Design of a High Frequency Transformer
for High Voltage Applications
Master’s thesis in Electric Power Engineering

Niharika Sharma
Vaghul Kramadhati Venkatagiri

DEPARTMENT OF ELECTRICAL ENGINEERING

CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden 2023
www.chalmers.se

www.chalmers.se




Master’s thesis 2023

Design of a High Frequency Transformer for High
Voltage Applications

Niharika Sharma & Vaghul Kramadhati Venkatagiri

Department of Electrical Engineering
Division of Electric Power Engineering

Chalmers University of Technology
Gothenburg, Sweden 2023



Design of a High Frequency Transformer for High Voltage Applications
Niharika Sharma & Vaghul Kramadhati Venkatagiri

© Niharika Sharma & Vaghul Kramadhati Venkatagiri, 2023.

Supervisor: Andreas Hjert, KraftPowercon Sweden AB
Examiner: Prof. Yuriy Serdyuk, Electrical Engineering

Master’s Thesis 2023
Department of Electrical Engineering
Division of Electric Power Engineering
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone +46 31 772 1000

Gothenburg, Sweden 2023

iii



Design of a High Frequency Transformer for High Voltage Applications
Niharika Sharma & Vaghul Kramadhati Venkatagiri
Department of Electrical Engineering
Chalmers University of Technology

Abstract
An Electrostatic Precipitator is an industrial device that filters exhaust gases from
process plants to release clean air into the atmosphere. For its functioning, a suit-
able high voltage transformer is required which operates under electrical stresses
induced by power electronic components. Existing transformers for this applica-
tion are bulky in size, expensive due to material requirement, and require complex
balancing circuitry to operate due to multiple cores. A "planar" transformer is there-
fore suggested as a suitable replacement. With the secondary consisting of multiple
PCBs with deposited traces layered around a single core, it is expected that the size
of the transformer can be reduced along with the system complexity and material
requirements.

This thesis introduces FEM (finite element method)-based simulation model for
designing the high-voltage high-frequency planar transformer. Implementation of
the model in COMSOL Multiphysics software is presented focusing on application
specific aspects such as AC effects at high frequencies, planar PCB design, insulation
standards, and the implementation of integrated rectification. In addition, the effect
of using dielectric barriers to mitigate high voltage stresses is evaluated based on
simulations of the electric fields in the transformer.

Furthermore, based on the results of the simulations, a prototype of the transformer
was assembled and its efficiency was evaluated. The measurements of core losses
were performed to identify limiting operational conditions. It was found that op-
erating the core close to the saturation magnetic flux density should be avoided
since this resulted in rapid heating and high steady-state temperatures in the tests.
The least losses were obtained by operating at nearly one-third of saturation flux
density. The efficiency of the transformer was calculated considering the measured
core loss, winding loss and rectification loss. The results of the performed simula-
tions and measurements indicate that a planar transformer can be considered as a
promising solution for replacement of traditional transformers used in electrostatic
precipitators.

Keywords: Planar transformer, high frequency, high voltage, PCB design, ESP

iv





List of Acronyms

Below is the list of acronyms that have been used throughout this thesis listed in
alphabetical order:

ESP Electrostatic Precipitator
FEM Finite Element Method
HV High Voltage
LV Low Voltage
OSE Original Steinmetz Equation
PA Polyamide
PCB Printed Circuit Board
PP Polypropylene
PVC Polyvinyl chloride
RF Radio Frequency
SiC Silicon Carbide
SRC Series Resonant Converter
ZCS Zero Current Switching
ZVS Zero Voltage Switching

vi





Contents

List of Acronyms vi

List of Figures xi

List of Tables xiii

1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Sustainability Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.7 Ethical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Principles of Transformer Design 5
2.0.1 Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.0.2 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.0.3 Lenz’s Law of Electromagnetic Induction . . . . . . . . . . . . 6
2.0.4 Transformer Efficiency . . . . . . . . . . . . . . . . . . . . . . 6
2.0.5 Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.0.6 Core Shape and Magnetizing Inductance . . . . . . . . . . . . 8

2.1 Series Resonant Converter . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Choice of core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Factors affecting the selection of core material . . . . . . . . . 11
2.3 Core Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Measurement of Core Loss . . . . . . . . . . . . . . . . . . . . 12
2.4 Winding Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 DC resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 AC resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.2.1 Skin effect . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2.2 Proximity effect . . . . . . . . . . . . . . . . . . . . . 15

2.5 Primary winding design . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 Steps for Litz wire design [14] . . . . . . . . . . . . . . . . . . 16

3 Design of Planar Secondary 19
3.1 PCB for Secondary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

viii



Contents

3.2 High Voltage Design Aspects . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Fields for AC and DC stresses . . . . . . . . . . . . . . . . . . 21

3.2.1.1 Conduction Fields . . . . . . . . . . . . . . . . . . . 22
3.2.1.2 Capacitive Fields . . . . . . . . . . . . . . . . . . . . 22

3.3 Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 HV Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Diode for Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 BYT78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Output Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Stray Capacitance and Optimization . . . . . . . . . . . . . . . . . . 26

4 Transformer Primary Calculations and Setup 29
4.1 Transformer Input and Output Specifications . . . . . . . . . . . . . . 29
4.2 Transformer Design Procedure . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Choice of Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Core Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.2 Core Packing . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.3 Measurement of Core Loss . . . . . . . . . . . . . . . . . . . . 33
4.3.4 COMSOL Simulations . . . . . . . . . . . . . . . . . . . . . . 34

4.3.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.4.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.4.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.4.4 Study . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.4.5 Equations . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 Primary Winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5.1 Primary Winding Geometry . . . . . . . . . . . . . . . . . . . 38
4.5.2 Primary Winding AC Resistance . . . . . . . . . . . . . . . . 39

5 Transformer Secondary Design, Calculations and Modeling Proce-
dure 41
5.1 Secondary Winding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 PCB for Secondary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.1 PCB and traces . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.2 PCB design calculations . . . . . . . . . . . . . . . . . . . . . 42

5.2.2.1 Dimensions of the trace . . . . . . . . . . . . . . . . 43
5.2.3 PCB turns distribution . . . . . . . . . . . . . . . . . . . . . . 44
5.2.4 PCB resistance measurement . . . . . . . . . . . . . . . . . . 45

5.3 COMSOL Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.1.1 Requirement of bobbin . . . . . . . . . . . . . . . . . 46
5.3.1.2 Comparison of Bobbin Materials . . . . . . . . . . . 47
5.3.1.3 Requirement of PCB coating . . . . . . . . . . . . . 48
5.3.1.4 Position of traces and PCB . . . . . . . . . . . . . . 49

5.3.2 Insulation distances between core and PCB . . . . . . . . . . . 50
5.3.2.1 Arrangement of the PCB stacks . . . . . . . . . . . . 51

5.3.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

ix



Contents

5.3.4 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3.5 Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3.6 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Diode Loss Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4.1 Threshold and Forward voltages, and Dynamic resistance . . . 54
5.4.2 Conduction loss calculation [30] . . . . . . . . . . . . . . . . . 54

6 Results and Discussions 59
6.1 Core and B-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1.1 Temperature Rise Test on the Core . . . . . . . . . . . . . . . 60
6.1.2 Core Loss Measurement . . . . . . . . . . . . . . . . . . . . . 60
6.1.3 Litz Wire dimensions . . . . . . . . . . . . . . . . . . . . . . . 61

6.2 PCB geometry and E-field . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1 Requirement of bobbin . . . . . . . . . . . . . . . . . . . . . . 61
6.2.2 Comparison of Bobbin Materials . . . . . . . . . . . . . . . . . 63
6.2.3 Requirement of PCB coating . . . . . . . . . . . . . . . . . . . 64
6.2.4 Insulation distances between core and PCB . . . . . . . . . . . 65
6.2.5 Arrangement of the PCB stacks . . . . . . . . . . . . . . . . . 66

6.3 Total Loss and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Conclusion 71
7.1 Future work and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 73

x



List of Figures

2.1 Applied square wave voltage and corresponding flux density . . . . . 7
2.2 Shell-type transformer core shape . . . . . . . . . . . . . . . . . . . . 8
2.3 Magnetic circuit of a Shell-type transformer . . . . . . . . . . . . . . 9
2.4 Flux path in a Shell-type transformer . . . . . . . . . . . . . . . . . . 10
2.5 Circuit diagram of the series resonant converter . . . . . . . . . . . . 10
2.6 Skin effect in a circular conductor . . . . . . . . . . . . . . . . . . . . 14
2.7 Proximity effect in a circular conductor . . . . . . . . . . . . . . . . . 15

3.1 Multi-layer transformer winding [16] . . . . . . . . . . . . . . . . . . 19
3.2 Disc/Pancake/Sandwich transformer winding [16] . . . . . . . . . . . 20
3.3 Capacitive and Conduction Fields . . . . . . . . . . . . . . . . . . . . 22
3.4 Standard output rectification . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Integrated/ Stage output rectification . . . . . . . . . . . . . . . . . . 26

4.1 Design method for the high frequency planar transformer . . . . . . . 30
4.2 Core suppliers and materials . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Core loss measurement setup . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Core geometry setup in COMSOL . . . . . . . . . . . . . . . . . . . . 35
4.5 BH interpolation of N87 ferrite . . . . . . . . . . . . . . . . . . . . . 35
4.6 Free Tetrahedral (Selective) meshing . . . . . . . . . . . . . . . . . . 36

5.1 Placement of secondary turns in different layers on a PCB . . . . . . 45
5.2 Geometry for secondary to compare E-field stresses with and without

a bobbin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Geometry for secondary to compare E-field stresses with and without

the presence of PCB coating . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Placement of PCB with respect to core to compare E-field stresses . . 49
5.5 PCB design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.6 PCB placement in core window . . . . . . . . . . . . . . . . . . . . . 50
5.7 Stacked PCBs with primary winding . . . . . . . . . . . . . . . . . . 51
5.8 BYT78 diode - plotted characteristics [19] . . . . . . . . . . . . . . . 53
5.9 PCB trace with diode bridge . . . . . . . . . . . . . . . . . . . . . . . 55
5.10 BYT78 conduction loss vs. TJ . . . . . . . . . . . . . . . . . . . . . . 56

6.1 B-field 2D plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Core loss versus Frequency . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 E-field with bobbin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xi



List of Figures

6.4 E-field without bobbin . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.5 Bobbin material PP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.6 Bobbin material PA12 . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.7 E-field surface plot with PCB coating . . . . . . . . . . . . . . . . . . 65
6.8 E-field surface plot without PCB coating . . . . . . . . . . . . . . . . 65
6.9 E-field with all traces on the top layer . . . . . . . . . . . . . . . . . 66
6.10 E-field line plot for the top PCB on LV side . . . . . . . . . . . . . . 67
6.11 E-field line plot for the bottom PCB on LV side . . . . . . . . . . . . 68

xii



List of Tables

2.1 Properties of Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Ohmic resistance of DC and high frequency AC in adjacent conductors 16

3.1 Maximum Voltage experienced for 20-20 turn PCB . . . . . . . . . . 24

4.1 Transformer specifications . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Core properties of N87 Ferrite . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Calculations for core at a frequency of 50 kHz . . . . . . . . . . . . . 33
4.4 Instruments used in core loss measurement . . . . . . . . . . . . . . . 33
4.5 N87 material properties used in COMSOL . . . . . . . . . . . . . . . 34
4.6 Mesh statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.7 PACK’s RUPALIT V155 Litz cable [13] . . . . . . . . . . . . . . . . . 39
4.8 Primary winding AC resistance . . . . . . . . . . . . . . . . . . . . . 39

5.1 Secondary turns and PCB stacks calculations . . . . . . . . . . . . . 41
5.2 Properties of PCB FR4 material . . . . . . . . . . . . . . . . . . . . . 43
5.3 Recommended trace widths for different trace thicknesses . . . . . . . 43
5.4 Track clearance required as per IPC2221B standards . . . . . . . . . 44
5.5 4 - wire testing for PCB trace DC resistance . . . . . . . . . . . . . . 45
5.6 Relative permittivities and conductivities of different dielectrics in

the geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.7 Properties of bobbin materials . . . . . . . . . . . . . . . . . . . . . . 48
5.8 VFMAX for the extreme temperature values . . . . . . . . . . . . . . 56
5.9 VTO and RD at the reference TJ values . . . . . . . . . . . . . . . . . 56

6.1 Temperature rise test of core . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Core loss measurement results . . . . . . . . . . . . . . . . . . . . . . 60
6.3 Average E-field in [kV/mm] across bobbin, oil, coating, and first trace 67
6.4 Results from the simulation of PCB stacks . . . . . . . . . . . . . . . 69
6.5 Calculated transformer losses . . . . . . . . . . . . . . . . . . . . . . 69

xiii



1
Introduction

This chapter presents the background, aim, problem description, scope, and outline
of the thesis. The sustainable and ethical aspects of the work are also discussed.

1.1 Background
Air pollution is one of the major health risk factors across the globe, and industrial
process plants contribute largely to this [1]. To reduce the impact of emissions from
these processes, electrostatic precipitators (ESP) are frequently used. An ESP is
a device that uses high electrostatic potential to remove dust and other fine par-
ticles from a moving air stream. ESPs have uses in both domestic and industrial
applications for air purification. The operation of an ESP is as follows; The high
electrostatic potential is applied to discharge electrodes, which are spike-shaped
wires, causing ionization of particles due to the high electric field caused by the
accumulation of charges at the sharp points. The electrostatic force will attract
the charged dust particles to several grounded collector plates. The voltage is then
turned off for a short period of time and the collector plates are rapped to remove
the collected dust. This dust will then be collected in hoppers placed below the
plates and clean gas is emitted [2]. The required high voltage is provided by a
power supply consisting of a suitable step-up transformer and rectifier unit.

The existing power supply uses a high voltage transformer that consists of a pri-
mary winding wound across several toroidal cores to achieve the required operat-
ing voltage. The cores are made of nanocrystalline material, which is expensive.
Each toroidal core contains a secondary winding, whose rectified outputs are series-
connected to obtain the final voltage. Due to such discrete magnetic circuits, each
core requires individual flux-balancing. The present transformer is a bulky device
of high cost, sophisticated construction, and complex circuitry. Overall, the trans-
former is the most expensive part of the converter. By redesigning, it is possible to
reduce the cost by carefully choosing the topology of the transformer; its core shape
and material.

An alternative design approach is to use a single core design, by packing multiple
ferrite cores into a singular magnetic circuit. This eliminates the need for flux-
balancing. The core is assembled such that there is a suitable window for stacking
multiple secondary windings, sharing the same primary flux. They are similarly
rectified at each stage, and series connected to achieve the required voltage. The

1



1. Introduction

secondary is in the form of traces, etched onto PCBs. This reduces complexity, size,
material requirement and hence cost, making the construction and assembly simple.

Reducing the transformer’s size, material and cost, by increasing the operating fre-
quency has limitations when it comes to high-voltage transformers [3]. Hence, a
proper insulation design for the transformer is necessary.

The switching converter that provides the adjustable high-frequency input voltage,
is made of Silicon Carbide (SiC) MOSFET modules; a new technology that reduces
switching losses and enables high switching frequencies. To efficiently utilize the SiC
modules, the transformer needs to be designed to operate at a higher frequency.

1.2 Aim
The aim of the thesis is to design a high frequency, planar transformer for a specific
high voltage application, the ESP, evaluate the feasibility of the design using FEM
software simulations, and validate its design theoretically.

1.3 Problem Description
The transformer currently used for supplying power to the ESP uses expensive
nanocrystalline cores, which are difficult to assemble due to their toroidal shape.
A machine is required to keep the windings tightly wound. This is both time-
consuming and cumbersome. There are 10 such toroidal cores, required to be wound
separately, increasing the total amount of copper.To keep them in balance, additional
compensation windings with a dedicated balancing circuitry are required for each
secondary. Several hundred turns of copper wire are used in each core. The bulki-
ness, cost and complexity make it desirable to look for alternatives.
By changing the topology of the transformer to planar, the multiple cores can be re-
duced to a single core design that carries multiple planar secondary windings. Using
a single core eliminates the need for flux balancing. It also reduces the overall size
of the core. The planar design reduces copper material usage. The core material is
chosen based on the redesigned frequency and insulation requirements.

The main challenge with the proposed design, because of reducing the size, is in-
sulation design. Since the output voltage requirement is the same, the electrical
insulation needs to be carefully considered while reducing overall size in the design.
The use of a core bobbin having high dielectric strength, conformal PCB (Printed
Circuit Board) coatings, suitable distances with a safe margin in regions experienc-
ing high voltage stresses are considered. Due to the increase in frequency and power
density, consideration of heating and core loss is necessary as well. The physical
design is equally important since sharp boundaries and points can experience high
voltage stresses.

The output is rectified and series connected from one PCB to the next PCB. This

2



1. Introduction

is known as “integrated rectification”. As opposed to standard rectification where
the total output is rectified, this method follows a stage-rectification process that
has the advantage of lowering alternating voltage stress. It can also reduce the high
winding capacitance. Another advantage is the possibility of using components with
lower voltage ratings [4].

1.4 Scope
In this thesis work, the magnetic and electrical aspects, including the insulation
system, are simulated and analyzed. The design is based on calculations and elec-
tromagnetic simulations in the FEM-based software, COMSOL Multiphysics. The
thermal aspects of the transformer are not considered in COMSOL. Assembly of
the prototype is not carried out as part of the thesis, however the core is physically
tested for optimal temperature distribution.

1.5 Outline of the Thesis
This thesis is divided into 7 chapters with several subsections. Chapter [1] includes
the background, problem description, aim and scope. This chapter also discusses
the sustainable and ethical aspects of the thesis work. Chapter [2] presents the basic
concepts of transformer design, procedure for selection of core and loss calculations.
Chapter [3] describes the high voltage design aspects considered for the PCB design,
an overview of diodes and their loss, and a comparison between the old and new
designs. It also includes a discussion on the effect of stray capacitance and methods
to minimize it. Chapter [4] is the methodology followed during the thesis work for
the selection of the primary winding; the calculations and COMSOL simulations
carried out to obtain an optimal design. Chapter [5] presents the steps followed for
design of the secondary, COMSOL modeling procedure, insulation requirements and
calculations. Chapter [6] presents the results obtained from the setups in chapters
4 and 5. These include B-field and E-field plots for different geometries with veri-
fication of materials, assembly, and insulation distances in regard to their effect on
E-field stresses. In Chapter [7], the results are discussed, and conclusions are drawn
to arrive at the most efficient design for the high frequency transformer. The future
work is also discussed in this chapter.

1.6 Sustainability Aspects
The United Nations adopted 17 sustainable development goals in 2015 to ensure
future needs are met without compromising present consumption [5]. The goals are
expected to be met by the year 2030 and encompass a wide range of social and
environmental issues.
With the energy sector being a major contributor to the climate change issues chal-
lenging the world, it is important to analyze the impact of new innovations on the
environment. This thesis work, while improving the efficiency of the transformer

3



1. Introduction

and reducing costs, also contributes to some of the sustainable goals as presented
below.

• Affordable and Clean Energy (goal 7)
The project aims at ensuring access to affordable, reliable, sustainable, and
modern energy. There are encouraging signs that the world is progressing
towards the goal of achieving affordable and clean energy as energy is becoming
more sustainable and widely available.
The new design of a planar transformer replaces the expensive cores of the
old design and reduces copper requirement for its secondary. This brings
down the cost of the transformer and makes it much more affordable. With
the introduction of the planar design, there is lower core loss, elaborated in
further chapters. The efficiency goes up, resulting in a reliable supply for the
ESP.

• Sustainable Consumption and Production (goal 12)
Electricity is hailed as one of the major ways in achieving sustainable develop-
ment goals. The transformer design introduced in this thesis reduces the size
and hence the material required. As described in the previous section, with
a reduction in loss, the ESP supply becomes more efficient and hence more
sustainable. With a future scope of mass production, this would cause the
ESP industry to have improved and more sustainable operation.

1.7 Ethical Aspects
A list of 10 ethical aspects is described by the Institute of Electrical and Electronics
Engineers (IEEE) [6], which forms a code of ethics that all members are expected
to follow. Some of the ideas covered in this thesis work include:

• Objectivity (2)
The goal is to compare and replace an existing transformer with a planar
transformer. By doing so, actual outcomes are compared by monitoring energy
loss and material savings.

• Carefulness (3)
It is critical to deliver the desired outcome, since this study will determine
if an entire line of transformers will require a change. It is necessary that
measurements are done carefully, and reliable data is produced.

• Unlawful conduct (4)
This project is only for Research and Development; no supplier is being pro-
moted or advertised. This is purely a study for potential replacement of an
existing product to a higher-efficiency one.

• Honesty (5)
The outcome of this study will greatly influence the company’s production for
the ESP industry; hence it is important that honest results are presented, and
not in the favor of the present transformer or the new one. If it reaches a
conclusion that introducing a planar topology into the required high voltage
system is not feasible, then it is presented with honest data describing why it
is not a good choice.

4



2
Principles of Transformer Design

This chapter introduces the basic concepts involved in the design of a transformer;
laws of electromagnetism, transformer equations, core and winding losses. It also
contains a description of the transformer used in a series resonant converter network.

To begin the design of a transformer, it is necessary to understand the principles
governing its working, namely; the laws of electromagnetism. Transformers are
based on Ampere’s Magnetic Circuit Law, and Faraday’s Law of Electromagnetic
Induction.

2.0.1 Ampere’s Law
The derivation follows from Maxwell’s equations for a linear, homogeneous, isotropic
medium. Ampere’s law relates current, and the magnetic field created by it, de-
scribed by Equation (2.1) [7].∮

C

~B · d~l = µ0[Ienc + ε0 ·
d

dt

∫
S

( ~E · ~n)ds] (2.1)

where, "B" [T] is the induced magnetic field vector, with the dot product of the
closed integral indicating the B field along the path "C".
"µ0" indicates the magnetic permeability [H/m] of free space, "Ienc" [A] indicates the
enclosed conduction current, and "ε0" indicates the electric permittivity [F/m] of free
space. The displacement current is expressed as the rate of change of "E", electric
flux [Wb] with respect to time "t". The electric field [V/m] is a vector, bounded by
the closed surface "S".

2.0.2 Faraday’s Law
Maxwell’s equation for Faraday’s law describes the relationship between the electric
field intensity and the rate of change of magnetic flux within a closed surface. In the
integral form, Faraday’s law equates the integral of electric field intensity around a
closed contour "C" to the rate of change of the magnetic flux that crosses the surface
"S" enclosed by "C", described by Equation (2.2) [7].∮

C

~E · d~l = − d

dt

∫
S

~B · ~nda (2.2)

5



2. Principles of Transformer Design

where, "E" [V/m] is the induced electric field circulating around the closed path "C",
caused by the changing magnetic field "B" [T] with respect to time. The negative
sign is explained by Lenz’s law [2.0.3].

2.0.3 Lenz’s Law of Electromagnetic Induction
Lenz’s law states that the direction of the magnetic field generated by the induced
electric field is such that it opposes the original magnetic field causing it. It is
explained using Equation (2.3) [7].

e = −dφB
dt

(2.3)

where, the induced e.m.f. "e" [V], and the changing magnetic flux "φB" have opposite
signs.

2.0.4 Transformer Efficiency
The efficiency of the transformer can be expressed as the ratio of the output power
to the input power, as described in Equation (2.4).

η = Pin − loss
Pin

· 100 (2.4)

2.0.5 Flux Density
When a magnetic field passes through a material, the measure of the material’s
resulting flux density to the applied magnetic field is given by its magnetic perme-
ability µ, described by Equation (2.5) [7].

~B = µ0µR ~H (2.5)

where "µR" is the relative magnetic permeability of the material.
The magnetic flux density of a transformer under a changing flux density is obtained
by relating Ampere’s law and Faraday’s law, as described in Equation (2.6).

e = N · dφ
dt

= N · Ae ·
~B
dt

(2.6)

where "φ" is the alternating flux in the core, "Ae" [m2] is the effective cross-sectional
area of the core, and "e" is the counter e.m.f. as described in Equation (2.3) to the
applied primary voltage "v", and in accordance to Kirchoff’s voltage law [7], v = e.

In the case of an applied square wave voltage, v remains constant until the polarity
changes, and the corresponding flux density is a triangular wave. When the voltage
changes polarities, flux density "B" switches to its maximum values from positive to
negative. Therefore, there are two peaks every half period, as illustrated in Figure
2.1 from [7].

6



2. Principles of Transformer Design

Figure 2.1: Applied square wave voltage and corresponding flux density

The average value of the applied voltage during an interval t = τ , where the flux
density varies from zero to its maximum value is |v|. The value of |v| is found by
integrating Equation (2.6), resulting in Equation (2.7).

|v| = 1
τ

∫ τ

0
v(t)dt = 1

τ
N · Ae

∫ Bmax

0
d ~B = 1

τ
N · Ae · Bmax (2.7)

The form factor "k" relates the obtained average value of |v| to the rms value of the
applied waveform as described in Equation (2.8).

k = Vrms

|v|
(2.8)

Combining Equation (2.7) and Equation (2.8) results in Equation (2.9),

Vrms = K
τ
T
· f · N · Bmax · Ae (2.9)

where "K " is the waveform factor defined in Equation 2.10

K = k
τ f (2.10)

The wave shape from the SRC (Series Resonant Converter) source is a square wave
as described in Figure 2.1, hence the duty cycle D is 0.5. To establish the value of
K for a square wave, the flux rises from zero to Bmax in time τ = T/4, therefore
resulting in τ/T = 0.25. The form factor k for a square wave is 1, since the average
value over the time τ is Vdc, and the rms value is Vdc. K is hence 1/0.25 = 4.
Applying the obtained K value in Equation (2.9), Equation (2.11) is obtained.

Vrms = 4 · f · N · Bmax · Ae (2.11)

7



2. Principles of Transformer Design

Rearranging Equation (2.11) to present in terms of Bmax, Equation (2.12) is ob-
tained.

Bmax = v

4 · f ·N · Ae
(2.12)

Equation (2.12) will be used for all calculations henceforth for values of the peak
flux density of the core to select the most optimal combination of Bmax and number
of cores to achieve the most efficient design [7].

2.0.6 Core Shape and Magnetizing Inductance
The core shape is an early design factor in the construction of the transformer. The
conventional shell-type transformer and its realization using multiple cores is shown
in Figure 2.2.

(a) Core shape cross-section (b) Realized with multiple cores

Figure 2.2: Shell-type transformer core shape

As the cross section area, core window distance, and other design factors are decided,
the transformer is assembled using smaller ferrite cores to obtain the desired size.
For effective transformer operation, a suitable B-field needs to be set up across the
cross-section of the core, and this is done by the alternating flux established by the
changing voltage across the primary, described in Equation (2.12).
The magnetizing current is a small portion of the primary current, which is always
drawn to keep the core magnetized. This is observed even when there is no secondary
load. The magnetizing inductance [H] can be calculated by Equation (2.13).

Lm = N2

Rc

(2.13)

where, "N" is the number of primary turns and "Rc" is the reluctance of the core.

The core’s reluctance Rc can be thought of as the magnetic equivalent of electrical
impedance, as magnetic flux is to electric current. Reluctance is the measure of the
impeding effect by a material to a magnetic field flowing through it. Reluctance
is given in Ampere-turn per Weber [AT/Wb] or inverse Henry [1/H], and can be
calculated using Equation (2.14).

8



2. Principles of Transformer Design

Rc = lc
µr · µ0 · Ae

(2.14)

where, "lc" is the length of the magnetic path in [m], "Ae" is the effective cross-
section area of the core [m2], and "µr" is the relative magnetic permeability of the
core material.

The m.m.f generated in the core’s magnetic path can be expressed in the form of
Equation (2.15).

mmf = N · I = φ ·RC (2.15)

where "N" is the number of primary turns and "I" is the primary current [A]. It can
also be equated to the product of "φ" which is the magnetic flux [Wb] generated, and
"RC", the core’s reluctance [AT/Wb]. The reluctance of a transformer’s core can be
derived by expressing the transformer as a magnetic circuit. Figure 2.3 expresses
the primary winding as an m.m.f. source.

Figure 2.3: Magnetic circuit of a Shell-type transformer

The established flux travels through the middle leg and splits into half (into parallel
magnetic paths) across each side leg, to add and return to the source. The path that
the flux takes is known as the Magnetic Path Length or Mean Magnetic Path. It is
the path of least reluctance that the flux takes, which is the mean circumference of
the core’s magnetic loop, normal to the current direction in the winding. Figure 2.4
illustrates the magnetic flux lines through the mean magnetic path of the core.

9



2. Principles of Transformer Design

Figure 2.4: Flux path in a Shell-type transformer

2.1 Series Resonant Converter
The transformer’s input power is fed using an SRC which is a DC-DC converter
that allows “soft-switching” operation due to topology. In this operation, switching
can occur when voltage and/or current values are zero (ZVS)/(ZCS) (Zero Volt-
age Switching/Zero Current Switching), thus significantly improving the converter’s
efficiency under optimal conditions [8].
In the case of the current design the power input is an industrial three-phase power
network that is rectified in the DC-link. The specific application is an ESP, that can
be described as a load in the form of a parallel RC network. The resonant network
has the inductance split into two inductors. The transformer is also included in the
resonant tank as a part of the circuit, as illustrated in Figure 2.5.

Figure 2.5: Circuit diagram of the series resonant converter

10



2. Principles of Transformer Design

2.2 Choice of core
The design of the primary starts with the choice of the core. The core is essential
to contain the magnetic field and provide a path for the flux. Mn-Zn ferrites are
the most common core materials used for power electronic devices. This material
has a high electrical resistivity which reduces Eddy Current losses in the core. The
selection of the type of ferrite core for transformers depends on several factors.
In practice, a transformer needs to be designed around a particular input waveform,
taking into consideration the amplitude, frequency and current. Additionally, there
will be changes in properties such as permeability during operation due to rise in
temperature. The transformer needs to be designed to operate over a range of values
for its parameters.

2.2.1 Factors affecting the selection of core material
• Magnetic permeability (µ): The permeability varies with temperature and

magnetic flux density. However, this relationship is not linear. It takes an
optimum value across a range of temperature and flux density. Ideally, maxi-
mum permeability is desirable at the operating point of the transformer.

• Temperature (T): The resistive heating, eddy currents and hysteresis loss grad-
ually raise the temperature of the core. Different cores have different optimal
temperatures. At a certain temperature, some materials will lose their mag-
netic properties. Additionally, temperature will cause the core’s permeability
to change. Hence cooling might be needed to prevent such a situation.

• Frequency (f ): Input frequency selection is important. Chosen correctly, it
will allow low core losses, reduce temperature and increase flux.
From Equation (2.12), an alternating voltage is required to establish a flux.
Since the input voltage is fixed at 500VRMS, frequency, primary turns and
cross-section area are the allowed variables;
– Increasing the cross-section area results in increased material cost.
– Increasing the number of primary turns results in increased secondary

turns (multiplied by the transformation ratio), requiring more Litz cable
length on the primary and increased number of copper volume by traces
on the secondary, as well as the increased volume occupied within the
core window.

– Increasing the frequency reduces core material requirement and hence
cost, while increasing loss in the converter and winding.

Losses are proportional to the frequency, but also to the flux. The decrease
due to the lower flux can have more influence than the increase due to high
frequency, hence increasing the frequency in this case will be beneficial.

• Core loss: Core loss is divided into hysteresis loss and eddy current loss;
– PHysteresis: proportional to “f” and “B”
– PEddy: proportional to “f2” and “B2”.

11



2. Principles of Transformer Design

In the current design, the use of a ferrite core reduces eddy currents substan-
tially due to its high electrical resistivity, unlike soft iron cores.

2.3 Core Loss
The core loss in a transformer core consists of two parts; Hysteresis and Eddy current
loss. Hysteresis loss is due to the continuous reversal of the magnetic domains, and
the heating it causes. Eddy current loss is caused by the alternating magnetic field
inducing circular currents within the core material itself.
To determine the core loss, usually in [mW/cm3] or [kW/m3] as a function of the
given peak flux density Bmax [T], the Original Steinmetz Equation (OSE), given by
Equation (2.16), is used

Pv = Kc · fα ·Bβ
max (2.16)

where "f" is the frequency in [kHz], and "Kc", "α", and "β" are constants that may
be found from the manufacturer’s datasheet, or found by curve fitting from existing
data.

2.3.1 Measurement of Core Loss
In the design of magnetic components, it is necessary to understand the complete
characteristics of the materials. However, there is an inadequacy in the availability
of data and validation models for high frequency applications. This can be due to
several factors such as the lack of reliable experimental methods, the complexity
of providing excitation, along with the changes in properties of the material with
varying parameters [9].

For the measurement of core loss, a suitable excitation source is provided. The
primary and secondary turns are kept equal and are tightly wound. The primary
winding is the exciting winding, whereas the secondary is more of a sensory winding.
This is necessary to avoid any voltage drop across primary, due to resistance, from
affecting the measurements [9].
To measure the core loss, electro-mechanical wattmeters were conventionally used.
However, these measurements are not reliable at high frequencies. For an accurate
core loss measurement, the setup with equal primary and secondary turns can be
used, and the waveforms of secondary voltage and primary current are then mul-
tiplied to find the power of the core. The average power loss is then measured by
finding the mean of the power in one cycle. To find the loss per unit volume, the
measured core loss can be divided by the core volume provided by the manufacturer.
This calculated loss is then plotted versus frequency, and the process is repeated for
different frequencies, voltages and flux density levels, arriving at an overall core loss
characterization [9].

12



2. Principles of Transformer Design

2.4 Winding Loss
In the case of high frequency transformers, besides DC loss, the copper winding
exhibits significant AC loss that needs consideration [7].

2.4.1 DC resistance
For a DC input, the conductor exhibits Ohmic I2R loss, and the resistance "R" is
given by the resistivity of the material "ρ" at the chosen temperature, its length "l"
and cross sectional area "A".

R = ρ · l
A

(2.17)

Copper’s resistance is frequently calculated for 1000C, as that would be typical
steady state temperature for a transformer during operation.
Electrical resistivity as a function of temperature Rρ[T ] is given by Equation (2.18).

Rρ(T )l/A = ρ(T0)[1 + α(T − T0)l/A] (2.18)

where "T0" is the initial reference temperature, "α" is the temperature coefficient of
conductivity of the material, "l" is the length of the conductor, and "A" is the cross
section of the conductor. The resistivity described in Equation (2.18) is calculated
from the standard resistivity for copper given at 200C with the parameters presented
in Table 2.1.

Table 2.1: Properties of Copper

Symbol Definition Expression Value Unit
T Conductor Temperature - 100 0C
T0 Initial Temperature - 20 0C
ρ20 Copper resistivity at 200C - 1,68E-8 Ωm
α Temperature coefficient of copper - 0,00386 -
ρ100 Copper resistivity at 1000C ρ20[1 + α(T − T0)] 2,2E-8 Ωm
σ100 Copper conductivity at 1000C (1/ ρ100) 4,55E+7 S/m

Hereafter, every calculation concerning copper material will be carried out with the
resistivity and conductivity as presented in Table 2.1.

2.4.2 AC resistance
While it is quite straightforward for DC, several effects are exhibited with AC,
namely; skin effect and proximity effect. Especially at high frequencies (it is appar-
ent at low frequencies as well, but their losses are significant as frequency increases),
skin effect takes place within a conductor, and proximity effect takes place between
two insulated, nearby conductors. The skin and proximity effects together result
in higher loss in conductors since they cause non-uniform distribution of current,
resulting in an increase of effective AC-resistance.

13



2. Principles of Transformer Design

2.4.2.1 Skin effect

An isolated conductor carrying AC generates a concentric, alternating magnetic
field, which induces Eddy currents. This cancels some of the currents at the center,
while increasing surface current. This is called skin effect and the overall result is
that current flows in a smaller annular area, as illustrated in Figure 2.6 [7].

Figure 2.6: Skin effect in a circular conductor

where, "J(r)" is the surface current density taken along the radius "r" from the center
of the conductor to the outer surface.
At higher frequencies, the current flows in an equivalent annular cylindrical portion,
of radial thickness “δ”, called skin depth. Deriving from Equation (2.1), Equation
(2.2) and Bessel’s equation, the skin depth is given by Equation (2.19).

δ =
√

1
π · f · µ0 · µR · σ

(2.19)

where "f" is input frequency in [Hz], "σ" is the conductor’s conductivity in [S/m],
and "µ" is the conductor’s magnetic permeability in [H/m].
While the DC resistance is fixed with the conductor’s design, the AC resistance
depends on the operating frequency of the application. A factor "ks" is defined as
RAC/RDC , this gives the approximation as described in Equation (2.20).

ks = 1 +
r0
δ0

4

48 + 0, 8 · r0
δ0

4 (2.20)

From Equation (2.19), it is visible that "δ" reduces with increasing frequency, reduc-
ing effective conduction area. Skin depth describes the depth up to which 67% of

14



2. Principles of Transformer Design

the maximum current density flows within that surface area.
To determine the diameter of the strands (d0) with regard to skin depth, the follow-
ing factors are considered,

• If d0 » δ, it is wasteful to use more conducting material since most of the
current would only flow in the skin depth from the surface.

• If d0 « δ, the conductor would exhibit mostly DC properties but at the cost of
small surface area, requiring more conductors to carry the same rated current.

The recommendation for diameter, from Litz wire manufacturers is to use "1/e"
times the “δ” value, where "e" is Euler’s constant [10].

2.4.2.2 Proximity effect

At high frequencies, conductors in close proximity induce Eddy currents in adjacent
conductors. This disturbs the uniform distribution of currents by concentrating
them towards the region that is furthest from the nearby conductors that induce
the Eddy currents. This is called proximity effect, it reduces effective conduction
area and hence increases the AC resistance.[7]. The COMSOL simulation shown
in Figure 2.7 describes the uniform current density due to AC and the varying
current density due to high frequency AC flowing in the same direction in adjacent
conductors.

Figure 2.7: Proximity effect in a circular conductor

15



2. Principles of Transformer Design

The variation in current density for high frequency AC increases the effective resis-
tance, as presented in Table 2.2 [11].

Table 2.2: Ohmic resistance of DC and high frequency AC in adjacent conductors

Frequency (kHz) Conductor 1 R(Ω) R% increase Conductor 2 R(Ω) R% increase
0,0 0,0053 171.5% 0,0053 171.5%
50,0 0,01439 171.5% 0,01439 171.5%

2.5 Primary winding design
To minimize the effects of AC, Litz wire is used. Litz wire is composed of multiple,
thin individual strands that are twisted together, with each strand insulated. If
a solid conductor of the same cross section is considered, high frequency AC will
cause the currents to gravitate to the surface of the conductor, hence not utilizing a
major portion of the actual conductor. With several smaller, insulated conductors
making up effectively the same cross section, the current in each cable is limited to
each strand’s surface, allowing a much greater effective conducting area. While this
reduces the skin effect, the chances of proximity effect increases as the strands are
bunched together. Hence, the method of transposing them at intervals evens the
magnetic field across all conductors, maintaining the uniformity of current density
[7].
With Litz wire proving beneficial to reduce loss, it is necessary to understand how
to effectively design the strands. Manufacturers of Litz wires provide different rec-
ommendations for the diameters of strands based on frequency; however these may
not always lead to optimal designs and can lead to high resistances and hence higher
loss [14]. From [14], a simple design approach was implemented in this thesis work.
This provides a method to calculate the number and diameter of strands needed
for the present application. Once the numbers are derived, the arrangement of the
strands in bundles can also be chosen.

2.5.1 Steps for Litz wire design [14]
• Skin depth calculation (δ)

Using Equation 2.19, the skin depth in [mm] is calculated for the required
frequency of operation. Based on this value, the diameter (dc) can be deduced
to be 1/e times δ, as mentioned in Section 2.4.2.1.

• Winding specifications
To calculate number of strands, the parameters of the winding required are the
available core window area (bc) in [mm] and the number of turns per section
(N). Since there is no interleaving, the number of turns here can be chosen as
the number of primary turns of the design.

• Calculation of number of strands
From these parameters, the number of strands can be calculated using Equa-

16



2. Principles of Transformer Design

tion (2.21), as derived in [14].

n = k
δ2 · bc
N

(2.21)

where, "n" is the effective number of strands, and "k" [mm−3] is a constant de-
rived in [14] based on the strand diameter and resistance factor for economical
designs.
Based on availability and manufacturer limitations, the number of strands can
be up to 25% greater or less than the calculated value of n.

• Choice of optimal number and diameter of strands
While finalizing the number of strands acceptable for the design, the calculated
cross-sectional area must be within 25 to 30% of the core window available.
This value must take into account the cost and performance for the required
application.

• AC loss factor
The AC loss coefficient of a Litz wire, given a sinusoidal current of known
frequency and amplitude can be found from Equation (2.22), derived in [15].

Fr = RAC

RDC

=
(

1 + π2ω2µ0
2N2n2d6

ck

768ρcb2
c

)
(2.22)

where, "Fr" is a factor relating DC resistance to the AC resistance.
• AC loss calculation

The winding loss due to AC is described by Equation (2.23), derived in [15].

Ploss = Fr · Iac2 ·Rdc (2.23)

• Choice of arrangement
The twisting arrangement in mainly advantageous to avoid a bundle-level prox-
imity effect. To avoid a bundle-level skin effect, the placement of the strands
must also be considered. The twisted strands must be arranged in multiple
levels with respect to their centres. Each group can consist of 5 or less twisted
sub-bundles to have a minimum skin effect, using strands with diameters much
smaller than the skin depth [14].

17





3
Design of Planar Secondary

In this chapter, the high voltage design factors are discussed, along with the choice
of planar coils for the secondary winding and their parameters. Characteristics of
the diode used and the output rectification method are also described.

3.1 PCB for Secondary
Unlike conventional wound transformers, the distinct development here is to use a
planar winding as the secondary. There are several types of transformer windings,
such as foil, multi-layer, and pancake/disc type. Foil type transformers are for low
voltages (not relevant to this study). The multi-layer type is most commonly found
in the medium voltage range, and pancake/disc/sandwich winding offers the best
design for high voltage and extra high voltage purposes [16].

• Multi-layer winding
The turns are laid next to each other, and as layers on top of the other, as
illustrated in Figure 3.1. Paper winding on the conductor, or an oil-proof
enamel coating serves as inter-turn insulation.

Figure 3.1: Multi-layer transformer winding [16]

The full voltage is found to occur at the narrow front side of the winding.

19



3. Design of Planar Secondary

Hence, it is used in the medium voltage range. This geometry offers excellent
packing hence most commonly found in domestic to medium voltage ranges.
Shortening the winding layer length and increasing radius causes a potential
grading in the axial direction.

• Disc/Pancake/Sandwich winding
This is the basis of the planar design. At a first approximation, the DC extreme
“V” is uniformly distributed over the winding layer length as illustrated in
Figure 3.2, hence it can be implemented for HV. Across the narrow front side,
there is a relatively low partial voltage.

Figure 3.2: Disc/Pancake/Sandwich transformer winding [16]

By gross comparison, disc winding is better when considering higher voltages. There
are however undesirable effects observed in disc winding, such as increased longi-
tudinal capacitance. Other factors of interest include leakage inductance and heat
removal. Leakage inductance is the result of insulation distances, gap widths be-
tween winding, layers and turns. It is mainly the distance between the primary
and secondary winding that results in a large leakage inductance. Compact designs
reduce Lleakage but they compromise insulation distances. Insulation design is hence
a crucial part of this study.
In conclusion, using a planar coil for this application would work best [16].

3.2 High Voltage Design Aspects
The design of the electrical insulation involves electric strength calculations, clear-
ance and creepage distances, breakdown of oil, among other factors. This design
must be based on semi-empirical relationships to consider all parameters that can
have an impact on the system [16].
Due to the reduction in the size of the transformer, the constraints related to the

20



3. Design of Planar Secondary

E-field stresses need to be addressed. In this planar transformer, there is a DC-field
with a small super imposed AC-ripple. It is the conductivity of the dielectric ma-
terials that determines the distribution of the E-field and this in turn determines
the design of the transformer secondary [16]. Hence an electric conduction model
is considered for the simulations as well. In the case of DC stresses, the electrical
conduction field is prevalent and in the case of AC stresses, it is the dielectric dis-
placement field. Maxwell’s Equations form the basis for the classification of these
stationary fields [16]. These are further discussed in the next section.
Under the sub-categories of the Maxwell’s equations, the material equations are rel-
evant to this study. These describe the impact of material properties on the electric
and magnetic fields. For insulation systems with a DC stress, the Maxwell equation
described in Equation 3.1 is the starting point.

J = σ · E (3.1)

where, "J" is the current density [A/m2] and "σ" is the material conductivity [S/m].

The transformer is being designed to generate 50kVDC at the output terminals. The
stresses in the secondary observed across its compact design make it necessary to
ensure that there is no possibility of inception and dielectric breakdown, and to
design for a suitable margin for distances.

3.2.1 Fields for AC and DC stresses
From Maxwell’s equations, fields are classified based on their time rate of change
into Static and Stationary, Quasi-Static and Non-stationary [16]. Since the field
quantities here are time invariant, the non-stationary field is not discussed.
In the case of a static electric field, it must be assumed that the material has zero
conductivity or is a perfect dielectric and this is not possible in reality. Hence, the
conduction field must be considered instead of an electrostatic field.
The electrical conduction and dielectric displacement (capacitive) fields are as de-
picted in Figure 3.3. The properties of oil and the bobbin influence the E-field
distribution across the modelled geometry and hence are indicated here. The mate-
rial for bobbin is a polymer plastic called Polypropylene (PP), motivated in further
chapters.

21



3. Design of Planar Secondary

Figure 3.3: Capacitive and Conduction Fields

3.2.1.1 Conduction Fields

A stationary or steady-state field, the field quantities here, E and J, are constant or
time-invariant. According to Equation (3.1), these fields are defined by a constant
J and a proportional E. When there is a DC stress, the distribution of the E-field is
influenced by a conduction current caused by the material conductivity that result
in the movement of charges. The relative permittivity of the material does not have
an impact on this field [16].
The E-field stress on oil can be calculated using the properties of the dielectric
interface between oil and the bobbin (PP). For the conduction field, the conductivity
is considered, as shown in Equation 3.2.

Eoil = U

doil + dPP · σoil

σP P

(3.2)

where, "U" is the output voltage [kV], "doil" and "dPP " are the distances [mm] of oil
and bobbin respectively, and "σoil" and "σPP " are the conductivities [S/m] of oil and
bobbin respectively.

3.2.1.2 Capacitive Fields

A quasi-stationary displacement field, the field quantities here can be slightly time-
varying. However, in insulating materials, the changes in the field are insignificant
when compared with the source field and can hence be neglected. For HV insulating
materials, the AC stress is of importance. If the conductivity is determined to be
relatively low and negligible, the displacement currents would be larger than the
conduction. For the capacitive field, the relative permittivities are considered to
study the E-field stresses, as in Equation 3.3.

Eoil = U

doil + dPP · εoil

εP P

(3.3)

22



3. Design of Planar Secondary

where, "εoil" and "εPP " are the relative permittivities of oil and bobbin respectively.
Depending on the condition described in Equation 3.4, the displacement or conduc-
tion currents would dominate the E-field distribution within the insulation system.

∂D

∂t
= ε0εr

∂E

∂t
<< J = σ · E (3.4)

Considering the materials used in the current design and the stress applied, the
conductivity cannot be neglected and hence the model for E-field must be based
on the electrical conduction field. Equation 3.4 would in this case be satisfied and
the conduction currents would dominate the displacement. The material with the
lowest conductivity would face the highest E-field, and as will be explained in further
chapters, this reduces the stress on the insulating oil and renders the assembly safe
from breakdowns [16].

3.3 Insulation
The stressed E-field regions define the voltages experienced across the materials.
The materials in the transformer should be capable of withstanding these stresses.
Here are some of the desired qualities of a good insulation [17].

• High Dielectric strength
• Low Dielectric constant

3.3.1 HV Insulation
The transformer is placed inside a tank containing the insulating oil to facilitate
cooling and provide safety while generating a high voltage output. The strength
of the insulating oil is largely affected by the gap width, along with the material
properties of the electrode surfaces and coatings used [16].
The barriers created by adding coatings, plastic bobbin, and plastic sheets between
PCBs, help in sub-dividing oil gaps. While the gap width reduces, introducing a
dielectric material increases dielectric strength. This is due to the formation of fiber
bridges preventing particle/charge carrier drift. These barriers also reduce stress on
the oil [16].
The shape of the electrode plays a major role in determining streamer ignition. Since
E-fields tend to concentrate at sharp geometries, it is usually recommended to keep
rounded conductors to reduce the chances of streamers being formed. This includes
the PCB traces’ edges and soldered connections that need to be rounded.
Coatings are applied over conducting, as well as non-conducting surfaces, also to
prevent streamer formation [18]. The quality of the oil and the coating of the
electrode makes a significant difference to the inception field strength. Streamer
formation is limited to the gap width, and the inception field strength for a discharge
to occur increases as the gap width reduces, Section 3.4.1.4 in [16]. The empirical
relation is described by Equation 3.5 and this can be used as a basic dimensioning
guideline for AC voltage test loading of oil gaps.

EBD = E0.(d)−a (3.5)

23



3. Design of Planar Secondary

where "d" [mm] is the gap distance, and "a" is a factor derived from the slope of
the discharge inception field strength standard plot, indicating the quality of oil and
insulation of electrodes, for mineral oil [16].
The oil gaps within the transformer tank in this study are designed according to
the insulation of the electrode and the saturation levels of the oil. Stresses of 5-10
kV/mm for AC voltage are permitted within these oil gaps to avoid a breakdown
[16]. This is dependent on the local field strengths within the transformer assembly.
To ensure the safety of the transformer setup and avoid breakdowns even after
several years of operation, the operating field strengths are much below the actual.
In practice, stresses on the oil gaps are restricted to 70% of their value. In this
study, the stress on the insulating oil is hence restricted to 7 kV/mm, considering
an AC stress of 10 kV/mm on the oil gaps. The materials and their properties are
chosen and dimensioned in such a way that the insulating oil is within the safety
limit of 7 kV/mm [16].
To obtain optimal distances from standards for insulation, some of the extreme
voltages and E-fields experienced in the transformer’s geometry were studied and
referenced with Equation 3.5 to ensure safe distances. These are presented in Table
3.1.

Table 3.1: Maximum Voltage experienced for 20-20 turn PCB

Between following points Vmax (kV)
Trace-to-trace 0,125

Inner trace to inner core leg 48,75
Outer trace to outer core leg 50

The use of barriers in oil gaps greatly increase the dielectric strength; the gap widths
are reduced and the drifting of particles and charge carriers over larger gaps is pre-
vented. Impregnated pressboard barriers causing subdivisions in the oil greatly in-
crease the dielectric strength because of these insulating boundaries. Hence, smaller
dimensions can be achieved for transformers, with reduced magnetic leakage flux,
weight, dimension and cost [16].

3.4 Diode for Rectification
The required output of the transformer is DC. Diodes can be used to rectify the
AC output of the secondary into DC. The diodes will be connected in the form of
a bridge rectifier. Slight conduction loss is observed in every diode. For light to no
loads, the output will be the peak of the sine wave, and for heavy loads, it will be

VDCout = 0.9 · VRMS (3.6)

For diode loss in the case of the current design, only conduction loss was considered,
since switching loss was found to be negligible.

24



3. Design of Planar Secondary

3.4.1 BYT78
BYT 78 is a fast switching diode having soft recovery characteristics and low reverse
current, well-suited for high voltage high frequency rectification. The sintered glass
case and spherical shape make it suitable for HV applications [19].
Due to its semiconducting properties, there will be a small voltage required by
the diode to start conducting in forward bias. Hence there is a voltage differ-
ence observed across the diode terminals in forward bias called “forward voltage”
VFORWARD > 0V.
This voltage difference across the diode causes conduction loss and is described using
Equation 3.7

PLOSS = IDIODE · VFORWARD (3.7)

3.5 Output Rectification
The rectification can be done in single or multiple steps. The process can be ex-
plained by considering a simple step-up transformer (turns in diagram do not reflect
actual number of turns) that uses only one core to get the required output. From
insulation standards, this cannot be achieved.
Splitting the secondary turns into planar PCBs gives the setup in Figure 3.4. This
is the case of standard rectification, which is the same as having a single multi-turn
coil and having the ends connected to the full-bridge rectifier.

Figure 3.4: Standard output rectification

Standard rectification works well at lower voltages and frequencies, however there
are limitations while considering the insulation requirements of high voltage trans-
formers. This can be addressed by integrated rectification, which reduces AC field
stresses across the coils and also prolongs the ageing of the insulation system [4].
The splitting and individual rectification of the secondary windings into separate,

25



3. Design of Planar Secondary

winding sections of desired geometry and proportions, which are later connected in
series is known as integrated rectification, as illustrated in Figure 3.5.

Figure 3.5: Integrated/ Stage output rectification

The diodes in integrated rectification can be of a lower rating (or lower in number)
since the voltage across each coil is lower. The DC output is connected to a load
through a low pass filter (capacitor CF ). Although the output voltage between the
two setups is not affected, it will impact the E-field distribution in the transformer.
The secondary with integrated rectification enables the separation of the E-field
into AC and DC components. The introduction of a DC field implies an associated
reduction of the AC field. This has proven to be beneficial since high magnitude
high frequency E-fields can give rise to partial discharge problems in the insulation
materials [4]. The separation is governed by the winding structure and the number of
winding sections. To consolidate these points, the benefits of integrated rectification
over standard rectification are:

• Reduced inter-layer field strengths
• Prolonged ageing of insulation system
• Reduction of AC stress in the windings (hence lower chances of partial dis-

charge)
• Major portion of the winding capacitance is moved to the DC side of the

rectifier
From these conclusions, the best choice in the case of a HV planar transformer is to
use integrated rectification.

3.6 Stray Capacitance and Optimization
Stray capacitance is an unavoidable, parasitic capacitance, due to parallel arrange-
ment of conductors. High frequency systems encounter this and with increasing
frequency, the losses increase. At extremely high frequencies, capacitors can act as
conductors. All circuit elements such as inductors and wire wound resistors have a

26



3. Design of Planar Secondary

capacitance due to the geometric arrangement of the conductor. At high frequen-
cies, it causes them to move further from their ideal characteristics [16].

Stray capacitance can also combine with stray inductance to give parasitic oscil-
lations. There are two stray capacitances in this transformer design; the self-
capacitance within the planar traces in a single PCB, and between the stacked
PCBs.
While this is a major issue in Radio Frequency (RF) circuits, the transformer will
operate at a maximum of 50kHz. Hence the losses are small but present. The pro-
portionality of capacitance with regards to physical dimensions can be expressed
using Equation 3.8.

C ∝ A

d
(3.8)

where, "A" is the area of parallel plate exposure [mm2] and "d" is the distance of
separation [mm].

A few effective methods to reduce stray capacitance are based on the fundamental
proportionality of capacitance [16].

1. Separation of wires (in case of traces, more plate spacing and less plate area)
2. Using guard rings
3. Using ground/power planes
4. Shielding between the input and output

The priority is to ensure electrical insulation and since the operating frequency is
low, it results in low stray capacitance effects. The design is meant to ensure that
there is no inception or chances of breakdown. Taking this into account, ways to
reduce Cstray are considered optimization and not elaborated in this thesis work.

27





4
Transformer Primary Calculations

and Setup

This chapter describes the methods followed for choice of core and primary winding,
simulations on COMSOL for B-field, and loss calculations.

4.1 Transformer Input and Output Specifications
The transformer’s input supply provided by the SCR system, and the output spec-
ifications are presented in Table 4.1.

Table 4.1: Transformer specifications

Parameter Value
Primary voltage 500Vrms
Primary current 50Arms
Secondary voltage 50kVDC
Secondary current 0.5ADC

Frequency 50kHz

4.2 Transformer Design Procedure
The design method followed for the high frequency, high voltage planar transformer
is illustrated in Figure 4.1. Each step is further elaborated in following sections,
with conditions and calculations.

29



4. Transformer Primary Calculations and Setup

Figure 4.1: Design method for the high frequency planar transformer

From the input parameters and the output desired, primary and secondary turns are
obtained. The permissible flux density is calculated with the limit of saturation, and
the number of cores are chosen. Based on required insulation, the core dimensions
are chosen, and the availability and ease of assembly leads to the choice of material
and shape respectively. Core and winding losses are calculated, and this can lead
to change in number of primary turns and Bmax values. From tests, the number
of primary turns were narrowed down to an optimal number. These are setup on
COMSOL and the B-field is simulated.
For the secondary, number of turns are calculated using Equation (5.1). From this,
the number of PCB stacks required can be calculated to obtain the required output
voltage. The secondary current, that the PCB must handle, gives the optimal trace
width. Insulation considerations here are the trace gap, clearance and creepage
distances, use of bobbin and PCB coating. To find the setup with the least E-field
stresses, the secondary is modelled for 40 turns.
To calculate the efficiency of the transformer, the core and winding losses of the

30



4. Transformer Primary Calculations and Setup

primary, the AC loss of the secondary, and the diode loss are considered.

4.3 Choice of Core
Considering the electrical requirements of the transformer’s core, the suppliers and
materials shown in Figure 4.2 are available.

Figure 4.2: Core suppliers and materials

For the transformer, the U-shaped core appears to be most suitable based on sup-
plier datasheets. The core material is made of ferrite. Based on a Manganese-Zinc
composition, it has a high resistivity that reduces eddy currents to a negligible value.
Since E-cores have a small window area, U-cores were assembled to form the shell-
type design. Based on design limitations and material availability, the standard core
shape U93/76/30 is chosen. This core has the largest cross-section area, hence re-
quiring fewer parallel cores. Certain assumptions, such as taking 2,3, and 4 primary
turns were considered as a starting point, after which the values were optimized to
obtain the number of parallel cores that returned the least losses for the specifica-
tions. For the chosen shape, core dimensions, temperature range and frequency, the
material N87, from the supplier TDK, has been found to be the most cost-effective
solution [11].

After the core is chosen, the flux density must be deduced based on the frequency
and core properties. Theoretically, it is advised to choose the operating Bmax value
at the “knee” of the curve which is the closest to saturation. However, on practical
implementation, it was found that operating close to Bsat causes excessive heating
due to the large volume and local hotspots, further accentuated due to several cores
making up the main core instead of a single solid core. This works well for cores
with smaller volume but otherwise the heating implications are severe and can lead

31



4. Transformer Primary Calculations and Setup

to thermal runaway. Hence it is preferred to keep the operating Bmax at a lower
value.

4.3.1 Core Calculations
The properties of the core U93/76/30 with the material N87 are presented in Table
4.2 from [11].

Table 4.2: Core properties of N87 Ferrite

Material Property of Core Value
Relative permeability 3600
Effective area (m2) 0,00084

Volume, per pair (cm3) 297
Length of magnetic path (m) 0,354

Saturation flux density, Bsat(T ) 0,39
Steinmetz coefficients

α 1,8392
β 2,9104
Kc 10,67

The Steinmetz coefficients "α" and "β" may be found from the manufacturer’s
datasheet. "Kc" can be calculated from the value of power loss "Pv", given by the
manufacturer, using Equation (2.16). For the core material N87, the coefficients,
presented in Table 4.2, are defined for a frequency of 50kHz.
It is also noted that, given the properties of N87 Ferrite, β > α. From Equation
(2.16), variation of Bmax largely affects core loss, hence it is desirable to minimize
Bmax by varying N and Ae. From factors described in Subsection [2.2.1], and the
window area of the chosen core, N = 4 primary turns was found to be the best
choice. Increasing the turns over N = 4 would mean that greater secondary turns,
or increased number of stacks would be required to maintain the transformer’s turn
ratio, which would not fit within the core window according to insulation standards.
Reducing the number of turns from N = 4 would require an increased cross-section
area, leading to increased material cost.
The magnetizing inductance is calculated for each combination of cores and flux den-
sity. For this, it is necessary to calculate the reluctance of the core. The equation
used to calculate the magnetizing inductance "Lm" and reluctance "Rc" are described
by Equations (2.13) and (2.14) respectively. Based on the equations and data, the
volume loss and magnetizing inductance are calculated as tabulated below. From
the calculations presented in Table 4.3, the core’s total reluctance "Rc", magnetizing
inductance "Lm", and core loss (Hysteresis) were found for 4 primary turns carrying
500Vrms, at 50kHz.

32



4. Transformer Primary Calculations and Setup

Table 4.3: Calculations for core at a frequency of 50 kHz

Parallel cores Bmax % of Bsat Core loss Volume loss Rc Lm
Units Tesla % mW/cm3 kW kH−1 mH
1 0,37 95,39 800,02 0,48 53,42 0,3
2 0,19 47,70 106,41 0,13 26,71 0,6
3 0,12 31,80 32,70 0,058 17,81 0,9
4 0,09 23,85 14,15 0,03 13,36 1,2
5 0,07 19,08 7,39 0,02 10,68 1,5

The calculations presented in Table 4.3 present the core loss for different number of
parallel cores for N = 4. With regards to the core’s temperature rise as presented
in Table 6.1, material cost, and operating frequency, the option of 3 parallel cores
was found to be most suitable.

4.3.2 Core Packing
The assembly of the cores also needs consideration. Since they are discrete units,
any gaps in assembly will mean there is air introduced into the magnetic path, and
the much weaker magnetic permeability of air will not let the transformer transfer
power and will cause hotspots between cores. Gluing them together would require
the material to have both high magnetic permeability as well as a long life, as the
core will operate at 1000C. Existing solutions within time and cost feasibility indicate
that the glue will lose its properties after several thousand hours of operation, hence
robust mechanical packing has been considered.

4.3.3 Measurement of Core Loss
As mentioned in Section [2.3.1], the core material can be characterised using the
core loss data. The core loss is plotted for different frequencies, voltages and flux
densities. Hence, to find the core loss, 12 pieces of the U93/76/30 core are assembled
as two E-cores to form a shell-type construction. The primary and secondary wind-
ings are wound on the middle leg with four turns each. From the manufacturer’s
data and the core dimensions, the cross-sectional area is found to be 5040 [mm2].
The equipment used for the measurements are presented in Table 4.4.

Table 4.4: Instruments used in core loss measurement

Instrument Model
Oscilloscope Tektronix MD04104C

Rogowski current probe CWT mini 3B
Differential probes Tektronix THDP0200

Heat camera FLIR E60

Figure 4.3 illustrates the core loss measurement setup. A variable 3-phase voltage
source is rectified and applied to a full bridge converter of variable frequency. The
output of the full bridge is connected to the primary winding with a capacitor (1,25

33



4. Transformer Primary Calculations and Setup

µF) in series. The primary current is measured with a Rogowski probe and the
secondary voltage is measured with a differential probe. Using a digital oscilloscope,
the two signals are obtained and the average power is calculated by multiplying the
signals and utilizing the built-in mean function.

Figure 4.3: Core loss measurement setup

Using this setup, the average core loss was measured for variations in frequency, flux
density and voltage, presented in Chapter 6.

4.3.4 COMSOL Simulations
The FEM-based software COMSOL Multiphysics 6.0 is used throughout this thesis
work to verify and support calculations. Under the core, B-field distribution and
magnetizing inductance are mainly simulated for the core’s dimensions.

4.3.4.1 Geometry

The core was built using the 3D geometry facility, with dimensions from the data
sheet [11]. The geometry is shown in Figure 4.4. The material properties of interest
are mainly the magnetic permeability of the cores and electrical conductivity of the
coils, presented in Table 4.5.

Table 4.5: N87 material properties used in COMSOL

Material property Value Unit
Electrical conductivity 1e-12 S/m
Relative permittivity 1
Magnetic field norm normH A/m

Magnetic flux density norm normB T

34



4. Transformer Primary Calculations and Setup

Figure 4.4: Core geometry setup in COMSOL

Since the ferrite core does not have a fixed magnetic permeability, it has been defined
as a function of the varying magnetization with respect to the applied B-field. The
values are graphed in the BH-curve by interpolation from the core material’s data
sheet [11], using COMSOL’s "BH-curve checker" application, as shown in Figure 4.5.

Figure 4.5: BH interpolation of N87 ferrite

4.3.4.2 Mesh

The “free tetrahedral” option allows us to choose the meshing degree for different
domains, hence domains such as air, and the coils were given normal meshing sizes.

35



4. Transformer Primary Calculations and Setup

The core was given a finer mesh, considering the filleted and sharp edges that can
cause changes in results, if not meshed properly. The meshed geometry is shown in
Fig 4.6.

Figure 4.6: Free Tetrahedral (Selective) meshing

The mesh statistics of the core geometry are presented in Table 4.6.

Table 4.6: Mesh statistics

Domain Mesh type Mesh size limits
Air Extra coarse 13,5 mm - 75 mm
Core Normal 2,7 mm - 8 mm

4.3.4.3 Physics

To study the B-field, the “magnetic fields” physics was chosen for simulation. The
“coil” option was used to identify the coil carrying the primary current. Only the
primary coil was considered since the objective here is to obtain B-field which is
independent of the load. “Effective BH curve” was the mode of coil induction to
account for the non-linear BH curve of the core. The entire geometry had a magnetic
insulation across its boundary to limit the simulation, with no initial value (initial
magnetic field = 0 Wb/m in all axes).

4.3.4.4 Study

A frequency sweep was performed for 50kHz in a period of 20 steps from 0Hz. The
magnetic flux in the core is in phase with the magnetizing current, which lags the
supplied current by 90 degrees. Since the flux is also alternating as it is in the case

36



4. Transformer Primary Calculations and Setup

of any transformer action, a frequency sweep helps in gauging its values as it varies
with time.
An additional study “Coil geometry analysis” was chosen for COMSOL to realise the
coil’s structure before the frequency sweep is performed. This feature enables us to
define a physical domain that is not necessarily "coil" shaped, and whose properties
are defined as a coil having a certain number of turns.

4.3.4.5 Equations

The governing equations that COMSOL uses to compute this model in frequency
domain study include Maxwell’s equation of Ampere’s law, Equation (2.1). This
explains how the alternating current generates a magnetic field around the current’s
axis. The current density is found using Equation (4.1).

J = σE (4.1)

where, "J" is the current density, expressed as the conductivity of the material "σ"
times the applied E-field.
The curl of the magnetic field at any point in space is equal to the obtained current
density at that point. This relates magnetic fields to moving charges described by
Equation (4.2).

~B = O× ~A (4.2)

where " ~A" is the tangential component of the magnetic potential. The Electric field
under frequency domain is described by Equation (4.3).

~E = −∂
~A

∂t
(4.3)

4.4 Boundary Conditions
Boundary conditions are constraints that act as limits to solving one or many dif-
ferential equations. Unlike initial value problems that have conditions defined for
one extreme, and are solved in time, the application of electromagnetism is limited
by spatial dimensions of the defined domain. It helps to constrain the behaviour
of fields at a boundary where two different media intersect. An improperly defined
boundary condition may lead to the convergence or divergence of an incorrect solu-
tion [12].
The boundary conditions for the electric current density defines the limits for the
electric current within the concerned domain, the surface region acting as an ideal
insulator. Only a surface charge can be contained at the boundary of the two media,
described by Equation (4.4).

~n ·
(
~D1 − ~D2

)
= ρs (4.4)

37



4. Transformer Primary Calculations and Setup

where, " ~D1" and " ~D2" are the perpendicular "~n" electric flux densities [C/m2] of both
media, and "ρs" is the boundary surface charge "C".
The boundary conditions for ~A contain the magnetic field within the concerned
domain, causing the magnetic field at the boundary to be tangential. It is described
by Equation (4.5).

~n ·
(
~B1 − ~B2

)
= 0 (4.5)

where, " ~B1" and " ~B2" are the perpendicular "~n" electric flux densities [C/m2] of both
media.

4.5 Primary Winding
For a DC input, the conductor exhibits I2R loss, and the resistance is given by the
resistivity of the material at that temperature, its length and cross section. The AC
resistance is given by a coefficient for skin effect and proximity effect. The primary
Litz winding is purchased from the manufacturer PACK [13].

4.5.1 Primary Winding Geometry
The primary winding was designed using litz wire, based on [14]. The benefit of
using litz wire as described earlier is to reduce AC effects. By reducing the diameter
calculated by several factors including operating frequency, the transmission almost
acts as DC, and the compensation for using lesser cross section is done by increasing
the number of such strands. Using the method described in Section [2.5], the litz
wire parameters were designed.

• Using Equation (2.19), the skin depth at 200C for copper at 50kHz was calcu-
lated to be 0.3185mm.

• The winding parameters "b" and "Ns" are chosen based on the core window and
number of primary turns respectively. The core window is 36mm, subtracting
3mm on either side for the bobbin, we have 30mm. The number of primary
turns is 4 which is chosen as "Ns".

• The constant "k" is described as a value taken depending on the strand diame-
ter. The diameter is chosen as δ/e. Within the skin depth, 67% of the current
density exists.
From the manufacturer’s recommendation, for the skin depth of 0.3mm, the
recommended diameter comes to 0.11mm. The corresponding "k" value is
1800, from [14].
Recommended number of strands is given by Equation (2.21). For the chosen
values, ne = 1215, PACK offers a bundle of 1260 strands which is closest to
the requirement. The distribution is 5 · 5 · 51; 5 bundles of 5 smaller bundles,
with each carrying 51 copper strands of 0,1mm diameter.

38



4. Transformer Primary Calculations and Setup

Table 4.7: PACK’s RUPALIT V155 Litz cable [13]

Cable parameters Value Unit
Copper area 10,013 mm2

DC Resistance 1,762 mΩ/m
No. of strands 1260 -

Diameter of each strand 0,1 mm
Type of cable P155 G1 Litz wire
Construction 5 ·5 · 51 -

4.5.2 Primary Winding AC Resistance
The AC resistance of the winding is found by multiplying the DC resistance provided
by the manufacturer, as presented in Table 4.7, with the AC loss factor found from
Equation (2.22). The cable AC resistance is the AC resistance at the transformer’s
operating temperature, multiplied by the cable’s length, as presented in Table 4.8.

Table 4.8: Primary winding AC resistance

Cable parameter Value Unit
Manufacturer provided DC resistance 1,762 mΩ/m

Sullivan loss coefficient for AC 2,32 -
Calculated AC resistance 4,087 mΩ/m
AC resistance for 1000C 5,362 mΩ/m

Cable length 5 Meters
Cable AC resistance 0,027 Ω

39





5
Transformer Secondary Design,

Calculations and Modeling
Procedure

This chapter describes the design procedure followed for the secondary of the trans-
former, including PCB design calculations and arrangement, E-field setup on COM-
SOL and diode loss calculations.

5.1 Secondary Winding
As described in Equation (5.1), the turns ratio can be expressed as the ratio of
respective voltages of the primary and secondary to the ratio of the respective num-
ber of turns. Therefore, the number of turns required for the secondary can be
calculated, as presented in Table 5.1.

Vs
Vp

= Ns

Np

(5.1)

where, "Vs" is the voltage output at the secondary, "Vp" is the voltage at primary,
"Ns" and "Np" are the number of secondary and primary turns respectively.
Stacked PCBs are used in the secondary to provide the required output. Based on
the requirement, number of stacks needed can also be calculated as presented in
Table 5.1. These stacks are a total of 10, split into 5 on either side of the primary.

Table 5.1: Secondary turns and PCB stacks calculations

Secondary parameters Value Unit
Vp 500 Vrms

Vs 50 kVDC
Voltage per stack 5 kVDC

Number of stacks needed 10 -
Number of turns for: Primary Secondary

3 30
4 40

41



5. Transformer Secondary Design, Calculations and Modeling Procedure

5.2 PCB for Secondary

This transformer design is meant to operate at 50kV. Since there is an idea on the
number of secondary turns required, the geometry of the winding, required gaps and
clearances need to be considered. The PCB design was first modelled in COMSOL
Multiphysics for optimization and validation, and further designed using the Altium
tool, for placing orders with suppliers, which is not elaborated here. The coils are
arranged on a PCB for the following reasons.

• Mechanical support offered by the PCB to hold the coils.
• PCB’s insulation qualities, which along with the coating would enclose the

windings’ E-field stresses.
• Easy to manufacture and assemble PCBs.

These together form the basis of the "planar" arrangement of the secondary windings.

5.2.1 PCB and traces
While modelling the secondary PCB, several designs were considered. The objective
is to obtain 50 kV from a set of stacked PCBs. Each PCB would have a planar
winding and it would fit in the core window, with a suitable number of turns and
a full-bridge rectifier across the output. Several such PCBs would be stacked and
their terminals would be connected in series to obtain the full output. This could
be done by having increased turns to have fewer stacks, or vice-versa as long as the
output voltage is reached.
The factors that would define the construction of the PCB are listed here.

• For the PCB, with each turn, there will be downsides such as stray capacitance,
fringe capacitance and parasitic capacitance.

• The increasing outward spiral will also increase length, causing an increase in
resistance (geometric series) with linear increase in turns.

• There will be a limit on the number of turns since the core-window is 36mm
for the U93/76/30 core.

• The trace width would have to be chosen for the amount of current and skin
depth needed, and inter-trace width would have to be calculated to prevent
inter-trace breakdown.

• Since the cores are grounded, the separation between the innermost (say d1 )
and outermost trace (say d2 ) would have to be sufficient.

• Inter-trace voltage difference would be Vmax · (1/total turns), but trace-core
voltage can reach Vmax during the peak.

5.2.2 PCB design calculations
The material used for the PCB in this design is an FR4 standard material and its
properties are as presented in Table 5.2 [20].

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5. Transformer Secondary Design, Calculations and Modeling Procedure

Table 5.2: Properties of PCB FR4 material

Properties Maximum value Unit
Glass Transition Temperature, Tg 160 0C

Ambient temperature, Ta 60 0C
Operating temperature, T0 105-130 0C

Dielectric strength 45 kV/mm
Dielectric constant 4,4 -
Surface resistivity 4 ·1018 Ω

where, the glass transition temperature "TG" represents the temperature at which
the solid material loses its rigidity and becomes flexible. "TO" describes the oper-
ating temperature. The dielectric strength and electric permittivity are useful in
calculating the stress experienced under an E-field. The surface resistivity addresses
its resistance to prevent creepage currents.

5.2.2.1 Dimensions of the trace

The trace design must consider the current handling capacity, insulation distances
between traces and from core to trace. The allowable AC and DC resistances, stray
capacitance and inductance values must also be considered.

• The first step was to consider the dimensions of the trace itself. The current
it would carry is 0,5Arms, at a frequency of 50kHz. Hence the cross-section
would have to be chosen such that it can handle 0,5Arms, and the width and
height would have to accommodate for skin depth.
The formula for calculating allowable current through an external trace is
taken from the IPC-2221 standards - section 6.2 [21]

I = 0, 048 · dT 0,44 · A0,725 (5.2)

where, "dT" is the temperature rise above ambient in [0C], "A" is the cross-
sectional area in [mils], "I" is the maximum current in [A]. Rearranging to get
the area as a function of current, we get -

A =
(

I

0, 048 · dT 0,44

) 1
0,725

(5.3)

To find the trace width required, the minimum required area is considered with
available trace thicknesses from the PCB manufacturer. The recommended
trace widths for given thicknesses at the ambient temperature Ta = 600C
under IPC-2221 standards are presented in Table 5.3, as derived from [22].

Table 5.3: Recommended trace widths for different trace thicknesses

Trace width Trace width
Trace thickness Tr = 200 C Tr = 400 C

70 µm 0,038 mm 0,025 mm
35 µm 0,076 mm 0,05 mm

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5. Transformer Secondary Design, Calculations and Modeling Procedure

However, while this gives an idea of the minimum cross section to handle the
ampacity, it is the length of the trace that determines the resistance, and
I2R loss heating that can cause drastic temperature rise. Once the length is
ascertained, the cross section is given a suitable margin to account for heating.

• The second step after finalizing the trace cross section is to decide the clearance
and creepage distances. These factors are important to ensure HV insulation.
Clearance is the shortest distance between each turn, and the voltage differ-
ence will be Vmax · (1/total turns). This is to prevent breakdown between
traces. This distance can be chosen for the dielectric (the worst case is to use
air to get the safest values). This distance will also have to be considered to
reduce creepage. Creepage occurs across the PCB layer’s surface.
Under the same IPC2221B standards (specifically 9592B rule for power con-
version devices), track clearance required for external layers is presented in
Table 5.4, as derived from [23].

Table 5.4: Track clearance required as per IPC2221B standards

Vpeak(V ) Uncoated (mm) Rule 9592B (mm) Coated (mm)
125 1,25 1,45 0,4

• The other distance to be considered is between the traces on each extreme to
the core since the outermost trace will be Vmax itself.

• After obtaining the least and safest distance between traces, and the minimum
core-trace distance, the turns can be modelled.

• After this, the resistances and other quantities can be calculated, Rac, Rdc, L,
C etc.

The minimum trace width offered by the PCB manufacturer PACK [13], is 0,254mm,
which is well-above the margin, and affordable for this application.

While it is desirable to keep the number of secondary turns low to reduce both loss
and ensure generous distances for insulation, operating the core at 2 and 3 primary
turns showed extreme heating for 50kHz. Since 4 primary turns indicated a steady
state temperature of 900C in air itself, the secondary now requires 40 turns to obtain
5kV per PCB. Simulations and calculations were done for a PCB design that had 40
turns split into two sets of 20 turns in the double layer "shifted" format, elaborated
in further sections.

5.2.3 PCB turns distribution
An effective method is to use double-layered windings on the PCB; having a trace
on the upper and lower portion, connected by "via" to complete the winding, and
the terminals could be traced to the extreme edges.
Using double layered windings greatly reduces number of turns that would have
to be squeezed on one side, and allows for a greater inter-trace gap. This also
reduces creepage and field strength across the PCB and air/oil, but also poses some
challenges. The parasitic capacitance greatly increases, and a proper geometry is

44



5. Transformer Secondary Design, Calculations and Modeling Procedure

needed to address this carefully. Considering the PCB’s thickness, the upper and
lower layers have to be arranged such that the stray capacitance is reduced [24].

5.2.4 PCB resistance measurement
The 4-wire method was used to measure the DC resistance of the PCB. The quan-
tities are presented in Table 5.5.

Table 5.5: 4 - wire testing for PCB trace DC resistance

Quantity Value Unit
Measured terminal voltage 7,4 VDC

Applied current 0,49 ADC
Ambient temperature 26 0C
Calculated resistance 15,1 Ω

Estimated resistance at 1000C 19,41 Ω

5.3 COMSOL Simulations
To assess the insulation distances required, the E-field distribution was verified for
the secondary. This includes PCB geometry with different trace widths and gaps,
presence of bobbin and PCB coating, placement of PCB with respect to the core,
and arrangement of PCB stacks. The E-field stresses were then simulated, across
transformer oil and PCB surface, to find the optimal insulation design for an output
of 50kV.
Along with this, two distinct models were designed in COMSOL; one with layers in
parallel and the other, with a planar offset in both x and y directions at lengths of
the trace width. On superimposing in the latter case, both traces will fill in the gaps
of the other. Comparing these two models, it was found that the shifted layers have
a lower parallel plate capacitance compared to the parallel layers. The difference in
geometries is illustrated in Figure 5.1.

(a) Parallel layers (b) Shifted layers

Figure 5.1: Placement of secondary turns in different layers on a PCB

45



5. Transformer Secondary Design, Calculations and Modeling Procedure

5.3.1 Geometry
The E-field simulations were carried out using a 2D geometry for simplicity. The
dimensions are not up to scale, however the distances are. The number of turns on
the secondary are chosen relative to the primary turns; for 4 primary turns, there
are 40 secondary, using Equation (5.1). The PCB and traces are placed at a distance
from the core, validated in further sections, along with bobbin, PCB coating and
transformer oil. The arrangement of the PCB stacks on either side of the primary
winding is also validated with respect to distances and E-field stresses.
The permittivity and conductivity of the materials are considered for E-field simu-
lations, and the corresponding properties used are presented in Table 5.6, [16], [18],
[20], [25], [26].

Table 5.6: Relative permittivities and conductivities of different dielectrics in the
geometry

Material Relative permittivity Conductivity (S/m)
Bobbin 2,3 10−16

Oil 2,2 10−13

Copper traces 1012 5,96 ·107

PCB (FR4) 4,4 10−11

PCB coating 4,4 10−11

Mylar sheet 2,3 10−11

5.3.1.1 Requirement of bobbin

In any conventional transformer, the core is surrounded by a plastic bobbin. The
purpose of this is mechanical, to hold the primary windings in place. But, it also
plays a role in electrical isolation between the core and the windings. With the
present case, a plastic bobbin made of polypropylene, with a relative permittivity
of 2,3 and conductivity of 10−16 [S/m], is used [16].
To plot the E-field and verify the need for bobbin in the present PCB design, the
geometries shown in Figure 5.2, were modelled on COMSOL. The traces are given
potentials with an offset of 125 V, starting from 48.7 kV at the first trace up to the
final trace of 50 kV, which is the output required from the PCB as per the design.
To compare the effect of the presence and removal of bobbin, the geometry is kept
intact, but the material properties are changed. In the first case, the geometry is
modelled for the conductivity of the bobbin as presented in Table 5.6 as in Figure
5.2a and for the next case, the bobbin area has the conductivity of oil as in Figure
5.2b.
The E-field stresses are then observed across oil and compared for both cases to
validate the requirement of the bobbin.

46



5. Transformer Secondary Design, Calculations and Modeling Procedure

(a) Transformer with bobbin

(b) Transformer without bobbin

Figure 5.2: Geometry for secondary to compare E-field stresses with and without
a bobbin

5.3.1.2 Comparison of Bobbin Materials

Thermoplastic polymers are used here since the bobbin must be capable of handling
high temperatures and have high mechanical strength. To decide on the most suit-
able thermoplastic, a comparison study was conducted between Polypropylene (PP)
and Polyamide (PA 12). The PA 12 material was used for the previous construction
(transformer with nanocrystalline cores) and was therefore considered for the new
planar transformer. Both materials have high thermal resistance and good mechan-
ical properties. However, polyamides have high water absorption and conductivity,
and exhibit higher losses. This makes the material less suitable for the high electrical
stress of the application. While comparing other plastics such as PP, polyethylene,
PVC and Polystyrene, PP has higher thermal resistance [16]. The motivation be-
hind the choice of bobbin material is to keep the E-field across the insulating oil less
than 7kV/mm, elaborated in Chapter 3. COMSOL simulations are carried out to
understand the impact of both the materials on E-field distribution across the PCB
geometry.

47



5. Transformer Secondary Design, Calculations and Modeling Procedure

The dielectric properties and conductivity for PP and PA 12 are listed in Table 5.7
[25].

Table 5.7: Properties of bobbin materials

Material Permittivity Dielectric Strength Conductivity
Units (Relative) kV/mm S/m

Polypropylene 2,3 23-25 10−18-10−16

PA 12 4,5 26-30 10−11

5.3.1.3 Requirement of PCB coating

To verify the need for the coating, COMSOL models were simulated with and with-
out PCB coating as shown in Figure 5.3.

(a) With PCB coating

(b) Without PCB coating

Figure 5.3: Geometry for secondary to compare E-field stresses with and without
the presence of PCB coating

48



5. Transformer Secondary Design, Calculations and Modeling Procedure

From the earlier section on insulation in Chapter 3, it is known that the stressed
E-field regions can cause breakdown of materials in the transformer. It is necessary
to have sufficient insulation and withstanding properties. With this consideration,
the PCB is coated with an insulating material of higher breakdown strength. This
helps reduce the stresses caused by high voltages on the PCB and the traces en-
closed within the coating. The E-field models here have been simulated