RESONANT CONVERTER FOR ALL METAL INDUCTION … · RESONANT CONVERTER FOR ALL METAL INDUCTION...

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell'Informazione

Corso di Laurea Magistrale in Ingegneria Elettrica

RESONANT CONVERTER FOR ALL METAL INDUCTION COOKERS:

ANALYSIS, MODELLING AND SIMULATIONS

Relator: Prof. Marco MAURI

Thesis of Master’s Degree by:

Alberto CALONACI Matr. n. 799207

Academic Year 2013-2014

Thanks

A thanks to professor Marco Mauri who helped me understand a topic of which I had a little

knowledge and solve all the problems that arose during the development of this thesis.

I would especially like to thank all the people that have supported me in these years, my friends

and above all my family that always pushed me to do better.

A special thank goes to Tommaso, all the confidences and the advices that we have shared

make him my closer fried, even if we live far apart.

I

Index

Index ........................................................................................................................................ I

Index of figures .................................................................................................................. IV

Chapter 1-Introduction .................................................................................................... 1

1.1-History of induction heating for cooking applications ........................................... 2

1.2-Induction cooking advantages ................................................................................ 5

1.2.1-Efficency ........................................................................................................... 5

1.2.2-Control .............................................................................................................. 6

1.2.3- Reduced consumptions .................................................................................... 7

1.2.4-Safety ................................................................................................................ 8

1.2.5-Ease and Adaptability Installation .................................................................... 9

1.2.6-Cooler Environment ......................................................................................... 9

1.2.7-Cleanliness ........................................................................................................ 9

1.2.8– Optional features ........................................................................................... 10

1.3-Induction cooking disadvantages.......................................................................... 11

1.3.1-Price ................................................................................................................ 11

1.3.2-Vessel size ...................................................................................................... 11

1.3.3-Electromagnetic radiation ............................................................................... 11

1.3.4-Impact on the domestic electric system .......................................................... 12

1.3.5-Specific Cookware .......................................................................................... 12

1.4-Cookware materials properties and all metal problem ......................................... 13

1.4.1-Common materials and how they compare .................................................... 13

1.4.2-Thermal conductivity ...................................................................................... 17

1.4.3-Heat capacity .................................................................................................. 18

1.4.4-Thermal diffusivity ......................................................................................... 19

1.4.5-Reactivity ........................................................................................................ 20

1.5-Principle of operation ........................................................................................... 22

1.6-Induction cookers' structure .................................................................................. 23

Chapter 2-Load-winding system model .................................................................... 25

2.1-Litz wire ................................................................................................................ 26

II

2.2-Circuital model of the load-winding system......................................................... 27

2.3–Winding resistance ............................................................................................... 33

2.3.1-Tourkhani-Viarouge model ............................................................................ 33

2.3.2-Acero-Hernandez-Burdio-Alonso-Barragan model ....................................... 40

2.4-Winding inductance .............................................................................................. 50

2.5-Acero-Burdio-Alonso-Barragan model for impedance calculation ...................... 58

2.5.1-Model without the ferrite disc ........................................................................ 59

2.5.2-Model with the ferrite disc .............................................................................. 66

Chapter 3-Induction cookers' power electronics ................................................... 73

3.1-EMC filter ............................................................................................................. 73

3.2-Rectifier circuits ................................................................................................... 75

3.3-Protection circuits ................................................................................................. 75

3.3.1-Series type resistance-capacitance (R-C) unpolarised snubber circuits ......... 76

3.3.2-R-C polarised snubber circuits ....................................................................... 77

3.3.3-R-C switch-on polarised snubber circuits ....................................................... 78

3.4-Inverter.................................................................................................................. 79

3.4.1-Inverter topologies .......................................................................................... 79

3.4.2-Control techniques .......................................................................................... 81

3.4.3-Half-bridge Inverter ........................................................................................ 82

3.4.4-Quasi-resonant converters .............................................................................. 86

3.4.4.1-Zero-current-switching quasi-resonant converters (ZCS-QRCs) ...... 89

3.4.4.1.1-Principle of operation ........................................................ 89

3.4.4.1.2-DC Voltage-Conversion Ratio........................................... 94

3.4.4.2-Zero-voltage-switching quasi-resonant converters (ZVS-QRCs) ..... 97

3.4.4.2.1-Principle of operation ........................................................ 98

3.4.4.2.2-DC Voltage-Conversion Ratio......................................... 101

3.4.4.3-Comparison of ZCS-QRC and ZVS-QRC....................................... 102

3.4.4.4-Gate-drive design ............................................................................. 103

3.4.5-Operation modes used with non-ferromagnetic pans ................................... 107

Chapter 4-Inverter losses ............................................................................................. 108

4.1-Conduction losses ............................................................................................... 108

4.2-Switching losses ................................................................................................. 110

4.3-Verification of the losses model ......................................................................... 114

III

4.3.1-Verification of the conduction losses model ................................................ 114

4.3.2-Verification of the switching losses model ................................................... 117

4.4-Dynamic losses model ........................................................................................ 122

4.5-Verification of the dynamic losses model .......................................................... 127

Chapter 5-Simulations .................................................................................................. 134

5.1- Third-harmonic operation mode model ............................................................. 138

5.1.1-Simulations and validation, ferromagnetic configuration ............................ 138

5.1.1.1- fsw ≠ fres ...................................................................................... 142

5.1.2-Simulations and validation, non-ferromagnetic configuration ..................... 144

5.1.2.1- fsw ≠ fres ...................................................................................... 147

5.1.2.2-Variable 𝑅𝑙𝑜𝑎𝑑 ................................................................................. 149

5.2-Frequency-doubler operation mode model ......................................................... 152

5.2.1-Simulations and validation, ferromagnetic configuration ............................ 155

5.2.1.1- fsw ≠ fres ...................................................................................... 158

5.2.2-Simulations and validation, non-ferromagnetic configuration ..................... 161

5.2.2.1- fsw ≠ fres ...................................................................................... 164

Chapter 6-Conclusions .................................................................................................. 167

References .......................................................................................................................... 168

IV

Index of figures

Figure 1.1-1933 Chicago world’s fair ........................................................................................ 3

Figure 1.2-First induction cooker presented at Chicago world fair ........................................... 3

Figure 1.3-Westinghouse’s Cool Top 2 .................................................................................. 4

Figure 1.4-Comparison of gas and induction cooker heat transfer ............................................ 6

Figure 1.5-Comparison of efficiency of different cooking techniques ...................................... 6

Figure 1.6-Philips control panel and display ............................................................................. 7

Figure 1.7-Examples that the heat is focused ............................................................................ 8

Figure 1.8-Example of safety device ......................................................................................... 8

Figure 1.9-Lincat induction cooker IH42 .................................................................................. 9

Figure 1.10-Induction cooker structure.................................................................................... 23

Figure 1.11-Induction cooker power electronics main blocks ................................................. 23

Figure 1.12-Electric model of the cookware ............................................................................ 24

Figure 2.1-Inductor system structure ....................................................................................... 25

Figure 2.2-Litz wire ................................................................................................................. 26

Figure 2.3-Circuital model of the load-winding system .......................................................... 27

Figure 2.4-Simplified R-L model ............................................................................................ 28

Figure 2.5-Models with more inductances and resistances ..................................................... 29

Figure 2.6-Comparison between the characteristics of the real inductance and resistance and

those obtained by the six parameters model, with variable frequency .................. 30

Figure 2.7-Comparison between the total impedance of two recipients with different sizes and

at variable frequency .............................................................................................. 31

Figure 2.8-Comparison between the total impedance of two recipients made by different

materials ................................................................................................................. 32

Figure 2.9-Single conductor, parameters ................................................................................. 34

Figure 2.10-Comparison between actual and approximated model ......................................... 37

Figure 2.11-Comparison between Dowell model and the new ................................................ 39

Figure 2.12-Magnetic field at a distance ρ from a strand carrying a current Is, in a litz wire . 42

Figure 2.13-A length 𝜆𝑐 of a litz wire ...................................................................................... 43

Figure 2.14-Losses in a single strand under a spatially uniform alternating magnetic field ... 44

Figure 2.15-Magnetic field H𝑜_𝑠 in a strand situated at ρ position in the wire ........................ 45

V

Figure 2.16-A planar inductor of n turns; R1 and Rv are the free-space and R2 is the load .... 46

Figure 2.17-Components of the external magnetic field 𝐻𝑜,𝑖(𝑟𝑗 , 𝑧𝑘) in a strand situated at

position (𝜌, 𝜃) inside the bundle for the i-turn of the planar inductor ................. 47

Figure 2.18-Total resistance of a winding and its components for a winding made of litz wire

with 0.4 mm diameter strands, at 30 kHz, and 25 ᵒC ........................................... 49

Figure 2.19-Total resistance of a winding and its components for a winding made of litz wire

with 0.4 mm diameter strands, at 30 kHz, and 250 ᵒC ......................................... 50

Figure 2.20-Magnetic flux lines created by a winding of the inductor .................................... 50

Figure 2.21-Two coils model ................................................................................................... 51

Figure 2.22-Geometry of the model......................................................................................... 52

Figure 2.23-Geometry of the conductor disposition ................................................................ 55

Figure 2.24-Experimental results (values in nH) ..................................................................... 56

Figure 2.25-Model with ferromagnetic substrate..................................................................... 57

Figure 2.26-Comparison between the measured inductance and resistance and the one

calculated with the model .................................................................................... 58

Figure 2.27-Model for inductance calculation ......................................................................... 59

Figure 2.28-Comparison between the measured and calculated resistance, as a function of

frequency, for different materials ........................................................................ 65

Figure 2.29-Comparison between the measured and calculated inductance, as a function of

frequency, for different materials ........................................................................ 66

Figure 2.30-Ferrite disc model ................................................................................................. 67

Figure 2.31-Comparison between the measured and calculated resistance and inductance, as

a function of frequency, for different materials with winding made by

one turn ................................................................................................................ 71

Figure 2.32-Comparison between the measured and calculated resistance and inductance, as

a function of frequency, for different materials and a spiral coil......................... 71

Figure 3.1-Induction cooker power electronics main blocks ................................................... 73

Figure 3.2-EMC filter .............................................................................................................. 74

Figure 3.3-Mono-phase Graetz bridge ..................................................................................... 75

Figure 3.4-R-C unpolarised snubber circuit ............................................................................. 76

Figure 3.5-R-C polarised snubber circuit................................................................................. 77

Figure 3.6- voltage and current trends for different values of the capacitance ........................ 78

Figure 3.7-Half-bridge and full bridge topology ..................................................................... 80

VI

Figure 3.8- Inverter half-bridge topology ................................................................................ 82

Figure 3.9-Working principle of the half-bridge inverter, phase 1 .......................................... 83

Figure 3.10-Working principle of the half-bridge inverter, phase 2 ........................................ 83



Figure 3.13-ZC resonant switches ........................................................................................... 88

Figure 3.14-Switching load line trajectory .............................................................................. 88

Figure 3.15-ZV resonant switches ........................................................................................... 89

Figure 3.16-Buck quasi-resonant converter ............................................................................. 90

Figure 3.17-Induction charging state ....................................................................................... 90

Figure 3.18-Resonant stage ...................................................................................................... 91

Figure 3.19-Waveforms of the buck resonant converter, half-wave mode .............................. 93

Figure 3.20-Waveforms of the buck resonant converter, full-wave mode- ............................. 93

Figure 3.21-Capacitor discharging stage ................................................................................. 93

Figure 3.22-Free-wheeling stage ............................................................................................. 94

Figure 3.23-DC voltage-conversion ratio for the buck resonant converter, for the half-wave

mode and full-wave mode .................................................................................... 95

Figure 3.24-Waveforms of the buck resonant converter, different load conditions ................ 96

Figure 3.25-Boost quasi-resonant converter ............................................................................ 98

Figure 3.26-Capacitor charging state ....................................................................................... 98

Figure 3.27-Resonant stage ...................................................................................................... 99

Figure 3.28-Waveforms of the boost resonant converter, half-wave mode ........................... 100

Figure 3.29-Waveforms of the boost resonant converter, full-wave mode ........................... 100

Figure 3.30- Inductor discharging stage ................................................................................ 100

Figure 3.31- Free-wheeling stage .......................................................................................... 101

Figure 3.32-DC voltage-conversion ratio for the boos resonant converter, for the half-wave

mode and full-wave mode .................................................................................. 102

Figure 3.33-Major characteristics of the ZCS and ZVS techniques ...................................... 103

Figure 3.34-Basic circuit diagram of the quasi-resonant gate drive ...................................... 106

Figure 3.35-Theoretical waveforms of the quasi-resonant gate drive at 𝑉𝐺 = 4 𝑉, 𝐿𝐺 =

120 𝑛𝐻, and 𝐶𝑖𝑠𝑠 = 1.8 𝑛𝐹 ................................................................................ 106

Figure 4.1-IGBT voltage and current curves during turn-off phase ...................................... 111

Figure 4.2-Voltage and current in the diode and the IGBT during turn-on phase ................. 113

Figure 4.3-Half-bridge inverter for conduction losses calculation ........................................ 115

VII

Figure 4.4-IGBT V-I characteristic ........................................................................................ 115

Figure 4.5-Diode V-I characteristic ....................................................................................... 116

Figure 4.6-𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 characteristics, function of the switching current .......................... 118

Figure 4.7-Comparison between 𝐸𝑜𝑓𝑓−𝑚𝑜𝑑 and 𝐸𝑜𝑓𝑓−𝑑𝑎𝑡𝑎𝑠ℎ𝑒𝑒𝑡 ........................................... 119

Figure 4.8-Comparison between 𝐸𝑜𝑓𝑓−𝑚𝑜𝑑𝑌 and 𝐸𝑜𝑓𝑓−𝑑𝑎𝑡𝑎𝑠ℎ𝑒𝑒𝑡 .......................................... 120

Figure 4.9-Characcteristics of 𝐼𝑟𝑟𝑚 and 𝑡𝑟𝑟, as a function of the switching current .............. 121

Figure 4.10-Comparison between 𝐸𝑜𝑛−𝑚𝑜𝑑 and 𝐸𝑜𝑛−𝑑𝑎𝑡𝑎𝑠ℎ𝑒𝑒𝑡 ............................................. 122

Figure 4.11-Blocks for the losses calculation ........................................................................ 123

Figure 4.12-Demux IGBTs and Diodes currents and voltages blocks ................................... 124

Figure 4.13-IGBT losses block .............................................................................................. 125

Figure 4.14-Eon Table block ................................................................................................. 125

Figure 4.15-Eoff Table block ................................................................................................. 126

Figure 4.16-Energy to Power block ....................................................................................... 126

Figure 4.17-Trigger logic circuit ............................................................................................ 127

Figure 4.18-Validation circuit ................................................................................................ 128

Figure 4.19-Model of the validation circuit in Simulink ....................................................... 128

Figure 4.20-Input voltage from the experimental results and the model ............................... 129

Figure 4.21-Input current from the experimental results and the model ............................... 130

Figure 4.22-Output voltage and current from the experimental results and the model ......... 131

Figure 4.23-Conduction losses, highlight of the regime trend ............................................... 132

Figure 4.24-Switching losses, highlight of the regime trend ................................................. 133

Figure 5.1-Schematic of half-bridge series resonant inverter topology ................................. 134

Figure 5.2-(a) Frequency-dependent inductance, (b) frequency-dependent resistance ......... 135

Figure 5.3-Configuration for ferromagnetic pans .................................................................. 136

Figure 5.4-Configuration for non-ferromagnetic pans ........................................................... 137

Figure 5.5-Simulink model, ferromagnetic configuration ..................................................... 138

Figure 5.6-Comparison between the output from the experimental results and our

simulation .............................................................................................................. 139

Figure 5.7-𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 breakpoints ..................................................................................... 140

Figure 5.8-Switching losses ................................................................................................... 141

Figure 5.9-Conduction losses................................................................................................. 141

Figure 5.10-𝑃𝑖𝑛 and 𝑃𝑜𝑢𝑡 at different frequencies .................................................................. 142

Figure 5.11- Losses at different frequencies .......................................................................... 143

VIII

Figure 5.12- Efficiency at different frequencies .................................................................... 143

Figure 5.13-Simulink model, non-ferromagnetic configuration ............................................ 144


simulation ........................................................................................................... 145

Figure 5.15-Switching losses ................................................................................................. 146

Figure 5.16-Conduction losses............................................................................................... 146


Figure 5.18-Losses at different frequencies ........................................................................... 148

Figure 5.19-Efficiency at different frequencies ..................................................................... 148

Figure 5.20-𝑃𝑖𝑛, fixed and variable 𝑅𝑙𝑜𝑎𝑑 .............................................................................. 150

Figure 5.21-𝑃𝑜𝑢𝑡, fixed and variable 𝑅𝑙𝑜𝑎𝑑 ............................................................................ 150

Figure 5.22-Switching losses, fixed and variable 𝑅𝑙𝑜𝑎𝑑 ........................................................ 151

Figure 5.23-Conduction losses, fixed and variable 𝑅𝑙𝑜𝑎𝑑 ...................................................... 151

Figure 5.24-Efficency, fixed and variable 𝑅𝑙𝑜𝑎𝑑 ................................................................... 152

Figure 5.25- Time-sharing high-frequency multiple-resonant soft-switching inverter ......... 153

Figure 5.26-A schematic system arrangement for all IH cooking appliances with selective

switching of load resonant capacitor 𝐶𝑐 ............................................................. 154

Figure 5.27- Simulink model, ferromagnetic configuration .................................................. 155


simulation ........................................................................................................... 156

Figure 5.29-𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 breakpoints ................................................................................... 157






Figure 5.35-Simulink model, non-ferromagnetic configuration ............................................ 161


simulation ........................................................................................................... 162





IX


1

Chapter 1-Introduction

The modern cooking environment display many technologies, most of which are aimed to

reduce the time needed to prepare food, but also to ease the most dangerous and tiring actions.

Cooking is without a doubt one of the principal activities in everyday life, thus it’s important

that it improves with the aid of new technologies.

In the past cooking was obtained via the combustion of natural gas, but, nowadays, the attention

has been drawn by the electric ways to cook.

Induction heating, a method already employed in the industrial field, in the heating and fusion

of metals, has been introduced also in the domestic environment, leading to numerous benefits

compared to traditional cooking techniques.

The aim of this project is to realize models of induction cookers able to efficiently heat

cookware made of both ferromagnetic and non-ferromagnetic materials (all metal induction

cookers).

In order to do that, first of all we need to analyse our load (the pan), how it can be modelled

and the changes it is subjected by the variation in the working conditions and the effects due to

the use of different materials.

Then we will see the components needed to make a working induction cooker, while focusing

on the most crucial part, the resonant inverter, its topologies, its working principle and control

techniques.

Also in the inverter we have the majority of losses, thus their reduction is an important aspect

in order to obtain an efficient energy transfer.

Because of this reason we will analyse the calculation of the losses in an IGBT and also develop

a dynamic calculation method in Simulink, in order to observe their variation in different

working condition while analysing our simulations.

In these two different techniques to achieve all metal induction cooking will be shown and

compared between themselves and with the experimental results present in literature, in order

to prove the validity of our models and evaluate the development of the all metal techniques.

In fact, in the field of induction cooking, efficiency has always been a major problem while

dealing with non-ferromagnetic cookware, thus the development of new resonant converters

able to efficiently transmit energy has always been a necessity.

2

1.1-History of induction heating for cooking applications [1]

The basic principle involved in induction heating is the law of induction developed by Michael

Faraday in 1831.

Faraday's work involved the use of a switched DC supply provided by a battery and two

windings of copper wire wrapped around an iron core and when the switch was closed a

momentary current flowed in the secondary winding, which could be measured by means of a

galvanometer.

On opening the switch a current again flowed in the secondary winding, but in the opposite

direction.

Hence Faraday concluded that, since no physical link existed between the two windings, the

current in the secondary coil must be caused by a voltage that was induced from the first coil,

and that the current produced was directly proportional to the rate of change of the magnetic

flux.

Another important principle behind induction heating is Resistive heating, that was first studied

by James Prescott Joule in 1841.

Joule immersed a length of wire in a fixed mass of water and measured the temperature rise

due to a known current flowing through the wire for a 30 minute period.

By varying the current and the length of the wire he deduced that the heat produced was

proportional to the square of the current multiplied by the electrical resistance of the wire: Q ∝

Ri2t

Early in the 20th century engineers started to look for ways to harness the heat-generating

properties of induction for the purpose of melting steel and in early 1900s for cooking

applications.

Induction cooking technology was introduced at Chicago in a 'World Fair" in 1933 as well in

the mid-1950s.

In those occasions, induction cooker demonstrations were held by General Motors in North

America, by the Frigidaire division of GM.

http://en.wikipedia.org/wiki/Minute

3

To demonstrate convenience and safety of induction cooking the induction cooker

demonstration was shown by placing a newspaper between Induction cooker's surface and the

pot while boiling a pot of water.

Figure 1.1-1933 Chicago world’s fair

Figure 1.2-First induction cooker presented at

Chicago world fair

4

But it never quite caught on and induction cooker productions were delayed for another few

more years and for subsequent 40 years the technology was used mostly in industrial

applications.

Modern developments and implementations of Induction cooking starts in early 1970s at the

R&D Center of Westinghouse Electric Corporation in USA.

That work was first put on public display at the 1971 National Association of Home Builders

convention in Houston, Texas, as part of the Westinghouse Consumer Products Division

display.

Those were named Cool Top 2 (CT2) Induction ranges and were priced at $1,500.

CT2 had four burners of about 1,600 watts each and the range top was a Pyroceram ceramic

sheet surrounded by a stainless-steel bezel, upon which four magnetic sliders adjusted four

corresponding potentiometers set below.

The electronic section was made in four identical modules cooled by fans and in each of the

electronics modules, the 240V, 60 Hz domestic line power was converted to between 20V to

200V of continuously variable DC by a phase-controlled rectifier.

That DC power was in turn converted to 27 kHz 30 A (peak) AC by two arrays of six paralleled

Motorola automotive-ignition transistors in a half-bridge configuration driving a series-

resonant LC oscillator, of which the inductor component was the induction-heating coil and its

load, the cooking pan.

Figure 1.3-Westinghouse’s Cool Top 2

http://en.wikipedia.org/wiki/National_Association_of_Home_Builders

http://en.wikipedia.org/wiki/Rectifier

http://en.wikipedia.org/wiki/Oscillator

5

Control electronics included functions such as protection against over-heated cook-pans and

overloads. Provision was made to reduce radiated electrical and magnetic fields.

Production took place in 1973 through to 1975 and stopped with the sale of Westinghouse

Consumer Products Division to White Consolidated Industries Inc.

Though induction cookers faded from the American consumer market, it continued to be

developed in Europe and Asia.

In 2000, European manufacturers made a breakthrough in insulating materials design for

integrating the electronics with the induction generator coils.

In 2009 Panasonic developed an all metal induction cooker that used a different coil design and

a higher operating frequency to allow operation with non-ferrous metal cookware.

The new technologies and the reduced fabrication costs enabled the manufacturers to produce

induction generators for far less than previously, with much more compact designs that were

inherently more reliable and as a result the market in Europe really took off.

In Asia a similar phenomenon has occurred, in fact huge numbers of Asian households are

switching to induction for their cooking due to the safer and cooler cooking environment it

provides.

With recent improvements in technology, induction cookers are now better than ever, while

cheaper manufacturing from China has reduced the cost of induction cookers to levels that are

being affordable to every household.

1.2-Induction cooking advantages

1.2.1-Efficency

Efficiency represents the ratio between the absorbed energy and the energy sent to the recipient

and is the main strength of induction hobs, in fact in induction cooking, energy is supplied

directly to the cooking vessel by the electro-magnetic field, thus almost all of the energy from

the source gets transferred to that vessel.

With gas or conventional electric cookers (including halogen), the energy is first converted to

heat and only then directed to the cooking vessel, thus a lot of that heat goes to waste heating

up the kitchen instead of heating up the food.

6

The direct transfer of energy to the cookware reduces to the minimum the dissipation of heat

and energy towards the neighboring environment, thus the energy efficiency of induction

cookers is higher compared to the other technologies available on the market.

As shown in Figure 1.6 induction heating can reach an efficiency of 90%, that compared to

traditional heating techniques is significantly higher.

1.2.2-Control

Thanks to power electronics and the management done by the control system, power can be

maintained constant during the supply and changed with precision, higher than the one that can

be obtained with other technologies.

90

5847

40

Induction Halogen Electric Gas

EfficencyEfficency [%]

Induction 90

Halogen 58

Electric 47

Gas 40

Figure 1.4-Comparison of gas and

induction cooker heat transfer

Figure 1.5-Comparison of efficiency of different cooking techniques

7

With induction cooking, adjusting the cooking heat will be faster and much more precise as

there is no need to guess if the flames are at the right output levels.

It is possible, for example, to set the power at the minimum level needed during boiling or in

order to prevent the food from getting attached, thus dressings (like oil or butter) are not needed.

Some models have special control features that allow power flux regulation in case of an

excessive heating of the cookware, thus increasing the safety during cooking.

The example of a Philips induction cooker is proposed in Figure 1.7.

1.2.3-Reduced consumptions

Induction cooking allows the reduction of the employed energy due to the short cooking times

and the high efficiency that allows the transmission to the cookware of almost all the energy

taken from the power grid.

This is obtained thanks to the use of power electronics and the control that can be implemented

on it, thus the cookers activate only when cookware is placed on a hob and automatically

disconnect when it is removed from it.

Figure 1.6-Philips control panel and display

8

1.2.4-Safety

From the safety point of view the adoption of induction cooking enables a significant reduction

of the dangers involved in cooking.

In fact, the control system is able to detect the size of the item placed on the hob, thus it

dismisses the heating of small metallic object (with a diameter smaller than 10 cm), like

cutleries erroneously placed on the cooking range.

Also, heating is focused only in the part of the hob assigned to the cookware positioning, thus

the possibility of burns is reduced.

Usually there is also a luminous or auditory warning in case that, when the container is

removed, the surface is still hot, as shown in the example in Figure 1.9.

Figure 1.7-Examples that the heat is focused

Figure 1.8-Example of safety device, the

lighten H shows that the surface is still hot

9

The removal of natural gas lead to ulterior advantages regarding safety, there are no gas leaks

or open flames, thus toxic substances released during combustion are eliminated.

1.2.5-Ease and Adaptability Installation

Induction cookers don’t require a specific positioning inside the kitchen, in fact they don’t need

to be linked to a fixed spot of a house electrical system, thus, it is possible to position the

kitchen area independently from the choices made during the construction of the residence.

Unlike most other types of cooking equipment, induction units are typically very thin in the

vertical, often requiring not over five centimetres of depth below the countertop surface.

When a cooking area is to be designed to allow wheelchair access, induction cookers make the

matter convenient and safe.

1.2.6-Cooler Environment

Traditional cooking methods usually waste the generated heat because it is not supplied directly

to the cookware, while, with induction cooking, heat is generated in the induction cookware

directly and will not disperse anywhere that is not in contact with it.

The stovetop will also be relatively cool except for the surface that is in contact with the

cooking vessel.

1.2.7-Cleanliness

Burning gas has by-products that are vaporized, but eventually condense on the surface

Figure 1.9-Lincat induction cooker IH42

H(mm) 115

D(mm) 654

W(mm) 600

10

somewhere in the vicinity of the cooktop.

Induction cooking eliminates such by-products.

Also, heating is focused only in a part of the hob, thus even if food comes out from a pan or a

pot, it doesn’t burn or attach itself to the cooker.

The totally smooth ceramic surface is a great advantage to cleanliness, it can be polished with

a simple wet cloth.

1.2.8– Optional features

Some induction cookers are characterized by optional features regarding both the control

system and physical characteristics of the hobs in order to please the consumer’s needs.

The most used characteristics are:

Bridge components, necessary to link two active circular zones, making it possible to

heat items with different shapes and sizes.

Heat sensors installed in the electronics in order to protect the cooker and the electronics

itself from high value of temperature dangerous for the user and the hob.

A booster function that allows a fast heating of the cookware, used most of all during

boiling.

Its function is to convey to a single hob a big portion of the energy taken from the net

and it needs to be limited in time so that the high level of power doesn’t damage the

electronics

The detectors of the cookware’s presence are used to interrupt or supply power only if

a pot is present, so that the hob doesn’t supply when the cookware is raised from the

active zone.

The detectors of the size of the metallic item placed on the hob are able to interrupt the

supply of power to objects that are not compatible or are erroneously placed on the

cooking range.

The “child lock” function that avoid the accidental activation of the cooker, forbidding

the use of commands by the user in the “locked” stage.

Luminous indicators of heat allows the user to locate the zones that are still hot after

the cookware is removed from the hob.

Timers in order to set up the cooking times, even in case of “automatic” heating.

11

Locking systems necessary to interrupt power in case that spills from the pot are

detected or time limits are reached during the use of a specific level of power.

1.3-Induction cooking disadvantages

1.3.1-Price

The purchase of an induction cooker represents an aspect that can lure the consumer away from

the purchase of this technology.

This initial investment is high due to the cost of the employed materials in the production of

the hobs, however, it has to be considered a long term investment.

Evaluating all the benefits regarding the consumption, by comparing induction cooking and

the traditional one, the conclusion can be reached that, the economic spread between the two

choices is acceptable, considering also all the advantages presented earlier.

In the close future, however, it is possible that the purchase cost will be highly reduced thanks

to the spread of this technology, making induction cooking even more competitive.

1.3.2-Vessel size

Cooking vessels at the extremes of size, the very small and the very large, occasionally raise

issues.

As stated before, because the auto-detect feature that all induction units have is meant to assure

that things, from cooking implements (such as metal tongs or spoons or ladles) to jewellery

(rings or bracelets), will not activate an element, the detectors are often set rather

conservatively, so much so that, on some units very small pots, pans or percolators will not be

detected (the usual minimum pot base size for activation is around 10 cm, depending on

particular unit).

1.3.3-Electromagnetic radiation

The induction cooker’s principle of operation is that a magnetic field created by the presence

of an alternating current in the copper coils is used to transfer energy.

This causes an exposure to an electromagnetic radiation due to some field lines that goes

through the neighboring environment without closing in the bottom of the pot.

12

Thus it is important that the producer confronts an analysis of the daily dose of electromagnetic

field that the user can withstand because of the use of this technology.

1.3.4-Impact on the domestic electric system

A disadvantage that preclude some users from the purchase of induction cookers is the quantity

of energy absorbed from the domestic electric system during operation.

In many countries is frequent the necessity to restrict the limit of kW/h deliverable to the

domestic net, with a resulting increase of the energy fee given by the managing authority of

the net and sometimes in adjustments in the electrical system in a residence.

This often discourage the transition to the new technology, even if, with the spread of the

awareness toward energetic efficiency, electromagnetic induction is dominating the market in

the new residences’ kitchen with higher levels of efficiency.

Also If you have an outage for some reason, you won’t be able to cook (unlike with gas).

1.3.5-Specific Cookware

The major drawback of induction cookers, regarding older models and the ones that are not

designed for an all metal application, is that they will only work with induction cookware made

by magnetic materials, such as stainless steel, iron or cast iron.

While they are easily available in stores, some people may not want to change from their

existing cookware set which they have built over the years and also those materials have some

disadvantages because of their intrinsic proprieties:

Cast iron has a high thermal capacity, thus is not suitable for fast heating, while with stainless

steel there can be problems with the activation of the inductive processes for some kinds of

metals.

For those reasons, some solutions have been implemented in order to use cookware composed

by low resistivity materials, such as copper and aluminium, the so called all metal induction

cookers.

13

1.4-Cookware materials properties and all metal problem

As outlined in the previous chapter, the output of the induction cooker depends from the

material whereof the cookware is made, thus its behaviour with different loads has to be

analysed.

1.4.1-Common materials and how they compare [2]

Now we will look at each of the common materials used in cookware.

Copper

Description Copper is a soft (scratches easily) but durable (will last a lifetime) material that

has great thermal properties.

Pros

High thermal diffusivity

With enough thickness, pans heat extremely evenly

Extremely responsive

Cons

Heavy

Extremely expensive

Copper surface can tarnish or scratch

Pan may cool too fast after removal from heat (due to extremely high

thermal conductivity)

Cooking directly on copper may result in undesirable levels of copper

intake

Best uses

When lined with tin, nickel, or stainless steel, excellent for all stovetop

uses.

Care

Hand wash with a non-abrasive detergent and hand dry

Regularly use polish on exposed copper to preserve shine

14

Aluminium

Description

Plain aluminium utensils are low-cost, light-weight, and thermally responsive,

but it's reactive. Teflon coated aluminium utensils are low-cost and both non-

stick & non-reactive.

Clad or lined aluminium has had stainless steel bonded to the interior of the

utensil to form a non-reactive surface.

Pros

Extremely low cost if plain or teflon lined; moderated priced when

anodized

Great thermal properties

Cons

Very expensive if stainless steel lined or clad

highly reactive to acid ingredients (and somewhat reactive to alkaline as

well)

Lower density may require thicker construction to increase heat capacity

Unless anodized or lined or clad with stainless steel, may warp under high

heat

Unless anodized or clad, aluminium is prone to scratching.

Best uses

Plain aluminium, good for non-acid foods(like boiling stock or cooking

pasta)

Coated aluminium ,excellent for all purposes if aluminium is fairly thick

Care Hand-wash with a mild detergent and washcloth or sponge.

Cast iron

Description

Cast iron is composed on iron, carbon (more than carbon steel), and trace

elements found in common clays.

The iron is melted down and poured into a sand or clay mold to form the utensil.

Enameled cast iron has a thin but durable nonreactive layer of glass fused to the

surface of the utensil.

15

Pros

Plain cast iron is low cost

Manufacturing process results in thick and dense cookware for

unparalleled heat capacity

Thickness also results in even heating

Cons

Enameled cast iron can be expensive (although some are moderately

priced)

High heat capacity means the utensil takes longer to heat up

Although extremely hard, can crack or fracture if dropped or thermally

shocked (pouring cold water into a hot pot)

Best uses

Traditional woks (plain cast iron), skillets, Dutch ovens

Southern cooking

Care

Plain cast iron should be seasoned before first use and as needed.

A seasoned utensil should receive minimal contact to soap or detergent.

Wash by soaking in warm water for a few minutes and repeatedly scrubbing with

salt and rinsing until salt remains white (usually one scrubbing is does it).

Dry with a cloth and heat over low heat briefly to evaporate all moisture. For

enameled cast iron, hand wash in hot soapy water.

Carbon steel

Description

Carbon steel contains less carbon than cast iron and is formed and pressed from

sheets instead of being casted.

It can be annealed (heating the metal until its molecular structure realigns to

alleviate internal stresses and then specially cooled to preserve the new structure)

to form blue steel (or black steel), a harder and less reactive material.

Carbon steel can also be enamel coated.

Pros

All variations are usually low cost

Fast seasoning process for carbon steel; enameled carbon steel and blue

or black steel does not need seasoning

16

Cons

Poor thermal properties means slow heat up and uneven temperatures.

Thin and light (this might be a pro for some people) which results in very

little heat capacity

Best uses Fry pans, woks

Care Should be seasoned before first use. Care for as if it was cast iron.

Stainless steel

Description

Mixing steel with chromium and nickel (18/8 stainless steel is 18% chromium

and 8% nickel while 18/10 has 10% nickel) produces a corrosion resistance steel

that is both hard and easy to maintain a shine.

Disks of copper or aluminium can be fused to the stainless steel cookware to

enhance its thermal properties. Stainless steel can also be used to line copper or

aluminium utensils as well as cladding aluminium or copper (see aluminium and

copper cookware summaries above).

Pros

Plain stainless steel and stainless steel with aluminium or copper disks are

low cost to moderately priced

Shiny surface makes it easy to see how your food is browning

Corrosion resistant and easy to clean

With a thick aluminium or copper disk or clad around a core, stainless

steel becomes one of the best materials to cook in (not just for its thermal

properties, but as well as durability, ease of care, and visual control of

cooking - all the benefits of stainless steel with very little of its

drawbacks)

Cons

Plain stainless steel: worst material to cook on (in terms of thermal

properties)

Salt may cause pitting over time unless added to boiling liquid

17

Best uses

Plain stainless steel: boiling water (steaming is okay) and non-cooking

related tasks (mixing bowls, storage containers, etc.)

Stainless steel with copper or aluminium disk: great for all purposes if

disk is well bonded and of a fair thickness

Care Hand wash with mild detergent. Use gentle abrasives as needed.

As seen in the previous tables, the ferromagnetic materials (copper and aluminium), compared

to the non-ferromagnetic ones (cast iron, stainless steel and carbon steel), have better thermal

properties, but a higher cost, while the opposite is true for the non-ferromagnetic materials.

Hence, in order to determine which material is better to use with induction cookers, we need

to analyse in detail their properties, as follows:

1.4.2-Thermal conductivity

The thermal conductivity of a material is how readily that material absorbs and releases energy.

When the fire or heating element of a range comes in contact to a pan, the energy from the heat

source is transmitted to the pan, thus the internal kinetic energy is increased of the pan

(commonly called "heating up").

The heated material then transmits the energy to nearby materials that are at a lower average

molecular kinetic energy level (at a lower temperature than the material).

The higher the thermal conductivity of the material, the faster it will heat up and also, the faster

the heated area will spread to unheated areas of the same piece of material.

So a pan made of a low thermally conductive material will take a longer time to reach cooking

temperatures.

In fact, materials with low thermal conductivity take longer to react to any change in

temperature, so the thermal response of the pan would also be slow.

In most cooking applications, it is desirable to have the utensil heat up quickly, not develop hot

spots, and react to changes we make to the range controls.

Materials with high thermal conductivity fulfil our needs because they transmit heat quickly

resulting in fast response to thermal changes and even distribution of the internal kinetic

energy.

Thermal conductivity of some common materials used in cookware:

18

Material Thermal conductivity

Copper 401 W/m*K

Aluminium 237 W/m*K

Cast Iron 80 W/m*K

Carbon steel 51 W/m*K

Stainless steel 16 W/m*K

1.4.3-Heat capacity

The amount of internal kinetic energy stored in a material can be referred to as its heat capacity.

This isn't the same thing as temperature, which is the average molecular kinetic energy within

the material.

While thermal conductivity describes the materials ability to absorb energy, heat capacity is

the amount of energy that is needed to raise or lower the temperature of the material.

The molecular composition of some materials is such that as they absorb energy, much of it

gets converted into potential energy and only a small amount increases the molecular kinetic.

Other materials, like most metals, increase their molecular kinetic energy readily and do not

store much of the absorbed energy as potential energy.

The heat capacity of a material is proportional to its mass, this means that cookware made of

materials with high heat capacity, will take longer to heat up, but will also have a significant

amount of energy stored up when it is hot.

When energy is pulled out of the material, the temperature of the material will lower slowly

when compared to materials with low heat capacity

Since heat capacity is a function of the mass of the material, density must be known to make

comparisons between cookware of different materials.

19

Material Specific Heat Density

Aluminium 910 J/kg*K 2600 kg/m3

Stainless Steel 500 J/kg*K 7500 - 8000 kg/m3

Carbon Steel 500 J/kg*K 7500 - 8000 kg/m3

Cast Iron 460 J/kg*K 7900 kg/m3

Copper 390 J/kg*K 8900 kg/m3

Looking at the table above, if you multiply specific heat with density, we'll find that the heat

capacity per unit volume of steel, cast iron, and copper are about 1.5 times that of aluminium.

This means, to achieve the same heat capacity in an aluminium pan as in stainless steel pan,

the aluminium pan needs to be 1.5 times as thick (assuming the other pan dimensions are the

same).

1.4.4-Thermal diffusivity

Thermal conductivity alone does not determine how fast the pan will heat up (and also how

evenly it will heat), in fact, the heat capacity plays a role in determining this as well.

That’s why thermal diffusivity of a material is calculated and it is simply the thermal

conductivity divided by the unit heat capacity.

Material Thermal diffusivity

Copper 120 * 10-6 m2/s

Aluminium 100 * 10-6 m2/s

Cast Iron 22 * 10-6 m2/s

Carbon Steel 14 * 10-6 m2/s

Stainless Steel 4.3 * 10-6 m2/s

It is clear, however, that the best performing materials (in terms of dishing out energy) are

copper and aluminium.

This leads to the final consideration: reactivity.

20

1.4.5-Reactivity

Not only do we have to concern ourselves with the thermal properties of materials, but we need

to make sure that the materials we use in our cookware do not harm us or adversely affect the

taste of our food.

For this reason, in addition to the high thermal diffusivity, we would also like a non-reactive

material.

Unfortunately, both copper and aluminium react readily to foods (Copper, when ingested in

quantity or consistently, can cause liver, stomach, and kidney problems as well as anaemia.

Also, aluminium has long been suspected of contributing to Alzheimer's disease).

Stainless steel, the least reactive of the materials used in cookware, also has the worst thermal

diffusivity.

In order to have cookware made of materials with high thermal diffusivity and low reactivity,

by combining the non-reactive surface of stainless steel with the thermal properties of copper

or aluminium, you get the best of both materials.

There are several variations on this theme: steel- or tin-lined copper, stainless steel with

aluminium or copper disk, stainless steel cladded aluminium, and stainless steel cladded

copper.

The table below summarizes the effectiveness of various material combinations (they are listed

in order from most effective to least):

Composition Comments

Copper with tin lining Highest response; tin lining can be finicky can be susceptible to

melting; copper exterior requires more care

Copper with stainless steel

lining

Copper exterior requires more care but imparts the utensil with

copper's excellent thermal properties

Aluminium with stainless

steel lining

Thick aluminium provides excellent thermal response to thin

steel interior

Copper fully clad by

stainless steel

Copper layer may be thinner than copper with stainless steel

lining; exterior and interior are durable and easy to maintain

21

Aluminium fully clad by

stainless steel

Aluminium layer may be thinner than aluminium with stainless

steel lining; exterior and interior are durable and easy to maintain

Aluminium with stainless

steel lining and copper

exterior

Same performance as cladded aluminium, but with the

difficulties in maintaining copper

Stainless steel with copper

disk

Curved edge of the bottom causes the disk to not come into full

contact with the complete bottom of the pan resulting in inferior

heat conduction as compared to cladded copper

Stainless steel with

aluminium disk Same as stainless steel with copper disk

As shown in the previous discussion using cookware made of copper or aluminium have some

advantages, but, if preventative measures aren’t taken, then some issues arise in the use of

those materials with induction cookers.

In fact, the values of the inductance and the resistance of the pan-inductor coupling depend on

the characteristics of the induction coil, the frequency of the current and the properties of the

pan.

The relationships between the inductance and the resistance with non-ferromagnetic pans

(𝐿𝑒𝑞𝑁𝐹 and 𝑅𝑒𝑞𝑁𝐹) and with ferromagnetic pans (𝐿𝑒𝑞𝐹 and 𝑅𝑒𝑞𝐹) for the same frequency are

shown in (1.1) and (1.2).

.

𝐿𝑒𝑞𝐹 = 2 𝐿𝑒𝑞𝑁𝐹 (1.1)

𝑅𝑒𝑞𝐹 = 10 𝑅𝑒𝑞𝑁𝐹 (1.2)

As shown in the equations, the resistance for non-ferromagnetic pans is too low, thus the

maximum output power and the load current are too high for an acceptable performance of the

inverter.

The high load current and output power, also, cause the increase of conduction and switching

losses of the switching devices, thus decreasing the efficiency.

22

Hence, while using non ferromagnetic pans, traditional operating conditions are not acceptable

for a domestic induction cooker because they exceed the ratings of the commonly used devices.

In order to decrease the maximum output power and the load current, and thus improve the

efficiency, when a non-ferromagnetic pan is detected, the induction cooker have to change its

operating conditions.

The aim of the studies conducted on the all metal structures is to find the operating conditions

and electronic configurations able to efficiently heat cookware made by any kind of material,

taking into account the problems previously presented.

1.5-Principle of operation

Induction cooking is completely different from all other heating technologies, it does not

involve generating heat which is then transferred to the cooking vessel, in fact, it makes the

cooking vessel itself the original generator of the heat.

The basic concept is that induction cooker works on electromagnetic induction principle,

it refers to the generation of an electric current by passing a metal wire through a magnetic

field.

In a basic induction heating setup, a solid state RF power supply sends an AC current through

an inductor and the part to be heated is placed inside the inductor.

The inductor serves as the transformer primary and the part to be heated becomes a short circuit

secondary, this causes high currents to flow through the cookware, these are known as eddy

currents.

Eddy Currents are closed loops of induced current circulating in planes perpendicular to the

magnetic flux, they normally travel parallel to the coil's winding and the flow is limited to the

area of the inducing magnetic field.

These eddy currents flow against the electrical resistivity of the metal, generating precise and

localized heat without any direct contact between the part and the inductor.

This heating occurs with both magnetic and non-magnetic parts, and is often referred to as the

"Joule effect", referring to Joule's first law, a scientific formula expressing the relationship

between heat produced by electrical current passed through a conductor.

Secondarily, additional heat is produced within magnetic parts through hysteresis, that is

internal friction that is created when magnetic parts pass through the inductor.

Magnetic materials naturally offer electrical resistance to the rapidly changing magnetic fields

within the inductor and this resistance produces internal friction which in turn produces heat.

23

In the process of heating the material, there is therefore no contact between the inductor and

the part, and neither are there any combustion gases, thus the material to be heated can be

located in a setting isolated from the power supply.

In order to take into account these phenomenon, the cookware can be modelled as a resistance

in series with an inductance, as will be shown and deeply analysed in the following chapter.

1.6-Induction cookers’ structure

The main blocks of a domestic induction cooker are outlined in Figure 1.11.

Figure 1.10-Induction cooker structure

Figure 1.11-Induction cooker power

electronics main blocks

24

The mains voltage is rectified and filtered, obtaining a DC bus, then the resonant inverter

supplies high-frequency current, between 20 and 100 kHz, to the induction coil.

The use of those frequencies allow the induction cooker to work at frequencies higher than

those audible by humans, thus the hobs result as silent as possible and, meanwhile, to avoid

that an excessive switching frequency shall favour an increase in losses.

The current, coming from the inverter, produces an alternating magnetic field, which causes

eddy currents and magnetic hysteresis in the material heating up the pan.

The half-bridge series resonant inverter is the most used topology because of the electrical

requirements of its components, its simplicity and its cost-effectiveness.

All these elements, briefly discussed, will be analysed in Chapter 3, dedicated to the power

electronics inside an induction cooker.

The inverter load consists of the pan and the induction coil and its impedance is modelled as

the series connection of an induction coil and a resistor, based on the transformer analogy,

and it is defined by the equivalent values of Leq and Req .

The values taken by these two parameters, will be deeply discussed in the following chapter,

dedicated to the analysis of the load-winding system model.

Figure 1.12-Electric model of the

cookware

25

Chapter 2-Load-winding system model

The whole inductor system, which comprises the inductor itself and the pot, works like a

transformer: the winding is the primary, the pot is the secondary, and the flux conveyors

(usually ferrite bars) work like part of the core.

A vetroceramic glass is placed between the winding and the pot, which creates the inductance

needed for the resonant inverter (Figure 2.1).

Traditionally the inductor was winded with solid wires, principally due to cost reasons.

Fortunately in traditional arrangements a considerable efficiency is achieved with solid wires

due to the relatively low operating frequencies (about 30 kHz).

By the other hand, in these systems an appreciable efficiency is reached only with

ferromagnetic pans.

However, nowadays, many users require the capability to heat non ferromagnetic pots, as a

consequence the market of the all metal appliances is increasing quickly.

The simplest way to increase the induction heating is to increase the frequency of excitation

currents, however it is well know that this also increases the AC losses in the windings and, as

a result, the induction heating efficiency is not increased appreciably over a critical frequency.

Figure 2.1-Inductor system

structure:

Pot (1)

Cooking surface (2)

Electric insulation (3)

Winding (4)

Flux conveyor (5)

Shielding (6)

26

Some all metal apparatus have been patented in which mainly two strategies are used: first,

higher operating frequencies are used and second, inductors are winded with litz wires.

2.1-Litz wire [3]

Litz wire is a type of cable used in electronics to carry alternating current (Figure 2.1.2).

The wire is designed to reduce losses in conductors used at frequencies up to about 1 MHz.

Those losses are caused by:

Skin effect: it consist in the transfer of current density to the external surfaces of the

conductor.

Proximity effect: In a conductor carrying alternating current, if currents are flowing

through one or more other nearby conductors, such as within a closely wound coil of

wire, the distribution of current within the first conductor will be constrained to smaller

regions.

The resulting current crowding is termed as the proximity effect.

This crowding gives an increase in the effective resistance of the circuit, which

increases with frequency.

Geometric effects: effects, caused by the winding geometry, that act on the magnetic

field imposed to the conductor.

It depends from the distribution of the flux lines and from the presence of the gap in the

nucleus.

Figure 2.2-Litz wire

http://en.wikipedia.org/wiki/Alternating_current

http://en.wikipedia.org/wiki/Wire

http://en.wikipedia.org/wiki/Alternating_current

http://en.wikipedia.org/wiki/Current_crowding

27

Litz wire consists of many thin wire strands, individually insulated and twisted or woven

together, following one of several carefully prescribed patterns often involving several levels

(groups of twisted wires are twisted together).

This winding pattern equalizes the proportion of the overall length over which each strand is

at the outside of the conductor.

Every strand is isolated through enamels or an application of polyurethane-nylon and arranged

in a way that all of the conductor section is filled, allowing the omogeneous disposition of the

current between different strands, even at high frequency.

The standard twist configuration is 12 twists per foot (TPF), but it is possible to obtain a higher

number of TPF by decreasing the diameter of the strands.

The AC losses in litz wires used in magnetic components have been analysed in different ways:

Using the Finite Element Analysis (FEA), applied principally to high frequency

transformers,

Using analytical models oriented to magnetic components.

2.2-Circuital model of the load-winding system [8]

The creation of a mathematical model able to accurately describe the load seen by the inverter

is necessity because of the role covered by the winding model in the energy transfer to the pot.

In order to obtain a simplified model of the load-winding system, one based on resistances and

inductances is used.

Figure 2.3-Circuital model of the load-winding system

28

It is represented by the circuit in Figure 2.3, in which a transformer with N turns is present, that

model the energy transfer between the primary winding and the load-cookware at the

secondary.

The model can be useful in the study of maximum functionality of the system and is

characterized by the following parameters:

Rp resistance of the primary side that models the winding one and causes its heating.

Rs resistance of the secondary side that models the load-pot one.

Lp leakage inductance of the secondary side.

Ls leakage inductance of the secondary side.

N number of turns in the winding.

Furthermore it is possible to go to a model made by an equivalent inductor and an equivalent

resistance only by summing the elements on the primary side to those on the secondary side

seen from the primary winding.

Req = 𝑅𝑝 + 𝑁2𝑅𝑠 (2.1)

Leq = 𝐿𝑝 + 𝑁2𝐿𝑠 (2.2)

Figure 2.4-Simplified R-L model

29

The limitation of the equivalent resistance and equivalent inductance, in Figure 2.4, is due to

the fact that it is employable for a fixed frequency only, thus it can’t be used with variable

frequencies.

Hence, this model is ineffective in dealing with the power estimation, unless the values of

resistance and inductance are recalculated.

In order to remove the limitation intrinsic into this kind of model, in recent years other circuits

characterized by more inductances and resistances (as the one presented in Figure 2.5) have

been developed, thus models truer to reality in front of frequency variations have been

obtained.

In it a new couple inductor-resistance are added, they represent magnetic losses and the flux in

the cookware.

However, this modification doesn’t consider the losses due to induced currents, because the

winding results completely immersed in the magnetic field created by it.

Hence, it is possible to include skin and proximity effects dependent from the frequency by

adding an additional R-L couple (𝑅2, 𝐿2).

Through an algorithm all the six parameters can be calculated starting from the experimental

waveforms of the impedance of a real winding with variable frequency.

As can be seen in Figure 2.6, the six parameters model has an error lower than 5% on the

measured peak to peak voltage and it is lower than 15% regarding the power values.

Figure 2.5-Models with more inductances and resistances

30

In order to obtain the two parameters is sufficient to acquire the values of voltage and current

on the load by using an oscilloscope and through an elaboration program to express them with

Fourier series.

Given the almost square shape of the input voltage waveform, only the odd terms of the series

will be considered, also only the first eight harmonics are used, because the value of higher

harmonics are negligible.

Hence, it is possible to obtain the magnitude and phase of the impedance through the ratio of

equations (2.3) and (2.4).

Figure 2.6-Comparison between the characteristics of the real inductance

and resistance and those obtained by the six parameters model, with

variable frequency.

31

In order to obtain the total impedance 𝑍𝑅𝐿𝐶 seen by the converter the resonant capacitance just

has to be added, thus from the equations (2.5) and (2.6) modified in this way, the desired

parameters can be calculated.

VRL(t) =A02∑A2j+1

∞

j=0

sin(2π(2j + 1)ft + ϕ2j+1) (2.3)

IRL(t) =B02∑B2j+1

∞

j=0

sin (2π(2j + 1)ft + φ2j+1 ) (2.4)

|ZRL[(2j + 1)f]| =A2j+1

B2j+1 (2.5)

∠|ZRL[(2j + 1)f]| = ϕ2j+1 − φ2j+1 (2.6)

Req(f) =|ZRLC(f)|

√1 + [tan(∠ZRLC(f))]2 (2.7)

Leq(f) =Req(f) tan(∠ZRLC(f)) +

12πfC

2πf (2.8)

Figure 2.7-Comparison between the total impedance of two recipients

with different sizes and at variable frequency

32

Two impedance waveforms are depicted in Figure 2.7, where the Z waveforms obtained with

two pots with diameter 21 cm and 15 cm, it can be seen that near the resonance frequency,

the amplitude of 𝑍𝑅𝐿𝐶 is smaller for a lower diameter of the pot.

The same comparison is feasible for two different materials, one ferromagnetic and the other

non-ferromagnetic in Figure 2.8.

In addition to the amplitude of Z near the resonant frequency, the latter shifts itself to higher

frequencies, thus it is necessary to adjust the electronics to the new load condition.

The use of litz wires, furthermore, required a revision of the models in use until a decade ago.

The Dowell model in the 1966, for example, although it consider a linear distribution of the

leakage field through the windings, he examined one made by foil conductors.

Some researches during the years tried to adapt the foil conductor models to round section

conductors, but this is based on the approximation that a layer made by round conductor is

equivalent to foil conductors with particular geometric expedients.

Later some models have been developed based on round conductors, but considering single

wire conductors they are not suited to develop a model involving litz wires.

Thanks to the study of Ferreira on the development of analytical model in order to determine

the losses in the conductors subjected to external leakage fluxes, it was possible to get to the

model developed by Tourkhani-Viarouge, used to calculate the losses in windings with litz

wire and thus their resistance.

Figure 2.8-Comparison between the total impedance of two recipients

made by different materials

33

2.3–Winding resistance

2.3.1-Tourkhani-Viarouge model [4]

This model starts from the analytical expressions obtained from Dowell’s researches that allow

to understand the distribution of the leakage flux regarding transformers or windings.

Considering the expression of the eddy current density developed for round and foil

conductors, the conductor is supposed straight and its length is very large with respect to its

cross section, it will have the expression (2.9) and starting from this equation, the power

dissipation per unit length in the conductor is calculated (2.10).

J̅ = γj32

(

I0̅πd0

J0 (γrj32)

J1 (γd02 j

32)+ 2H̅(

J1 (γrj32)

J0 (γd02 j

32)sinθ)

)

(2.9)

Where:

δ = √ρ

πfμ0 skin depth

γ = √2

δ

I0̅ amplitude of the current in the conductor (phasor quantity)

H̅ amplitude of the leakage field across the conductor (phasor quantity);

f frequency

d0 diameter of the conductor

ρ copper resistivity

μ0 air permeability

J0 and J1 Bessel functions

P =ρ

2∫ ∫ J ̅J̅∗r dr dφ =

I2ρ

√2πδd0

2π

0

r0

0

ψ1(ξ) −2√2πρ

δH2ψ2(ξ) (2.10)

In which:

34

ψ1(ξ) =ber(

ξ

√2)bei′(

ξ

√2)−bei(

ξ

√2)ber′(

ξ

√2)

ber′(ξ

√2)+bei2(

ξ

√2)

(2.11)

ψ2(ξ) =ber2(

ξ

√2)ber′(

ξ

√2)+bei′(

ξ

√2)bei2(

ξ

√2)

ber2(ξ

√2)+bei2(

ξ

√2)

(2.12)

berv(x) and beiv(x) in (2.1.4) are called Kelvin functions.

They and are the real and imaginary parts, respectively, of Jv (x e3πi

4 ), where x is real,

and Jv(z), is the vth-order Bessel function of the first kind.

As can be seen in equation (2.10), it necessary to calculate the leakage field H.

In the special case of stranded conductor windings, the conductor is submitted to an internal

(𝐻𝑖𝑛𝑡) and an external (𝐻𝑒𝑥𝑡 ) leakage field.

H̅ = H̅ext + H̅int = (Hext + Hintsinθ)y̅ + Hintcosθx̅ (2.13)

In the calculations related to Hext we suppose that it develops itself in the y direction only, that

is true in case of transformer windings and inductors with a distributed gap.

At position ∆x of the k-th layer the leakage field Hext is given by (2.14) and

it can be shown that the internal field acting at the position (r, θ) of each conductor

(Figure 2.8) can be expressed by the equation (2.17) developed by J. Lammeraner and

M.Stafl. [4]

Figure 2.9-Single conductor, parameters

http://en.wikipedia.org/wiki/Real_part

http://en.wikipedia.org/wiki/Imaginary_part

http://en.wikipedia.org/wiki/Bessel_function

35

Hence from (2.13), (2.16) and (2.17) we can obtain the distribution along a strand in position

(r, θ) of a conductor in the k-th layer (2.18).

Hext(∆x, k) = (k − 1)IL +IL∆x

dc (2.14)

∆x = rc + rcosθ (2.15)

Hext(k, (r, θ)) = IL (k −1

2+r

rc

cosθ

2) (2.16)

Hint(r) =I

2πrc2r (2.17)

H̅ = [IL (k −1

2+r

rc

cosθ

2) +

I

2πrc2rsinθ] y̅ +

I

2πrc2rcosθx̅ (2.18)

Where IL is the current by unit width of a layer and rc is the radius of the conductor.

(2.18) will be used in the power integral of the proximity effect losses in (2.10), thus the

density of eddy current losses in the (k-th, j-th) conductor can then be calculated with the

equation (2.19), that, when integrated, give us the power dissipation per unit length in the (k-

th, j-th) conductor of the winding (2.20).

dPkj

dS=P0β

πr02 =

I02ρβ

2√2π2δr03ψ1(ξ) −

2√2ρβ

δr0H2ψ2(ξ) (2.19)

Pkj = ∫ ∫dPkj

dSr drdθ

2π

0

rc

0

=

=I2ρ

√2πδN0d0{ψ1(ξ) − 2π

2N0β [(k −1

2)2

+1

16+

1

2π2] ψ2(ξ)} (2.20)

P =∑∑Pkj

m′

j=1

m

k=1

=NI2ρ

√2πδN0d0[ψ1(ξ) −

π2N0β

24(16m2 − 1 +

24

π2)ψ2(ξ)] (2.21)

36

(2.21) is obtained from (2.20) and takes into account every strand in the wire, in which:

β packaging factor

m number of layers

m′ number of conductors per layer

N number of conductors in the winding

From (2.21) we can obtain the value of the ac resistance per unit length of round Litz wire

windings.

RAC =2P

I2=

√2Nρ

πδN0d0[ψ1(ξ) −

π2N0β

24(16m2 − 1 +

24

π2)ψ2(ξ)] (2.22)

Rb =Nρ

πN0δ2 (2.23)

(2.23) is the base resistance, used in order to normalize (2.22), thus we obtain the normalized

resistance Kr.

Kr =√2

ξ[ψ1(ξ) −

π2N0β

24(16m2 − 1 +

24

π2)ψ2(ξ)] (2.24)

dKrdξ

=d

dξψ1(ξ) −

π2N0β

24(16m2 − 1 +

24

π2)d

dξψ2(ξ) = 0 (2.25)

By putting dKr

dξ= 0, it is possible to find a point of minimum for values of ξ lower than 1.4 and

in order to simplify the calculations the values of ψ1(ξ) and ψ2(ξ) can be approximated by the

first terms of their Taylor-series expansion.

ψ1(ξ) = 2√2 (1

ξ+

1

3 ∗ 28ξ3 −

1

3 ∗ 214ξ5 +⋯) (2.26)

37

ψ1(ξ) =1

√2(−

1

25ξ3 +

1

212ξ7 +⋯) (2.27)

In Figure 2.9, the variation of the actual model of ψ1(ξ) and ψ2(ξ) waveforms with variable ξ

is shown

It can be seen that, for, ξ < 2, the approximate model of ψ1(ξ) and ψ2(ξ) is very close to their

actual model, thus by using (2.25), (2.26) and (2.27), we can obtain the optimum value of ξ,

thus the minimum value of the normalized resistance Kr is given by (2.29).

ξop = 4 √3

1 +π2N0β 4 (16m2 − 1 +

24π2)

4 (2.28)

Krop =8

ξop2 (2.29)

In planning stage is possible to size the strands diameter in an optimal manner, thus we can

reduce the losses and the resistance of the winding.

Figure 2.10-Comparison between actual and approximated model

38

By using (2.29) together with the equation related to the number of strands (2.30) we can obtain

the optimum diameter d0op.

N0 =βdc

2

d02 (2.30)

d0op = √−b + √b2 + 12δ4

2 (2.31)

Where b =π2β2

4(16m2 − 1 +

24

π2) dc

2 and it is a function of dc and m.

In order to compare this model with the ones used previously, in particular with the Dowell

model, it’s possible to use the dc resistance of the winding as a comparison parameter, thus we

normalize the ac resistance of the two models by dividing them to (2.32), obtaining (2.33) for

the Tourkhani-Viarouge model and (2.34) for the Dowell one.

RDC =Nρ

πN0δ2ξ2

(2.32)

Kd =ξ

√2[ψ1(ξ) −

π2N0β

24(16m2 − 1 +

24

π2)ψ2(ξ)] (2.33)

Kd−Dowell = ξ [φ1(ξ) +m2 − 1

3φ2(ξ)] (2.34)

φ1(ξ) =sinh2ξ + sin2ξ

cosh2ξ − cos2ξ (2.35)

φ2(ξ) = 2sinh2ξ − sin2ξ

cosh2ξ + cos2ξ (2.36)

The diversity in the conductors geometry introduces the need to modify in a proper manner the

parameter 𝜉, thus that the height of the foil conductors will be considered.

39

𝜉 =ℎ

δ (2.37)

The comparison between the normalized resistances Kd and Kd−Dowell, functions of 𝜉, shows

that for 𝜉 < 1, thus in the dc domain, the behaviour between the two methods is almost the

same (As we can see from Figure 2.11).

In the ac domain (𝜉 ≫ 1), when the frequency increases, instead, we have a significant gap

between the two curves, due to the presence of a losses factor related to skin effect and

proximity effect.

Hence, this new model is more accurate in its representation of a reality than the Dowell one.

This result is given by the fact that, in the equations, a linear distribution of the leakage field

along the winding layers.

This is the cause of the non uniformity of the current density, that, in turn, causes the

dependence of the resistance to the frequency.

Figure 2.11-Comparison between Dowell model and the new one

40

2.3.2-Acero-Hernandez-Burdio-Alonso-Barragan model [5]

The use of the previously presented model is often complicated in the design stage, thus this

simplified model is proposed that uses numeric calculus in order to calculate the ac

resistance.

In the following disquisition, taking advantage of cylindrical symmetry, an analytical model

for loss calculations in litz wire planar windings is developed and experimentally validated.

The model, developed by J.Acero, P.J. Hernandez, J.M. Burdio, R.Alonso e L.A. Barragan, is

based on the separation of loss contributions and in the analytical field calculations allowed by

the planar inductor geometry.

The total power losses PT in the litz wire winding have three parts.

First, losses due to Joule effect by the carrying current in the wire, Pcond (traditionally Pcond

was associated with skin effect).

Second, losses due to the eddy currents induced in each strand due to its vicinity to other

strands, Pind_int (these losses are, in fact, internal proximity losses).

Third, losses due to the eddy currents induced in the whole winding by the magnetic flux

created by itself, Pind_ext (traditionally Pind_ext was associated with the proximity effect and

in each turn they are caused by the magnetic field created by the rest of turns of the winding).

Thus, 𝑃𝑇 is given by the following equations:

PT = Pcond + Pind_int + Pind_ext (2.38)

PT = (Rcond + Rind_int + Rind_ext)I2

2 (2.39)

Taking into account the magnitude of the peak inductor current I (assumed to be sinusoidal),

these losses can be associated to three different resistance contributions and therefore the total

losses are written as (2.39).

A model is developed for each contribution.

Conduction resistance

Let r0 the radius of a copper strand, σ = 5.8.107(Ωm)−1 its conductivity and μr = 1 its

relative magnetic permeability.

41

Rcond, resistance in the wire is calculated by the parallel of the frequency-dependent resistance

(skin resistance) of n0 strands.

Then its resistance per unit of length is given by (2.40).

Rcond = −1

n0

ξ

2πr0σ ϕcond(ξr0) (2.40)

ϕcond(ξr0) =ber(ξr0)bei

′(ξr0)−ber′(ξr0)bei(ξr0)

ber′2(ξr0)+bei′2(ξr0)

(2.41)

ξ = √ωμσ =√2

δ (2.42)

According to its equation Rcond decreases when the number of strands n0 increases.

Internal proximity resistance

Let I the magnitude of the peak total current in the litz wire winding.

The current in each strand is given by (2.43), then the magnetic field at a distance p from the

strand is obtained by Ampere's Law (2.44).

Is =I

n0 (2.43)

Hϕ(p) =Is2πρ

(2.44)

42

As it can be seen in Figure 2.12, this strand is surrounded by the rest of the strands of the

bundle.

Then eddy currents will be induced in the rest of strands and, as a consequence, losses will take

place in them.

Each strand surrounding a particular strand acts, in fact, like a load for this strand and therefore

as much strands surround it as much losses will be induced.

By the other hand the induced losses are associated to a resistance and, for each strand, its value

will depends on the number of surrounding strands.

If different strands have different resistance, as the losses are caused by the conduction current,

the result is a non-uniform current distribution and an inefficient use of the conduction section

of the wire.

The effect of the transposition can be seen in Figure 2.13.

In this figure a length 𝜆𝑐 of a litz wire of diameter ϕc is considered.

Figure 2.12- Magnetic field at a distance ρ from a

strand carrying a current Is, in a litz wire

43

For a given strand, its load changes depends on the longitudinal coordinate.

Regarding the complete bundle, all strands possess the same transposition and, in average, all

strands cause the same losses.

For this reason, in all strands are induced the same currents and therefore the internal proximity

losses will be the losses in a single strand multiplied by the number of strands (2.45).

The averaged losses induced in a single strand of radius r0 and conductivity σ under an

alternating magnetic field H0, as shown in Figure 2.14, are (by unit of length) given by

(2.46).

Figure 2.13- A length 𝜆𝑐 of a litz wire.

A strand and its nearer loads are showed for the different 𝜆 coordinates.

Coordinate system is also showed.

44

𝑃𝑖𝑛𝑑_𝑖𝑛𝑡 = 𝑛0𝑃𝑖𝑛𝑡_ℎ (2.45)

Punit lenght = −2πξr0𝐻0

2

σϕcond(ξr0) (2.46)

ϕind(ξr0) =ber2(ξr0)ber

′(ξr0)+bei′(ξr0)bei2(ξr0)

ber2(ξr0)+bei2(ξr0) (2.47)

2πρH𝑜_𝑠 =1

𝜋𝑟𝑐2πρ2 → H𝑜_𝑠(ρ) =

1

2𝜋𝑟𝑐2ρ (2.48)

According to (2.46) the losses in the strand depend on the squared magnetic field.

Considering the litz wire as solid, as it is shown in Figure 2.15, the magnetic field in a strand

situated at ρ position is calculated by Ampere's Law (2.48).

Figure 2.14-Losses in a single strand under a spatially

uniform alternating magnetic field

45

H𝑜_𝑠2 =

1

𝑟𝑐∫

𝐼2

4𝜋2𝑟𝑐4

𝑟𝑐

0

ρ2 𝑑ρ (2.49)

𝛽 =𝑛0𝑟0

2

𝑟𝑐2 (2.50)

Punit lenght = −2𝜋ξr0ϕind

√2 𝜎H𝑜_𝑠2 = −𝑛0

𝛽ξI2ϕind

6√2 𝜋𝑟0𝜎 (2.51)

Rind_int_unit lenght = −𝑛0𝛽ξϕind

3√2 𝜋𝑟0𝜎 (2.52)

The radial transposition of the strand produces different fields at different positions,

consequently its losses will be proportional to the average of the squared field (2.49).

The averaged losses in a strand per unit of length are obtained from (2.46) and (2.49), and also

modified by using the packaging factor 𝛽 is defined by (2.50).

The total losses will be the sum of the losses of no strands, thus (2.51) is obtained and provided

that I is the total current, the resistance due to internal proximity per unit of length can be

identified by (2.52).

Figure 2.15-Magnetic field H𝑜_𝑠 in a strand situated at ρ position in the wire

46

External proximity resistance

In a particular turn of a planar winding, in addition to the magnetic field caused by the

neighbouring strands, other field created by the rest of turns will induce eddy currents in a

strand and, therefore, its losses will be increased.

The internal proximity losses in a strand, calculated in the precedent analysis, are caused by

the conduction current and in this case the external proximity losses are caused by the external

field.

The induced losses by an alternating magnetic field in a strand are obtained by (2.46).

In this case, the field H0 in a particular turn is the field created by the rest of the turns.

This field can be calculated by different ways, numerically by using a Finite Element Analysis

as well as analytically.

The analytical solution corresponds to a unique turn and we have generalized it for n turns as

it is shown in Figure 2.16.

In Figure 2.16 an induction system comprising the planar inductor and the load is shown.

The planar inductor consists of n litz wire concentric tums, each one of them has a radius 𝑎1.

In the analytical model this inductor is ideal and composed of filamentary currents.

The load consists of a medium characterized by its conductivity Q, and its relative magnetic

permeability μr.

Figure 2.16-A planar inductor of n turns; R1 and Rv are the

free-space and R2 is the load

47

Unlike small high frequency transformers for switching power supplies, where geometry

effects may be very important, in planar inductors the field in each turn due to the rest of turns

is almost constant.

Therefore we apply a calculation method mainly based on the basis of the uniformity of the

magnetic field inside each strand (the amplitude of B is assumed to be constant in the whole

section of the strand).

This is mostly true as far as the diameter of the strand is small enough compared with the skill

depth.

However the presence of the load changes hardly the field distribution and forces to calculate

it at different points inside the bundle as it is shown in Figure 2.17.

The external field depends on the strand position (rj, zk).

As above mentioned, in a litz wire the strands have transposition inside the bundle, then in this

case the average field will be the same for all strands and we can consider all of them

equivalent.

Let a strand in the i-turn of the planar inductor as it is shown in Figure 2.17 and (rj, zk) the

possible coordinates of the strand inside the bundle.

Figure 2.17-Components of the external magnetic field

𝐻𝑜,𝑖(𝑟𝑗, 𝑧𝑘) in a strand situated at position (𝜌, 𝜃) inside the

bundle for the i-turn of the planar inductor

48

The average of the squared magnetic field over this strand is given by the equation (2.53),

where nj and nk are the number of points in which the field is calculated.

Ho,i2 =

1

njnk∑∑[Hor,i

2 (rj, zk) + Hoz,i2 (rj, zk)]

nk

k=1

nj

j=1

(2.53)

As all the strands ate equivalents the losses per unit of length for the i -turn of the inductor are

calculated according to (2.51).

Taking into account that the magnetic external field is different in each turn, each of them

situated at a radius ai and, taking into account its length, the losses caused by the external field

in the whole planar inductor are calculated as follows.

Pind_ext = −𝑛02𝜋ξr0ϕind

√2 𝜎∑[2𝜋aiHo,i

2 ]

𝑛

𝑖=1

(2.54)

But the external field in each turn is caused by the total current I in the inductor then we can

link these losses with a resistance Rind_ext.

Let h0,i the magnetic field generated by 1 A over the i-turn of the inductor.

Then the average of the square field is given by (2.55), thus for the whole winding the external

proximity resistance results (2.56).

Ho,i2 = ho,i

2 I2 (2.55)

Rind_ext = −n0√2𝜋2𝜉𝑟0𝜙𝑖𝑛𝑑

𝜎∑[aiho,i

2 ]

n

i=1

(2.56)

Total resistance calculation

According to (2.40), (2.52) and (2.56), and taking into account the total length of the whole

winding, we can calculate the total resistance as

49

RT = −1

n0[ξϕcond

√2r0σ∑[ai]

n

i=1

] − n0 [√2𝛽ξϕind3𝑟0𝜎

∑[ai]

n

i=1

+√2𝜋2𝜉𝑟0𝜙𝑖𝑛𝑑

𝜎∑[aiho,i

2 ]

n

i=1

] (2.57)

In (2.57) the terms are divided by the square brackets into two groups: the first one denoted as

conduction resistance which origin is the losses caused by the current conduction and the

second one denoted induction resistance which origin is the induced losses.

As we can see, in this equation, the conduction resistance decreases if the number of strands

n0, is increased.

However the induction resistance is proportional to the number of strands and as a

consequence, for each geometry and number of tums it exists an optimum number of strands.

The frequency dependence of the total resistance is twofold.

First according to (2.42), the parameter 𝜉 has frequency dependence.

Second, the magnetic field created by 1 A of current, h0 depends strongly of the frequency as

a result of the load influence.

Figure 2.18-Total resistance of a winding and its components for a winding

made of litz wire with 0.4 mm diameter strands, at 30 kHz, and 25 ᵒC

50

2.4-Winding inductance

The determination of the system inductance is one of the main features that characterize the

study of the load, it is crucial in the planning phase in order to obtain the desired operation.

One of the principal problems related to the winding used in the induction cookers is the

absence of the core that allows the magnetic flux to flow through the winding, thus a significant

leakage is originated.

Hence is important to design the winding in an accurate way, such that the leakage is reduced.

In order to do that is important to create an accurate model of the system able to determine an

indicative coefficient, thus we need to take into account the entity of the forces in play, the

disturbances in the system to obtain a clear idea about the operation in resonant zone.

Figure 2.19-Total resistance of a winding and its components for a winding

made of litz wire with 0.4 mm diameter strands, at 30 kHz, and 250 ᵒC

Figure 2.20-Magnetic flux lines created by a winding of the inductor

51

In order to proceed with inductance calculations it is possible to use two different methods, the

first one based on Maxwell equations, the second based on the knowledge of the total energy

of the system.

This method, often preferred for computational reasons, is applicable in a few cases, when

there is a prototype allowing the calculation of the total energy.

In order to calculate the value of the inductance a simplification of the system is often used.

It consists in seeing the load as the composition of two superimposed and parallel coils, the

first one represents the inductor winding and the second one, fictitious, made by the currents

circulating in the bottom of the pot.

By using this simplification, it is possible to determine the coefficients of auto and mutual

inductance while considering a single central thread for every coil and applying to it the

formulas to calculate the mutual impedance.

The geometry of the winding, however, don’t allow this approximation because the ratio

between width and height of the winding section is not negligible.

Hence it is necessary to consider every thread by integrating the formula related to the mutual

inductance of a thread for the whole area that is considered.

Figure 2.21-Two coils model

52

In order to do that we have to add a magnetic substrate, which introduces eddy current losses

and takes into account the non-uniformity of the current density distribution, to the coil.

These considerations create the basis for the model conceived by W.G. Hurley and M.C. Duffy

that allow us to obtain a model closer to reality compared with the ones created in the past.

This model can be extended to a winding with some turns per layer and also few multiple

layers, taking into account the insurgence of capacities at high frequency, that can introduce an

unwanted resonance.

We start from Maxwell general formula of the mutual inductance between two filaments, in

which 𝐽1 is a Bessel function of the first kind, 𝑎, 𝑟 are the filament radii shown in Figure 2.22

and 𝜇0 is the permeability of free space.

The solution of (2.58) can be written in terms of 𝐾(𝑓) and 𝐸(𝑓), the elliptic integrals of first

and second kind respectively and 𝑓 is given by (2.59), thus we obtain (2.60).

𝑀 = 𝜇0𝜋𝑎𝑟∫ 𝐽1(𝑘𝑟)𝐽1(𝑘𝑎)𝑒−𝑘|𝑧| 𝑑𝑘 (2.58)

∞

0

𝑓 = √4𝑎𝑟

𝑧2 + (𝑎 + 𝑟)2 (2.59)

Figure 2.22-Geometry of the model

53

𝑀 = 𝜇0√𝑎𝑟2

𝑓 |(1 −

𝑓2

2)𝐾(𝑓) − 𝐸(𝑓)| (2.60)

The voltage induced in a filament at (𝑟, 𝜏1) in coil 1 due to the current in an annular section 𝑑𝑎

x 𝑑𝑟, at radius a in coil 2 is given by (2.61), where M is the mutual inductance between the

filaments at (𝑟, 𝜏1) and (𝑎, z + 𝜏2) and 𝐽 is the current density.

The total voltage at (𝑟, 𝜏1) due to all the current in coil 2 is obtained by integrating (2.61) over

the cross-section of coil 2, thus obtaining (2.62).

𝑑𝑉 = 𝑗𝜔𝑀𝐽(𝑎)𝑑𝑎 𝑑𝜏2 (2.61)

𝑉(𝑟) = 𝑗𝜔𝜇0𝜋𝑟∫ ∫ ∫ 𝑎𝐽1(𝑘𝑟)𝐽1(𝑘𝑎)𝐽(𝑎)𝑒−𝑘|𝑧+𝜏2−𝜏1| 𝑑𝑎 𝑑𝜏2 𝑑𝑘 (2.62)

𝑎2

𝑎1

−ℎ22

−ℎ22

∞

0

The power transferred to the annular segment at(𝑟, 𝜏1) due to coil 2 is given by (2.63) and the

total power transferred to coil 1 (2.64) is found by integrating it over its cross-section.

Hence it is necessary to consider the non uniformity of the current density 𝐽 in the cross-section

due to the difference of length between the external and internal edge.

In order to do that is necessary to write the formula that describe the distribution of 𝐽 as (2.65).

The total power transferred from coil 1 to coil 2 is calculated with (2.68) by using (2.67),

calculated through the total current given by equation (2.66).

𝑑𝑃 = 𝑉(𝑟)𝐽(𝑟)𝑑𝑟 𝑑𝜏1 (2.63)

𝑃 = 𝑗𝜔𝜇0𝜋∫ ∫ ∫ ∫ 𝑟𝐽1(𝑘𝑟)𝐽1(𝑘𝑎)𝐽(𝑟)𝐽(𝑎)𝑒−𝑘|𝑧+𝜏2−𝜏1| 𝑑𝑎 𝑑𝜏1 𝑑𝜏2 𝑑𝑘 (2.64)

𝑎2

𝑎1

𝑟2

𝑟1

ℎ22

−ℎ22

∞

0

𝐽 =𝐾

𝑟 (2.65)

𝐼 = ℎ∫ 𝐽(𝑟)𝑑𝑟 (2.66)𝑟2

𝑟1

54

𝐽 =𝐼

ℎ𝑟 ln (𝑟2𝑟1) (2.67)

𝑃 = 𝑗𝜔𝜇0𝜋𝐼1𝐼2

ℎ1 ln (𝑟2𝑟1) ℎ2 ln (

𝑎2𝑎1) ∫ 𝑆(𝑘𝑟2, 𝑘𝑟1)𝑆(𝑘𝑎2, 𝑘𝑎1)∞

0

𝑄(𝑘ℎ1, 𝑘ℎ2)𝑒−𝑘|𝑧|𝑑𝑘 (2.68)

In which:

𝑆(𝑘𝑥, 𝑘𝑦) =𝐽0(𝑘𝑥) − 𝐽0(𝑘𝑦)

𝑘 (2.69)

𝑄(𝑘𝑥, 𝑘𝑦) =

{

2

𝑘2[cosh (𝑘

𝑥 + 𝑦

2 ) − cosh (𝑘

𝑥 − 𝑦

2 )] 𝑧 >

ℎ1 + ℎ22

2

𝑘(ℎ +

𝑒−𝑘ℎ − 1

𝑘) 𝑧 = 0, 𝑥 = 𝑦 = ℎ

(2.70)

The power can be written in a general way with equation (2.71) where 𝑀12 is the mutual

inductance between the two coils (2.72) ad can be calculated by equating (2.71) and (2.68):

𝑃 = 𝑣2𝑖2 = 𝑗𝜔𝑀12𝐼1𝐼2 (2.71)

𝑀12 =𝜇0𝜋


𝑎2𝑎1) ∫ 𝑆(𝑘𝑟2, 𝑘𝑟1)𝑆(𝑘𝑎2, 𝑘𝑎1)∞

0

𝑄(𝑘ℎ1, 𝑘ℎ2)𝑒−𝑘|𝑧|𝑑𝑘 (2.72)

The obtained equations depends from many parameters related to the coil planar geometry and

from the distance of the windings.

55

Usually in order to calculate auto and mutual inductance z is replaced by the Geometric Mean

Distance (GMD) between the coils.

In the case of self inductance, z is replaced by the GMD of the coil from itself

𝐺𝑀𝐷 = 0.2235(w + h) (2.73)

In the case of mutual inductance, by using the model in Figure 2.23, it is possible to obtain the

coefficients 𝑀12 and 𝑀13, in case of 𝑧 ≠ 0, modifying the value of z with the GMD

In order to calculate the mutual inductance between two inductors at the same z, instead, as in

the case of 𝑀14, we just need to set z equal to zero, thus the radial dimensions are expressed by

(2.74).

The mutual inductance (2.75) is then obtained by intermediating the mutual inductances

between the filaments of every section, each of which conducts half the total current.

𝑟12 = 𝑅 (1 +ℎ2

24𝑅2) ± √

𝑤2 − ℎ2

12 (2.74)

𝑀14 =𝑀𝑎𝑐 +𝑀𝑎𝑑 +𝑀𝑏𝑐 +𝑀𝑏𝑑

4 (2.75)

Where 𝑎 and 𝑏 are the filaments in a transversal section, while 𝑐 and 𝑑 are those in the other

section.

In order to validate the model an experimental device has been constructed with the dimensions

shown in Figure 2.22, subjecting it to the following conditions:

Figure 2.23-Geometry of the conductor

disposition

1-Self Inductance 𝐿1

2-Mutual Inductance 𝑀12,𝑀13, 𝑧 ≠ 0

3-Mutual Inductance 𝑀14, 𝑧 = 0

56

Measurement carried out at 10 kHz to avoid high frequency effects.

Finite Element Analysis carried out until 10 kHz.

Numerical evaluation of the mutual inductance

Maxwell formula related to the mutual inductance is applied on filaments placed at a

distance equal to the geometric distance between internal and external radius of the coil.

𝑟0 = √𝑟1𝑟2 (2.76)

Maxwell formula related to the mutual inductance is applied on filaments placed at a

distance equal to the geometric distance between the center of sections.

The results presented in Figure 2.24 show that the geometric mean (GM) approximation gives

the most accurate estimation of the inductance.

In order to obtain values closer to reality a ferromagnetic substrate is inserted to simulate the

presence of the pot as shown in Figure 2.25.

Figure 2.24-Experimental results (values in nH)

57

The mutual impedance between the two filamentary circular concentric turns of Figure 2.25 is

given by (2.77), where M is the mutual inductance which would exist in the absence of the

substrate and is the same as (2.58).

𝑍𝑠 is the additional impedance due to the presence of the substrate, as shown in equation (2.78),

that can be modified by considering the inverse proportionality of the current density with

respect to the radius and, by applying the same procedure used for the model without the

substrate, we obtain (2.81)

𝑍 = 𝑗𝜔𝑀 + 𝑍𝑠 (2.77)

𝑍𝑠 = 𝑅𝑠 + 𝑗𝜔𝐿𝑠 = 𝑗𝜔𝜇0𝜋𝑎𝑟∫ ∫ 𝐽1(𝑘𝑎)𝐽1(𝑘𝑟)𝑎2

𝑎1

∞

0

𝜑(𝑘)𝑒−𝑘(𝑑1+𝑑2)𝑑𝑘 (2.78)

𝜑(𝑘) =𝜇𝑟 −

𝜂𝑘

𝜇𝑟 +𝜂𝑘

(2.79)

𝜂 = √𝑘2 + 𝑗𝜔𝜇0𝜇𝑟𝜎 (2.80)

𝑍𝑠 =𝑗𝜔𝜇0𝜋


𝑎2𝑎1) ∫ 𝑆(𝑘𝑟2, 𝑘𝑟1)𝑆(𝑘𝑎2, 𝑘𝑎1)∞

0

𝑄(𝑘ℎ1, 𝑘ℎ2)𝜑(𝑘)𝑒−𝑘(𝑑1+𝑑2)𝑑𝑘 (2.81)

The frequency dependent mutual impedance formula (2.81) takes full account of eddy current

losses in the substrate.

Figure 2.25-Model with ferromagnetic substrate

58

The resistive component of 𝑍𝑠 represents the substrate losses and the reactive component of 𝑍𝑠

represents the enhanced inductance due to the reflected field of the magnetic substrate.

As shown in Figure 2.26, there is very good agreement between the simulated and calculated

results which establishes the validity of the proposed formula in predicting the effect of a

magnetic substrate on the inductance and on the losses in a planar magnetic device.

The most salient feature of Figure 2.26 is that the inductance remains essentially flat up to 1

MHz and rapidly decreases for higher frequencies, while the resistance behaves in a opposite

way due to the variation of the distribution of the current density.

2.5-Acero-Burdio-Alonso-Barragan model for impedance

calculation

After the analysis of the main two parameters of the system model, it is necessary to take into

account the coupling between the winding and the cookware.

For that reason is possible to determine an equation able to describe, at the same time, both the

resistance and the inductance of the system.

Figure 2.26-Comparison between the measured inductance

and resistance and the one calculated with the model

59

Hence, two approaches developed by J.Acero, J.M. Burdio, R.Alonso e L.A. Barragan are

presented, allowing us to find an analytical model that include the dependency from the

frequency, the electromagnetic parameters and the geometrical dimensions of the system.

Those two methods are very similar, the only difference lies in the presence or the absence of

the ferrite disc placed under the inductance winding.

2.5.1-Model without the ferrite disc [6]

Figure 2.27 shows the schematic diagram of an induction heating system consisting of a n-turn

planar winding loaded by a linear, homogeneous and isotropic medium characterized by its

magnetic permeability 𝜇 = 𝜇0𝜇𝑟 and its electrical conductivity 𝜎.

The load is assumed to be infinite in the 𝑧 > 0 direction.

The winding consists of n filamentary circular concentric currents, representing a flat spiral

coil.

Figure 2.27-Model for inductance calculation

60

The system possesses cylindrical symmetry, where 𝑎1 and 𝑎2 are the inner and outer winding

radii, respectively.

The winding is placed at a distance 𝑧 = 𝑑 from the load interface.

In the magnetostatic case it is possible to write the Maxwell equations (2.82) and (2.83), that

based on the system geometry can be expressed in cylindrical coordinates.

∇x𝐻 = 𝐽𝜑 (2.82)

∇x𝐸 = −𝜕𝐵

𝜕𝑡 (2.83)

𝐸𝑟 = 0 , 𝐸𝑧 = 0 ,𝜕𝐸𝜑

𝜕𝜑= 0 (2.84)

𝐻𝜑 = 0 , 𝜕𝐻𝑟𝜕𝜑

= 0 , 𝜕𝐻𝑧𝜕𝜑

= 0 (2.85)

By considering a sinusoidal current equal to (2.86) inside the coil in the position 𝑧 = 𝑑1 = 𝑑

and, by applying the relations written above, we obtain (2.87), (2.88) and (2.89) for the air path

(𝑧 > 0) and (2.90), (2.91) and (2.92) for the magnetic path (𝑧 < 0).

𝐼𝜑(𝑡) = 𝐼𝜑𝑒𝑗𝜔𝑡 (2.86)

𝜕𝐻𝑟𝜕𝑧

−𝜕𝐻𝑟𝜕𝑧

= 𝐼𝜑𝛿(𝑟 − 𝑎)𝛿(𝑧 − 𝑑) (2.87)

𝜕𝐸𝜑

𝜕𝑧= 𝑗𝜔𝜇0𝐻𝑟 (2.88)

1

𝑟

𝜕(𝑟𝐸𝜑)

𝜕𝑟= −𝑗𝜔𝜇0𝐻𝑧 (2.89)

𝜕𝐻𝑟𝜕𝑧

−𝜕𝐻𝑟𝜕𝑧

= 𝜎𝐸𝜑 (2.90)

𝜕𝐸𝜑

𝜕𝑧= 𝑗𝜔𝜇0𝜇𝑟𝐻𝑟 (2.91)

1

𝑟

𝜕(𝑟𝐸𝜑)

𝜕𝑟= −𝑗𝜔𝜇0𝜇𝑟𝐻𝑧 (2.92)

Equation (2.91) is due to the presence of magnetic substrate and the fact that the electric field

has only the component along the coordinate 𝜑.

61

If we cancel H from the relations related to the path in air we obtain equation (2.93) and, by

applying the Fourier-Bessel transform, we get (2.94).

𝜕2𝐸

𝜕𝑧2+𝜕2𝐸

𝜕𝑟2+1

𝑟

𝜕𝐸

𝜕𝑟=𝐸

𝑟2+ 𝑗𝜔𝜇0𝐼𝜑𝛿(𝑟 − 𝑎)𝛿(𝑧 − 𝑑) (2.93)

𝜕2𝐸

𝜕𝑧2= 𝑘2𝐸∗ + 𝑗𝜔𝜇0𝐼𝜑𝑎𝐽1(𝑘𝑎)𝛿(𝑧 − 𝑑) =

= 𝑘2𝐸∗ + 𝑗𝜔𝜇0𝐼𝜑𝛿(𝑧 − 𝑑)∫ 𝛿(𝑟 − 𝑎)𝐽1(𝑘𝑟)𝑟 𝑑𝑟∞

0

(2.94)

The solution has the form given by (2.95).

Above the coil the field goes to the infinity, thus it becomes (2.96).

In the region between the coil and the surface (2.95) can be written as (2.97).

𝐸∗ = 𝐴𝑒−𝑘𝑧 + 𝐵𝑒𝑘𝑧 (2.95)

𝐸∗ = 𝐴𝑒−𝑘𝑧 𝑧 ≥ 𝑑 (2.96)

𝐸∗ = 𝐶𝑒−𝑘𝑧 + 𝐵𝑒𝑘𝑧 0 ≤ 𝑧 ≤ 𝑑 (2.97)

If the same procedure is applied to the equations regarding the ferromagnetic path, we obtain

(2.99) and, taking into account that the electric field tends to zero as 𝑧 goes to infinity, the

solution of (2.99) is given by (2.100).

𝜕2𝐸

𝜕𝑧2+𝜕2𝐸

𝜕𝑟2+1

𝑟

𝜕𝐸

𝜕𝑟=𝐸

𝑟2+ 𝑗𝜔𝜇0𝜇𝑟𝜎𝐸 (2.98)

𝜕2𝐸

𝜕𝑧2= 𝑘2𝐸∗ + 𝑗𝜔𝜇0𝜇𝑟𝜎𝐸

∗ (2.99)

𝐸∗ = 𝐷𝑒𝜂𝑧 (2.100)

With 𝜂 = √𝑘2 + 𝑗𝜔𝜇0𝜇𝑟𝜎.

Then by applying the boundary conditions to the equations it is possible to find the coefficients,

thus we consider:

62

The continuity of the electric field E at the boundary (z=0):

𝐵 + 𝐶 = 𝐷 (2.101)

The continuity of the radial component of the magnetic field H at the boundary, thus

we get (2.102), from which it is possible to obtain, by considering the general solutions

of the transform of the Fourier-Bessel integral seen previously, equation (2.103).

𝜕𝐸

𝜕𝑧= 𝑗𝜔𝜇0𝜇𝑟𝐻𝑟 (2.102)

𝐷 = 𝜇𝑟𝑘

𝜂 (𝐵 − 𝐶) (2.103)

The continuity of the electric field E in the plane of the coil (z=d), thus:

𝐴𝑒−𝑘𝑑 = 𝐵𝑒𝑘𝑑 + 𝐶𝑒−𝑘𝑑 (2.104)

The condition at the boundary of the magnetic field H in the plane of the coil (z=d),

thus:

𝜂 ̅x (𝐻+̅̅ ̅̅ − 𝐻−̅̅ ̅̅ ) = 𝐾𝑓 (2.105)

Where 𝜂 ̅represents the unitary vector perpendicular to the plane of the coil and the superficial

current density at the boundary is given by (2.106).

If we value the second condition at the boundary it is possible to obtain the terms 𝐻+ and 𝐻−,

thus we get (2.107).

𝐾𝑓 = ∫ 𝐼𝜑𝛿(𝑟 − 𝑎)𝛿(𝑧 − 𝑑)𝑑𝑧 =𝑑+

𝑑−

𝐼𝜑𝛿(𝑟 − 𝑎) (2.106)

𝑘𝐴𝑒−𝑘𝑑 − 𝑘(𝐵𝑒𝑘𝑑 + 𝐶𝑒−𝑘𝑑) = 𝑗𝜔𝜇0𝐼𝜑𝑎 𝐽1(𝑘𝑎) (2.107)

63

By putting in a system the acquired equations, it is possible to determine the coefficients A,B,C

and D, thus we can find the relation (2.108), where 𝜙(𝑘) =𝜇𝑟−

𝜂

𝑘

𝜇𝑟+𝜂

𝑘

.

Using the inverse transform of the Fourier-Bessel integral we obtain equation (2.109), valid for

the electric field E in the air, and, by considering the i-th coil in a position (r,z) with

−𝑑 ≤ 𝑧 < 0, we get (2.110).

𝐸∗ = −𝑗𝜔𝜇0𝐼𝜑𝑎

2𝑘[𝑒−𝑘|𝑧−𝑑| + 𝜙(𝑘)𝑒−𝑘(𝑧+𝑑)]𝐽1(𝑘𝑎) (2.108)

𝐸 = −𝑗𝜔𝜇0𝐼𝜑𝑎

2∫ [𝑒−𝑘|𝑧−𝑑| + 𝜙(𝑘)𝑒−𝑘(𝑧+𝑑)]𝐽1(𝑘𝑎)𝐽1(𝑘𝑟)𝑑𝑘∞

0

(2.109)

𝐸𝜙,𝑖(𝑟, 𝑧) = −𝑗𝜔𝜇0𝐼𝜑𝑎

2∫ 𝑒−𝑘𝑑[𝑒−𝑘𝑧 − 𝜙(𝑘)𝑒𝑘𝑧]𝐽1(𝑘𝑎)𝐽1(𝑘𝑟)𝑑𝑘∞

0

(2.110)

Given the intent to calculate the total electric field of the winding, thus determining the load

impedance, it is necessary to verify the linearity of the problem, taking into account all the

elements that can introduce any non-linearity, like magnetic hysteresis and saturation.

These phenomena, in the case of induction cooking applications, aren’t a problem, in fact the

containers are usually made by soft magnetic materials, characterized by a weak residual

magnetization.

The total electric field in position (r,z) is given by (2.111), in which n is the number of turns.

From this equation it is possible to obtain the voltage induced between the terminals of the

winding as the sum of the difference of potential at the terminals of each turn and we get (2.112)

and if we assume the electric field constant we obtain (2.113).

𝐸𝜙(𝑟, 𝑧) =∑𝐸𝜙,𝑖(𝑟, 𝑧)

𝑛

𝑖=1

(2.111)

𝑉 = −∮𝐸𝜙 𝑑𝑙 = −∑∫ 𝐸𝜙(𝑟 = 𝑎𝑗 , 𝑧 = −𝑑)𝑎𝑗 𝑑𝜙 (2.112)2𝜋

0

𝑛

𝑗=1

𝑉 = −∑∑∫ 𝐸𝜙(𝑟 = 𝑎𝑗 , 𝑧 = −𝑑)𝑎𝑗 𝑑𝜙 = − ∑∑2𝜋𝑎𝑗𝐸𝜙,𝑗,𝑖

𝑛

𝑖=1

𝑛

𝐽=1

(2.113)2𝜋

0

𝑛

𝑖=1

𝑛

𝐽=1

64

We define the geometric function 𝑇(𝑘) and by combining it with the previous equations we

get (2.115).

𝑇(𝑘) =∑𝑎𝑖2𝐽12(𝑘𝑎𝑖)

𝑛

𝑖=1

+ 2∑∑𝑎𝑖𝑎𝑗𝐽1(𝑘𝑎𝑖)𝐽1(𝑘𝑎𝑗) (2.114)

𝑛

𝑖=1

𝑛

𝐽=1

𝑉 = 𝑗𝜔𝜇0𝜋𝐼𝜙∫ [1 − 𝜙(𝑘)𝑒−2𝑘𝑑]𝑇(𝑘)𝑑𝑘 (2.115)∞

0

By dividing (2.115) by the circulating current in the winding, it is possible to obtain the

impedance Z, from which we get the equivalent resistance and inductance due to the

contribution of the winding (𝑅0, 𝐿0) and of the material (∆𝑅, ∆𝐿) , equal to the real and

imaginary part of Z respectively.

𝑍 = 𝑅𝑒𝑞 + 𝑗𝜔𝐿𝑒𝑞 =𝑉

𝐼𝜙= 𝑗𝜔𝜇0𝜋∫ [1 − 𝜙(𝑘)𝑒−2𝑘𝑑]𝑇(𝑘)𝑑𝑘 (2.116)

∞

0

𝑅𝑒𝑞 = 𝑅0 + ∆𝑅 = 𝑅0 + 𝜔𝜇0𝜋∫ 𝜙𝑖(𝑘)𝑒−2𝑘𝑑𝑇(𝑘)𝑑𝑘 (2.117)

∞

0

𝐿𝑒𝑞 = 𝐿0 + ∆𝐿 = 𝜇0𝜋∫ 𝑇(𝑘)𝑑𝑘∞

0

− 𝜇0𝜋∫ 𝜙𝑟(𝑘)𝑒−2𝑘𝑑𝑇(𝑘)𝑑𝑘 (2.118)

∞

0

If the inductor winding is made by Litz conductors and we neglect the proximity effect because

of the strand diameter, then it is possible to demonstrate that the first term of the resistive

component is given by (2.119), in which 𝜉 is the inverse of the skin depth.

Assuming the conductor as round, with a diameter 𝜙0 = 2𝑟0, and the power equal to P, the

component due to the load is given by (2.120).

𝑅0 =ξ

𝜎r0

𝑏𝑒𝑟(ξr0)𝑏𝑒𝑖′(ξr0) − 𝑏𝑒𝑟′(ξr0)𝑏𝑒𝑖(ξr0)

𝑏𝑒𝑟′2(ξr0) + 𝑏𝑒𝑖′2(ξr0)

∑𝑎𝑖 (2.119)

𝑖

∆𝑅 =2𝑃

𝐼𝜙2 (2.120)

65

From Figure 2.28 we can observe the characteristics of R as functions of the frequency and

their dependence to the skin effect and the different behaviour in case of different materials.

Regarding the inductive component of the impedance, instead, the contribution 𝐿0 due to the

winding is given by (2.121), where the first and the second term represents the sum of the self-

inductance of every turn 𝐿𝑖𝑖 and the mutual-inductance 𝑀𝑖𝑗 between two coplanar turns,

respectively.

Considering a strand with a diameter equal to 𝜙0 and that the distance from the centre of the

coil 𝑎𝑖 will be greater than the radius of the conductor 𝑟0, 𝐿𝑖𝑖 and 𝑀𝑖𝑗 can be expressed by

(2.122) and (2.123) respectively, in which 𝐾(𝑥) and 𝐸(𝑥) are elliptic integrals.

𝐿0 = 𝜋𝜇0 [∑𝑎𝑖2∫ 𝐽1

2(𝑘𝑎𝑖)𝑑𝑘∞

0

𝑛

𝑖=1

+ 2∑∑∫ 𝑎𝑖𝑎𝑗𝐽1(𝑘𝑎𝑖)𝐽1(𝑘𝑎𝑗)∞

0

𝑛

𝑖=1

𝑛

𝐽=1

𝑑𝑘] (2.121)

𝐿𝑖𝑖 = 𝜇0𝑎𝑖 [ln (8𝑎𝑖𝑟0) − 2] (2.122)

𝑀𝑖𝑗 = 𝜇0√𝑎𝑖𝑎𝑗 2

𝑥[(1 −

x2

2)𝐾(𝑥) − 𝐸(𝑥)] (2.123)

Figure 2.28-Comparison between the measured and calculated

resistance, as a function of frequency, for different materials

66

2.5.2-Model with the ferrite disc [7]

In order to be as faithful as possible to reality, it is necessary to develop a model that takes into

account the presence of the ferrite disc placed under the inductor winding.

Its function is to improve the coupling between the winding and the load and to shield the

electronic components.

The structure obtained by adding the ferrite disc, made by four regions, is shown in Figure 2.30

Figure 2.29-Comparison between the measured and calculated

inductance, as a function of frequency, for different materials

67

Hence, a model is proposed, characterized by n circular concentric currents regarding the planar

winding, inserted between two means that are linear, homogeneous and isotropic.

We approximate the system in a quasi-static way, given the small dimensions of the involved

waveforms compared to their length, thus it is possible to consider as valid the differential

equation (2.124), taken from the Maxwell equations as in the previous discussion, but written

in terms of the potential magnetic vector A.

By using this parameter and the Hankel transform, we obtain the general solution (2.125),

where 𝜂𝑘 = √𝛽2 + 𝑗𝜔𝜇𝑟𝑘𝜇0𝜎𝑘.

We indicate with 𝛽 the integral core, 𝐽1 the first order Bessel function of the first kind and with

B and C the coefficients dependent from the boundary conditions, we obtain the expressions

describing A for each region.

The value of 𝜙𝑘 in the equations from (2.126) to (2.129) depends from the integration variable

𝛽, the frequency and the proprieties of the material, according to the relation (2.130).

Taking advantage of the fact that the system is linear, it is possible to use the superposition of

the effects principle, thus we obtain equation (2.131), through which we can find the value of

the total 𝐴𝜙(𝑟, 𝑧), where p is the number of turns in the winding.

Figure 2.30-Ferrite disc model

68

∇2�̅� = −𝜇𝐽 ̅ = −𝜇(𝐽�̅�𝑜𝑏𝑖𝑛𝑎 − 𝑗𝜔𝜎�̅�) (2.124)

𝐴𝜙(𝑟, 𝑧) = ∫ [𝐵𝑘(𝛽)𝑒−𝜂𝑘𝑧 + 𝐶𝑘(𝛽)𝑒

𝜂𝑘𝑧]∞

0

𝐽1(𝛽𝑟)𝛽 𝑑𝛽 (2.125)

𝐴1𝜙 =𝜇0𝐼𝜙𝛼

2∫ 𝑒−𝜂1(𝑧−𝑑)

(1 − 𝜙1)(1 + 𝜙4𝑒−2𝛽ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)

𝐽1(𝛽𝛼)𝐽1(𝛽𝑟) 𝑑𝛽 (2.126)∞

0


2∫ 𝑒−𝛽𝑧

(1 + 𝜙1𝑒−2𝛽(𝑑−𝑧))(1 + 𝜙4𝑒

−2𝛽ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝐽1(𝛽𝛼)𝐽1(𝛽𝑟) 𝑑𝛽 (2.127)

∞

0


2∫ 𝑒−𝛽𝑧

(1 + 𝜙1𝑒−2𝛽𝑑)(1 + 𝜙4𝑒

−2𝛽(ℎ+𝑧))

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)

𝐽1(𝛽𝛼)𝐽1(𝛽𝑟) 𝑑𝛽 (2.128)∞

0


2∫ 𝑒−𝜂4(𝑧+ℎ)

(1 + 𝜙1𝑒−2𝛽𝑑)(1 − 𝜙4)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝑒−𝛽ℎ𝐽1(𝛽𝛼)𝐽1(𝛽𝑟) 𝑑𝛽 (2.129)

∞

0

𝜙𝑘(𝛽) =𝛽𝜇𝑟𝑘 − 𝜂𝑘𝛽𝜇𝑟𝑘 + 𝜂𝑘

(2.130)

𝐴𝜙(𝑟, 𝑧) = ∑𝐴𝜙,𝑝(𝑟, 𝑧) (2.131)

𝑛

𝑝=1

The voltage at position 𝑧 = 0 with A constant along the coil is given by (2.132), in which

𝐴𝜙,𝑝(𝑎𝑞, 0) indicates the vector of the potential generated in the pth coil at position q.

69

In order to calculate the impedance Z, we divide equation (2.132) by the circulating current

and we impose 𝑧 = 0 in one of the relations of A related to the regions affected by the coil,

thus zone 2 and zone 3 (equations (2.127) and (2.128)).

Hence we obtain (2.132), in which 𝑇(𝛽) represents a geometric function, expressed by (2.133).

In order to find the equations describing 𝑅𝑒𝑞 and 𝐿𝑒𝑞 we use the same method as the one used

in the case without the ferrite disc.

𝑉 = 𝑗𝜔∮𝐴𝜙 𝑑𝑙 = 𝑗𝜔∑∫ 𝐴𝜙(𝑟 = 𝑎𝑞 , 𝑧 = 0)𝑎𝑞𝑑𝜙 = 𝑗𝜔∑∑𝑎𝑞𝐴𝜙,𝑝(𝑎𝑞, 0) (2.132)

𝑛

𝑝=1

𝑛

𝑞=1

2𝜋

0

𝑛

𝑞=1

𝑍 = 𝑅𝑒𝑞 + 𝑗𝜔𝐿𝑒𝑞 =𝑉

𝐼𝜙= 𝑗𝜔𝜇0𝜋∫

(1 + 𝜙1𝑒−2𝛽𝑑)(1 + 𝜙4𝑒

−2𝛽ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝑇(𝛽)𝑑𝛽 (2.133)

∞

0

𝑅𝑒𝑞 = 𝑅0 + ∆𝑅 =

= 𝑅𝑒 {𝑗𝜔𝜇0𝜋∫ 𝑇(𝛽)𝑑𝛽∞

0

} +

+𝑅𝑒 {𝑗𝜔𝜇0𝜋∫𝜙1𝑒

−2𝛽𝑑 + 𝜙4𝑒−2𝛽ℎ + 2𝜙1𝜙4𝑒

−2𝛽(𝑑+ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝑇(𝛽)𝑑𝛽

∞

0

} (2.134)

𝐿𝑒𝑞 = 𝐿0 + ∆𝐿 =

= 𝐼𝑚 {𝑗𝜔𝜇0𝜋∫ 𝑇(𝛽)𝑑𝛽∞

0

} +

+𝐼𝑚 {𝑗𝜔𝜇0𝜋∫𝜙1𝑒

−2𝛽𝑑 + 𝜙4𝑒−2𝛽ℎ + 2𝜙1𝜙4𝑒

−2𝛽(𝑑+ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝑇(𝛽)𝑑𝛽

∞

0

} (2.135)

The resistance 𝑅0, as in the previous case, is calculated considering the skin effect and without

neglecting the proximity effect starting from equation (2.136).

Hence, considering 𝑅0 as made by the contributions of induction (2.138) and conduction

(2.139), we can describe it using (2.137), in which the magnetic field (2.140) is given by the

sum of (2.141) and (2.142).

70

𝑃𝑤 =1

2(𝑅𝑐𝑜𝑛𝑑 + 𝑅𝑖𝑛𝑑)𝐼𝜙

2 =1

2𝑅0𝐼𝜙

2 (2.136)

𝑅0 = 𝑅𝑐𝑜𝑛𝑑 + 𝑅𝑖𝑛𝑑 =2ξ

𝜎𝜙0𝜙𝑐𝑜𝑛𝑑∑𝑎𝑖 +

4𝜋2ξr0

√2𝜎𝜙𝑖𝑛𝑑∑[𝑎𝑖(𝐻0,𝑖

𝑇 )2] (2.137)

𝑛

𝑖=1

𝑛

𝑖=0

𝜙𝑖𝑛𝑑(ξr0) =𝑏𝑒𝑟2(ξr0)𝑏𝑒𝑟

′(ξr0) − 𝑏𝑒𝑖′(ξr0)𝑏𝑒𝑖2(ξr0)

𝑏𝑒𝑟2(ξr0) + 𝑏𝑒𝑖2(ξr0) (2.138)

𝜙𝑐𝑜𝑛𝑑(ξr0) =𝑏𝑒𝑟(ξr0)𝑏𝑒𝑖

′(ξr0) − 𝑏𝑒𝑟′(ξr0)𝑏𝑒𝑖(ξr0)

𝑏𝑒𝑟′2(ξr0) + 𝑏𝑒𝑖′2(ξr0)

(2.139)

�̅�(𝑟, 𝑧) = 𝐻𝑧(𝑟, 𝑧)𝑒�̅� + 𝐻𝑟(𝑟, 𝑧)𝑒�̅� (2.140)

𝐻𝑧(𝑟, 𝑧) =𝐼𝜙

2∫ 𝛽𝑒−𝛽𝑧

(1 + 𝜙1𝑒−2𝛽(𝑑−𝑧))(1 + 𝜙4𝑒

−2𝛽ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝜓(𝛽)𝐽0(𝛽𝑟) 𝑑𝛽 (2.141)

∞

0

𝐻𝑟(𝑟, 𝑧) =𝐼𝜙

2∫ 𝛽𝑒−𝛽𝑧

(1 + 𝜙1𝑒−2𝛽(𝑑−𝑧))(1 + 𝜙4𝑒

−2𝛽ℎ)

1 − 𝜙1𝜙4𝑒−2𝛽(𝑑+ℎ)𝜓(𝛽)𝐽1(𝛽𝑟) 𝑑𝛽 (2.142)

∞

0

Where 𝐽0 is the zero order Bessel function of the first kind and 𝜓(𝛽)is a function dependant

from the geometry of the winding.

The resistive component due to the induced currents is expressed by ∆𝑅.

Given the presence of the ferrite disc, theoretically, the induced current in it should be taken

into consideration, but they are negligible due to the almost null conductivity of the ferritic

material.

71

As we can see from the comparison between Figure 2.31 and 2.32, the contribution given to

∆𝑅 by the number of turns is considerable.

Noticeable is the increase in the proximity effect as the working frequency increases.

In order to calculate the inductance is important to consider the presence of the ferrite disc, in

fact the component ∆𝐿 is due to, not only, the ferromagnetic load, but also the disc itself.

This change in respect to the case without the ferrite disc is positive because the variation of

∆𝐿 is reduced.

Figure 2.31- Comparison between the measured and calculated resistance and inductance, as a

function of frequency, for different materials and a winding made by one turn

Figure 2.32- Comparison between the measured and calculated resistance and inductance, as a

function of frequency, for different materials and a spiral coil

72

Comparing the curves obtained with the model and the ones obtained in an experimental way,

we can conclude that the model developed by J.Acero, J.M. Burdio, R.Alonso e L.A. Barragan

is the closer to reality.

73

Chapter 3-Induction cookers’ power electronics [9]

Now we will analyse the power electronics constituent of an induction cooker, briefly

introduced in section 1.6 and shown in Figure 3.1.

3.1-EMC filter [10]

Electromagnetic compatibility (EMC) is the branch of electrical sciences which studies the

unintentional generation, propagation and reception of electromagnetic energy with reference

to the unwanted effects (Electromagnetic interference, or EMI) that such energy may induce.

The goal of EMC is the correct operation, in the same electromagnetic environment, of

different equipment which use electromagnetic phenomena, and the avoidance of any

interference effects.

In order to achieve this, EMC pursues two different kinds of issues:

Emission issues are related to the unwanted generation of electromagnetic energy by

some source, and to the countermeasures which should be taken in order to reduce

such generation and to avoid the escape of any remaining energies into the external

environment.

Figure 3.1-Induction cooker power

electronics main blocks

74

Susceptibility or Immunity issues, in contrast, refer to the correct operation of

electrical equipment, referred to as the victim, in the presence of unplanned

electromagnetic disturbances.

In order to reduce EMI inside the induction cooker, an EMC filter has to be connected to the

mains.

In fact although circuits may be well screened to prevent any signal radiated or being picked

up by the circuit itself, there are always interconnections to and from the electronics circuit.

These wires themselves can conduct unwanted signals into and out of the unit. If the unit is to

be able to meet its electromagnetic compatibility, EMC requirements and pass its EMC

testing, it is necessary to reduce the levels of unwanted signals that can enter or leave the unit

via its interconnections.

The idea is that the interfering signals generally have a frequency above that of the signals

normally travelling along the wire or line, thus by having what is termed a low pass filter as

the EMC filter, only the low frequency signals are allowed to pass, and the high frequency

interference signals are removed.

The EMC filters may categorised into two main types. One is where the unwanted energy is

absorbed by the EMC filter. The other type of filter rejects the unwanted signal and in this

case it is reflected back along the line.

For EMC filtering applications, the absorptive type is preferred.

These EMC filters can be in one of a variety of formats, for more exacting requirements,

these may need to be made up from a number of components.

In case of induction cookers the EMC filter is composed by a capacitance in parallel with an

inductance, as shown in the picture below.

Figure 3.2-EMC filter

75

3.2-Rectifier circuits

These are circuits fundamental for the inverter’s functioning and enable to modify the voltages

and the currents circulating in the circuit, thus the correct functioning of the system can be

achieved.

In order to convert the alternating current into continuous current, the use of static converters

is common and the most used solution is the Graetz bridge.

Graetz bridges are composed by diodes and take advantage of their blocking capacity in order

to reverse the negative half-wave of the input sinusoid from the net.

Hence the output signal have a continuous component, due to the unipolar nature of the rectified

wave, and a pulsating component.

The simplicity of the structure and the cheapness of the used components aided the spread of

the Graetz bridge in numerous applications.

The use of the rectifier allows us to approximate the mains, the EMC filter and the rectifier

altogether to a DC source.

This approximation will be taken into account while dealing with the simulations in Chapter 5.

3.3-Protection circuits [12]

The protection circuits (snubbers) are additional circuits used in order to reduce the

elctrodynamic strain on a device or a semiconductor during the commutation transient.

Figure 3.3-Mono-phase Graetz bridge

76

They allow us to maintain the values of voltage or current inside the safe operating area

(SOA) stated by the manufacturer.

Furthermore the snubbers can be used as energy recovery circuits, thus they increase the

efficiency of the system and limit the commutation losses.

In induction cookers protection circuits are imperative, in fact the topological configuration

of the used circuit, along with the inductive load subjected to abrupt interruptions, generates

peaks of current that are otherwise unbearable for the system.

In summary the protection circuit can be used to:

Limit the voltage applied to the device during the switching off transients.

Limit the value of dv

dt during the switching off transients or when a direct voltage is

applied when a device is reverse biased.

Limit the current applied to the device during the switching on transients.

Limit the value of di

dt during the switching on transients.

Modify the commutating characteristic of a device.

3.3.1-Series type resistance-capacitance (R-C) unpolarised snubber circuits

They are used in the protection of diodes and thyristors by limiting the maximum voltage and

its variation.

The snubber circuits are made by a resistance with a low value in series with a small capacity.

Those two components are dimensioned in such a way that an erroneous activation of the

uncontrolled electronic components is prevented.

Figure 3.4-R-C unpolarised snubber circuit

77

3.3.2-R-C polarised snubber circuits

They are used in order to limit the overvoltage applied to a device during the switch-on

phases, thus they are made by using a resistance, a condenser and a diode connected as shown

in the picture below.

The dampening state allows to evade rough peaks of current in the switch, decreasing its

value in a linear way.

In the capacitor a current circulates equal to:

ics = I0 − ic (3.1)

In which I0 is the current circulating in the device before the transient and is modelled as an

ideal generator in Figure 3.7.

It is accountable for the charge on the capacitor, thus for the increase of the voltage Vcs

applied to C.

Hence, Vcs increases until it reaches the input voltage value that allows the current going

through the capacitor to slowly nullify.

Figure 3.5-R-C polarised snubber circuit

78

Figure 3.6 shows some voltage and current trends for different values of the capacitance.

3.3.3-R-C switch-on polarised snubber circuits

The switch-on snubbers are made by a resistance in series with an inductor and can modify

the switch-on characteristic of the controlled switch, thus limiting the overcurrent during this

phase of the commutation.

They are employed in order to reduce the commutation losses at high frequency and to limit

the maximum reverse current due to the diode restoration.

Those snubbers can be inserted in series to both the device and their freewheeling diode.

In both cases, the sought gain is obtained, in fact the operation is based on the reduction of

the voltage applied to the switch by the increasing current due to the electric potential

difference between the terminals of the snubber’s inductance 𝐿𝑆.

Sometimes is sufficient to use a single freewheeling diode placed in antiparallel with the

inductive load or the switch.

This allows to evade the development of an overvoltage on the device, thus the current can

flow through an alternative path.

In this way, the energy stored in the load can be dissipated in the components, thus protecting

the switch.

The disadvantage of this configuration is due to the fact that the diode allows the flow of

current in the load, thus it requires a circuit dedicated to a fast interruption.

Furthermore, at high frequency, energy have to be dissipated in the diode and in the parasitic

capacitors in order not to damage the device.

Figure 3.6- Low C 𝐶𝑠 =𝐼0𝑡

2𝑉𝑑 High C

79

Hence, these circuits entail a power loss and a higher number of components, thus the future

trend will be to remove them if devices, able to withstand bigger strains or topologies able to

reduce the electrical stress, will be achieved.

3.4-Inverter

The inverter task is to turn a continuous voltage into a sinusoidal alternating voltage with

adjustable frequency and amplitude.

The continuous voltage comes from the alternating voltage supplied by the mains rectified by

a diode bridge.

Given the symmetric configuration, the process proves to be reversible, thus the flux can move

in both directions.

When the energy flux moves to the load, the product between voltage and current is positive,

while when the flux goes to the net this product is negative.

This last characteristic is used most of all in order to recover kinetic energy during the braking

phase, when the motor function as a generator.

Regarding the induction cooker, however, it’s impossible to convert thermal energy of the

cookware in electrical energy moving towards the source.

In the market there are also inverters that work with impressed current, called CSI (Current

Source Inverter), that unlike the inverter with impressed voltage VSI (Voltage Source Inverter),

they have a continuous current at the entrance.

They are only used with high power AC motors.

3.4.1-Inverter topologies

It is possible to classify the inverters through their topology, thus based on the disposition and

quantity of the circuit’s components.

The most used topologies in various applications are the ones called half-bridge and full-bridge,

shown in Figure 3.7.

80

As we can observe the two configurations differ by the number of branches, characterized by

the series of two controlled switches, each one have a diode in antiparallel.

The difference between those two topologies regards the presence of an additional branch in

the full-bridge topology, thus it is possible to use more control techniques than the half-bridge

configuration.

Upstream of the switches two resonant condensers of equal value both in the first and the

second configuration.

They have a high enough capacity, so that it is possible to consider the voltage, between the

link point “O” and the clamp N, constant during every commutation period.

Hence, the voltage value is equal to 1

2Vd, where Vd is the voltage applied to the continuous

side.

The presence of those two condensers has the advantage that the converter will have a higher

power factor at the line entrance.

Also, every condenser, ideally equal to each other, supports only half of the current going

through the coil in every commutation state.

Depending on the current direction, in case that the controlled switch 𝑇+ is closed, then the

current flows through 𝑇+ and 𝐷+, instead, if 𝑇− is closed than the current flows through 𝑇− and

𝐷−.

Hence, the condensers are really connected in parallel towards the current route and “O” is a

point with a potential intermediate between the bridge’s positive output and the negative one.

Also, the presence of capacitors allow us to filter the continuous component, because it doesn’t

flow through the two condensers that are in parallel on the route of i0.

Hence, the saturation problem regarding the transformer’s primary windings, that in induction

cooking application ideally represents the excitation winding operating on the load, is avoided.

Figure 3.7-Half-bridge and full bridge topology

81

3.4.2-Control techniques

It is also possible to classify the VSI in three categories based on the control techniques

implemented on them:

Pulse width modulation inverter

They are inverters that exploit the pulse width modulation (PWM) in order to control the

switches, thus, by varying the duration of the conduction phases, they allow to obtain the

desired value of the output voltage.

Through a suitable control of the pulse width controlling the command of the valves, it is

possible to turn the input continuous voltage into an output sinusoidal wave.

The inverter has to regulate both the amplitude and the frequency of the output voltage.

Square wave inverter

In this type of inverter, the control is devoted only to the regulation of the output voltage

frequency.

The amplitude regulation is obtained by regulating the input voltage amplitude.

The name of this type of inverter is due to the waveform of the output voltage, that is similar

to the square wave.

Single-phase inverter with voltage cancellation

This type of converters have characteristics that lump together those of the previously discussed

inverters.

The amplitude and the frequency of the output voltage are in fact directly regulated by the

converter that creates an output voltage similar to a square wave.

The input voltage is constant and the valves aren’t controlled through a PMW technique.

82

3.4.3-Half-bridge Inverter

The half bridge topologies the most used by the designers of induction cookers, taking into

account the previously listed advantages.

The symmetry both of the operation modes and of the topological level allows us to simplify

and optimize at best the working principle without excessive costs for the producer.

In Figure 3.8 the half bridge configuration in which the inductor-resistance model,

representing the load-pot winding and discussed in the previous chapters, is inserted, is

represented.

This topology’s operation can be described through four phases, each of which represents the

particular logical state of the controlled switches:

Figure 3.8-Inverter half-bridge topology

83

Phase 1

In this phase the IGBT 1 is on, the current circulates from the source in the switch and then in

the load.

Downstream of the load the current splits in two equal parts in the condensers.

Phase 2

The second phase starts when 𝑆1 switches off.

Figure 3.9-Working principle of the half-bridge inverter, phase 1


84

As a result, the voltage at the node common to the two IGBTs, diminishes until its cancellation.

The current circulating in the load doesn’t immediately go to zero because of the inductor and

flows into the diode 𝐷2.

The upper diode 𝐷1 is in an interdicted state, thus the current circulates through the upper

resonant capacitor, thus charging the DC bus capacitor.

The current previously circulating starting from the source, in this phase, reverse its direction

of movement.

Because of the turn off of the IGBT when the voltage and the current are different from zero

in this phase, it is characterized by high commutation losses, that can be partially mitigated by

inserting voltage snubbers in parallel with the device.

Phase 3

The third phase starts when the IGBT 𝑆2 switches on.

It turns on when the voltage is equal to zero due to the fact that the diode 𝐷2 is in conducting

state, this type of turn on without losses on the devices is called soft switching and, in this case,

since the voltage nullifies the voltage-current product, is defined as ZVS (Zero Voltage

Switching).


85

The IGBT’s change of state and the zero-crossing of the load current enable a new inversion

of the direction of the current and the discharge of the DC bus capacitor, that has been charged

in the previous phase.

The same happens for the resonant converters, due to the fact that the currnt flows in the

opposite direction, they release the charge collected in the previous phases through the load.

Phase 4

In this last phase, the turn off of the lower IGBT happens, in the same way as the turn off of 𝑆1

in phase 2.

The diode 𝐷1 allows the circulation of current that charge again the DC bus by going through

it and circulating, in the same direction as the one in phase 3, through the load and the resonant

converters.

After this phase the commutation process restarts from phase 1, in a cyclical manner.

The power obtained by this type of converter can be calculated with equation (3.2).

P0 =∑𝑅𝑒𝑞𝐼0 𝑟𝑚𝑠2

∞

ℎ=1

=∑𝑅𝑒𝑞𝑉0 𝑟𝑚𝑠2

𝑅𝑒𝑞2 + 2𝜋ℎ𝑓𝑠𝑤𝐿𝑒𝑞 −1

2𝜋ℎ𝑓𝑠𝑤𝐶𝑟

=

∞

ℎ=1

=∑𝑅𝑒𝑞𝑉𝑚𝑎𝑖𝑛𝑠2 /(𝜋ℎ)2

𝑅𝑒𝑞2 + 2𝜋ℎ𝑓𝑠𝑤𝐿𝑒𝑞 −1

2𝜋ℎ𝑓𝑠𝑤𝐶𝑟

∞

ℎ=1

(3.2)


86

Where 𝑅𝑒𝑞 and 𝐿𝑒𝑞 refers to the load inductor-pot, 𝑓𝑠𝑤 is the switching frequency and ℎ is the

considered harmonic.

The maximum value of power is the one corresponding to the resonant frequency 𝑓0.

Hence it follows that the load resistance has to satisfy equation (3.3).

𝑅𝑒𝑞 ≤2𝑉𝑚𝑎𝑖𝑛𝑠

2

𝜋2𝑃0 𝑚𝑎𝑥 (3.3)

In order to obtain the value of the resonant capacitance, thus obtaining the desired resonant

frequency, equation (3.4) can be used.

𝐶𝑟 =1

𝐿𝑒𝑞(2𝜋𝑓𝑠𝑤)2 (3.4)

Regarding the sizing of the snubber capacitors, used to reduce the losses, in the inductive

operating zone (in which the switching frequency is higher than the resonant one), it is enough

to choose a capacitance value thus that the snubbers can be charged and discharged during the

downtimes between the transistors’ activation phases by means of the turn off currents.

Their values have to be limited in relation to the values of the resonant capacitors, so that they

don’t interfere with the calculations aimed to obtain the resonant frequency of the system.

3.4.4-Quasi-resonant converters [14-18]

The PWM technique processes power by interrupting the power flow and controlling the duty

cycle, thus, resulting in pulsating current and voltage, while the resonant technique processes

power in a sinusoidal form.

The power switches are often commutated under zero-current ("soft" turn-off) but switches are

turned on with an abrupt increase of device current ("hard" turn-on).

87

In cases where resonant converters operate above the resonant frequency, the switches are

turned off abruptly (forced or hard turn-off) but turned on softly.

Compared with PWM converters, the switching losses and stresses of resonant converters are

reduced; however, the conduction loss is generally increased since the sinusoidal current

produces higher rms current.

Due to circuit simplicity and ease of control, the PWM technique has been used predominantly

in today's power electronics industry, particularly, in low-power applications.

Resonant technology, although well-established in high-power SCR motor drives and

uninterrupted power supplies, has not been widely used in low power dc-dc converter

applications due to its circuit complexity.

With available devices and circuit technologies, PWM converters have been designed to

operate generally with a 30-50 kHz switching frequency.

In this frequency range, the equipment is deemed optimal in weight, size, efficiency, reliability,

and cost.

In certain applications where high-power density is of primary concern, the conversion

frequency has been chosen as high as several hundred kilohertz.

Accompanying the higher switching frequency, the switching stresses and losses are increased

and, furthermore, the presence of leakage inductances in the transformer and junction

capacitances in the semiconductor devices causes the power devices to inductively turn-off and

capacitively turn-on.

As the semiconductor device switches off an inductive load, voltage spikes induced by the

sharp 𝑑𝑖

𝑑𝑡 across the leakage inductances produce increased voltage stress and noise.

On the other hand, when the switch turns on at a high voltage level, the energy stored in the

device's output capacitance, 1

2𝐶𝑉2, is dissipated internally when the device is switched on.

Furthermore, turn on at high voltage levels induces a severe switching noise through the

capacitor coupled into the drive circuit, which leads to significant noise and instability in the

drive circuit.

The detrimental effects of parasitic elements become more pronounced as the switching

frequency is increased.

To improve switching behaviour of semiconductor devices in power processing circuits, two

techniques were proposed.

The first is the zero-current-switching (ZCS) technique.

88

By incorporating an LC resonant circuit, in which the inductor 𝐿𝑟 is in series with the switch

𝑆1, the current waveform of the switching device is forced to oscillate in a quasi-sinusoidal

manner, therefore, creating zero-current-switching conditions during both turn-on and turn-

off.

By simply replacing the power switches in PWM converters with the proposed resonant

switch, a family of quasi-resonant converters (QRCs) has been derived.

This new family of circuits can be viewed as a hybrid of PWM and resonant converters.

QRCs utilize the principle of inductive or capacitive energy storage and transfer in a manner

similar to PWM converters, and their circuit topologies also resemble those of PWM

converters.

However, an LC tank circuit is always present near the power switch and is used, not only to

shape the current and voltage waveforms of the power switch, but also, to store and transfer

energy from the input to the output in a manner similar to the conventional resonant converters.

For off-line as well as dc-dc converter applications, the zero-current-switching technique is

very effective up to 1-2 MHz, since it can eliminate turn-off switching loss and switching

stresses.

Figure 3.13-ZC resonant switches

Figure 3.14-Switching load line trajectory.

Conventional PWM switching (A) and ZC resonant switching (B).

89

The load-line trajectory of PWM switching behaviour, shown as trajectory A in Figure 3.14,

traverses across the high-stress region in which the device is subjected to simultaneous high

voltage and high current; whereas the load-line trajectory of a resonant switch, shown as

trajectory B, moves along the axes.

Since no simultaneous high voltage and high current are exerted on the switch device, the

switching stresses and losses are minimal.

The second technique proposed is zero-voltage switching (ZVS).

By using an LC resonant network, in which the capacitor 𝐶𝑟 is in parallel with the switch 𝑆1,

the voltage waveform of the switching device can be shaped into a quasi-sine wave, such that

zero-voltage conduction is created for the switch to turn on and turn off without incurring any

switching loss.

This technique eliminates the turn-on loss associated with the parasitic junction capacitances.

Through the establishment of the zero-current switching technique a large family of

ZCS-QRCs have been derived.

Similarly, the zero-voltage switching technique has led to the discovery of a large family of

ZVS-QRCs.

Furthermore, the duality relationship which exists between the zero-current-switching and the

zero-voltage-switching techniques provides a framework permitting knowledge transfer from

one converter family to the other.

3.4.4.1-Zero-current-switching quasi-resonant converters (ZCS-QRCs)

3.4.4.1.1-Principle of operation

In order to describe the principle of operation, a buck quasi-resonant converter, as shown in

Figure 3.16, is employed.

Figure 3.15-ZV resonant switches

90

To analyse the steady-state circuit behaviour, the following assumptions are made:

1) 𝐿0 ≫ 𝐿r

2) Output filter 𝐿0 − C0 and the load are treated as a constant current sink.

3) Semiconductor switches are ideal, thus no forward voltage drop in the on-state, no leakage

current in the off-state, and no time delay at both turn-on and turn-off.

4) Reactive elements of the tank circuit are ideal.

We define the following variables:

1) Characteristic impedance: Zn = √𝐿𝑟

𝐶𝑟

2) Resonant angular frequency: 𝜔 =1

√𝐶r𝐿r

3) Resonant frequency: 𝑓𝑛 =𝜔

2𝜋

A switching cycle can be divided into four stages.

We suppose that before S1 is turned on, the diode 𝐷0 carries the output current 𝐼0 and resonant

capacitor voltage 𝑉𝐶 is clamped at zero.

At the beginning of a switching cycle, 𝑡 = 𝑇0, 𝑆1 is switched on.

Phase 1, Induction charging state [𝑇0, 𝑇1]

Input current 𝐼𝑙 rises linearly and is governed by the state equation (3.5).

The duration of this stage, 𝑇𝑑1(= 𝑇1 − 𝑇0), can be solved with boundary conditions

Figure 3.16-Buck quasi-resonant converter

Figure 3.17-Induction charging state

91

𝐼𝑙(0) = 0 and 𝐼𝑙(𝑇𝑑1) = 𝐼0, thus we obtain equation (3.6).

𝐿𝑟𝑑𝑖𝑙𝑑𝑡

= 𝑉𝑙 (3.5)

𝑇𝑑1 = 𝐿𝑟𝐼0𝑉1 (3.6)

Phase 2, Resonant stage [𝑇1, 𝑇2]

At time 𝑇1, when the input current rises to the level of 𝐼1, 𝐷0 is commutated off and the amount

of current 𝑖(𝑡) − 𝐼0 is now charging 𝑉𝐶, as can be seen from Figure 3.18.

The state equations are given by (3.7) and (3.8), with the initial conditions (3.9) and (3.10).

Therefor we obtain equations (3.11) and (3.12).

𝐶𝑟𝑑𝑉𝐶𝑑𝑡

= 𝑖𝑙(𝑡) − 𝐼0 (3.7)

𝐿𝑟𝑑𝑖𝑙𝑑𝑡

= 𝑉𝑙 − 𝑉𝐶𝑟(𝑡) (3.8)

𝑉𝐶𝑟(0) = 0 (3.9)

𝐼𝑙(0) = 0 (3.10)

𝑖𝑙(𝑡) = 𝐼0 +𝑉𝑙𝑍𝑛sin(𝜔𝑡) (3.11)

Figure 3.18-Resonant stage

92

𝑉𝐶𝑟(𝑡) = 𝑉𝑙(1 − cos(𝜔𝑡)) (3.12)

If a half-wave resonant switch is used, switch 𝑄1 will be naturally commutated at time 𝑇𝑎 when

the resonating input current 𝑖𝑟(𝑡) reduces to zero, as shown in Figure 3.19.

On the other hand, if a full-wave resonant switch is used, current 𝑖𝑙(𝑡) will continue to oscillate

and feed energy back to the source 𝑉𝑙 through the antiparallel diode 𝐷1, as shown in Figure

3.20.

Current through 𝐷1 again oscillates to zero at time 𝑇𝑏.

The duration of this stage, 𝑇𝑑2(= 𝑇2 − 𝑇1), can be solved from (3.11) by setting 𝑖𝑙(𝑇𝑑2) = 0.

Thus:

𝑇𝑑2 =𝛼

𝜔 where 𝛼 = 𝑠𝑖𝑛−1 (−

𝑍𝑛𝐼0

𝑉𝑙) (3.13)

𝜋 ≤ 𝛼 ≤3𝜋

2 and 𝑇2 = 𝑇𝑎 for half-wave mode (3.14)

3𝜋

2≤ 𝛼 ≤ 2𝜋 and 𝑇2 = 𝑇𝑏 for full-wave mode (3.15)

At time 𝑇2, 𝑉𝐶𝑟 = 𝑉𝐶𝑏 can be solved from (3.12), thus we obtain (3.16).

𝑉𝐶𝑟(𝑇𝑑2) = 𝑉𝐶𝑏 = 𝑉𝑙(1 − cos(𝛼)) (3.16)

93

Phase 3, Capacitor discharging stage [𝑇2, 𝑇3]

Since switch 𝑆1 is off at time 𝑇2, 𝐶𝑟 begins to discharge through the output loop and 𝑉𝐶 drops

linearly to zero at time 𝑇3, as shown in Figure 3.19 and 3.20.

The state equation during this interval is given by (3.17).

The duration of this stage, 𝑇𝑑3( = 𝑇3 − 𝑇2), can be solved with the initial condition

𝑉𝐶𝑟(0) = 𝑉𝐶𝑏, thus we obtain (3.18).

Figure 3.19-Waveforms of the buck

resonant converter, half-wave mode

Figure 3.20-Waveforms of the buck

resonant converter, full-wave mode

Figure 3.21- Capacitor discharging stage

94

𝐶𝑟𝑑𝑉𝐶𝑟𝑑𝑡

= 𝐼0 (3.17)

𝑇𝑑3 =𝐶𝑟𝑉𝑐𝑏𝐼0

=𝐶𝑟𝑉𝑙(1 − cos(𝛼))

𝐼0 (3.18)

Phase 4, Free-wheeling stage [𝑇3, 𝑇4]

The output current flows through diode 𝐷0.

The duration of this stage is 𝑇𝑑4(= 𝑇4 − 𝑇3), and 𝑇𝑑4 is given by (3.19), where 𝑇𝑠 is the

period of a switching cycle.

Typical circuit waveforms, as shown in Figure 3.21 and 3.22, clearly demonstrate the zero-

current switching property.

𝑇𝑑4 = 𝑇𝑠 − 𝑇𝑑1 − 𝑇𝑑2 − 𝑇𝑑3 (3.19)

3.4.4.1.2-DC Voltage-Conversion Ratio

The output voltage 𝑉0 can be solved by equating the input energy per cycle 𝐸𝑖 and the output

energy per cycle 𝐸0.

From (3.7), (3.12) and (3.18), given the values of 𝐼0 and 𝑇𝑠, 𝑇𝑑1, 𝑇𝑑2 and 𝑇𝑑3 can be solved

from (3.7),(3.8)-(3.16) and (3.18).

Hence we can obtain the value of 𝑉0 through (3.22) and, by defining 𝑥 =𝑉0

𝑉𝑙 and 𝑟 =

𝑅

𝑍𝑛, (3.22)

can be written as (3.23).

𝐸𝑖 = Vl [∫ 𝑖𝑙(𝑡)𝑑𝑡𝑇1

𝑇0

+∫ 𝑖𝑙(𝑡)𝑑𝑡𝑇2

𝑇1

] (3.20)

Figure 3.22- Free-wheeling stage

95

𝐸0 = V0𝐼0𝑇𝑠 (3.21)

𝑉0 =𝑉𝑙 (

𝑇𝑑12 + 𝑇𝑑2 + 𝑇𝑑3)

𝑇𝑠 (3.22)

𝑥 − (1

2𝜋) (

𝑓𝑠𝑓𝑛) [(

𝑥

2𝑟) + sin−1 (−

𝑥

𝑟) + (

𝑟

𝑥)(1 + 𝑠𝑖𝑔𝑛√1 − (

𝑥

𝑟)2

)] = 0 (3.23)

We define sin−1 (−𝑥

𝑟) = 𝛼 then:

𝜋 ≤ 𝛼 ≤3𝜋

2→ 𝑠𝑖𝑔𝑛 = +1 for half-wave mode (3.24)

3𝜋

2≤ 𝛼 ≤ 2𝜋 → 𝑠𝑖𝑔𝑛 = −1 for full-wave mode (3.25)

Voltage-conversion ratios for the buck resonant converter are plotted in Figure 3.23 for the

half-wave mode and full-wave mode, respectively.

Figure 3.23- DC voltage-conversion ratio for the buck resonant

converter, for the half-wave mode and full-wave mode

96

It can be seen that the voltage-conversion ratio in the half-wave mode is very sensitive to load

variation, while in the full-wave mode the voltage-conversion ratio is almost independent of

load variation.

This can be understood by examining the waveforms of 𝑖𝑙 and 𝑉𝐶 in Figure 3.24.

Under heavy load 𝑖𝑙 is offset by a large amount of 𝐼0 and the negative portion of the resonant

inductor current, which is the current flowing through the antiparallel diode, is small.

When the load is light, 𝑖𝑙 is offset by a small amount of 𝐼0 and the reverse current through 𝐷1

is increased.

This behaviour can be simply stated in the following: as the power switch is turned on, energy

is transferred from the source to the resonant tank.

When the load demand is light, a large portion of the tank energy is returned to the source.

When the load is heavy, most of the tank energy is transferred to the load and only a small

portion of that tank energy is returned to the source.

Consequently, the antiparallel diode regulates the tank energy such that the voltage-conversion

ratio remains constant as the load is varying.

Figure 3.24-Waveforms of the buck resonant converter.

Upper waveforms: heavy load. Lower waveforms: light load.

97

For the half-wave mode, the excessive tank energy cannot be returned to the source when the

load channel is reduced.

Consequently, the operating frequency has to be reduced to regulate the output voltage.

It should be noted that the characteristic shown in Figure 3.23, regarding the full-wave mode,

is exactly the same as that of the PWM buck converter, provided that the horizontal axis is

replaced by the duty cycle ratio.

This conclusion can be extended to other converter topologies as well.

This simply implies that the QRCs operating in full-wave mode have the same dc conversion

characteristics as their PWM counterparts, while capable of achieving zero-current turn-on and

zero-current turn-off.

An examination of the input current waveform 𝑖𝑙 of Figure 3.24 reveals that 𝑖𝑙 contains a dc

component of 𝐼0 and an ac component of 𝑉𝑠

𝑍𝑛.

The ac component is fixed for a given input and characteristic impedance 𝑍𝑛 and the dc

component 𝐼0 is simply the load.

To maintain an ac component greater than the dc component, an upper bound exists on the load

current above which the zero-current switching property will be lost.

𝐼0 ≤𝑉𝑠𝑍𝑛 (3.26)

3.4.4.2-Zero-voltage-switching quasi-resonant converters (ZVS-QRCs)

One of the fundamental limitations of ZCS-QRCs for very high-frequency operation is the

problem of capacitive turn-on loss, whereas this technology is free of turn-off switching

losses.

The energy stored in the device's output capacitance, 1

2𝐶𝑉2, during the off-state is dissipated

inside the device when the device is switched on.

At high input voltage the capacitive turn-on loss is significant and the 𝑑𝑣

𝑑𝑡 during turn-on

further introduces a severe switching noise through the capacitor which is coupled into the

drive circuit.

While not severe in lower switching frequencies, the capacitive turn-on loss becomes the

dominating factor when the switching frequency is raised to the megahertz range.

98

The zero-voltage switching technology is proposed to alleviate the losses during turn-on, thus

enabling the quasi-resonant converters to operate at a much higher frequency.

3.4.4.2.1-Principle of operation

In order to describe the principle of operation, a boost quasi-resonant converter, as shown in

Figure 3.25, is employed.

For simplicity, the converter is treated as a constant current source 𝐼𝑙, supplying power to a

constant voltage sink 𝑉0.

In steady-state operation, a complete switching cycle can be divided into four stages starting

from the moment 𝑆1 turns-off.

Suppose, before 𝑆1 is turned off, it carries the input current 𝐼𝑙.

Diode 𝐷0 is off and no current is flowing into the voltage load 𝑉0.

At time 𝑇0 𝑆1 turns off and the input current 𝐼0 is diverted into capacitor 𝐶𝑟.

The following description summarizes the circuit operation during each of the four stages.

Phase 1, Capacitor charging state [𝑇0, 𝑇1]

𝑆1 turns off at 𝑇0.

Current 𝐼𝑙 flows into 𝐶𝑟, the voltage across 𝐶𝑟, 𝑉𝐶𝑟, rises linearly.

Figure 3.25-Boost quasi-resonant converter

Figure 3.26-Capacitor charging state

99

At time 𝑇1, 𝑉𝐶𝑟 reaches 𝑉0 and diode 𝐷0 starts conducing.

Phase 2, Resonant stage [𝑇1, 𝑇2]

𝐷0 turns on at 𝑇1, thus a portion of 𝐼𝑙 starts to flow into 𝑉0.

In the half-wave mode of operation, when 𝑉𝐶 drops to zero at 𝑇𝑎 it is clamped at the zero value

by the antiparallel diode 𝐷1 which carries the reverse current.

While in full-wave mode, 𝑉𝐶 continues to oscillate to a negative value and return to zero at

time 𝑇𝑏.

For half-wave mode, the end of this stage, 𝑇2 is equal to 𝑇𝑎 and for full-wave mode, it is equal

to 𝑇𝑏.

The waveforms for half-wave mode and full-wave mode can be seen in Figure 3.30 and 3.31

respectively.

Figure 3.27-Resonant stage

100

Phase 3, Inductor discharging stage [𝑇2, 𝑇3]

After 𝑇2, current 𝑖𝐿𝑟 drops linearly and reaches zero at time 𝑇3.

Normally, in the half-wave mode of operation, transistor 𝑄1 shall turn on after 𝑉𝐶𝑟 drops to

zero at 𝑇𝑎 and before the current through 𝐷1 drops to zero at 𝑇𝑐.

Otherwise,𝑉𝐶𝑟 will begin to recharge and 𝑄1 will lose the opportunity to turn on under the zero-

voltage condition.

Figure 3.30- Inductor discharging stage

Figure 3.28-Waveforms of the boost

resonant converter, half-wave mode

Figure 3.29-Waveforms of the boost

resonant converter, full-wave mode

101

In the full-wave mode of operation, 𝑄1 shall turn on between 𝑇𝑎 and 𝑇𝑏 , when diode 𝐷1 is

blocking the negative voltage.

Phase 4, Free-wheeling stage [𝑇3, 𝑇4]

At 𝑇3, the entire input current 𝑖𝑙 flows through 𝑄1.

𝐼𝑄1 remains constant until 𝑄1 turns off at 𝑇4.

Notice that the voltage waveform of 𝑉𝐶𝑟 contains a dc component of 𝑉0 and an ac component

of 𝑍𝑛𝑖𝑙.

Since 𝑖𝑙 is in proportion to the load current when 𝑉𝑙 and 𝑉0 are fixed, the peak value of 𝑉𝐶𝑟

increases as the load current is increased.

Furthermore, to maintain a larger ac component than dc component, a lower bound on load

current exists below which the zero-voltage switching property will be lost.

The waveform of the current through 𝑄1 is somewhat square, and its peak value is the same as

that of 𝑖𝑙.

This results in a lower rms value of the switch current and the conduction loss is kept minimal.

3.4.4.2.2-DC Voltage-Conversion Ratio

The dc voltage-conversion ratio, 𝑉0

𝑉𝑙, as a function of load resistance and switching frequency,

can be derived as discussed in 3.4.4.2.2.

The conversion ratio for the boost ZVS-QRC is plotted in Figure 3.32 for half-wave mode and

full-wave mode, respectively.

Figure 3.31- Free-wheeling stage

102

It can be seen that the voltage-conversion ratio in full-wave mode is insensitive to load variation

and is certainly more desirable.

However, for full-wave mode, a series diode is required to provide a reverse-voltage blocking

capability.

Consequently, the energy stored in the junction capacitances of the semiconductor switch is

trapped during off time and is dissipated internally after the switch turns on.

Therefore, full-wave mode suffers from capacitive turn-on losses and 𝑑𝑣

𝑑𝑡 noise, as are ZCS-

QRCs, and is not practical for very high-frequency operation.

3.4.4.3-Comparison of ZCS-QRC and ZVS-QRC

A duality relationship exists between ZCS-QRCs and ZVS-QRCs.

For example, the boost ZVS-QRC is the dual of the buck ZCS-QRC.

A comparison of the waveforms shown in Figures 3.19, 3.20 and 3.28, 3.29, clearly displays

the duality relationship between them.

Figure 3.32- DC voltage-conversion ratio for the boos resonant

converter, for the half-wave mode and full-wave mode

103

In fact, the dual properties between them are not only true qualitatively, but also are true

quantitatively.

For example, the voltage-conversion ratio of the boost ZVS-QRC can be derived from that of

the buck ZCS-QRC simply by applying the duality principle.

The duality relationships also exist between any given converter in the ZCS-QRC family and

its counterpart in the ZVS-QRC family.

Figure 3.33 summarizes the major characteristics of the zero current switching and zero-voltage

switching techniques.

3.4.4.4-Gate-drive design

A gate drive is an important part of any high-frequency converter design and there are several

problems associated with high-frequency gate drives.

Switching devices in the gate drive

To reduce the dynamic losses in a power MOSFET, turn-on and/or turn-off times should be

minimized.

For a given MOSFET this can be achieved only by increasing the charging/discharging gate

current during turn-on/turn-off.

Figure 3.33-Major characteristics of the ZCS and ZVS techniques

104

Increased charging/discharging gate current requires larger (i.e., slower) gate-drive devices.

Since the switching devices in the gate-drive circuit should be substantially faster and easier to

drive than the power MOSFET, there are practical limitations of the maximum gate current

achieved during turn-on/off.

Parasitic inductances

To achieve rapid change of the gate current during turn-on/off, the inductance in the gate-

current loop should be as low as possible.

Therefore, the devices used in the gate-drive, as well as the power MOSFET, should have

minimum lead inductances.

Packaging of the devices and layout of the circuit are important factors affecting the switching

speed.

Surface mount and thick-film hybrid technologies can help to reduce parasitic inductances.

Power dissipation in the gate-drive circuit

Turning a MOSFET on/off requires charging/discharging the MOSFET's input capacitance

𝐶𝑖𝑠𝑠.

In general, 𝐶𝑖𝑠𝑠 is nonlinear.

Although in the following discussion 𝐶𝑖𝑠𝑠 is assumed to be constant, the conclusions are

identical for a non linear 𝐶𝑖𝑠𝑠.

In most high-speed gate-drive circuits, the charging/discharging of 𝐶𝑖𝑠𝑠 , is achieved by

connecting the gate through lower impedance switches to positive and negative potentials,

𝑉𝑂𝑁 and 𝑉𝑂𝐹𝐹, respectively.

Every time the MOSFET is turned on/off the energy, given by equation (3.27), is dissipated in

the gate-drive circuit.

Power dissipation in the gate-drive circuit is proportional to the switching frequency, expressed

by (3.28).

𝐸𝐺 =1

2(𝑉𝑂𝑁

2 + 𝑉𝑂𝐹𝐹2 )𝐶𝑖𝑠𝑠 (3.27)

105

𝑃𝐺 = 2𝑓𝑠𝐸𝐺 (3.28)

Switching speed can be increased by increasing the differential voltage, 𝑉𝑂𝑁 − 𝑉𝑂𝐹𝐹. However,

this would substantially increase power dissipation in the gate-drive circuit.

Therefore, 𝑉𝑂𝑁 should be only as high as necessary to saturate the MOSFET and 𝑉𝑂𝐹𝐹 should

be sufficiently low to cut it off.

From (3.28) it can be seen that an increased switching frequency results in an increased power

dissipation in the gate drive.

Thus, gate-drive devices should have appropriate power ratings and often need heat sinks.

A higher power rating of a device usually implies lower switching speed, while the presence

of heat sinks usually contributes to an increase in wiring inductance in the gate-drive circuit.

Therefore, reduction of power dissipation in the gate-drive circuit is essential to increase its

speed.

Power dissipation in the gate-drive circuit can be reduced by charging and/or discharging 𝐶𝑖𝑠𝑠

using a resonant technique.

Theoretically, an inductance introduced in series with the gate and appropriately resonant with

𝐶𝑖𝑠𝑠 could reduce power dissipation in the gate drive, but unfortunately, as mentioned, any

inductance in series with the gate inevitably reduces switching speed.

As an example, a novel, quasi-resonant, gate-drive circuit suitable for high-frequency quasi-

resonant converters is proposed.

It is observed in QRCs that fast switching is critical either during turn-on or during turn-off,

but not during both.

For example, in the ZVS-QRCs operating in half-wave mode, the turn-off time is critical

because the switching loss, caused by the nonzero product of the drain-to-source voltage and

the drain current, occurs only during turn-off.

In ZCS-QRCs, turn-on speed is more critical.

Therefore, in ZVS-QRCs, the fall time of the gate-to-source voltage should be minimized to

achieve fast turn-off.

Rise time of the gate-to-source voltage, however, is not as critical since turn-on occurs during

the conduction period of diode 𝐷𝑆 which occupies a substantial portion of the switching period.

Thus fast turn-off can be achieved in a conventional, dissipative manner, while turn-on can be

obtained using a resonant technique.

106

Such operation of the gate drive should theoretically reduce power dissipation in the gate-drive

circuit by half.

A circuit implementation of the quasi-resonant gate drive is shown in Figure 3.34, it consists

of a single supply voltage, one switch, a diode, and a resonant inductor.

The operation of the circuit is as follows.

When switch 𝑆𝐺 is on, the MOSFET is off and voltage 𝑉𝐺 is applied to inductance 𝐿𝐺 .

During this stage, which lasts as long as the fixed off-time of the MOSFET, current in the

inductance builds up and eventually reaches 𝐼𝐿𝐺0 as shown in Figure 3.35.

Figure 3.34-Basic circuit diagram of the quasi-resonant gate drive

Figure 3.35-Theoretical waveforms of the quasi-resonant gate drive

at 𝑉𝐺 = 4 𝑉, 𝐿𝐺 = 120 𝑛𝐻, and 𝐶𝑖𝑠𝑠 = 1.8 𝑛𝐹

107

When 𝑆𝐺 turns off, 𝐿𝐺 and 𝐶𝑖𝑠𝑠 form a resonant circuit and 𝑉𝐺𝑆 increases in a resonant fashion.

When current 𝑖𝐿𝐺 reaches zero, resonance is stopped by diode 𝐷𝐺 .

Since turn-on is achieved through the resonance of 𝐿𝐺 and 𝐶𝑖𝑠𝑠, the power dissipation in the

gate drive is reduced approximately by half compared to conventional gate-drive circuits.

The dual network of Figure 3.34 would be suitable for ZVS-QRCs.

3.4.5-Operation modes used with non-ferromagnetic pans

As stated in Chapter 2, non-ferromagnetic materials, compared to the ferromagnetic ones, are

characterized by lower values of resistance and inductance, thus it exists the risk of surpassing

the limits sustainable by the devices.

In addition to that, the resonant frequency shifts itself to higher frequencies, thus it is necessary

to adjust the electronics to the new load condition.

Hence we use different operation modes when dealing with non-ferromagnetic cookware, in

order to maximize the efficiency and prevent the appearance of values too high, frequency

doubler or trippler operation modes.

These will be analysed in Chapter 5, while dealing with the Simulink simulation of two

exemplifying models.

108

Chapter 4-Inverter losses

The losses calculations in the inverter is a very important matter concerning the efficiency of

the system, in fact, because of the voltages and currents going through those devices, they are

a relevant part of the system’s total losses.

Inverters are characterized by two kind of losses: the first one refers to the losses during the

conduction phase, originating from the current flow during this phase, the residual voltage on

the device and its intrinsic resistance.

The aim of this chapter is to obtain a model able to represent the conduction and switching

losses, in order to identify a general method for the calculation of the losses in an inverter.

We start by using static calculations in order to obtain the values of the losses energies through

the data contained in the datasheet of the device, as presented in [20], then we use those results

to develop a dynamic model for the losses calculation with Simulink.

4.1-Conduction losses

The conduction phase can be divided into two distinct phases: the first one refers to the

conduction of the IGBT, while the second refers to the reverse conduction due to the diode in

antiparallel with the previous device.

The conduction losses estimated on the pair IGBT-diode in terms of lost energy is given by

(4.1), in which the energy lost during a generic phase can be determined with equation (4.2).

𝐸𝑙𝑜𝑠𝑠−𝑐𝑜𝑛𝑑 = 𝐸𝐼𝐺𝐵𝑇 + 𝐸𝑑𝑖𝑜𝑑𝑒 (4.1)

𝐸𝑙𝑜𝑠𝑠 = ∫ 𝑉(𝑡)𝐼(𝑡) 𝑑𝑡 (4.2)𝑡𝑒𝑛𝑑

𝑡𝑠𝑡𝑎𝑟𝑡

Where 𝑉(𝑡) and 𝐼(𝑡) are the voltage and the current during the interval [𝑡𝑠𝑡𝑎𝑟𝑡 , 𝑡𝑒𝑛𝑑].

Hence we obtain that, for the conduction phase of the IGBT, we need to take into account the

presence of the current circulating into the device and the residual voltage between collector

and emitter, thus we obtain (4.3).

109

The voltage on the device can be approximated with the first order function (4.4), thus it is

possible to approximate the IGBT conduction energy with (4.5).

𝐸𝐼𝐺𝐵𝑇 = ∫ 𝑉𝐶𝐸(𝑡)𝐼𝐶𝐸(𝑡) 𝑑𝑡 (4.3)𝑡𝑜𝑓𝑓

𝑡𝑜𝑛

𝑉𝐶𝐸 = 𝑉𝐶𝐸𝑜𝑛 + 𝑅𝑜𝑛𝐼𝐺𝐵𝑇𝐼𝐶𝐸 (4.4)

𝐸𝐼𝐺𝐵𝑇 = (𝑉𝐶𝐸𝑜𝑛𝐼𝐶𝐸,𝑚 + 𝑅𝑜𝑛𝐼𝐺𝐵𝑇𝐼𝐶𝐸,𝑟𝑚𝑠2 )(toff − ton) (4.5)

In this equation two dissipation factors are present: the first one is due to the contemporary

presence of residual voltage on the device and current flowing through the device and the

second one is due to the internal resistance in which the current flows during the conduction

phase.

The power lost by the IGBT during the conduction phase is given by (4.6), in which 𝑓 is the

working frequency of the device.

𝑃𝑐𝑜𝑛𝑑,𝐼𝐺𝐵𝑇 = 𝐸𝐼𝐺𝐵𝑇 ∗ 𝑓 (4.6)

This line of reasoning can be also applied to the losses during the diode conduction phase, thus

we obtain (4.7).

For an approximated calculation it is possible to refer to the data given by the manufacturers

of semiconductor devices, thus we get equation (4.8), where 𝑉𝑜𝑛𝐷 is the residual voltage on the

diode, 𝑅𝑜𝑛𝐷 is the conduction resistance of the diode and 𝐼𝑑𝑖𝑜𝑑𝑒 is the effective value of current

flowing into the diode.

𝐸𝑑𝑖𝑜𝑑𝑒 = ∫ 𝑉𝑑𝑖𝑜𝑑𝑒(𝑡)𝐼𝑑𝑖𝑜𝑑𝑒(𝑡) 𝑑𝑡 (4.7)𝑡𝑜𝑓𝑓

𝑡𝑜𝑛

𝐸𝑑𝑖𝑜𝑑𝑒 = (𝑉𝑜𝑛𝐷𝐼𝑑𝑖𝑜𝑑𝑒,𝑚 + 𝑅𝑜𝑛𝐷𝐼𝑑𝑖𝑜𝑑𝑒,𝑟𝑚𝑠2 )(toff − ton) (4.8)

𝑃𝑑𝑖𝑜𝑑𝑒,𝐼𝐺𝐵𝑇 = 𝐸𝑑𝑖𝑜𝑑𝑒 ∗ 𝑓 (4.9)

110

The power lost during the conduction phase of the diode is given by (4.9), in which 𝑓 is the

working frequency of the device, the same as the IGBT one.

4.2-Switching losses

The model used is based on the plottage of the trends of current and voltage during the

switching phases, thus we can calculate the involved energies in an analytical way.

As stated before, the switching losses can be studied by separating the turn on and turn off

phase of the IGBT, thus we can observe the differences between the two stages in which this

device works.

During the turn off phase the switch opens, the current flowing goes to zero, while the voltage

goes from an almost null value to the value of the supply voltage.

In case of an ideal switch, the current and voltage commutation times would be equal to zero,

but in case of a real switch they are different from zero, thus losses are created.

In addition to the losses due to the commutation times unequal to zero, during the turn off

phase, a relevant contribution is given by the tail of current, a phenomenon caused by the

physics of the IGBT.

In fact the IGBT is a device with the aim of combining the control characteristic of a MOSFET

and the capacity of a BJT.

The problem is that, during the turn off, the MOSFET is characterized by turn off speed higher

than that of the BJT, thus the result is a rapid decrease in the flowing current, due to the

MOSFET turn off, but not a complete zeroing caused by the fact that the BJT is slower.

Hence it follows that the lost power, due to the product between the tail of current still present

and the voltage that have reached its maximum value.

111

The model of the turn off phase is shown in Figure 4.1, in which:

𝐼𝑑 is the current circulating in the device during the conduction phase

𝐼𝑏 is the initial value of the tail of current and is equal to 10% 𝐼𝑑

𝑉𝑑 is the supply voltage

𝑡𝑟 is the voltage rise time, from the zero value to the supply value

𝑡𝑓 is the current fall time, from its maximum value to its 10%.

It represents the time needed for the MOSFET to turn off.

𝑡𝑡 is the depletion time of the tail of current and represents the time needed for the BJT

to turn off.

During this interval the voltage goes to its maximum value. By using these parameters we can

calculate the energy lost during this phase, thus we obtain (4.10), in which the contribution

given by the tail of current is (4.11).

𝐸𝑜𝑓𝑓 =1

2𝐼𝑑𝑡𝑟𝑉𝑑 + [

1

2(𝐼𝑑 − 𝐼𝑏)𝑡𝑓 + 𝐼𝑏𝑡𝑓] 𝑉𝑑 +

1

2𝐼𝑏𝑡𝑡𝑉𝑑 =

1

2[𝐼𝑑(𝑡𝑟 + 𝑡𝑓) + 𝐼𝑏(𝑡𝑡 + 𝑡𝑓)]𝑉𝑑

Figure 4.1-IGBT voltage and current curves during turn-off phase

112

=1

2[𝑡𝑟 + 𝑡𝑓 +

1

10(𝑡𝑡 + 𝑡𝑓)] 𝑉𝑑𝐼𝑑 (4.10)

𝐸𝑜𝑓𝑓−𝑡𝑎𝑖𝑙 =1

2𝐼𝑏𝑡𝑡𝑉𝑑 =

1

20𝐼𝑑𝑡𝑡𝑉𝑑 (4.11)

The phenomenon of the tail of current is one of the most difficult to model because of its high

variability due to the parameters involved with it, like the working temperature and the lifetime

of the electric charges.

Often the manufacturer, because of the difficulties introduced by this phenomenon, neglect its

presence.

If during the IGBT turn off the tail of current assume an important role, during the turn on

phase we need to pay particular attention to the phenomenon of reverse recovery.

First of all, it needs to be clarified that, during this phase, the IGBT turns on and the diode in

antiparallel with it turns off at the same time.

While, during the IGBT turn off phase, the diode turning on doesn’t give any contribution, in

this phase, the diode is subjected to reverse recovery.

It depends from the physics of the device and it is represented by the current and the reverse

recovery time.

These two quantities are the current and the time necessary to reinstate the equilibrium between

the electric charges present in the device.

The values of current and reverse recovery time are linked to the dimensions and the

characteristics of the diode, but are easier to model compared to the tail of current.

The model of the turn off phase is shown in Figure 4.2, in which:

𝐼𝑑 is the current circulating in the diode during the conduction phase

𝑉𝑑 is the supply voltage

𝐼𝑟𝑚 is the maximum reverse recovery current

ICE is the conduction current of the IGBT

𝑡𝑟𝑖 is the time needed for the current to reach the value 𝐼𝐶𝐸 + 𝐼𝑟𝑚

𝑡𝑓𝑣 is the time needed for the voltage to go from the supply voltage value to zero

𝑡𝑟𝑟 is the reverse recovery time, equal to 𝑡𝑠 + 𝑡𝑓, recovery rise and fall time respectively

113

From the model, we can see that the reverse recovery phenomenon modify the transient of the

current flowing in the IGBT, it reaches a peck value, tied to the reverse recovery current, before

reaching its equilibrium value.

The reverse recovery energy is given by (4.12), in which the choice of 1

2 instead of

1

4 is due to

the hypothesis of working in the limit condition with 𝑉𝑑 constant.

The next step consist in the calculation of the IGBT turn-on losses.

If we don’t consider the effect of reverse recovery, the turn-on losses would be the ones written

in (4.13).

By adding the losses due to the diode inverse recovery (4.14) to (4.13) we obtain the IGBT

total turn-on losses (4.15).

The total contribution given by the reverse recovery in this phase is given by (4.16).

Figure 4.2-Voltage and current in the diode and the IGBT during turn-on phase

114

𝐸𝑟𝑟 =1

2𝐼𝑟𝑟𝑚𝑡𝑠𝑉𝑑 +

1

2𝐼𝑟𝑟𝑚𝑡𝑓𝑉𝑑 =

1

2𝐼𝑟𝑟𝑚𝑡𝑟𝑟𝑉𝑑 (4.12)

𝐸𝑜𝑛 =1

2(𝑡𝑟𝑖 − 𝑡𝑠)𝑉𝑑𝐼𝐶𝐸 +

1

2𝑡𝑓𝑉𝑑𝐼𝐶𝐸 =

1

2(𝑡𝑟𝑖 − 𝑡𝑠 + 𝑡𝑓)𝑉𝑑𝐼𝐶𝐸 (4.13)

∆𝐸𝑜𝑛 = 𝐸𝑟𝑟 =1


𝐸𝑜𝑛−𝑡𝑜𝑡 = 𝐸𝑜𝑛 + ∆𝐸𝑜𝑛 =1

2(𝑡𝑟𝑖 − 𝑡𝑠 + 𝑡𝑓)𝑉𝑑𝐼𝐶𝐸 +

1


𝐸𝑟𝑟−𝑡𝑜𝑡 = ∆𝐸𝑜𝑛 + 𝐸𝑟𝑟 = 𝐼𝑟𝑟𝑚𝑡𝑟𝑟𝑉𝑑 (4.16)

4.3-Verification of the losses model [20]

The objective of this section is to validate the models introduced previously, in such a way that

they can be used in the calculation of the inverter losses and obtain a reliable evaluation of the

powers involved.

For this purpose an IGBT developed by Mitsubishi, the PM50CL1B120, will be used.

4.3.1-Verification of the conduction losses model

Regarding the conduction losses, the equations previously presented can be verified by using

a half-bridge inverter and a sinusoidal generator as a load, as shown in Figure 4.3.

115

Where [G1] and [G2] are the control signals given to the two IGBTs, a square wave with duty-

cycle equal to 50% for 𝐼𝐺𝐵𝑇2 and an opposite signal for 𝐼𝐺𝐵𝑇3.

The data used in order to calculate the conduction losses can be obtained from the datasheet of

the IGBT and inserted into the Simulink model.

The characteristic of the IGBT is shown in Figure 4.4.

Figure 4.3-Half-bridge inverter for conduction losses calculation

Figure 4.4-IGBT V-I characteristic

116

From this curve we can extract:

𝑉𝐶𝐸𝑜𝑛 = 0.7 𝑉

𝑅𝐼𝐺𝐵𝑇𝑜𝑛 = 34 𝑚𝛺

For the diode instead:

From this curve we can extract:

𝑉𝑜𝑛𝐷 = 0.5 𝑉

𝑅𝑜𝑛𝐷 = 55 𝑚𝛺

Figure 4.5-Diode V-I characteristic

117

The use of a sinusoidal current generator downstream of the power supply is justified by the

fact that it makes possible to study various working conditions.

The characteristics that we can control are the amplitude of the sinusoid and the phase of the

current waveform.

By modifying the phase, the conduction losses can be analysed not only in resonant condition,

where only the IGBT conducts, but also in inductive and capacitive working conditions, in

which the diode and the IGBT conduce partly.

The validation happens in the following way: given a certain working condition, the losses

during the simulation are obtained by calculating the instantaneous power, obtained with the

product between the current and voltage, and we use the mean function in order to mean over

the period of the instantaneous power.

4.3.2-Verification of the switching losses model

Regarding the turn-on and turn-off energies, we compare the results given by the equations

presented previously with the waveforms taken from the datasheet shown in Figure 4.6.

118

Among these characteristics, we consider the ones at 125°𝐶, in fact it can be assumed that the

device works for most of the time at high temperature.

We resume the equation of the turn-off energy (4.10).

𝐸𝑜𝑓𝑓 =1

2[𝑡𝑟 + 𝑡𝑓 +

1

10(𝑡𝑡 + 𝑡𝑓)] 𝑉𝑑𝐼𝑑 (4.10)

In this equation the parameters 𝑡𝑓 and 𝑡𝑡 are strictly related to the current, the temperature and

the physics of the device, since that the characteristics of these parameters aren’t described,

they are chosen based on the ranges set in the datasheet and thus for the following working

conditions:

𝑉𝑑 = 𝑉𝐶𝐶 = 600 𝑉

𝐼𝑑 = 𝐼𝑠𝑤 = 30 𝐴

We chose the following parameters:

Figure 4.6-𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 characteristics, function of the switching current

119

𝑡𝑡 = 600 𝑛𝑠

𝑡𝑓 = 100 𝑛𝑠

𝑡𝑟 = 140 𝑛𝑠

The results obtained as the switching current changes are shown in the following table:

The main consequence of the approximation of the 𝐸𝑜𝑓𝑓 characteristic to a straight line is that,

in the model, we obtain values smaller for smaller switching currents and higher values for

higher switching currents.

Since in the datasheet the characteristic of turn-off energy is fully represented, we can introduce

an adaptation factor Y, in order to obtain values closer to the real ones.

In order to obtain the value of Y as a function of the switching current, we use a third grade

polynomial (4.17), in which:

𝑃1 = −1.3 ∗ 10−5

Figure 4.7-Comparison between 𝐸𝑜𝑓𝑓−𝑚𝑜𝑑 and 𝐸𝑜𝑓𝑓−𝑑𝑎𝑡𝑎𝑠ℎ𝑒𝑒𝑡

120

𝑃2 = 0.0017

𝑃3 = −0.0826

𝑃4 = 2.274

𝑌 = 𝑃1𝐼𝑠𝑤3 + 𝑃2𝐼𝑠𝑤

2 + 𝑃3𝐼𝑠𝑤 + 𝑃4 (4.17)

By using this adaptation factor we obtain the following results:

After introducing the factor Y, we can see that the turn-off energy values of the model is closer

to the real values taken from the datasheet.

It needs to be clarified that Y can be used only when the manufacturer reports the entire

characteristic of the turn-off energy and not only its value for particular working conditions.

The same procedure applied to turn-off energy can be used for the turn-on energy, of which we

resume the equation.

𝐸𝑜𝑛−𝑡𝑜𝑡 = 𝐸𝑜𝑛 + ∆𝐸𝑜𝑛 =1

2(𝑡𝑟𝑖 − 𝑡𝑠 + 𝑡𝑓)𝑉𝑑𝐼𝐶𝐸 +

1


Figure 4.8-Comparison between 𝐸𝑜𝑓𝑓−𝑚𝑜𝑑𝑌 and 𝐸𝑜𝑓𝑓−𝑑𝑎𝑡𝑎𝑠ℎ𝑒𝑒𝑡

121

The values of 𝑡𝑠 and 𝑡𝑓 are linked by the relation 𝑡𝑟𝑟 = 𝑡𝑠 + 𝑡𝑓 and its characteristic, togheter

with the 𝐼𝑟𝑟𝑚 one, is shown on the datasheet.

The results obtained are shown in the following table:

Figure 4.9-Characcteristics of 𝐼𝑟𝑟𝑚 and 𝑡𝑟𝑟, as a function of the switching current

122

With this method we obtain values close to the datasheet ones and it is not even necessary the

introduction of an adaptation factor, because the parameters vary with the working conditions.

4.4-Dynamic losses model

The aim of this section is to use the calculation presented previously in this chapter in order to

develop a dynamic model using Simulink.

Figure 4.10-Comparison between 𝐸𝑜𝑛−𝑚𝑜𝑑 and 𝐸𝑜𝑛−𝑑𝑎𝑡𝑎𝑠ℎ𝑒𝑒𝑡

123

In Figure 4.11 we can see the Simulink blocks that we will use to calculate the losses in our

models.

QD1 and QD2 are signals coming from the measurement port of the blocks IGBT/Diode on

the same branch, they are the voltages and currents of the IGBTs and, by sending them to the

Demux block, we can separate the current flowing into the diode from the one flowing into the

IGBT.

Figure 4.11-Blocks for the losses calculation

124

As we can see from Figure 4.12, first of all we use a demux to separate the voltage and the

current belonging to the first and second IGBTs on a branch and then through the current

separator, we divide the current into the two contributions, the one belonging to the IGBT and

the one belonging to the diode.

In order to do so we use the current separator, we employ the saturation block, thus we have

the positive current that will go to the IGBT losses calculation block and we invert the sign of

the negative one that will go to the diode losses calculation block.

Current Separator

Figure 4.12-Demux IGBTs and Diodes currents and voltages blocks

125

First of all we need to use a selector to separate the voltages and the currents.

Then we need to use the 2-D Lookup Table to dynamically find the values of the energies, by

using the values that we obtained with the static calculations as breakpoints (as can be seen in

Figure 4.14) and their linear interpolation.

At the input of the 𝐸𝑜𝑛 table we use the memory block to obtain the voltage at the previous

time step, thus we assure to find an intersection of the current and the voltage different from

zero.

Figure 4.13-IGBT losses block

Figure 4.14-Eon Table block

126

For the calculation of the 𝐸𝑜𝑓𝑓 we use the same line of reasoning, but we use the memory block

to obtain the current at the previous time step, instead of the voltage, as shown in Figure 4.15.

The next step is to convert the energy into power, through the Energy to Power block (Figure

4.15).

Figure 4.15-Eoff Table block

Figure 4.16-Energy to Power block

127

This block have two inputs, the energy and the trigger, the logic circuit shown in Figure 4.17,

used to separate the power lost during turn-on stage and turn-off state.

The upper logic circuit in the previous figure is used for the 𝐸𝑜𝑛 trigger, in fact it determines

when the current is greater than zero after passing through the zero, thus when the current is

rising during the turn-on phase.

The lower circuit, instead, determines when the current is decreasing during the turn-off phase,

in facts it detects when the current is passing through the zero while in the previous time

interval it was greater than zero.

Regarding the Energy to Power block second input, the energy is multiplied by the

corresponding trigger, so that the power lost during turn-on is calculated only during the turn-

on phase and we do the same for the power lost during turn-off phase.

Inside the block we use the trigger to create a counter in order to determine the number of

switchings during the simulation.

This number will be divided by the duration of the simulation and, by multiplying this value

for the energy, we obtain the power.

Regarding the conduction power losses we multiply the resistance, calculated by using the

curve of the collector-emitter voltage and collector current (Figure 4.4), and we multiply it for

the square of the current.

4.5-Verification of the dynamic losses model

In order to verify the validity of our dynamic model we use the data given by the static losses

Figure 4.17-Trigger logic circuit

128

calculation and the circuit shown in Figure 4.18, of which we have the experimental data that

we can compare to the one obtained from the Simulink model.

The values of the circuit’s parameters are the following:

𝑉𝐷𝐶 = 230 𝑉

𝐿4 = 600 𝜇𝐻

𝐶4 = 0.106 𝜇𝐹

Figure 4.18-Validation circuit

Figure 4.19-Model of the validation circuit in Simulink

129

𝐶3 = 4.4 𝜇𝐻

𝐶1 = 𝐶2 = 1.54 𝜇𝐹

𝑅1 = 𝑅2 = 20 𝑚𝛺

𝐿1 −𝑀 = 𝐿2 −𝑀 = 41.3 𝜇𝐻

𝑀 = 14.4 𝜇𝐻

𝑅𝑙𝑜𝑎𝑑 = 13 𝛺

We choose the switching frequency equal to the resonant one, thus 𝑓𝑠𝑤 = 20 𝑘𝐻𝑧.

First of all we compare waveforms and peak values at the input and output of the load,

shown in the following figures, and then we will analyse the values of power and efficiency.

Input

Figure 4.20-Input voltage from the experimental results and the model

130

The peak values of the current and voltage are:

𝑉1_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝑉1_𝑚𝑜𝑑𝑒𝑙 = 230 𝑉

𝐼1_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝐼1_𝑚𝑜𝑑𝑒𝑙 = 24 𝐴

Figure 4.21-Input current from the experimental results and the model

131

Output

The peak values of the current and voltage are:

𝑉2_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝑉2_𝑚𝑜𝑑𝑒𝑙 = 285 𝑉

𝐼2_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝐼2_𝑚𝑜𝑑𝑒𝑙 = 22 𝐴

Power and efficency

In order to calculate the mean power we multiply the voltage and the current and then we

use the Matlab function mean to obtain the desired value.

After this we divide the mean value of the output power by the one of the input power, thus

we obtain the efficiency.

Figure 4.22-Output voltage (red) and current (black) from the

experimental results and the model

132

𝑃𝑖𝑛_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 3400 𝑊 𝑃𝑖𝑛_𝑚𝑜𝑑𝑒𝑙 = 3460 𝑊

𝑃𝑜𝑢𝑡_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 3140 𝑊 𝑃𝑜𝑢𝑡_𝑚𝑜𝑑𝑒𝑙 = 3177 𝑊

𝜂𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 92% 𝜂𝑚𝑜𝑑𝑒𝑙 = 92%

As we can see from these results, the values of the power from the model are higher than

those of the simulation, this is the effect of the initial transient present in the model that

increase the initial value of every parameter calculated in Simulink, thus the mean value

increases.

Regarding the power lost in the inverter, we observe the waveforms we obtain from our

model (Figure 4.23 and 4.24), related to the losses in a branch, thus to obtain the total power

lost, we need to multiply them by two.

Figure 4.23-Conduction losses, highlight of the regime trend

133

Hence the peak regime value of the losses is the following:

𝑃𝑙𝑜𝑠𝑠,𝑖𝑛𝑣𝑒𝑟𝑡𝑒𝑟_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 44.5 𝑊 𝑃𝑙𝑜𝑠𝑠,𝑖𝑛𝑣𝑒𝑟𝑡𝑒𝑟_𝑚𝑜𝑑𝑒𝑙 = 40.76 𝑊

From these results we can see that, even though our model is rather accurate, the losses we

obtain with the simulation are slightly lower than the ones from the experimental results, this

can be attributed to the fact that, the values in our simulation are lower than the ones used

to calculate the breakpoints in the Table block, thus the linear interpolation give us smaller

values of energy compared to the real ones.

Figure 4.24-Switching losses, highlight of the regime trend

134

Chapter 5-Simulations

In this chapter we will present two models, representing two different methods used to achieve

an efficient induction cooking with both ferromagnetic and non-ferromagnetic cookware.

The simulations related to these models, made with Simulink, will be compared to experimental

data, present in literature, in order to validate our dynamic losses calculation method, and

between themselves, as a mean to show the development of all metal induction cooking.

First of all we will analyse the behaviour of these circuits at resonant frequency and then we

will compare those results with the ones obtained at different frequencies, in the capacitive

zone (𝑓𝑠𝑤 < 𝑓𝑟𝑒𝑠) and the inductive zone (𝑓𝑠𝑤 > 𝑓𝑟𝑒𝑠).

5.1- Third-harmonic operation mode model

This model depicted in [21] and [22] employs a modified half-bridge series resonant inverter

topology that can heat ferromagnetic and non-ferromagnetic pans.

This inverter topology has two topological configurations with two operation modes: first

harmonic operation mode (FHOM) and third-harmonic operation mode (THOM), ensuring an

acceptable performance even with non-ferromagnetic cookware.

Figure 5.1- Schematic of half-bridge series resonant inverter topology

135

As we can see from Figure 5.1, according with what have been stated in Chapter 2, the pan-

inductor coupling is modelled as a resistance 𝑅𝑒𝑞 in series with an inductance 𝐿𝑒𝑞.

These parameters’ values depend on the characteristics of inductor, the frequency of the load

current and the material of the pan.

Figure 5.2(a) shows that the inductance is approximately constant with the frequency and that

the inductance for non-ferromagnetic (𝐿𝑒𝑞𝑁𝐹) pans is about a half of the inductance for

ferromagnetic (𝐿𝑒𝑞𝐹) pans for the same frequency.

Figure 5.2 (b) shows that the resistance increases with the frequency.

The resistance for ferromagnetic (𝑅𝑒𝑞𝐹) pans is about ten times of the resistance for non-

ferromagnetic (𝑅𝑒𝑞𝑁𝐹) pans for the same frequency.

The resonant frequency (𝑓𝑟) can be calculated as (5.3) and from this formula we can obtain the

value of the resonant capacitor (5.4).

Figure 5.2- (a) Frequency-dependent inductance, (b) frequency-dependent resistance

136

𝐿𝑒𝑞𝐹 = 2 𝐿𝑒𝑞𝑁𝐹 (5.1)

𝑅𝑒𝑞𝐹 = 10 𝑅𝑒𝑞𝑁𝐹 (5.2)

𝑓𝑟 =1

2𝜋√𝐿𝑒𝑞𝐶 (5.3)

𝐶 =1

4𝜋2𝑓𝑟2𝐿𝑒𝑞

(5.4)

The requirement of the inverter to achieve the THOM is that the value of the resonant capacitor

must be changed when the load is a non-ferromagnetic pan.

For this reason the inverter has some changes for heating non-ferromagnetic pans.

The inverter topology is a modified half-bridge series resonant inverter with an additional relay

to change the value of the resonant capacitor.

As we can see from Figure 5.3, in this configuration the additional relay is closed, thus the

resonant capacitor is 𝐶𝑓 = 𝐶1 + 𝐶2.

A typical value of the switching frequency for maximum output power for ferromagnetic pans

is 23 kHz.

Figure 5.3- Configuration for ferromagnetic pans

137

The resistance is approximately 3 𝛺 and the inductance is 40 𝜇𝐻 for this frequency.

The resonant capacitor calculated with (5.4) is 1200 𝑛𝐹 (𝐶1 = 300𝑛𝐹 and 𝐶2 = 900𝑛𝐹) for

this load and 𝑓𝑟 = 23 𝑘𝐻𝑧.

The output power at resonance is 3500 W with 𝑉𝑖 = 230 𝑉, so the rms load current is 34 A.

The inverter with the previous resonant capacitor can heat a non-ferromagnetic pan, although

in other conditions of operation.

The resonant frequency calculated with (5.1) and (5.3) is 32 kHz.

The resistance for this frequency is approximately 0.3 𝛺 and the inductance is 20 𝜇𝐻.

The value of the maximum output power for resonant frequency is 36 kW, so the rms value of

load current is 345 A.

These conditions of operations are not acceptable for a domestic induction cooker, due to the

current exceeds the ratings of the devices that composed the inverter.

Regarding these results one conclusion can be extracted: the resistance for non-ferromagnetic

pans is low, and the maximum output power and the load current are high for an acceptable

performance of the inverter.

In order to increase this resistance and decrease the maximum output power and the load

current, we use the third-harmonic operation mode (THOM).

As shown in Figure 5.4, if the detected pan is non-ferromagnetic the relay is opened and the

resonant capacitor is 𝐶1 = 300 𝑛𝐹 (𝐶𝑁𝐹).

Figure 5.4- Configuration for non-ferromagnetic pans

138

In order to obtain an efficient working condition with a non-ferromagnetic pan we also need to

change the inverter operation mode.

The third-harmonic operation mode (THOM) is based on obtaining an inductor current whose

frequency is the third harmonic to the switching frequency (𝑓𝑠).

The principal characteristics of this operation mode are: first, a higher resistance due to it

increases when the frequency of the inductor current increases, with the THOM this frequency

is three times the frequency for the FHOM.

The resistance for the THOM is 0.4 𝛺, when the frequency of the inductor current is 69 𝑘𝐻𝑧.

Second, the rms value of output voltage is three times less than the value for the FHOM.

The maximum output power is 2980 W for the THOM, this power is similar to the power for

ferromagnetic pans.

Now we will see the simulations and the comparison between them and the experimental data

shown in the literature.

For the ferromagnetic configuration, a pan made of ferromagnetic steel has been chosen, for

the non-ferromagnetic one, an aluminium pan has been used.

In case of this model we are not given some of the values of the parameters used to obtain the

experimental results, thus, for those, we choose reasonable values, some of which has been

shown previously in this chapter.

5.1.1-Simulations and validation, ferromagnetic configuration

Figure 5.5- Simulink model, ferromagnetic configuration

139

Data:

𝑉𝐷𝐶 = 230 𝑉

𝐶1 = 300 𝑛𝐹

𝐶2 = 900 𝑛𝐹

𝐶𝑒𝑞𝐹 = 𝐶1 + 𝐶2 = 1200 𝑛𝐹

𝑅𝑒𝑞𝐹 = 3 𝛺

𝐿𝑒𝑞𝐹 = 40 𝜇𝐻

𝑅𝐼𝐺𝐵𝑇 = 34 𝑚𝛺

Regarding the IGBT we choose one suitable for this kind of application, the PM50CL1B120

manufactured by Mitsubishi.

The phase delay between the IGBTs’ gate signals is 180ᵒ.

The switching frequency is equal to the resonant one and its value is 22.5 𝑘𝐻𝑧.

The output current and voltage are shown in Figure 5.6, due to the reasons stated before, the

peak values are different, but we can still compare the waveforms.

The peak value of current we obtain from the simulation is 46.75 𝐴.

Figure 5.6- Comparison between the output from the

experimental results and our simulation

140

Regarding the power:

𝑃𝑖𝑛 = 3406.4 W

𝑃𝑜𝑢𝑡 = 3247.2 W

Thus we obtain an efficiency equal to 95%, while in the experimental results this configuration

reaches a maximum value of efficiency equal to 93%.

Our value is nonetheless acceptable, in fact we have to take into consideration that all the

elements in our circuit are ideal, thus higher values of power and efficiency are understandable.

For the calculation of the losses in the IGBTs, we use the method depicted in Chapter 4 and, in

the Tables for the calculation of 𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓, we insert the energies calculated in a static way

through the data in the PM50CL1B120 datasheet as breakpoints.

For the calculation of the conduction losses we use the IGBT’s internal resistance 𝑅𝐼𝐺𝐵𝑇 .

Figure 5.7- 𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 breakpoints

141

At regime the peak values for the switching and the conduction losses are:

Figure 5.8- Switching losses

Figure 5.9- Conduction losses

142

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 3.8 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 74.34 𝑊

𝑃𝑙𝑜𝑠𝑠𝑒𝑠 = 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 + 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 78.14 𝑊

5.1.1.1- 𝒇𝒔𝒘 ≠ 𝒇𝒓𝒆𝒔

𝑓𝑠𝑤 = 20.5 𝑘𝐻𝑧 𝑓𝑠𝑤 = 21.5 𝑘𝐻𝑧 𝑓𝑠𝑤 = 23.5 𝑘𝐻𝑧 𝑓𝑠𝑤 = 24.5 𝑘𝐻𝑧

𝑃𝑖𝑛 = 2928 𝑊 𝑃𝑖𝑛 = 3248.4 𝑊 𝑃𝑖𝑛 = 3397.1 𝑊 𝑃𝑖𝑛 = 3229.9 𝑊

𝑃𝑜𝑢𝑡 = 2722.8 𝑊 𝑃𝑜𝑢𝑡 = 3126 𝑊 𝑃𝑜𝑢𝑡 = 3220.5 𝑊 𝑃𝑜𝑢𝑡 = 3024.5 𝑊

𝜂 = 93.8% 𝜂 = 94% 𝜂 = 94.8% 𝜂 = 94.2%

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 12.8 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 5.8 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 13.8 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 18.7 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 69.8 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 73.9 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 71.2 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 65.9 W

Figure 5.10- 𝑃𝑖𝑛 and 𝑃𝑜𝑢𝑡 at different frequencies

143

From the figures above we can see that, compared to the values at resonant frequency, both in

the inductive and capacitive zones:

The power, both at the input and the output, decreases.

The conduction losses decrease more in the inductive zone.

The switching losses increase more in the inductive zone.

The efficiency is lower in the capacitive zone than in the inductive one.

Figure 5.11- Losses at different frequencies

Figure 5.12- Efficiency at different frequencies

144

5.1.2-Simulations and validation, non-ferromagnetic configuration

Data:

𝑉𝐷𝐶 = 230 𝑉

𝐶1 = 300 𝑛𝐹

𝐶2 = 900 𝑛𝐹

𝐶𝑁𝐹 = 𝐶1 = 300 𝑛𝐹

𝑅𝑒𝑞𝑁𝐹 = 0.4 𝛺

𝐿𝑒𝑞𝑁𝐹 = 20 𝜇𝐻


The same assumption used for the ferromagnetic configuration are employed in this one.

The switching frequency is equal to 22.3 𝑘𝐻𝑧, thus the frequency of the inductor current is

67 𝑘𝐻𝑧, due to third-harmonic operation mode that triple the frequency of the current in the

load.

The output current and voltage are shown in Figure 5.14.

Figure 5.13- Simulink model, non-ferromagnetic configuration

145

The peak value of current we obtain from the simulation is 47.65 𝐴.


𝑃𝑖𝑛 = 1182.1 W

𝑃𝑜𝑢𝑡 = 828.8 W

In order to obtain power values closer to those obtained with a ferromagnetic load and to the

maximum achievable, we need to increase the supply voltage.

In case 𝑉𝐷𝐶 = 380 𝑉, the peak of the current in the load is 115.5 𝐴 and:

𝑃𝑖𝑛 = 3226.8 W

𝑃𝑜𝑢𝑡 = 2262.4 W

We obtain an efficiency equal to 70%, close to that of the experimental result.

This result makes it obvious that the efficiency, even if it stays in an acceptable range, is lower

for the non-ferromagnetic pans.

This is due to the THOM, in fact, it allows the current to not surpass the limits of the devices

in the circuit, but the increase in frequency generates higher losses in the IGBTs, as shown in

Figure 5.14- Comparison between the output from

the experimental results and our simulation

146

the following Figures.

At regime the peak values for the switching and the conduction losses are:

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 56.45 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 165.5 𝑊

𝑃𝑙𝑜𝑠𝑠𝑒𝑠 = 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 + 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 221.95 𝑊



147

The losses in the IGBTs are almost tripled due to the tripled frequency.

5.1.2.1- 𝒇𝒔𝒘 ≠ 𝒇𝒓𝒆𝒔


𝑃𝑖𝑛 = 466.1 𝑊 𝑃𝑖𝑛 = 1582.5 𝑊 𝑃𝑖𝑛 = 421.1 𝑊 𝑃𝑖𝑛 = 191.6 𝑊

𝑃𝑜𝑢𝑡 = 318.6 𝑊 𝑃𝑜𝑢𝑡 = 985.7 𝑊 𝑃𝑜𝑢𝑡 = 273.8 𝑊 𝑃𝑜𝑢𝑡 = 134.5 𝑊

𝜂 = 64.8% 𝜂 = 62.3% 𝜂 = 66.48% 𝜂 = 66.9%

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 62.8 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 59 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 52 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 45 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 71.7 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 229.45 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 57.7 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 49.9 W


148

From the figures above we can see that, compared to the values at resonant frequency:

The power, both at the input and the output, decreases except in the capacitive zone

near the resonant frequency.

The conduction losses decrease because the current decreases except in the capacitive

zone near the resonant frequency.

Figure 5.18-Losses at different frequencies


149

The switching losses decrease going from the capacitive to the inductive zone.

The efficiency is lower in the inductive zone then the capacitive one.

5.1.2.2-Variable 𝑹𝒍𝒐𝒂𝒅

In the previous section we analysed the behaviour of the circuit at different switching

frequencies while maintaining the resistance of the load fixed.

Now we consider the variation of the resistance due to the changing frequency, by using the

Acero-Hernandez-Burdio-Alonso-Barragan model presented in section 2.3.2, thus the

following equation:

RT = −1

n0[ξϕcond

√2r0σ∑[ai]

n

i=1

] − n0 [√2𝛽ξϕind3𝑟0𝜎

∑[ai]

n

i=1

+√2𝜋2𝜉𝑟0𝜙𝑖𝑛𝑑

𝜎∑[aiho,i

2 ]

n

i=1

]

In which 𝑟0 = 1 𝑚𝑚 and 𝑛0 = 55.

In this way we can compare the values obtained from the simulations in case of fixed and

variable resistance.


𝑅𝑙𝑜𝑎𝑑 = 0.371 𝛺 𝑅𝑙𝑜𝑎𝑑 = 0.388 𝛺 𝑅𝑙𝑜𝑎𝑑 = 0.414 𝛺 𝑅𝑙𝑜𝑎𝑑 = 0.425 𝛺

𝑃𝑖𝑛 = 452.9 𝑊 𝑃𝑖𝑛 = 1580.6 𝑊 𝑃𝑖𝑛 = 439.4 𝑊 𝑃𝑖𝑛 = 198.6 𝑊


𝜂 = 63% 𝜂 = 61.6% 𝜂 = 67.22% 𝜂 = 68.19%

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 63.8 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 61 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 51.8 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 =

44.77 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 72.1 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 236 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 57.4 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 29.9 W

From these results we can see that the variations in the input and output power are very small,

while for the other parameters, above all the efficiency, are more apparent, as highlighted in

the following figures.

150

Figure 5.20-𝑃𝑖𝑛, fixed and variable 𝑅𝑙𝑜𝑎𝑑

Figure 5.21-𝑃𝑜𝑢𝑡, fixed and variable 𝑅𝑙𝑜𝑎𝑑

151

Figure 5.22-Switching losses, fixed and variable 𝑅𝑙𝑜𝑎𝑑

Figure 5.23-Conduction losses, fixed and variable 𝑅𝑙𝑜𝑎𝑑

152

Hence we can conclude that:

The resistance in the capacitive zone is lower than the fixed one, equal to the one at resonant

frequency and the input and output power are lower.

The losses are higher, thus the efficiency is lower.

In the inductive zone, the opposite happens, thus we have higher power, lower losses and a

higher efficiency than the ones in case of a fixed load resistance.

5.2-Frequency-doubler operation mode model

This model depicted in [23] and [24] employs a time-sharing high-frequency multiple-resonant

soft-switching inverter that can heat ferromagnetic and non-ferromagnetic pans.

This high-frequency multi-resonant inverter can efficiently operate under the conditions of load

resonant frequency changing mode, frequency doubler mode (100 kHz) for low resistivity

metal IH loads and fundamental switching frequency mode (50 kHz) for high resistivity metal

IH loads, respectively.

Figure 5.24-Efficency, fixed and variable 𝑅𝑙𝑜𝑎𝑑

153

In order to adapt and match for various metallic pans and vessels, the latest development of

high-frequency inverters which include selective resonant frequency changing principle, as

well as frequency tripler or frequency doubler are effective power solutions.

However, the higher switching frequency, the more switching losses are generated for high

frequency power conversion processing.

The proposed high-frequency resonant inverter is expected to reduce switching losses by not

only frequency doubler operation, but also soft switching power conversion principle.

As stated before, in order to heat cookware made by different materials, it is effective to change

the load resonant current frequency 𝑓𝑟.

This can be done by changing the value of the series compensated capacitor 𝐶𝑐, as shown in

Figure 5.21.

Figure 5.25- Time-sharing high-frequency multiple-resonant soft-switching

inverter

154

In the case of low resistivity metal materials, such as copper or aluminium, the load resonant

current frequency 𝑓𝑟 is to be designed for frequency doubler operation mode, thus 𝑓𝑟 = 𝑓𝑜𝑢𝑡 =

2𝑓𝑠𝑤.

Here 𝑓𝑜𝑢𝑡 is output frequency.

On the other hand, in the case of high resistivity metals such as magnetic-stainless steel or iron,

𝑓𝑟 is designed to be almost same as 𝑓𝑠𝑤 (𝑓𝑟 = 𝑓𝑜𝑢𝑡 = 𝑓𝑠𝑤).

The changing mechanism of the series compensated capacitor 𝐶𝑐 is actually implemented by

detecting the effective value 𝐼𝑅(𝑟𝑚𝑠) of the load resonant current 𝑖𝑅.

The output power regulation of the proposed time-sharing high-frequency inverter can be

achieved by phasor angle control for link resonant capacitor currents (𝑖𝑅(𝑄1) or 𝑖𝑅(𝑄2)) which

is phase-shifted PWM in the fixed switching frequency and fixed duty factor 𝐷 = 0.5.

Now we will see the simulations and the comparison between them and the experimental data

shown in the literature.

For the ferromagnetic configuration, a pan made of magnetic stainless steel has been chosen,

for the non-ferromagnetic one, a copper pan has been used.

In case of this model we are given more data compared to the previous model, thus we can

expect closer results.

Figure 5.26- A schematic system arrangement for all IH cooking

appliances with selective switching of load resonant capacitor 𝐶𝑐

155

5.2.1-Simulations and validation, ferromagnetic configuration

Data:

𝑉𝐷𝐶 = 200 𝑉

𝐶1 = 𝐶2 = 16 𝑛𝐹

𝐿1 = 𝐿2 = 32 𝜇𝐻

𝐶𝑐 = 800 𝑛𝐹

𝑅0 = 23 𝛺

𝐿0 = 300 𝜇𝐻


Regarding the IGBT we use the one exploited in the experimental application, the

CM50DY-24H, manufactured by Mitsubishi.

Figure 5.27- Simulink model, ferromagnetic configuration

156


The switching frequency is equal to the resonant one and its value is 50 𝑘𝐻𝑧.

The output current is shown in Figure 5.17.

Having all the data regarding the experimental part we can compare both its waveforms and its

values.

The peak value of current we obtain from both the simulation and the experimental data is

13.6 𝐴.


𝑃𝑖𝑛 = 2396.2 W

𝑃𝑜𝑢𝑡 = 2360.7 W

Thus we obtain an efficiency equal to 98.5%, while in the experimental results this

configuration reaches a maximum value of efficiency equal to 96.7%.

For the same reason taken into account for the previous model, this value of efficiency is

acceptable.

For the calculation of the losses in the IGBTs, we use the method depicted in Chapter 4, as we

have done for the previous model, we insert the energies calculated in a static way through the

data in the CM50DY-24H datasheet as breakpoints.

The problem is that this datasheet doesn’t give us the energy curves, thus we can’t calculate an



157

adaptation factor, so we expect values different than the experimental ones for the switching

losses.

The values of the breakpoints are the following:

As always, for the calculation of the conduction losses we use the IGBT’s internal resistance

𝑅𝐼𝐺𝐵𝑇.


Figure 5.29- 𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 breakpoints

158

In literature we are given the values of the switching and conduction losses for this model, thus

we can compare them.

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 3.7 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔_𝑚𝑜𝑑𝑒𝑙 = 2 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛_𝑚𝑜𝑑𝑒𝑙 = 48 𝑊

𝑃𝑙𝑜𝑠𝑠𝑒𝑠_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 51.7 𝑊 𝑃𝑙𝑜𝑠𝑠𝑒𝑠_𝑚𝑜𝑑𝑒𝑙 = 50 𝑊

Hence there is a small difference in the IGBTs’ total losses, the ones in the model are a little

lower, but that was expected because of the previously explained reasons.

5.2.1.1- 𝒇𝒔𝒘 ≠ 𝒇𝒓𝒆𝒔

𝑓𝑠𝑤 = 48 𝑘𝐻𝑧 𝑓𝑠𝑤 = 49 𝑘𝐻𝑧 𝑓𝑠𝑤 = 51 𝑘𝐻𝑧 𝑓𝑠𝑤 = 52 𝑘𝐻𝑧

𝑃𝑖𝑛 = 1846.7 𝑊 𝑃𝑖𝑛 = 2110.6 𝑊 𝑃𝑖𝑛 = 2652 𝑊 𝑃𝑖𝑛 = 2707 𝑊



159

𝜂 = 97.8% 𝜂 = 98.4% 𝜂 = 97% 𝜂 = 95.6%

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 2.1 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 2 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 2.1 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 2.3 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 50 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 48.4 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 49 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 49.12 W


Figure 5.33- Losses at different frequencies

160


The power, both at the input and the output, increases going from the capacitive to the

inductive zone.

The conduction losses increase both in capacitive and inductive zones.

The switching losses are very similar in every zone.



161

5.2.2-Simulations and validation, non-ferromagnetic configuration

Data:

𝑉𝐷𝐶 = 200 𝑉

𝐶1 = 𝐶2 = 16 𝑛𝐹

𝐿1 = 𝐿2 = 32 𝜇𝐻

𝐶𝑐 = 17.4 𝑛𝐹

𝑅0 = 2.6 𝛺

𝐿0 = 220 𝜇𝐻



Figure 5.35- Simulink model, non-ferromagnetic configuration

162

The switching frequency is equal to 50 𝑘𝐻𝑧, thus the frequency of the inductor current is

100 𝑘𝐻𝑧, due to frequency-doubler operation mode that, as the name implies, double the

frequency of the current in the load.

The output current is shown in Figure 5.31.

The peak value of current we obtain from both the simulation and the experimental data is

24.5 𝐴.


𝑃𝑖𝑛 = 1443.4 W

𝑃𝑜𝑢𝑡 = 1343.4 W

In order to obtain power values closer to those obtained with a ferromagnetic load and to the

maximum achievable, we need to increase the supply voltage.

In case 𝑉𝐷𝐶 = 250 𝑉, the peak of the current in the load is 30.5 𝐴 and:

𝑃𝑖𝑛 = 2255.3 W

𝑃𝑜𝑢𝑡 = 2099 W

We obtain an efficiency equal to 93.3%, while in the experimental results this configuration



163

reaches a maximum value of efficiency equal to 92.6%, they are very similar.

The doubled frequency increases the losses but not as much as the frequency tripler and having

a smaller power going through the non-ferromagnetic configuration, thus the losses are very

close to the ones with ferromagnetic pans.



164

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 4.3 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔_𝑚𝑜𝑑𝑒𝑙 = 2.3 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛_𝑚𝑜𝑑𝑒𝑙 = 45 𝑊

𝑃𝑙𝑜𝑠𝑠𝑒𝑠_𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 = 49.3 𝑊 𝑃𝑙𝑜𝑠𝑠𝑒𝑠_𝑚𝑜𝑑𝑒𝑙 = 47.3 𝑊

Hence there is a small difference in the IGBTs’ total losses, the ones in the model are a little

lower, but that was expected because of the previously explained reasons.

We can also see that with the smaller parameters in the non-ferromagnetic configuration the

difference between the real losses and the model ones increases compared to the

ferromagnetic configuration.

This is due to the accentuation of the effects of the linearization during the energy calculation

and the lack of an adaptation factor that varies with the current.

Comparing the two models between themselves, we can see that the frequency-doubler

operation mode seems to be a better solution than the third-harmonic mode, in fact we get

smaller losses and a very high efficiency, very close to the one obtained with ferromagnetic

pans.

5.2.2.1- 𝒇𝒔𝒘 ≠ 𝒇𝒓𝒆𝒔

𝑓𝑠𝑤 = 48 𝑘𝐻𝑧 𝑓𝑠𝑤 = 49 𝑘𝐻𝑧 𝑓𝑠𝑤 = 51 𝑘𝐻𝑧 𝑓𝑠𝑤 = 52 𝑘𝐻𝑧

𝑃𝑖𝑛 = 475.3 𝑊 𝑃𝑖𝑛 = 911.9 𝑊 𝑃𝑖𝑛 = 4594 𝑊 𝑃𝑖𝑛 = 2909.8 𝑊


𝜂 = 90.1% 𝜂 = 91.8% 𝜂 = 92% 𝜂 = 93.2%

𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 1.6 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 1.5 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 8.7 𝑊 𝑃𝑠𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔 = 7.4 𝑊

𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 33.2 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 32.2W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 83.7 W 𝑃𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 80 W

165


Figure 5.40-Losses at different frequencies

166


The power, both at the input and the output, decreases in the capacitive zone, while it

highly increases in the inductive one.

The conduction losses decreases in the capacitive zone and increase in the inductive

zone.

The switching losses decrease a little in the capacitive zone, while they increase in the

inductive one.


Figure 5.41-Efficiency at different frequencies

167

Chapter 6-Conclusions

Induction cookers are nowadays an established reality in the market of many countries in the

world, above all in those in which the cost of electricity is lower compared to ours.

Nonetheless even in Italy they are spreading on the market, employed in the most advanced

kitchens that are responsive to the problem of energy efficiency.

The advantages in terms of safety and cleanliness are remarkable, but also the design and

energy conservation represents a strong selling point for this technology.

The difference in cost, compared with traditional technologies, is still higher, but in time it will

became lower thanks to the increase in the number of produced models and the resulting

decrease in the production cost due to the spread of this new technology in the market.

This project has the objective to create a model for all metal induction cooking starting from

experimental data taken from the literature.

Initially the model of the load has been presented, with all the problems related to the use of

cookware different materials and operative conditions.

Then we showed and analysed the typical structure of an induction cooker, with particular

attention to the characteristics of the resonant inverter, the most crucial component regarding

the study of this technology.

Lastly we have shown two all metal induction cooker’s structures present in literature and

compared them with our models created with Simulink, focusing on the analysis of the power,

efficiencies and losses given by the dynamic losses calculations that we developed.

From the results shown in Chapter 5, we can see that we have better working conditions at

resonant frequency and also that, having to increase the frequency in order to operate induction

cookers with non-ferromagnetic pans, we obtain better results with a frequency doubler

operation mode than with frequency tripler.

In fact the losses in the IGBT have a remarkable impact on the system total losses and they

increase with frequency, thus the frequency doubler operation mode allow us to obtain an

higher efficency, very close to the one with ferromagnetic materials.

168

References

[1] Alfred Mühlbauer, “Hystory of induction heating and melting”

[2] http://www.cookingforengineers.com/article/120/Common-Materials-of-Cookware

[3] J. Acero, J.M. Burdio, L.A. Barragin, D. Puyal, R. Alonso; “Frequency-Dependent

Resistance in Litz- Wire Planar Windings for All-Metal Domestic Induction Heating

Appliances”

[4] F. Tourkhani, P. Viarouge, “Accurate Analytical Model of Winding Losses in Round

Litz Wire Windings”, IEEE Transactions on Magnetics”

[5] Jesús Acero, Pablo J. Hernández, José M. Burdío, Rafael Alonso, Luis A. Barragán;

“Simple Resistance Calculation in Litz-Wire Planar Windings for Induction Cooking

Appliances”

[6] J. Acero, J.M. Burdio, L.A. Barragin, D. Puyal, R. Alonso;” Analytical Equivalent

Impedance for a Planar Circular Induction Heating System”

[7] J. Acero, R. Alonso, L.A. Barragan, C. Carretero, O. Lucia, I. Millan, J.M. Burdio,

“Domestic Induction Heating Impedante Modeling Includind Windings, Load, and Ferrite

Substrate”

[8] A. Beato, C. Bocchiola, S. Frattesi; “Modelling and Design of the Half-Bridge Resonant

Inverter for Induction Cooking Application”

[9] I. Millań,J.M. Burdıó,J. Acero,O. Lucıá,S. Llorente; “Series resonant inverter with

selective harmonic operation applied to all-metal domestic induction heating”

[10] Clayton R. Paul; “Introduction to Electromagnetic Compatibility”

[11] Valery Vodovozof;”Introduction to Electronic Engineering”

[12] http://www.engr.usask.ca/classes/EE/443/notes/Snubber_Circuits.pdf

[13] Abdus Sattar, “Insulated Gate Bipolar Transistor (IGBT) Basics”

[14] K. Liu, “High frequency quasi-resonant converter techniques”

[15] K. Liu and F. C. Lee, “Resonant switches-A unified approach to improve performances

of switching converters”

[16] K. Liu, R. Oruganti, and F. C. Lee, “Resonant switches-Topologies and characteristics”

[17] E. J. Miller, “Resonant switching power conversion”

[18] F. C. Lee, “High-Frequency Quasi-Resonant Converter Technologies”

169

[19] R.W. Erickson, “Fundamentals of Power Electronics”

[20] A. Pozzi, “Studio e analisi dell’efficenza di un inverter impiegato in un sistema di

trasmissione di energia a distanza”

[21] I. Millán, D. Puyal, J.M. Burdío, J. Acero; “Resonant Inverter Topology for All-Metal

Domestic Induction Heating”

[22] I. Millán, D. Puyal, J.M. Burdío, J. Acero; “Series resonant inverter with selective

harmonic operation applied to all-metal domestic induction heating”

[23] Takayuki Hirokawa, Eiji Hiraki , Toshihiko Tanaka, Masayuki Okamoto, Mutsuo

Nakaoka; “The Practical Evaluations of Time-Sharing High-Frequency Resonant Soft-

Switching Inverter for All Metal IH Cooking Appliances”

[24] Takayuki Hirokawa, Eiji Hiraki , Toshihiko Tanaka, Masayuki Okamoto, Mutsuo

Nakaoka; “A Novel Type Time-Sharing High-Frequency Resonant Soft-Switching

Inverter for All Metal IH Cooking Appliances”

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