Eddy Current Losses in the

IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2016 ,

Eddy Current Losses in the

Vicinity of Reactor Air Gaps due to Fringing Flux

VALDEMAR STENLUND

Eddy Current Losses in the Vicinity of Reactor Air Gaps due to Fringing Flux

Valdemar Stenlund

A Master Thesis Report written in collaboration with the Department of Electromagnetic Engineering at the School of Electrical Engineering, KTH

and

AQ Electric Suzhou Co., Ltd. and AQ Trafo AB

Supervisors: Christer Eriksson at AQ Trafo AB, He Dongxia at AQ Electric Suzhou Co., Ltd. and Prof. G¨ oran Engdahl at KTH.

Examiner: Prof. Martin Norgren at KTH

August, 2016

TRITA - EE 2016:126

Abstract

The objective of this report is to investigate the fringing flux around the air gap of a high frequency reactor and what correlation it has with losses, air gap length and frequency. A computer model is made using a finite element analysis software and a prototype reactor is built and tested on to verify the model. Variables such as air gap length, ripple frequency and current are changed in order to investigate different relationships.

Results show that the computer model is sufficiently valid, and that trends

regarding core losses versus frequency and air gap length correlates with

theory. From test and simulation results conclusions are made for making

designers aware of different measures for mitigating unwanted fringing flux

effects.

Sammanfattning

Syftet med denna rapport ¨ ar att unders¨ oka det magnetiska l¨ ackfl¨ odet runt luftgap hos h¨ ogfrekventa reaktorer och vad det har f¨ or korrelation med f¨ orluster, luftgapsdimensioner och frekvens. En datormodell som anv¨ ander finita elementanalys ¨ ar skapad och en prototyp ¨ ar byggd och testad p˚ a f¨ or att kunna verifiera datormodellen. Variabler s˚ asom luftgapsl¨ angd, rip- pelfrekvens, och str¨ ommar ¨ ar ¨ andrade f¨ or unders¨ oka olika f¨ orh˚ allanden.

Resultaten visar att datormodellen ¨ ar tillr¨ ackligt giltig och trender be- tr¨ affande k¨ arnf¨ orluster, frekvens och luftgapsdimensioner korrelerar med teorin. Fr˚ an test och simulationer slutsatser g¨ ors f¨ or att belysa olika

˚ atg¨ arder som kan implementeras f¨ or att mildra effekterna som har att

g¨ ora med l¨ ackfl¨ ode runt luftgap.

Acknowledgement

I would like to thank my sponsor companies AQ Electric Suzhou Co., Ltd. in China and AQ Trafo AB in Enk¨ oping for helping me both with their compe- tence and with the facilities which made it possible for me to design and build a prototype reactor and for laboratory measurements. My supervisor in China, He Dongxia, has been an indispensable source of experience when designing my first own reactor. Christer Eriksson, who was my supervisor in Sweden, has also been of great help when making calculations for my design.

At KTH my supervisor Prof. G¨ oran Engdahl has been of invaluable help with his fast e-mail response time and profound knowledge regarding electromag- netism and inductive components.

Last but not least Anna Juhasz at COMSOL Support has been helping me

navigate through COMSOL’s many complicated functions.

Contents

1 Summary 1

2 Introduction 2

3 Theory 2

3.1 Reactor . . . . 2

3.1.1 Application . . . . 3

3.2 Magnetic Material . . . . 3

3.2.1 Saturation of Magnetic Material . . . . 4

3.2.2 Non-Oriented Silicon-Steel . . . . 4

3.3 Introducing Air Gap . . . . 5

3.3.1 Theoretical Optimisation of Air Gap Length . . . . 7

3.4 Flux Fringing . . . . 9

3.5 Winding Losses . . . . 10

3.5.1 Skin Effect . . . . 11

3.5.2 Proximity Effect . . . . 12

3.6 Reactor Core Losses . . . . 12

3.6.1 Eddy Current Losses and Cauer Modelling . . . . 13

3.6.2 Hysteresis Losses . . . . 15

3.6.3 Complex Permeability . . . . 16

3.7 Heat Transfer . . . . 17

4 Reactor Prototype 18 4.1 Magnetic Core . . . . 18

4.2 Winding Design . . . . 21

4.3 Other Constructional Parts . . . . 24

4.4 Finished Prototype . . . . 26

4.5 Measurement Set-up . . . . 27

4.5.1 DC Bias with AC Ripple Test . . . . 27

4.5.2 High Frequency Test . . . . 28

5 Computer Model 30 5.1 Modelling the Magnetic Core . . . . 30

5.2 Modelling the Winding . . . . 32

6 Results and Discussion 32 6.1 Calculations . . . . 33

6.1.1 Winding Losses . . . . 33

6.1.2 Eddy Current Losses . . . . 34

6.1.3 Complex Permeability . . . . 35

6.1.4 Data-Sheet Power Loss . . . . 36

6.1.5 Fringing Flux Consideration . . . . 37

6.2 Test and Simulation Results . . . . 37

6.2.1 Winding Resistance . . . . 37

6.2.2 Inductance Comparison . . . . 38

6.2.3 Saturation Test . . . . 38

6.3 Fringing Flux . . . . 40

6.4 Power Loss . . . . 42

6.4.1 No Air Gap Power Loss Comparison . . . . 42

6.4.2 Measurements with Air Gap . . . . 43

6.4.3 Simulations with Air Gap . . . . 47

6.4.4 Power Loss Comparison . . . . 48

7 Conclusions 48

8 Future Work 49

References 50

Appendix

List of Figures

1 Illustration of magnetic path lengths in a core. . . . 5

2 Exemplification of how one can theoretically optimise the air gap length. . . . 8

3 Contour plot of magnetic flux density in a core. . . . 9

4 One type of Litz wire for use in high frequency reactor winding. . 12

5 A laminate with eddy currents flowing in subregions [1]. . . . 13

6 Circuit for determining eddy currents. . . . 14

7 Example of B-H loops showing energy into and out of a core [2]. 15 8 2D schematic with front view (a) and side view (b) of the core. Dimensions are in mm. . . . 18

9 Distribution of the silicon in a laminate. . . . . 19

10 DC magnetisation curve and DC permeability curve, supplied by manufacturer [3]. Magnetic flux density is shown as peak value. . 20

11 Core losses for different frequency and flux density, supplied by manufacturer [3]. Magnetic flux density is shown as peak value. . 20

12 Flow chart for determine number of winding turns. . . . 22

13 Parallel connected Litz wires. . . . 23

14 Cross section front view of the core and winding. . . . 24

15 Top view showing core and double layered winding. . . . 24

16 3D drawing of the reactor made in SolidWorks. . . . 25

17 Completely assembled prototype and air gap pieces with different thickness. . . . 26

18 Partly dissembled prototype. . . . . 27

19 DC bias with AC ripple test circuit schematic. . . . . 28

20 Frequency converter used in power loss measurements. . . . 29

21 Comsol 3D model with one coil removed to visualise air gap. . . 30

22 BH-curve implemented in Comsol for the DC bias simulation. . . 31

23 Currents through the resistors R

to R

. . . . 35

24 Data-sheet, see Fig. 11, losses vs frequency for constant magnetic flux density and no air gap. . . . 37

25 Verification of computer model, comparing inductance with dif- ferent air gap lengths. . . . 38

26 Incremental inductance test for different air gap lengths. The RMS AC ripple current is about 6 A. . . . 39

27 3D model showing magnetic flux density distribution and direc- tion of the field. . . . 40

28 Fringing magnetic flux in the vicinity of air gap. . . . . 41

29 Plot showing decreasing fringing magnetic flux outside the air gap. 42 30 Measured power loss vs AC current for different air gap lengths. 44 31 Air gap length vs power loss for two different frequencies. . . . . 45

32 Magnetic flux density vs air gap length for two different frequencies. 45 33 Measured losses vs air gap length for two frequencies. . . . . 46

34 Simulated losses vs air gap length for two frequencies. . . . 47

35 Comparison of loss vs air gap length for two frequencies. . . . 48

List of Tables

2 Summary of some advantages and disadvantage for air gapped

cores [4]. . . . 7

3 Reactor core parameters. . . . . 19

4 Winding parameters. . . . . 21

5 Winding design parameters. . . . . 24

6 Model input variables. . . . . 32

7 Output variables. . . . . 33

8 Table of inductance, layer thickness and resistance for s = 0.1 mm and k = 1.5. . . . . 34

9 Complex permeability calculation results. B

= 0.22 T and H

= 42 A/m. . . . 36

10 Power loss comparison when having no air gap, power in W. . . 43

Notation Unit Description

R

Ω AC winding resistance P

W AC winding resistance loss A

m

Air gap cross sectional area R

1/H Air gap reluctance

θ

Angle which B delays H

ω rad Angular frequency

A

m

Coil conductor cross sectional area

U V Coil voltage

ρ

Ωm Conductor resistivity

δ m Conductor skin depth

k - Constant for determine subregion thickness ρ

Ωm Copper resistivity

A

m

Core cross sectional area

D kg/m

Core density

P

W Core eddy current losses P

W Core hysteresis losses

m kg Core mass

P

W Core power loss (P

+ P

) R

1/H Core reluctance

ρ

Ωm Core resistivity

V m

Core volume

W

m

Core window area

I A Current

J A/m

Current density

R

Ω DC winding resistance P

W DC winding resistance loss l

m Effective magnetic path length µ

- Effective permeability

l

m Electrical air gap length

a - Factor for determine subregion thickness K

- Fill factor for copper allocation

f Hz Frequency

µ

- Imaginary part of complex permeability

L H Inductance

s

m Lamination subregion thickness

s m Lamination thickness

l

m Magnetic core length

W J Magnetic energy

H A/m Magnetic field strength

Φ Wb Magnetic flux

B T Magnetic flux density

µ H/m Magnetic material permeability

µ

- Magnetic material relative permeability

0

ρ

Ωm Magnetic material resistivity B

T Magnetic saturation level

F

At Magneto-motive force

M LT m Mean-turn-length of winding P

W Measured power loss

l

m Mechanical air gap length

N - Number of conductor turns in the winding Φ

Wb Peak magnetic flux

B

T Peak magnetic flux density

Z

Ω Reactor impedance

µ

- Real part of complex permeability p W/m

Specific loss per m

from data-sheet

l

m Total conductor length

R

1/H Total magnetic reluctance

µ

H/m Vacuum permeability

1 Summary

The operating frequency in systems where air gapped reactors are present is expected to increase in order to reduce the overall size and weight of power converters [5]. However, as the trend of increasing the switching frequency con- tinues, the effects it has on magnetic components becomes a greater concern [6]. High frequency introduces unwanted effects that hamper a reactor design.

The effects include skin and proximity effects in the windings, increase in core losses and fringing flux [7]. This project focuses on fringing flux that occur in the vicinity of the air gap of the reactor core. The fringing flux induces eddy currents around the air gap and in the core itself. The induced eddy currents in turn adds to the overall losses of the component and can create localised over- heating which can be disastrous for the equipment. It is therefore important to design magnetic components in such a way to minimise these additional losses.

It is critical to investigate how additional losses due to fringing flux depends on the air gap dimensions and frequency.

A model of an air gapped reactor using a Finite Element Method (FEM) soft- ware is introduced. To be able to verify the model a prototype reactor is built and tested with high frequency excitation. A typical application for the pro- totype reactor could be in a DC-DC converter with a bias DC current and a superimposed ripple current with a frequency of tens of kilohertz. After test- ing, the simulation results are compared with experimental results. Different parameters are varied, for example air gap length and frequency, to be able to see if there are any relations between losses and parameter values. When vary- ing parameters such as air gap length and frequency for both the model and the prototype one can draw conclusions on how the air gap losses depend on different component variables.

The main goal of this project is to obtain a deeper understanding of the losses for an air gapped reactor for high frequencies, especially losses due to fringing flux. This will hopefully provide a tool for high frequency reactor designers to minimise the losses and to give an accurate numerical value for the air gap losses due to fringing flux when varying component parameters.

In the first section the reader is provided with some introduction and back-

ground of the project. The next section includes some important phenomena

that occur within a reactor with an air gap. Subsequently the FEM model built

in the software COMSOL Multiphysics

, henceforth referred to as Comsol, is

presented. Afterwards the building of the reactor prototype is explained as well

as the measurement set-ups. The section after shows the simulation results to-

gether with the test results. A comparison follows to determine the validity of

the computer model. Interpretation and discussion of the results will finalise

the report. Some conclusions of the results and future work is presented lastly

in the report.

2 Introduction

Air gapped cores are extensively used in power reactors [8]. One of the reasons to have air gapped reactors is to increase the reluctance of the core, thereby enable it to store more energy before saturation. A negative effect when having air gaps is that the magnetic flux fringes around the edges of the gap. This is an unwanted phenomenon in which the fringing magnetic flux induces eddy current in conductor windings, in the magnetic core and in other material in the vicinity of the air gap. The increase in temperature due to eddy currents are related to frequency and could potentially lead to hot spots in nearby material which exceeds the thermal limits required by a specification [9]. The non-uniform magnetic flux that is the fringing flux could lead to dangerously high concentra- tion of losses. These losses are a significant part of the total losses of the reactor.

The problem of the increasing losses of reactors with air gaps becomes more evident when using higher frequencies. Higher frequency of the current in the windings leads to a faster varying magnetic field thus increasing losses due to eddy current effects and hysteresis. The reason why the demand of high fre- quency wound components is increasing is because it enables manufacturers to downsize the component yet keeping the same power output. For example in a Switch-Mode Power-Supply (SMPS) the reactor accounts for 25% of the vol- ume and 30% of the weight [10]. The reason is because slower switching supplies must store more energy per cycle, the result is that the core size is smaller for high frequencies and bigger for low frequencies.

The power dissipated in a reactor originates from two sources: winding losses and losses associated with the reactor core. Accurate determination of these losses require direct measurements or FEM simulation software, however it is possible to estimate the losses just by looking at data supplied by core and winding suppliers together with power supply parameters [11]. An important approximation regarding core loss values supplied by manufacturers is that no fringing flux is taken into account. So this method is not applicable in this project.

3 Theory

This section will give the reader some background theory of some areas that preferably should be understood in order to appreciate the modelling and sim- ulations in later sections.

3.1 Reactor

A reactor is sometimes also called a choke or an inductor, all three refers to the

same type of component, but the usage is somewhat application specific. In this

project, the component is referred to as reactor. The reactor is an electromag-

netic device, with the primary purpose of introducing inductive reactance into

a circuit. It manages this by storing magnetic energy by presenting a negative

field voltage when the current in the windings increases and then supplies addi-

tional voltage when the current decreases, thus releasing the stored energy. How

quickly the current changes and the current amplitude influences the reactors efficiency. Reactors with higher frequency can be made smaller than reactors with low frequency, while keeping the same rated power.

The reactor has an inductance of one henry (H) when it presents one volt for one ampere for one period of a second. The structure of the reactor depends on what application it is meant for, but the common denominator for all reactors are that they consist of a conductor, such as a wire, usually wound into a coil.

In this project only reactors that has a coil surrounding magnetic material are dealt with.

3.1.1 Application

A typical application in which the reactor that is presented in this report could be used for is in an inductor filter. The filter, which also consists of a capacitor, blocks higher frequencies hence removing the AC component of the rectified output, but allows the DC component of the current to reach the load. The impedance of a reactor is

Z

= j2πf L, (3.1)

where Z

is the impedance and L is the inductance of the reactor respectively.

A typical application for a high frequency reactor is in traction applications.

For example as a battery charger filter inductor in a battery charger module, where the input to the battery should be as smooth as possible, i.e. no ripples in the DC current component.

3.2 Magnetic Material

The magnetic core in any wound component plays an important role in raising the magnetic flux produced by a current that is flowing in the coil winding. The choice of magnetic material however is not elementary. Nowadays high power density magnetic devices are required in various applications, for in example DC-DC converters, in order to downsize the component. The reason why the compactness of the reactor is so important is because the reactor is often the largest device in the electrical circuit, due to its magnetic property. The correct choice of magnetic material can reduce the size of the reactor significantly, while keeping the same power density [12]. There are many different ferromagnetic materials for the designer to choose from, most common in transformers and re- actors are soft magnetic materials. This kind of material have higher magnetic permeability thus conducts magnetic field easier, this will reduce losses dur- ing the magnetisation cycle [13]. The choice depends on application and there are many parameters that influence the core size, for example; operating cur- rent, frequency, air gap topology and current ripple. For the minimum reactor size the optimum material design is with small or distributed air gaps and high saturation flux, as stated in [14] and [15]. It is always desirable to have high sat- uration level in the magnetic material, this is more explained in the next section.

A reactor does not necessarily need a magnetic core. It can be for instance

an air core inductor. However the benefit of having a magnetic core is that

the flux tends to follow the path of higher permeability, magnetic material have

higher value than air, in doing so more flux is contained within the magnetic

circuit. That is preferable due to stray flux can be harmful and induce eddy currents in other material which is in the vicinity of the component. Another benefit is that air gaps can be implemented thus enable designers to vary more parameters to achieve specific electrical requirements, for instance a specific value of inductance.

3.2.1 Saturation of Magnetic Material The definition of inductance, L, of a long solenoid is

L = N Φ

I , (3.2)

where N is number of turns, Φ is the magnetic flux and I is the current. The increase in flux in (3.2) provided by the ferromagnetic material of the core is not a linear factor. A very important variable in magnetic materials is the relative permeability, µ

, which determine how much magnetic flux density, B, is obtained in the core for an externally applied magnetic field, H. The constitutive relation between B and H is

B = µ

µ

H, (3.3)

where µ

is the permeability of free space. µ

is not always constant but change when the flux increases. This is because for example iron, which is a ferro- magnetic material, is composed of many magnetic domains that act like tiny permanent magnets which are all randomly directed, effectively cancelling each other, so there is almost no magnetic field. However when one applies an ex- ternal magnetic field, for example which is produced from a current in a coil winding, the magnetic domains align with the external field, and thus adding to- gether to create a larger magnetic field. The more the external field is increased the more magnetic domains will align, but at some point when the external field is strong enough it can not align any more magnetic domains in the material, simply because there are no left, in other words the material is saturated. This means that at saturation a large increase in current in the windings will neg- ligibly increase the magnetic flux density in the material. So the value of the inductance will decrease with increase magnetic saturation, see (3.2) when cur- rent increases but the flux stays constant. Different ferromagnetic material has different magnetic saturation levels and it puts practical limits on the maximum magnetic field achievable. One does not want saturation due to the fact that closer to saturation the cores operate non-linearly which means that the induc- tance vary with the change in drive current and it can also cause harmonics and distortions.

3.2.2 Non-Oriented Silicon-Steel

In this project the magnetic material of choice is non-oriented silicon-steel. Its of type 10JNHF600 and is a newly developed core material from the Japanese company JFE. This steel is specially made to form certain magnetic properties, for example high permeability, low hysteresis and linear permeability. The core itself is made of stacked laminations each only a fraction of a millimetre thick.

In a non-oriented electrical steel the magnetic properties are the same in any

direction in the plane of one sheet. One reason why silicon is added is due to the low electrical conductivity of silicon thus increasing the resistivity of the steel which leads to a decrease in induced eddy currents. Alloys typically have a silicon content of 3.5 %. The silicon distribution inside a laminate can be seen in Fig. 9. The reason why the proportion of silicon is not higher is because when adding more silicon the material becomes brittle and easy to crack. However 10JNHF600 has a silicon content of up to 6.5 % [16], which makes it a good choice for high-power-density high frequency reactors.

3.3 Introducing Air Gap

The problem of saturation, especially for applications which require a large DC current, can be alleviated by inserting a gap in the magnetic core which can be of air or any other non-magnetic material, both are henceforth referred to as air gap. The air gap in reactors allows a higher DC current before magnetic saturation, thus increasing the effectiveness of the reactor. The comparably low permeability of the air increases the reluctance of the magnetic circuit thus decreasing the flux and therefore limits the excessive flux produced by a high DC current in the winding. Also because the air gap is a non-magnetic material it is immune to saturation and so its reluctance does not change with increase of flux. Its reluctance depends only in its length, l

, and its cross-sectional area, A

, both can be seen as stable parameters. The reluctance of the air gap, R

, and the magnetic core, R

, are approximately

R

= l

µ

A

and R

= l

µ

µ

A

, (3.4)

where µ

is the permeability for vacuum and µ

is the relative permeability of the ferromagnetic material. l

can also be called the effective length of the magnetic core, l

, which is the distance that the magnetic flux travels to complete a closed circuit. In this case the effective core length is represented by the blue lines in Fig. 1. The red lines are represents the shortest path and the green lines the longest.

Figure 1: Illustration of magnetic path lengths in a core.

An approximate effective core length can be calculated as follows l

= 2(b + a) + π( A − a

2 ), (3.5)

with the parameters seen in Fig. 1 and where B − b = A − a. Henceforth the effective magnetic path length, l

, is referred to as core length with the notation l

.

Most often the cross-sectional area for the core and the air gap are the same, A

= A

= A. Combining R

and R

in series yields the total reluctance of the magnetic core

R

=

lcore

µr

+ l

µ

A . (3.6)

As can be seen in (3.6) for a big value for µ

the term

r

can be neglected and the total reluctance depends only on l

and A. The main purpose of increasing the reluctance, by adding an air gap, is to be able to store more magnetic energy, W , according to

W = 1

2 R

Φ

, (3.7)

where Φ is the magnetic flux in the core. Up until now one might think that adding an air gap does not have any downside, one can just increase the re- luctance by adding bigger air gaps. This of course has other effects. When the reluctance of the entire component is increased it means that the effective permeability, µ

, is decreased according to

µ

= µ

1 +

core

µ

. (3.8)

The effective magnetic permeability, also called apparent magnetic permeability, is a term used is when dealing with air gapped cores. With the air gap present, a greater magnetomotive force is required to obtain the same flux density, than it would have if there was no air gap. That is to say the total permeability of the core decreases. The effective permeability is simply a value that takes in to account the decrease in permeability that the air gap produces.

The decrease in effective permeability leads to a decrease in inductance. How- ever inductance also depend on number of turns in the winding, N , according to

L = µ

µ

N

A l

, (3.9)

where l

is the effective magnetic path length. This means that when adding an air gap in the magnetic circuit one also have to add number of turns to keep the same inductance value. But one has to be careful when increasing number of turns because it will lead to that the magneto-motive force (mmf), F

, will increase which leads to higher flux and closer to saturation.

F

= N I. (3.10)

But as can be seen in (3.9) the inductance is increased by the square of number

of turns whereas the magneto-motive force is proportional to N . So it is still

advantageous to increase N . But increasing number of turns will take up extra space so one might have to increase the size of the reactor.

Table 2 shows some advantages and disadvantages when implementing an air gap in a magnetic core similar to the one shown in Fig. 1.

Advantages Disadvantages

Higher values of mmf before saturation More turns to obtain inductance Reduced core losses Increased losses in the windings Inductance constant for larger DC current Increased leakage inductance µ

capable of being changed Increased fringing fields

Table 2: Summary of some advantages and disadvantage for air gapped cores [4].

If operation with very large currents is required then the magnetic field becomes stronger and the air gap might not be able to store all the magnetic energy. A method often implemented is to enlarge the air gap so it can store more energy.

A common design to achieve this is to have a reactor with a winding around just a magnetic rod. The magnetic field must then go through the surrounding air to enclose the magnetic circuit. Thus making the air gap length similar to that of the rod. If even higher current is required, then it can be cost beneficial to design an reactor with no magnetic material, a so called air-core reactor with the entire magnetic circuit is just one big air gap.

3.3.1 Theoretical Optimisation of Air Gap Length

As said before µ

= µ

 µ

so the air gap accounts for almost all the reluc- tance and therefore stores almost all field energy. One can draw analogy with electric circuits, where the resistance dissipates almost all energy. There is of course a limit of how much energy that may be stored, which depends on core type. When designing reactors with air gaps one can find the optimum air gap length, l

, that delivers maximum energy [4]. If l

is too small then there is risk of excess core flux, on the other hand if l

is too large then the space for the winding might run out due to more turns are required for compensate the decrease in inductance. A good approach of how to find the ultimate air gap length is to consider the reactor energy, W , which varies with the air gap length.

Consider two cases with each a limiting parameter. The first case can be called the mmf, F

, limited case. This assumes that the core flux can take any value dependent on the gap, even though the total magneto-magnetic force, F

, is fixed by the area of the winding opening. The reactor energy is

W = 1 2

F

R

. (3.11)

Inserting (3.6) in (3.11) and solving for l

yields l

= µ

AF

2W − l

µ

. (3.12)

As seen in (3.12) the energy is an inverse function of the air gap length. Now consider another case where the flux is limited, here the flux is set to the peak value which the magnetic material can support. The energy variation is now different

W = 1

2 Φ

R

. (3.13)

Inserting (3.6) in (3.13), using Φ = BA and solving for l

yields l

= 2µ

W

B

A − l

µ

. (3.14)

Here the energy stored is linearly related to the air gap length. The term

r

in (3.12) and (3.14) can usually be omitted due to the large value of µ

. To exemplify this optimising procedure one can plot the functions (3.12) and (3.14) with values taken from a typical ferrite magnetic core called RM7, see data sheet [17], where B

= 0.3 T, A = 44 mm

, l

= 30 mm, µ

= 1500 and F

= 43 At. RM is an abbreviation for Rectangular Module. The result is shown in Fig. 2, where W

is the point where the two curves are crossing.

Figure 2: Exemplification of how one can theoretically optimise the air gap length.

The cross hatched curve in Fig. 2 represents the flux limited case and the solid

curve represents the mmf limited case. If the air gap length is chosen to be on

the left side of the crossing point in Fig. 2 then it means that there will be an

excess of core flux. If the length is chosen to be on the right of the crossing

means that there will be an excess of winding area. The graph in Fig. 2 shows

that the best choice of air gap length is 0.16 mm with the energy of 0.28 mJ. The

major disadvantage of this method to determine the optimised air gap length

is that it neglects fringing flux, which for high frequency operation especially

increases losses. This phenomena is explained in the next section.

3.4 Flux Fringing

A quite big disadvantage when having air gapped reactors is fringing flux. Fring- ing in this project refers to a phenomena that occurs when having an air gap in a magnetic core that is subjected to a magnetic flux. Fringing flux is the flux that deviates from the intended path in the magnetic circuit which are straight flux lines through the gap. Instead the flux bulges away from the air gap. For a clearer understanding see Fig. 3 where the fringing flux are the bulging contour lines that are indicated.

Figure 3: Contour plot of magnetic flux density in a core.

Flux fringing is generally an unwanted effect which come into existence due to the huge reluctance of the air gap compared to the magnetic core. The larger the air gap is, the wider the flux fringes.

The reasons why fringing flux is so important to consider when designing mag- netic components with air gaps is because it can increase the total losses of the device. This is because the AC magnetic field induces eddy currents according to Faraday’s law of induction. Eddy currents can be induced in both nearby conductors as well as in the core itself. The induced eddy currents forms closed loops in planes perpendicular to the magnetic field and is proportional to the rate of changing field. Furthermore according to Lenz’s law the eddy currents in turn creates a magnetic field that opposes the magnetic field that produced it. A widely used method of reducing eddy currents in the core is to have the core made of thin sheets of magnetic material, called laminations. By doing this one restrict the eddy currents to circulate in wide arcs, due to the insulation gap between the laminations.

Another method to reduce the losses associated with fringing flux is to dis-

tribute the air gaps in the reactor core to many small gaps, thus making the

fringing from each air gap deviate less [15]. A way to distribute the gaps even

more scarcely is to use special magnetic material which are made of a certain

percentage of non-magnetic material. This makes the effective permeability

much lower but the fringing is reduced because the air gaps are uniformly dis-

tributed within the core. The disadvantage is that these cores are much more expensive due to the manufacturing costs. Another disadvantage is that the magnetic flux density distribution becomes non-uniform in the core [18]. The flux density changes according to magnetic reluctance along the flux path, and therefore reducing the utilisation of the magnetic core.

A third method is to place a sheet of material with low magnetic permeability, for example a copper sheet [19] between the winding and the air gap, effectively acting as a flux barrier. The purpose is to build another path for the magnetic flux so it becomes more uniform in the winding window, thus reducing eddy current losses in the winding [20].

For design engineers it is vital to consider the fringing flux when determine how large the air gap should be in order to obtain the correct inductance value for a given design. The inductance is related to the reluctance and the air gap accounts for almost all the reluctance, which is approximately

R

= l

µ

A

. (3.15)

The reluctance depends on the cross-sectional area of the air gap, A

. When fringing occurs the magnetic flux is shared between the air gap in the core and also in the neighbouring volume outside of the core. One can imagine that the cross-sectional area in the air gap must increase because of the bulging flux lines. As a result the reluctance will decrease due to fringing flux and thus the inductance will increase. To account for this one can increase the air gap length. There are several empirical equations suggested in literature for correcting the calculations regarding this effect. For example McLyman [21]

suggest the following equation

F = 1 + l

√ A

· ln( 2h

l

), (3.16)

where F is called the fringing flux factor by which the inductance is increased because of the fringing. l

is called the electrical air gap and is the air gap calculated to obtain a specific inductance without regarding fringing flux. A

is the cross-sectional area of the core and h

is the height of the coil winding.

The fringing phenomena means that when designing air gapped reactors one has to increase the air gap by a factor F in order to obtain the same inductance value.

l

= F · l

, (3.17)

where l

is the real mechanical air gap which is implemented in the reactor.

3.5 Winding Losses

The winding losses in power reactors consists of two types of losses, namely

DC loss and AC loss. The DC loss arise because of the DC resistance of a

winding conductor. In reactors where a large DC bias current is applied the DC

resistance becomes more important. The DC current is evenly distributed in

the conductor, so this resistance can be reduced by increasing the cross sectional

area, A

, of the conductor according to R

= ρ

l

A

, (3.18)

where ρ

is the resistivity of the material, often copper, and l

is the total length of the current carrying conductor. For operation at low frequency the DC re- sistance makes up almost all the winding losses in the conductor [22]. However when increasing the operating frequency the total winding losses increases dras- tically. It increases because of two phenomena, skin effect and proximity effect, which are explained in the two following subsections.

3.5.1 Skin Effect

An unwanted effect when increasing the frequency of the AC current is that the skin effect within the conductors of the winding increases and thus contribute to the overall additional loss of the magnetic component. Skin effect occurs when an AC current becomes distributed within a conductor such that the current density decreases with larger depths in the conductor. The largest density is near the surface. It causes the effective AC resistance to increase with frequency, because when frequency increases the skin depth decreases, thus reducing the effective cross sectional area where the current is flowing.

A cause of skin effect is the alternating magnetic field, produced by the AC current flowing through a conductor, which creates an electric field which op- poses the change in current intensity, this is called counter electromotive force (back EMF) and is strongest at the center of the conductor, thus forcing the current to flow near the surface of the conductor wall [23]. The back EMF becomes stronger when frequency is increased, thus forcing the current further away from the center, resulting in an increase of AC resistance and thus ulti- mately increases the winding losses.

For high frequency wound components wires with isolated strands are imple- mented in order to decrease the losses due to skin effect [24]. A common type of wire is the Litz wire, see Fig. 4, and is used for mitigating the skin effect for higher frequencies. The Litz wire in this project is made of 16 bundles wo- ven spirally, each bundle consists of 50 stranded individually insulated copper wires, each with the diameter of 0.16 mm. This fashion makes the total current distribute evenly, so that the the magnetic field acts equally on all the wires.

The result is that the bundle of strands does not suffer the same AC resistance

due to that the skin effect has little effect on each of the thin strands.

Figure 4: One type of Litz wire for use in high frequency reactor winding.

3.5.2 Proximity Effect

Another effect that complicates the design of high frequency high efficient reac- tors is the proximity effect. When an AC current is flowing in a conductor an alternating magnetic field is created around it. When the conductor is part of a wound coil then the alternating magnetic field induces eddy currents in nearby conductors. These induced eddy currents alter the total current distribution in conductors nearby, some areas of the conductor will have a greater density of current which in turns leads to an increase in AC resistance. If the magnetic field alternates faster, i.e. the exciting frequency is raised, then the proximity effect becomes greater which leads to higher temperatures due to AC resistance.

This increased AC resistance adds to the total power losses of the reactor and can generate undesirable heating.

As well as for mitigating skin effect, wires made of isolated strands, like the Litz wire, can be used to alleviate the proximity effect [24]. With the principle that when thinner but more strands of wire are used the current density distri- bution is less affected by the magnetic field. Another method of reducing the proximity effect is to wind the coil in layers.

3.6 Reactor Core Losses

It is difficult to accurately predict the reaction of a magnetic core when exciting it at high frequency due to the non-linearity of the magnetisation curve. The performance of the magnetic component is strongly linked to which core material is used, because different material have different magnetisation curves and core losses [25]. Accurate core loss calculations becomes especially difficult when operating in DC bias conditions, because the magnetic properties like core loss density and permeability will vary when DC pre-magnetisation is present [26].

The core losses include eddy current losses and hysteresis losses. Core losses

obtained from data sheets of core materials is often a good approximation,

however measurements are performed without consideration of leakage magnetic

flux [5]. This makes it difficult for reactor designers to perform accurate loss

calculations when introducing air gaps. Therefore it is of interest to evaluate

core losses and localised overheating in air gapped reactors under high frequency excitation.

3.6.1 Eddy Current Losses and Cauer Modelling

Time varying magnetic fields give rise to induced voltage even where it is not desirable. Because of this phenomena the induced voltage inside the conductive magnetic material give rise to unwanted induced currents, called eddy currents.

This leads to an increase in losses and a decrease in the core’s overall perfor- mance, especially for higher frequencies. It is of great importance to understand and predict this phenomena in order to optimise the performance of the induc- tive component. As mentioned previously one can mitigate the eddy current losses in the core by having it built up by thin sheets (laminations). Between the laminations insulation is applied to prevent eddy current to form loops be- tween the laminations.

It is computationally unreasonable to model all laminations in a core in a FEM model, because there are too many laminations. Instead one can use a model called Cauer. If one make the assumption that the flux is flowing parallel along the laminates one can consider only one laminate with the thickness of one lam- ination but with the cross sectional area of the core. This wide thin laminate is then divided into a number of subregions seen in Fig. 5.

Figure 5: A laminate with eddy currents flowing in subregions [1].

Further assumptions are made that the current, I

= N I, circumscribes the core where I

is lumped together in each turn to form a current loop. Each subregion has a different thickness, denoted s

, where the subscript i represents the number of the subregion. The aim now is to model each subregion as a resistance-inductance element. The inductance represents the reluctance of the core and the resistance represents the core losses in the magnetic domain. The circulating currents seen in Fig. 5 can be seen as adding to the total current I

and is flowing through inductances to create the flux. For example in the third subregion starting from outside going inward, the flux would be

Φ

= L

(I

+ I

+ I

). (3.19)

Here an assumption is made that the current in one region does not affect the flux in the same region. The inductance in (3.19) are calculated by Ampere’s law where the integration path is in this case the entire length of the core, l

, and N I is the total current.

H

l

= I

+ I

+ I

+ ... + I

. (3.20) If one makes another assumption that there is a linear relation between applied magnetic field and magnetic flux density one obtains the flux in each subregion as

Φ

= µ

µ

A

H

, (3.21) where µ

is the permeability of the core. If equations (3.19), (3.20) and (3.21) are combined as well as adding multiple turns, N , and solving for L, following equation is derived

L

= N

µ

µ

A

l

, (3.22)

where A

is the area of subregion i. This area is calculated as A

= As

2

s , (3.23)

where A is cross sectional area of the core, s

is the thickness of a subregion and s is the lamination thickness. The factor two is added because we only consider half of the area, i.e. subregions from the outer to the middle of the laminate.

The voltage induced give rise to eddy currents in subregions with resistance R

. It is calculated same as the wire resistance R = ρlA

, where ρ is the re- sistivity of the material, l is the length of the conducting material and A is the cross sectional area of the material. In the Cauer modelling case this translates to

R

= ρ

2

l

s

, (3.24)

where l

is the effective magnetic path length of the core. The electrical circuit analogy is then ready to be drawn, see Fig. 6.

Figure 6: Circuit for determining eddy currents.

What is left now is to determine number and thickness, s

, of the subregions.

Following an example from [1] 11 subregions are chosen. The subregion thickness

is different in order to compensate for the eddy currents that are larger in the outer regions. The thickness is calculated as

s

= ak

, (3.25)

where k is a constant between 1-4 and a is a factor that must satisfy following equation

s 2 = X

i

ak

= X

i

s

. (3.26)

After the different thickness’s, s

, are calculated for the subregions the induc- tance and resistance of each layer can be obtained using the equations presented in this section. Thereafter the circuit shown in Fig. 6 can be implemented in a circuit simulation software in order to obtain the currents in every sub layer, hence the eddy current losses can be obtained.

Not only the core is subjected to eddy currents, other conductive materials in the vicinity of the magnetic field will also experience induced eddy currents.

This could for example be winding conductors and mounting straps [27] . De- pending on frequency and conductivity of the material the eddy current induced will generate resistive heating that could be too much for the material to handle, in the worst case the material will melt.

3.6.2 Hysteresis Losses

Hysteresis loss stem from the energy it takes to align the tiny magnetic dipoles, explained in section 3.2.1. When having alternating current in the windings the magnetic field will change polarity at the same rate as the exciting frequency.

Every time the magnetic field changes polarity the magnetic material is mag- netised in that same polarity. Once the dipoles are reoriented, it takes some energy to redirect them to the opposite direction, this is the hysteresis loss.

When an alternating magnetic field is applied to the material, the magnetisa- tion will trace out a loop, called a hysteresis loop which is shown in Fig. 7.

The area of the loop in the first quadrant in Fig. 7 corresponds to the energy dissipated from the core.

Figure 7: Example of B-H loops showing energy into and out of a core [2].

The magnetic flux density first follows the initial magnetisation curve and in- creases until it reaches an asymptote, the saturation, then when the applied H -field decreases (because of the sinusoidal current in the windings) B takes a different curve. The reason why there can be a B-field at zero field strength is because of the memory of the magnetic material which will retain some mag- netisation. This phenomenon is called remanence. The width of the loop is called magnetic coercivity and is a measure of how well the magnetic material can withstand external magnetic field without becoming demagnetised. The hysteresis loop is traversed during each cycle of an alternating H -field, very similar to the curve in Fig. 7.

3.6.3 Complex Permeability

A difficulty when dealing with a high frequency applied magnetic field to mag- netic material is that the permeability of the material is most often non-linear, this can also be seen in Fig. 7. It means that it is much more difficult to know the magnetic flux density from the applied magnetic field. This in turns means it is difficult to investigate core losses because the losses are related to the magnetic flux density in the core. A method for dealing with this is by in- troducing complex permeability, where the magnetic permeability of the core is represented by a complex permeability value. The real part of the complex value relates to the frequency dependent energy stored in the core and the imaginary part is related to the losses [28]. For high frequencies the magnetic flux density, B, and the magnetic field strength, H, will react to each other with some lag time [29]. These quantities can therefore be written as phasors

H = H

e

and B = B

e

, (3.27) where θ is the delay of B from H. Permeability is defined as the ratio of the magnetic flux density to the magnetic field strength, this yields

µ

µ

= B

H = B

e

H

e

= B

H

e

. (3.28)

Translation from polar form to rectangular form using Euler’s formula the fol- lowing is obtained

µ

µ

= B

H

cosθ − j B

H

sinθ = µ

(µ

− jµ

), (3.29) where the real part is a measure of how inductive the component is (normal permeability) and the imaginary part is the loss component. The ratio of the real and imaginary part is called loss tangent

tanθ = µ

µ

, (3.30)

which gives a measure on how much power is lost versus how much is stored.

The delay between B and H is also known as the hysteresis angle which makes the hysteresis loop shift by an angle θ relative to the H-axis.

From the core material’s data-sheets one can obtain the complex permeabil-

ity. By knowing the specific lamination thickness, frequency and magnetisation

the total core losses can very simplified be described by complex permeability.

One can start deriving the complex permeability by expressing the added coil voltage in terms of magnetic field density

U = N A

jωB, (3.31)

where B = µ

µ

H and if complex permeability is considered one gets

B = µ

(µ

− jµ

)H. (3.32) Inserting (3.32) in (3.31) and also replace H using H =

core

the following can be written

U = h

jω µ

µ

AN

l

+ ω µ

µ

AN

l

i

I, (3.33)

where the first term corresponds to the impedance, jωL, and the second term can be considered as a frequency dependent resistance R(ω). Now if one multi- plies the second term with the current I the total resistive loss is obtained

P

= ω µ

µ

AN

l

I

= N

I

l

Al

ωµ

µ

= H

V ωµ

µ

. (3.34) Dividing the left side with V to get specific loss per m

, p, and solving for µ

one obtains

µ

= p

H

µ

2πf . (3.35)

As predicted (3.35) shows that the loss component of the complex permeability is frequency dependent. The H -field is obtained by looking at the DC magneti- sation curve in the manufacturer’s data-sheet. The magnetic flux density should here be the AC part of the magnetisation. This magnetic flux density can be calculated by knowing the DC current, DC magnetisation and peak-to-peak AC current, I

.

B

= B

I

I

. (3.36)

From the B and H-values the real part of the complex permeability is also ob- tained. The power loss per kilogram is obtained from the high frequency core loss curve. Here the magnetic flux density that should be used is the peak value of B

from (3.36). Now the imaginary part of the complex permeability can be obtained for different frequencies. All magnetic property curves can be seen in data-sheet [3].

The power density used in (3.35) which is taken from manufacturer’s data- sheet is obtained when measuring total core losses. That means that the power density includes both the eddy current losses and the hysteresis losses. So when using this method of complex permeability to describe core losses, one can not include Cauer model in the calculations, which only analyse eddy current in the laminations.

3.7 Heat Transfer

Losses arising due to high frequency magnetic fields in reactors can often lead

to an unacceptable temperature rise [30]. It is an important limiting factor one

has to consider when maximising the throughput power of a reactor.

Since fringing flux induces eddy current on the surface of nearby conductors, the total ohmic losses of the reactor winding increases due to temperature rise in the conductors [19]. That is why it is very important to reduce the extent of the region where fringing flux occurs. The temperature rise in the reactor is closely linked to the power loss, so when decreasing the power loss the temperature will also decrease.

4 Reactor Prototype

This reactor is designed for research purposes and is therefore not subject to specific electrical requirements from second part. The purpose of this prototype is to be able to verify the computer model through experiments. This section explains how the prototype is constructed and how the measurements are set up.

4.1 Magnetic Core

Figure 8: 2D schematic with front view (a) and side view (b) of the core.

Dimensions are in mm.

The core is made of stacked non-oriented electrical-steel laminations, each lam- ination is 0.1 mm thick. All laminations are stacked in the direction indicated by the red arrow seen in Fig. 8(b). Each lamination has a silicon content of 6.5

% near the outer surface of the sheet, see Fig. 9, in order to reduce the current

conductivity.

Figure 9: Distribution of the silicon in a laminate.

The main reason why grading the silicon in the manner shown in Fig. 9 is be- cause the induced eddy currents are larger at the surface plane of the laminate.

Therefore increasing the resistivity at that location will reduce the induced eddy currents.

The core is supplied by the company JFE Steel Corporation and is specialised for high frequency operation. For more detailed information about the material see Table 3, data-sheet [16] and magnetic property curves [3]. The reactor core is built up by two yokes and ten blocks. Two legs of five blocks each are placed as seen in Fig. 8. The reason for having many blocks is to be able to distribute the air gap in order to avoid large fringing of the magnetic flux. For the purpose of this study a larger fringing flux is wanted in order to study it. Because of this the blocks are glued together except for where the air gap is located.

Notation Value Unit Description

s 0.1 mm Core lamination thickness

W

3600 mm

Core window area

A

2394 mm

Cross sectional area of core ρ

0.57 µΩm Core lamination resistivity B

1.88 T Saturation magnetisation level D 7530 kg/m

Core density

V 1058 cm

Core volume

m

7.97 kg Total core mass

l

0.42 m Effective magnetic path Table 3: Reactor core parameters.

Even though the data supplied from magnetic core suppliers are estimations and

tests most often performed with a standard so-called Epstein frame, the numeri-

cal values can give designers an insight on different aspects such as permeability

and core losses. Fig. 10 and Fig. 11 shows two valuable graphs showing the

DC magnetisation curve and high frequency core loss curve respectively. Both

of these shows peak values and are needed when calculating the correct com-

plex permeability, described in section 3.6.3, for using in the computer model

to describe the core losses.

Figure 10: DC magnetisation curve and DC permeability curve, supplied by manufacturer [3]. Magnetic flux density is shown as peak value.

Figure 11: Core losses for different frequency and flux density, supplied by

manufacturer [3]. Magnetic flux density is shown as peak value.

4.2 Winding Design

The winding material is chosen to be Litz wire because its superior benefits when operating at high frequency. Each Litz wire is electrically insulated with Nomex paper. Winding parameters are shown in Table 4.

Notation Value Unit Description

d

0.16 mm Single wire diameter A

0.02 mm

Single wire area

n

800 - Number of wires in one Litz

A

16.1 mm

Litz copper area (no Nomex) w

3.3 mm Conductor width including Nomex h

9.7 mm Conductor height including Nomex α

0.0772

C

Temperature coefficient at 20

C ρ

167.8 µΩm Copper resistivity at 20

C

Table 4: Winding parameters.

To start off with the design of the winding topology one can determine the max current density, J

, that is allowed to flow in the conductor. Because this is a DC reactor so a large DC current will flow in the winding and it will lead to resistive heating which cannot be too high. For a non forced-air cooled reactor the rule-of-thumb for copper is

J

< 3 · 10

A/m

. (4.1) If one uses this method then the max current allowed is

I

≤ J

A

= 48.3 A. (4.2) Now one has to determine if this current is suitable for the reactor in terms of number of winding turns. One can look at the following relation between current, air gap length, magnetic flux density and number of turns

N = B

l

µ

I

. (4.3)

B

in this equation can be determined from B

that is a supplied numerical value from the core supplier. One wants to operate the reactor well below the saturation level in order to avoid unnecessary non-linearity. In this case B

is chosen to be 0.5B

, which is the total flux density from both the DC and the AC current. Furthermore from (4.3) it is evident that for a fixed value of l

and a lower current more turns are needed in order to obtain B

. To determine how much of the available core window area that can be allocated with copper winding one can introduce a window utilisation factor, denoted K

, that is a factor between 0-1. The constraint here is

K

W

≥ N A

, (4.4)

where W

is the actual size of the window area, N is number of conductor turns and A

is the wire copper area. Typical factors that influence K

is that round wire does not pack perfectly, winding technique, insulation, air channel and bobbin. To be certain the number of turns will fit in the core window, the utilisation factor is chosen to be K

= 0.15, which is a reasonable figure for similar components.

Figure 12: Flow chart for determine number of winding turns.

With the method depicted in Fig. 12, with an approximated air gap length of

l

= 2 mm, the number of turns is calculated to N = 31 turns. When testing the

prototype the air gap length is varied, and for larger air gaps the current must

be higher. To further ensure that the current will not cause any unnecessary

heating in the winding, two Litz wires are connected parallel together with same

input, effectively doubling the cross sectional area where the current will flow,

see Fig. 13, where the Nomex insulation is shown in grey. This is done to

decrease the current density in the winding hence decrease the temperature rise

caused by resistance. Now the max current is double of that when using only

one wire.

Figure 13: Parallel connected Litz wires.

When connecting two Litz wired in parallel as seen in Fig. 13 the space between the yokes are quickly filled up. In this design this is solved by adding another winding layer. Fig. 14 shows the core and the winding in a cross section front view and Fig. 15 shows a top view of the core and the double layered winding.

The number of turns for the new conductor set-up is now half of the previous

calculated number. To finally be able to fit the number of turns in the core win-

dow and between the yokes the number of turns are determined to be 14. How

the conductor is wound is clearly seen in Fig. 14, where 3.5 turns are wound in

the first layer followed by another 3.5 turns added to complete the second layer,

these layers together amount to 7 turns. Two identical coils are constructed

and then mounted on the core legs. The coils are then connected to form a

series connection. For being certain that winding will fit when constructing the

prototype some over dimensioning has been done, see Table 5 for the winding

parameters.

Figure 14: Cross section front view of the core and winding.

Figure 15: Top view showing core and double layered winding.

Notation Value Unit Description

N 14 - Number of conductor turns

I

96.6 A Maximal rated current allowed in conductor h

19.4 mm Height of conductor (including Nomex) A

32.2 mm

Cross sectional area of conductor l

3.94 m Total conductor length

M LT 281.6 mm Mean-turn-per-length m

1.19 kg Total conductor mass

Table 5: Winding design parameters.

4.3 Other Constructional Parts

The core legs are supported by bars made of reinforced plastic with holes for

screws to make sure the structure is stable. The holes for the upper part of

the bar that goes along the leg are shaped rectangularly for being able to vary the air gap length. For changing the air gap length, the upper yoke is simply lifted up, then different thickness of rectangular shaped non-magnetic air gap pieces can be inserted inside the core, see Fig. 18 where the top yoke has been dismantled. The upper and lower support bars are made of steel. All other constructional parts that are located close to the air gap are made of non- magnetic material. The winding is wound on a bobbin which is also made from reinforced plastic. The completed prototype is drawn in the computer aided design tool SolidWorks, see Fig. 16. From this 3D model the 2D schematics for all parts can be extracted for the manufacturing process.

Figure 16: 3D drawing of the reactor made in SolidWorks.

4.4 Finished Prototype

Fig. 17 shows the finalised prototype ready for testing. The output conductor of one coil is connected to the input conductor of the other coil, thus connection the coils in series. This connection makes sure the current is flowing in the correct direction to make the magnetic flux flow around the core. The brown wires in Fig. 17 are thermocouples for measuring the temperature. They are located between the layers close to the air gap on both sides of the coil.

Figure 17: Completely assembled prototype and air gap pieces with different thickness.

The thickness of the air gap pieces shown in the lower part of Fig. 17 ranges

from left to right in steps of 0.5 mm from 0.5 to 2.5 mm. Depending on desired

air gap length, pieces of different lengths are stacked together to form the total

air gap. Before inserting the air gap pieces in the core, the total length is

measured with a high precision calliper, in order to be certain that the air gap

pieces adds up to the desired length. The constructional design of the reactor

makes it easy to change the air gap length. To insert the air gap pieces in the

core, the top part of the core is simply unmounted as seen in Fig. 18, the air

gap pieces are then carefully placed inside the core. The top part of the core is

then put back, and the prototype is then ready for testing.

Figure 18: Partly dissembled prototype.

4.5 Measurement Set-up

Two different tests are to be executed on this prototype. The first is with a low frequency ripple current superimposed on a varying DC bias current. The second is with a varying high frequency current without DC bias current. For both tests the air gap length is varied.

4.5.1 DC Bias with AC Ripple Test

The circuit schematic seen in Fig. 19 is commonly used for incremental induc-

tance test, where the DC current is increased to verify that the inductance is

stable. The AC voltage is tuned so that desired ripple current is flowing in the

AC side of the test circuit. After desired AC current is achieved the switch

is closed in order to apply the DC component. The DC current is thereafter

increased in steps to verify that the inductance is stable for a specified current

range.