Dual-Loop Linear Controller for LLC Resonant Converters · Dual-Loop Linear Controller for LLC Resonant Converters by Franco Degioanni Ing., Universidad Nacional Cordoba, 2014 A THESIS

Dual-Loop Linear Controller forLLC Resonant Converters

by

Franco Degioanni

Ing., Universidad Nacional Cordoba, 2014

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in

The Faculty of Graduate and Postdoctoral Studies

(Electrical & Computer Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

June 2018

c⃝ Franco Degioanni 2018

The following individuals certify that they have read, and recommend to the Faculty of

Graduate and Postdoctoral Studies for acceptance, the thesis entitled:

Dual-Loop Linear Controller for LLC Resonant Converters

submitted by Franco Degioanni in partial fulfillment of the requirements for

the degree of Master of Applied Science

in Electrical and Computer Engineering

Examining Committee:

Dr. Martin Ordonez

Supervisor

Dr. William Dunford

Supervisory Committee Member

Dr. Alireza Nojeh


Additional Examiner

Additional Supervisory Committee Members:



ii

Abstract

For the last years, LLC resonant converters have gained wide popularity in a large number of

domestic and industrial applications due to their high-efficiency and power density. Common

applications of this converter are battery chargers and high efficiency power supplies, which

require tight output voltage regulation. In traditional PWM converters, closed-loop con-

trollers based on small-signal models are typically implemented to achieve zero steady-state

error and minimize the effects of disturbances at the output. However, traditional averaging

techniques employed in PWM converters cannot be applied to LLC’s and highly complex

mathematical models are required. As a consequence, designing linear controllers for this

type of converter is usually based on empirical methods, which require high-cost equipment

and do not provide any physical insight into the system.

The implementation of current-mode controllers has been vastly developed for PWM con-

verters. Employing an inner current loop and outer voltage loop has shown numerous ad-

vantages, such as, tight current regulation, over-current protection, and ample bandwidth.

However, this control architecture is not commonly implemented in LLC resonant converters,

and conventional single voltage loop controllers are employed.

This work proposes a simple and straightforward methodology for designing linear con-

trollers for LLC resonant converters. A simplified second order equivalent circuit is developed

and employed to derive all the relevant equations for designing proper compensators. A dual-

loop control scheme including an inner current loop and outer voltage loop is proposed. The

iii

Abstract

implementation of the dual-loop configuration provides improved closed-loop performance

for the entire operational range.

The theoretical findings are supported by detailed mathematical procedures and validated

by simulation and experimental results.

iv

Lay Summary

Power converters enable the control of power flow from different energy sources to diverse

electrical devices, playing a fundamental role in most electrical systems. The ever-increasing

popularity and complexity of electrical devices in day-to-day applications has increased the

requirements of power converters. There is a large variety of power converters, and the LLC

resonant converter has gained popularity in different applications due to its high efficiency.

Usually, power converter applications require constant voltage and current at the output,

and control techniques are applied to achieve this requirement. However, the control design

procedure for LLC resonant converters requires highly complex mathematical analysis or

empirical methodologies. This work introduces a simple and straightforward methodology

for designing controllers for such complex topology as the LLC converter. In addition, a new

control scheme is proposed enabling significant advantages beyond conventional approaches.

v

Preface

This work is based on research performed at the Electrical and Computer Engineering de-

partment of the University of British Columbia by Franco Degioanni, under the supervision

of Dr. Martin Ordonez.

A first version of this work was presented in IEEE Energy Conversion Congress and

Exposition (ECCE), 2017 [1].

An extended version of this work was published in IEEE Transaction on Power Electron-

ics [2].

As the first author of these publications, the author of this thesis developed the theoretical

contribution, the simulation models, and performed the experimental results. The author

received advice and technical support from Dr. Ordonez and members of his research team

vi

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Lay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

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

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Resonant Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 LLC Resonant Converter . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 LLC Resonant Converter Control . . . . . . . . . . . . . . . . . . . . 7

1.3 Contribution of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

vii

Table of Contents

2 Analysis of the LLC Resonant Converter . . . . . . . . . . . . . . . . . . . 12

2.1 LLC Resonant Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 DC Gain Characteristics of the LLC Resonant Converter . . . . . . . 14

2.2 Numerical Identification of the Most Challenging Operating Condition . . . 16

2.2.1 Extended Describing Function Method . . . . . . . . . . . . . . . . . 17

2.2.2 Eigenvalues Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Simulation and Experimental Validation . . . . . . . . . . . . . . . . 21

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Large-Signal Model of the LLC Resonant Converter . . . . . . . . . . . . 26

3.1 Average Model of the LLC Resonant Converter . . . . . . . . . . . . . . . . 27

3.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2 Experimental No Load Start-Up . . . . . . . . . . . . . . . . . . . . 35

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Dual-Loop Controller for LLC Resonant Converter . . . . . . . . . . . . . 39

4.1 Small-Signal Model of the LLC Resonant Converter . . . . . . . . . . . . . 40

4.2 Dual-Loop Controller for the LLC Resonant Converter . . . . . . . . . . . . 42

4.3 Controller Design Procedure and Simulation Results . . . . . . . . . . . . . 45

4.3.1 Controller Design Procedure . . . . . . . . . . . . . . . . . . . . . . 46

4.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

viii

Table of Contents

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Appendices

A Extended Describing Function Matrices . . . . . . . . . . . . . . . . . . . . 71

ix

List of Tables

2.1 Normalized LLC Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Start Up Experimental Parameters . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 LLC Parameters - Control Design . . . . . . . . . . . . . . . . . . . . . . . . 46

x

List of Figures

1.1 Block Diagram Power Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Block Diagram of Resonant Converters . . . . . . . . . . . . . . . . . . . . . 5

2.1 Full-Bridge LLC Resonant Converter . . . . . . . . . . . . . . . . . . . . . . 13

2.2 DC Gain characteristics LLC Resonant Converter . . . . . . . . . . . . . . . 15

2.3 Eigenvalues of the Normalized LLC Resonant Converter . . . . . . . . . . . . 22

2.4 Simulation open-Loop Response of the LLC Resonant Converter . . . . . . . 23

2.5 Experimental open-Loop Response of the LLC Resonant Converter . . . . . 23

3.1 Operating Modes of the LLC Resonant Converter at Resonant Frequency . . 28

3.2 Start-up Response of the LLC Resonant Converter . . . . . . . . . . . . . . 31

3.3 Large-Signal Average Model of the LLC Resonant Converter . . . . . . . . . 33

3.4 LLC Converter vs Average Large-Signal Model . . . . . . . . . . . . . . . . . 34

3.5 Experimental Start Up LLC1 and LLC2 . . . . . . . . . . . . . . . . . . . . . 36

3.6 Experimental Start Up LLC3 and LLC4 . . . . . . . . . . . . . . . . . . . . . 37

4.1 Small-Signal Circuit of the LLC Resonant Converter . . . . . . . . . . . . . 41

4.2 Control Frequency-to-Output Voltage Transfer Function . . . . . . . . . . . 42

4.3 Proposed Dual-Loop Controller for the LLC Resonant Converter . . . . . . . 43

4.4 Simplified Transfer Functions of the LLC Resonant Converter . . . . . . . . 44

4.5 Simplified Voltage Loop Transfer Function . . . . . . . . . . . . . . . . . . . 45

4.6 Compensated Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 47

xi

List of Figures

4.7 Closed-loop Bode Plot of the Designed Controller . . . . . . . . . . . . . . . 48

4.8 Closed-loop Bode Plot for Different Gains . . . . . . . . . . . . . . . . . . . 49

4.9 Load Steps Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.10 Voltage Reference Step Simulation . . . . . . . . . . . . . . . . . . . . . . . 51

4.11 Experimental Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.12 Experimental Reference Step for Dual-Loop Controller . . . . . . . . . . . . 54

4.13 Experimental Current Load Step for Dual-Loop Controller . . . . . . . . . . 55

4.14 Experimental Current Load Step for Dual-Loop Controller . . . . . . . . . . 56

4.15 Experimental Current Load Step for Conventional Voltage Controller . . . . 57

4.16 Experimental Current Load Step for Conventional Voltage Controller . . . . 58

xii

Acknowledgements

I would like to thank my supervisor Dr. Martin Ordonez, for accepting me as a part of

his research team. His contagious enthusiasm, continuous support, enormous dedication and

great mentorship made my Master’s program a rich and fascinating experience.

I would also like to thank my lab mates, all the members in Dr. Ordonez’s research team.

Their comments, suggestions, corrections and discussions have made my time in the lab a

fruitful and enjoyable experience beyond technical aspects.

I must thank the University of British Columbia, the Faculty of Graduate and Postdoctoral

Studies and the Electrical and Computer Engineering Department for the opportunity and

support received during this time. Particularly, I would like to thank the professors of the

courses I took as part of the program for a valuable knowledge transferred.

I feel specially grateful to my parents Americo and Celina, my brothers Jose and Marco,

and all of my family for their support throughout my life. I also feel the need to express my

deepest gratitude to my girlfriend Sofia for her support in my decision to pursue graduate

studies, and her constant encouragement without which this work would not have been

possible. Special thanks to my closest friends in Canada for their help and time shared with

me, and to my friends in Argentina for being in touch and making me feel they are close.

Last but not least, I would like to thank you for taking the time to read this work.

xiii

To all those I love.

xiv

Chapter 1

Introduction

1.1 Motivation

Power converters are the necessary systems to provide the interface among different electrical

loads and sources, either direct current (DC) or alternating current (AC). They are present

in most of today’s electrical devices, and their power levels vary according to the application

ranging from low power levels, such as LED lighting, to higher power applications such as

battery chargers for electric vehicles. During the last decades, the increasing complexity and

growing popularity of electrical loads in day-to-day applications has significantly increased

the requirements for high-quality power converters, and the design of reliable and efficient

power stages has become critical to the performance of the entire system.

Due to the latest advances of renewable energy sources, power systems entirely based on

DC, such as DC microgrids [3], data centers [4] and residential buildings [5] are commonly

seen nowadays, and they are predicted to grow in popularity. A typical distributed power

system diagram is shown in Fig. 1.1. Depending on the type of power source, front-end power

converters require AC-DC or DC-DC conversion to create a DC bus employed to interconnect

all sources and loads. Each DC load then requires a DC-DC converter to take energy from

the DC bus. DC-DC stages are among the most important parts of the energy system, since

they are required for both front-end and back-end applications.

DC-DC power converters can be sub-categorized into two main groups: Pulse Width

Modulation (PWM converters) and Resonant converters. The early development of PWM

1

1.1. Motivation

Electrical

Grid

AC-DC

Converter

PV Panel

DC-DC

Converter

DC Bus

Batery

PackDC LOADDC LOAD

DC LOADDC LOAD

DC LOADDC LOAD

Figure 1.1: General block diagram of a DC power system. The power can be supplied fromdifferent sources (either AC or DC). DC-DC converters provide the interface between the DCbus and the DC loads.

converters have resulted in them being the preferred DC-DC configuration during the last

decades. However, the growing use of DC loads demands power stages with higher efficiency,

increased power density and improved thermal management which are not usually achieved

by conventional PWM topologies. Although great efforts have been made to achieve these

new specifications for PWM converters, resonant converters are an attractive alternative.

Resonant topologies are able to meet these requirements by operating at higher switching

frequencies, achieving higher efficiency and power density in comparison with PWM convert-

ers.

Power conversion systems usually demand tight voltage regulation at the output which is

traditionally achieved by employing closed-loop controllers based on small-signal models [6,

7]. The optimization of these control loops is critical for achieving higher efficiency, better

performance, smaller size and lower costs. The small-signal models employed in PWM-based

2

1.2. Literature Review

power converters are relatively simple to derive, facilitating the design and implementation

of the control loop. Due to the high nonlinear nature of resonant converters and their many

different operation modes, the traditional mathematical modelling techniques employed in

PWM converters cannot be applied. Several methodologies have been proposed to derive the

small-signal model for resonant converters. However, the existing methods are usually overly

complex and difficult to apply in practical applications.

Moreover, most of the current literature focuses on developing single voltage loop con-

trollers for resonant converters. The implementation of dual-loop control schemes, employing

an inner current loop and outer voltage loop has shown numerous advantages in traditional

PWM topologies. As a result, there is room for exploring and analyzing the implementation

of dual-loop controllers in resonant converters.

Among the different types of resonant converters, LLC’s are usually preferred due to

their higher efficiency, wide output voltage regulation and increased power density compared

with other resonant topologies. As a result, the work presented in this thesis provides a

comprehensive analysis, modelling, and control of LLC resonant converters using a dual-loop

control strategy.

1.2 Literature Review

DC-DC converters have been widely used in a variety of applications for many years, such as

telecommunication equipment [8], cell phone power supplies [9], laptop battery chargers [10],

and DC motor drives [11]. Nowadays, DC-DC converters are also employed for renewable

energy applications like photovoltacis [12, 13] and DC microgrids [14], among others [15].

In all these applications, high efficiency and high power density requirements are the main

key requirements. Resonant converters are able to meet these requierments by operating at

3


higher switching frequencies [16], while maintaining high efficiency [17] in comparison with

PWM converters.

The main characteristics of resonant converters are summarized in the following para-

graphs, and particularly the advantages of the LLC resonant converter are explored. More-

over, an extensive review of the principal contributions as well as the latest advances in the

modellling and controller design of resonant converters is presented, highlighting the known

issues that give place to the present work.

1.2.1 Resonant Converters

A DC-DC resonant converter can be constructed by using the main stages shown in Fig. 1.2.

The switch network produces a square wave voltage which is applied to the input terminals

of the resonant tank network. As a consequence, the tank current is essentially a sinusoidal

waveform. By adjusting the switching frequency (fs), closer to, or further from, the tank res-

onant frequency (fr), the magnitude of the current can be controlled. Finally, the sinusoidal

resonant tank output current is rectified and filtered to supply a DC load [18]. A variety of

resonant tank networks can be employed, obtaining different output characteristics. Three of

the most common resonant tank configurations are illustrated in Fig. 1.2. The resonant tank

in the series resonant converter is formed by one capacitor and one inductor connected in

series, while in the parallel resonant converter, the capacitor is connected in parallel with the

rectifier network [19]. Diverse series-parallel resonant converter converters can be obtained

by combining the resonant tanks of the series and parallel converters, with one combination,

the LLC resonant converter, shown in Fig. 1.2. In some configurations, a high frequency

voltage transformer is placed between the resonant tank and the rectifier to scale down/up

the output resonant tank voltage and current.

The main advantage of resonant converters is their high efficiency and power density. The

efficiency of power converters is mainly determined by the losses in the semiconductor ele-

4


Switch

Network

Switch

Network

Resonant

Tank

Resonant

Tank

Rectifier

Network

Rectifier

Network

Output

Filter

Output

FilterVin Vo

+

-

RL

fs

Cs Cp

Ls CsLs

Lp

Ls

Series

Resonant

Tank

Series

Resonant

Tank

Parallel

Resonant

Tank

Parallel

Resonant

Tank

LLC

Resonant

Tank

LLC

Resonant

Tank

Figure 1.2: General block diagram of resonant converters.

ments, and it can be divided into conduction losses and switching losses. Switching losses

can be minimized by making the turn-on and/or turn-off transitions of the various converter

semiconductor elements at zero crossings of the resonant converter quasi-sinusoidal wave-

forms [21]. These techniques are known as zero-current switching (ZCS), and zero-voltage

switching (ZVS) [20]. Resonant converters are able to guarantee ZVS and ZCS operation

at different conditions, reducing switching losses and enabling operation at higher switching

frequencies than comparable PWM converters. Moreover, the reduction of switching losses

though ZVS reduces electromagnetic interference (EMI) generated by the converter [22].

As documented in [23], the series and parallel resonant topologies have several limiting

factors which make them a less-popular choice for practical applications. For the series

resonant converter, light-load operation requires a very wide range of switching frequencies

in order to retain output voltage regulation. Compared to the series resonant converter, the

parallel resonant topology does not require a wide range of switching frequencies to maintain

output voltage regulation. However, at high input voltage conditions, the converter shows

worse conduction losses, and higher turn-off currents.

5


1.2.2 LLC Resonant Converter

The LLC resonant converter combines the advantages of both, series and pararallel topolo-

gies. It can work in a wide output voltage range with relatively small range variation of

the switching frequency, offering wider output regulation than series and parallel resonant

converters [24–26]. This converter has been introduced in a large number of applications such

as battery chargers [27, 28], photovoltaic applications [29, 30], fuel-cells [31], high-efficiency

power supplies [32] and DC microgrids [33], among many other interesting applications [34–

36].

As illustrated in Fig. 1.2, the resonant tank of the LLC converter is composed of one series

inductor, one series capacitor and one parallel inductor. One of the practical advantages of

the LLC converter is that the magnetic components can be easily integrated into a high

frequency transformer [37]. The LLC resonant converter achieves ZVS conditions at the

input switches and ZCS conditions at the output rectifier, thus achieving [42], achieving

higher efficiency levels, and the maximum efficiency point is obtained at resonant frequency

operation.

Methods based on the First Harmonic Approximation (FHA) are usually employed to

find the transfer ratio of the converter [18, 38, 39]. The FHA approach is based on the

assumption that the power transfer from the source to the load through the resonant tank

is completely associated to the fundamental harmonic of the Fourier expansion of the cur-

rents and voltages involved. The harmonics of the switching frequency are then neglected

and the tank waveforms are assumed to be purely sinusoidal at the fundamental switching

frequency [40, 41].

6


1.2.3 LLC Resonant Converter Control

Most of the aforementioned applications require tight voltage and current regulation at the

output. For this reason, a feedback loop must be implemented in order to achieve the desired

performance and guarantee stability at the output. The design of the control loops is usually

based on small-signal models and linear controllers are commonly implemented.

In traditional PWM converters, small-signal models are well known and have been de-

veloped and analyzed for most of the control methods. Since the natural frequency of the

output filter is much lower than the switching frequency, averaging techniques are usually

employed to derive the mathematical models [43–46]. For single-loop (voltage mode) con-

trol, the averaging concept was first proposed, and then represented in state-space. These

methods provide a continuous-time small-signal model which is accurate up to one half of the

output ripple frequency. PWM converters also employ dual-loop (current-mode) control in

order to achieve higher bandwidths. The averaging concept has been also extended to obtain

models for this dual-loop configuration [47].

However, the scenario is completely different for resonant converters because some of the

state variables (currents and voltages in the resonant tank) do not contain DC components,

and contain large components of the switching frequency and its harmonics. Therefore,

obtaining small-signal models is not as straightforward as in PWM converters, since conven-

tional averaging methods cannot be directly applied. Due to the oscillatory nature of some

of the state variables, the switching frequency interacts with the resonant frequency of the

resonant tank. This interaction is usually known as the beat frequency dynamics [48], and it

cannot be investigated by using traditional averaging concepts which neglect the switching

frequency component and its harmonics.

Diverse methods have been proposed to derive small-signals models for resonant convert-

ers. A sample-data method is applied in [49–51]. This approach provides discrete small-signal

7


models which are a set of diverse difference equations with the switching period as the sam-

pling interval. The discrete model captures the inherent sampling nature and is able to predict

the small-signal behavior of resonant converters up to the switching frequency. However, the

sample-data analysis technique shows only equations that need to be solved numerically and

does not provide much physical insight for practical implementation.

In [52], an effort to extend the state-space averaging technique to modelling resonant

converters is presented. This method proposes state-space analysis without linear ripple ap-

proximation; instead only considering the DC component (as it is done for PWM converters)

and the amplitude of the switching frequency harmonics. A continuous-time small-signal

model was derived for a series resonant converter and the obtained results for the control-

to-output voltage transfer function are shown to be very accurate in comparison with the

experimental results. However, the method is based on matrix equations and it requires

numerical computation in order to obtain the small-signal model.

An approach using the harmonic balance technique is proposed in [53]. This method

describes the system in frequency-domain as a set of nonlinear equations that describe the

harmonics of the system. This set of equations is solved by selecting the harmonics at the

frequency of the stimulus and the transfer function can be obtained. Furthermore, a gener-

alized approach is proposed in [54]. However, both methods require numerical calculations

to obtain the corresponding transfer function and they do not provide closed form solutions.

In [55], the load presented to the AC side of the rectifier is modeled as a time vary-

ing resistor. Using conversion matrix techniques and an iterative procedure the magnitude

and phase of the time varying impedance can be obtained. Also, communication theory is

employed in [56] to derive the small-signal model for an LLC resonant converter. The ap-

proaches described here are also based on the solution of complex mathematical analysis and

they are not straightforward to apply in practical applications.

8


The Extended Describing Function (EDF) method was introduced in [57] to derive small-

signal models for resonant converters. The describing function technique is extended to a

more generalized multi-variable case. This method combines the time-domain and frequency

domain analysis and extracts the model by dividing modulated waveforms into sine and

cosine waveforms. This approach has been widely employed in many different applications

to extract the small-signal model of resonant converters [58–60].

Although the aforementioned models generated accurate results, they required complex

mathematical analysis instead of using circuit representation. Some attempts have been

made in order to simplify the analysis and derive small-signal equivalent circuits for resonant

converters. An equivalent circuit is obtained in [61] by applying the extended describing

function concept. Despite the fact that the obtained model is a fifth order system, numerical

solutions show that using a lower order system is accurate enough to model the behavior

of the converter [54]. Moreover, the transfer functions are still derived based on numerical

solutions and no explicit analytical solutions are provided. A simplified third order system is

provided in [62, 63]. This equivalent circuit provides analytic expressions and accurate results.

However, the obtained expressions are still complex to evaluate and hard to implement.

The high mathemical complexity involved requires empirical methodologies, such as software

simulation [64] and hardware measurements [65], as a means to obtain the frequency response

of LLC resonant converters. Although, these approaches provide accurate results, they do

not provide general closed form expressions.

The previous modelling methodologies are usually employed to design a single voltage

control loop, rather than a dual-loop control scheme. The implementation of current-mode

controllers has been vastly explored for PWM Buck and Boost converters [66–69], but there

is not much research done for LLC resonant converters. The benefits of employing current-

mode in LLC converters is shown [70, 71]. Employing some interesting empirical methods

the transfer functions are obtained and described. However, in order to augment the under-

9

1.3. Contribution of the Work

standing of LLC converters, closed-form expressions that describe the system behavior for

different parameters are needed and have yet to be studied.

1.3 Contribution of the Work

This work introduces valuable theoretical concepts to the field of control for LLC resonant

converters as well as useful practical application of the ideas developed:

• Identification and analysis of the most challenging condition from a modelling and

control point of view.

• A large-signal model of the LLC resonant converter operating at resonant frequency.

• The derivation of a linearized model to obtain all the required transfer functions for

designing control loops.

• Implementation of a dual-loop control scheme with inner rectified-current loop and

outer output-voltage loop.

As a result, an effective and straightforward methodology is achieved. The proposed con-

troller is formed by an inner current loop and outer voltage loop employing the averaged

rectified current and output voltage as feedback signals respectively. The application of a

dual-loop scheme enables significant advantages, as detailed in this work. Those advantages

include tight current regulation, over-current protection, and ample controller bandwidth.

Moreover, the implementation of an inner-current loop enables desirable closed-loop per-

formance over a wide range of operating conditions. In this way, this work introduces an

accurate linear model for LLC converters valid for the most challenging control condition,

the application of dual-loop controllers to successfully compensate the complex LLC resonant

converter topology, and a useful tool that enables simple and straightforward linear controller

design.

10

1.4. Thesis Outline

1.4 Thesis Outline

This work is organized as follows;

• In Chapter 2 the main equations and characteristics of the LLC resonant converter

are analyzed and the dynamic behavior of the LLC resonant converter is studied. The

Extended Describing Function method is employed to obtain the linearized model of

the converter. A parametric analysis of a normalized LLC converter is performed to

study the pole displacement at different operating conditions.

• In chapter 3 the average large-signal model of the LLC converter operating at resonant

frequency is introduced. Equations are provided to obtain the equivalent dynamic

inductance of the equivalent circuit. Simulation and experimental results are provided

to validate the proposed model.

• In chapter 4 the proposed dual-loop control scheme is introduced. A simplified small-

signal model is derived to obtain all the required transfer functions to design the con-

trollers. A control design procedure example is shown, and the conpensator for both

inner and outer loop are designed. The performance of the controller is verified by

simulations and experimental results at different operating points.

• Finally, in Chapter 5 a summary and conclusions of this work are presented, along with

some details of future research ideas.

11

Chapter 2

Analysis of the LLC Resonant

Converter

Power supplies with higher efficiency and power density are highly desired in power electronics

applications. Resonant converters are able to achieve those requirements with low switching

losses enabling higher frequency operation. In particular, the LLC converter stands out for

its wide output voltage regulation achieving soft-switching over the entire operating range.

As a usual practice, modelling efforts aim to model the dominant behaviors of the system,

while neglecting other insignificant phenomena. Simplified terminal equations of the compo-

nent elements are used, and many aspects of the system response are neglected altogether.

The resulting simplified model yields physical insight into the system behavior, simplifying

the analysis procedure. Thus, the modelling process involves use of approximations to ne-

glect less significant complicated phenomena, in an attempt to understand what is the most

important.

In this chapter the operation principles and conventional modelling techniques for LLC

resonant converters are analyzed. First, the DC gain characteristics in steady-state operation

of the converter are studied. In the second part of the chapter, the small-signal behavior of the

LLC resonant converter at different operating point is analyzed. The Extended Describing

Function method is developed and explained. A numerical analysis of the eigenvalues of the

system is performed. Finally, simulation and experimental results are provided to validate

the theoretical analysis.

12

2.1. LLC Resonant Converter

2.1 LLC Resonant Converter

The schematic circuit of the LLC resonant converter is shown in Fig. 2.1. The square wave-

form generated by the full-bridge inverter is applied to the resonant tank composed by three

resonant elements, the resonant series inductor (Lr), the series resonant capacitor (Cr) and

the parallel magnetizing inductor (Lm). The filtered current is scaled by a the turns ratio

n = np

nsfor the high frequency transformer and rectified by the output rectifier with filter

capacitor to obtain a DC voltage at the output.

From the schematic circuit shown in Fig. 2.1, the nonlinear equations that describe the

behavior of the converter can be expressed as

LrdiLr(t)

dt= Vin − vCr(t)− n sign (iLr(t)− iLm(t)) vo(t) (2.1)

Crdvc(t)

dt= iLr(t) (2.2)

LmdiLm(t)

dt= n sign (iLr(t)− iLm(t)) vo(t) (2.3)

Codvo(t)

dt= n|iLr(t)− iLm(t)| −

vo(t)

RL

(2.4)

S4

S3

S2

S1

Vin vinv

D1

D2

Co voRL

Cr Lr

Lm

iLr iLm

irec

ns

ns

np

Figure 2.1: Full-Bridge LLC Resonant Converter.

13


The state variables are defined in function of the series inductor current (iLr), series

capacitor voltage (vc), parallel magnetizing inductor (iLm) and the output capacitor voltage

(vo).

2.1.1 DC Gain Characteristics of the LLC Resonant Converter

The gain characteristics of the converter that relate the output voltage and the switching

frequency are mainly defined by the filter nature of the resonant tank. Methods based on

First Harmonic Approximation are usually employed to find the steady-state characteristics

of resonant converters. The procedure may be summarized as

• Represent the input square-wave voltage and current with their fundamental compo-

nents, ignoring all the higher-order harmonics.

• Ignore the effects from the output capacitor, assuming constant voltage at the output

during one switching cycle.

• Refer the obtained secondary-side variables to the primary side of the transformer.

By following these steps, the nonlinear behavior of resonant converters is approximated

by a linear model excited by an effective sinusoidal input source. The simplified model can

be solved by using conventional circuit theory and the expression that relates the output

voltage in function of the switching frequency is derived. This expression is well-known in

the literature and is given by

Mv =nVo

Vin

=f 2n(m− 1)√

(m(f 2n − 1))2 + f 2

n(f2n − 1)2(m− 1)2Q2

(2.5)

With

14


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Overload

Gain

No LoadNo Load

Region II

ZVS

Region II

ZVS

Region I

ZVS

Region I

ZVS

Region III

No ZVS

Region III

No ZVS

M v

Δ

f s,max

f s,min

Lighter

Load

Lighter

Load

QQ11QQ22QQ33QQ44

M =1vM =1v

Normalized Switching Frequencyfsfr

Norm

aliz

ed D

C G

ain

V o V in

n

Figure 2.2: DC Gain characteristics LLC resonant converter.

Q =1

Rac

√Lr

Cr

, Rac =8

π2n2RL, fs =

fsfr, fr =

1

2π√LrCr

, m =(Lr + Lm)

Lr

(2.6)

By using equation (2.5), the normalized voltage gain vs normalized switching frequency

characteristic of the LLC resonant converter is plotted in Fig. 2.2. As shown, the maximum

voltage gain varies according to the load condition. For lighter loads (Q decreases), the peak

gain is higher and it moves to lower frequencies, reaching fr2 = 1

2π√

(Lm+Lr)Cr

for no load

condition. On the other hand, at resonant frequency fr, unitary gain is achieved for all load

conditions including short-circuit operation.

15

2.2. Numerical Identification of the Most Challenging Operating Condition

The DC gain characteristics can be divided in the three main regions shown in Fig. 2.2.

In order to guarantee ZVS, the converter must operate within Region I and II. When the

converter’s switching frequency is above resonance (buck region), the voltage gain is lower

than one. Around resonance, the gain of the converter becomes close to one, and for frequen-

cies below resonance the gain becomes higher than one (boost mode). Therefore, the output

voltage can be controlled by varying the switching frequency of the main switches.

The operating range of the LLC converter depends on the load condition as shown in

Fig. 2.2. However, when the switching frequency is fr, converter’s gain characteristics are

independent of the load condition. Therefore, operation around resonant frequency is usually

desired to minimize the switching frequency variation at different load. Moreover, at resonant

frequency operation the converter achieves the highest efficiency operating point.

2.2 Numerical Identification of the Most Challenging

Operating Condition

Due to the nonlinear characteristics of the system, operation at different switching frequen-

cies leads to different transient responses. In this section, the extended describing function

method is applied to analyze the small-signal behavior of the converter around different op-

erating conditions. Identifying the most challenging condition from a control point of view

and deriving a simplified model at that point is a handy and common strategy to design

linear compensators.

16


2.2.1 Extended Describing Function Method

The describing function analysis is a technique commonly employed to study frequency re-

sponse of nonlinear systems. It is an extension of linear frequency response analysis. In

linear systems, transfer functions depend only on the frequency of the input signal. However,

in nonlinear systems, when a specific class of input signal such as sinusoidal is applied to a

nonlinear element, the system can be represented by a function that depends not only on

frequency but also on input amplitude. This function is referred to as a describing func-

tion. In order to use sinusoidal-input describing function analysis the system must satisfy

the following conditions:

• Nonlinearities are time-invariant.

• Nonlinearities do not generate any sub-harmonic as a response to the sinusoidal input.

• The system filters out the harmonics generated by the nonlinearities.

As resonant converters satisfy all these conditions, the EDF method has been widely used

to model the behavior of these converters.

Harmonic Approximation

Due to the resonant tank filter characteristics, the current through the resonant inductor

and the voltage in the resonant capacitor are nearly sinusoidal waveforms. They can be

approximated by their fundamental components, whereas the current and voltage at the

output can be approximated by their DC terms. By using FHA, the AC state variables

may be defined as a combination of sine and cosine components in function of the switching

frequency. The resonant currents and the resonant voltage are approximated by:

iLr(t) = irs(t) sin(ωst)− irc(t) cos(ωst) (2.7)

17


iLm(t) = ims(t) sin(ωst)− imc(t) cos(ωst) (2.8)

vc(t) = vcs(t) sin(ωst)− vrc(t) cos(ωst) (2.9)

The derivatives are given by

diLr(t)

dt=

(dirs(t)

dt+ ωsirc(t)

)sin(ωst)−

(dirc(t)

dt− ωsirs(t)

)cos(ωst) (2.10)

diLm(t)

dt=

(dims(t)

dt+ ωsimc(t)

)sin(ωst)−

(dimc(t)

dt− ωsims(t)

)cos(ωst) (2.11)

dvc(t)

dt=

(dvcs(t)

dt+ ωsvcc(t)

)sin(ωst)−

(dvcc(t)

dt− ωsvcs(t)

)cos(ωst) (2.12)

The terms irs and irc represents the sine and cosine components of the current in the

resonant tank. The same concept can be applied for im(t) and vc(t). This decomposition

introduces two states for each ac variable, resulting in a seventh-order dynamic model.

Extended Describing Function Method

As stated before, the EDF method combines time domain and frequency domain analysis,

extracting the model by dividing modulated waveforms into sine and cosine components.

The nonlinear terms can by approximated either by the fundamental component or the DC

component. The EDF’s are defined as

F1(d, Vin) =4Vin

π(2.13)

F2(iss, isc, Vo) =4

π

issipvo (2.14)

F3(iss, isc, Vo) =4

π

iscipvo (2.15)

F4(iss, isc) =2

πip (2.16)

18


ip =√(irs − ims)2 + (irc − imc)2 (2.17)

where Vin is the input voltage and d is the duty cycle (fixed 50%), iss and isc are the sine

and cosine components of the secondary of the transformer, and ip is the current flowing in

the primary of the transformer defined by equation (2.17). From equations (2.13) - (2.16),

the nonlinear terms of the state equations (2.1) - (2.4) can be approximated by

Vin = F1(d, Vin) sin(ωst) (2.18)

sign(iLr − iLm)vo(t) = F2(iss, isc, Vo) sin(ωst)− F3(iss, isc, Vo) cos(ωst) (2.19)

is = F4(iss, ics) (2.20)

The current is is the one flowing at the secondary of the transformer. Substituting the quasi-

sinusoidal waveforms and the nonlinear terms of the state equations by their approximations,

the continuous state equations are obtained as follows

Lrdirsdt

=4Vin

π− Lrωsirs − vcs −

4n

π

(irs − ims)

ipvo (2.21)

Lrdircdt

= Lrωsirc − vcc −4n

π

(irc − imc)

ipvo (2.22)

Crdvcsdt

= −Crωsvcc + irs (2.23)

Crdvccdt

= Crωsvcs + irc (2.24)

Lmdims

dt= −Lmωsimc +

4n

π

(irs − ims)

ipvo (2.25)

Lmdimc

dt= Lmωsims +

4n

π

(irc − imc)

ipvo (2.26)

Codvodt

=2n

πip −

n

RL

vo (2.27)

19


This set of equations describes the approximated large-signal model of the LLC resonant

converter. For obtaining the steady-state operating point, the time derivatives must be

set to zero. By defining the operating point (RL, ωs), the equation can be solved and

obtain the steady-state vector X = [Irs, Irc, Ims, Imc, Vcc, Vcs, Vo]. However, this model

continues having some nonlinear terms arising from the product of two or more time-varying

quantities. The system can be linearized around one operating point. The linearized model

can be expressed in a state space representation x(t) = Ax(t) + Bu(t), where A and B are

the Jacobian matrices of the system given by

Aij =∂f(x(t), u(t))

∂xj(t)

∣∣∣∣xo,uo

(2.28)

Bij =∂f(x(t), u(t))

∂uj(t)

∣∣∣∣xo,uo

(2.29)

The aforementioned expressions and derivations are highlighted in detail in Appendix.

Equation (2.28) and (2.29) are matrices of dimension 7× 7 and 7× 1 respectively.

2.2.2 Eigenvalues Analysis

The eigenvalues (λi) of matrix A represent the poles of the linearized model around a quiescent

operating point, and their position in the complex plane describes the dynamic behavior

of the system. Due to the high order of the system, obtaining analytical expression are

hard to derive. A numerical analysis is employed to evaluate the dynamic behavior of the

converter. To gain generality, the subsequent analysis is done using the normalized LLC

resonant converter shown in Table 2.1, employing the following base quantities von, Zo =√Lr

Cr, ion = vo

Zoand fn = 1

2π√LrCr

.

The location of the eigenvalues of the normalized converter is plotted in Fig. 2.3 for

different switching frequencies. As shown in the figure, the eigenvalues go through different

20


Table 2.1: Normalized Parameters

Parameter Value

Vin 1

Lr12π

Cr12π

Lm 3Lr

fr 1

n 1

lines of constant damping (ζ) and natural frequency (ωn) as fs changes. In LLC resonant

converters, the resonant tank elements are usually small compared with the output filter

parameters in order to achieve high switching frequencies with low ripple components at the

output voltage. Therefore, the dynamics of the resonant tank are much faster and it is only

defined by the resonant tank elements. On the other hand, there is a low frequency component

determined by the interaction between the output filter capacitor and the equivalent dynamic

impedance of the resonant tank. From the figure, a pair of dominant poles can be identified.

Since the modelling efforts focus on characterizing the converter’s low frequency dynamics,

the spotlight of this work is put in the dominant poles.

The displacement of the dominant poles for different load conditions is detailed in Fig.

2.3(b). As illustrated in the figure, the eigenvalues show the lowest damping ratio at resonant

frequency. When the load is decreased, the poles move over lines of constant ωn while

crossing diverse ζ lines. At constant switching frequency the minimum damping is obtained

for minimum load and operating at resonant frequency.

2.2.3 Simulation and Experimental Validation

The numerical analysis performed shows the minimum damping operation happens at the

resonant frequency. This behavior can be also observed in the time domain response of the

21


Im(λ )iIm(λ )iIm(λ )iIm(λ )i

Constant ζ Lines

(a) (b)

ζmin

fr

Constant ω Linesn

R =1nR =1nR = 2.5nR = 2.5n

R =5nR =5nR =10nR =10n

Light Load

Dominant Poles

for Different

Loads

Re(λ )iRe(λ )i

ffss

ffss

f s>f r

f s<f r

Dominant Poles

Re(λ )iRe(λ )i

Figure 2.3: Eigenvalues of the normalized LLC resonant converter of Table 2.1 (a) eigenvaluesdisplacement for different operating conditions defined by the fs, a pair of dominant pole isidentified. Note that the resonant frequency shows the less damped operating condition andthis is a critical point for control compensation. (b) Displacement of the dominant poles fordifferent load conditions, the damping factor decreases with the load.

converter at different operating conditions. The output voltage response for a frequency step

around different operating conditions is shown in Fig. 2.4.

The voltage response for the normalized resonant converter around different operating

conditions after applying a 2% frequency step is illustrated in Fig. 2.4 (a) to (c). As expected,

the transient response at different operating conditions shows distinctive characteristics. The

voltage displays an oscillatory behavior below and around the resonant point. At resonant

operation, the oscillations show the larger relative overshoot and settling time, meaning

smaller damping factor. At higher frequencies, the damping factor increases, and the output

has an over-damped response and no oscillations are produced.

22


1.16

1.18

1.2

1.22

0 200 400

v onv on

v onv on

v onv on

tntn tntn tntn

Below Resonance

0 200 4000.98

1

1.02

1.04Resonance

0 200 4000.77

0.79

0.81

0.83Above Resonance

(b) (c)(a)

Figure 2.4: Output voltage response of the normalized LLC converter under a small frequencystep (2%). (a) Below resonance, (b) around resonance, (c) above resonance.

2%2%

vo,around resonance f( )r

vvo,below resonanceo,below resonance 0.7f0.7f(( ))rr

vo,above resonance 1.3f( )r

frequency

Figure 2.5: Experimental transient response under a small frequency step (2%) around dif-ferent operating points.

An experimental response of an LLC resonant converter platform is shown in Fig. 2.5.

The dynamic response of the output voltage after applying a small switching frequency step

23

2.3. Summary

(2%) at three different operating conditions can be observed. For operation above resonant

frequency, the over-damped behavior can be observed and the output does not show overshoot

or oscillations. At lower frequency operating points, as resonance and below resonance, the

output voltage shows an overshoot and oscillatory response. However, the minimum damping

factor is achieved at resonant frequency operation, showing larger relative percentage of

overshoot and longer oscillations.

An AC ripple component associated with the high frequency poles of matrix A can be

seen in Fig. 2.5. For different switching frequencies, these high frequency poles are moving

in the complex plane; as such, the voltage ripple has different amplitudes depending on the

operating point. When operating below the resonant frequency, the output voltage of the

converter is larger and shows a maximum voltage ripple of 500 mV. However, when operating

at higher frequencies, the ripple component is reduced (approximately 100 mV for operation

at 1.3 fr). For all operating points, regardless of the switching frequency, the voltage ripple

component is less than 5% of the output voltage and can be neglected in the dynamic analysis

of the system. Therefore, the assumptions in Section 2.2.1 and the analysis in Section 2.2.2

are validated.

2.3 Summary

In this chapter the main equations of the LLC resonant converter were introduced and an-

alyzed. The extended describing function method was performed to obtain a mathematical

model of the LLC resonant converter. The steady-state equations of the system were lin-

earized around different operating conditions. Due to the high mathematical complexity, a

numerical analysis was performed in the normalized domain to identify the dynamic behav-

ior of the LLC converter. The pole displacement of the system was analyzed under different

frequency and load operating condition. Simulations and experimental results were provided

24

2.3. Summary

to verify the time domain response of the converter. A pair of complex dominant poles were

identified. The low frequency component of the system is determined by the position of

those dominant poles. The poles reach the minimum damping operation when the converter

operates at resonant frequency, defining this as the most challenging condition from a control

point of view.

25

Chapter 3

Large-Signal Model of the LLC

Resonant Converter

The behavior of the LLC resonant converter operating around resonance is defined by a pair

of dominant poles. However, the EDF method requires highly complex mathematical analysis

and numerical calculation to study the converter’s response for practical applications. The

use of equivalent circuits is a physical and intuitive approach which enables the application

of the well-known circuit analysis techniques. Focusing on the low frequency components

of the waveform is helpful to achieve usable models for the converter, by focusing on these

dominant features, the behavior of the converter can be modeled as a reduced order system.

In this chapter, an average large-signal model of the LLC resonant converter operating at

resonant frequency is derived. The obtained second order circuit predicts the low frequency

behavior of the system when it is operating at resonant frequency. The dynamic impedance

of the resonant tank is modeled as an equivalent inductor defined by the parameters of the

resonant tank and output filter. It is shown that the average natural response of the system is

similar to a rectified LC circuit and the characteristic equations can be found. The obtained

circuit can be directly solved to calculate the average output voltage and rectified current.

Finally, simulation and experimental captures are provided to verify the accuracy of the

obtained equivalent model.

26

3.1. Average Model of the LLC Resonant Converter

3.1 Average Model of the LLC Resonant Converter

At resonant frequency operation, the LLC converter has only two operation modes depending

on the position of the full-bridge switches. A simplified equivalent circuit for each one of the

operating modes is as shown in Fig. 3.1. When the switches S1 and S4 are turned on

(structure I ), the voltage Vin−nVo is applied to the resonant tank and the converter behaves

as shown in Fig. 3.1 (a). On the other hand, when S2 and S3 are conducting (structure II ),

the resonant tank voltage is defined by nVo − Vin as shown in Fig. 3.1 (b). As the converter

operates at resonant frequency, the interval of time of each substructure is defined by

tk = kπ√

LrCr, with k = 0, 1, 2, 3, 4... (3.1)

The converter operates in Structure I during half cycle k (tk < t < tk+1) and it switches to

Structure II during half cycle (k + 1) (tk+1 < t < tk+2). The converter is switching between

structure I and II at every switching action defined by fr.

From Fig. 3.1 (a), structure I can be defined by the following differential equations

Vin = LrdiLr(t)

dt+ vCr(t) + nvo(t) (3.2)

nvo(t) = Lmdim(t)

dt(3.3)

iLr(t) = CrdvCr(t)

dt(3.4)

irec(t) =Co

n

nvo(t)

dt(3.5)

irec(t) = n(iLr(t)− iLm(t)) (3.6)

By solving (3.2) - (3.5) the solutions for iLr(t), iLm(t), vCr(t), vo(t) and irec(t) are obtained.

In practice, Co >> Cr, and simplified current and voltage solutions including only one

27


(b)

(a)

nvO

CrLr

LmVin im

irecn

n2

iLr vCrVin vO

CrLr

Lmim CO

iLr vCr

D1

D2S4

S1 S3

S2

Vin vO

CrLr

Lmim

iLr vCr

S4

S1 S3

S2

D1

D2

vinv

CrLr

LmVin im

irecn

n2

iLr vCrnvO

CO

COCO

vinv

n:1

n:1

Figure 3.1: Operating modes of the LLC resonant converter at resonant frequency: a) Struc-ture I, switch S1 and S4 are ON, the applied voltage in the resonant tank is positive. b)Structure II, S2 and S3 are ON, the applied voltage in the resonant tank is negative.

sinusoidal component of frequency ωr =√LrCr are obtained as

iLm(t) = iLm(tk) +nvo(tk)

Lm

t (3.7)

iLr(t) = iLr(tk) +

(Co

Co + n2Cr

) (Vin − nvo(tk)− vCr(tk)

Zo

)sin(ωrt) +

nvo(tk)

Lm

t (3.8)

vCr(t) = vCr(tk) +

(Co

Co + n2Cr

)(Vin − nvo(tk)− vCr(tk))(1− cos(wrt)) +

nvo(tk)

2Lm

t2 (3.9)

vo(t) = vo(tk) +

(nCr

Co + n2Cr

)(Vin − nvo(tk)− vCr(tk))(1− cos(wrt)) (3.10)

irec(t) = n

(Co

Co + n2Cr

)(Vin − nvo(tk)− vCr(tk)) sin(wrt) (3.11)

28


On the other hand, the differential equations for structure II are given by

− Vin = LrdiLr(t)

dt+ vCr(t)− nvo(t) (3.12)

− nvo(t) = Lmdim(t)

dt(3.13)

iLr(t) = CrdvCr(t)

dt(3.14)

irec(t) = −Co

n

nvo(t)

dt(3.15)

irec(t) = −n(iLr(t)− iLm(t)) (3.16)

By solving (3.12) through (3.16), the current and voltages in the converter when it is oper-

ating in structure II are given by

iLm(t) = iLm(tk+1)−nvo(tk+1)

Lm

t (3.17)

iLr(t) = iLr(tk+1) +

(Co

Co + n2Cr

)(−Vin + nvo(tk+1)− vCr(tk+1)

Zo

)sin(ωrt)

− nvo(tk+1)

Lm

t (3.18)

vCr(t) = vCr(tk+1) +

(Co

Co + n2Cr

)(−Vin + nvo(tk+1)− vCr(tk+1))(1− cos(wrt))

− nvo(tk+1)

2Lm

t2 (3.19)

vo(t) = vo(tk+1)−(

nCr

Co + n2Cr

)(−Vin + nvo(tk+1)− vCr(tk+1))(1− cos(wrt)) (3.20)

irec(t) = −n

(Co

Co + n2Cr

)(−Vin + nvo(tk+1)− vCr(tk+1)) sin(wrt) (3.21)

29


As the converter switches between structure I and II, the initial conditions for each sub-

interval are defined by the final state in the previous structure and they need to be solved

in a cycle-by-cycle manner. As the converter is operating at resonance, the length of each

sub-interval is defined by Ts =1

2fr, and (3.9)-(3.20) can be expressed in discrete domain as

vCr(k + 1) = vCr(k) + 2

(Co

Co + n2Cr

)(Vin − nvo(k)− vCr(k)) +

nvo(k)

8Lmf 2r

(3.22)

vo(k + 1) = vo(k) + 2

(nCr

Co + n2Cr

)(Vin − nvo(k)− vCr(k)) (3.23)

vCr(k+2) = vCr(k+1)+2

(Co

Co + n2Cr

)(−Vin+nvo(k+1)−vCr(k+1))− nvo(k + 1)

8Lmf 2r

(3.24)

vo(k + 2) = vo(k + 1)− 2

(nCr

Co + n2Cr

)(−Vin + nvo(k + 1)− vCr(k + 1)) (3.25)

The start-up response of the LLC converter is illustrated in Fig 3.2. The response pre-

sented by the low frequency components of the rectified current and output voltage resembles

a sinusoidal waveform similar to those of a second order LC circuit. The discrete values of

the output voltage at every switching action can be predicted by (3.22) - (3.25). This low

frequency response of the converter can be modeled by a second order system by neglecting

the high frequency dynamics. In order to find the closed-form time that includes the discrete

solutions, the variables vCr(k), vo(k), vCr(k + 1) and vo(k + 1) are redefined as x1(k), x2(k),

x3(k) and x4(k) respectively, and vc and vo can be rewritten in an matrix form as shown

in (3.27). Defining

a =Co

Co + n2Cr

; b =Cr

Co + n2Cr

; (3.26)

30


irecirec

SS

t

t

t

Average

Response

Average

Response

LLC

Converter

LLC

Converter

LLC

Converter

LLC

Converter

Iterative

Solution

Iterative

Solution

vo

n2Vin

T2

s

T

2

eq

Figure 3.2: Start-up response of the LLC resonant converter at resonant frequency. Thediscrete equations predict the dynamic response of the converter at each switching action.The low frequency response can be modeled by a second order system by neglecting the highfrequency dynamics.

x1(k + 1)

x2(k + 1)

x3(k + 1)

x4(k + 1)

=

0 0 1− 2a −2na

0 0 −2nb 1− 2n2b

1− 2a 2na 0 0

2nb 1− 2n2b 0 0

x1(k)

x2(k)

x3(k)

x4(k)

+

a

nb

−a

nb

2Vin (3.27)

31


The discrete expression (3.27) can be solved by applying the Z-transform. Solving for vo

the output voltage expression in Z domain is given by the following

x2(z) = vo(z) = Vin2nbz(z + 1)

(z − 1) [z2 + 2 (n2b− a) z + (2a− 1)](3.28)

The Z transform for a cosine waveform of amplitude A and period T with DC component

is given by the following

Z (A(1− cos(ωkT ))) =(1− cos(ωT ))(z + 1)z

(z − 1)(z2 − 2z cos(ωT ) + 1)(3.29)

According to (3.28) and (3.29), and considering that Co >> Cr, the output voltage can

be expressed as

vo(k) =Vin

n(1− cos(ωeqTsk)) (3.30)

with Ts =1

2fr. By Comparing (3.30) and (3.28) the following expression is obtained

cos

(ωeq

1

2fr

)=

Co

Co + n2Cr

(1− n2Cr

Co

)(3.31)

The low frequency behavior of the converter can be modeled by the equivalent circuit

shown in Fig. 3.3. The inductor Leq represents the impedance of the resonant tank at

resonant frequency, whereas the diodes D1 and D2 model the output rectifier. The natural

frequency of the average model is defined by ωeq =n√

LeqCo, and Leq can be derived from (3.31)

as

Leq =n2Cr

Co

π2Lr

cos−1

Co

Co+n2Cr

1−n2Cr

Co

2 (3.32)

32

3.2. Model Validation

D

Io vo

Leq

Vin

irec

Co

n:1

Figure 3.3: Simplified large-signal equivalent circuit of the LLC resonant converter thatmodels the average behavior of the converter operating at resonant frequency.

as in practice, Co >> Cr and (3.32) can be approximated by Taylor series as

Leq(Cr)|Cr=0 =

(1

4+

n2Cr

6Co

− n4C4r

60C2o

+8n6C2

r

945Co

− ...

)π2Lr

∼=π2

4Lr (3.33)

This equation shows the simplified mathematical expression synthesis of the equivalent

inductor introduced in Fig. 3.3. This equivalent inductance represents the dynamic equivalent

impedance of the resonant tank described in Chapter 2. The interaction between Leq and

Co can be represented as a second order circuit that enables modelling the low frequency

dynamic behavior of the LLC resonant converter operating at resonant frequency.

3.2 Model Validation

In this section, the derived large-signal model is compared and verified with the LLC resonant

converter. Simulation results and experimental captures are provided to validate the proposed

model. First, simulations of the LLC resonant converter and its average model are analyzed.

Secondly, experimental start-up responses of the LLC converter are shown. The obtained

measurements are consistent with the calculated times obtained by using the average model,

it is able to predict the low frequency behavior of the LLC resonant converter.

33


3.2.1 Simulations

The start-up transient response of the normalized LLC resonant converter and its equivalent

average circuit of are shown in Fig. 3.4. The figure illustrates the output voltage and rectified

current when the converter is operating at resonant frequency. As shown, the model is able

to predict low frequency behavior of the LLC converter. Both voltage and current show a low

frequency sinusoidal waveform characteristic of LC circuits. The model predicts the behavior

neglecting the switching ripple in the output voltage and the resonant current in the rectifier.

The natural frequency of the converter is defined by the interaction between the resonant

tank, modeled as an equivalent inductor given by (3.33), and the output filter capacitor Co.

The equation that defines this fundamental frequency is given by

feq =n

2π√LeqCo

(3.34)

Teq

0

1

2

0

1

20

4

8

0

4

8

0 10 20 30 40 50 0 10 20 30 40 50

Output Voltage Rectified Current

v on

i recn

i recn

vo LLC Converter irec LLC Converter

vo Average Model

irec Average Model

v on

tn tn

The low frequency

component is predicted

by the average model

Figure 3.4: Comparison between LLC Converter and Average Large-Signal Model

34


As it is shown in Fig. 3.4, the dynamic behavior of both the LLC converter and the average

model are the same. Moreover, the current through Leq resembles the average behavior of

the rectified current.

3.2.2 Experimental No Load Start-Up

Different experimental start-up responses of the LLC operating at resonant frequency and

no load are presented in Fig. 3.5 - 3.6. The captures show the transient response for different

resonant tank and output filter parameters described in Table 3.1. The obtained responses

shows the predicted second order average behavior. However, as expected, the rising times for

the output voltage and rectified current are different depending on the converter’s parameters.

During start-up transient at no load condition, the rising time of the output voltage in a

rectified LC circuit is half of the natural period of the system. In this case, it is given by

Teq

2=

1

2feq=

π

n

√LeqCo (3.35)

As shown in Fig. 3.5 and 3.6, for different set of output capacitor and resonant tank

parameters, the start up response has a fundamental sinusoidal component. The measured

Table 3.1: Start Up Experimental Parameters

Parameter LLC1 LLC2 LLC3 LLC4

Lr 81µH 81µH 81µH 39.7µH

Cr 32nF 32nF 32nF 18.9nF

Co 16.5µF 32µF 50µF 39.6µF

n 5.33 5.33 5.33 7.8

Leq 199.86µH 199.86µH 199.86µH 97.95µH

Teq 67.7µs 94.27µs 117.84µs 50.17µs

35


(a)

(b)

TT

22

eqeq

TT==

22

eqeq Calculated 34 μs Calculated 34 μs

Measured 37 μs Measured 37 μs

TT

22

eqeq

TT==

22



Figure 3.5: Start-up LLC resonant converter at resonant frequency. Parameters: (a) Full-Bridge, LLC1. (b) Full-Bridge, LLC2.

36


(a)

(b)

TT

22

eqeq

TT==

22



TT

22

eqeq

TT==

22


Measured 27.5 μs Measured 27.5 μs

Figure 3.6: Start-up LLC resonant converter at resonant frequency. Parameters: (a) Full-Bridge, LLC3. (b) Half-Bridge, LLC4.

37

3.3. Summary

Teq for each tank-output filter configuration is similar to the calculated values employing

(3.35).

3.3 Summary

In this chapter, a large-signal average model of the LLC resonant converter was derived. The

obtained model predicts the low frequency component of the convert when it is operating

at resonant frequency. The dynamics of the resonant tank are modeled as a equivalent

inductance Leq that interacts with the output filter to define the low frequency response of

the converter. The obtained circuit can be directly solved to calculate the average output

voltage and rectified current. Simulation and experimental captures were provided to verify

the derived analytical model.

38

Chapter 4

Dual-Loop Controller for LLC

Resonant Converter

The ever increasing demand to provide tight regulation at the output voltage and current re-

quires the implementation of closed-loop controllers. Most of the work found in the literature

has focused on developing single-loop controllers for resonant converters. Meanwhile, employ-

ing an inner-current-loop and outer-voltage-loop is a widely adopted method to control PWM

converters. One of the main advantages of current mode control is its simpler dynamics. The

transfer function current-to-voltage contains one less pole than the control-to-output-voltage

transfer function. Moreover, the addition of current measurement is required for protection

against excessive currents during transient and fault conditions. Employing a similar scheme

for a resonant converter may enable such advantages as those mentioned above.

This chapter introduces the proposed dual-loop control scheme, employing an inner current

loop and outer voltage loop. First, a simplified small-signal model of the LLC resonant

converter valid for the most challenging condition from a control point of view is derived.

Then, the analysis of the required transfer functions for designing the dual-loop controller

is performed. A controller design procedure example is shown, and proper compensators for

both current and voltage loop, are designed to guarantee stable operation. Finally, simulation

and experimental results of the closed-loop system under different operating conditions are

provided to verify the performance of the controller.

39

4.1. Small-Signal Model of the LLC Resonant Converter

4.1 Small-Signal Model of the LLC Resonant

Converter

Usually small-signal models are employed to predict how the variations in the control variables

affect the output. Conventional perturbation and linearization techniques assume that an

averaged voltage or current consists of a constant component and a small-signal AC variation

around the DC component. Therefore, the large-signal average model developed in Chapters 2

and 3 can be extended to obtain a linearized model of the power plant. Obtaining reliable

and accurate small-signal models can be employed to derive the required transfer function

for designing linear controllers.

Before applying conventional linearization techniques, the dynamic gain of the model needs

to be obtained. This gain represent how much is the variation of the output voltage and

current when a small perturbation is applied in the control variable. For the LLC converter,

the control variable is the switching frequency. Therefore, the dynamic relationship between

the output voltage and the switching frequency can be found as the slope of the DC gain

voltage characteristics given by (2.5). Taking the partial derivative, the slope is found as

∂M

∂fn= −

Lrfn(2LmLrf2n + 2L2

m(f2n − 1) + LrQ

2(f 6n − f 2

n))

L3m

[(f 2n

(Lr

Lm+ 1

)− 1

)2

+ L2rQ

2f2n(f

2n−1)2

L2m

] 32

(4.1)

As the model has been derived assuming resonant operation, fr = fs, and fn = 1. The

dynamic small-signal gain can be derived as

kf =∂Vo

∂fs= −

8

π

Vin

n

Lm

Lr

1

fr(4.2)

40

4.1. Small-Signal Model of the LLC Resonant Converter

RL

~

sCo1

irec;avg

sLeqn2fs

~

kfn

vo~

Figure 4.1: Small-Signal Equivalent Circuit of the LLC resonant converter at resonant fre-quency. This linearized model is employed to derive the transfer functions for designing theproposed dual-loop controller.

Linearizing the average model shown in Fig. 3.3, the small-signal model around the reso-

nant frequency can be obtained as shown in Fig. 4.1. The output voltage to control frequency

transfer function (Gvf ) can be obtained from the linearized model as

Gvf (s) =vo(s)

fs(s)=

kf

1 + s Leq

RLn2 + s2LeqCo

n2

(4.3)

The Bode plot of the transfer function Gvf is illustrated in Fig. 4.2. The dynamic response

of the LLC resonant converter resembles the small-signal model of a Buck converter. The

switching to natural frequency ratio of a buck converter is critical to achieve high bandwidth

controllers. In case of LLC converters. the ratio between the resonant and the double pole

frequencies is defined by

rfc =frfeq

=π

2n

√Co

Cr

(4.4)

which is dependant on the converter parameters.

The control to output voltage transfer function was derived in this section based on the

small-signal model shown in Fig. 4.1. The same methodology can be employed to obtain

the control to rectified current transfer function, and the rectified current to output voltage

transfer function required for designing a dual-loop controller.

41

4.2. Dual-Loop Controller for the LLC Resonant Converter

Frequency [Hz]

0

180

Phas

e [d

eg]

R1R2R3

Mag

nit

ude

[dB

]rrfcfcrrrfcfcfc

rrffeqeqff

==ffrrfffrrrffeqeqfffeqeqeq

rrfcfc

Gv0

Figure 4.2: The control-frequency to output-voltage transfer function of LLC resonant con-verter presents a second order frequency response.

4.2 Dual-Loop Controller for the LLC Resonant

Converter

In open-loop operation, in presence of disturbances, the output voltage may show oscillations,

overshoot and steady-state error. A closed-loop controller can be employed to minimize these

issues. In this section, a dual-loop control scheme employing an inner current and an outer

voltage loop is proposed for the LLC resonant converter and illustrated in Fig. 4.3. Imple-

menting a dual-loop scheme enables significant advantages such as tight current regulation

and over-current protection. As shown in the figure, the desired output voltage is employed

as a set-point for the outer loop, whereas the inner-loop current reference is commanded

by the voltage-loop compensator. The averaging of the rectified current is included in the

42


Outer Voltage Loop

Inner Current LoopLLC

ev irecirec

irecirec

Gv(s) Gvi(s)Gif(s)Gi(s)

Avg

vovref

ei f iref

Figure 4.3: The proposed control scheme for the LLC resonant converter. The control designprocedure is simplified by employing the linearized model of the LLC power converter.

feedback path of the inner loop, and the current loop compensator adjusts the switching

frequency in order to track the average current reference.

The control frequency to rectified current transfer function is given by

Gif (s) =irec,avg(s)

fs(s)=

kf

(1RL

+ sCo

)1 + s Leq

n2RL+ s2LeqCo

n2

(4.5)

As it can be observed in the equation, the transfer function has a double pole located at

ωeq = n√LeqCo

, and a zero defined by the interaction of the output capacitor and the load.

The gain kf is negative as a consequence of the negative on the DC characteristics of the LLC

converter, which translate into a −180◦ phase delay of the DC component. The Bode plot

of (4.5) shown in Fig. 4.4 (a) includes the discretization effect given by the averaging block

and controller sampling frequency (fs = 2fNy). As the control loop bandwidth is desired to

be between feq and fNy, the ratio between these frequencies (rfc2) is critical for the controller

design as shown in Fig. 4.4 (a).

Provided enough bandwidth difference between the two control loops, the inner loop can

be considered as a controlled current source simplifying the loop dynamics to a first order

43


22

rrffc2c2

fr1

fr2

Gvi3Gvi3

Gvi2Gvi2

Gvi1Gvi1

fz3 fz2 fz1

R1

R2

R3Gvi3Gvi3 Gvi2Gvi2 Gvi1Gvi1

22

rrffc2c2

fNy2

feq fNy1

-360

0

Frequency [Hz] Frequency [Hz]

Phas

e [d

eg]

Mag

nit

ude

[dB

]

-90

-45

0

Phas

e [d

eg]

Mag

nit

ude

[dB

]

Same f Same f eqeq

Different f Different f rr

(a) (b)

Frequency-to-Current Current-to-Voltage

Figure 4.4: (a) Control frequency-to-average rectified current transfer function consideringtwo resonant frequencies. The ratio between fr and feq is critical in the controller design.(b) Average rectified current-to-output voltage transfer function. The outer voltage loop hasbeen simplified.

system as shown in figures 4.4 (b) and 4.5. The rectified current to output voltage transfer

function is then given by

Gvi =vo(s)

irec,avg(s)=

11RL

+ sCo

(4.6)

Once the transfer functions have been defined for both inner and outer loops, linear con-

trollers can be implemented by employing conventional techniques. In this way, reducing the

44

4.3. Controller Design Procedure and Simulation Results

(b)(b)

(a)(a)

Outer Voltage Loop

Inner Current Loop

ev Gv(s) Gvi(s)Gi CL(s) vovref

iref irec

RLsCo1

vo~iref

Figure 4.5: Simplified equivalent circuit of the outer loop for designing the voltage controller:a) circuit model, b) closed-loop block control diagram. The implementation of an innercurrent loop simplifies the voltage loop controller design.

LLC resonant converter dynamics to a second order system enables simple and straightfor-

ward controller design procedure.

4.3 Controller Design Procedure and Simulation

Results

The small-signal model derived in the previous chapter is used in the following sections

to design each control loop. Adding a feedback loop can cause an otherwise well-behaved

circuit to exhibit oscillations, overshoot, and other undesirable behavior. Usually, lead-lag

compensators are used to improve the phase margin and extended the bandwidth of the

feedback loop, while proportional integral controllers are used to increase the low-frequency

loop gain. This leads to better rejection of disturbances and small steady-state error.

45


4.3.1 Controller Design Procedure

First, the current loop must be designed. The control-to-rectified current transfer func-

tion (4.5) was illustrated in Fig. 4.4 (a). According to the parameters in Table 4.1, in this

case the position of the double pole (feq) is 2.3 kHz, with a fr2of 50 kHz. Obtaining larger

bandwidths than feq guarantees that the current-loop is able to attenuate the oscillations

due to the effect of the double pole. On the other hand, this is a sample data system and the

maximum theoretical bandwidth is defined by the Nyquist frequency. Therefore, the com-

pensator employed in the inner current-loop must be designed to satisfy a loop bandwidth

feq andfr2. A bandwidth of 10 kHz is selected to satisfy this condition.

The second step is designed the outer voltage loop controller. As it was explained in

Chapter 4, the implementation of an inner current loop simplifies the voltage loop transfer

function, and the rectified current-to-output voltage transfer function (4.6) is employed.

However, in this case, the compensator must be designed to achieve a much lower bandwidth

than the inner-loop, which leads to a selection of a 1 kHz frequency. In this way, the dynamics

of the current loop can be neglected when analyzing the behavior of the whole system.

Table 4.1: LLC Parameters - Control Design

Parameter Value

Vin 120 V

Lr 81 µH

Cr 32 nF

Lm 227 µH

fr 98 kHz

Po 150 W

n 5.33

Co 660 µF

feq 2.3 kHz

46


Proportional Integral (PI) controllers are employed as compensators for each control loop.

The compensated open-loop Bode plots are shown in Fig. 4.6 (a) and 4.6 (b) for the inner

and outer loops respectively. As it can be observed the desired bandwidths are achieved with

phase margins (ϕm) of 57◦ and 79◦ respectively which guarantee stable operation.

To verify the design of the current and voltage controllers, the dynamics of both inner and

outer control loops are included in the closed-loop response of Fig. 4.7 and 4.8. Designing

the outer loop bandwidth ten times slower than the inner-loop enables a −3dB bandwidth

-135

-180

-90

-45

0

45

80

60

40

20

-20

0

150 W 15 W

101

100

102

103

104 fr

22

f = 2.3 kHzeq

Frequency [Hz] Frequency [Hz]

Ph

ase

[deg

]M

agn

itu

de

[dB

]

GciGciGfi1Gfi1

GciGciGfi2Gfi2

GciGciGfi1Gfi1

GciGciGfi2Gfi2

-50

0

50

100

-180

-150

-120

-90

φm=79°φm=79°

ΔBW = 1 kHzΔBW = 1 kHz

150 W 15 W

101

100

102

103

104

GcvGcvGiv1Giv1

GcvGcvGiv2Giv2

GcvGcvGiv2Giv2

GcvGcvGiv1Giv1

φm=57°φm=57°

ΔBW = 10 kHzΔBW = 10 kHz

Compensated Current Loop Compensated Voltage Loop

(a) (b)

Figure 4.6: (a) Compensated control-frequency to rectified-current transfer function. Thebandwidth loop is selected to attenuate the double-pole at feq. (b) Compensated rectified-current to output-voltage transfer function. The bandwidth of the voltage loop is ten timeslower than the inner-loop bandwidth.

47


of 1.5 kHz closed-loop bandwidth as shown in Fig. 4.7. As mentioned before, the relative

distance between control loops is critical to guarantee closed-loop stability. A parametric

analysis of the closed-loop Bode response is performed in Fig. 4.8 for different voltage control

loop bandwidths. As shown in the figure, the closed-loop −3 dB bandwidth varies with the

gain of the voltage controller. Larger bandwidth in the outer loop enables higher closed-loop

bandwidths, however designing the outer loop with large gains may create oscillations due

to the interaction between both control loops.

-360

-180

0

10

0

-10

-20

-60

-40

3 dB3 dB

ΔBW = 1.5 kHzΔBW = 1.5 kHz

101

100

102

103

104 fr

2Frequency [Hz]

Phas

e [d

eg]

Mag

nit

ude

[dB

]

150 W 15 W GCL1

GCL1

GCL2GCL2

GCL2GCL2

GCL1GCL1

Figure 4.7: Closed-loop Bode plot of the proposed dual-loop control scheme for the LLCresonant converter of Table 4.1. The obtained closed-loop 3 dB bandwidth is 1.5 kHz. Theouter voltage loop gain is adjusted to be obtain a voltage bandwidth ten time slower thanthe inner current loop to prevent the interaction between control loops.

48


20

10

0

-10

-30

-20

-360

-270

-180

-90

0

0.1 Kv0.3 Kv 0.6 Kv 1 Kv 1.5 Kv 3.5 Kv5.5 Kv

Control Loops

Interference

Low

Bandwidth

101

100

102

103

104 fr

2Frequency [Hz]

Phas

e [d

eg]

Mag

nit

ude

[dB

]

GCLkGCLk

GCLkGCLk

Figure 4.8: Different closed-loop bode plots of the proposed dual-loop scheme for differentouter loop bandwidths. Larger gains in the voltage loop enable higher bandwidth and fasterresponses in the closed-loop system. However, large gains may create oscillation betweenboth control loops.

4.3.2 Simulation Results

In this part, the LLC resonant converter using the parameters of Table 4.1 is simulated under

different operating conditions implementing the designed dual-loop controller.

Simulation results of the proposed dual-loop scheme under different load steps and volt-

age reference set-points are shown in Fig. 4.9 and compared with the open-loop responses

(constant fs). The upper plot illustrates the response of vo for a current load step from 1 to

7 A. As expected, the closed-loop system response shows no oscillations and zero steady-state

error under the applied disturbances at different operating points.

49


18

22

26

30

34

38

60

80

100

120

140

160

0 2 4 6 80

4

8

ref v = 34 V

ref v = 30 V

ref v = 26 V

ref v = 22 V

ref v = 18 V

119 KHz

98 KHz102 KHz

145 KHz

88 KHz

79 KHz

73 KHz

83 KHz

76 KHz

70 KHz

f[K

Hz]

si

[A]

Load

v[V

]o

Unregulated Open-Loop

Tight Closed-Loop

Similar Responses

Over a Wide Range of

Operating Conditions

Wide Range of

Frequencies}6 A6 A

t [ms]

Figure 4.9: Simulation results for the proposed dual-loop controller for the LLC converterof Table 4.1 under a load step for different operating points. The closed-loop system is stableunder a wide range of operating conditions below and above the resonance.

The second part of Fig. 4.9 illustrates the variation of the switching frequency during

the transients. As shown, the dual-loop controller is able to compensate the output over a

50


18

20

22

24

26

28

30

32

v[V

]o

Voltage

Reference

i 1 [A]Load

i 3.5 [A]Load

i 7 [A]Load

Tight Closed-Loop

0 5 10 15 20 2570

80

90

100

110

120

f[K

Hz]

s

Switching Frequency

t [ms]

Wide Range of

Frequencies}Zero Steady-State

Error

No Overshoot

Figure 4.10: Simulation results for the proposed dual-loop controller for the LLC converter ofTable 4.1 under a voltage reference step for different load conditions. The controller operatesin a wide range of switching frequencies.

wide range of frequencies above and below resonance. It can be observed that the controller

continues operating properly for reference set-points that reach 34 and 18 V with switching

frequencies ranging from 70 to 150 kHz.

The output voltage behavior under a set-point step is illustrated in Fig. 4.10. The figure

shows a comparison between different loading condition for closed-loop operation, and illus-

trates how the switching frequency must change in order to track the changing set-point. As

shown, the closed-loop system is stable under a wide range of operating points below and

above the resonance.

51

4.4. Experimental Results

4.4 Experimental Results

A 150 W prototype of a full-bridge LLC resonant converter was implemented employing the

parameters given by Table 4.1. The dual-loop controller, inner current and outer voltage is

implemented based on the design of Chapter 4.3. The experimental prototype of the full-

bridge LLC resonant converter is shown in Fig. 4.11. The proposed controller is implemented

in a Texas Instrument digital microcontroller for the series C2000 (MSP320F28335 DSP).

The output of the prototype is connected to a electronic load Chroma 63204 configured in

constant current load mode and, at the input, a DC power supply Sorensesn SGA 400/38 is

connected. Experimental results for the transient response of the 150 W prototype under a

different conditions are presented in Fig. 4.12 - Fig. 4.16. All the signals were captured by a

Teledyne Lecroy WaveRunner 604Zi oscilloscope.

Inverter Board

Control Board Input VoltageInput Voltage

Output (to load)Output (to load)

Resonant Tank

Rectifier and

Output Filter

Figure 4.11: 150 W full-bridge LLC resonant converter experimental prototype.

52

4.4. Experimental Results

The experimental captures showing the dynamic response of the LLC converter with the

developed dual-loop controller current load step operating at different voltage set-points are

shown in Fig. 4.12 and 4.13. The response of the converter for a voltage reference of 18 V

is shown in Fig. 4.12 (a). As shown, the setting time is 2.2 ms and 1.2 ms for a load current

step-down and step-up, while the overshoot is 1.2 V and 1.5 ms respectively. For a set-point

of 22 V, the converter operates around the resonant point, as shown in Fig 4.12 (b) with

settling time and overshoot values of 60 mV and 1.2 ms respectively. The behaviour of

the closed loop system with the converter operating below fr is illustrated in 4.12 (a) and

4.12 (b). As it can be observed, both cases show similar settling time and overshot values.

The implementation of an inner-current loop improves the dynamic of the system over a wide

range of switching frequencies.

The dynamic response of the experimental platform under a voltage set-point step is

shown in Fig. 4.14. The capture illustrates the output voltage and switching frequency of

the converter for three different loading conditions. As shown, the output voltage reaches

the reference point with no overshot and no oscillations for all loading condition . During the

step-up, the output shows similar responses for all loading conditions. However, during step-

down, lighter loads take longer time to settle due to the converter entering in discontinuous

conduction mode.

A conventional voltage mode controller has been implemented to compare the dynamic

response of the dual-loop controller. The experimental results for load transient response of

the 150 W prototype are shown in Fig. 4.15 - 4.16. As observed, the dual-loop control shows

superior transient behavior over a wide range of operating conditions. Apart from reduce

the order and simplify the voltage loop, the inner current loop minimizes the variation of the

small-signal characteristics of the converter at different operating conditions.

53

4.5. Summary

4.5 Summary

This chapter introduced a dual-loop controller for LLC resonant converters. The considera-

tions for the analysis and design of both inner and outer loop were presented. A small-signal

model of the LLC resonant converter operating at resonant frequency was derived from the

average large-signal model by applying conventional linearization techniques. The small-

signal model is a second order system with a double pole located at the resonant frequency

defined by Leq and Co. The small-signal model was employed to derived proper transfer

functions for the design of compensators in a simple and straightforward manner. Finally,

a design controller example was performed, and simulation and experimental results at dif-

ferent operating conditions were provided to validate the stability and performance of the

closed-loop system.

ref v =20 V

ref v =30 V

vo,iload = 7 A

vo,iload = 3.5 Avo,iload = 1 A

fs,iload = 7 A

fs,iload = 3.5 A

fs,iload = 1 A

Figure 4.12: Experimental capture reference step from 20 V to 30 V for different load currents.C1: output voltage (iload = 7 A), C2: frequency variation (7 A), M1: output voltage (iload =1 A), M2: frequency variation (1 A), M3: output voltage (iload = 3.5 A), M4: frequencyvariation (3.5 A), C3: voltage reference step.

54

4.5. Summary

(a)

(b)

Proposed Dual-Loop Controller


ref v =18 V

ref v =22 V

vo

iavg

irec

iload

1.2 V1.2 V

2.2 ms2.2 ms

1 ms1 ms

1.5 V1.5 V

500 mV500 mV

1.2 ms1.2 ms

vo

iavg

irec

iload

1 ms1 ms

600 mV600 mV

6 A6 A

6 A6 A

The response is

consistent accros

different condtions

The response is

consistent accros

different condtions

Figure 4.13: Experimental capture current load step from 1 A to 7 A for reference voltage18 V. C1: output voltage, C2: average current from the microcontroller, C3: rectifiedcurrent, C4: load current.

55

4.5. Summary

(a)

(b)

600 mV600 mV

1.4 ms1.4 ms

1 ms1 ms

600 mV600 mV

1.8 ms1.8 ms

1 ms1 ms

700 mV700 mV

ref v =26 V

vo

iavg

irec

iload 6 A6 A

The response is

consistent accros

different condtions

The response is

consistent accros

different condtions

vo

iavg

irec

iload 6 A6 A

ref v =32 V




56

4.5. Summary

(a)

(b)

ref v =18 V2.6 ms2.6 ms

2.3 V2.3 V

The transient

performance is

impaired at different

conditions

900 mV900 mV

2.2 ms2.2 ms

1.2 ms1.2 ms

1 V1 V

The transient

performance is


conditions

vo

iavg

irec

iload 6 A6 A

vo

iavg

irec

iload 6 A6 A

1.4 V1.4 V

1.8 ms1.8 ms

Conventional Single-Loop Controller

Conventional Single-Loop Controller


57

4.5. Summary

(a)

(b)

The transient

performance is


conditions

The transient

performance is


conditions

ref v =26 VConventional Single-Loop Controller

vo

iavg

irec

iload 6 A6 A

ref v =30 V

vo

irec

iload 6 A6 A

1.2 mV1.2 mV

3 ms3 ms

1.4 ms1.4 ms

1.1 V1.1 V

1.4 V1.4 V

1.2 V1.2 V

iavg

4.2 ms4.2 ms

1.2 ms1.2 msConventional Single-Loop Controller


58

Chapter 5

Conclusion

5.1 Summary

This work developed a simple and straightforward compensator design methodology for LLC

resonant converters. In this approach, a numerical analysis of the converter’s dynamic be-

havior was performed, and the resonant frequency operation was identified as the most chal-

lenging condition from a control design point of view. An accurate large-signal average model

and a simplified linearized model of the converter operating at resonant frequency were pro-

posed. An effective dual-loop scheme with inner current loop and outer voltage loop was

implemented employing the output voltage and output rectifier current as feedback signals.

This dual-loop scheme enabled numerous advantages, such as, tight current regulation and

over-current protection. The implementation of an inner current loop minimizes the varia-

tions of the small-signal characteristics of the converter at different operating points, thus

achieving similar transient responses over a wide range of frequencies.

The effectiveness of the proposed dual-loop scheme was verified by simulation and exper-

imental results by using a 150 W full-bridge LLC resonant converter. The obtained results

show that the implementation of a dual-loop configuration provides the desired closed-loop

performance for the entire operational range. Finally, this work introduces a useful tool that

enables a simple and straightforward procedure for designing linear controllers for a complex

topology: the LLC resonant converter.

59

5.2. Future Work

5.2 Future Work

The concept developed in this work provides an original contribution to the design of con-

trollers for LLC resonant converters. This work could be extended to develop models of

the converter at different operating points, including additional effects such as parasitic ele-

ments and industry standard filters, and designing and implementing adaptive controllers to

improve the performance of the system.

The simplified controller design methodology could be extrapolated to other topologies.

The implementation of a dual-loop scheme opens the possibility of extending the control

strategy to several converters working in parallel. The whole system could be controlled by

implementing an individual inner current loop in each converter and a main voltage loop to

provide the current reference of each individual stage.

60

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Appendix A

Extended Describing Function

Matrices

The Jacobian matrices (2.28) and (2.29) are calculated from equations (2.21)- (2.27). For

simplification, the equations have been redefined as f1, f2, f3, f4, f5, f6 and f7 respectively.

The matrix A is formed by the following elements

a11 =∂f1(x, t)

∂irs= − 4n

πLr

Vo

(1

ip− (irs − ims)

2

i3p

)(A.1)

a12 =∂f1(x, t)

∂irc= −ωs −

4n

πLr

Vo

((ims − irs)(imc − iirc)

i3p

)(A.2)

a13 =∂f1(x, t)

∂vcs= − 1

Lr

, a14 =∂f1(x, t)

∂vcc= 0 (A.3)

a15 =∂f1(x, t)

∂ims

=4nVo

πLr

(1

ip− (ims − irs)

2

i3p

)(A.4)

a16 =∂f1(x, t)

∂imc

=4nVo

πLr

((ims − irs)(imc − irc)

i3p

)(A.5)

a17 =∂f1(x, t)

∂Vo

=4nVo

πLr

(ims − irs)

ip(A.6)

a21 =∂f2(x, t)

∂irs= ωs +

4nVo

πLr

((imc − irc)(ims − irs)

i2p

)(A.7)

a22 =∂f2(x, t)

∂irc= −4nVo

πLr

(1

ip+

(imc − irc)2

i3p

)(A.8)

71

Appendix A. Extended Describing Function Matrices

a23 =∂f2(x, t)

∂vcs= 0, a24 =

∂f2(x, t)

∂vcc= − 1

Lr

(A.9)

a25 =∂f2(x, t)

∂ims

= −4nVo

πLr


i3p

)(A.10)

a26 =∂f2(x, t)

∂imc

=4nVo

πLr

(1

i2p+

(imc − irc)2

i3p

)(A.11)

a27 =∂f2(x, t)

∂Vo

=4nVo

πLr

((imc − irc)

ip

)(A.12)

a31 =∂f3(x, t)

∂irs=

1

Cs

, a32 =∂f3(x, t)

∂irc= 0, a33 =

∂f3(x, t)

∂vcs= 0 (A.13)

a34 =∂f3(x, t)

∂vcc= −ωs, a35 =

∂f3(x, t)

∂ims

= 0, a36 =∂f3(x, t)

∂imc

= 0, a37 =∂f3(x, t)

∂Vo

= 0

(A.14)

a41 =∂f4(x, t)

∂irs= 0, a42 =

∂f4(x, t)

∂irc=

1

Cs

, a43 =∂f4(x, t)

∂vcs= ωs (A.15)

a44 =∂f4(x, t)

∂vcc= 0, a45 =

∂f4(x, t)

∂ims

= 0, a46 =∂f4(x, t)

∂imc

=1

Cs

, a47 =∂f4(x, t)

∂Vo

= 0 (A.16)

a51 =∂f5(x, t)

∂irs=

4nVo

πLr

(1

ip− (ims − irs)

2

i3p

)(A.17)

a52 =∂f5(x, t)

∂irc=

4nVo

πLr


i3p

)(A.18)

a53 =∂f5(x, t)

∂vcs= 0, a54 =

∂f5(x, t)

∂vcc= 0 (A.19)

a55 =∂f5(x, t)

∂ims

= −4nVo

πLr

(1

ip− (ims − irs)

2

i3p

)(A.20)

a56 =∂f5(x, t)

∂imc

= −ωs +4nVo

πLr


i3p

)(A.21)

a57 =∂f5(x, t)

∂Vo

= −4nVo

πLr

((ims − irs)

ip

)(A.22)

a61 =∂f6(x, t)

∂irs= −4nVo

πLr


i3p

)(A.23)

72


a62 =∂f6(x, t)

∂irc= frac4nVoπLr

(1

ip− (imc − irc)

2

i3p

)(A.24)

a63 =∂f6(x, t)

∂vcs= 0, a64 =

∂f6(x, t)

∂vcc= 0 (A.25)

a65 =∂f6(x, t)

∂ims

= ωs +4nVo

πLr


i3p

)(A.26)

a66 =∂f6(x, t)

∂imc

=4nVo

πLr

(1

ip− (imc − irc)

2

i3p

)(A.27)

a67 =∂f6(x, t)

∂Vo

= −4nVo

πLr

((imc − irc)

ip

)(A.28)

a71 =∂f7(x, t)

∂irs= − 2n

πCo

(ims − irs)

ip(A.29)

a72 =∂f7(x, t)

∂irc=

2n

πCo

(imc − irc)

ip(A.30)

a73 =∂f7(x, t)

∂vcs= 0, a74 =

∂f7(x, t)

∂vcc= 0 (A.31)

a75 =∂f7(x, t)

∂ims

=2n

πCo

(ims − irs)

ip(A.32)

a76 =∂f7(x, t)

∂imc

= − 2n

πCo

(imc − irc)

ip(A.33)

a77 =∂f7(x, t)

∂Vo

= − n

CoRL

(A.34)

The matrix B is compounded by the following elements

b11 =∂f1(x, t)

∂ωs

= −irc, b11 =∂f2(x, t)

∂ωs

= irs (A.35)

b31 =∂f3(x, t)

∂ωs

= −vcc, b41 =∂f4(x, t)

∂ωs

= vcs (A.36)

b51 =∂f1(x, t)

∂ωs

= −imc, b61 =∂f6(x, t)

∂ωs

= ims (A.37)

73


b71 =∂f7(x, t)

∂ωs

= 0 (A.38)

74

Documents

Dual-Loop Linear Controller for LLC Resonant Converters · Dual-Loop Linear Controller for LLC Resonant Converters by Franco Degioanni Ing., Universidad Nacional Cordoba, 2014 A THESIS