1=9 and

N2=7 presents a smaller angular coefficient with respect to the other configuration. As a

consequence, it is possible to state that the former presents a better behaviour in case of not

perfect alignment.

• The frequency influences the system regarding the resistance. The resistance is proportional to

the frequency value due to the skin effect.

• The case with the vertical coils is the best configuration analysed in terms of performance of the

system. This guarantees the highest coupling coefficient (Table 3.13)

TABLE 3.13: RESULTS BEST CONFIGURATION

Airgap [cm] L1 [µH] L2 [µH] M [µH] K R1 [mΩ] R2 [mΩ]

0.7 13.08551 14.39672 11.31386 0.8243 34.61935 47.613616

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____________________________________________________ Chapter 4: STABILITY ANALYSIS

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Capitolo 4 : STABILITY ANALYSIS

4.1 INTRODUCTION

After the analysis of several geometries, the “two vertical coils” geometry results the best in terms of

coupling coefficient, achieving a value of 0.82; in the next step, the stability of this configuration has

been analysed. Supposing a SS compensation topology (Figure 4.1), keeping the same features of the

first model (see Paragraph 3.3.5), the number of turns of both coils have been varied in order to carry out

which configuration presents the most stable behaviour in case of frequency variations. The assumption

done is that the resistances of the coils remain constant with the frequency; the resistances considered

are at 85 kHz that coincides with the design frequency. The frequency interval analysed varies from 68

kHz to 104 kHz.

FIGURE 4.1: SS COMPENSATION TOPOLOGY

The study of the stability has been based on the concepts presented in the Chapter 1. The capacitances

are evaluated with the equations (1.28) and (1.29). The principal terms taken into account for the analysis

are the primary and secondary quality factor; in particular the relation between them:

𝑞𝑝 >

4 𝑞𝑠3

4 𝑞𝑠2 − 1

(4.1)

As already said, this relation states the stability condition of the system.

For realizing the analysis, the circuit has been solved in a proper software, where the main terms have

been evaluated and the plotted. Rearranging the inequality (4.1) it is possible to obtain the (4.2); this has

been applied in the software as stability function.

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𝑞𝑝 −

4 𝑞𝑠3

4 𝑞𝑠2 − 1

> 0 (4.2)

First of all, the attention has been focused on the configuration that presents the same number of turns in

the primary and in the secondary circuit. This showed an unstable behaviour. After that, the number of

turns of the transmitter has been increased, showing an improvement on the stability. Finally, the number

of turns of the primary coil has been increased till the most stable condition is achieved.

The circuit studied presents a battery of power PBATT=250 W and voltage VBATT=36 V; this can be

considered purely ohmic. The rectifier before the battery is ideal; as a consequence, the input power and

the output are equal. Moreover, considering the equation (1.97), the resistance of the load (RL),

considering the AC/DC converter and the battery, results equal to 4.2 Ω.

The unknown variable in this analysis is the input voltage V1. It is evaluated using the software fixing

the power at the load at 250 W.

4.2 TWO VERTCAL COILS N1=N2=8

The results of the first simulation (see Figure 3.26) are shown in the Table 4.1.

TABLE 4.1: RESULTS "TWO VERTICAL COILS N1=N2=8" MODEL

With these values the quality factors and the stability function result:

TABLE 4.2: STABILITY FACTORS OF "TWO VERTICAL COILS N1=N2=8" MODEL

qp qs Stability function

0.8043 1.8298 -1.1732

Focusing on the stability function, it results lower than zero. As a consequence, an unstable behaviour is

supposed.

In order to provide to the load a power of 250 W, the input voltage of the system has to be equal to 46.8

V.


13.086 14.397 11.314 0.8243 34.619 47.614


__________________________________________________________________________________ 69

The plots of the main electrical parameters, with respect to the frequency, are shown in the following

figures.

FIGURE 4.2: PRIMARY CURRENT OF "TWO VERTICAL COILS N1=N2=8" MODEL

FIGURE 4.3: VOLTAGE ON PRIMARY CAPACITOR OF "TWO VERTICAL COILS N1=N2=8" MODEL


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FIGURE 4.4: SECONDARY CURRENT OF "TWO VERTICAL COILS N1=N2=8" MODEL

FIGURE 4.5: VOLTAGE ON SECONDARY CAPACITOR OF "TWO VERTICAL COILS N1=N2=8" MODEL


__________________________________________________________________________________ 71

FIGURE 4.6: LOAD VOLTAGE OF "TWO VERTICAL COILS N1=N2=8" MODEL

FIGURE 4.7: LOAD POWER OF "TWO VERTICAL COILS N1=N2=8" MODEL


__________________________________________________________________________________ 72

Looking at the plots, it is easy to see the unstable behaviour of the system in case of frequency changes.

When the pulsation moves from the nominal condition, the nominal values of the electric parameters

assume too high values. This could cause damages and risks during the normal operation of the system.

4.3 TWO VERTICAL COILS N1=16 AND N2=6

Due to the high instability of the previous case, the following simulation has been made on a system with

16 turns in the primary coil and 6 in the secondary (Figure 4.8). The results of this model are present in

the table 4.3. The figure 4.9 shows the 3D distribution of the magnetic field.

FIGURE 4.8: TWO VERTICAL COILS N1=16 AND N2=6


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FIGURE 4.9: 3D BFIELD IN LOGARITHMIC SCALE OF "TWO VERTICAL COILS N1=16 AND N2=6"

MODEL

TABLE 4.3: RESULTS OF "TWO VERTICAL COILS N1=16 AND N2=6" MODEL

The input current (V1) to apply in order to have a power at the load of 250 W results equal to 52.2 V.

With this configuration the stability parameters results:

TABLE 4.4: STABILITY FACTORS "TWO VERTICAL COILS N1=16 AND N2=6" MODEL


1.6448 1.1952 0.196

The stability function results higher than zero but still close to it. Due to some approximations done in

the calculations for evaluating the stability function, it is not perfectly accurate. As a consequence, for

positive values, but really close to the zero, the system is still not stable. This is demonstrated in the

following figures.


33.325 9.404 12.609 0.7132 45.891 29.340


__________________________________________________________________________________ 74

FIGURE 4.10: PRIMARY CURRENT OF "TWO VERTICAL COILS N1=16 AND N2=6" MODEL

FIGURE 4.11: VOLTAGE ON PRIMARY CAPACITOR OF "TWO VERTICAL COILS N1=16 AND N2=6"

MODEL


__________________________________________________________________________________ 75

FIGURE 4.12: SECONDARY CURRENT OF "TWO VERTICAL COILS N1=16 AND N2=6" MODEL

FIGURE 4.13: VOLTAGE ON THE SECONDARY CAPACITOR OF "TWO VERTICAL COILS N1=16 AND

N2=6" MODEL


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FIGURE 4.14: LOAD VOLTAGE "TWO VERTICAL COILS N1=16 AND N2=6" MODEL

FIGURE 4.15: LOAD POWER "TWO VERTICAL COILS N1=16 AND N2=6" MODEL


__________________________________________________________________________________ 77

Even though the system is still unstable, the values assumed by the electrical parameters in not nominal

conditions are smaller than the previous configuration. The over-currents and over-tensions are closer to

the nominal values. Since increasing the number of turns of the primary coil has led closer to the stability

of the system, the best idea is keep going with this trend.


The next analysis performed is on a configuration that presents 25 turns in the primary coil and 5 in the

secondary (Figure 4.16). The 3D plot of the magnetic field is showed in the Figure 4.17.



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FIGURE 4.17: BFIELD IN LOGARITHMIC SCALE OF "TWO VERTICAL COILS N1=25 AND N2=5"

MODEL

The results of the simulation are shown in the table 4.5.

TABLE 4.5: RESULTS OF "TWO VERTICAL COILS N1=25 AND N2=5" MODEL

Increasing the turns of the primary circuit, its self-inductance decreases. In the same way, decreasing the

turns of the secondary coil, its self-inductance decreases. However, the mutual inductance remains close

to the previous values. The variation of L1 in more significant than the variation of the self-inductance

of the secondary circuit. As a consequence, the coupling factor decrease, reaching a value of 0.59 that

still is greater than the other geometries analysed.

On the other hand, this configuration is really close to the stability condition. The primary quality factor

is pretty higher than the secondary, as a consequence the stability function results higher than zero (Table

4.6).

TABLE 4.6: STABILITY FACTORS OF "TWO VERTICAL COILS N1=25 AND N2=5" MODEL


3.0264 0.9357 1.7167


58.735 7.362 12.357 0.5943 70.775 25.373


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The value of the stability function suggests a stable behaviour of the system. Actually, looking at the

plots, the system presents a good behaviour in case of frequency variation. However, the system still

presents conditions where the nominal values are overcome.

FIGURE 4.18: PRIMARY CURRENT OF "TWO VERTICAL COILS N1=25 AND N2=5" MODEL


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FIGURE 4.19: VOLTAGE OF PRIMARY CAPACITOR OF "TWO VERTICAL COILS N1=25 AND N2=5"

MODEL

FIGURE 4.20: SECONDARY CURRENT OF "TWO VERTICAL COILS N1=25 AND N2=5" MODEL


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FIGURE 4.21: VOLTAGE OF SECONDARY CAPACITOR OF "TWO VERTICAL COILS N1=25 AND N2=5"

MODEL

FIGURE 4.22: LOAD VOLTAGE OF "TWO VERTICAL COILS N1=25 AND N2=5" MODEL


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FIGURE 4.23: LOAD POWER OF "TWO VERTICAL COILS N1=25 AND N2=5" MODEL


Since the previous configuration was really close to the stability, in this simulation it has been considered

a system with 30 turns in the primary coil and 4 in the secondary (Figure 4.24).



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FIGURE 4.25: 3D BFIELD IN LOGARITHMIC SCALE OF “TWO VERTICAL COILS N1=30 AND N2=4”

MODEL

The results of this simulation are shown in the table 4.7.

TABLE 4.7: SIMULATION RESULTS OF “TWO VERTICAL COILS N1=30 AND N2=4” MODEL

Even though the coupling coefficient keeps decreasing, the stability factors present values that suggest

the stability of the system (Table 4.8).

TABLE 4.8: STABILITY FACTORS OF “TWO VERTICAL COILS N1=30 AND N2=4” MODEL


5.0355 0.6693 3.521


77.403 5.266 10.997 0.5447 86.003 19.752


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FIGURE 4.26: PRIMARY CURRENT OF “TWO VERTICAL COILS N1=30 AND N2=4” MODEL

FIGURE 4.27: VOLTAGE ON PRIMARY CAPACITOR OF “TWO VERTICAL COILS N1=30 AND N2=4”

MODEL


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FIGURE 4.28: SECONDARY CURRENT OF “TWO VERTICAL COILS N1=30 AND N2=4” MODEL

FIGURE 4.29: VOLTAGE ON SECONDARY CAPACITOR OF “TWO VERTICAL COILS N1=30 AND N2=4”

MODEL


__________________________________________________________________________________ 86

FIGURE 4.30: LOAD VOLTAGE OF “TWO VERTICAL COILS N1=30 AND N2=4” MODEL

FIGURE 4.31: LOAD POWER OF “TWO VERTICAL COILS N1=30 AND N2=4” MODEL

From the plots, it is possible to confirm the stable behaviour of the system in case of frequency variations.

This is the most stable configuration analysed; the values of the electrical parameters are almost always


__________________________________________________________________________________ 87

lower than the nominal values. The worst case is for the voltage on the primary capacitor, here it is

achieved a peak of 1.06 VC1/VC1_0.

This system, in order to work with the supposed load, needs an input voltage of 45.8 V. In this way, the

power supplied to the load is equal to 250 W.

4.6 DISCUSSION OF RESULTS

All the simulations performed show that last configuration represents the best option from a stability

point of view, even though it presents the lowest coupling factor.

Focusing on the table 4.9, the self-inductance of the primary circuit increases with the increase of its

number turns, while the self-inductance of the secondary circuit decreases with the decrease of its number

of turns. Basing on these results, it is possible to state that the self-inductances are proportional to the

turns of the winding.

The variation of the turns of the primary circuit is more significant with respect to the variation of the

secondary. As a consequence of this, the increase of the L1 is more notable than the decrease of L2; while

the mutual inductance presents small variations. This leads to the reduction of the coupling coefficient.

The values of the resistances of the coils, obviously, increase with the dimensions of the coils.

TABLE 4.9: COMPARISON OF ELECTRICAL VALUES OF THE DIFFERENT GEOMETRIES

For what concerns the stability, with each simulation the system has got closer to a stable behaviour, till

the last system that is stable. This is proved by the quality factors and stability function. The latter,

starting from a negative value, keep increasing improving the stability of the ICPT system.

N1-N2 L1 [µH] L2 [µH] M [µH] K R1 [mΩ] R2 [mΩ]

8-8 13.086 14.397 11.314 0.8243 34.619 47.614

16-6 33.325 9.4039 12.609 0.71321 45.891 29.340

25-5 58.735 7.362 12.357 0.5943 70.775 25.373

30-4 77.403 5.266 10.997 0.54471 86.003 19.752


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TABLE 4.10: COMPARISON OF STABILITY FUNCTIONS OF THE DIFFERENT GEOMETRIES

N1-N2 Stability function

8-8 -1.1731

16-6 0.196

25-5 1.7167

30-4 3.521

Lastly, in the Figure 4.32, it is represented the response of the load power to the frequency variation

considering the different configurations.

FIGURE 4.32: COMPARISON OF PL IN THE DIFFERENT CONFIGURATIONS

__________________________________________________________________________________ 89

Conclusion

The simulations performed have shown the behaviour of the WPT system depending on the physical and

geometrical features.

The circular geometries with horizontal disposition present good performance and they are able to

manage significant air gaps. The principal result that can be deducted for this geometry is the relation

between the dimensions of the coils and the coupling coefficient. It increases with the number of turns

of the coils, improving the performances of the system. Furthermore, the circular geometry presents good

performance also in case of misalignment. These features allow a certain degree of freedom for the

mechanical coupling between charging station and device. To conclude, it is possible to state that for

applications where a really close and strict displacement is not permitted, this geometry is the best choice.

On the other hand, where it is possible a strict mechanical coupling, the geometry suggested by the author

increases the performance of the system. Considering the application for electrical bike, this geometry

can be considered perfectly suitable. The best case considered presents a coupling coefficient of 0.82.

However, this configuration presents an unstable behaviour in case of frequency variation. Looking for

the stability of the system, it is guaranteed for a configuration that achieves a coupling factor of 0.54

with an efficiency higher than the 95%.

The configuration proposed in this work could result interesting in bike sharing applications. This sector

presents a continuous development, supporting the sustainable mobility. Even though a strict mechanical

coupling is still needed, the terminals are not exposed avoiding risk of faults and degradation. Moreover,

this configuration could be easily coupled to a locking system. Lastly, adopting bi-direction WPT system,

the charging station could be seen as a storage system, able to support the grid in case of need.

__________________________________________________________________________________ 90

_________________________________________________________________________ References

__________________________________________________________________________________ 91

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