Abstract Transformer Design for Dual Active Bridge

Home , Copper loss, Leakage inductance, Pavel Yablochkov, Planar transformer

ABSTRACT

TRANSFORMER DESIGN FOR DUAL ACTIVE BRIDGE CONVERTER

by Egor Iuravin

Power transformers have a long history which takes root in 19th century when Michael Faraday introduced the definition of the electromagnetic induction. In the beginning of the 1960s, there was a tendency to increase the frequency of switch mode power supplies. This thesis provides the detailed summary of operating principles, design, simulation and experimental analysis of high frequency power transformers. The focus of this research is to find an optimal transform solution for the DAB converter operating at a power level of 2 kW. Furthermore, the investigation will be carried out to estimate and measure the contact loss of a transformer.

TRANSFORMER DESIGN FOR DUAL ACTIVE BRIDGE CONVERTER

A Thesis

Submitted to the

Faculty of Miami University

in partial fulfillment of

the requirements for the degree of

Master of Science in

Computational Science and Engineering

Egor Iuravin

Miami University

Oxford, Ohio

2018

Advisor: Dr. Mark J. Scott

Reader: Dr. Haiwei Cai

Reader: Dr. Dmitriy Garmatyuk

This Thesis titled

TRANSFORMER DESIGN FOR DUAL ACTIVE BRIDGE CONVERTER

Egor Iuravin

has been approved for publication by

College of Engineering and Computing

and

Department of Electrical and Computer Engineering

______Dr. Mark J Scott

______Dr. Haiwei Cai

______Dr. Dmitriy Garmatyuk

Table of Contents 1. Introduction ...... 2 1.1 The history of the transformers ...... 2 1.2 Dual Active Bridge Converter ...... 3 1.3 Thesis objectives ...... 5 1.4 List of the chapters ...... 5 2. Transformer basics ...... 7 2.1 Transformer operation ...... 7 2.1.1 Ideal Transformer ...... 9 2.1.2 Real Transformer ...... 9 2.2 High-frequency transformers overview ...... 10 2.3 Magnetic Core types and Materials ...... 13 2.3.1 Saturation ...... 13 2.3.2 Magnetic Core Materials ...... 14 2.3.3 Ferrite Core Shapes ...... 16 2.4 Winding Types ...... 17 2.4.1 Litz Wire ...... 17 2.4.2 Printed circuit board ...... 18 2.4.3 Copper sheet...... 19 2.5 High-frequency transformers design ...... 19 3. Magnetizing inductance and parasitic components ...... 26 3.1 Leakage and magnetizing inductance ...... 26 3.1.1 Leakage inductance determination ...... 28 3.1.2 How to change the leakage inductance in the transformer ...... 29 3.1.3 Magnetizing inductance determination...... 29 3.1.4 How to change the magnetizing inductance ...... 30 3.2 Winding Capacitance ...... 31 3.2.1 Determination of the winding capacitance ...... 32 3.2.2 How to change the winding-to-winding capacitance ...... 33 3.3 Contact Resistance ...... 33 4. Transformer loss ...... 35 4.1 Winding loss ...... 35 4.1.1 Determination of the winding resistance ...... 39 4.2 Core loss ...... 40 4.2.1 Modified Steinmetz equation (MSE)...... 41 4.2.2 Generalized Steinmetz equation (GSE) ...... 41 4.2.3 Improved GSE method (IGSE)...... 42 4.2.4 Natural Steinmetz equation (NSE) ...... 42 4.2.5 Waveform coefficient Steinmetz equation (WCSE) ...... 42 4.2.6 Determination of the core loss ...... 44 5. Simulation and Experimental Results ...... 45 5.1 Transformers Design ...... 45 5.2 Simulation ...... 47 5.3 Experimental results ...... 49 5.4 Comparison and Cost ...... 53 6. Conclusion and future work ...... 55 Appendix ...... 60

iii

List of Tables

Table I. Characteristic of the main magnetic core materials ...... 15 Table II. Initial data of the power transformers ...... 45 Table III. transformer’s cores data ...... 46 Table IV. Loss coefficients ...... 48 Table V. Comparison of the results ...... 52 Table VI. Comparison of the Transformer’s cost ...... 53

List of Figures Figure 1.1: Faraday's ring transformer [4] ...... 2 Figure 1.2: Voltage waveforms of the dual active bridge converter ...... 4 Figure 1.3: Dual active bridge converter...... 5 Figure 2.1: The electric diagram of the transformer...... 8 Figure 2.2: The ideal transformer circuit...... 9 Figure 2.3: The real transformer equivalent circuit...... 9 Figure 2.4: The model of the planar transformer...... 11 Figure 2.5: Integrated Transformer ...... 13 Figure 2.6: Hysteresis loop of the magnetic material ...... 14 Figure 2.7: EI magnetic core shape...... 16 Figure 2.8: UI magnetic core shape...... 16 Figure 2.9: Planar E magnetic core shape...... 17 Figure 2.10: The wires which are used in the transformer: (a) litz wire and (b) solid wire...... 18 Figure 2.11: The printed circuit board...... 18 Figure 2.12: The typical copper sheet which is used as a winding for the planar transformer. ....19 Figure 2.13: Mean Length Turn of the wire wound transformer...... 21 Figure 2.14: Mean Length Turn of the planar transformer...... 21 Figure 2.15: The E core’s dimensions for determining the cross-section area 퐴푐...... 22 Figure 2.16: Dimensions of the core needed to find surface (퐴푡) and window (푊푎) areas...... 22 Figure 2.17: Dimensions of the PCB/Copper Sheet trace...... 23 Figure 2.18: Surface area versus total power loss...... 25 Figure 3.1: Transformer’s circuit with leakage inductance...... 26 Figure 3.2: Main and leakage fluxes distribution in a transformer...... 27 Figure 3.3: Voltage spikes in a transformer due to the leakage inductance...... 27 Figure 3.4: Measurement of the leakage inductance by doing short-circuit test...... 28 Figure 3.5: (a) Wire wound, (b) integrated and (c) planar transformers’ winding arrangement ...29 Figure 3.6: Measurement of the magnetizing inductance with LCR Meter...... 30 Figure 3.7: Equivalent circuit of the transformer with parasitic capacitance ...... 31 Figure 3.8: Current spikes due to the parasitic capacitance in the transformer ...... 32 Figure 3.9: Measurement of winding-to-winding capacitance using LCR meter ...... 33 Figure 3.10: The connection point between transformer and board which causes the contact resistance...... 34 Figure 3.11: (a) Current density distribution in the winding and (b) ohmic losses distribution in the winding...... 34 Figure 4.1: Eddy currents distribution in the conductor ...... 36 Figure 4.2: Proximity effect in the conductors. The currents flow (a) in the opposite and (b) in the same direction...... 36 Figure 4.3: Current distributions in the wire: (a) solid wire and (b) litz wire...... 38 Figure 4.4: MMF distributions in the windings: (a) non-interleaving arrangement and (b) interleaving arrangement...... 38 Figure 4.5: Conversion of the round conductors’ layer to the foil conductor...... 39 Figure 4.6: Winding arrangements of the transformer: (a) one layer and (b) multilayer...... 39 Figure 4.7: Square and sinusoidal waveforms of: (a) voltage and (b) flux...... 43 Figure 4.8: Electrical scheme to measure the core loss...... 44 Figure 5.1: (a) Primary and (b) secondary PCB windings...... 46 Figure 5.2: (a) Planar, (b) wire-wound and (c) integrated transformers 3D models ...... 47 Figure 5.3: Flux density distribution. (a) Planar, (b) wire-wound and (c) integrated transformers...... 48 Figure 5.4: Core loss versus frequency. (a) Integrated, (b) planar and (c) wire wound transformers ...... 49

Figure 5.5: The physical transformers: (a) wire wound, (b) planar and (c) integrated...... 49 Figure 5.6: The test set up: LCR meter and wire wound transformer...... 50 Figure 5.7: Test setup ...... 51 Figure 5.8: Transformers’ waveforms. (a) Integrated (b) wire wound (c) planar ...... 51 Figure 5.9: Secondary winding (a) with and (b) without connectors...... 52 Figure 5.10: Comparison chart...... 54

Abstract

Power transformers have a long history that takes root in 19th century when Michael Faraday introduced the definition of the electromagnetic induction. Beginning in the 1960s, there was a tendency to increase the frequency of switch mode power supplies. Many research institutions and organizations were trying to invent power converters that were able to operate at a frequency range of 20 kHz and higher. The main advantages of high frequency operation are the significant reduction of the component’s size in the power converters, low weight, low cost and high performance. This tendency has led to the invention of various high frequency power converter topologies such as resonant converters (e.g. LLC) and the dual active bridge converter (DAB). These converters consist of two power stages separated by a high-frequency power transformer. Since that time, the design of the transformers, which can operate at a high frequency, has become a major challenge for the engineers. This is because as the size of the components becomes smaller, the parasitic parameters such as leakage inductance, parasitic capacitance and ac resistance, begin to significantly effect the converter performance. This thesis provides a detailed summary of the operating principles of high frequency power transformers. It includes a discussion on their design and simulation, and it is supported with experimental analysis. Three transformers are compared in a DAB converter: a wire wound, a planar and an integrated one. 3-D models of the transformers are built and simulated in ANSYS’s Maxwell. These results are compared to those obtained from measurements taken from the real transformers. This includes measurements from an LCR meter and those obtained from experiments with a DAB converter. The findings illustrate the benefits of using simulation software to perform transformer design prior to manufacturing. The thesis includes a comparison between wire wound, planar and integrated structures in terms of leakage inductance, parasitic capacitance, winding and core loss. The focus of this research is to find an optimal transform solution for the DAB converter operating at a power level of 2 kW. Furthermore, the investigation will be carried out to estimate and measure the contact loss of a transformer.

1. Introduction 1.1 The history of the transformers The history of the transformer dates to the 1831 when English physician Michael Faraday conducted his experiments with two coils wound on the iron ring core as shown in the Fig. 1.1. He wound coils on the opposite sides of the ring which makes it the first power transformer in the world. During the experiments, Faraday observed the phenomenon which is now called the electromagnetic induction [1], [2].

Figure 1.1: Faraday's ring transformer [4]

One year later, in 1832, the phenomenon of self-inductance was invented by American scientist Joseph Henry. He found that with higher current interruption, the fast change of the magnetic flux, induces a voltage in the coil. In 1845, German mathematician Franz Ernst Neumann developed the mathematical formulation of the electromagnetic induction. Twenty years later, in 1865 the English scientist James Clerk Maxwell published his work “Dynamical Theory of the Electromagnetic field” [3]. In this paper, he provided four equations describing the behavior of the magnetic field, those equations are known today as Maxwell’s equations of electromagnetism. In 1876, Russian engineer Pavel Yablochkov invented a lighting system which consisted of a set of induction coils with primary windings connected to an AC source. The secondary windings were connected to several arc lamps called “electric candles”.

In 1882, French chemist Lucien Gaulard and British scientist John Dixon proposed the system of single phase AC current distribution [2]. The system contained the iron core, one primary coil and six secondary coils which can be considered as the first multiple output transformer. It was demonstrated both in 1882, and at the Turin International Exhibition in 1884 that the maximum distance that an ac line could transfer current was 34 km [1]. Three years later, Hungarian engineer Otto Titusz Blathy invented the first transformer with a toroidal core. Blathy was an engineer at Ganz & Co. the company that entered the railway industry in 1878. In four years, Blathy together with two other engineers: Karoly Zipernowsky and Miska Deri proposed transformers with two different winding designs, finally called ZBD transformers. In the first version, the winding was wound around the core and in the second one, the winding was surrounded by the core. Nowadays, these designs are called core and shell types, respectively. In 1885, George Westinghouse, American entrepreneur and engineer, purchased the patent of ZBD transformer and started production in the United States. One of the requirements was to create alternative design based on the same principles. Later, Westinghouse hired American engineer William Staneley who in 1886, based on Blathy’s work, introduced the first transformer with laminated core which provide cheaper construction and better isolation for high voltage application. In 1889, Russian engineer Mikhail Dolivo-Dobrovolsky invented the first three-phase transformer for General Electricity Company in Germany. Two years later, Serbian scientist Nikola Tesla invented the Tesla coil – the first air core transformer which handle the high voltages at high frequencies. [4]

1.2 Dual Active Bridge Converter The dual active bridge (DAB) converter is an isolated bidirectional converter that can transfer power in two directions as required by the system [5]. Fig. 1.3 shows the basic circuit topology of the DAB converter. It consists of two H bridges with their AC terminals connected to the primary and secondary windings of a high frequency transformer. The operational principle of the DAB is based on phase shift control: the outputs of two voltage bridges are phase shifted from each other by the phase shift angle 휙. Then, by changing the angle, the amount of power transferred between two bridges can be controlled. The output power 푃표 of the dual active bridge converter can be found as [6], [7]:

2 푉𝑖 휙 푃표 = ∗ 푑훿 ∗ (1 − ) (1) 푋퐿 휋

푉표 where 푋퐿 = 휔퐿푙푘, 푑 = , 푉𝑖 is the input voltage, 푉표 is the output voltage, 휔 = 2휋푓 is the 푁푉𝑖 switching frequency, 퐿푙푘 is the leakage inductance referred to the primary side of the transformer,

N is the turns ratio and 훿 is the phase shit between two bridges. The DAB voltage waveforms are shown in Fig. 1.2.

Figure 1.2: Voltage waveforms of the dual active bridge converter

The main advantages of the DAB converter are high efficiency, small components size, low device stresses and low switching losses, zero-voltage switching capability, small number of components and bidirectional power flow. DAB converters can be used in different areas such as automotive, renewable energy, and smart grid applications. It has the capability of stepping the voltage up or down by changing the turns ratio of the transformer. DAB converters reduce the number of components since the output inductor filter is not required. Galvanic isolation is provided by a high frequency transformer. In addition, high power densities with high efficiency can be achieved. On the contrary, the drawbacks of DAB converter are the presence of high circulating current in the transformer, which increases losses and not being able to take advantage of the ZVS feature at light loads [7], [8]. The key component of the DAB is the high frequency transformer whose internal leakage inductance (Lg) can be used as the main component for transferring energy; therefore, it is a crucial factor in designing the transformer. Leakage inductance therefore determines the maximum transferable power for a given switching frequency. Zero-voltage switching can be achieved using the internal leakage inductance of the transformer [9], [10].

Figure 1.3: Dual active bridge converter. 1.3 Thesis objectives In the thesis, the following goals are to be determined: • Evaluate the performance of the finite element analysis (FEA) software in terms of the accuracy of extracted parameters (e.g. leakage inductance, core loss, parasitic capacitance). It will be done by comparing those parameters with the results obtained from mathematical expressions and measurements. • Explore a new transformer topology, referred to as an integrated transformer, and compare it to the planar and wire wound transformer. • Determine the optimal transformer solution for the DAB application at a power level of 2 kW. Some of the criterion being the size, low cost, high efficiency, and maintaining the appropriate levels of leakage inductance. • Explore the methods of simulating the contact resistance in the connection point between transformer’s terminal and voltage source.

1.4 List of the chapters Chapter 2 provides the operating principle of the transformer and lists the winding and magnetic core types. An overview of the existing high frequency transformers is included. Finally, it covers the detailed design procedure of three different transformers. Chapter 3 discusses the parasitic components existing in the transformers, such as leakage inductance and inter winding capacitance. It provides the definition of those parameters, their analytical expressions and a number of ways to measure them. Chapter 4 gives the definition of various types of losses existing in the transformer. Moreover, it provides with several ways of how to find those losses. Such methods as Modified Steinmetz Equation, Generalized Steinmetz Equation and Improved Generalized Steinmetz Equation are provided.

Chapter 5 contains the design process for the three power transformers; it is based on the equations from the chapter 2. Also, it contains a section on simulation where both transformers are built and tested to extract all necessary parameters. Finally, several experiments are performed to validate the simulation results and analytical calculations. Chapter 6 contains the conclusion about what steps have been completed in this proposal and provides the future work about next steps. The Appendix contains the Matlab code, which includes all calculations described in this thesis.

2. Transformer basics This chapter provides detailed explanation about transformer structures. It describes fundamental principles of the magnetics such as Faraday’s and Ampere’s laws. The chapter also contains the information about various types of windings, magnetic cores, magnetic materials and their purposes. Furthermore, it explains the differences between ideal and real transformers and provides an overview of existing high-frequency structures. Finally, detailed high-frequency transformer design process is provided.

2.1 Transformer operation The transformer’s operation principle is based on the Faraday’s Law, which states that electromagnetic force (푣(푡)) induced in the circuit equals to the negative rate of the time-variated magnetic flux (Φ(t)) existing in the contour [11]: 푑Φ(t) 푣(푡) = −푁 , (2) 푑푡 The modified version of the eq. (2) was proposed by James Clerk Maxwell and called Maxwell–Faraday equation: 푑 ∫ 퐸⃗ ∗ 푑푙 = − ∫ 퐵⃗ ∗ 푑푆 , (3) 푙 푑푡 푆 where 퐸⃗ is the electric field, 퐵⃗ is the magnetic field, l is the length of the conductor and S is the area of the contour. The transformer is an electrical machine which can be used for increasing or decreasing of the output voltage with respect to the input (step-up or step-down transformers), current measurements (current transformer), or galvanic isolation between two parts of the circuit. The transformer usually consists of the magnetic core, which can be made of various materials (ferrite, electrical steel) and that have different shapes (EI, EE, UU, UI, Planar cores etc.), and two or more windings, which are usually made of copper. The current 퐼푝 flowing through the primary winding generates the magnetic flux in the core. The flux lines flowing through the magnetic core induces the voltage 푉푠 (푡) in the secondary winding of the transformer as shown in the Fig. 2.1.

The load connected to the secondary side cause the current 퐼푠 to flow through the circuit. Therefore, in transformers the power is transferred without physical connection between conductors.

Figure 2.1: The electric diagram of the transformer.

Another important law in electromagnetism is Ampere’s Law. Ampere’s Law is one of Maxwell’s four equations which describe the theory of electromagnetism [12]. It states that the magnetic field, distributed through the closed conduction path is equals to the electric current that encloses in this path:

퐻⃗⃗ ∗ 푑푙 = 퐽 ∗ 푑푆, (4) ∫푙 ∫푆 where 퐻⃗⃗ is the magnetic field, and 퐽 is the current density. If the current flows through the wire wounded on the magnetic core, then Ampere’s law has the following form [11]:

퐻⃗⃗ ∗ 푑푙 = 퐽 ∗ 푑푆 = 푁퐼, (5) ∫푙 ∫푆 where N is the number of turns in the core and I is the current flowing in the wire. The product of N*I equals to magnetomotive force (MMF). MMF can be found using following equation:

푀푀퐹 = 푁 퐼 = Φ ℜ, (6) where Φ is the magnetic flux and ℜ is the reluctance of the core. The reluctance of the core is calculated as: 푙 ℜ = 푒 , (7) 휇퐴푐 where 푙푒 is the magnetic length path of the transformer, 휇 is the permeability of the core material and 퐴푐 is the core’s cross-section area. Since the transformer in the Fig. 2.1 has only two winding, applying Ampere’s law to it results in the following expressions:

푀푀퐹 = 푁푝퐼푝 + 푁푠퐼푠 , (8)

Φ ℜ = 푁푝퐼푝 + 푁푠퐼푠 . (9) It can be noticed that magnetic circuits are analogous to the electric circuits. For example, resistivity (ℜ) is the magnetic equivalent of the resistance and magnetic flux (Φ) is the equivalent of the electric current. In the next two section, (8) will be used to explain the difference between “Ideal” and “Real” transformers.

2.1.1 Ideal Transformer The ideal transformer is an electric machine where the input power is equal to the output. The ideal transformer is shown in Fig.2.2.

Figure 2.2: The ideal transformer circuit.

Since the ideal transformer is lossless, the reluctance of the magnetic core is zero, thus the MMF is also equal to zero:

푁푝퐼푝 + 푁푠퐼푠 = 0. (10)

According to the Faraday’s law the primary 푣푝 and secondary 푣푠 voltages can be obtained as follows: 푑Φ(t) 푣 = 푁 , (11) 푝 푝 푑푡 푑Φ(t) 푣 = 푁 , (12) 푠 푠 푑푡 where 푁푝 and 푁푠 are the number of turns for primary and secondary windings respectively. By equating the fluxes from equations (11) and (12), the turns ratio of the transformer n can be derived: 푣 푁 푛 = 푝 = 푝 . (13) 푣푠 푁푠

2.1.2 Real Transformer Unlike the ideal transformer, in the real one the input power does not equal to the output power. Such transformers have a number of parasitic elements which result in the various forms of losses.

Figure 2.3: The real transformer equivalent circuit.

In the Fig. 2.3, 푅푝 and 푅푠 are the equivalent resistances for the primary and secondary windings, respectively. 푅푐 represents the equivalent core loss resistance. 퐿푝 and 퐿푠 are equivalent leakage inductances for the primary and secondary windings, respectively. 퐿푚 represents the magnetizing inductance, 퐶푃 and 퐶푠 are intra-winding capacitances of primary and secondary windings, respectively. 퐶푝푠 is called inter-winding capacitance. In a real transformer, the reluctance is not equal to zero and the flux of the magnetic core is equal to: 푁 퐼 +푁 퐼 Φ = 푝 푝 푠 푠. (14) ℜ Applying the Faraday’s law to the real transformers yields the following equations:

푑i 푑i 푣 = 퐿 푝 + 퐿 푠 , (15) 푝 푝 푑푡 푝푠 푑푡 푑i 푑i 푣 = 퐿 푠 + 퐿 푝 , (16) 푠 푠 푑푡 푝푠 푑푡 where 퐿푝 and 퐿푠 are self-inductances of the transformer and 퐿푝푠 is the mutual inductance between primary and secondary windings.

2.2 High-frequency transformers overview In the beginning of 20th century, low frequency transformers had a number of limitations being used as traditional power transformers such as relatively large size and high-power losses [13]. In 21st century, the high-frequency transformers have become more popular because of their relatively small size, as compared to the low frequency devices, and ability to work with high- frequency switch-mode power supplies. The main areas of application of high frequency transformers are automotive and telecom industries. One of the first high frequency transformers was the conventional wire wound structure. Unlike the low frequency devices, high frequency wire wound transformer has multiple strands winding called litz wires which are used to reduce the influence of skin and proximity effects on the transformer. Moreover, ferrite has been used as a magnetic core material because it has a comparative low core loss at high frequencies [14]. This structure has several drawbacks such as inability to fully integrate it to the power stage and relatively high contact resistance. Planar transformer was invented in the 1980s [15], and in recent years its popularity has dramatically increased due to the numerous advantages of this structure. The planar transformer is where the traditional wire windings are replaced by printed circuit boards (PCB). Since the conductors are thin copper foil, the operating frequency of such transformers are not limited by skin effect. The advantages of this topology include the low profile, excellent thermal characteristics, and easy integration to the power converter circuit [16]. A planar transformer is shown in Fig. 2.4. However, the planar transformer also has the number of drawbacks such as cost,

low maintainability (if damaged, the entire power stage needs to be replaced) and low range in which the leakage inductance can be controlled. For the past decades, a lot of work has been done in the area of high-frequency transformer design. In [9], the authors developed the wire wound high frequency transformer for the dual active bridge (DAB) converter that operates with switching frequency of about 500 kHz. During the study, they found that since the converter operates at such high frequencies, to achieve zero voltage switching the required leakage inductance should be approximately 3uH. Hence, there is no more need in the lumped inductance that is usually used to maintain ZVS.

Figure 2.4: The model of the planar transformer.

In 2009, Zhang et. al [17] proposed the design of the planar transformer for aerospace application. The presented design results in a 36.3 % reduction in size of the device when compared to the previously used transformer. Moreover, the core loss decreased from 2 W to 1.59 W and winding loss was reduced by 1.19 W using interleaving layers method. In [18], the authors presented a design optimization procedure for high-frequency transformer employed in dual active bridge converter. The main goal of the research was to show that leakage inductance, phase-shifted angle, skin and proximity effects have to be calculated properly to minimize core loss. Furthermore, the study also demonstrated that the leakage inductance required for zero-voltage switching can be implemented without using additional inductor. Fei et. al [19] proposed the novel model of the planar matrix transformer employed in LLC converter which includes the flux density reduction and design optimization. Matrix transformer represents the traditional magnetic core divided into the four core structures with primary winding connected in series and secondary winding connected in parallel. Distributing the secondary

current with multiple cores helps to increase the output current capability of the transformer. The presented structure results in a decrease in flux density, equal to half that of the original model. Since the core loss is mainly determined by flux density, it was also decreased by 40 %. Moreover, the proposed design methodology helps to decrease the number of PCB layers from 12 to 4 and significantly reduce the cost of the transformer and its parasitic capacitance. Lee et. al [20] presented a design optimization procedure for three phase dual active bridge converter to increase power density and efficiency. The design optimization involves the changing of transformer parameters such as the number of turns and the current density and then checking the transformer performance using FEA tool. Moreover, the authors created the objective function to estimate transformer efficiency, using magnetizing current, core loss, copper loss and total weight as function variables. The study shows that after all changes were made, two experimental prototypes demonstrated the efficiency of 98.95% and 99.01% respectively. Yan et. al [21] proposed a new transformer structure for 1.2 kW LLC topology. New transformer has fully interleaved winding structure, and novel integration technique where the rectifier and capacitors are directly integrated to the secondary winding to avoid increasing leakage inductance and ac resistance. The transformer model has been built and tested. The experiments show the total efficiency is equal to 96 %. Chattopadhyay et. al [22] provides the analysis of three limb high frequency transformer for DAB converter. The multi-limb transformers can be used to decrease inter winding capacitance in the transformer. In this paper, such aspects as design and simulation of the three-limb transformer has been discussed, also the equation for the leakage inductance and inter winding capacitance are provided. As it can be seen from the papers provided above, in case of the high frequency transformers, the main research was mostly concentrated on traditional wire wound and planar transformers. In this proposal, a relatively new transformer structure, the integrated transformer, is analysed along with the wire wound and planar ones. The integrated transformer was proposed in [23]. It consists of UI or UU magnetic cores and two windings where the primary is made of a litz wire and the secondary one is made of a copper sheet. The proposed structure can help to control the leakage inductance (either make it very small or relative high by changing the position of the secondary side) of the transformer and can be easily integrated to the power stage. It also has better efficiency due to higher contact area and hence lower contact resistance. The integrated transformer is shown in Fig. 2.5. The drawbacks of this topology can be a higher cost as compared to the wire wound transformer and relatively small number of cores’ sizes available at the market.

This thesis is manly focused on finding the optimal solution for isolated dc/dc converters in terms of cost, size, efficiency, design complexity and repair flexibility. Three different transformer’s topologies are designed and simulated using finite element analysis (FEA) tool. Then, the simulation results are compared to the measurement ones of the physical transformers to verify the accuracy of the extracted parameters. At the end, the transformers are compared to each other in terms of various factors (e.g. cost, efficiency) to determine the better solution for the dual active bridge converter’s application.

Figure 2.5: Integrated Transformer

2.3 Magnetic Core types and Materials The magnetic core is the key element in the transformer. It is used to contain the magnetic flux and create the magnetic paths for the flux to be distributed [26]. When choosing the magnetic core, several parameters such as permeability, saturation flux, core loss, core size, core weight and ability to operate at high frequencies should be taken into consideration.

2.3.1 Saturation Saturation is an important parameter of the magnetic core – the state when further increasing of the magnetic field does not cause the magnetic flux to increase. As a result, the permeability of the core falls down to zero and the winding becomes a short circuit. The relationship between magnetic field intensity (H) and magnetic flux density (B) is called the hysteresis loop and is shown in the Fig. 2.6 The maximum value of B is called saturation flux density (퐵푠푎푡). It can be noted that magnetic materials, which can be easily magnetized or demagnetized are called soft magnetics. In the contrary, those that are difficult to demagnetized are called hard magnetics [27]. The parameter 퐵푟푒푚 in Fig. 2.6 represents the flux which remains

in the core after excitation has been removed and called remanence flux density. The magnetizing force required to demagnetize (i.e. bring the remanence flux back to zero) the core is called coercivity and is denoted as −퐻푐. The saturation can be controlled by selecting appropriate size of the magnetic core. The detailed explanation of how to choose the magnetic core is given in the chapter 3.

Figure 2.6: Hysteresis loop of the magnetic material

The relationship between the magnetic field intensity and magnetic flux density is called permeability: 퐵 휇 = . (17) 퐻 Permeability represents the ability of the magnetic material to conduct the magnetic flux and shows how easy the core can be magnetized.

2.3.2 Magnetic Core Materials The selection of the magnetic materials depends on several factors such as cost, size, performance and the application. Nowadays, there are four materials that are typically used for magnetic cores production [26]: • Air. The material that does not have a core loss because of the lack of the physical core. The main drawback is that it is a poor conductor of the magnetic flux due to its low permeability. • Electric steel. Mostly used for the low frequency transformers. It is produced by stacking together the number of the silicon steel lists. This process is called lamination. Using laminated cores can significantly reduce the eddy currents effect. The main reason for that is because in the laminated structure, the eddy currents are distributed in each particular thin metal sheet and therefore reducing their magnitude. Silicon steel

has a large saturation flux density and good permeability which makes it widely used for the low frequency transformers. • Powder materials. The first nickel iron powder cores were produced in 1918 [26]. Powder cores are made of compressed powder materials such as molypermalloy and powdered iron. One of the advantages of the powder materials is their relatively low price which make them widely used for both low and high frequency applications. • Ferrite. The ferrite material was synthesized by Yogoro Kato and Takeshi Takei, researchers from Tokyo Institute of Technology in 1930 [30]. It eventually led to the foundation of the TDK – one of the largest manufacturers of the ferrite products in the world. Ferrites are the ceramic materials which are the combination of iron oxide and one of the metallic elements such as oxide, carbonate, manganese, zinc, nickel, magnesium or cobalt [26]. The two major categories of the soft ferrite are manganese- zinc (MnZn) and nickel-zinc (NiZn). Ferrite maintain excellent performance at high frequencies and have a high permeability (from 700 to 10000 [28], [29]). The main disadvantage of the ferrite core is low saturation flux density (~ 0.4 T). However, in the high frequency range the value of flux density is usually significantly lower than saturation flux. Since the ferrite is the best suit for the high frequency application, it is used for all transformers described in this proposal. Table I provides the characteristics of some magnetic materials [26].

TABLE I. CHARACTERISTIC OF THE MAIN MAGNETIC CORE MATERIALS Material name Composition Flux density (T) Permeability (흁) Magnesil SiFe 1.5 – 1.8 1500 Supermendur CoFeV 1.9 – 22 800 Silicon steel SiFe 1.5 – 1.8 1500 Orthonol NiFe 1.42 – 1.58 2000 Sq. Permalloy NiFeMo 0.66 – 0.82 1200 – 100000 Iron powder Fe 0.5 – 1.4 4 – 125 Amorphous 2605–SC FeBFe 1.5 – 1.6 3000 Amorphous 2714 –A CoSiFe 0.5 – 0.58 20000 Ferrite (MnZn) MnZn 0.3 – 0.5 700 – 15000 Ferrite (NiZn) NiZn 0.3 – 0.5 15 – 1500

2.3.3 Ferrite Core Shapes EI Core EI core is shown in the Fig. 2.7. The main advantage of EI core is their relatively low price and simple bobbin winding. There are several modifications of E core such as EP (the combination of pot and E cores), EER, EFD (core with long center leg, usually employed for low profile transformers), ETD (core with cylindrical center leg) [14]. Ideal for the three and one phase high frequency power transformers.

Figure 2.7: EI magnetic core shape.

UI Core UI or UU cores are popular in power electronics application because the long legs of U core maintain low leakage inductance. Windings can be wound on both legs of the core.

Figure 2.8: UI magnetic core shape.

Planar E Core As it is mentioned in the name, this type of the core is used for the planar transformer. The main difference from general “E” core is low profile and hence lower window length which makes them very popular in power electronic applications since it can be integrated directly to the power stage.

Figure 2.9: Planar E magnetic core shape.

2.4 Winding Types Operating at high frequencies requires less number of turns as compared to low frequency. The main reason for that is because the ac winding resistance dramatically increase at high frequency. Therefore, it is necessary to use as small number of turns as possible to minimize winding losses. As it was mentioned above, high frequencies transformers are exposed to dramatically increased ac winding losses, which can be a huge problem and cause low efficiency of the device. Depending on the application, there are several types of high frequency transformer windings, which are widely used in industry. In this section, the description for each type of the windings is provided.

2.4.1 Litz Wire The term “Litz” came from the German’s word “Litzendraht” and means braided/stranded wire [24]. Litz wire consists of the number of small twisted strands isolated from each other and collected together. The litz and solid wires are shown in Fig. 2.10 (a) and 2.10 (b) accordingly. The diameter of each strand is usually smaller than a skin depth which means that litz wire does not suffer from skin effect losses. As for the proximity effect, in case of litz wires it can be divided into two categories [25]: the bundle- and the strand-level. The bundle level effect relates to the current circulating in paths involving multiple strands, it can be reduced by twisting the strands. The strand-level is related to the current circulating in each particular strand, to control the strand-level loss, the optimal number of strands can be calculated.

(a) (b) Figure 2.10: The wires which are used in the transformer: (a) litz wire and (b) solid wire.

2.4.2 Printed circuit board The planar transformer was invented in 1980s and during the next 10 years became very popular among the designers and manufacturers because of their thermal characteristics, low profile and cheap and simple production. The windings in the planar transformer are implemented on the printed circuit board (PCB) which, depending on the needs, can be one or multiple layers. The typical PCB winding for the planar transformer is shown in Fig. 2.11.

Figure 2.11: The printed circuit board.

Along with the litz wires, the PCB winding has also advantage of reduced skin effect because the thickness of each trace is made such that it is smaller than or equal to a skin depth. To better cope with the proximity effect, the interleaving method can be implemented when designing the PCB.

2.4.3 Copper sheet. Copper sheets can also be used when building the planar transformer. Especially for the secondary side, which usually has just one turn. The copper sheet has the same advantages as the PCB in terms of proximity and skin effect minimization. Moreover, the cost and production time of the copper sheet is usually lower those for the PCB. The typical copper sheets used in the planar transformers is shown in Fig. 2.12.

Figure 2.12: The typical copper sheet which is used as a winding for the planar transformer.

2.5 High-frequency transformers design Before starting to build the 3D transformers model for simulation, the transformer design process should be performed. The basic design includes the calculation of area product and then choosing appropriate transformer core, determining the area of primary and secondary winding, calculating the number of turns on the primary and secondary sides, calculating the copper and core loss. In this chapter, a step-by-step design process is provided for three different transformer types: wire wound, planar and integrated. In general, the design procedure for all transformers is the same, except the part where the winding area needs to be determined. In case of wire wound structure, it is enough to find the area of the wire and then determine the gauge number. At the same time, for the planar transformer, one more calculation needs to be done in order to find the length of the PCB trace. The whole design structure is based on the equations from [26]. Below is the initial data that is usually provided to designer:

• Input voltage, 푉𝑖푛

• Output voltage, 푉표푢푡

• Output power, 푃표푢푡 • Frequency, 푓 • Current density, J • Efficiency, 푛

• Magnetic flux density, 퐵푚 • Core material (ferrite, electrical steel and so on) • Core configuration (U, E cores)

When designing a high-frequency transformer, one of the main parameters that should be taken into consideration is the skin depth (훿), which is the distance below surface of the conductor where the current is distributed. If the diameter of the conductor exceeds the maximum wire diameter that corresponds to the skin depth (퐷푠푘𝑖푛_푑푒푝푡ℎ), it causes large copper loss in the transformer’s winding. The skin depth is calculated using the following relationship: 6.62 훿 = ∗ 퐾 [푐푚] , (18) √푓 where 푓 is an operating frequency of the transformer and 퐾 is skin depth coefficient, which equals 1 for the copper.

Then, the diameter of the single strand in wire diameter (퐷푠푘𝑖푛_푑푒푝푡ℎ) and the appropriate wire area (퐴푤) that correspond to the skin depth is determined by:

퐷푠푘𝑖푛_푑푒푝푡ℎ = 2 ∗ 훿 [푐푚] , (19)

휋∗퐷 2 퐴 = 푠푘𝑖푛_푑푒푝푡ℎ [푐푚2]. (20) 푤 4

Once the wire gauge is found, the wire area (퐴푤) can be used as the maximum allowed single wire strand area in case of litz wire and the diameter of the single strand in wire (퐷푠푘𝑖푛_푑푒푝푡ℎ) can be used as a maximum thickness of the printed circuit board (PCB) and copper sheets for planar and integrated transformers.

A transformer’s output current, (퐼표푢푡) is another important parameter in a transformer.

Using 퐼표푢푡, the wire area for the secondary side can be determined. The output current can be found as follows:

푃표푢푡 퐼표푢푡 = [퐴]. (21) 푉표푢푡

The apparent power of the transformer 푃푡 is basically the sum of the input (푃𝑖푛) and output

(푃표푢푡) power of the transformer: 푃 푃 = 푃 + 푃 = 푃 + 표푢푡 [푊]. (22) 푡 표푢푡 𝑖푛 표푢푡 푛 One of the main parameters that determines the power handling ability is the area product

(퐴푝), which is the product of window area 푊푎 (window area is described later in the chapter) and core area 퐴푐:

퐴푝 = 푊푎 ∗ 퐴푐 . (23) It is used by engineers to determine the appropriate core size. Equation (22) can be rewritten as follows:

4 푃푡∗10 4 퐴푝 = [푐푚 ], (24) 퐾푓∗퐵푚∗푓∗퐾푢∗퐽

where, 퐾푓 is waveform coefficient (equals 4 for square waveform and 4.44 for sinusoidal waveforms), 퐵푚 is the maximum flux density of the core, J is current density and 퐾푢 is a window utilization factor.

The maximum flux density 퐵푚 can be calculated by [31], [34]: 1 2 2 훽+2 8 휌휆 퐼푡표푡푎푙 푀퐿푇 1 퐵푚 = [10 ∗ ∗ 3 ∗ ] [T], (25) 2퐾푢 푊푎퐴푐푀푃퐿 훽∗퐾푓푒 −6 where, 휌 = 1.724 ∗ 10 is the resistivity, 휆 = 퐷 ∗ 푇 ∗ 푉𝑖푛 is applied volt-seconds, D is the duty cycle, T is the period, 퐼푡표푡푎푙 is the total current, 퐾푢 is the winding fill factor, 훽 is the core loss exponent and 퐾푓푒 is the core loss coefficient.

Once 퐴푝 is found, the transformer core can be selected from the manufacturer’s datasheet [28], [29]. Moreover, the following core parameters need to be taken from the datasheet for further calculations: mean length turn (MLT), core area (퐴푐), window area (푊푎), surface area (퐴푡), magnetic path length (MPL) and core weight. Since not all parameters listed above are given in the datasheet, they need to be calculated. The MLT of the wire wound and planar structures can be found as follows [18]:

푀퐿푇푤𝑖푟푒 = 2 ∗ (푎 + 푐) + 푏 ∗ (2 + 휋), (26)

푀퐿푇푝푙푎푛푎푟 = 2푏 + 2푐 + 2.82푎, (27) where a, b and c are core dimensions that can be found in the Fig. 2.13 and 2.14

Figure 2.13: Mean Length Turn of the wire wound transformer.

Figure 2.14: Mean Length Turn of the planar transformer.

One of the main parameters in the datasheet is cross-sectional area of the core. It is used to calculate important parameters such as leakage inductance and magnetizing inductances. It can be found by the following equation:

퐴푐 = 퐶 ∗ 퐷, (28) where C and D are the core dimensions which are shown in the Fig. 2.15.

Surface area (퐴푡) of the transformer core can be found using equations (29) – (32), dimensions of the core are shown in the Fig. 2.16 퐸푛푑 = 퐻푒𝑖푔ℎ푡 ∗ 퐿푒푛푔푡ℎ, (29) 푇표푝 = 퐿푒푛푔푡ℎ ∗ 푊𝑖푑푡ℎ, (30) 푆𝑖푑푒 = 퐻푒𝑖푔ℎ푡 ∗ 푊𝑖푑푡ℎ, (31)

퐴푡 = 2 ∗ (퐸푛푑) + 2 ∗ (푇표푝) + 2 ∗ (푆𝑖푑푒). (32)

Figure 2.15: The E core’s dimensions for determining the cross-section area 퐴푐.

Figure 2.16: Dimensions of the core needed to find surface (퐴푡) and window (푊푎) areas.

The window area of the transformer can be calculated as follows:

푊푎 = 퐴 ∗ 퐵. (33)

The number of primary (푁푝) and secondary (푁푝) turns can be found by using Faraday’s law:

4 푉𝑖푛∗10 푁푝 = , (34) 퐾푓∗퐵푚∗푓∗퐴푐

푁푝∗푉표푢푡 푁푠 = . (35) 푉𝑖푛

Primary current (퐼푝) of the transformer can be found in the following:

푃표 퐼푝 = [퐴], (36) 푉𝑖푛∗푛 where 푛 is the desired efficiency of the transformer. The next important parameters which need to be calculated are the wire area of primary (

퐴푤퐵(푝)) and secondary (퐴푤퐵(푠)) conductors. They can be calculated by using equations (37) and (38): 퐼 퐴 = 푝 [푐푚2], (37) 푤퐵(푝) 퐽 퐼 퐴 = 표푢푡 [푐푚2]. (38) 푤퐵(푠) 퐽 Designers should be careful when choosing the current density value (J). For example, if the current density is chosen to be very high, it will cause additional winding loss. After wire the areas are calculated, those values are used to find the appropriate wire gauge number (AWG), which can be selected from the table given in [32]. Since a high-frequency transformer is being designed, as opposed to a low frequency transformer, it is highly recommended to use litz wires instead of solid wires. From [25], [26] and [33] it is known that the litz wires reduce skin effect losses and consequently winding loss because they consist of multiple isolated strands and each strand has the diameter which is equal or smaller than 퐷푠푘𝑖푛_푑푒푝푡ℎ. In the case of the planar and integrated transformers, by knowing the gauge number, the trace length of the copper sheet/PCB can be determined using the following equation: 푊𝑖푟푒 푎푟푒푎 퐿푒푛푔푡ℎ 표푓 푡ℎ푒 푡푟푎푐푒 = . (39) 푆푘𝑖푛 푑푒푝푡ℎ

Figure 2.17: Dimensions of the PCB/Copper Sheet trace.

The primary (푅푝) and secondary ( 푅푠 ) DC winding resistances are used to calculate copper/ windings loss in the transformer. They can be calculated as follows: 휇Ω 푅 = 푀퐿푇 ∗ 푁 ∗ ∗ 10−6 [Ω], (40) 푝 푝 푐푚 휇Ω 푅 = 푀퐿푇 ∗ 푁 ∗ ∗ 10−6 [Ω], (41) 푠 푠 푐푚 휇Ω where is the winding resistance of the conductor, which can be obtained from [32]. 푐푚

The final stage of the design is the determination of the loss existing in the transformer.

These losses are broken up into two components: core losses and winding losses. The primary (푃푝) and secondary (푃푝) winding loss and total winding loss (푃푤𝑖푛푑𝑖푛푔) can be found as follows: 2 푃푝 = 퐼푝 ∗ 푅푝 [푊], (42) 2 푃푠 = 퐼표 ∗ 푅푠 [푊], (43)

푃푤𝑖푛푑𝑖푛푔 = 푃푝 + 푃푠 [푊]. (44) 푚푊 The core loss density in milliwatts per cubic centimetre ( ) can be found as: 푐푚3 푚푊 = 푎 ∗ 푓푥 ∗ 퐵 푦. (45) 푐푚3 푚 Coefficients a, x and y depend on the magnetic material which is used for the transformer core and can be derived from the core loss curve in the manufacturers datasheet. The core loss of the transformer can be found by multiplying core loss density by core volume, (푉푐표푟푒): 푚푊 푃 = ∗ 푉 [푊]. (46) 푐표푟푒 푐푚3 푐표푟푒

The total loss of the transformer (푃휖) can be calculated as the sum of winding and core loss:

푃휖 = 푃푤𝑖푛푑𝑖푛푔 + 푃푐표푟푒 [푊]. (47) The watts per unit area, (Ψ) is used to estimate the core loss density and can be calculated by dividing total loss by the surface area:

푃휖 푊 휓 = [ 2] . (48) 퐴푡 푐푚 Finally, the last part that has to be done when designing the transformer is a temperature rise calculation. There are two allowable temperature rises above the ambient temperature: 25 and 50 degrees Celsius. The figure below shows the relationship between total surface area of the core (퐴푡) and total power loss (푃휖) for both 20C and 50C.

Surface Area versus Total Power Loss 103 20°C Rise

50°C Rise

)

(

, a

e 2

r 10

u S

101 10-1 100 101 102

Figure 2.18: Surface area versus total power loss.

The temperature rise of the transformer (푇푟), can be found as follows:

푃휖 0.826 푇푟 = 450( ) (49) 퐴푡 If it turns out that the calculated temperature rise is higher than desired one, the design process needs to be started from the beginning by picking another core with a higher surface area.

Conclusion In this chapter, the difference between real and ideal transformer was described. An overview of high-frequency transformer topologies was provided, and a new previously unexplored, transformer structure was presented. Various magnetic core materials, core shapes and winding types have been discussed. Finally, all necessary equations for the transformer design have been provided.

3. Magnetizing inductance and parasitic components It can be known from the chapter 2 that in real transformer, there are number of parasitic elements, which make a serious impact on its performance. Those elements are leakage inductance, parasitic capacitance and contact resistance. The purpose of this chapter is to provide detailed analysis about each element. The analysis incudes all necessary equations and discussion of methods that help to change or measure each parasitic element.

3.1 Leakage and magnetizing inductance There is no doubt that a real transformer, as opposed to an ideal version, contains a number of parasitic elements that can cause abnormal operation of the power device. However, in some cases some amount these parasitic elements is required. One of those elements is the leakage inductance. Fig. 3.1 shows the equivalent circuit of the real transformer. The correct selection of the leakage inductance can help to avoid higher switching losses and achieve zero-voltage switching condition.

Figure 3.1: Transformer’s circuit with leakage inductance.

Where 푅푝 and 푅푠 are equivalent resistances for the primary and secondary windings, 푅푐 represents the equivalent core loss resistance, 퐿푝 and 퐿푠 represents the leakage inductances of primary and secondary sides, respectively. In a transformer, in addition to the main flux flowing through the core, there is also leakage flux that escapes from the core and flows through only one winding. This is results in an additional inductance that appears to be in series with the primary and secondary windings. This is called leakage inductance. The sum of leakage and magnetizing inductances represents self-inductance of the transformer. Fig. 3.2 shows the magnetizing flux flowing inside the core and the leakage flux escaping from the core.

Figure 3.2: Main and leakage fluxes distribution in a transformer.

The leakage inductance is a parasitic element in the transformer and energy stored in leakage inductance can be a cause of voltage spikes as depicted in Fig. 3.3 [26]. As a result, spikes increase the switching losses and decrease the efficiency of the transformer [35]. However, despite this fact, it can be used to achieve ZVS in the DAB converter. Furthermore, if the leakage inductance is high enough, it can be used instead of additional lumped inductor [9]. However, the leakage inductance should not be very high because it will introduce additional conduction losses and limit the amount of power that can be transferred [16]. At the same time, it should not be very low because in this case ZVS will not be achieved.

Figure 3.3: Voltage spikes in a transformer due to the leakage inductance.

3.1.1 Leakage inductance determination The leakage inductance can be found as follows: o calculated using the equations (49) for the planar transformer or (50) for the wire wound and integrated structures o obtained with the help of the FEM simulation tool (ANSYS Maxwell – Magnetostatic/Eddy Current solver). Maxwell computes the energy stored in the magnetic field. Then, using the equation (48), the leakage inductance can be calculated. o measured by means of the LCR meter as shown in Fig. 3.4. It can be done by shorting the secondary winding and connecting the LCR meter to the primary side.

Figure 3.4: Measurement of the leakage inductance by doing short-circuit test.

The leakage inductance referred to the primary, can be found from energy stored in the magnetic field [5]: 1 1 퐸푛푒푟푔푦 = ∫ 퐵 ∗ 퐻푑푉 = ∗ 퐿 ∗ 퐼2 , (50) 2 푉 2 푙푒푎푘 푝 where V is the total effective volume and 퐼푝 is the primary current. The leakage inductance of the planar transformer can be calculated as [5], [9]:

2 휇0푙푤푁푝 ∑ 푥 퐿푙푘_푝푙푎푛푎푟 = 2 ∗ ( + ∑ 푥Δ), (51) 푏푤푀 3 where, 휇0 is permeability of free space, 푙푤is the length of one turn, 푏푤 is the width of the turn, ∑ 푥 is the sum of all section layer heights (windings), ∑ 푥Δ is the sum of the distances between layers (insulation, air), M is the number of insulation layers.

The leakage inductance of the wire wound (퐿푙푘_푤𝑖푟푒) and integrated (퐿푙푘_integ) transformers can be found as [18], [26]:

4휋∗푀퐿푇∗푁2 ∑ a +푎 퐿 = 푝 (푐 + 푝푟𝑖푚푎푟푦 푠푒푐표푛푑푎푟푦) ∗ 10−9, (52) 푙푘_푤𝑖푟푒 푏 3

4휋∗푀퐿푇∗푁2 ∑ b +b 퐿 = 푝 (푐 + 1 2) ∗ 10−9, (53) 푙푘_integ 푎 3 where, 푎 is the winding lengths, 푏 is the winding build and 푐 is the insulation thickness.

(a) (b)

3.1.2 How to change the leakage inductance in the transformer In the DAB converter, depending on operating frequency, different values of leakage inductance might be required to achieve ZVS. As a result, sometimes there is a need to reduce or increase the leakage inductance. There are multiple methods by which it can be done in a transformer: a) change distance between primary and secondary windings b) interleave of primary and secondary windings. c) choose the core that encloses the windings d) insert magnetic shunt between primary and secondary windings [36] In case of using methods (a) and (b) for decreasing the leakage inductance, parasitic capacitance problem should be taken in consideration. It is known that the leakage inductance and parasitic capacitance have an inverse relationship, so that the closer the primary and secondary windings are to each other, the higher parasitic capacitance that occurs in transformer and vice versa [26] As a result, there is a trade-off between leakage inductance and parasitic capacitance.

3.1.3 Magnetizing inductance determination

As compared to the leakage inductance, the magnetizing inductance 퐿푚 is related to the flux that links to the core of the transformer. The magnetizing inductance is found as follows:

o calculated using equations (51) for planar transformers or (52) for wire wound and integrated structures o obtained using FEM simulation tool (ANSYS Maxwell – Magnetostatic/Eddy Current solver) o measured by LCR meter as shown in Fig. 3.6. It could be done by performing open circuit test of the transformer. The magnetizing inductance of the transformer can be found using the self-inductance equation [28]:

퐴푐 2 퐿 = 퐿푚 + 퐿푙푒푎푘 = 휇0휇푟 푁푝 , (54) 푙푒 where, 퐴푐is cross section area of the transformer and 푙푒 is magnetic path length Since the leakage inductance is usually negligibly small, we can use eq. (54) for magnetizing inductance calculation. For example, when leakage inductance is large enough to make some effect on the system, it can be subtracted from L to obtain more accurate value of 퐿푚. Alternatively, the magnetizing inductance can be calculated using following equation [29]:

퐴 ∗푁2 퐿 = 퐿 [휇퐻], (55) 푚 1000 where 퐴퐿 is an inductance factor – the value that obtained by manufacturer and usually given in the datasheet, N is the number of turns.

Figure 3.6: Measurement of the magnetizing inductance with LCR Meter.

3.1.4 How to change the magnetizing inductance From the equations (54) and (55), it can be noticed that there are number of ways to change the magnetizing inductance: o change the number of turns

o use another core with different cross-sectional area 퐴푒 and magnetic path length 푙푒

o use different core material and as a result with different relative permeability [휇푟] o change or introduce an air-gap to the core

3.2 Winding Capacitance Nowadays, when the transformer can operate at high frequencies, the problems associated with a parasitic capacitance are presented. The value of the capacitance depends on the windings arrangements, thickness of isolation and the relative permittivity of the dielectric material. Moreover, the parasitic capacitance is reversely proportional to the leakage inductance – another parasitic element presented in the transformer. As a result, the main challenge for the designer is to establish a trade-off between both parasitic components. Fig. 3.7 shows the main capacitances existing in the transformer.

Figure 3.7: Equivalent circuit of the transformer with parasitic capacitance

All capacitances existing in the transformer can be divided on 3 categories:

o Self-capacitance of the primary side (퐶푝)

o Self-capacitance of the secondary side (퐶푠)

o Capacitance between windings (퐶푝푠)

퐶푃 and 퐶푠 are also called intra-winding capacitance and 퐶푝푠 is called inter-winding capacitance [41]. The capacitance between windings is the main concern in the transformer design because it distorts the current waveforms and induced current spikes which reduce the efficiency and reliability of the transformer [16], [26], [42]. Moreover, the height of the spike is independent form the load. Fig.3.8 shows the effect that parasitic capacitance makes on the current waveforms. As a result, this part of the thesis is mostly concentrated on finding the winding-to-windings capacitance. Although in this thesis, capacitances presented in the transformers are considered as parasitic elements, in resonant converters they can be used as resonant elements in order to reduce the size of the power stage [44].

Figure 3.8: Current spikes due to the parasitic capacitance in the transformer

3.2.1 Determination of the winding capacitance The winding-to-winding capacitance can be determined as follows: • using FEM simulation tool (ANSYS Maxwell - Electrostatic solver) • measuring the capacitance by shortening primary and secondary sides and connect them to the LCR meter as shown in Fig 3.9. • calculated using equations (56) and (57) The winding-to-winding capacitance for the wire wound transformer can be calculated as follows [45]:

4휀0휀푟푟푎푣푒푁푎푣푒푀퐿푇 퐶푝푠 = , (56) 푎−휋푟푎푣푒 where, 휀0 is a permittivity of free space, 휀푟 is a relative permittivity, 푟푎푣푒 is an average wire radius, 푁푎푣푒 is an average number of turns and 푎 is a distance between windings. Winding-to-winding capacitance for the planar transformer can be calculated as follows [37]: 푙∗휔 퐶푝푠 = 휀0휀푟 , (57) ℎΔ where, 푙 is the total length of the conductor, 휔 is the width of the conductor and ℎΔ is the distance between conductive plates. To obtain the winding-to-winding capacitance with FEM simulation method, Electrostatic Solver of ANSYS Maxwell can be used to compute static electric fields. After the simulation is finished, the resulting capacitance matrix has the following form [43]: 퐶 + 퐶 −퐶 퐶 = [ 푝 푝푠 푝푠 ] (58) −퐶푝푠 퐶푠+퐶푝푠

It can be seen that by using (55) we can extract all parasitic capacitances presented in the transformer.

Figure 3.9: Measurement of winding-to-winding capacitance using LCR meter

3.2.2 How to change the winding-to-winding capacitance As it was discussed above, the winding-to-winding capacitance can cause a huge current spike. In some cases, for example in LLC converter, it can be used to replace the lumped capacitor [44]. As a result, sometimes it is necessary to change the value of the parasitic capacitance in the transformer. It can be implemented using the following options [16]:

- increase the distance between primary and secondary windings - change the number of turns on each layer (in case of planar transformer) - change the number of intersections between primary and secondary windings - increase the wire insulation,

- use an insulation material with smaller permittivity 휺풓 3.3 Contact Resistance In addition to the ac and dc resistances, windings of the power transformers also suffer from what is called contact resistance. Contact resistance is the resistance between two connectors as shown in Fig.3.10.

Figure 3.10: The connection point between transformer and board which causes the

contact resistance.

The main problem with this type of resistance is that the area at the connection point is much smaller than the actual wire area, while there is a difference in area, the currents flowing through the wire and connection point are still the same. Therefore, it will cause additional winding losses in the transformer. The current density distribution and ohmic losses at the connection point and at the rest of the wire are shown in Figs. 3.11 (a)-(b). From these figures, it can be seen that both current density and ohmic losses are much higher at the connection point compared to the rest of the wire.

(a) (b) Figure 3.11: (a) Current density distribution in the winding and (b) ohmic losses distribution in

the winding.

Also, the contact resistance can be even higher when the connection between transformer and board is poor. An example of this situation is when a bolt that holds two connectors together is not adequately tightened. In this thesis, two out of three transformers might suffer from contact resistance issues: the wire wound and planar transformers. However, the challenges with planar transformer can be solved with integration of the transformer’s PCB to the converter’s PCB.

Consequently, it would also limit the flexibility of the power stage so that if the transformer doesn’t meet the requirements we would need to make a new board. The best way to decrease the contact resistance is to increase the contact area between two terminals. This idea was implemented in the integrated transformer where the secondary side was replaced with the copper sheet which has higher connection area as compared to the wire- wound transformer. As a result, current on the secondary side is distributed equally across the surface of the conductor. Conclusion In this chapter, detailed explanation of the leakage inductance, parasitic capacitance and contact resistance has been provided. All necessary equations, the number of ways that can help to change the values of parasitic elements have been discussed. Also, it was noticed that the parasitic capacitance and the leakage inductance are dependent on each other and decreasing of one parameter results in increasing of another.

4. Transformer loss The transformer loss is another is a major concern in high frequency power devices since they may reduce the total efficiency. As a result, the loss problem requires designers to be very careful when choosing magnetic cores and windings for the transformer. In an ideal transformer, the amount of transferred power is only limited by saturation flux of the magnetic core. On the contrary, in a real transformer, in addition to the flux limitation, transformer losses are introduced. The transformer loss can be divided into 2 categories: core loss and winding (copper) loss. The big advantage of high-frequency operation is the size reduction of the magnetic components. However, when the frequency increases, there are additional problems such as skin and proximity effect. In this chapter, detailed analysis of the core and winding losses is provided. The analysis includes the detailed description of how to calculate, measure and simulate the core and winding losses.

4.1 Winding loss When designing the power transformer, one of the main challenges is the calculation of the ac winding resistance because in practice it is very difficult to accurately determine the exact value of it. The main analytical methods for calculation of ac resistance are described in [47], [48], [50], [51]. In high frequency transformers, the winding losses become a significant part of the total loss and, as a result, should be explored thoroughly. It is known that with higher frequencies, conduction losses increase due to the eddy current effects, which results in a lower efficiency of

the transformer. When the current flows in the conductor, the alternating magnetic field existing in the winding induces eddy currents in the conductor, which creates a field that cancels the field created by the original current. The eddy current effect is shown in Fig 4.1. There are two type of eddy current effects existing in the transformer windings: • The skin effect • The proximity effect The skin effect is a tendency of the AC current to be distributed inside the conductor so that the current density near the surface is higher than one in the center. The value that represents the area of the conductor with high current concentration is called skin depth. Skin effect increases as frequency increases, and this causes higher winding losses in the transformer. The proximity effect has similar results as the skin effect, but it is created by the current that flows in an adjacent conductor. The current in adjacent conductor creates a time-varying field and induce the circulating Eddy currents inside the second conductor. As a result, in the second conductor, the Eddy currents cause the current density to be distorted and concentrated mostly in one side of the conductor.

Figure 4.1: Eddy currents distribution in the conductor

(a) (b) Figure 4.2: Proximity effect in the conductors. The currents flow (a) in the opposite and

(b) in the same direction.

The skin and proximity effects make the current density in the conductor uniform, which increases the winding resistance in the transformer. The skin effect or the relationship between ac and dc resistances can be found as follows [16], [46], [47]: 푅 휉 sinh 휉+sin 휉 푎푐 = . (58) 푅푑푐 2 cosh 휉−cos 휉 Where, 푑 휉 = 푠푡푟푎푛푑 , (59) 푤𝑖푟푒 √2∗훿 ℎ 휉 = , (60) 푝푙푎푛푎푟 훿 where 훿 is skin depth, h the thickness of the conductor’s trace and d is a diameter of the round conductor. The best practice to reduce the skin effect is to use litz wire where each strand will have a diameter, which is less than or equal to the skin depth. Fig. 4.3 shows the current distribution in solid and litz wires at high frequencies. Proximity effect (퐺) of an infinite foil conductor can be calculated in the following [46]: sinh 휉+sin 휉 퐺 = 휉 . (61) cosh 휉−cos 휉 Based on the equations (56) and (59), the expression of the ac resistance of m-th layer can be found as [39], [46]: 휉 sinh 휉+sin 휉 sinh 휉+sin 휉 푅 = 푅 ∗ [ + (2푚 − 1)2 ∗ ], (62) 푎푐−푚 푑푐−푚 2 cosh 휉−cos 휉 cosh 휉−cos 휉 where m is the number of winding layers of wire wound transformer. In the case of a planar transformer m represents the following the ratio [16]: 퐹(ℎ) 푚 = , (63) 퐹(ℎ)−퐹(0) where F(h) and F(0) are MMFs at the limits of a layer.

The DC resistance of m-th layer (푅푑푐−푚) can be found as [37]: 푙 푅 = 휌 , (64) 푑푐−푚 푤푡 where 휌 is the resistivity of the conductor, l is the length of m-th layer, w is the width of the trace and t is the thickness of the trace. For the litz wires, ac resistance equation has the following form [33]:

2 2 √2∗휌∗푁푝 ber(휉)∗beip( 휉)−bei(휉)∗berp( 휉) 휋 푁0푃푓 2 24 푅푎푐 = [ 2 2 − ∗ (16푚 − 1 + 2) ∗ 휋∗훿∗푁0∗푑푠푡푟푎푛푑 beip ( 휉)+berp ( 휉) 24 휋 ber ( 휉) berp( 휉)−푏푒𝑖 ( 휉)∗beip( 휉) 2 2 ], (65) bei2( 휉)+ber2( 휉) where 푑푠푡푟푎푛푑 is the diameter of the strand, 푁0 is the number of strands in the wire, bei and ber are the Kelvin functions.

(a) (b) Figure 4.3: Current distributions in the wire: (a) solid wire and (b) litz wire.

In (62), the first and second parts represent the skin and proximity effects respectively. It is also known that in high frequencies, the proximity effect can be much higher than a skin effect, depending on the number of layers [1]. To decrease the influence of the proximity effect on the winding losses, interleaving windings should be used as shown in Fig 4.4. From (62), it can be seen that proximity effect increases with increasing of the number of layers in the transformer. Therefore, a good way to reduce the influence of the proximity effect is to decrease the number of layers by placing the winding in the narrow long core as shown in Fig. 4.5.

(a) (b) Figure 4.4: MMF distributions in the windings: (a) non-interleaving arrangement and (b)

interleaving arrangement.

Equations (62), and (64) use the approach where a round conductor is replaced with square conductor with the same area. Next, all square conductors in a layer are collected together to create the conductor foil winding which makes it convenient for further analytical calculations [46], [47]. Fig. 4.5 shows the complete procedure. At the same time, to make sure DC resistance of the original winding the same as that of the foil conductor, the porosity factor (휂) was introduced [46], [48]. The porosity factor can be found as follows [46]: 푁푎 휂 = , (66) 푏 where N is the number of square conductors in the layer, a is the width of each conductor and b is the width of the foil conductor which basically equals to the width of the core window.

Figure 4.5: Conversion of the round conductors’ layer to the foil conductor.

(a) //(b) Figure 4.6: Winding arrangements of the transformer: (a) one layer and (b) multilayer.

The porosity factor is used in skin depth calculation and consequently in 휉 [46], [47]: 1 훿 = , (67) √휋휇휂휎푓 where 휇 is the permeability, 휎 is the conductivity and f is the operating frequency. In fact, the porosity factor can provide a good approximation in case when conductors are located close to each other [49]. To summarize, winding losses can be decreased using the following procedures: • Use the litz wires instead of solid ones in case of the wire wound transformer and PCBs/ copper sheets with the thickness of trace not exceeding the value of 2 ∗ 훿 (where 훿 is the skin depth) in case of the planar and integrated structures. • Decrease the number of winding layers. To do that, use a long narrow magnetic core. • Use interleaving winding arrangements. It helps to reduce the proximity effect.

4.1.1 Determination of the winding resistance There are the number of ways to obtain transformer’s winding resistance:

• measure using LCR meter • obtain using FEM simulation tool [37], [51], [52] (in the low frequencies the value obtained represents DC resistance and the value obtained in the high frequencies represents the AC resistance) • calculate using equation (5) Equation (60) is based on the assumption that the field is one-dimensional and leakage flux run parallel to the conductive layers [46], [47]. Therefore, it is highly recommended to use FEM tool for getting more accurate values of the winding resistance.

4.2 Core loss It is known that transformers as power transferring devices are subjected to the loss inside the magnetic core. It happens when some amount of power transferred from the primary to the secondary windings is dissipated as heat or noise which cause the lower efficiency of the power device. This is what is called core loss. In the literature, there are the number of methods proposed that help to determine the core loss: • Hysteresis model. This model consists of differential equation that describes the behavior of ferromagnetic material. • Loss separation approach [53]. It divides the core losses in three sub-categories: hysteresis loss, eddy current loss and excess eddy-current loss. The main disadvantage of this approach is huge calculation error which varies between 200% - 2000% [54]. • Empirical approach [55], [56]. The first two methods require a lot of computations and measurements and hence they are time consuming. In the contrary, the empirical method is based on the Steinmetz equation [55], the simple equation which requires only parameters provided by manufacturers. All those advantages make the Steinmetz equation widely used among designers. The Steinmetz equation represents the core loss density in various units such as watts per pound, watts per kilogram, milliwatts per gram, and milliwatts per cubic centimeter [16], [26]: 훼 훽 푃푣 = 퐾 ∗ 푓 ∗ 퐵 , (68) where f is an operating frequency of the transformer, B is a flux density, and K, 훼 and 훽 are coefficients which are provided by manufacturers directly or can be derived from the core loss/flux density graphs which can also be found in manufacturer’s datasheets [28], [29]. Since 푃푣 represents the core loss density in various units as it was explained above, the value of the core loss in watts can be obtained by multiplying 푃푣 with mass density.

From the equation (24), it can be seen that frequency is inversely proportional with the flux density and coefficient 훽 is usually higher than 훼, core loss will decrease with increasing frequency. At the same time, high frequency causes winding losses to increase. It means that there is a tradeoff between core and winding losses. The main problem related with the equation (68) is that it can only be used for the transformer that operates with the sinusoidal waveforms [53]. To calculate the core loss for non- sinusoidal waveforms, several methods are presented in this section.

4.2.1 Modified Steinmetz equation (MSE). Since the original Steinmetz equation was originally designed for sinusoidal excitation, the method which works for non-sinusoidal cases was introduced in [54]. It assumes that time-varying 푑퐵(푡) magnetic induction ( ) is directly related to the core loss [56]. Therefore, the frequency 푑푡 parameter f in the original equation (68) was replaced by equivalent frequency 푓푒푞:

2 푇 푑퐵(푡) 2 푓 = ∫ ( ) 푑푡, (69) 푒푞 ∆퐵2휋2 0 푑푡 the loss equation in this case can be rewritten as:

훼−1 훽 푃푣 = (퐾 ∗ 푓푒푞 ∗ 퐵푚)푓푟 , (70) where, 푓푟 is fundamental frequency of the non-sinusoidal waveforms. Experiments show that, in case of non-sinusoidal waveforms, proposed method has better accuracy as compared to original Steinmetz equation especially in the cases when duty cycle becomes higher than 75%.

4.2.2 Generalized Steinmetz equation (GSE) It has been introduced in [57] and in addition to the MSE it considers the instantaneous value of the flux density B(t). In this case, core loss can be found as follows [56], [57]:

1 푇 푑퐵(푡) 훼 푃 = ∫ 푘 | | |퐵(푡)|훽−훼푑푡 , (71) 푣 푇 0 1 푑푡 퐾 푘1 = 2휋 , (772) 훼−1 | |훼| |훽−훼 (2휋) ∫0 푐표푠휃 푠𝑖푛휃 푑휃 where 휃 is the phase angle of the sinusoidal waveforms and coefficients K, 훼 and 훽 are determined the same way as for the original Steinmetz equation. Experiment with triangle waveform with variable duty cycle shows that GSE method is more accurate at duty cycle 50% while MSE works better at duty cycle of 95%. Next experiment was performed with the voltage source as the sum of two sinusoids. This experiment shows that MSE can err by 57 % where GSE gives an error of only 5%. The problem related with GSE approach is that it is based on the hypothesis about instantaneous power dissipation and does not consider minor hysteresis loops.

4.2.3 Improved GSE method (IGSE) The problem with the previous method was that GSE assumes that core loss is only 푑퐵(푡) dependent of instantaneous flux density and time-varying magnetic induction ( ) which is not 푑푡 valid when flux waveform starts containing minor hysteresis loops [60], [61]. As a result, the reaction of a material to various magnetic variation will vary depending on its history. [61] In the improved general Steinmetz equation method [60], the instantaneous flux density was replaced by peak-to-peak value:

1 푇 푑퐵(푡) 훼 푃 = ∫ 푘 | | |∆퐵|훽−훼푑푡 (73) 푣 푇 0 1 푑푡 퐾 푘1 = 2휋 (74) 훼−1 | |훼 훽−훼 (2휋) ∫0 푐표푠휃 2 푑휃 where K, 훼 and 훽 are the core loss coefficients that can be obtained from manufacturer’s datasheet. Experimental results show that IGSE almost match to the experimental data obtained from flux waveforms composed of two harmonically related sinusoids. The main limitation for IGSE method is that there is infinite number of possible non-sinusoidal waveforms which make it impossible to test IGSE for all of them.

4.2.4 Natural Steinmetz equation (NSE) NSE was introduced in [58], two years after GSE approach was proposed, and basically looks similar to it:

∆퐵 훽−훼 푘 푇 푑퐵(푡) 훼 푃 = ( ) 푁 ∫ | | 푑푡 (75) 푣 2 푇 0 푑푡 퐾 푘푁 = 2휋 (76) 훼−1 | |훼 (2휋) ∫0 푐표푠휃 푑휃 looking at the equations (71) and (72) it can be noted that the only difference between them and (75), (76) is the location of the elements.

4.2.5 Waveform coefficient Steinmetz equation (WCSE) Waveform coefficient Steinmetz equation approach was described in [62]. It is based on the method that is trying to correlate sinusoidal waveforms with non-sinusoidal one with the same peak magnetic induction by calculating the area of the flux waveforms. The magnetic core loss using the correction factor FWC can be found as: 훼 훽 푃푣 = 퐹푊퐶 ∗ 퐾 ∗ 푓 ∗ 퐵 (77) where, 푊 1/2 퐹푊퐶 = 푠푞 = (78) 푊푠𝑖푛 2/휋 4 푇/4 4퐵∗푡 1 푊 = ∫ ( ∗ 푑푡) = (79) 푠푞 푇∗퐵 0 푇 2

1 푇/2 2 푊 = ∫ (퐵 ∗ sin 휔푡 ∗ 푑푡) = (80) 푠𝑖푛 푇∗퐵 0 휋 where 푊푠푞 and 푊푠𝑖푛 are the waveform factors of triangular and sinusoidal flux waveforms respectively. Fig.4.7. (a) shows the sinusoidal and square voltage waveforms and Fig 4.7. (b) shows corresponding triangular and sinusoidal flux waveforms. From the equation (77), it can be seen that the correction factor is applied to the original Steinmetz equation to obtain the expression for any non-sinusoidal waveforms. The study in [56] explores all the core loss calculation methods described in this section and found that the use of original Steinmetz equation for the transformers which are excited with the square voltage waveforms gives the 45% error. The study also shows that the best core loss calculation methods are the MSE and the IGSE (error less than 14 %). However, IGSE copes better with a wider variety of voltage waveforms [55]. Moreover, it is known that in practice, IGSE approach is the most widely used among the designers for the best core loss determination. The main reason is that it requires the same data as needed for the original Steinmetz equation. The equation (73) can be rewritten as follows [16]:

2훼−1 훼 훽 훽−훼+1 푃푣 = 2 ∗ 퐾 ∗ 푓 ∗ 퐵푠푞푚 ∗ 퐷 (81) where D is the duty cycle, 퐵푠푞푚 is the peak flux density when the square waveform is with 50% duty cycle:

푉∗푇 퐵푠푞푚 = (82) 4푁∗퐴푒 where N is the number of turns, 퐴푒 is the cross-section area of the core, T is the period of the waveform.

(a) (b) Figure 4.7: Square and sinusoidal waveforms of: (a) voltage and (b) flux.

There are several ways to decrease the core loss: • Increase the number of turns. It will decrease the maximum flux density in the transformer, but it is also increase the winding loss

• Using several cores stacked together, this method can also decrease the maximum flux density and hence increase the winding loss because of additional length of the winding.

4.2.6 Determination of the core loss Magnetic core loss of the transformer can be determined as follows: • measure [11], [63], [64] using power converter and digital oscilloscope as shown in Fig. 4.8. By running the open circuit test at full power, the temperature rise of the core is measured and then, based on the temperature’s value, the core loss can be calculated. • obtain using FEM simulation. • calculate using equation (68) for sinusoidal and (81) for square voltage waveforms

Figure 4.8: Electrical scheme to measure the core loss.

Conclusion In this chapter, detailed information about core and winding losses has been provided. Different methods of winding and core loss determination have been discussed and optimal solution for each type of loss has been found. Moreover, the chapter also describes the ways of measuring core and winding losses.

5. Simulation and Experimental Results To verify all information contained in chapters 2 – 4, experiments with the physical transformers are performed. In this chapter, three transformer structures: wire wound, integrated and planar are implemented. It includes the transformer design, simulation and measurements of various parameters such as the leakage inductance, winding resistance, parasitic capacitance, contact resistance, temperature rise and core loss.

5.1 Transformers Design Before starting the design of the transformer, the input data should be determined. The input data basically depends on the power capabilities of the output device which is connected to the transformer’s output. The operating frequency of the transformer depends on the input voltage source. In case of a power converters, the operating frequency is usually equal to the switching frequency of the power device. As it was mentioned in the chapter two, one of the best magnetic materials which can deal with high frequencies is the power ferrite. To bring to the minimum the winding loss, it was decided to use 8:1 turns ratio. The initial data for the transformers is given in the table II.

TABLE II. INITIAL DATA OF THE POWER TRANSFORMERS

Input voltage, 푉𝑖푛 400V

Output voltage, 푉표푢푡 48V

Output current, 퐼표푢푡 41.6A Frequency, 푓 200 kHz Efficiency, 푛 0.98

Flux density, 퐵푚 0.2 Core material Ferrite Core configuration EE, Planar E, UU cores Turns ratio of the transformer 8:1

The design process is based on the equations (18) – (47) from the chapter 2. The main challenge at this step is to choose the appropriate core size. Choice of core size is vital as an undersized core can cause a high temperature rise and fast saturation of the magnetic material as the current increases. Conversely, choosing a large core can cause the system to be cumbersome, which is not desired since the main advantage of the high-frequency operation is the size reduction of all power components. After the design is done, three transformer cores - 0P46527EC for the wire wound, 0R49938EC for the planar and U46/40/28 for the integrated structures are chosen from the manufacturer datasheet [28], [29]. The core’s data are given in the tables III, IV and V.

TABLE III. TRANSFORMER’S CORES DATA Core name 0P46527EC 0R49938EC U46/40/28

4 Area product, 퐴푝, 푐푚 25 46 40 Mean length turn, MLT, cm 12 20.4 11

2 Cross-section area, 퐴푐, 푐푚 5.4 5.44 3.92

2 Window area, 푊푎, 푐푚 5.37 9.57 9.18

2 Surface area, 퐴푡, 푐푚 156.74 190 142.68 Magnetic path length, 푀푃퐿, cm 14.7 14.8 18.2

Core weight, 푊푡푓푒, 푔푟푎푚푠 410 400 364

The planar transformer consists of a planar core 0R49938EC [28] from Magnetics Inc. and two printed circuit boards (primary and secondary) [65], [66]. The PCB windings are shown in the Fig. 5.1. The wire wound transformer consists of the EE core 0P46527EC [1], primary winding has AWG 12 and secondary winding has AWG 8 [32]. For both windings, type 2 litz wire was used to reduce the skin effect [67]. The integrated transformer has AWG 12 for the primary side and 1 mm thick copper sheet for the secondary side. The cores that were selected have an area product that is higher than calculated one due to the temperature rise issues.

(a) (b) Figure 5.1: (a) Primary and (b) secondary PCB windings.

5.2 Simulation The transformers were built and simulated using ANSYS Maxwell v.17 FEM tool [68]. Such parameters of the power transformers as the leakage and the magnetizing inductances, the winding-to-winding capacitance, the winding resistance and the core loss are found and analyzed. Moreover, the image plots of the current, core loss and flux densities are obtained to observe transformer’s behavior under the assigned conditions. For the most part of the simulation, the eddy-current solver has been used [70], [71]. The only exception is the simulation of the winding- to-winding capacitance where the electrostatic solver has been used [69]. Based on the data given in the previous section, a 3D Finite Element Method (FEM) transformer models were built using ANSYS Maxwell simulation software. Simulations were performed under full load [10], using the frequency sweep from 150kHz to 200kHz in Eddy Current solver. Fig. 5.2 shows the 3D models of planar, wire-wound and integrated transformers, respectively. In all models, the primary side and secondary are shorted to provide a conduction path for current to flow. This condition is a requirement of the simulation software that was used.

(a) (b) (c) Figure 5.2: (a) Planar, (b) wire-wound and (c) integrated transformers 3D models

After the simulation is done, the leakage inductance is found by substituting the energy stored in the magnetic field using the following equation [37], [69]: 2∗푒푛푒푟푔푦 퐿푙푒푎푘푎푔푒 = 2 (83) 퐼푝 The magnetizing inductance was found in solution tab after the simulation by choosing L, R option in “type” drop down window. Fig. 5.3 demonstrates the image plots of the magnetic flux for planar, wire wound and integrated transformers respectively. They were obtained to demonstrate that cores are not saturated under full load condition.

(a) (b)

(c)

Figure 5.3: Flux density distribution. (a) Planar, (b) wire-wound and (c) integrated

transformers.

It can be seen, that maximum flux in the cores under 200 kHz frequency are 0.012 T, 0.02 T and 0.015T for planar, wire wound and integrated transformers respectively that is less that saturation flux density of the ferrite core which is equal to 0.47 T.

TABLE IV. LOSS COEFFICIENTS Material K 훼 훽

Ferrite R 3.53 1.42 2.88

Ferrite P, Ferroxcube 3.2 1.46 2.75

Core loss of the transformer can also be computed using Maxwell. Before starting the simulation, Steinmetz coefficients for each particular core should be added to the material’s properties. Those coefficients are usually provided by manufacturers and shown in table VI [28]. Fig. 5.4 (a) – (c) show the core loss versus frequency for integrated, planar and wire wound transformers accordingly. From the figures, it becomes clear that core loss decreases with higher frequency. The main reason for that is because frequency and flux density are reversely

proportional to each other. Therefore, when frequency increases the flux density decreases and since the flux density usually has higher power value, it results the core loss to be decreased.

Planar Transformer. Core Loss vs Frequency Integrated Transformer. Core Loss vs Frequency 17 18 Calculation Calculation 17 16 Simulation Simulation Measurement 16 Measurement 15

W 14

s s

14 s s

o 13

13 e

r r

o 12

o C

C 12

11 11

10 10

9 9 150 155 160 165 170 175 180 185 190 195 200 150 155 160 165 170 175 180 185 190 195 200 Frequency, kHz Frequency, kHz (a) (b)

Wire Wound Transformer. Core Loss vs Frequency 16 Calculation 15 Simulation Measurement 14

W 13

s s

o 12

e r

o 11 C

8 150 155 160 165 170 175 180 185 190 195 200 Frequency, kHz (c) Figure 5.4: Core loss versus frequency. (a) Integrated, (b) planar and (c) wire wound

transformers

5.3 Experimental results In this section, the series of measurements using HIOKI IM 3536 LCR meter are performed to compare the measured values with those which are obtained by analytical expressions and FEM simulation. The physical wire wound, planar and integrated transformers are built and shown in Fig. 5.5 (a) 5.5 (b) and 5.5 (c) respectively.

(a) (b) (c) Figure 5.5: The physical transformers: (a) wire wound, (b) planar and (c) integrated.

Figure 5.6: The test set up: LCR meter and wire wound transformer.

Next, to show the real measurements of the inductance, AC and DC resistance and capacitance, physical transformers were built using the specifications obtained in the design section. Inductance, resistance and capacitance measurements were performed using an LCR meter. The leakage inductance was measured by doing short circuit test while magnetizing inductance was measured using the open circuit test as shown in Fig. 3.4 in chapter 3. Winding-to winding capacitance is measured by shorting primary and secondary sides and connected them to an LCR meter as shown in Fig. 5.6. The test setup for core loss measurement is shown in Fig. 5.7. Test setup consists of the high voltage bridge, oscilloscope Tektronix MSO 4034 and transformer. The converter is running at the full power and, after a while, when the core’s temperature is stabilized the temperature rise is measured. Next, using equation (49) the core loss of the power transformer can be calculated. The transformers’ waveforms are shown in the Fig. 5.8. The value of the contact resistance was also obtained for all transformers. The simulation of the resistance was implemented in two steps using ANSYS Maxwell software. First, the model of the winding was simulated without connectors to get the value of the wire’s DC resistance as shown in Fig. 5.9 (a). Second, as shown in Fig. 5.9 (b), the winding was modeled with connectors added to it to get the combined DC resistance of the wire and connectors. By subtracting the wire’s DC resistance from the combined one, the value of the contact resistance can be obtained. The measurement process of the contact resistance was also performed in two steps. The first part involves the measurement of the full DC resistance of the winding. The primary side was an open circuit, while the secondary side of the transformer was connected to the board and supplied with 10 A of current. Then, the wire DC resistance was measured using an LCR meter and is subtracted from the full resistance obtained in step 1. Following this procedure leads to obtaining the contact resistance between the board and transformer.

Figure 5.7: Test setup

(a) (b)

Figure 5.9: Secondary winding (a) with and (b) without connectors.

Table V represents the values of leakage and magnetizing inductances, parasitic capacitance, AC resistance, core loss and contact resistance which were obtained by 3 different methods: simulation, calculation analytical equations, measurements of the physical transformer with a LCR meter. From the tables, it can be seen, that almost all values the difference between analytical expression, simulation and real device’s measurement is in acceptable range. From the table, it can be noticed that all transformers have almost similar magnetizing and leakage inductances. The wire wound transformer has higher core loss in comparison with planar and integrated ones. As for the contact resistance, the integrated transformer is having the lowest value of resistance among all three devices.

TABLE V. COMPARISON OF THE RESULTS Wire wound Integrated Planar Parameter Simulation Analytical Measurements Simulation Analytical Measurements Simulation Analytical Measurements Magnetizing 530 휇퐻 550 휇퐻 520 휇퐻 418.6 휇퐻 422.4 휇퐻 400 휇퐻 614 휇퐻 627.84 휇퐻 630 휇퐻 inductance Leakage 13.4 휇퐻 13.57 휇퐻 13.1 휇퐻 14 휇퐻 11 휇퐻 13 휇퐻 12.39 휇퐻 12.37 휇퐻 12.24 휇퐻 inductance (high side) Inter- 5 pF 7 pF 4 pF 15 pF 11 pF 25 pF winding capacitance AC 0.027 훺 0.03 훺 0.02 훺 0.0128 훺 0.0115 훺 0.0124 훺 0.11 훺 0.116 훺 0.15 훺 resistance Core loss 8.57 W 7.94 W 9.3 W 10.3 W 9.6 W 10.1 W 9.84W 10.35 W 11.71 W

Contact 0.13 m훺 0.18 m훺 0.005 m훺 0.001 m훺 0.12 m훺 0.2 m훺 Resistance Temperature 29.6 C 30 C 33.4 C 35 C 27.9 C 31 C Rise

5.4 Comparison and Cost When choosing the transformer, one of the main factors along with the efficiency is the cost. In this section, the transformers are compared in terms of their cost. The table below shows the cost of each transformer.

TABLE VI. COMPARISON OF THE TRANSFORMER’S COST Transformer type Price of the core set Price of the winding Final price of the transformer Planar $ 6.84 $ 15.16 $ 22 Wire-wound $ 5 $ 7.94 $ 12.94 Integrated $ 9.56 $ 6.84 $ 16.14

It can be seen from the table, that the cheapest topology is the wire wound transformer. As for the integrated transformer, it is 4 dollars cheaper than the planar one. From the first glance, it might look like a small difference but when it come to the mass production, the difference of four dollars per transformers becomes significant. Figure 5.10 shows the comparison of all transformers in terms of their cost (1- high, 5 - low), size, complexity (repair, manufacturing), flexibility (full integration capability, ability to control the leakage inductance) and efficiency. In the “size” category, the wire and integrated both received 4 points while planar only received 2 since it has some design constrains which don’t allow to pick the smaller planar core for the given conditions. In “complexity” category, planar transformer also received lowest score because it involves more complicated manufacturing process as compared to other topologies. Integrated transformer received the highest score in the “flexibility” because of its integration capability and high leakage inductance’s regulation range. In the “efficiency” category, the integrated transformer also received the highest score because of the lack of the contact resistance. As a result, from the graph it can be seen that the integrated transformer demonstrates better results as compared to wire wound and planar.

Comparison Chart

Wire Wound Planar Integrated

Cost 5 4 3 2 Size Complexity 1 0

Flexibility Efficiency

Figure 5.10: Comparison chart.

Conclusion.

In this chapter, the simulations of three different transformer structures: wire wound, planar and integrated were performed. 3D models of both transformers were built and simulated under full load condition. Such parameters as leakage and magnetizing inductances, parasitic capacitance, core loss, winding and contact resistances were extracted. Next, three physical transformers were created, and all required parameters were measured. Then, simulation results were compared to analytical expressions and experimental data obtained from LCR meter and core loss test set up. Comparison showed that experimental, simulation and analytical results almost corresponded to each other.

6. Conclusion and future work In this thesis, the detailed analysis of the high-frequency transformers has been presented. It started with an introductory chapter which contains the short history of the transformer, section about dual active bridge converter and its operation principle. In the second chapter, the operational principle of the magnetics and review of various high-frequency transformers existing in the market have been described. The detailed analysis of parameters which effect the transformer efficiency has been discussed. A review of all parasitic elements presented in transformer was provided in the chapter 3. The review included all necessary equations which can be used to find those parasitic and Chapter four contains the review of several methods which can be used to determine the core loss of the transformer. At the end, the most optimal solution – IGSE method was selected to be later used in the experimental section. In the chapter five, three high-frequency structures – wire wound, planar and integrated transformers have been designed and tested. In addition to the physical transformer, 3D computer models were built and simulated using ANSYS Maxwell software package. Next, such important transformer parameters as the leakage and the magnetizing inductances, the winding-to-winding capacitance, core loss, the winding and contact resistances were derived using three different methods: FEM simulation, analytical expressions and measurements of physical devices. Then, the comparison between all three methods has been performed. Experiments showed that results in all three cases are about the same and prove that FEM software is very useful tool for transformer’s analysis and simulation. Using FEM tools might prevent inappropriate work the physical transformer, verify design accuracy and model its behavior under various condition. Finally, all transformers were compared between each other in terms of cost, complexity, flexibility and efficiency. The comparison shows that integrated transformer demonstrated the best results as compared to the other two structures. Future work will be concentrated mostly on: • finding the better way for optimization of the power transformers • exploring the impact of the transformer’s parasitic capacitance on the conducted EMI • testing different magnetic materials to find the best solution for high-frequency application

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Appendix close all clear all clc %Transfomer Design - Wire Transformer Vin = 400; %Input voltage [V]; Vo = 48; %Output voltage [V]; Io = 41.6; %Output current [A]; f = 200000; %Frequency [Hz]; n = 0.98; %Efficiency [%]; t = 50; %Teperature rise [C]; Bm = 0.2; % Flux density [T]; Core_material = 'Ferrite';% Core material: Ferrite; Core_shape = 'EE core'; ku = 0.15; %Window utilization factor; kf = 4; % Waveform coefficient 4.44 for a sin wave and 4.0 for a square wave; J=500; %Current density

%This part is for temperature rise D=0.5; Vin1 =400; Np1=8; Ac1 = 0.00054; fprintf('Initial data:\n - Input voltage = %i V\n - Output voltage = %i V',Vin,Vo); fprintf('\n - Output current = %i A\n - Frequency = %i Hz\n - Efficiency = %0.2f ',Io,f,n); fprintf('\n - Teperature rise = %i C \n - Flux density = %0.2f T\n - Current denisty = %i A/cm',t,Bm,J); fprintf('\n - Core material = %s \n - Core shape = %s',Core_material,Core_shape); fprintf('\n - Window utilization factor = %0.1f \n - Waveform coefficient = %i',ku,kf);

%Step 1. Calculate skin depth fprintf('\n\nStep 1. Calculate skin depth (e):') e = 6.62/sqrt(f); fprintf ('\n - Skin depth, e = %0.4f', e); Dawg=2*e; fprintf ('\n - Wire diameter, Dawg = %0.4f', Dawg); Aw=pi*Dawg^2/4; fprintf ('\n - Bare wire area in cm^2, Aw = %0.5f', Aw);

AWG_strand = 'AWG #26'; AwB_strand = 0.00129; Res_strand = 1339; fprintf('\n - AWG number for single strand = %s ', AWG_strand); fprintf('\n - Wire area in cm^2, AwB_strand = %.5f ', AwB_strand); fprintf('\n - Resistance, Res_strand = %.1f \n', Res_strand);

%Step 2. Calculate the transformer output power Po fprintf('Step 2. Calculate the transformer output power (Po):') Po = 2000; fprintf ('\n - Output power, Po = %i', Po);

%Step 3. Calculate the apparent power Pt: fprintf('\nStep 3. Calculate the apparent power (Pt):') Pt = Po+Po/n; fprintf('\n - Apparent power, Pt = %0.1f', Pt);

%Step 4. Calculate the area product Ap: fprintf('\nStep 4. Calculate the area product (Ap):') %Ap = ((Pt*10^4)/(kf*Bm*f*ku*kj))^x; Ap=(Pt*10^4)/(kf*Bm*f*ku*J); fprintf('\n - Area product, Ap = %0.3f', Ap);

%Step 5. Using Ap value obtained in step 4, select core from datasheet: fprintf('\nStep 5. Using Ap value obtained in step 4, select core from datasheet:') Core_name = 'Core E65/32/27-3C90'; Apn = 40; % Area product, cm^4 MLT = 9; % Mean length turn, cm

Ac = 5.4; % Iron area, cm^2 Wa = 5.37; % Window area, cm^2 MPL = 14.7; % Magnetic path length, cm Wtfe = 410; % Core weight, grams Vcore = 7.9e-5; % Core volume in m^3

Height=6.56; Length=6.5; Width=2.74;

End=Height*Length; Top=Length*Width; Side=Height*Width; At = 2*End+2*Top+2*Side; % Surface area, cm^2 fprintf('\n - Core name: %s', Core_name); fprintf('\n - Area product [cm^4], Apn = %.3f ', Apn); fprintf('\n - Mean length turn [cm], MLT = %.1f ', MLT); fprintf('\n - Iron area [cm^2], Ac = %.2f ', Ac); fprintf('\n - Window area [cm^2], Wa = %.2f ', Wa); fprintf('\n - Surface area [cm^2], At = %.1f ', At); fprintf('\n - Magnetic path length [cm], MPL = %.1f ', MPL); fprintf('\n - Core weight [grams], Wtfe = %.2f ', Wtfe); fprintf('\n - Core volume [m^3], Vcore = %.8f ', Vcore);

%Step 6. Calculate Np, the number of primary turns: fprintf('\nStep 6. Calculate the number of primary turns (Np):') %Np = (Vin*10^4)/(kf*Bm*f*Ac); Np=8; fprintf('\n - The number of primary turns, Np = %i ', round(Np));

%Step 7. Calculate the primary current Ip: fprintf('\nStep 7. Calculate the primary current (Ip):') Ip = Po/(Vin*n); fprintf('\n - Primary current, Ip = %0.3f', Ip);

%Step 8. Current density J: fprintf('\nStep 8. Current density (J):') %J = (kj)*(Apn^y); %J=(Pt*10^4)/(kf*Bm*f*ku*Apn); fprintf('\n - Current density, J = %0.2f', J);

%Step 9. Calculate the bare wire size Aw(B) for the primary: fprintf('\nStep 9. Calculate the bare wire size Aw(B) for the primary:') AwB = (Ip)/J; %AwB=Ip*sqrt(Dmax)/J; Where Dmax is a duty cycle fprintf('\n - Bare wire size, AwB = %0.3f', AwB);

%Step 10. Select a wire size from the wire table. Remember: If the wire % area is not within 10%, take the next smallest size. Also record micro-ohms per centimeter from % column 4. fprintf('\nStep 10. Select a wire size from the wire table:') AWG = 'AWG #16'; AwBn = 0.0331; Res = 52.11; Snp=AwBn/AwB_strand; %New_res=Res_strand/Snp; That value basically equals to #12 AWG res from %table; fprintf('\n - AWG of primary wire = %s ', AWG); fprintf('\n - Bare wire size in cm^2, AwBn = %.3f ', AwBn); fprintf('\n - Resistance, Res = %.3f ', Res); fprintf('\n - Required number of primary strands, Snp = %.3f ', Snp);

%Step 11. Calculate the primary winding resistance (Rp): fprintf('\nStep 11. Calculate the primary winding resistance (Rp):'); Rp = MLT*Np*Res*(10^-6); fprintf('\n - Primary winding resistance, Rp = %f', Rp);

%Step 12. Calculate the primary copper loss Pp:

fprintf('\nStep 12. Calculate the primary copper loss (Pp):'); Pp = (Ip^2)*Rp; fprintf('\n - Primary copper loss, Pp = %0.3f',Pp);

%Step 13. Calculate the number of secondary turns (Ns): fprintf('\nStep 13. Calculate the number of secondary turns (Ns):'); Ns = (Np*Vo)/Vin; fprintf('\n - Number of secondary turns, Ns = %i', round(Ns,0));

%Step 14. Calculate the bare wire size AwBs for the secondary: fprintf('\nStep 14. Calculate the bare wire size AwBs for the secondary:'); AwBs = (Io)/J; fprintf('\n - Bare wire size for the secondary, AwBs = %0.3f', AwBs);

%Step 15. Select a wire size from the wire table. If the wire area is not % within 10%, take the next smallest size. Also record micro-ohms per centimeter from column 4. fprintf('\nStep 15. Select a wire size for secondary from the wire table:'); AWG = 'AWG #8'; AwBsn = 0.0837; Res2 = 20.6; Ssp=AwBsn/AwB_strand; fprintf('\n - AWG of secondary wire = %s ', AWG); fprintf('\n - Bare wire size in cm^2, AwBn = %.3f ', AwBsn); fprintf('\n - Resistance, Res2 = %.3f ', Res2); fprintf('\n - Required number of secondary strands, Ssp = %.3f ', Ssp);

%Step 16. Calculate the secondary winding resistance (Rs): fprintf('\nStep 16. Calculate the secondary winding resistance (Rs):'); Rs = (MLT)*Ns*Res2*(10^-6); fprintf('\n - Secondary winding resistance, Rs = %f', Rs);

%Step 17. Calculate the secondary copper loss Ps: fprintf('\nStep 17. Calculate the secondary copper loss (Ps):'); Ps = (Io^2)*Rs; fprintf('\n - Secondary copper loss, Ps = %0.3f', Ps);

%Step 18. Calculate the transformer total copper loss: fprintf('\nStep 18. Calculate the transformer total copper loss (Pcu):'); Pcu = Pp + Ps; fprintf('\n - Transformer total copper loss, Pcu = %0.3f', Pcu);

%Step 19. Calculate the transformer regulation, Œ±. fprintf('\nStep 19. Calculate the transformer regulation, (Œ±):'); alpha=Pcu*100/Po; fprintf('\n - Transformer regulation, Œ± = %0.3f', alpha);

%Step 20. Calculate the milliwatts per gram, mW/g: fprintf('\nStep 20. Calculate core loss in milliwatts per gram (mW/g) and in watts per cubic meter W/m^3:'); Bsqr = (Vin1*1/f)/(4*Np1*Ac1); B1 = Vin1/(4*Ac1*f*Np); %Core loss coefficients alpha = 1.46; beta = 2.75; Kfe = 3.2;

Wm3_1 = 2^(2*alpha-1)*Kfe*(f^alpha)*(B_third_equation^beta)*D^(beta-alpha+1); fprintf('\n - mW/g = %0.3f', mWg); fprintf('\n - W/m^3 = %0.3f', Wm3); fprintf('\n - W/m^3 (B1) = %0.3f', Wm3_1);

%Step 21. Calculate the core losses Pfe: fprintf('\nStep 21. Calculate the core losses (Pfe):'); Pfe = mWg*Wtfe*10^(-3); Pfe2 = Wm3*Vcore; Pfe3 = Wm3_1*Vcore; fprintf('\n - Pfen = %0.5f', Pfe); fprintf('\n - Pfen (Wm3) = %0.5f', Pfe2); fprintf('\n - Pfen (Wm3_1) = %0.5f', Pfe3);

%Step 22. Calculate the total losses (Psum): fprintf('\nStep 22. Calculate the total losses (Psum):'); Psum = Pcu + Pfe3; fprintf('\n - Total losses, Psum = %0.3f', Psum);

%Step 23. Calculate psi, the watts per unit area: fprintf('\nStep 23. Calculate psi, the watts per unit area:'); psi = Psum/At; fprintf('\n - psi [watts/cm^2] = %0.4f', psi);

%Step 24. Calculate total window utilization, Kut: fprintf('\nStep 24. Calculate total window utilization, (Kut):'); Kup=Np*Snp*AwB_strand/Wa; Kus=Ns*Ssp*AwB_strand/Wa; Kut=Kup+Kus; fprintf('\n - Total window utilization Kut = %0.4f\n', Kut);

fprintf('\n - B_original = %0.4f \n', B1); fprintf('\n - Bsqr = %0.4f \n', Bsqr); fprintf('\n - B_third_equation = %0.4f \n', B_third_equation);

% Temperture rise deltaT = (Pfe3*1000/At)^0.833; fprintf('\n - Temperature rise = %0.4f \n', deltaT);

% Integrated Transfomer Design Vin = 400; %Input voltage [V]; Vo = 48; %Output voltage [V]; Io = 41.6; %Output current [A]; f = 200000; %Frequency [Hz]; n = 0.98; %Efficiency [%]; t = 50; %Teperature rise [C]; Bm = 0.3; % Flux density [T]; Core_material = 'Ferrite'; Core_shape = 'UU core'; ku = 0.4; %Window utilization factor; kf = 4; % Waveform coefficient 4.44 for a sin wave and 4.0 for a square wave; J_p = 500; % Primary current density J_s = 1000; % Secondary current density

%Flux density D=0.5; Vin1 = 400; Np1=8; Ac1 = 0.000392; fprintf('Initial data:\n - Input voltage = %i V\n - Output voltage = %i V',Vin,Vo); fprintf('\n - Output current = %0.1f A\n - Frequency = %i Hz\n - Efficiency = %0.2f ',Io,f,n); fprintf('\n - Teperature rise = %i C \n - Flux density = %0.2f T\n - Current denisty = %i A/cm',t,Bm,J_p); fprintf('\n - Core material = %s \n - Core shape = %s',Core_material,Core_shape); fprintf('\n - Window utilization factor = %0.1f \n - Waveform coefficient = %i',ku,kf);

%Step 2. Calculate the transformer output power Po fprintf('\nStep 2. Calculate the transformer output power (Po):') %Po = Vo*Io; Po=2000; fprintf ('\n - Output power, Po = %i', Po);

%Step 3. Calculate the apparent power Pt: fprintf('\nStep 3. Calculate the apparent power (Pt):') Pt = Po+Po/n; fprintf('\n - Apparent power, Pt = %0.1f', Pt);

%Step 4. Calculate the area product Ap: fprintf('\nStep 4. Calculate the area product (Ap):') %Ap = ((Pt*10^4)/(kf*Bm*f*ku*kj))^x; Ap=(Pt*10^4)/(kf*Bm*f*ku*J_p); fprintf('\n - Area product, Ap = %0.3f', Ap);

%Step 5. Using Ap value obtained in step 4, select core from datasheet: fprintf('\nStep 5. Using Ap value obtained in step 4, select core from datasheet:') Core_name = 'Core U46/40/28'; Apn = 36; % Area product, cm^4 MLT = 9.5; % Mean length turn, cm Ac = 3.92; % Core cross sectional area, cm^2 Wa = 9.18; % Window area, cm^2 MPL = 18.2; % Magnetic path length, cm Wtfe = 364; % Core weight, grams Vcore = 7.9e-5; % Core volume in m^3 Height=3.95*2; Length=4.6; Width=2.8; End=Height*Length; Top=Length*Width; Side=Height*Width; At = 2*End+2*Top+2*Side; % Surface area, cm^2 fprintf('\n - Core name: %s', Core_name); fprintf('\n - Area product [cm^4], Apn = %.3f ', Apn); fprintf('\n - Mean length turn [cm], MLT = %.1f ', MLT); fprintf('\n - Iron area [cm^2], Ac = %.2f ', Ac); fprintf('\n - Window area [cm^2], Wa = %.2f ', Wa); fprintf('\n - Surface area [cm^2], At = %.1f ', At); fprintf('\n - Magnetic path length [cm], MPL = %.1f ', MPL); fprintf('\n - Core weight [grams], Wtfe = %.2f ', Wtfe);

%Step 7. Calculate the primary current Ip: fprintf('\nStep 7. Calculate the primary current (Ip):') Ip = Po/(Vin*n); fprintf('\n - Primary current, Ip = %0.3f', Ip);

%Step 8. Current density J: fprintf('\nStep 8. Current density (J):') %J = (kj)*(Apn^y); %J=(Pt*10^4)/(kf*Bm*f*ku*Apn); fprintf('\n - Current density of primary side, J_p = %0.2f', J_p); fprintf('\n - Current density of secondary side, J_s = %0.2f', J_s);

%Step 9. Calculate the bare wire size Aw(B) for the primary: fprintf('\nStep 9. Calculate the bare wire size Aw(B) for the primary:') AwB = (Ip)/J_p; %AwB=Ip*sqrt(Dmax)/J; Where Dmax is a duty cycle fprintf('\n - Bare wire size, AwB = %0.3f', AwB);

%Step 10. Select a wire size from the wire table. Remember: If the wire % area is not within 10%, take the next smallest size. Also record micro-ohms per centimeter from % column 4. fprintf('\nStep 10. Select a wire size from the wire table:') AWG = 'AWG #16'; AwBn = 0.0131; Res = 131.7; Snp=AwBn/AwB_strand; %New_res=Res_strand/Snp; That value basically equals to #12 AWG res from %table; fprintf('\n - AWG of primary wire = %s ', AWG); fprintf('\n - Bare wire size in cm^2, AwBn = %.4f ', AwBn); fprintf('\n - Resistance, Res = %.2f ', Res); fprintf('\n - Required number of primary strands, Snp = %.3f ', Snp);

%Step 12. Calculate the primary copper loss Pp: fprintf('\nStep 12. Calculate the primary copper loss (Pp):'); Pp = (Ip^2)*Rp; fprintf('\n - Primary copper loss, Pp = %0.3f',Pp);

%Step 14. Calculate the bare wire size AwBs for the secondary: fprintf('\nStep 14. Calculate the bare wire size AwBs for the secondary:'); AwBs = (Io)/J_s; fprintf('\n - Bare wire size for the secondary, AwBs = %0.3f', AwBs);

%Step 15. Select a wire size from the wire table (Table 6.1), column 2. If the wire area is not % within 10%, take the next smallest size. Also record micro-ohms per centimeter from column 4. fprintf('\nStep 15. Select a wire size for secondary from the wire table:'); AWG = 'AWG #11'; AwBsn = 0.0417; Res2 = 41.32; Ssp=AwBsn/AwB_strand; fprintf('\n - AWG of secondary wire = %s ', AWG); fprintf('\n - Bare wire size in cm^2, AwBn = %.3f ', AwBsn); fprintf('\n - Resistance, Res2 = %.3f ', Res2); fprintf('\n - Required number of secondary strands, Ssp = %.3f ', Ssp);

%Step 17. Calculate the secondary copper loss Ps:

fprintf('\nStep 17. Calculate the secondary copper loss (Ps):'); Ps = (Io^2)*Rs; fprintf('\n - Secondary copper loss, Ps = %0.3f', Ps);

%Core loss coefficients alpha = 1.46; beta = 2.75; Kfe = 3.2;

% alpha = 1.42; % beta = 2.88; % Kfe = 3.53;

% This flux density equation if from "Fundamental of Power Electronics" book ro = 1.724*10^(-6); lambda = D*(1/f)*Vin1; ku=0.18; Itotal = Ip+Io/8; B_third_equation = (10^8*ro*lambda^2*Itotal^2*MLT/(2*ku*Wa*Ac^3*MPL*beta*Kfe))^(1/(beta+2)); x = lambda/(2*Np*Ac1); mWg = 4.855*10^(-5)*(fâlpha)*(B1^beta); Wm3 = Kfe*(fâlpha)*(B1^beta); Wm3_1 = 2^(2*alpha-1)*Kfe*(fâlpha)*(B_third_equation^beta)*D^(beta-alpha+1); fprintf('\n - mW/g = %0.3f', mWg); fprintf('\n - W/m^3 = %0.3f', Wm3); fprintf('\n - W/m^3_1 = %0.3f', Wm3_1);

%Step 21. Calculate the core losses Pfe: fprintf('\nStep 21. Calculate the core losses (Pfe):'); Pfe = mWg*Wtfe*10^(-3); Pfe2 = Wm3*Vcore; Pfe3 = Wm3_1*Vcore; fprintf('\n - Pfen = %0.6f', Pfe); fprintf('\n - Pfen (Wm3) = %0.3f', Pfe2); fprintf('\n - Pfen (Wm3_1) = %0.5f', Pfe3);

%Step 22. Calculate the total losses (Psum): fprintf('\nStep 22. Calculate the total losses (Psum):'); Psum = Pcu + Pfe; fprintf('\n - Total losses, Psum = %0.3f', Psum);

%Step 23. Calculate psi, the watts per unit area: fprintf('\nStep 23. Calculate psi, the watts per unit area:'); psi = Psum/At; fprintf('\n - psi, [watts/cm^2] = %0.4f', psi);

%Step 24. Calculate total window utilization, Kut: fprintf('\nStep 24. Calculate total window utilization, (Kut):'); Kup=Np*Snp*AwB_strand/Wa; Kus=Ns*Ssp*AwB_strand/Wa; Kut=Kup+Kus;

fprintf('\n - Total window utilization, Kut = %0.4f\n', Kut); fprintf('\n - B1 = %0.4f \n', B1); fprintf('\n - Bsqr = %0.4f \n', Bsqr); fprintf('\n - B_third_equation = %0.4f \n', B_third_equation);

% Temperture rise deltaT = (Pfe3*1000/At)^0.833; fprintf('\n - Temperature rise = %0.4f \n', deltaT);

%Transfomer Design - Planar Transformer Vin = 400; %Input voltage [V]; Vo = 48; %Output voltage [V]; Io = 41.6; %Output current [A]; f = 200000; %Frequency [Hz]; n = 0.98; %Efficiency [%]; t = 50; %Teperature rise [C]; Bm = 0.3; % Saturation Flux density [T]; Core_material = 'Ferrite'; Core_shape = 'Planar EE core'; ku = 0.4; %Window utilization factor; kf = 4; % Waveform coefficient 4.44 for a sin wave and 4.0 for a square wave; J=1000; %Current density A/cm

%Flux density D=0.5; Vin1 = 400; Np1=8; %Ac1 = 5.40/10000; Ac1 = 0.00054; fprintf('Initial data:\n - Input voltage = %i V\n - Output voltage = %i V',Vin,Vo); fprintf('\n - Output current = %i A\n - Frequency = %i Hz\n - Efficiency = %0.2f ',Io,f,n); fprintf('\n - Teperature rise = %i C \n - Flux density = %0.2f T\n - Current denisty = %i A/cm',t,Bm,J); fprintf('\n - Core material = %s \n - Core shape = %s',Core_material,Core_shape); fprintf('\n - Window utilization factor = %0.1f \n - Waveform coefficient = %i',ku,kf);

%Step 2. Calculate the transformer output power Po fprintf('\nStep 2. Calculate the transformer output power (Po):') Po = Vo*Io; fprintf ('\n - Output power, Po = %i', Po);

%Step 3. Calculate the apparent power Pt: fprintf('\nStep 3. Calculate the apparent power (Pt):') Pt = Po+Po/n; fprintf('\n - Apparent power, Pt = %0.1f', Pt);

%Step 4. Calculate the area product Ap: fprintf('\nStep 4. Calculate the area product (Ap):') %Ap = ((Pt*10^4)/(kf*Bm*f*ku*kj))^x; Ap=(Pt*10^4)/(kf*Bm*f*ku*J);

fprintf('\n - Area product, Ap = %0.3f', Ap);

%Step 5. Select an area product Ap from the specified core section, and record the appropriate data: fprintf('\nStep 5. Using Ap value obtained in step 4, select core from datasheet:') Core_name = 'Core 0R49938EC'; Apn = 46; % Area product, cm^4 MLT = 20; % Mean length turn, cm Wa = 9.57; % Window area, cm^2 Ac = 5.44; % Core area, cm^2 MPL = 14.8; % Magnetic path length, cm Wtfe = 400; % Core weight, grams Vcore = 7.98e-5; % Core volume in m^3

Height=4.06; Length=10.2; Width=3.75; End=Height*Length; Top=Length*Width; Side=Height*Width; At = 2*(End)+2*Top+2*Side; % Surface area, cm^2 fprintf('\n - Core name: %s', Core_name); fprintf('\n - Area product [cm^4], Apn = %.3f ', Apn); fprintf('\n - Mean length turn [cm], MLT = %.1f ', MLT); fprintf('\n - Iron area [cm^2], Ac = %.2f ', Ac); fprintf('\n - Window area [cm^2], Wa = %.2f ', Wa); fprintf('\n - Surface area [cm^2], At = %.1f ', At); fprintf('\n - Magnetic path length [cm], MPL = %.1f ', MPL); fprintf('\n - Core weight [grams], Wtfe = %.2f ', Wtfe); fprintf('\n - Core volume [m^3], Vcore = %.8f ', Vcore);

%Step 7. Calculate the primary current Ip: fprintf('\nStep 7. Calculate the primary current (Ip):') Ip = Po/(Vin*n); fprintf('\n - Primary current, Ip = %0.3f', Ip);

%Step 8. Current density J: fprintf('\nStep 8. Current density (J):') %J = (kj)*(Apn^y); %J=(Pt*10^4)/(kf*Bm*f*ku*Apn); fprintf('\n - Current density, J = %0.2f', J);

%Step 10. Select a wire size from the wire table. Remember: If the wire % area is not within 10%, take the next smallest size. Also record micro-ohms per centimeter from % column 4. fprintf('\nStep 10. Select a wire size from the wire table:') AWG = 'AWG #20'; AwBn = 0.00518; Res = 333.1; Snp=AwBn/AwB_strand; Trace_length = AwBn/(3*0.00347); %New_res=Res_strand/Snp; That value basically equals to #12 AWG res from %table; fprintf('\n - AWG of primary wire = %s ', AWG); fprintf('\n - Length of the primary trace in cm = %0.3f ', Trace_length); fprintf('\n - Bare wire size in cm^2, AwBn = %.3f ', AwBn); fprintf('\n - Resistance, Res = %.3f ', Res);

fprintf('\n - Required number of primary strands, Snp = %.3f ', Snp); fprintf('\n - Trace length cm, Trace_length = %.3f ', Trace_length);

%Step 12. Calculate the primary copper loss Pp: fprintf('\nStep 12. Calculate the primary copper loss (Pp):'); Pp = (Ip^2)*Rp; fprintf('\n - Primary copper loss, Pp = %0.3f',Pp);

%Step 15. Select a wire size from the wire table. If the wire area is not % within 10%, take the next smallest size. Also record micro-ohms per centimeter from column 4. fprintf('\nStep 15. Select a wire size for secondary from the wire table:'); AWG = 'AWG #11'; AwBsn = 0.0417; Res2 = 41.32; Ssp=AwBsn/AwB_strand; Trace_length_s = AwBsn/e; fprintf('\n - AWG of secondary wire = %s ', AWG); fprintf('\n - Bare wire size in cm^2, AwBn = %.3f ', AwBsn); fprintf('\n - Length of the secondary trace in cm = %0.3f ', Trace_length_s); fprintf('\n - Resistance, Res2 = %.3f ', Res2); fprintf('\n - Required number of secondary strands, Ssp = %.3f ', Ssp);

%Step 17. Calculate the secondary copper loss Ps: fprintf('\nStep 17. Calculate the secondary copper loss (Ps):'); Ps = (Io^2)*Rs; fprintf('\n - Secondary copper loss, Ps = %0.3f', Ps);

%Core loss coefficients alpha = 1.46; beta = 2.75; Kfe = 3.2;

% alpha = 1.42; % beta = 2.88; % Kfe = 3.53;

% This flux density equation if from "Fundamental of Power Electronics" book ro = 1.724*10^(-6); lambda = D*(1/f)*Vin1; ku=0.15; Itotal = Ip+Io/8; B_third_equation = (10^8*ro*lambda^2*Itotal^2*MLT/(2*ku*Wa*Ac^3*MPL*beta*Kfe))^(1/(beta+2)); x = lambda/(2*Np*Ac1); mWg = 4.855*10^(-5)*(fâlpha)*(B1^beta); Wm3 = Kfe*(fâlpha)*(B1^beta); Wm3_1 = 2^(2*alpha-1)*Kfe*(fâlpha)*(B_third_equation^beta)*D^(beta-alpha+1); fprintf('\n - mW/g = %0.3f', mWg); fprintf('\n - W/m^3 = %0.3f', Wm3); fprintf('\n - W/m^3_1 = %0.3f', Wm3_1);

%Step 21. Calculate the core losses Pfe: fprintf('\nStep 21. Calculate the core losses (Pfe):'); Pfe = mWg*Wtfe*10^(-3); Pfe2 = Wm3*Vcore; Pfe3 = Wm3_1*Vcore; fprintf('\n - Pfen = %0.6f', Pfe); fprintf('\n - Pfen (Wm3) = %0.3f', Pfe2); fprintf('\n - Pfen (Wm3_1) = %0.5f', Pfe3);

%Step 22. Calculate the total losses (Psum): fprintf('\nStep 22. Calculate the total losses (Psum):'); Psum = Pcu + Pfe3; fprintf('\n - Total losses, Psum = %0.3f', Psum);

%Step 23. Calculate psi, the watts per unit area: fprintf('\nStep 23. Calculate psi, the watts per unit area:'); psi = Psum/At; fprintf('\n - psi [watts/cm^2] = %0.4f', psi);

fprintf('\n - B1 = %0.4f \n', B1); fprintf('\n - Bsqr = %0.4f \n', Bsqr); fprintf('\n - B_third_equation = %0.4f \n', B_third_equation);

% Temperature rise deltaT = (Pfe3*1000/At)^0.833; fprintf('\n - Temperature rise = %0.4f \n', deltaT);

% The leakage inductance equation Lleak1=400^2/(2000*2*pi*300000)*(1)*(pi/2)*(1-pi/(2*pi));

% Surface Area versus Total Power Loss Psum = logspace(-1,2,30); At_1 = zeros(30); At_2 = zeros(30); Psi_1 = 0.03; Psi_2 = 0.07; for i = 1:30 At_1(i)= Psum(i)/Psi_1; At_2(i)= Psum(i)/Psi_2; end

a = loglog(Psum,At_1,Psum,At_2,'--','LineWidth',2); legend({'20^{\circ}C Rise','50^{\circ}C Rise'},'Location','Northwest','FontSize',15); title('Surface Area versus Total Power Loss','FontSize',15); ylim([10,1000]) xlabel("Total Power Loss, Pœµ (W)",'FontSize',15) ylabel("Surface Area, At(cm^2)",'FontSize',15); set(gca,'fontsize',20); set(findall(gca, 'Type', 'Line'),'LineWidth',2); grid on;

% Leakage and magnetizing inductance calculation for integrated transformer

MLT=13; % mean length turn [centimeter] Np=8; % the number of primary turns a = 1.8; b1=0.4; b2 = 0.2; c=1.8; u0=4*pi*10^(-7); %Permeability of free space ur=2215; %Relative permeability Ae=392/1000000; %core's cross section area [meter^2] MPL=182/1000; %magnetic path length [meter] AL = 6600; %inductance factor [nH/T^2]

L_leakage=4*pi*MLT*Np^2/a*(c+(b1+b2)/3)*10^(-9); L_uH=L_leakage*10^6; fprintf("The leakage inductance, Lleak = %0.2f uH\n",L_uH);

Lm=u0*ur*Ae*Np^2/MPL; Lmu = Lm*10^6; Lm2 = Np^2*AL/1000; fprintf("The magnetizing inductance, Lm = %0.2f uH\n",Lmu); fprintf("The magnetizing inductance (using inductance factor), Lm = %0.2f uH\n",Lm2);

% Leakage and magnetizing inductances calculation for planar transformer u0=4*pi*10^(-7); %Permeability of free space ur=2300; %Relative permeability (ferrite R) lw=37.5/1000; % depth of the magnetic core [meter] bw=36/1000; % width of the core window [meter] kp=4; % number of turns per layers in the primary side Np=8; % number of primary layers hp=0.01041/100; %thickness of primary layer [meter] hdp=1.2/1000; %separation distance between primary layers Ae=540/1000000; %core cross section area [meter^2] le=148/1000; % magnetic path length [meter] Ns=1;%number of secondary turns hs=0.01041/100;%thickness of secondary layer hds=0; %separation distance between secondary layers Al = 9810; % inductance factor

%Magnetizing inducance %Lm=u0*ur*Ae*Np^2/le; %Lmu=Lm*10^6; Lm = Np^2*Al/1000; fprintf("The magnetizing inductance of the planar transformer, Lm = %f uH\n", Lm); % Leakage inductance % Lleak=u0*lw*kp^2*Np^2/(3*bw)*(((2*(hp+hdp)*Np-3*hdp+hdp/Np)+6*hdp)+((2*(hs+hds)*Ns- 3*hds+hds/Ns)+6*hds)); % Lleaku=Lleak*10^6; % fprintf("The leakage inductance of the planar transformer, Lleak = %f uH\n", Lleaku);

% Alternative way of leakage inductance calculation length_turn = 17/100; width_turn = 5/1000;

hdd = 18/1000; Lleak_alt = u0*length_turn*Np^2/(width_turn*2^2)*((2*hp+hs)/3+hdd); Lleak_alt_uH = Lleak_alt*10^6; fprintf("The leakage inductance of the planar transformer, Lleak = %f uH\n", Lleak_alt_uH);

% Leakage and magnetizing inductance calculation for wire wound transformer

MLT=8.1; % mean length turn [centimeter] Np=8; % the number of primary turns a1=3; a2=0.5; b=0.8; c=0.5; u0=4*pi*10^(-7); %Permeability of free space ur=1860; %Relative permeability Ae=540/1000000; %core's cross section area [meter^2] MPL=147/1000; %magnetic path length [meter] AL = 8600; %inductance factor [nH/T^2]

L_leakage=4*pi*MLT*Np^2/b*(c+(a2+a1)/3)*10^(-9); L_uH=L_leakage*10^6; fprintf("The leakage inductance, Lleak = %0.2f uH\n",L_uH);