Dynamical profile of switched-mode converter - fact or fiction?

Mikko Hankaniemi

Dynamical Profile of Switched-Mode Converter – Fact or Fiction?

Tampere University of Technology. Publication 687

Mikko Hankaniemi

Dynamical Profile of Switched-Mode Converter – Fact or Fiction?

Thesis for the degree of Doctor of Technology to be presented with due permission for public examination and criticism in Rakennustalo Building, Auditorium RG202, at Tampere University of Technology, on the 30th of November 2007, at 12 noon.

Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2007

ISBN 978-952-15-1863-8 (printed) ISBN 978-952-15-1902-4 (PDF) ISSN 1459-2045

This thesis proposes a dynamical profile for a switched-mode DC-DC converter. The developed concept and definition of the dynamical profile is independent of the topology, conduction mode and control principle, as long as the regulated quantity remains the same. The profile consists of certain transfer functions that describe the dynamical properties of a single converter. The basis of the dynamical profile for the voltage-output converter is the modified g-parameter set and for the current-output converter the modified y-parameter set, respectively. In addition, two special admittance parameters that are important in the interaction analysis are also introduced. These parameters, forming the dynamical profile, mainly define how a switched-mode converter would behave as a part of an interconnected system and how it would affect the other subsystems. Consistent formalisms for evaluating the stability and performance of a converter imposed by the load and supply interactions are provided. It is shown that the interactions are mainly reflected via the open-loop parameters. The dynamical profile can be derived in two distinct ways; analytical modeling methods can be used or the transfer functions that characterize the profile can be measured.

The existence of the dynamical profile, for the voltage-output converters, is demonstrated by developing the profile for a buck converter with different control principles. Operations in discontinuous and continuous conduction modes are also discussed. It is noticed that the control method and operation mode strongly affect the dynamical properties. It is verified both analytically and experimentally that these properties can be easily deduced by studying the parameters of the profile. The dynamical profile for the current-output converters is also proposed. The profile can be derived by using conventional modeling methods or from the corresponding voltage-output-converter profile by applying duality. It is discovered that the dynamics of a current-output converter are totally different than in the corresponding voltage-output converter. The prevailing assumption has been that the current-output converter has a peculiar characteristic of increased gain crossover frequency, when using a low impedance load. This phenomenon is addressed to be due to a wrong control design and the use of a resistive load as the initial load.

The prevailing method seems to be to use the resistive load in modeling and analyzing switched-mode converters. However, the true nominal load for the voltage- output converter is a constant current sink and for the current-output converter a pure voltage source. Illustrative examples are provided, which explicitly show the adverse effect of the resistive load hiding the real dynamical profile.

As a conclusion, the introduced concept of the dynamical profile provides valuable tool and framework in analyzing and ensuring the performance and stability of a switched-mode converter, or any electrical device, as a part of a larger system. Its use can significantly save the design and prototype testing times and peculiar phenomena can usually be avoided. Several examples and aspects presented in the thesis explicitly prove that the unique dynamical profile of any given converter is a fact not a fiction.

This work was carried out at the Institute of Power Electronics at Tampere University of Technology (TUT) during the years 2004 – 2007. The research was funded by TUT, Finnish Funding Agency for Technology and Innovation (TEKES), Efore Oyj, Salcomp Oyj and Patria Oyj. Their contributions are greatly appreciated. Financial supports in the form of personal grants from Nokia Foundation, Foundation of Technology, Emil Aaltonen Foundation and Ulla Tuominen Foundation are also greatly acknowledged.

I want to express my deepest gratitude to Professor Teuvo Suntio for supervising the thesis and providing interesting research topics. It has been a pleasure to work under his guidance and many fruitful conversations with him on the topic (and also off the topic) have inspired me. Matti Karppanen, M.Sc., deserves special thanks for the help in the lab and answering numerous questions. Professor Mummadi Veerachary and Dr. Vesa Tuomainen reviewed the thesis and their constructive comments and recommendations that improve the quality of the text in this thesis are greatly acknowledged. I would also like to thank Antti Hankaniemi, Lic.Phil., for finding and correcting the grammatical errors in the manuscript of this thesis.

I wish to thank my parents and sisters for their support and love, not only during my studies, but throughout my whole life. A year ago, when I was starting to write this thesis, I faced an unpredictable event in my own family. Without the help of my parents, sisters, other relatives, friends and colleagues it would not have been possible to complete this thesis in such a disciplined manner as I have now done. Thanks for being there for me! Above all, my beloved daughter Amilia deserves very special thanks. Playing with you at home and the sunshine in your eyes always make me feel happy, even if the skies are gray!

Tampere, October 2007

Mikko Hankaniemi

Abstract... iii

Preface ...v

Contents ... vii

List of Publications ... ix

Author’s Contribution...x

List of Notations, Symbols and Abbreviations... xi

1 Introduction ...1

1.1 Background and Motivation ...1

1.2 Voltage-Output Converter ...5

1.3 Current-Output Converter...6

1.4 A Review of Existing Methods to Analyze the Performance and Stability and Model Switched-Mode Converters...7

1.5 Structure of the Thesis ...12

1.6 Summary of Scientific Contributions ...14

2 Dynamical Profile...16

2.1 Definition of the Dynamical Profile ...16

2.1.1 Two-Port Representation ...17

2.1.2 Load and Supply Interaction Formalisms...23

2.2 Internal and Input-Output Stability...27

2.3 Discussion...29

3 Dynamical Review...32

3.1 Modeling of Switched-Mode Converter...32

3.1.1 Steady-State Operation ...33

3.1.2 State-Space Averaging...35

3.2 Effect of Control Principle...41

3.3 Interaction Analysis ...61

3.3.1 Load Interactions ...62

3.3.2 Supply Interactions ...66

3.3.3 Double Interactions...72

3.4 Effect of Load Resistance ...74

4 Experimental Evidence...78

4.1 Mixed-Data Method...79

4.1.1 Mixed-Data Control Design ...80

4.1.2 Mixed-Data Nominal Model...81

4.2 Measured Internal Loop-Gain...82

4.3 Load Interactions ...84

4.4 Supply Interactions ...87

5.1 Derivation of Dynamical Profile ...93

5.1.1 State-Space Averaging...94

5.1.2 Applying Two-Port Representation...96

5.1.3 Load Interactions ...100

5.1.4 Supply Interactions ...100

5.2 Dynamical Issues and Experimental Evidence...102

6 Conclusions ...107

6.1 Summary of Papers...107

6.2 Final Conclusions and Main Contributions ...110

6.3 Future Topics ...114

References...115

Appendices ...123

Appendix A...123

Appendix B...126

Appendix C...127

Appendix D...128

Publications ...131

The thesis is based on the following publications, which are referred to as P1, P2, P3, P4, P5, P6, P7, P8, P9, P10 and P11 in the text.

[P1] T. Suntio, M. Hankaniemi, ‘‘Unified small-signal model for PCM control in CCM: unterminated modeling approach,’’ HAIT Journal of Science and Engineering, vol. 2, issues 3-4, pp. 452 – 475, 2005.

[P2] T. Suntio, M. Hankaniemi, M. Karppanen, ‘‘Analysing dynamics of regulated converters,’’ IEE Proc. Electric Power Applications, vol. 153, issue 6, pp. 905 – 910. November 2006.

[P3] M. Hankaniemi, M. Karppanen, T. Suntio, ‘‘Load imposed instability and performance degradation in a regulated converter,’’ IEE Proc. Electric Power Applications, vol. 133, issue 6, pp. 781 – 786. November 2006.

[P4] M. Hankaniemi, M. Sippola, T. Suntio, ‘‘Load-impedance based interactions in regulated converters,’’ in Proc. IEEE International Telecommunications Energy Conference, Berlin, Germany, 2005, pp. 569 – 573.

[P5] M. Hankaniemi, M. Sippola, T. Suntio, ‘‘Characterization of regulated converters to ensure stability and performance,’ in Proc. IEEE International Telecommunications Energy Conference, Berlin, Germany, 2005, pp. 533 – 538.

[P6] M. Hankaniemi, T. Suntio, M. Karppanen, ‘‘Load and supply interactions in VMC-buck converter operating in CCM and DCM,’’ in Proc. IEEE Power Electronics Specialists Conference, Jeju, Korea, 2006, pp. 2768 – 2773.

[P7] M. Hankaniemi, M. Karppanen, T. Suntio, ‘‘Converter sensitivity to load imposed instability and performance degradation,’’ in Proc. IEEE Power Electronics Specialists Conference, Jeju, Korea, 2006, pp. 2674 – 2679.

[P8] M. Hankaniemi, M. Karppanen, T. Suntio, ‘‘EMI-filter interactions in a buck converter,’’ in Proc. International Power Electronics and Motion Control Conference, Portoroz, Slovenia, 2006, pp. 54 – 59.

[P9] M. Hankaniemi, M. Sippola, T. Suntio, ‘‘Analysis of the load interactions in constant-current-controlled buck converter,’’ in Proc. IEEE International Telecommunications Energy Conference, Providence, USA, 2006, pp. 343 – 348.

[P11] M. Hankaniemi, M. Karppanen, T. Suntio, A. Altowati, K. Zenger,

‘‘Source-reflected load interaction in a regulated converter,’’ in Proc.

Annual Conference of the IEEE Industrial Electronics Society, Paris, France, 2006, 2893 – 2898.

Author’s Contribution

The author planned and carried out the experimental tests, and was responsible for finding the internal profile in [P1]. The modelling of the peak-current-mode controlled converter in [P1] is done by the first author. In [P2], the author participated in the theoretical analysis and was responsible for performing the experimental evidence together with the co-authors.

Publications [P3] - [P11] were mainly contributed by the author of this thesis. The author performed the theoretical analysis as well carried out the measurements.

Professor Teuvo Suntio, the supervisor of the thesis, gave valuable and constructive comments regarding publications [P3] - [P11]. M.Sc. Matti Karppanen helped with the experiments and measurements.

Abbreviations

NOTATIONS

xˆ Small-signal component of x dx

dt Time derivate of x

y x G

G ^{Partial derivate of} y with respect to x x Absolute value, i.e. magnitude of x

x Time averaged value of x

‘x Angle of x

argx Argument of x

x Vector x x₁ ... x_n

° Degree SYMBOLS

A System matrix

B Input matrix

C Capacitor

C Output matrix

C Control variable

ˆ

c Perturbed control variable

D Input-output matrix

D Diode

D Averaged duty-ratio

Dc Averaged complementary duty-ratio (i.e. 1D)

d Instantaneous duty-ratio

dc Complement of the instantaneous duty-ratio (i.e. 1d)

dˆ Perturbed duty-ratio

eo Voltage-source load in current-output converter

fres Resonant frequency

Ga Control gain

Gcc Controller transfer function Gci Control-to-input transfer function Gco Control-to-output transfer function Gio Forward transfer function

Gse Sensor gain

( )

G s Transfer function

Hi Output-current-feedback gain

I Average current

Iin Averaged input current IL Averaged inductor current

o MAX

I Maximum output current

Io Averaged output current

i Instantaneous current

iin Instantaneous input current iL Instantaneous inductor current io Instantaneous output current

j Imaginary unit

jo Output current sink

L Inductor

Lij Transfer function of the subsystem L L Subsystem (matrix)

L s( ) Loop gain

LCO Loop gain of the current-output converter LVO Loop gain of the voltage-output converter

Mc Compensation ramp

M D( ) Conversion ratio

Q Switch

r Magnitude of a complex number

rC Equivalent series resistance of capacitor C

( ) ds on

r Dynamic on-time resistance of the switch Q

R Resistor

Req Equivalent load resistance Rs Output current sensing resistor

1

Rs Equivalent inductor current sensing resistor

2

Rs Equivalent output current sensing resistor Sij Transfer function of the subsystem S

S Subsystem (matrix)

s Laplace variable

Toi Output-to-line current transfer function Ts Length of switching period

, 1 off off

t t Switch off-time

off2

t Switch off-time when inductor current is zero (in DCM)

ton Switch on-time

u Instantaneous voltage

uc Output (voltage) of voltage controller

,

uc CO Output (voltage) of voltage controller in current-output converter

uC Instantaneous (output) capacitor voltage uin Instantaneous input voltage uL Instantaneous inductor voltage uo Instantaneous output voltage ur Instantaneous reference voltage

U Average voltage

Uin Averaged input voltage

UL Averaged inductor voltage

Uo Averaged output voltage

UE Equivalent voltage

Um PWM-sawtooth waveform amplitude, i.e. the PWM-gain

U Maximum output voltage

in oc

Y Open-circuit input admittance

in sc

Y Short-circuit input admittance Yin_f Ideal input admittance

,

z z Intermediate parameters

1, 2, 3

f f f

Z Z Z Filter output impedances

1 2 3

, , ,

L L L L

Z Z Z Z Load impedances

Zo Output impedance

ZS Source impedance

f Infinity

[ Damping factor

Zn Undamped natural frequency (rad/s)

M ^{Phase in radians}

SUBSCRIPTS

n Integer number

off Off-time

-c Closed-loop

-dcm Refers to the VMC-DCM converter

-o Open-loop

-ocf Refers to the PCMC-OCF converter -pcmc Refers to the PCMC converter SUPERSCRIPTS

L Load-affected transfer function S Supply-affected transfer function -i Refers to the current-output converter -v Refers to the voltage-output converter ABBREVIATIONS

AC Alternative current

AC-DC AC-to-DC rectifier

CC Constant current

CCM Continuous conduction mode

CO Current-output

DC-DC DC-to-DC converter

DCM Discontinuous conduction mode DPA Distributed power architecture DPS Distributed power system

EET Extra element theorem

EMI Electromagnetic interference FRA Frequency response analyzer

GM Gain margin

IEEE Institute of Electrical and Electronics Engineers IVFF Input-voltage feedforward

LTI Linear time-invariant

NRO Negative resistor oscillation

OCF Output-current feedforward

PCMC Peak-current-mode control PI Proportional-integral control

PID Proportional-integral-derivative control

PM Phase margin

POL Point of load

PWM Pulse width modulation

RHP Right half plane

S1 Subsystem 1

S2 Subsystem 2

TUT Tampere University of Technology

SSA State-space averaging

VMC Voltage-mode control

VO Voltage-output

This chapter provides the basis for the thesis. Essential background information and fundamental issues behind the topic of the thesis are presented. The basic operating principles of switched-mode converters are also discussed. An extensive literature review of the previous work related to the topic is presented, pointing out several prevailing ambiguities. Finally, the main contributions and short summaries of the following chapters are presented.

1.1 Background and Motivation

Electronic power processing and conversion have interested researchers and engineers since the 17^th century. Nowadays, various electrical apparatuses are used and the consumption of electrical energy is increasing year by year. However, the electrical energy has to be produced somewhere and somehow, and at the same time the amount of produced energy has to be consumed. Between the points of production and consumption different kinds of electronic power processing and conversion methods are needed. Generators, transmission lines, transformers, AC-DC rectifiers and DC-DC converters are the core components in the electric power distribution.

This thesis concentrates on the switched-mode DC-DC converters, which are usually close (in physical and conceptual sense) to the end user or application.

Switched-mode converters have replaced the linear regulators in the modern DC-DC conversion. The history of switched-mode converters dates back to the mid-60s, when the active power switches started to replace the mechanical switches and relays [1].

While the linear regulators are quite simple and have low efficiency, switched-mode converters have a nonlinear nature due to the switching action, and hence, they are more complicated to analyze and model. On the other hand, they are usually smaller,

in modeling and analyzing the dynamics and input-filter interactions are still quite relevant and important, although scientific research has been carried out for more than 30 years. Although the evolution of design and modeling of switched-mode converters, and power electronics in general, have been rapid since the 70s there are still some open questions and misunderstandings waiting for an answer and a correction.

uin

Supply system

48V

Isolated bus converter

8-16V

POL-converters 1-12V IBA

Fig. 1.1. Distributed power system.

In a modern electronic device (e.g. telecom power supply) various DC-voltage and DC-current levels are usually required. To power these devices distributed power systems/architectures (DPSs/DPAs) are widely employed [6]-[9]. An intermediate bus architecture (IBA), shown in Fig. 1.1 inside the dashed line, has become the most used DPA in new applications [10]. The IBA consists of an isolated bus converter, which produces an intermediate bus voltage (8 – 16 V) and several point-of-load (POL) converters. Usually, an EMI filter has to be placed before every power stage and a storage battery may be connected to the system after the front-end rectifier in order to provide energy to the load system during the power outages. It is obvious, that the system shown in Fig. 1.1 is complicated both from a dynamical and design viewpoint. In order to design a stable system with adequate performance margins the functioning of each building block in the system has to be known.

Each DC-DC converter in a DPS, or in any application, is always a part of an interconnected system. This actually means that the source and/or load system may

significantly affect the stability of an individual converter, and hence, the stability of the entire system. Therefore, an important and interesting question arises: How to perform the interaction analysis to ensure stability and adequate performance of the converter and the whole system? The canonical dynamical profile and interaction formalism presented in this thesis will answer this question and provide a powerful facility to analyze the performance and stability of DC-DC converters.

The terms performance, stability and also the crossover frequency or the bandwidth continuously appear in this thesis. Therefore, it is necessary to define the meaning of these terms in the scope of the thesis in order to avoid confusion. The terms performance and stability can be addressed to both the time and frequency domains.

The time domain performance is typically studied by means of a step response (i.e. a transient response in switched-mode converters), where a step change is introduced into the reference signal and the output of the system is monitored. Typical characteristics of the step response are the rise time, over shoot and settling time.

However, the classical step response analysis incorporates the disturbance signal (i.e.

the step) into the reference signal, which is typically constant and even physically unavailable in the modern converters. The transient response analysis of the switched- mode converters is typically done by introducing the step change e.g. into the load current or input voltage and the output voltage is monitored. From a dynamical viewpoint, this approach does not give the same performance characteristics as the classical step response method. The time domain performance is not discussed in this thesis, but the frequency domain performance is often considered. The frequency domain performance relates to the loop gain ( )L s of the converter. The performance of the converter is judged by means of the gain margin (GM) and phase margin (PM).

In the bode plot, the GM can be expressed as the vertical distance of the loop gain magnitude from the unity gain (i.e. 0 dB) at the frequency, where the phase is -180°.

Consequently, the PM is defined as the phase of the loop gain at the unity gain frequency added with 180°. To guarantee adequate performance the GM and PM are typically required to be at least 6 dB and 45°, respectively. The instability occurs if the GM < 0 dB or the PM < 0°. The bandwidth of the system and the (gain) crossover frequency f_c are sometimes confusingly defined in power electronics. According to the system theory, the bandwidth is related to the sensitivity or complementary sensitivity function providing two different definitions of the bandwidth. The gain crossover frequency f_c is naturally the frequency, where the gain of the system is unity (i.e. 0 dB). In this thesis, the gain crossover frequency f_c is used, when

An extensive review of the existing methods to analyze and model the dynamics of a switched-mode converter will be given in Section 1.3. However, it is worth mentioning already at this point that most of the existing modeling and analyzing methods do not reveal the true internal dynamics of a single converter. This is mainly due to the wrong initial modeling with a resistive load, which may hide the internal dynamics of the converter. The internal dynamics for e.g. a voltage-output converter can be derived by using a constant-voltage source at the supply side and a constant- current sink as a load. Illustrative examples of the effect of a wrong initial load will be given later.

New converter topologies and control methods are continuously developed and published in academia and industry, but the focus seems to be only on certain advantages of these new topologies. The dynamical issues and sensitivities for interactions are not usually discussed. So, usually the performance of these new topologies or control methods as a part of an interconnected system, like one shown in Fig. 1.1, is unknown.

It has been found out during the research that the true nature of switched-mode converters relates to the frequency domain. By studying certain frequency responses (i.e. transfer functions) it is easy to conclude the possible sensitivities for the load and/or supply interactions as will be shown e.g. in Chapters 3 and 4. However, only time domain simulations and measurements are usually presented e.g. in converter manufacturers’ datasheets [12]-[15]. If frequency responses are presented like in [12]

only the magnitudes are shown but not the phase plots, which are just as important as the magnitudes. Again, the performance of these commercial converters as a part of an interconnected system, like one shown in Fig. 1.1, remains unknown.

The concept of the dynamical profile of the switched-mode power supply is introduced in this thesis. It provides a straightforward method and a physical insight into the converter internal or the nominal dynamics. By analyzing the dynamical profile, the dynamical properties (i.e. load and supply sensitivities and insensitivities and control-loop stability) can be concluded by analyzing certain transfer functions in the frequency domain. Consequently, the stability and performance of a converter with known and analyzed dynamical profile as a part of the interconnected system can be easily derived. Chapters 2 and 3 will discuss more in detail the concept of dynamical profile and how to derive it.

1.2 Voltage-Output Converter

The basic configuration of a voltage-output (VO) converter, illustrated in Fig. 1.2, consists of a power stage, load and supply systems and a control circuitry. The power stage contains inductor(s), capacitor(s), switches and their parasites. If electrical isolation is needed, a transformer may be included in the power stage. Depending on the power-stage-circuit structure (i.e. topology), the output voltage U_o is either smaller or larger than the input voltage U_in. The output voltage is adjusted by means of the control circuit by typically applying a pulse width modulation (PWM) to control the power stage switch on-time t_on. In Fig. 1.2, the switch on-time t_on is independent variable forming a control scheme called a voltage-mode control (VMC) or direct duty-ratio control. Other control schemes can be formed by taking a feedback or feedforward e.g. from the input voltage, inductor current or load current.

The control circuitry includes an error amplifier or a controller and a PWM- comparator, which controls the switch on-time t_on. The nominal load for the voltage- output converter consists of a current sink j_o. The impedance Z_L in Fig. 1.2 connected parallel to the constant-current sink represents the non-ideal load system.

Respectively, the impedance Z_S at the supply side represents the effects of the non- ideal source.

+-

Power stage

Supply system Load system

Uref

+_

uc

Um ^controller

Control circuit

ZL

ZS +

-

iin i_o

+ -

iL

ton

uin u_C u_o j_o

Fig. 1.2. Basic configuration of voltage-output converter.

The basic steady-state analysis of switched-mode converters relates to the principles called an inductor volt-second balance and capacitor amp-second balance [18].

According to the inductor volt-second balance the net change of the inductor voltage over one switching period T_s is zero. The capacitor amp-second balance states that the net change of the capacitor current over one switching period T_s is zero. The steady-state and small-signal modeling and analysis are discussed more in detail in Chapter 3.

1.3 Current-Output Converter

Instead of controlling the output voltage U_o, the output current I_o is regulated in current-output (CO) converters. Current-output converters are typically used in applications, where an overload protection is needed. The need for current limiting arises e.g. in DPS system if a storage battery is connected to the system and charged/discharged, and hence, large currents are drawn. Fig. 1.3 shows the typical output-voltage/current characteristics of a converter, where the constant voltage and current controls are needed. When the output current is lower than the maximum set level, the converter is in the voltage-output mode and regulates only the output voltage. Consequently, when the maximum allowed output current is reached the converter enters into the current-output mode and starts regulating the output current.

uo

o MAX

U -

VO

io o MAX

I ^-

CO

Fig. 1.3. Typical output-voltage/current characteristics of a converter in applications where constant voltage and current control is needed.

The constant-current limiting can be accomplished either by using only an output current as the feedback signal or both the output voltage and current in cascade, where the inner loop is the voltage loop and the outer loop is the current loop [16].

The overload protection can also be accomplished by using a modified constant

power limiting, which is typically implemented in such a way that the reference of the current loop is gradually increased along the decrease in the output voltage until the maximum defined output current is reached after which the limiting scheme follows the constant-current scheme [16], and [17]. The study of the cascade operation and the modified-constant-power limiting are left out of this thesis and only the mode, where the output current is used as the feedback signal, is considered.

The basic configuration of the current output converter is shown in Fig. 1.4. The only difference compared to the voltage-output converters is the load system, which consists of an ideal voltage source e_o in series with the load impedanceZ_L. However, the internal dynamical profile and system interactions are totally different in the current-output converters. The corresponding dynamical profile, dynamics and interactions are discussed in detail in Chapter 5.

+- ^eo

Power stage

Supply system Load system

Uref

+_

c CO, m u

U ^controller

Control circuit

uo

ZS +

-

iin i_o

+ -

iL

ton

uC

R is o

Rs

+-

uin

ZL

Fig. 1.4. Basic configuration of current-output converter.

1.4 A Review of Existing Methods to Analyze the Performance and Stability and Model Switched-Mode Converters

As it was stated earlier, the scientific research on the topic started in the 70s. The basic topologies (i.e. buck, boost and buck-boost) were analyzed and modeled for the first time in [19]. Averaged and linearized general power-stage models were also

state-space averaging (SSA) modeling technique, which produces both continuous- time steady state and dynamic linearized models. The basic idea behind the SSA is to average the switch on- and off-time state-space equations over one switching period.

Circuit averaging and hybrid modeling were considered as alternative modeling methods that give the same canonical circuit model as the SSA method. The SSA has become popular since its introduction basically due to its simple and clear methodology. It is commonly known that the SSA gives accurate open-loop models up to half the switching frequency, when the converter operates in continuous- conduction-mode (CCM) under direct-duty-ratio control or VMC [20]. The canonical circuit model introduced in [4] was argued for being a useful tool for analyzing small- signal dynamics of switched-mode converters regardless of the topology. However, it contains a resistive load as well as parasitic loss elements in the duty ratio dependent generators hiding the true internal dynamics.

In spite of the practicality and simplicity of the SSA, several other modeling approaches have also been developed. In circuit averaging, the voltage and current waveforms are averaged instead of averaging the state equations as in SSA [19], and [21]. Average models for the PWM-switch were introduced in [22] and [23].

Linearization of the averaged circuit and PWM-switch yields appropriate small-signal models of the converter. It is clear that the ripple information is lost in averaging.

However, the averaged models are also usable in time domain simulations and transient analyses. Switched circuits can be equally used, if the ripple information is needed [21]. Because the switching action is actually discrete, a sampled-data modeling has been proposed. The basis of the sampled data modeling is presented in [21]. The modeling is based on the continuous time state-space model and the standard matrix exponential expression for linear time-invariant (LTI) systems.

According to [21], [24] and [25], the prevailing method is to derive the discrete-time model from the continuous-time state and switching equations. In [26], the discrete- domain model is derived from the corresponding model in the Laplace-domain by using z-transformation. The sampled-data modeling is derived in [27]-[29] by using a discrete-time state-space model. The sampled-data modeling typically involves tedious calculations and therefore it is not widely adopted. However, the sampled- data models might become useful, when digital controllers replace the analog controllers.

It should be clear that the true internal or nominal dynamics can be derived from the power stage model and from the control circuit model. However, it seems that the definition of the nominal power stage or model is not clear among the scientists and engineers. There are numerous examples of modeling and analyzing switched-mode

DC-DC converters with a resistive load (these are only example papers, not the complete list: [4] , [18] and [30]-[35]). The actual load is very seldom a pure resistor but should be treated as an external system not included in the nominal model. The seminal paper [4] actually uses a resistive load when introducing the SSA method and the canonical equivalent circuit. It is obvious that including the load resistor in the so- called canonical model might lose the information of the nominal dynamics. Even the fundamental power electronics text books such as [36] and [18] use the resistive load in their analyses and provide incorrect information for the reader. So, what type of load should be used to get the nominal dynamics? A voltage-output DC-DC converter is known to have current source input and voltage source output ports [P5], so the natural nominal load connected to the voltage source output port is obviously a current-sink. Consequently, the nominal load for a current-output converter with a current source output port is a pure voltage-source. In spite of the prevailing technique to use the resistive load, a few attempts to define the nominal or general load have been presented. A general load impedance is treated as an alternative for the resistive load in [37]. First it is stated that the load can be seen as a current source, but later the load is replaced with the general load impedance. The idea of the general load (impedance) is actually correct, but the authors seem to lack the understanding of the true nominal dynamical behavior of switched-mode converters. It is explicitly stated in [38] that the nominal load refers to the use of either a resistive or dc current sink load. It is true that the internal output impedance can be measured either by using a resistive or current sink load, but when measuring or analyzing e.g. the loop gain this does not apply. It seems that the authors of [38] are confused with the terminology of the nominal dynamics or nominal load and provide vague information.

An approach known as an unterminated modeling was introduced in [39] treating a converter as a stand-alone module without considering the load impedance, but using a current sink load. The unterminated modeling method has been applied in [40]-[42]

for studying the load interactions. In [43], the method was successfully used for analyzing the input filter interactions. The unterminated model was derived in [44] by first constructing the models with the load resistor R and then letting Ro f. However, the most convenient way of getting the unterminated model is to use the constant-current-sink load as an initial load system as it was done e.g. in [P4] and [P5]. As a significant contribution of this thesis, it was found that the derivation of the nominal dynamics, and hence, the dynamical profile by using the correct initial load is a starting point for understanding the behavior of switched-mode converters under various conditions.

which may deteriorate the performance of the converter. The input filter interactions were first studied by Middlebrook in the seminal paper [3]. A converter with an input filter was modeled and design criteria for input filter were developed. The derivation of the input-filter-affected transfer functions were based on the method known later as an extra element theorem (EET) [18], and [45]-[47]. The EET provides a tool to analyze the change of transfer functions, when impedance is added to the network.

However, the EET involves tedious calculations, and therefore, may not be suitable for practical usage. The load and input filter interactions can be easily concluded from a two-port linear circuit representation of the converter with load and supply (e.g.

filter impedance) impedances [48] and [43]. The two-port modeling technique based on g-parameters [49] is reviewed and discussed more in detail in Chapter 2. The input-filter interactions have been under extensive research since the Middlebrook’s paper. It has been noticed that different topologies and control methods have different sensitivities for instability or performance degradation due to the input filter or supply impedance [33], and [50]-[56]. Obviously, the converter dynamics are also affected by the load. The load interactions have also been studied in various papers such as in [38]-[42], [44], [57], and [58]. The load interaction formalism is simpler to understand than the corresponding supply side formalism. In Chapter 2, it will be shown that the performance of a converter may be deteriorated if the load impedance and the open-loop output impedance of the converter overlap. Although, the supply and load-side-interaction formalisms are different, the stability of the converter with load and/or supply system can be concluded from the impedance ratio known as a minor-loop gain [3], [39] and [41]. If studying the supply interactions, the minor-loop gain is defined as the ratio of the supply impedance (e.g. input filter output impedance) and the closed-loop input impedance of the converter. The corresponding minor-loop gain at the load side is the ratio between the closed-loop output impedance of the converter and the load impedance. In order to guarantee the stability, the minor loop gain must satisfy the Nyquist stability criterion [3], and [59].

Various forbidden regions in the complex half plane, out of which the minor loop gain should stay, have been presented in the literature [41], and [60]-[63]. It is claimed that these forbidden regions provide certain phase (PM) and gain margins (GM) for the interconnected system. However, the PM and GM of the minor loop very seldom coincide with the corresponding margins in the load or supply-affected loop gain of the converter [P3] and [P7]. This means that the performance of the converter may be drastically deteriorated even if the corresponding margins of the minor loop gain are adequate. The minor-loop-gain analysis is only suitable for ensuring the stability but the performance and true margins should always be checked from the true “major” loop gain of the converter.

A typical method to analyze the converter performance in the time domain is a transient response analysis. There are numerous papers claiming that the higher the crossover frequency f_c of the loop gain is the faster is the transient response [64]- [68]. According to the classical control theory this is true, because the reference is step-changed. However, in switched-mode converters the reference is usually kept constant but the load current is changed introducing a transient into the output voltage. In [P2], [69] and [70], it was demonstrated that even if the peak-current- mode controlled (PCMC) converter and PCMC converter with an output current feed- forward (OCF) have the same loop gain, the PCMC-OCF converter has a considerably faster transient response. The reason for this is the smaller open-loop output impedance of the PCMC-OCF converter. Therefore, the transient response actually relates to the open-loop output impedance of the converter and its behavior.

This was also noticed in [71], where a larger closed-loop output impedance of the PCMC converter at lower frequencies compared to the VMC converter yielded also a longer settling time. The transient response as a function of time using inverse Laplace transformation of the closed-loop output impedance, when a certain load step change occurs, was computed in [72]. The transient response of a parallel RLC-circuit was considered in [67] and [68] to mimic the transient response of a switched-mode DC-DC converter. The results of the above analyses seem to be, however, a bit unreliable because of the simplifications made in the analyses. The true relation between the frequency and time domain still seems to be fuzzy and needs further research in order to put the relation in a correct mathematical from. Actually, the challenge is the complex structure (i.e. the numerator and denominator are high-order polynomes in the Laplace variable s) of the closed-loop output impedance making the computation of the inverse Laplace transformation a difficult and tedious task.

Nevertheless, the relation can be implicitly studied as it was discussed above (i.e.

larger closed-loop output impedance at lower frequencies Æ longer settling time and small impedance Æ fast response). In addition, the amount of peaking in the closed- loop output impedance at the phase or gain crossover frequency dictates the PM and GM of the converter loop gain [72]: The peaking is related to sensitivity function (i.e.

1/ 1L s( ) ), which is a elementary part of the equation of the closed-loop output impedance by definition [11]. The peaking in the sensitivity function due to a low PM or GM would naturally be observable in the transient response.

The first basic courses on power electronics at universities are typically based on fundamental text books such as [18], [36] and [73]. They all present the fundamentals of switched-mode power conversion, but in [18] the study is done more in detail.

principles are discussed with the resistive load. The text book [18] is maybe the most often used introductory level book on switched-mode converters, but it loses the point of presenting the true canonical model (both steady-state and small-signal) and dynamical issues by incorporating the resistive load into the models. The contents of these fundamental text books may explain the reason, why the prevailing technique still strictly relies on the use of the resistive load in the analyses both among academia and industry.

Chapter 5 of this thesis is solely dedicated to the current-output converters, which are typically used in battery-powered applications. The modeling and analysis of the current output converter in [74]-[76] are based on the use of resistive load, although the real load typically consists of a back-up battery with low internal impedance [77].

The peculiar behavior of the increasing crossover frequency in the loop gain with battery-type load observed e.g. in [74] and [79] was shown to be due to the use of wrong initial load (i.e. resistive) in [P9].

1.5 Structure of the Thesis

The thesis contains six chapters. The main contributions and short summaries of the following chapters are:

Chapter 2: Dynamical Profile

The basic concept of the dynamical profile is presented. The chapter reviews the papers [P2], [P3], [P5], [P7], and [P11]. It is found that the g-parameter set characterizes effectively the dynamical properties of a switched-mode converter. In addition, the interaction formalism introduces two special admittance parameters. The two-port representation and the load and supply interaction formalisms are presented.

Both open- and closed-loop operations are considered. It is shown that the open-loop parameters reflect the interactions, and therefore, their behavior is the main interest.

The chapter concentrates only on the voltage-output converters, but the main idea behind the dynamical profile applies also to the current-output converters, which are studied in Chapter 5. It is found out that the true internal dynamics can be derived by using a pure voltage source at the input and a constant current sink at the output acting as a load. The prevailing technique to use a passive resistor as the initial load is argued for being a conservative and erroneous approach. Finally, an internal and input-output-stability formalism is presented yielding to an impedance ratio (i.e.

minor-loop gain) between the interfaces of two subsystems from which the stability can be verified.

Chapter 3: Dynamical Review

This chapter starts with introducing the basic principles in modeling a switched-mode converter. First, the steady-state analysis is derived by applying the volt- and ampere- second balances and then the SSA method is used to compute the dynamical profile of a VMC-buck converter operating in CCM. The effect of a control principle on the converter dynamics is discussed by using VMC-CCM, VMC-DCM, PCMC and PCMC-OCF as examples. It is found out that the dynamical properties can be sometimes deduced directly from the analytical model, but it is preferred to analyze also the frequency responses. The open-loop profiles of the three control modes are analyzed and based on the internal dynamical profiles, the load and supply interactions are effectively predicted and analyzed. It is demonstrated that the load interactions are reflected via the open-loop output impedance. Even though the loop gains of the PCMC and PCMC-OCF converters are the same, the transient responses are found out to be different (i.e. PCMC-OCF provides superior response compared to PCMC). This difference can be addressed to the small output impedance of the PCMC-OCF converter. The load and supply interactions are studied by using LC- circuits as the load and supply impedances. The study of this chapter is mainly related to papers [P1]-[P8] and [P11].

Chapter 4: Experimental Evidence

The basic procedures of analyzing the performance and stability that was introduced in Chapter 3 are verified in practical situations. A buck converter with VMC and PCMC control modes are studied. The VMC converter is designed to operate in CCM and DCM. A frequency response analyzer is used to measure the g-parameter set.

Some of the measured parameters are compared to the analytical model and it is observed that they match well with each other. The non-idealities making the measurements and predictions slightly to differ are also discussed. In cases, where the injection signal goes through the modulator, it is observed that the phase starts to lag more than predicted due the modulator circuit and the sinusoidal injection signal.

A mixed-signal method, which uses both analytical and experimental data, is also introduced. This method can be used e.g. in control design by measuring the transfer function between the control signal and output voltage at the open loop and then designing the controller based on the measurement. The method is also useful in defining the internal profile in some cases. Furthermore, it can be applied to calculate various non-idealities so that they can be taken into account and included in the dynamical profile. The study of this chapter is mainly related to papers [P1]-[P8] and

Chapter 5: Current-Output Converters

The dynamical issues of the current-output converters are discussed. The chapter is based on papers [P9] and [P10]. It is shown that the dynamical profile can be derived from the corresponding voltage-output converter by applying duality. A SSA modeling example is also introduced for computing the dynamical profile. It is discovered that the basis of the profile is the modified y-parameter set. It is demonstrated that the most convenient way of presenting the dynamical profile is to use the corresponding parameters of the voltage-output converter, because they are usually well known and available. The peculiar phenomenon of the increasing crossover frequency, when having a low impedance load e.g. a battery-back, is shown to be due to the wrong initial modeling with a resistive load. Therefore, the nominal load for the current-output converters is an ideal voltage source (i.e. low impedance load). Experimental evidence is provided to illustrate the effect of the increasing crossover frequency on the loop gain, when changing from a resistive load to a low impedance load. The practical load giving the desired result is proposed to be a parallel connection of a resistor and a capacitor, where the capacitor provides the dynamical short circuit and recovers the internal behavior at higher frequencies.

Chapter 6: Conclusions

This chapter concludes the thesis. A short summary of each paper is given. The final conclusions of the work behind the thesis are put together. The scientific contributions of the thesis are also given and discussed in detail. Finally, potential future research topics that have arisen during the work are presented.

1.6 Summary of Scientific Contributions

The main scientific contributions of the thesis can be listed and summarized briefly as follows:

x The concept of dynamical profile is proposed and its existence is proven.

x It is shown that every converter has its unique dynamical profile which characterizes its dynamical features.

x Open-loop parameters are shown to be the main facility to study the interactions and sensitivities to them.

x The nominal loads of the voltage- and current-output converters, invoking the nominal dynamics, are stated to be a constant-current sink and a pure voltage source, respectively.

x The dynamical profile of the current-output converter is presented for the first time. It fully explains the observed peculiar behavior.

The thesis will definitively show that the existence of the dynamical profile is a fact not a fiction.

The concept of dynamical profile is introduced and its capability of revealing the internal dynamics of a switched-mode DC-DC converter is shown. The two-port representation is reviewed and argued for being the most useful method for deriving the dynamical profile, yielding the true internal and canonical model of a switched- mode converter. The load and supply interaction and internal stability formalisms of an interconnected system are also presented in a consistent way. The focus of the chapter is mainly on the voltage-output converters. The dynamical profile of current- output converters is discussed in Chapter 5.

2.1 Definition of the Dynamical Profile

A dynamical profile defines the dynamical properties of a switched-mode DC-DC converter. Due to the observed frequency-domain nature, the proposed profile consists of various open and closed-loop transfer functions. These transfer functions are defined as the relation between certain voltage(s) and/or current(s) characterizing the dynamics of a given converter. It is important to note that the interest is particularly in the internal dynamics, which means that the interfaces between the converter and interconnected subsystems should be explicitly defined. The definition of the internal dynamical profile includes the following:

x For a given topology, the internal dynamical profile consists only of the corresponding power stage components and the necessary control circuitry.

x Additional components, such as input capacitors and extra output filters should be considered as external subsystems.

x The dynamical profile can be derived by using an ideal voltage source at the input and a constant-current sink load at the output in the case of voltage-

output converters and an ideal voltage source both at the input and output in the case of current-output converters.

x At closed loop, the additional components or subsystems inside the feedback loop may adversely change the internal dynamical profile and their effect should be considered.

Basically, there are two ways of extracting the internal dynamical profile; an analytical model of the converter can be derived or the corresponding transfer functions can be measured by using a frequency response analyzer (FRA). Typically, a practical converter contains some non-idealities, which cannot be modeled accurately. This implies that by measuring the transfer functions the “real” dynamical profile could be obtained. However, sometimes it may be impossible to measure every transfer function or the internal dynamics directly (i.e. a resistive load has to be used). In these cases the measurement and analytical data can be used together to compute the internal dynamical profile. This technique is known as a mixed-data method and it will be introduced in Chapter 4. Obviously, if the model and measurements are known to be in a good agreement, it is reasonable and convenient to use the model to study the dynamical profile analytically and guarantee the functioning, stability and performance of the converter before constructing the entire prototype or starting the mass production. Measurements can be used to verify the analytical results.

The rest of this chapter will concentrate on showing the capability of the internal dynamical profile. It can be proven that every converter topology, conduction mode and control method produces different dynamical profiles. The control method and conduction mode may significantly change the dynamical properties, although the chosen topology can retain some common dynamical features. However, if the dynamical profiles under different control methods or conduction modes and the source and load subsystems are known, the most suitable converter can be easily chosen for the application as well as the stability and performance of the interconnected system can be guaranteed.

2.1.1 Two-Port Representation

Almost any electrical system can be seen as a “black box” consisting of input and output ports. Two-port networks are typically used to represent such “black boxes”

[43], [44], [48], [51], [52], [81] and [82], but the idea of describing the true internal dynamical profile has not been explicitly presented earlier. A two-port model for a voltage-output converter is shown in Fig. 2.1 inside the dashed line. The input port of the model is a Norton equivalent circuit and the output port is a Thevenin equivalent circuit implying that the two-port model constitutes of g-parameters [49]. The use of g-parameters is well justified, because their existence is always guaranteed [49]. The input port of the model in Fig. 2.1 corresponds to

ˆ_{in in o in}ˆ _{oi o o}ˆ _ciˆ

i Y u T i G c (2.1)

at open loop (i.e. Y_N Y_{in o} and ˆi_N T_{oi o o} iˆ G c_ciˆ) and the output port to ˆ_{o io o in}ˆ _{o o o}ˆ _coˆ

u G u Z i G c (2.2)

at open loop (i.e. Z_T Z_{o o} and ˆu_T G_{io o in} uˆ G c_coˆ). The minus-sign before the term

o o oˆ

Z i is due to the direction of the output current. It is worth noting that here open loop refers to a situation where the outer feedback loop is disconnected (i.e. the output voltage feedback path is disconnected in the case of voltage-output converters). Inner feedback or feed-forward loops are connected in the case of the open loop. The “hat” over the variables represents the small-signal component of the variable. The general control variable is denoted by cˆ.

+_

ˆin

i

ˆ_in

u uˆ_o ˆ

io

ˆ_T u

ZT

YN ˆ

iN

_

+

ˆo

i

+_

cˆ

Fig. 2.1. Two-port model of voltage-output converter.

The equations for input in (2.1) and output in (2.2) can be equally presented in a matrix form yielding

ˆ ˆ

ˆ

ˆ ˆ

in

in o oi o ci

in

o

io o o o co

o

Y T G u

i i

G Z G

u c

ª º

ª º ª º « »

« » «¬ »¼ « »

¬ ¼ «¬ »¼

(2.3)

+_

ˆin

i

ˆ_in u

Yin o- ˆ

oi o o

T - i G c_ciˆ G_{io o in}- uˆ

coˆ G c

Zo o-

ˆ_o

u ˆ

io

+_

_

+

ˆo

i

+_

cˆ

Fig. 2.2. Two-port model of open loop voltage-output converter with g-parameters.

It should be noted that the general form of the g-parameter set typically consists only of four parameters (see e.g. [49]), but here the general control variable ƙ is also included in the parameter set. The set in (2.3) is represented as the corresponding two-port circuit in Fig. 2.2 and the corresponding open-loop parameters in (2.1), (2.2) and (2.3) are denoted as follows:

Yin o = input admittance, i.e. ^ˆ ˆ

in in

i u

Toi o = reverse or output-to-input current transfer function, i.e. ^ˆ ˆ

in o

i i

Gci = control-to-input current transfer function, i.e. ^ˆ ˆ iin

c

Gio o = forward, input-to-output, line-to-output transfer function or audiosusceptibility, i.e. ˆ

ˆ

o in

u u Zo o = output impedance, i.e. ˆ

ˆ

o o

u i

Gco = control-to-output transfer function, i.e. ˆ ˆ uo

c

An equivalent way of representing (2.3) and the model in Fig. 2.2 is to use control- block diagrams. The block diagrams for the open-loop voltage-output converter are shown in Fig. 2.3 describing the output and input dynamics. The block diagrams are useful, when deriving the closed-loop dynamical profiles as will be shown next.

ˆo

i

ˆ_o u

cˆ ˆ_in

u G_{io o}-

Gco

Zo o-

a)

ˆo

i

ˆin

i

cˆ ˆ_in

u Y_{in o}-

Gci

Toi o-

b)

Fig. 2.3. Control-block diagrams for voltage-output converter at open loop: a) output dynamics and b) input dynamics.

The closed-loop dynamical model of the voltage-output converter can also be presented as a two-port network as shown in Fig. 2.4. The equivalent g-parameter set at closed loop can be expressed by

ˆ ˆ

ˆ

ˆ ˆ

in

in c oi c ci c

in

o

io c o c co c

o

r

Y T G u

i i

G Z G

u u

ª º

ª º ª º « »

« » «¬ »¼ « »

¬ ¼ «¬ »¼

(2.4)

+_

ˆin

i

ˆ_in u

Yin c- ˆ

oi c o

T - i G_{ci c r}- uˆ

io c inˆ G - u

co c rˆ G - u

Zo c-

ˆ_o

u ˆ

io

+_

_

+

ˆo

i

+_

Fig. 2.4. Two-port model of closed-loop voltage-output converter with g-parameters.

The general description for the converter loop gain L_VO is typically given by

VO se cc a co

L G G G G (2.5)

where G_seis the output-voltage sensor gain, G_cc is the controller transfer function and Ga is the control gain (note: G_a is the gain between ˆu_co and cˆ as it is shown in Fig.

2.5. For the VMC converters it is the PWM generator gain, but for instance for the PCMC converters it is 1/R_s, where R_s is the current sensing resistor. Because G_a might be different for certain converters (i.e. control modes), it is named as “control gain”). The loop gain can be used to study the stability and performance of the converter. It should be noted that the controller design and also the loop gain are strongly affected by the behavior of G_co.

ˆo

i

cˆ ˆ_in

u G_{io o}-

Gco

Zo o-

Open-loop

Ga

Gcc

Gse

ˆ_o u

ˆ_co u

ˆ_r Closed-loop u

a)

ˆo

i

ˆin

i

cˆ ˆ_in

u Y_{in o}-

Gci

Toi o-

Open-loop

Ga

Gcc

Gse uˆ_o ˆ_co

u

ˆ_r Closed-loop u

b)

Fig. 2.5. Control-block diagrams for voltage-output converter at closed loop: a) output dynamics and b) input dynamics.

The control-block diagrams are efficient for computing the dynamical closed-loop profile consisting of the open-loop parameters. According to Fig. 2.5 a) and [18], the closed-loop output voltage ˆu_o can be given by

ˆ ˆ ˆ ˆ

1 1 1

io o o o cc a co

o in o r

se cc a co se cc a co se cc a co

G Z G G G

u u i u

G G G G G G G G G G G G

(2.6)