TVE-E; 19003

Examensarbete 15 hpJuni 2019

High Frequency Transformer

Implementation of prototype

David BergmanVincent JanssonNiklas Hermansson

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

High Frequency Transformer

David Bergman, Vincent Jansson, Niklas Hermansson

Since its invention in 1885 by Otto Bláthy, Miksa Déri and Károly Zipernowsky, transformers have become an important cornerstone of the electrical infrastructure we have today. They are found most prominently in any machinery or device that requires a different level of voltage or current than a general grid can supply, such as computers, motors or even cars. In the case of this project, the transformer was originally intended to be connected to a resonating H-bridge which supplies the primary coil with high frequency voltage pulses to be converted into a higher voltage transferred to a rectifier unit. Because of the level of frequency supplied, the transformer was required to be constructed with a different type of core and cable for the winding. When it became clear that the cable couldn't be supplied in time, the focus shifted towards constructing a prototype instead. The prototype was designed to generate a certain amount of leakage inductance while subjected to a short circuit test. After a couple of attempts, the group managed to construct a transformer whose leakage inductance was well within range of the specifications. The finished transformer prototype was delivered and the group had thus successfully constructed what is to be used as a template for further transformers of the same type.

A special thanks to ScandiNova Systems AB for initiating this project and giving us the opportunity to participate, and to Per Nilsson, Per Benkowski and Klas Elmqvist for mentoring us along the way.

ISSN: 1654-7616, TVE-E; 19003 Examinator: Mikael BergkvistÄmnesgranskare: Mats EkbergHandledare: Per Nilsson

Bachelor Thesis in Electrical Engineering

High Frequency Transformer

Authors: David Bergman, Niklas Hermansson, Vincent Jansson

UppsalaJune 9, 2019

Abstract

Since its invention in 1885 by Otto Bláthy, Miksa Déri and Károly Zipernowsky, transformers havebecome an important cornerstone of the electrical infrastructure we have today. They are found mostprominently in any machinery or device that requires a different level of voltage or current than a generalgrid can supply, such as computers, motors or even cars. In the case of this project, the transformer wasoriginally intended to be connected to a resonating H-bridge which supplies the primary coil with highfrequency voltage pulses to be converted into a higher voltage transferred to a rectifier unit. Becauseof the level of frequency supplied, the transformer was required to be constructed with a different typeof core and cable for the winding. When it became clear that the cable couldn’t be supplied in time,the focus shifted towards constructing a prototype instead. The prototype was designed to generate acertain amount of leakage inductance while subjected to a short circuit test. After a couple of attempts,the group managed to construct a transformer whose leakage inductance was well within range of thespecifications. The finished transformer prototype was delivered and the group had thus successfullyconstructed what is to be used as a template for further transformers of the same type.

A special thanks to ScandiNova Systems AB for initiating this project and giving us the opportunity toparticipate, and to Per Nilsson, Per Benkowski and Klas Elmqvist for mentoring us along the way.

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Table of Contents1 Introduction 3

1.1 General Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Illustration of 25 kW Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 52.1 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Non-ideal Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Induction in a Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Approximate Model for Leakage Inductance by Volume . . . . . . . . . . . . . . . . . . . . . 72.6 Skin Effect and Litz Wiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Circuit Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Implementation 93.1 Alternatives for Core and Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Calculation of Air-gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Construction of Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Prototype 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Prototype 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Result 134.1 Prototype 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Prototype 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Final Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Discussion 16

6 Conclusion 17

7 References 18

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1 Introduction

1.1 General StatementsThe initial purpose of the project was to design and construct a transformer capable of, among otherspecifications discussed later on, isolating high frequency voltage pulses sent through the primary windingcoil of up to 125 kHz. Before the construction however, it was clear that the company could not receive thenecessary cable required for winding isolation within the time frame of this project. Because of this, anothertype of cable was used for the winding of the primary and secondary coils and the purpose was insteadshifted towards getting a leakage inductance within range of their specifications between 13-17 µH.

1.2 BackgroundOne of the main components in the high voltage modulators the company has developed is a resonantH-bridge known as a Capacative Charging Power Supply, abbreviated CCPS. A part of the CCPS unit isa transformer capable of transforming high frequency pulses of 800V to 1400V DC. The company recentlycreated a 25 kW capacity transformer and is now in need of a smaller 10kW variant with a greater leakageinductance than the previous one.

1.3 SpecificationsThe core used in both the 25 kW and the 10 kW versions is a Manganese-zinc ferrite core designed towithstand high frequency changes of magnetic flux. The 25 kW transformer is assembled as two U shapedcores compressed together where each U core has the dimensions 93 x 76 x 30 [mm]. The 10kW version isassembled as a U shaped core compressed together with an I shaped core with the dimensions 93 x 28 x 30[mm]. Additional specifications for the 25 kW transformer are listed below in table 1.

Table 1: Specifications of the 25 kW transformer

Primary turns 20Secondary turns 40Voltage 800 [V]Period Time 4 [µs]Core cross area 8.4 [cm2]Flux swing 0.3 [T]Leakage inductance 6.6 [µH]Outer diameter of primary cable 6.2 [mm]Outer diameter of secondary cable 3.3 [mm]Primary RMS current 70 [A]Secondary RMS current 35 [A]Air gap 47 [cm3]

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The general design between the 25kW and 10kW transformers, such as the ferrite type core, winding currentdensity, turnover number, flux swing, input voltage and frequency remain the same. The overall differenceis the reduction of size of the transformer core, winding current and total diameter of the winding cable aswell as the increase of the air gap and subsequently the leakage inductance of the 10kW variant.

1.4 Illustration of 25 kW TransformerFigure 1 and figure 2 shows an illustration of the 25 kW transformer. Figure 1 displays that the 25 kWtransformer has a core consisting of two U shaped cores merged together. Both the coil of the primary andthe secondary side are winded on both sides of the core with half of the turns on each side. The secondarycoil is wound closest to the core. The primary side is separated from the secondary side with a hollowrectangular plastic bobbin creating an air-gap between the primary coil and the core.

Figure 1: Illustration of the 25 kW transformer

Figure 2: Cross section illustration of the 25 kW transformer

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2 Theory

2.1 TransformerA transformer is an electrical device that converts and transfers electrical energy. In its simplest way, atransformer consists of two windings where each creates a conducting coil. The coils are usually wound arounda ferromagnetic core and is therefore known as an iron-core transformer. The coil delivering electric poweris called the primary winding, and the one receiving electric power the secondary winding. A time-varyingflux is generated in the magnetic core by applying a time-varying alternating current (AC) through theprimary winding. Most of the flux flows through the core and the secondary winding induces a time-varyingelectromotive force or EMF (voltage). The induced secondary winding obtains the same frequency as theprimary. A current will flow through the second winding if it is connected to a load, and the power is thentransferred from the primary side, to the secondary side.

The induced EMF in a coil is proportional to the number of turns in a coil, therefore it is possible to have ahigher voltage across the secondary winding than the applied voltage. This is called a step-up transformer.

Faraday’s law of induction states that the magnetic flux in the core induces an EMF in the primary windingthat opposes the applied voltage. Similarly, the secondary winding induces an EMF from the magnetic fluxin the core opposite to that of the primary winding in accordance with Lenz’s law. The equations for theinduced EMF, for the primary winding ε1 and the secondary winding ε2, can be seen below in equations 1and 2, where N1 and N2 are the number of turns around the core for the primary and secondary winding,and Φ is the magnetic flux. An illustration of a transformer can be seen in figure1 3 below. [1]

ε1 = N1dΦ

dt(1)

ε2 = −N2dΦ

dt(2)

Figure 3: Illustration of a simple transformer with a load connected to the secondary winding

For an ideal transformer, the induced EMF on the primary and secondary side are equal to the correspondingterminal voltage. Equations 1 and 2 states that the emf is proportional to the number of turns in the coil.Therefore the ratio of primary to secondary induced EMF is equal to the ratio of primary to secondary turns.This equation is displayed in equation 3 below. The ratio of primary to secondary turns is often called thea-ratio or turnover number and is displayed below in the same equation. [1]

a =v1v2

=ε1ε2

=N1

N2(3)

1Figure taken from reference [1]

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2.2 The Non-ideal TransformerIn reality a completely ideal transformer can never be built as there will always be losses between the powerinput and the power output. The various losses in a transformer can be used to develop an equivalent circuitfor the non-ideal transformer. Each winding in the transformer has some resistance, usually small butincluded regardless. By inserting a lumped resistance R1 and R2, in series with the transformers primaryand the secondary winding, a more realistic transformer circuit is created. The inclusion of the windingresistance dictates that the power input must be greater than the power output, therefore the efficiency ofthe transformer is also less than 100%. The resistance also contributes to that the terminal voltage is notexactly equal to the induced EMF.

Another loss in the transformer is the leakage flux. Not all the flux created by the winding confines itselfto the magnetic core. Some of the flux completes its path through the surrounding medium. The leakageflux created by each winding are not linked, so the primary leakage flux restricts itself to the primary sideas well as for the secondary side. The leakage flux is modeled with one winding on each side, in series withthe transformer jX1 and jX2. Although the leakage flux is small in comparison with the total flux, it doeshave an impact of the overall performance of a transformer.

To complete the non-ideal transformer equivalent circuit, the transformers finite permeability and corelosses need to be taken into account. Therefore, the primary winding will draw some current when thesecondary side is open, known as the excitation current. The excitation current is often assumed as the sumof the core-loss current and the magnetizing current. The magnetizing component of the excitation currentis responsible for setting up the mutual flux in the core. The loss is modeled by a magnetizing reactancejXm1 in parallel with the transformer allowing current to flow through it even if the secondary side is open.

The core-loss component Rc1 of the excitation current accounts for the magnetic loss in the transformer, thisincludes the hysteresis and eddy-current losses. Since the excitation current is the sum of the core-loss- andthe magnetizing current, the core-loss is modeled in parallel with the magnetizing reactance. The completeequivalent circuit of the non-ideal transformer is displayed below in figure2 4. [1]

Figure 4: Illustration of a complete transformer with losses

2Figure taken from reference [1]

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2.3 Induction in a CoilFaraday’s law of induction states that the induced electromotive force (EMF) in a closed loop is the numberof turns in the loop multiplied with the flux change the loop experiences. The equation is displayed belowin equation 4:

ε = −N dΦ

dt(4)

The magnetic flux is defined as the surface integral of the magnetic flux density which is displayed in equation5 below:

Φ =

∫s

−→B ·−→ds (5)

If the flux is homogeneous across the entire surface and perpendicular to the surface, then equation 5 canbe simplified. The simplified equation is displayed below in equation 6:

Φ = BA (6)

By merging equations 4 and 6, an equation for the flux swing can be derived. The equation is displayedbelow in equation 7:

ε = −NAdBdt

=> ∆B =ε · TA ·N

(7)

Here, ∆B is the flux swing, A is the cross-area of the field, ε is the EMF, T is the period time for the fluxswing and N is the number of turns. [1]

2.4 Current DensityThe current density ρ is obtained by dividing the applied current I with the cross area A of the material thecurrent is flowing through. The equation of the current density is displayed below in equation 8: [2]

ρ =I

A(8)

2.5 Approximate Model for Leakage Inductance by VolumeThe AL value is called the inductance factor and is a value for the inductance per turn in a magnetic circuit,where L is the total inductance in the windings and N is the number of turns that in the windings. Theequation for the AL value is displayed below in equation 9: [3]

AL =L

N2(9)

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2.6 Skin Effect and Litz WiringSkin effect is the result of an inhomogeneous alternating current density being mostly distributed near thesurface of a conductor based on a variety of factors. These factors are included in equation 10 below where δis the skin effects conductor depth (in meters), % is the resistivity of the conductor, f is the frequency of theconductors current, µr is the relative magnetic permeability of the conductor, µ0 is known as the vacuumpermeability constant (4π·10−7 H/m), εr is the relative permittivity of the conductor material and finallyε0 is known as the vacuum permittivity constant (8.85·10−12 F/m): [4]

δ =

√2%

2πfµ0µr·√√

1 + (2πf%ε0εr)2 + 2πf%ε0εr (10)

In order to mitigate the resistive and capacitive losses that occur due to skin effect, copper litz wire wasoriginally intended to be used as a conductor for the winding. Litz wire consists of multiple strands of copperwire woven together in a bundle and individually insulated from each other. The skin effect is reduced wheneach individual conductor strand diameter is less than the skin effect conductor depth (as displayed inequation 10 above) thus causing the individual strands to not experience a significant skin effect loss.

2.7 Circuit ResonanceElectrical circuit resonance is a phenomenon that occurs when the resulting impedance between the inductanceand capacitance in an RLC AC circuit cancel each other out at a certain resonance frequency, forming asimple harmonic oscillation. The main purpose of utilizing resonance is mainly for tuning and filtratingbecause of the occurrence at a given frequency for a particular inductance and capacitance of inductors andcapacitors respectively, depending on weather they are coupled in series or parallel with each other. Theequation for resonance at a specific frequency is displayed in equation 11 below where L is the circuits totalinductance and C is the circuits total capacitance: [5]

fh =1

2π√LC

(11)

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3 ImplementationThe assigned project task is to develop a model for a 10 KW transformer similar to a 25 kW transformertemplate already in use. To complete this task, specifications for the 25 kW transformer were required alongwith parameters for the 10 kW transformer that were supposed to be adjusted compared to the the 25 kWtransformer.

3.1 Alternatives for Core and CablesIn order to implement a model for the 10 kW transformer, the first step was to find suitable alternatives forthe core of the transformer and the cable that made up the windings of the transformer. The ferrite coreand cable were chosen to satisfy the requirements. The model would also ideally be smaller than the 25 KWversion to be efficient in terms of volume. The cable was chosen so that the current density in the windingwould be the same for the 10 kW version as for the 25 kW version. Since the current density in the windingshould remain equal and the current is known for both transformers the cross area can be calculated for the10 kW transformer.

I10kWA10kW

=I25kWA25kW

(12)

A10kW =I10kW ·A25kW

I25kW(13)

A210kW =

35 · 30, 2

70= 15.10mm2 (14)

With the cross area of the of the cable determined, the diameter of the circular cables could in turn becalculated accordingly.

d =

√A · 4π

(15)

d =

√15.1 · 4π

= 4.39mm (16)

When the diameter of the cables were known, alternatives for what core should be used was researched.When the core was to be chosen it had to be chosen so that the cables were able to fit on the transformer,the flux density (Tesla) swing had to remain the same as for the 25 kW transformer.

Five different core shapes were used to create templates, but only one of them was selected to furtherbe created into prototypes with two different winding arrangements tested for that core. An illustration ofthe core that was used is displayed in figure 5. This core is similar to the core for the 25 kW transformeras it has the same cross area for the magnetic flux. The 25 kW transformer consists of two U shaped coresmerged together. So the idea to choose this type was to replace one of the U cores with an I shaped core thusreducing the overall size, which is possible if the winding cable do not occupy to much space. An illustrationof the core is displayed in figure 5 below.

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Figure 5: Illustration of the 10 kW ferrite core and the associated dimensions

In figure 5 the core used for the transformer is seen. It consists of one U-core and an I-core merged together,the U-core is the same type as is used in the 25 kW transformer. The I-core has the same cross area as theU-core. In figure 5 it can be seen that there is 48 mm of space for the cables to fit in to on each side of the core.

Since the same cross area is used for the 25 kW transformer, the same number of windings are neededto create the same flux density swing according to equation 7. This means that 20 turns were to be woundon the primary side split into two layers occupying a width W of approximately 44 mm each on the mergedcores long side as displayed in the formula below:

Wcables =N · dcables

2=

20 · 4, 385

2= 43.85mm (17)

The winding arrangements were attempted in two methods. In the first prototype all primary turns wereplaced on one side and all secondary turns were placed on the opposite side. On the second prototype thesame arrangement was performed as for the 25 kW transformer, where half of the windings of the primaryside were put on one side along with half of the secondary windings. The rest of the windings were placedon the opposite side of the core.

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3.1.1 Calculation of Air-gap

The design for prototype 2 was to include an air-gap between the primary and secondary winding. To obtaina theoretical value of what the volume of the air-gap should be, the leakage inductance was assumed to beproportional to the air-gap volume between the primary and secondary windings. With that assumption aformula could be set up using the values for the leakage inductance criteria and the volume of the air gapfrom the 25 kW transformer.

AL,25kW

AL,10kW=V25kWV10kW

(18)

V10kW =AL,10kW · V25kW

AL,25kW(19)

The AL value for the 25 kW transformer was calculated with equation 9 using the value for its leakageinductance and number of turns. The AL value for the 10 kW transformer was calculated using the leakageinductance demanded and its number of turns.

By inserting the known values the needed volume of the air-gap to obtain a leakage inductance of approximately14 µH was calculated to be approximately 100 cm3.

V10kW =(0, 035 · 10−6) · 47 · 10−6

0, 0165 · 10−6= 100cm3 (20)

3.2 Construction of Prototypes3.2.1 Prototype 1

The design of prototype 1 was chosen to be built first because of the more complicated design involvingthe air gap in prototype 2. The primary and secondary winding were wound around each plastic bobbin.Each layer of winding was sealed with kapton tape to isolate them from each other. The cable ends weresoldered, on just the tip, to remove the isolation in order to be connected with the inductance meter. Thebobbins were placed around the core and the core was held together with cable ties. To measure the leakageinductance of the transformer, a short-circuit test was applied. The secondary winding of the transformerwas short circuited, and the primary winding was connected to an inductance meter. The measured leakageinductance of prototype 1 was greater than the specifications allowed, so the construction of the secondprototype began.

3.2.2 Prototype 2

For prototype 2, the same ferrite core was used while all other parts from prototype 1 were removed. 20 turnsof the secondary were wound around each bobbin, and then sealed with kapton tape. The air-gap for thisprototype was calculated using equation 19. The number of turns of the primary were equal to that of the 25kW transformer combined with the leakage inductance being roughly doubled, meant that the air-gap wascalculated to be almost twice as high compared to the 25 kW transformer, thus 100 cm3. The air-gap wascreated and held in place by inserting plastic support blocks between the primary and secondary windings.10 turns of the primary were wound around each bobbin and sealed with kapton tape. The bobbins were thenplaced around the core held together with cable ties. The transformer went through the same measurementprocedure as prototype 1. The volume of the created air-gap between the transformer core and primarywinding was calculated by approximating it as an elliptic cylinder wound around a cuboid. An illustrationof the area of the air-gap is displayed below in figure 6 where the height of the elliptic cylinder was 3,80 cm.

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The winding of the primary side had to be performed twice to obtain the satisfactory result for the leakageinductance as the first attempt resulted in a leakage inductance lower than specified. The air-gap was slightlyincreased for the second attempt and when the short-circuit test was applied, it resulted in a satisfying valuefor the leakage inductance.

Figure 6: Illustration of the created air-gap of the transformer and the associated dimensions

The finished transformer is to be lowered in an aluminum container for cooling and shielding purposes.After lowering the prototype in the aluminum container, the environment of the aluminum made the leakageinductance partially decrease. Because of this, the transformers leakage inductance needed to be increasedslightly. This was accomplished by increasing the air-gap volume marginally using the same method asbefore. The final construction of the transformer was now completed.

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4 Result

4.1 Prototype 1The first prototype with the entire primary and secondary winding on opposite sides, had a leakage inductanceof 131,12 µH. The constructed transformer during a short-circuit test, is displayed below in figure 7.

Figure 7: Prototype 1 transformer

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4.2 Prototype 2The second prototype with half the primary and secondary winding on opposite sides, had a leakageinductance of 14,77 µH. The air-gap of the transformer was 93,9 cm3. The built transformer during ashort-circuit test is displayed below in figure 8.

Figure 8: Prototype 2 transformer

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4.3 Final PrototypeThe final prototype had a leakage inductance of 13,92 µH, and the air-gap was 135,5 cm3. The finishedtransformer is displayed below in figure 9.

Figure 9: Final transformer prototype

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5 DiscussionThe purpose of the project was to design and construct a transformer with properties specified by thecompany ScandiNova Systems AB. After about a month of theoretical research, different proposals werepresented to the company and the decision of creating two of those were made.

The first prototype constructed and tested had a high leakage inductance. That is because the primaryand secondary coils were wound around opposite sides of the transformer. Some of the magnetic flux createdby the primary winding flows through the surrounding medium, and then back through the primary. Whenhalf of the primary and the secondary coils were wound around same sides separated by an air-gap, theleakage inductance became significantly lower. Here, the magnetic flux flowing through the medium alsogoes through the secondary coil because the secondary is surrounded by the primary winding. A higher degreeof the magnetic flux will now flow through which subsequently leads to a reduction of the leakage inductance.

When the transformer was lowered in an aluminum container, the leakage inductance decreased further.This is because the surrounding aluminum shields and reflects more of the flux back into the transformercircuit than air and thus lowering the inductance loss. It was also discovered that twining each side of theprimary winding with each other decreased the inductance marginally, though it was not as important of afactor compared to the container and had little to do with the overall construction of the transformer.

Throughout the project, we had regular meetings with the organizers regarding theory and general progressof the project as a whole. This led to us never having a communication issue with them as we frequentlykept them updated and informed. The meetings were always productive as they also informed us aboutspecifications and the corresponding theories behind them.

A part of the constructional arrangements was to perform it directly after the calculations based on trialand error instead of, for instance, simulation using COMSOL. This because it was much more time efficientand the results were more concrete by building it in practice instead of having to learn COMSOL from thebeginning and simulating the circuit in advance.

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6 ConclusionIt can be concluded that the model for the first prototype built gave a significantly higher leakage inductancethan the model for the second prototype. Where the first prototype had all primary windings around oneside of the core and the secondary windings around the opposite side of the core, the second prototype hadhalf of the windings of both primary and secondary on each side of the core and an air-gap between thewindings.

It can also be concluded that the method for calculating the optimal air-gap volume for prototype 2 workedwell. The method was based on the assumption that the leakage inductance is proportional to the volume ofthe air-gap between the core and the primary windings. The calculated volume was used as the transformerwas assembled and the resulting leakage inductance was close enough to the ideal value so only smallmodifications were needed to get a satisfying result for the leakage inductance.

Furthermore, only two iterations had to be performed for the air-gap, where a small adjustment of thefirst attempt gave the result required. It can therefore be concluded that the model used worked well, sincethe theoretical estimations gave a result close enough so that it could be optimized with minor adjustments.

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7 References[1] Guru, Bhag S.; Hizirogu, Hüseyin R.; 2001, Electric Machinery and Transformers. 3rd Edition. UK:Oxford University Press, ISBN: 9780195138900

[2] Nordling, Carl; Osterman, Jonny; 2006, Physics Handbook for science and engineering. 8th Edition.Studentlitteratur AB, ISBN: 9789144044538

[3] Encyclopedia magnetica, accessed 5 april 2019<http://www.encyclopedia-magnetica.com/doku.php/al_value>

[4] Vander Vorst, Andre; Rosen, Arye; Kotsuka, Youji; 2006, RF/Microwave Interaction with BiologicalTissues, John Wiley and Sons, Inc., ISBN: 9780471732778

[5] Blanchard, Julian; 1941, History of electrical resonance, accessed 16 May 2019, ISSN: 0095-9197<https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6773104>

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