Analysis and Design of a 500 W DC Transformer by Tadakazu ...

Analysis and Design of a 500 W DC Transformer

by

Tadakazu Harada

B.S., Tokyo University of Agriculture and Technology, 2006

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirement for the degree of

Master of Science

Department of Electrical, Computer, and Energy Engineering

2014

This thesis entitled:

Analysis and Design of a 500 W DC Transformer

written by Tadakazu Harada

has been approved for the Department of Electrical, Computer, and Energy Engineering

Dragan Maksimovic

Robert W.Erickson

Date

The final copy of this thesis has been examined by the signatories, and we

find that both the content and the form meet acceptable presentation standards

of scholarly work in the above mentioned discipline.

iii

Abstract:

Tadakazu Harada (M.S., Electrical, Computer, and Energy Engineering)

Analysis and Design of a 500W DC Transformer

Thesis directed by Prof. Dragan Maksimovic

Fuel consumption and environmental concerns motivate new engineering

developments in the automotive industry. Vehicle electrification techniques, such as

drivetrain hybridization and replacements of various mechanical components with

electronic components require highly efficient switched-mode power converters.

Analysis and design of a DC transformer (DCX) converter configuration are

addressed in this thesis, including component design and efficiency optimization

techniques. A 500 W prototype converter is constructed and tested in two operating

modes: as a full-bridge DCX or as a dual active bridge DCX. It is shown that the

DCX converter has the potential to achieve small size and high efficiency over wide

ranges of operating conditions.

iv

Acknowledgements

I would like to thank my thesis advisor, Professor Dragan Maksimovic, for his

support and encouragement. I did not have enough knowledge about power

electronics before starting my studies at the University of Colorado at Boulder.

Thanks to my advisor’s support, I was able to keep up my motivation and to gain

valuable knowledge in the area.

Additionally, I would like to thank Professors Robert Erickson and Khurram

Afridi, who have helped me learn and better understand analysis, modeling and

design techniques in power electronics.

I am grateful to the Colorado Power Electronics Center (CoPEC) and Toyota

Motor Corporation, and I would like to acknowledge my fellow CoPEC students Hua

Chen, Hyeokjin Kim, Beom Seok Choi, and Fan Zhang for their help and support,

which I will never forget.

Finally, I would like to thank my family for their support throughout my

graduate studies and throughout my life.

Contents

Chapter

1 Introduction ..............................................................................................................1

1.1 Hybrid System ................................................................................................2

1.1.1 Toyota Hybrid System .....................................................................2

1.1.2 Power Control Unit ..........................................................................3

1.1.3 Intelligent Power Module ................................................................4

1.2 Selection of DC-DC Converters .....................................................................4

1.2.1 Soft Switching Technology..............................................................4

1.2.2 Three-Level Converter .....................................................................6

1.2.3 Proposed DC-DC Converter Architecture .......................................7

2 Power Stage Design of DC Transformer and Dual Active Bridge ........................10

2.1 Semiconductor Device Loss Model ..............................................................10

2.2 Transformer Loss Model...............................................................................14

2.2.1 Core Loss .......................................................................................14

2.2.2 Eddy Currents in Winding Conductors ..........................................14

2.2.3 Copper Loss ...................................................................................15

2.2.4 Accurate Prediction of Ferrite Core Loss

with Nonsinusoidal Waveform ......................................................17

2.3 Transformer and Tank Inductor Design Optimization..................................18

2.4 DC Blocking Capacitor .................................................................................21

vi

3 State Plane Analysis ...............................................................................................23

3.1 Review of State Plane Analysis ....................................................................23

3.2 State Plane Analysis Applied to the Dual Active Bridge .............................25

3.2.1 Primary ZVS Condition in DAB....................................................26

3.2.2 DAB State Plane Solution ..............................................................29

3.3 Analysis of DAB Operation Modes ..............................................................31

3.3.1 Mode1 : Operation with 𝛽 > 0 and Primary ZVS .........................32

3.3.2 Mode2 : Operation with 𝛽 > 0 and Hard-Switched Primary ........33

3.3.3 Mode3 : Operation with 𝛽 < 0 and Soft-Switched Primary .........35

3.4 Experimental Results ....................................................................................36

4 Experimental Results .............................................................................................39

4.1 Device Loss Analysis ....................................................................................39

4.2 Phase-Shift Optimization in DAB ................................................................43

4.3 Analysis of Optimal Efficiency Trajectory ...................................................45

4.4 Experimental Results: Conversion Ratio and Efficiency ..............................47

5 Summary and Future Work ....................................................................................49

5.1 DAB Design and Component Selection .......................................................49

5.2 State Plane Analysis of DAB Converter .......................................................50

5.3 Conclusion ....................................................................................................50

5.4 Future Work ..................................................................................................51

Bibliography ................................................................................................................53

Tables

Table

2.1 DAB Transformer Implementation Details ........................................................19

3.1 Correspondence table with DAB ..................................................................21

Figures

Figure

1.1 Toyota Hybrid System ....................................................................................2

1.2 Structure of the 3rd

generation PCU (Power Control Unit).............................3

1.3 Buck-boost SAZZ boost converter .................................................................5

1.4 Coupled-inductor converters ...........................................................................6

1.5 Three level boost converter with IGBT ..........................................................7

1.6 Proposed composite converter system architecture ........................................8

1.7 Operating modes .............................................................................................9

2.1 Switching loss simulation model ..................................................................11

2.2 MOSFET energy loss of the simulated data .................................................13

2.3 Transformer loss analysis .............................................................................19

2.4 Current waveform of tank inductor ...............................................................20

2.5 Dual active bridge converter .........................................................................22

2.6 Experimental waveforms of the prototype ....................................................22

3.1 Time-domain waveforms and state planes ....................................................29

3.2 State plane of the DAB in Mode1 .................................................................32

3.3 State plane of the DAB in Mode2 .................................................................34

3.4 State plane of the DAB in Mode3 .................................................................36

ix

3.5 Prototype DAB converter .............................................................................37

3.6 Comparison between theory and experiments in the state plane ..................38

4.1 Loss analysis of full bridge DCX and DAB DCX ........................................41

4.2 Characteristics of reverse diode and MOSFET loss analysis .......................42

4.3 Optimize phase shift between primary and secondary .................................43

4.4 Prototype experimental waveforms of the DAB ...........................................44

4.5 Efficiency contours of DAB .........................................................................46

4.6 Comparison of analytical ..............................................................................47

4.7 Prototype experimental waveforms of the full bridge and DAB DCX ........48

5.1 Prototype experimental switching transitions ..............................................51

1

Chapter 1

Introduction

Rising numbers of vehicles in the world cause increased air pollution induced by CO2 emissions.

Therefore, automotive industries focus on improvements of fuel efficiency technologies. Typical

technologies for gasoline mileage improvement include reductions in size and weight of the vehicle,

improvements in engine efficiency and powertrain components, reductions in air resistance, and increased

vehicle electrification, including use of high-efficiency switching power converters. It is expected that

auxiliary electric power consumption in vehicles will increase up to 2 kW to 3 kW in the future, because

mechanical components will be replaced by electronic components [1]. For example, the electric power

steering system using a drive motor operated from 12V increased the steering load capabilities [2]. The

impact of the total power loss in the electricity consumption can not be ignored in future. Furthermore,

drivetrain electrification techniques, such as hybridization, can lead to substantial improvements in fuel

economy. These approaches also rely on high-efficiency power electronics components.

In this chapter, hybrid vehicle systems are reviewed, and the need for high-efficiency, small-size

boost DC-DC converter is identified. This provides motivation for the work presented in this thesis,

which is focused on analysis and design of a DC transformer (DCX) converter, which can be used as a

building block in automotive applications.

2

1.1 Hybrid system

In recent years, environmentally conscious vehicles, such as hybrid vehicles (HV), fuel cell

vehicles (FCV), and electric vehicles (EV), have been developed to reduce the effects on the environment.

In particular, HV’s have spread most rapidly across the world. The basic HV system is reviewed in the

following section.

1.1.1 Toyota Hybrid system

A popular hybrid system is the Toyota Hybrid System II (THS-II) shown Fig. 1.1 [3]. This

system consists of two motors, two inverters, and a boost DC-DC converter. The bus voltage is boosted

by the DC-DC converter from 200 V obtained from a nickel-metal hydride battery voltage to 650 V [3].

The driving motor output is 60kW. The boost DC-DC converter power rating is 25kW. The high voltage

can reduce the size of motor and improve the system efficiency. Additionally, the bus voltage can be

controlled depending on the load conditions.

Figure 1.1: Toyota Hybrid System

3

1.1.2 Power Control Unit

In the 3rd

generation Toyota hybrid system, the maximum voltage of the boost converter increased

from 500 V to 650 V. Therefore, the volume of the power control unit, which is 11.2 liters, is reduced by

37% compared to the previous model. The power control unit (PCU) which consists of the inverters for

the generator and the motor, and the boost DC-DC converter, is shown in Figure1.2. A water cooling

system has been employed in the PCU.

Figure 1.2: Structure of the 3rd

generation PCU (Power Control Unit) [4]

4

1.1.3 Intelligent Power Module

Intelligent power module (IPM) of Fig. 1.2 consists of the high voltage power module and the

control circuit which contains the gate drivers. The power module is implemented using Insulated Gate

Bipolar Transistors (IGBT) and Free Wheeling Diodes (FWD). An important issue facing the design of

the power module for automotive applications is related to the thermal management. The thermal design

affects the size of device, thereby reducing cost. Therefore, the water cooling system is applied to the

IPM.

The most important issue facing the design of a high power hybrid system is the size of the boost

DC-DC converter between the battery and the inverters. In particular, the boost converter requires a large

capacitor and a large inductor. One approach to reducing the size of the capacitor and the inductor is to

increase the switching frequency. However, as the switching losses increase with increasing switching

frequency, it becomes difficult to keep the converter efficiency high. As a result, there is a tradeoff in the

selection of switching frequency. In this thesis, the design goal of small size and high efficiency is

addressed through considerations of converter topology and soft-switching techniques.

1.2 Selection of DC-DC Converter

Candidate DC-DC converter topologies in a hybrid system are reviewed in this section.

1.2.1 Soft Switching Technology

An approach to improving the DC-DC converter efficiency is to attempt to mitigate some of the

switching losses.

Soft switching techniques include zero-voltage switching (ZVS) or zero-current switching (ZCS).

In a ZVS transition, a switch is turned on at a time when the voltage across its terminals is approximately

zero, so that turn-on switching losses are ideally zero. In a similar manner, ZCS can be described as a

5

switching transition when a switch is turned off at a time when the switch current is approximately zero,

so that turn-off switching losses are ideally zero.

One soft switching circuit is the snubber-assisted zero-voltage transition/zero-current transition

(SAZZ) converter described in [5] and shown in Fig. 1.3. The main switches Sm1, Sm2, and the reactor

L1 are the same as in the conventional boost converter. The auxiliary snubber circuit is implemented as

the auxiliary active switches Sa1, Sa2, Diode D1-4, capacitors C1, C2 and reactor L2. In general, soft

switching approaches usually adopt auxiliary circuits to achieve soft switching. However, there are

additional conduction and switching losses in the auxiliary circuit.

Figure 1.3: Buck-boost SAZZ boost converter [5]

Another approach is based on the interleaving converter modules, i.e. operating converter

modules in parallel, with appropriate phase shifts between the modules, as shown in Fig. 1.4 [6]. This

approach can reduce the input and the output capacitor voltage ripples, thereby allowing reduced

capacitor. Additionally, the switching frequency of the switches in a two-phase interleaved converter is

one half of the inductor current frequency, so that the switching losses can be reduced. A disadvantage of

this approach is that additional transformer introduces additional size and loss.

6

Figure 1.4: Interleaved coupled-inductor converters [6]

1.2.2 Three-Level Converter

Three-level converters ([7], [8]) can significantly reduce inductor size. The inductor current ripple

compared to the traditional boost converter is significantly reduced. The voltage stress of the switching

devices is also one half of the voltage stress in the conventional boost converter. As a result, the switching

loss of each device is reduced. Lower-voltage devices, e.g. 600V MOSFETs can be applied in the three-

level converter even though the output voltage exceeds 600 V. The MOSFET switches have potentials to

reduce switching losses further. However, an extra capacitor is needed with large RMS current rating,

which implies increased capacitor size.

7

Figure 1.5: Three level boost converter with IGBTs; MOSFET switches can also be used in this

configuration

1.2.3 Proposed DC-DC Converter Architecture

A new boost composite converter architecture consisting of a boost converter, a buck converter,

and a DC transformer (DCX) as shown in Fig. 1.6 has recently been introduced [9].The system allows

600V MOSFETs to be used as switches, with advantages in terms of low on resistance and low switching

loss. The DCX behaves in a manner similar to an ideal transformer, where the primary and secondary

winding voltages and currents are related by the number of turns. The DCX can achieve very high

efficiency, but the voltage conversion ratio between the primary and the secondary is fixed. Based on the

hybrid system, the output voltage can be controlled by demand of power from the motor in a range from

200 V to 650 V. Therefore, a candidate boost DC-DC system needs to be able to adjust the conversion

ratio 𝑀 = 𝑉𝑜𝑢𝑡 𝑉𝑖𝑛⁄ . The boost converter together with the DCX can be configured to allow adjustment of

𝑉𝑜𝑢𝑡. In this case, the system conversion ratio is 𝑀 = 𝑁𝐷𝐶𝑋 + 𝑀𝑏𝑜𝑜𝑠𝑡. However, when a system

8

conversion ratio 𝑀 less than 𝑁𝐷𝐶𝑋 + 1 is required, the DCX should be shut down, and the boost module

should produce the required output voltage. In this transition from the operating mode with only the boost

module to the operating mode with the DCX and the boost module together, an abrupt voltage change

may occur.

One approach to mitigate the voltage discontinuity is to control the DCX output voltage. Another

buck converter is added before the DCX, so that the DCX input voltage can be controlled smoothly. In

this architecture, the system conversion ratio is given by 𝑀 = 𝑀𝑏𝑢𝑐𝑘𝑁𝐷𝐶𝑋 + 𝑀𝑏𝑜𝑜𝑠𝑡. This system can

efficiently achieve boost function over the required ranges of input and output voltages by selecting the

appropriate operating mode.

Figure 1.6: Proposed composite converter system architecture [9]

9

Figure 1.7: Operating modes in the composite converter

The DCX module is a key component in the composite system architecture. In the rest of this thesis,

analysis and design of a scaled (500W) DCX prototype are addressed, together with experimental results

comparing the DCX efficiency under various operating conditions.

10

Chapter 2

Power Stage Design of DC Transformer and Dual Active Bridge

In general, a DC-DC converter consists of capacitors, magnetics, and semiconductor devices.

Ideally, the converter efficiency is 100%. However, losses exits under real operating conditions. The

efficiency of a converter is defined as

𝜂 =𝑉𝑜𝑢𝑡𝐼𝑜𝑢𝑡

𝑉𝑖𝑛𝑉𝑖𝑛 (2.1)

In general, the converter is designed to maximize converter efficiency to the extent possible, and to

conform to requirement specifications, which include ranges of input and output voltages, as well as

output power. The DC-DC converter configuration considered in this chapter is the DC transformer

introduced in Chapter 1.

2.1 Semiconductor Device Loss Model

Although the DC transformer can achieve zero-voltage switching, it still has some switching loss.

The switching loss can be estimated based on simulations of detailed device models, or by empirical

observations. With zero-voltage switching, the switching loss is estimated as

𝑃𝑠𝑤𝑄,𝑠𝑜𝑓𝑡 ≅ 0.5𝑃𝑠𝑤𝑄,ℎ𝑎𝑟𝑑 (2.2)

where 𝑃𝑠𝑤𝑄,ℎ𝑎𝑟𝑑 is the transistor hard-switching loss.

11

The soft switching loss of the rectifier diode is approximated using the following empirical model

𝑃𝑠𝑤𝑄,𝑠𝑜𝑓𝑡 ≅ 𝛼𝑉𝐷𝐼𝐷𝑡𝑟𝑟𝑓𝑠 (2.3)

where 𝑉𝐷 is the voltage across diode, 𝐼𝐷 is the current through diode,𝑡𝑟𝑟 is the recovery time of the diode,

and 𝑓𝑠 is the switching frequency. In the considered DCX prototype, the resulting losses are modeled with

𝛼 = 0.15.

The device model of Infineon IPW65R660CFD CoolMOS is provided on the Infineon website

[10]. The simulation circuit is shown in Fig. 2.1. The gate drive voltage is 10V. The circuit is simulated

with different voltage and current values, and the energy loss during the switching transition is extracted.

The hard-switching loss of the body diode is approximated as the energy loss during high side MOSFET

turn-on transition, and switching loss of the transistor is approximated as the energy loss during high side

MOSFET turn-off transition.

Figure 2.1: Switching loss simulation model

The switching loss model of the MOSFET and the body diode are given by

𝐸𝑠𝑤_𝑀𝑂𝑆𝐹𝐸𝑇 = 𝐾𝑠𝑤_𝑀𝑂𝑆𝐹𝐸𝑇(𝐼𝑖𝑛 + ∆𝐼)𝑎𝑀(𝑉𝑜𝑢𝑡)𝑏𝑀 (2.4)

𝑃𝑠𝑤_𝑏𝑜𝑑𝑦_𝑑𝑖𝑜𝑑𝑒 = 𝐾𝑠𝑤𝐷(𝐼𝐿 − ∆𝐼𝐿)𝑎𝐷𝑉𝑏𝐷 (2.5)

12

The parameters in equations (2.6) and (2.7) are calculated by using Matlab curve fitting tool.

𝐾𝑠𝑤_𝑀𝑂𝑆𝐹𝐸𝑇 = 11.01𝐸 − 7, 𝑎𝑀 = 1.905, 𝑏𝑀 = 1.223 (2.6)

𝐾𝑠𝑤_𝐷 = 28.7𝐸 − 5, 𝑎𝐷 = 0.9248, 𝑏𝐷 = 1.068 (2.7)

The simulation data and the switching model are compared in Fig. 2.2.

13

(a)

(b)

Figure 2.2: Energy loss based on simulation data compared with the loss predicted by the model, for (a),

MOSFET and (b) diode.

14

2.2 Transformer Loss Model

2.2.1 Core Loss

The core loss 𝑃𝑓𝑒 can be approximated based on the core material datasheet.

∆𝐵 =𝜆

2𝑛𝐴𝑐 (2.8)

𝑃𝑓𝑒 = 𝐾𝑓𝑒(∆𝐵)𝛽𝐴𝑐𝑙𝑚 (2.9)

where 𝜆 is the core primary flus linkage, 𝐴𝑐 is the equivalent core cross-section area, 𝑛 is the primary

number of turns, and 𝑙𝑚 is the equivalent core magnetic path length. The waveform of flux density ∆𝐵 is

triangular in the DCX converter, so ∆𝐵 can be decomposed into a series of sinusoidal waveform by

Fourier series expansion,

𝑓(𝑥) =8

𝜋2 ∑ 𝑠𝑖𝑛 (𝑛𝜋2 )

𝑠𝑖𝑛(𝑛𝑥)𝑛2

∞𝑛=1

(2.10)

2.2.2 Eddy Currents in Winding Conductors

Eddy currents cause additional power losses in winding conductors. The length 𝛿 is the skin

depth. The resistivity 𝜎 is 1.724 ∙ 10−6 𝛺𝑐𝑚 at room temperature. The skin effect can be modeled by

calculating the effective area of copper, where the skin depth 𝛿 is given by

𝛿 = √𝜌

𝜋𝜇𝑓 (2.11)

15

A layer of winding conductors consists of 𝑛𝑙 turns. The layer can be approximately modeled as a foil,

where the winding porosity 𝜂 increases the effective resistivity and therefore increases the effective skin

depth,

𝛿′ =𝛿

√𝜂 (2.12)

The winding porosity 𝜂 is given by

𝜂 = √𝜋

4𝑑

𝑛𝑙

𝑙𝑤 (2.13)

𝜑 is defined as the ratio of the effective foil conductor thickness ℎ to the effective skin depth 𝛿′.

𝜑 =ℎ

𝛿′= √𝜂√

𝜋

4

𝑑

𝛿 (2.14)

2.2.3 Copper Loss

Copper loss includes both DC and AC components. DC copper loss is given by the DC current in

the winding. The DC copper loss is

𝑃𝐶𝑢_𝐷𝐶 = 𝐼𝑟𝑚𝑠2 𝑅 (2.15)

When multiple conductors are packed into layers, the magnetomotive force may increase from layer to

layer, and the corresponding eddy currents may increase significantly. This effect is known as the

proximity effect [11]. Interleaving the winding between primary and secondary layers in the transformer

can reduce the copper losses due to the proximity effect. The basic analysis of interleaving the winding is

reviewed here based on [12]. The quantity 𝑚 is the ratio of the magnetomotive force ℱ(ℎ).

𝑚 = ℱ(ℎ)

ℱ(ℎ) − ℱ(0) (2.16)

16

For a 1:2 transformer, if we interleave the primary and secondary winding layers in the

secondary/primary/secondary manner, the magnetomotive force will cancel each other, and the result is

equivalent to a single layer winding.

In the DCX prototype, Litz wire, which consists of many thin strands of wire, is applied in the

designs of the reactor and the transformer. The Litz wire can reduce the losses associated with the skin

and proximity effects. However, in Litz wire, the current among the strands is actually not evenly

distributed. To model this effect, we assume all the strands are packed in a square. Therefore for a total of

𝑁 strands, the equivalent number of layers in the primary is √𝑁𝑝𝑟𝑖𝑚𝑎𝑟𝑦. If we define

𝑀 = √𝑁𝑝𝑟𝑖𝑚𝑎𝑟𝑦𝑚 (2.17)

then the ac loss to dc loss ratio 𝐹𝑅 can be found by the classical Dowell’s equation

𝐹𝑅 =𝑃𝑝𝑟𝑖

𝑃𝑝𝑟𝑖,𝑑𝑐= 𝜑 [𝐺1(𝜑) +

2

3(𝑀2 − 1)(𝐺1(𝜑) − 2𝐺2(𝜑))] (2.18)

The functions 𝐺1(𝜑) and 𝐺2(𝜑) are

𝐺1(𝜑) =𝑠𝑖𝑛ℎ(2𝜑) + 𝑠𝑖𝑛(2𝜑)

𝑐𝑜𝑠ℎ(2𝜑) − 𝑐𝑜𝑠(2𝜑) (2.19)

𝐺2(𝜑) =𝑠𝑖𝑛ℎ(𝜑) 𝑐𝑜𝑠(𝜑) + 𝑐𝑜𝑠ℎ(𝜑) 𝑠𝑖𝑛(𝜑)

𝑐𝑜𝑠ℎ(2𝜑) − 𝑐𝑜𝑠(2𝜑) (2.20)

The waveform of input current 𝐼𝑖𝑛 is an approximately square-wave waveform, and therefore 𝐼𝑖𝑛 can be

described by Fourier series expansion,

𝑓(𝑥) = ∑4

𝜋

1

2𝑛 − 1𝑠𝑖𝑛{(2𝑛 − 1)𝑥}

∞

𝑛=1 (2.21)

17

The total copper loss is given by

𝑃𝑐𝑢_𝑝𝑟𝑖𝑚𝑎𝑟𝑦 =𝐼𝑖𝑛

2

2𝑅𝑝𝐹𝑅 (2.22)

Where 𝐼𝑖𝑛 is the amplitude of each harmonic according to Fourier expansion in (2.22).

2.2.4 Accurate Prediction of Ferrite Core Loss with Nonsinusoidal

Waveforms

Section 2.3.1 has shown methods for Fourier analysis applied to the calculation of core loss. The

generalized Steinmetz equation (GSE) [13] provides a more accurate prediction of core losses in the

presence of nonsinusoidal waveforms,

𝑃𝑓𝑒(𝑡) = 𝑘1 |𝑑𝐵

𝑑𝑡|

𝛼

|𝐵(𝑡)|𝛽−𝛼 (2.23)

The GSE approach takes into account only the instantaneous flux density. The improved generalized

Steinmetz equation (iGSE) [14,15] considers the flux density over a period.

𝑃𝑓𝑒(𝑡) = 𝑘𝑖 |𝑑𝐵

𝑑𝑡|𝛼

(∆𝐵)𝛽−𝛼 (2.24)

where 𝛥𝐵 is the peak-to-peak flux density. The equations for time-average loss are

𝑃𝑓𝑒̅̅ ̅̅ =

1

𝑇∫ 𝑘𝑖 |

𝑑𝐵

𝑑𝑡|

∝

(∆𝐵)𝛽−𝛼𝑑𝑡𝑇

0

(2.25)

𝑘𝑖 =𝑘𝑓𝑒

(2𝜋)𝛼−1 ∫ |𝑐𝑜𝑠 𝜃|𝛼2𝛽−𝛼𝑑𝜃2𝜋

0

(2.26)

18

2.3 Transformer and Tank Inductor Design Optimization

A designer makes an effort to minimize the loss and the size of the magnetics, and there are

variable design parameters such as core materials, core shape, number of turns, wire size. A useful

method for the examination of magnetics design is the 𝐾𝑔 method [16]. This approach can optimize the

wire size of the windings for minimized copper loss.

𝐾𝑔 ≥𝜎𝐿2𝐼𝑚𝑎𝑥

2

𝐵𝑚𝑎𝑥2 𝑅𝐾𝑢

(2.27)

𝐾𝑔 =𝐴𝑐

2𝑊𝐴

(𝑀𝐿𝑇) (2.28)

where (MLT) is the mean-length-per-turn of the winding. The expression for 𝐾𝑔 is a function of the core

components. In (2.27), the quantities 𝐼𝑚𝑎𝑥, 𝐵𝑚𝑎𝑥, L, 𝐾𝑢, R, and 𝜌 are given specifications. The quantity

𝐾𝑔 which refers to magnetics core datasheet is made large enough to satisfy (2.27). One should note that

the Kg method strictly applies only to the design of inductors with negligible core or proximity losses.

The core material TSF-50ALL from TSC international is chosen to implement the DAB converter

transformer. 50ALL is a ‘flat-line’ material, which means that the material properties such as core loss,

permeability, and saturation flux density, do not change much as temperature varies. A winding fill factor

𝐾𝑢 = 0.4 is assumed given the PQ core shape and Litz wire used for the windings.

The designer will optimize the total power loss between core and copper losses in magnetic

device. A large number of turns in the devices can reduce the core loss and peak flux density, but longer

primary and secondary windings also increase winding resistance and conduction losses. In this tradeoff,

Fig. 2.3 shows the result of total power loss with the PQ35/35 core. Copper loss is linearly-increasing

depending on the number of primary turns, because of the increasing winding resistance. In contrast,

increasing the number of turns results in lower peak ac flux density, so that the core loss is reduced, as

shown by (2.25). The primary number of turns to minimize total loss at 300W output power is around 57

turns. For the prototype construction, the number of turns selected was somewhat lower, 44 turns, because

of practical limitations imposed by the core window area, and the fact that further benefits of increasing

19

the number of turns to the theoretically optimal value are relatively small. More turns cannot provide a

significant benefit. Table 2.1 summarizes the DAB transformer design.

Figure 2.3: Core loss, copper loss and total loss vs. the number of turns in the primary at 300W output

power

Table 2.1: DAB Transformer Implementation Details

Core material TSF-50ALL

Core shape PQ35-35-00

Winding Primary : Litz wire 330 strands #44

Secondary : Litz wire 175 strands #44

Turns Primary : 44

Secondary : 82

Air gap 0.159mm

Core loss 0.927 W@300W

Copper loss 0.85 W@300W

20

Simulation of the DAB circuit is performed in LTspice. The trapezoidal (near square wave)

inductor current minimizes the rms current, which reduces the copper loss in both the transformer and the

inductor. The copper loss is calculated by the root-mean-square current 𝐼𝑟𝑚𝑠 through the tank inductor.

The inductor current waveform at 300 W output power is plotted in Fig. 2.4. for 3 different inductance

values. In this thesis, the tank inductor is optimized at 300 W, and the inductance of 40 H is chosen,

which results in the desirable trapezoidal current waveform. The tank inductor is implemented using a

PQ20/16 core, as documented in Table 2.1

(a)

(b)

(c)

Figure 2.4: Current waveform of tank inductor at (a) 60 H, (b) 40 H, and (C) 35 H. I(L3) is the

current of the inductor on the primary side. I(L2) is the secondary-side current.

21

Table 2.2: Specified tank inductor design

Core material TSC 50ALL

Core shape PQ20-16-00

Winding #16 AWG

Turns 9

Air gap 0.127mm

Inductance 40uH

Core loss 1.02 W@300W

Copper loss 0.083 W@300W

2.4 DC Blocking Capacitor

The behavior of the dual active bridge (DAB) circuit of Fig. 2.4 converter is studied in the

following chapter. At this point, the choice of the DC blocking capacitor is addressed. In practical

implementation, the switches may not operate with exact 50%duty cycles. The duty cycle mismatch may

result in a DC voltage presented across the transformer. This voltage may cause the transformer

magnetizing current to increase, which reduces efficiency, and may even result in core saturation. To

prevent this, a DC blocking capacitor is required to remove the DC voltage from transformer. However,

this extra DC blocking capacitor introduces an extra resonance with the transformer magnetizing

inductance. The resonant frequency between the magnetizing inductance of transformer 𝐿𝑀 and DC-

blocking capacitor 𝐶𝑏𝑙𝑜𝑐𝑘 is

𝑓0 =1

2𝜋√𝐿𝑀𝐶𝑏𝑙𝑜𝑐𝑘

(2.29)

When the blocking capacitor is 1.2𝜇𝐹, the resonant frequency is 1.89𝑘𝐻𝑧 in Fig. 2.6(a). The inductor

current loses the sharpness in the waveform. Therefore, a circuit is required to prevent high-Q resonance.

In the case of the blocking capacitor 50𝜇𝐹, the resonant frequency is 293𝑘𝐻𝑧 as illustrated by the

waveforms in Fig. 2.6(b).

22

Figure 2.5: Dual active bridge converter

(a) 1.2uF DC Blocking Capacitor

(b) 50uF DC Blocking Capacitor

Figure 2.6: (a)(b) Experimental waveforms of the prototype where the upper waveform is the tank

inductor current [1.0A/div], the secondary waveform is the drain-to-source voltage of a bridge switch

[50V/div], and the lower waveforms is the gate signal.

23

Chapter 3

State Plane Analysis

State plane presents a useful mathematical tool to analysis of switched second order

systems with resonance. On the state plane, the states of the system (capacitor voltage and

inductor current) are normalized and plotted against each other. If a proper normalization

factor is chosen, and if it is assumed that the system losses can be neglected, the state

trajectory is composed of only circular and linear segments. The state-plane trajectory

graphically represents the behavior of switched resonant circuits. As a result, we can

analyze the boundary condition of soft switching from a geometric viewpoint.

Based on the work presented in [17], the state plane analysis is reviewed in this

chapter, and applied to the DAB converter.

3.1 Review of State Plane Analysis

In state plane analysis [18,19], a normalized approach is an effective method for analyzing

resonant circuit behavior. First, suitable base voltage and base current are selected for normalization of

the converter waveforms. The base voltage is

𝑉𝑏𝑎𝑠𝑒 = 𝑉𝑔 (3.1)

24

The base current follows from this selection as

𝐼𝑏𝑎𝑠𝑒 =𝑉𝑏𝑎𝑠𝑒

𝑅0 (3.2)

𝑅0 = √𝐿𝑙

𝐶𝑝 (3.3)

𝜔0 =1

√𝐿𝑙𝐶𝑝

(3.4)

The notation for normalized unit-less waveforms is defined by

𝑚𝑝(𝑡) =𝑉𝑝(𝑡)

𝑉𝑏𝑎𝑠𝑒 (3.5)

𝑗𝑙(𝑡) =𝑖𝑙(𝑡)

𝐼𝑏𝑎𝑠𝑒 (3.6)

The DCX has behavior equivalent to that of an ideal transformer, where the turns ratio of the transformer

equals to the voltage conversion ratio of the converter,

𝑛2

𝑛1=

𝑉𝑜𝑢𝑡

𝑉𝑔 (3.7)

The ratio 𝑛1 𝑛2⁄ is equal to the turns ratio 𝑛𝑡. The conversion ratio is then defined as 𝑀𝑁 = 𝑉𝑜𝑢𝑡 𝑛𝑡𝑉𝑔⁄ ,

and equations describing normalized voltage and current waveforms are given by

𝑚𝑝(𝑡) = −𝐽𝑝𝑘 𝑠𝑖𝑛 𝜔0𝑡 − (𝑀𝑁 − 1) 𝑐𝑜𝑠 𝜔0𝑡 − 1 (3.7)

𝑗𝑙(𝑡) = −𝐽𝑝𝑘 𝑐𝑜𝑠 𝜔0𝑡 − (𝑀𝑁 − 1) 𝑠𝑖𝑛 𝜔0𝑡 (3.8)

It follows that

(𝑚𝑝(𝑡) + 1)2

+ 𝑗𝑙(𝑡) = 𝐽𝑝𝑘2 + (𝑀𝑁 − 1)2 (3.9)

25

which represents a circle in the state plane. The radius of the circle is,

𝑟1 = √𝐽𝑝𝑘2 + (𝑀𝑁 − 1)2 (3.10)

The circle shows how the energy is ringing between the inductor and the capacitor. The center of the

circle is at (𝑚𝑝, 𝑗𝑙) = (−1,0). The correlation can be visualized as geometric parameters of the 𝑚𝑝(𝑡) −

𝑗𝑙(𝑡) state plane trajectory.

3.2 State Plane Analysis Applied to the Dual Active Bridge

The benefit associated with the use of state plane analysis is a simplification of resonant interval

solutions. The state planes of the DAB converter for both the primary and the secondary resonances are

shown in Fig. 3.1 [20]. Fig. 3.1 (a) shows the gate-drive waveforms, and the corresponding primary and

secondary voltages. The system variables and their corresponding normalized quantities are listed in

Table 3.2. The state plane trajectory is composed of both linear and circular segment, because the DAB

converter operates with resonant intervals (Ⅰ&Ⅲ) and non-resonance intervals (Ⅱ&Ⅳ). Because DAB is

a multi-resonant converter, the state plane of the DAB converter has 𝑚𝑝 − 𝑗𝑙 state plane, which describes

the behavior of inductor of current and primary voltage, as well as 𝑚𝑠 − 𝑗𝑠 state plane, which describes

the behavior of inductor of current and secondary voltage. In this section, the converter conversion ratio is

considered to be approximately 𝑀𝑁 = 1.

𝑚𝑠(𝑡) =𝑣𝑠(𝑡)

𝑉𝑜𝑢𝑡 (3.11)

𝑗𝑠(𝑡) =

𝑖𝑙(𝑡)𝑛𝑡

𝑅𝑜′

𝑉𝑜𝑢𝑡

(3.12)

26

where a prime (′) indicates the normalization of the secondary parameters.

(𝑅𝑜′)2 = 𝑛𝑡

2𝐿𝑙

𝐶𝑠 (3.13)

which represents the secondary capacitance 𝐶𝑠 and the tank inductance reflected through the transformer

turns ratio to the secondary side in normalization.

3.2.1 Primary ZVS Condition in DAB

The trajectory of the state plane for the primary side transition is in the counter-clockwise

direction, while the trajectory of the state plane for the secondary is in the clockwise direction. In the

primary side transition, the counter-clockwise direction indicates that the magnitude of inductor current is

decreasing while the DAB converter primary side commutes. Therefore, sufficient inductor energy is

necessary to achieve primary ZVS, while the secondary ZVS condition is independent of the energy

stored in the inductor at the starting point of resonant transition. If the inductor energy is large enough to

charge/discharge all four primary-side switch capacitances 𝐶𝑝 across the input voltage, ZVS can be

accomplished,

1

2𝐿𝑙𝐼𝑝𝑘

2 >4

2𝐶𝑝𝑉𝑔

2 (3.14)

This equation simplifies to

𝐼𝑝𝑘 > 2𝑉𝑔√𝐶𝑝

𝐿𝑙 (3.15)

27

A minimum power 𝑃𝑚𝑖𝑛 at which primary ZVS will be achieved can be calculated. From the primary

state plane of Fig. 3.1, full ZVS is achieved if the radius of the resonance 𝑟1 > 2, where 𝑟1 is found in

(3.8). 𝑀𝑁 can be approximated as 1. In this case, the ZVS condition in the normalized state plane is given

by

𝐽𝑝𝑘 > 2 (3.16)

For the secondary, ZVS is always achieved.

Table 3.2: Correspondence table with DAB between time domain and primary state plane

Time Domain 𝑖𝑙(𝑡) 𝑣𝑝(𝑡) 𝑣𝑠(𝑡) 𝑣𝑔 𝐼1 𝐼2 𝐼𝑝𝑘 𝑡1 𝑡2 𝑡3 𝑡4

Primary State Plane 𝑗𝑙(𝑡) 𝑚𝑝(𝑡) 𝑚𝑠(𝑡) 1 𝑗1 𝑗2 𝐽𝑝𝑘 𝛼 𝛽 𝛿 𝜁

28

(a)

(b)

29

(c) (d)

Figure 3.1: Time-domain waveforms of the resonant transition of the primary and secondary (a) in the

DAB converter, the DAB schematic (b), primary (c) and secondary (d) state planes.

3.2.2 DAB State Plane Solution

Subinterval I: In subinterval I, ZVS is achieved during resonant transition of the primary full

bridge in the primary state plane of Fig. 3.1 (c). The length of the transition can be expressed by the

resonant angle 𝛼.

𝑡1 =𝛼

𝜔0

(3.17)

30

The angle 𝛼 is solved form the state plane by noting that the resonant radius 𝑟1 = 𝐽𝑝𝑘. Therefore, the

angle 𝛼 is given by

𝛼 = 𝑠𝑖𝑛−1 (2

𝐽𝑝𝑘) (3.18)

Fig. 3.1 (c) is applied to use geometric approach to finding the inductor current at the end of subinterval I.

𝐽1 can be solved by the triangle theorem between 𝑟1 and the 𝑚𝑝 axis.

Subinterval II: The transition of subinterval II is linear in both primary and secondary, because

𝑣𝑝 and 𝑣𝑠 are constant. The inductor voltage is 𝑉𝑔 + 𝑉𝑜𝑢𝑡 𝑛𝑡⁄ = 2𝑉𝑔 during the course of the subinterval.

Therefore, 𝑡𝑠 can be represented by

𝑡2 =𝐿𝑙

2𝑉𝑔

(𝐼1 + 𝐼2) (3.19)

Normalizing by multiplying both side by 𝜔0, and (3.3) and (3.4) is applied to (3.19)

𝛽 =1

2(𝐽1 + 𝐽2) (3.20)

Subinterval III: The transition of subinterval III in the secondary state plane takes the same

behavior of subinterval I in the primary state plane. The same approach of (3.18) can be applied to

𝛿′ = 𝑠𝑖𝑛−1 (2

𝐽𝑝𝑘′) (3.21)

and the initial current is calculated by geometrical arguments.

𝐽2′ = √𝐽𝑝𝑘

′2− 4 (3.22)

31

(3.18) is reflected to the secondary state plane.

𝛿 = 𝑛𝑡√𝐶𝑠

𝐶𝑝𝑠𝑖𝑛−1 (√

𝐶𝑠

𝐶𝑝

2𝑛𝑡

𝐽𝑝𝑘) (3.23)

and the initial current

𝐽2 = √𝐽𝑝𝑘2 − 4𝑛𝑡

𝐶𝑠

𝐶𝑝 (3.24)

Subinterval IV: In the transition of subinterval IV, the segments in both primary and secondary

may draw straight line. The sum of the four subintervals should be equal to half-period the switching

period,

𝑡1 + 𝑡2 + 𝑡3 + 𝑡4 =𝑇𝑠

2 (3.25)

The normalization of subinterval IV as 𝜍 = 𝑡4 𝜔0⁄ is given by

𝜍 =𝜋

𝐹− 𝛼 − 𝛽 − 𝛿 (3.26)

3.3 Analysis of DAB Operation Modes

In order to control the optimal efficiency trajectory for DAB, the state plane analysis of previous

section have to consider converter operation with 𝑉𝑜𝑢𝑡 ≠ 𝑛𝑡𝑉𝑔. The DAB operation modes have some

boundaries which are determined by the magnitude of phase shift and the dead times. Furthermore, there

are ZVS boundaries of both primary and secondary side devices. A new approach is developed for the

more general case. In this section, 𝑀𝑁 is defined as a variable.

𝑀𝑁 =𝑉𝑜𝑢𝑡

𝑛𝑡𝑉𝑔 (3.27)

32

3.3.1 Mode 1: Operation with 𝜷 > 𝟎 and primary ZVS

In Mode 1, the subinterval II has length greater than 0, and the peak current 𝐼𝑝𝑘 is sufficient to

achieve ZVS. When 𝛽 is assumed to be close to 0, the primary state plane trajectory is shown Fig. 3.2.

Figure 3.2: The DAB converter in Mode 1 in the normalized primary state plane

By analysis of the state plane geometry, the average output current can be found. First, two

triangles relating the radius 𝑟1 give the value of 𝐽1

𝐽1 = √𝐽𝑝𝑘2 − 4𝑀𝑁 (3.28)

The conduction angle of the primary resonance is given by

𝛼 = 𝑐𝑜𝑠−1 (1 −(𝐽𝑝𝑘 − 𝐽1)

2+ 4

2𝐽𝑝𝑘2 + 2(1 − 𝑀𝑁)2

) (3.29)

33

The waveform of the current is linear during intervals II and IV, therefore the time interval can be solved

by the slope of the current,

𝑡2 = 𝐿𝑙

𝐼1 + 𝐼2

𝑉𝑔 +𝑉𝑜𝑢𝑡𝑛𝑡

(3.30)

𝑡4 = 𝐿𝑙

𝐼𝑝𝑘 − 𝐼2

𝑉𝑔 −𝑉𝑜𝑢𝑡𝑛𝑡

(3.31)

which are normalized to obtain

𝛽 =𝐽1 + 𝐽2

1 + 𝑀𝑁 (3.32)

𝜍 =𝐽𝑝𝑘 − 𝐽2

1 − 𝑀𝑁 (3.33)

The switching period is 𝑇𝑠 = 2(𝑡𝛼 + 𝑡𝛽 + 𝑡𝛾), which remains the same as in the simplified case

considered previously

𝜋

𝐹= 𝛼 + 𝛽 + 𝜍 (3.34)

with 𝐹 = 𝑓𝑠 𝑓0⁄ . Finally, by applying charge balance to the output capacitor and integrating the charge

transferred to the output, the average output current can be found as

𝐽 =⟨𝑖0⟩

𝐼𝑏𝑎𝑠𝑒=

𝐹

𝑛𝑡𝜋(2 +

𝐽𝑝 + 𝐽2

2𝜍 +

𝐽𝑝 − 𝐽2

2𝛽) (3.35)

3.3.2 Mode 2: Operation with 𝜷 > 𝟎 and hard-switched primary

As the current 𝐼𝑝𝑘 available to achieve ZVS of primary devices decreases, the energy stored in the

inductor also decreases. As a result, hard switching of the primary devices will occurs. The boundary

34

condition of ZVS for the primary-side devices is found from the state plane of Fig. 3.2. If the radius of the

resonance 𝑟1 < 1 + 𝑀𝑁, the condition is substituted into (3.28)

𝐽𝑝𝑘 < √4𝑀𝑁 (3.36)

Figure 3.3: The DAB converter in Mode 2 as normalized state plane

At the end of the resonant transition, the primary devices are hard-switched. The state plane of

operation is shown in Fig. 3.3. 𝑀1 is defined as the normalized voltage at the end of the resonant interval.

Below 𝐽1 = 0, 𝑀1 is found by examining the triangle formed between 𝑟1 and 𝑚𝑝 axis as

𝑀1 = √𝐽𝑝𝑘2 + (1 − 𝑀𝑁)2 − 𝑀𝑁 (3.26)

Equations (3.32), (3.33), and (3.34) are unchanged, and the equations for ∝ and 𝐽 are given by

𝛼 = 𝑐𝑜𝑠−1 (1 −𝐽𝑝𝑘

2 + (1 + 𝑀1)2

2(𝑀1 − 𝑀𝑁)2 ) (3.27)

𝐽𝑜𝑢𝑡 =𝐹

𝑛𝑡𝜋(1 + 𝑀1 +

𝐽𝑝𝑘 + 𝐽2

2𝜍 −

𝐽2

2𝛽) (3.28)

35

3.3.3 Mode 3: Operation with 𝜷 < 𝟎 and primary soft-switched primary

In Mode 3, the primary devices are operated with ZVS. Because the power flow still remains

positive, the secondary devices are hard-switched during the dead time of the primary devices.

The state plane for Mode 3 is shown in Fig. 3.4. To ensure consistency with the other operation

mode, two new angles are defined as

𝛾 = 𝛼 + 𝛽 (3.29)

∅ = −𝛽 (3.30)

Furthermore, (1 − 𝑀2) is the normalized value of the voltage 𝑣𝑝 when the secondary devices are

switched. The equations for 𝛾 and 𝐽 become

𝛾 = 𝑐𝑜𝑠−1 (1 −(𝐽𝑝𝑘 + 𝐽2)

2+ (1 + 𝑀2)2

2(1 − 𝑀𝑁)2 + 2𝐽𝑝𝑘2 ) (3.31)

∅ = 𝑐𝑜𝑠−1 (1 −(𝐽1 + 𝐽2)2 + (1 − 𝑀2)2

2(𝑀𝑁 − 𝑀2)2 + 2𝐽22 ) (3.32)

and from two radii

𝐽22 = (1 − 𝑀𝑁)2 + 𝐽𝑝𝑘

2 − (𝑀2 + 𝑀𝑁)2 (3.33)

𝐽12 = −(1 − 𝑀𝑁)2 + 𝐽2

2 + (𝑀𝑁 − 𝑀2)2 (3.34)

the equation for the 𝜁 interval is modified

𝜁 =𝐽𝑝𝑘 + 𝐽1

1 − 𝑀𝑁 (3.35)

36

The normalization of switching frequency is defined as before, 𝐹 = 𝑓𝑠 𝑓0⁄ . The normalized switching

period equation becomes

𝜋

𝐹= 𝛾 + 𝜙 + 𝜁 = 𝛼 + 𝜁 (3.36)

Finally, the averaging of the output current yields

𝐽𝑜𝑢𝑡 =𝐹

𝑛𝑡𝜋(2𝑀2 +

𝐽𝑝𝑘 − 𝐽1

2𝜁) (3.37)

Figure 3.4: State plane of the DAB in Mode 3

3.4 Experimental Results

The proposed DAB converter was constructed with the transformer, as described in Table 2.1,

and 40 H tank inductor, as described in Table 2.2, implemented on the transformer primary. Infineon

37

IPW65R660CFD CoolMOS devices are used for all primary and secondary transistors. Four MOSFETs

are used for each bridge. Fig. 3.5 shows the 500 W prototype converter.

The analytical and the measured state plane trajectories are compared in Fig. 3.7. The measured

data come from oscilloscope waveform. There are surges in the state plane in Fig. 3.7 (a) and (f). Since

the 40 H tank inductor is designed for trapezoidal current waveform at 300 W, the inductor current in

interval IV has a positive slope under the conditions away from 300 W.

Figure 3.5: Prototype DAB converter

38

(a) (b)

(c) (d)

(e) (f)

Figure 3.6: Comparison between theory and experiments in the primary state plane trajectories at (a) 300-

W, (c) 400 W, and (e) 500 W. The secondary state plane trajectories are (b) 300 W, (d) 400 W, and (f)

500 W.

39

Chapter 4

Experimental Results

Analysis and design of the DAB converter prototype are presented in Chapter 3, focusing on the

300 W operating point. However, the vehicle application presents the converter with a range of operating

points. The loads vary widely and rapidly. Therefore, the operation of the DAB converter should be

considered across a wide range of output power levels. The converter should be able to accomplish high

efficiency under all conditions.

One approach to extending converter operation to a wide range of output powers is to use the

converter phase shift as a control variable. According to the load power requirements, the phase shift

between the primary and the secondary full bridge needs to be optimized.

In this chapter, efficiency analysis of the DAB is compared to experimental measurements

obtained in the prototype converter. Additionally, a control method of the optimized phase shift is

described and tested in experiments.

4.1 Loss Analysis

The full bridge DCX and DAB DCX converter prototype is implemented using eight MOSFET

transistor. Both of the circuit topologies are exactly same. In the full bridge DCX, only the primary

MOSFETs are actively controlled, while the secondary-side MOSFETs are kept off. The secondary-side

body diodes perform the rectification function. In the DAB case, both the primary-side and the secondary-

side MOSFETs are actively controlled, with a phase shift between the two bridges used as a control

variable.

40

A comparison between the full bridge DCX and the DAB DCX converter loss at 200 W is given

in Fig. 4.1. This loss analysis includes both the primary and secondary transistor device and magnetic

component loss. A difference between the full bridge DCX and DAB DCX is the conduction loss on

secondary side. The full bridge DCX has diode conduction and switching losses due to secondary

MOSFET intrinsic body diodes. In the DAB DCX, the MOSFET conduction losses compared to the full

bridge DCX increase because the output current flows through the MOSFET on-resistance on the

secondary. The total device losses of the DAB DCX can be reduced. To verity this, a comparison of

characteristics of reverse diode with MOSFET is shown in Fig. 4.2 (a). The plot of intrinsic body diode

conduction is obtained from the datasheet specification of the forward characteristics of the body diode,

while the plot of MOSFET conduction drop is linear because the MOSFET on-resistance is constant. The

analytical predictions of 𝑃𝐷𝑖𝑜𝑑𝑒_𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 and 𝑃𝑀𝑂𝑆𝐹𝐸𝑇_𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 are given by

𝑃𝐷𝑖𝑜𝑑𝑒_𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 2𝐼𝐷(𝑉𝐹 + 𝐼𝑑𝑅𝑑) (4.1)

𝑃𝑀𝑂𝑆𝐹𝐸𝑇_𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 2𝐼𝑑2𝑅𝑜𝑛 (4.2)

These relationships can be converted to conduction losses in the devices in Fig. 4.2 (b). The total device

loss in a wide load range can be improved by the secondary MOSFET conduction. The DAB DCX will

otherwise exhibit the same behavior of the full bridge DCX across the full load range. Therefore, both full

bridge DCX and DAB DCX are capable of achieving the same performance in the full load range.

41

Figure 4.1: Loss analysis of full bridge DCX and DAB DCX at 200W

0

1

2

3

4

5

6

7

Full Bridge Dual Active Bridge

Loss

[W

]

Diode Pcond

Diode Psw

MOSFET Pcond

MOSFET Psw

Ind Cu

Ind Fe

Transformer Cu

Transformer Fe

42

(a)

(b)

Figure 4.2: In (a), characteristics of body diode and MOSFET loss analysis of full bridge DCX and DAB

DCX. In (b), the secondary-side conduction losses are compared.

43

4.2 Phase-Shift Optimization in DAB

If primary and secondary dead times are constant with normalized values 𝛼0 and 𝛿0, the total

phase shift is [21]

𝛷𝑎𝑏 = 𝛼 + 𝛽 + 𝛿 (4.3)

Analysis of DAB operating modes presented in Chapter 3 indicates that a controller must adjust both the

converter primary dead time 𝛼 and the total phase shift 𝛷𝑎𝑏 to the specified values at each output power.

Fig. 4.3 shows the efficiency of DAB as a function of phase-shift time 𝑡2. Example waveforms are given

in Fig. 4.4. For the optimal controller, the optimized dead time to maximize the efficiency is nonlinear,

and a controller requires measurements of the output power to find the appropriate dead time for each

power. By the use of voltage sensing at the input port and output port, a controller can calculate the

conversion ratio 𝑀𝑁.

Figure 4.3: Efficiency as a function of phase shift between primary and secondary at 300 W

95.8

96

96.2

96.4

96.6

96.8

97

97.2

0 100 200 300 400 500

Effi

cie

ncy

[%

]

Phase Shift [ns]

44

(a) Phase shift 233 ns

(b) Phase shift 266 ns

(c) Phase shift 300ns

Figure 4.4: Experimental waveforms of the DAB converter prototype operating at 300W with dead time 𝛼

45

4.3 Analysis of Optimal Efficiency Trajectory

In order to maximize converter efficiency across a full range of operating points, a prototype

DAB converter is examined. Losses are calculated to produce the analytical efficiency contour plots of

Fig. 4.6 (a), while the experimental efficiency contour is plotted in Fig. 4.6 (b). The analysis model

considers only the square-wave approximated waveforms of the inductor current. Therefore, the analytical

efficiency contour plot does not match the experimental efficiency contour when the DAB is operated

with extreme phase shift so that the inductor current becomes a triangle wave. However, the optimal

trajectories in the experiment are similar to the presented analytical predictions. On this trajectory, the

primary dead time 𝛼 may be approximated as constant. The optimal trajectories show a linear relationship

between 𝛷𝑎𝑏 and 𝑀𝑁.

𝑀𝑁 = 𝐾𝛷𝑎𝑏 + 𝐶 (4.4)

The dead times 𝑡1 and 𝑡3 are constants. Therefore, the controller can adjust only 𝑡2 by sensing values for

𝑉𝑔 and 𝑉𝑜𝑢𝑡. The optimal trajectories for the power range 350 𝑊 ≤ 𝑃𝑜𝑢𝑡 is nonlinear. Therefore, non-

optima phase shift can lead to significantly reduced DAB DCX efficiency, while potential gains with

phase shift optimization are relatively small. One approach to maintain high efficiency is to switch to full

bridge DCX operation at full load.

46

(a)

(b)

Figure 4.5: Efficiency contours of the DAB converter based on (a) experimental results and (b) analytical

efficiency predictions. The blue line indicates the highest efficiency trajectory.

47

4.4 Experimental Results: Conversion Ratio and Efficiency

Conversion ratio and efficiency are measured experimentally using Fluke 45 multimeter to

measure input current and output voltage, and Agilent 34411A mutlimeter for output current and input

voltage. A comparison between the experimental efficiencies and the predicted efficiencies is given in

Fig. 4.7. The phase shift in the DAB DCX is manually optimized. Gate-drive losses are not included in

measured or calculated efficiency values, but are estimated to be approximately 173 mW for the full

bridge DCX and approximately 216 mW for the DAB DCX.

The DAB DCX analytical efficiency prediction is slightly better than Full Bridge DCX. However,

the DAB DCX experimental efficiency at 100 W is not as good as the prediction. The green waveforms

indicate the inductor current in Fig. 4.8. In the DAB DCX, the inductor current flows through secondary

MOSFETs. As a result, the inductor current waveform becomes a triangle wave. On the other hand, the

full bridge DCX at 100 W features discontinuous conduction modes of operations. As a result, the full

bridge DCX operates with lower RMS currents and lower losses at light load.

Figure 4.6: Comparison of analytical calculated (analytical) efficiency and experimentally measured

efficiency for the prototype operates as the full-bridge DCX, or as the DAB DCX.

90

91

92

93

94

95

96

97

98

0 100 200 300 400 500

Effi

cie

ncy

[%

]

Pout [W]

Full Bridge Exp

DAB Exp

Full Bridge Cal

DAB Cal

48

(a)

(b)

Figure 4.7: Prototype experimental waveforms of the full bridge DCX (a), and the DAB DCX converter

(b), operating at 100W.

49

Chapter 5

Summary and Future Work

This thesis focuses on analysis and design of a “DC transformer” (DCX) DC-DC converter,

which exhibits high efficiency, and small size suitable for automotive applications. The DCX is a building

block in a new DC-DC converter architecture described in [9]. This work details the analysis, design,

control method, and implementation of zero-voltage switching in the DCX operated as a dual active

bridge (DAB) converter. Additionally, from the 500 W prototype DAB converter, techniques for

component selection, converter design, and optimal efficiency trajectory are examined. Contributions of

the thesis are summarized as follows.

5.1 DAB Design and Component Selection

A loss model of the DAB converter which exhibits optimal efficiency at 300W output power is

proposed. Design parameters for optimization of converter efficiency include transformer design and tank

inductance value. The proposed optimization indicates that the converter tank inductance is selected to

place the ZVS boundary at a target operating power. The ZVS boundary condition is defined by the tank

inductor stored energy which is required to commutate the primary-side switches with zero-voltage

switching (ZVS). Therefore, the optimal tank inductance is related with the primary device selection and

converter operating point.

50

5.2 State Plane Analysis of DAB Converter

Motivated by the simplified analysis, the sate plane analysis techniques presented in [20, 21] are

reviewed in Chapter 3. This approach can visualize all resonant intervals in two dimensions, and derive

the ZVS boundary condition from a geometric viewpoint. The resulting converter solution can be applied

to achieve high efficiency design. First, the direct analysis of ZVS intervals yields the ZVS boundary

condition in terms of the output power level. Second, the ZVS condition can be used to program optimal

phase-shift between primary and secondary at each output power.

5.3 Conclusions

Through comparisons between the full bridge DCX and the DAB DCX, it is found that the DAB

DCX can improve the efficiency if MOSFET conduction loss is significantly lower than the intrinsic body

diode conduction loss. Additionally, a high step-down DAB converter can be expected to significantly

improve efficiency because the secondary-side devices conduct the high secondary currents.

The detailed state plane analysis is extended to analyze the converter across different operating

modes. The loss model is used to solve for a method of varying the output power. The converter needs to

response to changes in output power so that the highest efficiency can be achieved at each power level. It

is shown how it is possible to track the optimal phase shift through sensing the input voltage and the

output voltage. This optimal trajectory is slightly modified to obtain a proposed trajectory over the full

load range. In this case, non-optimum phase shift can lead to significantly reduced DAB DCX efficiency,

while potential gains with optimum phase shift are relatively small. One approach to maintaining high

efficiency is to switch from the DAB DCX to the full bridge DCX.

At very light loads, DAB DCX efficiency is reduced because of additional conduction and

switching losses. The primary switching transitions are shown in Fig. 5.1. The additional switching losses

are accrued by full or partial hard switching.

51

(a) (b)

(c) (d)

Figure 5.1: Prototype experimental switching transitions of the full bridge DCX (a) (b), and the DAB

DCX converter operating at 100 W (c) (d).

5.4 Future Work

Further research has been identified in the process of completing this thesis. First, this thesis

relied on the assumption of a linear value 𝐶𝑝 which can be used to model the parasitic energy storage

behavior of the transistor devices. However, the actual behavior of this parasitic element is nonlinear,

which affects resonant transitions. Therefore, using the equivalent linear capacitors model developed in

[22] can improve accuracy in the state plane analysis of DAB converter.

Second, the state plane analysis of the full bridge DCX considers only an ideal semiconductor

model. This model could be improved considerably. The switching circuit analysis technique based on the

non-ideal diode characteristics, reverse recovery and junction capacitance, can be applied to state plane

analysis of the full bridge DCX as in [23].

52

Finally, it is expected that these extensions to analysis and design techniques will permit

efficiency improvements and more accurate determination of the optimal efficiency trajectory under

varying output powers.

53

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