Dual-Loop Linear Controller for LLC Resonant Converters · Dual-Loop Linear Controller for LLC...
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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
Supervisory Committee Member
Additional Examiner
Additional Supervisory Committee Members:
Supervisory Committee Member
Supervisory Committee Member
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
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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
1.2. Literature Review
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
1.2. Literature Review
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. Literature Review
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. Literature Review
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
1.2. Literature Review
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
1.2. Literature Review
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
2.1. LLC Resonant Converter
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
2.1. LLC Resonant Converter
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. Numerical Identification of the Most Challenging Operating Condition
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
2.2. Numerical Identification of the Most Challenging Operating Condition
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
2.2. Numerical Identification of the Most Challenging Operating Condition
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
2.2. Numerical Identification of the Most Challenging Operating Condition
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
2.2. Numerical Identification of the Most Challenging Operating Condition
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
2.2. Numerical Identification of the Most Challenging Operating Condition
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
2.2. Numerical Identification of the Most Challenging Operating Condition
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
3.1. Average Model of the LLC Resonant Converter
(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
3.1. Average Model of the LLC Resonant Converter
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
3.1. Average Model of the LLC Resonant Converter
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
3.1. Average Model of the LLC Resonant Converter
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
3.1. Average Model of the LLC Resonant Converter
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. Model Validation
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
3.2. Model Validation
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
3.2. Model Validation
(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
eqeq Calculated 47 μs Calculated 47 μs
Measured 52 μs Measured 52 μs
Figure 3.5: Start-up LLC resonant converter at resonant frequency. Parameters: (a) Full-Bridge, LLC1. (b) Full-Bridge, LLC2.
36
3.2. Model Validation
(a)
(b)
TT
22
eqeq
TT==
22
eqeq Calculated 58 μs Calculated 58 μs
Measured 61 μs Measured 61 μs
TT
22
eqeq
TT==
22
eqeq Calculated 25 μs Calculated 25 μs
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
4.2. Dual-Loop Controller for the LLC Resonant Converter
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
4.2. Dual-Loop Controller for the LLC Resonant Converter
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. Controller Design Procedure and Simulation Results
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
4.3. Controller Design Procedure and Simulation Results
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
4.3. Controller Design Procedure and Simulation Results
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
4.3. Controller Design Procedure and Simulation Results
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
4.3. Controller Design Procedure and Simulation Results
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
4.3. Controller Design Procedure and Simulation Results
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
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
Proposed Dual-Loop Controller
Proposed Dual-Loop Controller
Figure 4.14: 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.
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
impaired at different
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
Figure 4.15: 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.
57
4.5. Summary
(a)
(b)
The transient
performance is
impaired at different
conditions
The transient
performance is
impaired at different
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
Figure 4.16: 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.
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
((imc − irc)(ims − irs)
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
((ims − irs)(imc − irc)
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
((ims − irs)(imc − irc)
i3p
)(A.21)
a57 =∂f5(x, t)
∂Vo
= −4nVo
πLr
((ims − irs)
ip
)(A.22)
a61 =∂f6(x, t)
∂irs= −4nVo
πLr
((imc − irc)(ims − irs)
i3p
)(A.23)
72
Appendix A. Extended Describing Function Matrices
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
((imc − irc)(ims − irs)
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
Appendix A. Extended Describing Function Matrices
b71 =∂f7(x, t)
∂ωs
= 0 (A.38)
74