li dissertation hong - fernuniversität hagen · dissertation hong li. 1 introduction 3 in...

Reducing Electromagnetic Interference

in DC-DC Converters with Chaos Control

DISSERTATION

zur Erlangung des akademischen Grades

DOKTOR-INGENIEURIN

der Fakultat fur Mathematik und Informatik

der FernUniversitat in Hagen

von

Hong Li, M.Sc.

Changzhi/China

Hagen 2009

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Abstract

Electromagnetic Interference (EMI) resulting from high rates of changes of voltage and current, im-pairing other devices’ performance and harming human being’s health, has become a major concern indesigning direct current (DC-DC) converters for a long time due to the increasingly wide applicationsof various electrical and electronic devices in industry and daily life. Thus, the question of how toreduce the annoying, harmful EMI has to be faced by scientists and engineers.Normally, EMI is handled by appending a properly tuned filter to reduce it within low frequencybands, referring to conducted EMI, or dealt with by electromagnetic shielding when it is within highfrequency bands, referring to radiated EMI. However, as a filter is restricted in a narrow frequencyband, it is not applicable to a much broader EMI frequency band alone. Therefore, multiple filtersshould be employed, increasing the difficulty of design. In addition, the affixed filter circuits not onlyincrease cost, but also imply an increase of size and weight, rendering a product to lack portability.Electromagnetic shielding is the process of limiting the penetration of electromagnetic fields intoa space, by blocking them with a barrier made of conductive material. Typically, it is applied toenclosures, separating electrical devices from the ‘outside world´, and to cables, separating wires fromthe environment, through which the cables run. Shielding is an effective but expensive solution forEMI suppression. Moreover, in practice there are many leak sources on the enclosures. Therefore,both approaches are not perfect solutions of EMI suppression.Due to the pseudo-random and continuous spectrum characteristics of chaos, more recently the EMIproblem has been tackled by the spread spectrum approach employing chaos control. However, thereexist two prominent problems still unsolved: one is that the ripples of the output waveforms are muchbigger than those with periodically running DC-DC converters, degrading DC power supplies; and theother one is that the parameter design of DC-DC converters becomes difficult due to the variationalfrequency under chaos control. Trying to fight these two problems, this dissertation is to improve theconventional chaos control approaches and to propose some new strategies of chaos control for EMIsuppression.Two kinds of control approaches will be proposed in this dissertation. One is a novel chaotic peakcurrent mode control via parameter modulation, which cannot only reduce EMI but also suppressoutput ripples easily; the other one is to combine chaos control with the most important and commoncontrol method in DC-DC converter, i.e., pulse width modulation (PWM) control, to form a novelchaos-based PWM control, named chaotic PWM control. This chaotic PWM control has the advan-tages of being easy to design, of applicability in various DC-DC converters, and of flexibility to reacha trade-off between output ripple and EMI. Therein, the chaotic carrier plays a key role in generatingchaotic signals, which is designed both in digital and analogue ways, providing two alternative choicesfor different applications in practice. Moreover, a chaotic soft switching PWM control is put forward,which combines soft switching with chaotic PWM due to the fact that the soft switching technique isto switch on and off at zero current or zero voltage to alleviate the high rates of changes of voltageand current, to reach a better effect for EMI reduction and to reduce the power loss as well. Fur-thermore, the proposed EMI control approaches are simulated and implemented in hardware. Theexperiments are of great significance to verify the theoretical results and simulations, especially forfuture marketing.To this end, some theoretical concerns about the calculation of the invariant density of a chaoticmapping in a peak current mode boost converter, parameter estimation, ripple estimation, and aboutstability analysis in a chaotic PWM DC-DC converter are also addressed in this dissertation, providingtheoretical explanation and verification for the simulation and experimental results, and a guidelinefor systems design. Finally, one of the modern spectral estimation method, viz., the Prony method,is employed to replace the conventional fast Fourier transform in estimating the spectra of chaoticsignals, providing more accurate results.

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Contents

1 Introduction 11.1 EMI and EMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 EMC Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Conventional EMI Suppression Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 EMI Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Electromagnetic Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Soft Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.4 Random Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 About this Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Chaos Control of EMI 102.1 Chaos in DC-DC Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.3 Chaos Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Approaches of Chaos Control for EMI Suppression . . . . . . . . . . . . . . . . . . . . 132.2.1 Chaos Control via Parameter Modulation . . . . . . . . . . . . . . . . . . . . . 132.2.2 Chaotic PWM Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Chaotic Peak Current Mode Boost Converters 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Chaotic Current Mode Boost Converter Model . . . . . . . . . . . . . . . . . . . . . . 213.3 Characteristics of the Chaotic Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Bifurcation and Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . 243.3.3 EMC Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Chaotic Pulse Width Modulation 334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 CPWM with Varying Carrier Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 CPWM with Varying Carrier Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 374.4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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5 Analogue Chaotic PWM 435.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Analogue Chaotic Carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.1 Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2.2 Chaotic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Analogue Chaotic PWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3.1 A Boost Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4.1 Chua’s Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 A Chaotic Soft Switching PWM Boost Converter 566.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2 Circuitry and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2.1 Circuit Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.2 Chaotic Soft Switching PWM Control . . . . . . . . . . . . . . . . . . . . . . . 60

6.3 Simulations and Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7 Invariant Densities of Chaotic Mappings 667.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.2 1-D Mapping for a Boost Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.3 Invariant Density of a Chaotic Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 687.4 Eigenvector Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.5 Invariant Density of the Boost Converter’s Chaotic Mapping . . . . . . . . . . . . . . 697.6 Examples of Applying Invariant Densities . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.6.1 Power Spectral Density of a DC-DC Converter’s Input Current . . . . . . . . . 707.6.2 Average Switching Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.6.3 Parameter Design with Invariant Density . . . . . . . . . . . . . . . . . . . . . 74

7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8 Stability of a Chaotic PWM Boost Converter 768.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768.2 Chaotic PWM Boost Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.3 Estimation of the Mean State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 778.4 Ripple Estimation of the Input Current . . . . . . . . . . . . . . . . . . . . . . . . . . 808.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.5.1 Two Operation Modes of the Boost Converter . . . . . . . . . . . . . . . . . . . 828.5.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9 Chaotic Spectra Analysis Using the Prony Method 859.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.2 Prony Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869.3 Deriving the Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879.4 Chaotic Spectral Estimation of DC-DC Converters . . . . . . . . . . . . . . . . . . . . 899.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

10 Conclusion 93

References 96

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Chapter 1

Introduction

With the rapid development and application of electrical and electronic devices and products,electromagnetic interference (EMI) has become a major problem annoying scientists and engi-neers. What is EMI? How do people control EMI? What and how can we do to fight EMI?These questions are to be answered first in this chapter.

1.1 EMI and EMC

The recent six decades have witnessed a rapid and tremendous advance in power electronics.A broad range of electronic products has come forth and is widely applied in industry andhuman daily life, such as computers, wireless communication devices, electrical motors, electricvehicles and so on. Most of them, e.g., laptop computers and cellular telephones, are suppliedor charged by direct current (DC). Therefore, AC-DC and DC-DC converters are necessary toconvert the alternating current (AC) supplied out of sockets to the DC required. Thus, DC-DC converters play a very important role in portable electronic devices, which are primarilysupplied with power from batteries. Such electronic devices often contain several subcircuitswith their own voltage requirements different to the ones provided by batteries or externalsupplies. Additionally, the voltage of a battery declines as its stored power drains away. DC-DC converters provide a means to maintain voltage from a partially lowered battery voltage,thereby saving space instead of using multiple batteries to accomplish the same task.

The electrical and electronic devices that carry rapidly changing electrical currents constitutea source of EMI, while some natural objects and phenomena, such as sun and northern lights,are other sources as shown in Figure 1.1. EMI is an unwanted disturbance that affects electricalcircuits due to either electromagnetic conduction or electromagnetic radiation emitted from anexternal source. The disturbance may interrupt, obstruct, or otherwise degrade or limit theeffective performance of circuits.

For example, we all know that the use of mobile telephones is forbidden on board of an airplanebecause of possible interferences with the aircraft’s communication and navigation systems.Recent events regarding cellular telephones include that of a Northwest Airlines flight whichwas diverted because of suspicious telephone use by passengers, and a British Airways flightthat had to return to Heathrow 90 minutes after take-off, because nobody confessed to haveused a cellular telephone even though crew members heard a telephone ringing, which causedconsiderable fear among passengers and crew and created severe flight delays. Two further ex-amples are an electrical wheelchair suddenly veering due to radio and microwave transmissions,and an infant apnea monitor failing to alarm because of the ambient electromagnetic fields[62, 73].

In terms of frequency bands, EMI is categorised as conducted EMI and radiated EMI, which

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Figure 1.1: Typical electromagnetic environment

are illustrated in Figure 1.2. Conducted EMI is caused by the physical contact of conductors asopposed to radiated EMI, which is caused by induction (without physical contact of conductors),depending on the frequency of operation. That is to say, for lower frequencies EMI is causedby conduction and, for higher frequencies, by radiation.

The conducted EMI, normally having frequencies between 10kHz and 30MHz, can be furtherclassified into common mode (CM) noise and differential mode (DM) noise in terms of differentdirections of conduction.

Common Mode Noise is conducted through all lines in the same direction, and always existsbetween any power line and ground.

Differential Mode Noise is conducted through all lines in inverse directions, and alwaysexists between power lines.

Figure 1.2: EMI coupling modes

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In converters, DM currents flow in and out of the power supplies via the power leads and theirsources (or loads), and are totally independent of any grounding arrangements. Consequently,no DM current flows through the ground connections. On the other hand, CM currents flowin the same direction either in or out of the power supplies via the power leads and return totheir sources through the lowest available impedance paths, which are invariably the groundconnections. Even if the ground connections are not deliberate, CM currents flow throughparasitical capacitors or parasitical inductors to the ground, as Figure 1.2 shows.Empirically, at frequencies below approximately 5MHz, the noise currents tend to be predom-inantly DM, whereas at frequencies above 5MHz the noise currents tend to be predominantlyCM [67].Converters also generate radiated EMI emissions normally with frequencies between 30MHzand 1GHz. Radiated EMI appears in the form of electromagnetic waves that “radiate” into theimmediate atmosphere directly from a circuitry and its interface leads. The circuitry and itsinterface leads can liken themselves to a transmitting antenna for this radiated EMI, as shownin Figure 1.2.Radiated EMI can contain electric and magnetic fields. The strength of the electric fieldis proportional to the circuit voltage, operation frequency, and “the effective length of theantenna”. The strength of the magnetic field is proportional to the circuit current, operationfrequency, and “the effective area of the antenna loop”. Since the circuit parameters andoperation frequency are fixed for a converter’s operation characteristics, the only variable factoris the length of the power line, or the enclosed loop area of the power line’s return path.Therefore, it can be seen that radiated EMI can be minimised by physically locating the noise-generating source as close to its source and load as possible. However, mechanics rarely permitsuch a compact assembly.Normally, EMI can be estimated by measuring the power spectral density (PSD), which de-scribes how the power of a signal or time series is distributed with frequency, such as theexample given in Figure 1.3. More information about PSD can be found in [55].

Figure 1.3: A triangle waveform and its power spectrum

According to Figure 1.3, it is obvious that the spectrum consists of the operation frequency andits harmonics. If the harmful harmonics of input and output signals are not filtered in convert-

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ers, they can corrupt the power sources and interfere with the operation of other equipmentrunning from the same sources. Radiated EMI noise will also be generated and interfere withthe operation of adjacent equipment, which gives rise to important electromagnetic compatibility(EMC) problems.EMC is defined as the ability of an apparatus to function satisfactorily in its electromagneticenvironment without introducing intolerable electromagnetic disturbance to other apparati inthe same environment. EMC includes two issues to achieve the defined ability.

Emission Emission issue is related to the unwanted generation of electromagnetic energy, andto the countermeasures which should be taken in order to reduce such generation and toavoid the escape of any remaining energies into the environment.

Susceptibility Susceptibility or immunity issue, in contrast, refers to the correct operation ofelectrical equipment in the presence of unplanned electromagnetic disturbances.

1.2 EMC Standards

As mentioned above, power electronic devices, including converters, are of great benefit tohuman beings and are widely applied in our daily life. Unfortunately, the widespread useof power electronic products, at the same time, causes the serious EMI problem. Facing theharmful interference, international communities have agreed on standard regulations, i.e., EMCstandards, which are supposed to ensure unimpeded systems in the electromagnetic environmentto comply with regulatory requirements. Here, some basic information on EMC standards forconverters is listed.

Generic EMC Standard A top-level standard for a type of equipment encompasses specificbasic standards in its references. The currently relevant standard for power supplies is[ EN61204-3: 2000] . This covers the EMC requirements for power supply units with DCoutput(s) of up to 200V, at power levels up to 30kW, and operating from AC or DCsource voltages of up to 600V. The abbreviation EN refers to Euro Norm or Europeanstandard. Europe has led the field in establishing standards for EMC and many otherareas, which have been adopted worldwide with some local deviations.

List of Basic Standards The relevant basic standards mentioned in EN61204-3 are: EN55022and EN55011 for conducted and radiated electromagnetic interferences emitted by powersupplies. The FCC has set similar standards in the USA. It is expected that EN55022 willbecome a worldwide standard as CISPR22. There are two levels for the emission limits,Class A and Class B. Class B is normally required, and puts a lower limit on allowedemissions. Particular aspects of EMC are addressed in the standard EN61000 as follows:

EN61000-4-2 Immunity to electrostatic discharge

EN61000-4-3 Immunity to radiated radio frequencies

EN61000-4-4 Immunity to fast transient voltages on input lines

EN61000-4-5 Immunity to lightning surges on input lines

EN61000-4-6 Immunity to conducted radio frequencies

EN61000-4-8 Immunity to power frequency magnetic fields

EN61000-4-11 Immunity to damage from input line voltage reductions

EN61000-3-2 Limits to the harmonic currents that can be taken from the input lines

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EN61000-3-3 Limits to the voltage fluctuations that the power supply can cause to theline input voltage

Performance Criteria In immunity testing, there are four classes by which passing or failureare assessed, viz., Class A: no loss of function or performance due to the testing, Class B:temporary loss of function or performance, self-recoverable, Class C: loss of function orperformance which needs intervention to restore, and Class D: permanent loss of functionor performance due to damage, always representing a failure.

1.3 Conventional EMI Suppression Techniques

Many methods have been proposed to suppress EMI of converters. Among them, EMI filteringis the most common and oldest approach, which is used to reduce conducted EMI to satisfylow-frequency EMC standards. For meeting high-frequency EMC standards, electromagneticshielding is usually employed, which is to reduce radiated EMI. Both methods can well suppressEMI, but at the same time increase cost and weight, rendering products to lack portability.In order to meet the stricter international EMC standards and the requirements for electronicproducts to be lighter, smaller, and cheaper, some new EMI suppression techniques should beproposed and field-tested, for instance, the soft switching technique and random modulation.In the sequel, these four methods will be introduced, respectively.

1.3.1 EMI Filtering

Converters are a source of EMI due to pulsating input currents and rapidly changing voltagesand currents [11]. An EMI filter is normally appended at the input side of a converter.

Since conducted EMI is made up of CM noise and DM noise, an EMI filter consists of twofunction blocks as shown in Figure 1.4: Cx and differential choke are used to filter the DMnoise, while Cy and common choke filter the CM noise.

Figure 1.4: EMI filter

EMI filters are effective to suppress conducted EMI for converters, but also have some short-comings, for instance, their volume is too huge for some products, not only the noise but also theuseful signals may be suppressed, and any EMI filter is designed for a special narrow frequencyband, only, unable to work on the entire broad frequency band.

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1.3.2 Electromagnetic Shielding

Electromagnetic shielding is the process of limiting the penetration of electromagnetic fieldsinto a space, by blocking them with a barrier made of conductive material as shown in Fig-ure 1.5. Typically, it is applied to enclosures, separating electrical devices from the ‘outsideworld´, and to cables, separating wires from the environment the cables run through. Electro-magnetic shielding used to block radio-frequency electromagnetic radiation is also known as RF(Radio Frequency, about 3KHz to 300GHz) shielding. It is worth to notice that electromagneticshielding is an effective but expensive solution for suppressing EMI. On the other hand, theremay exist many leak sources, such as intake, display window, socket in real shield, degradingthe effectiveness of EMI shielding.

Figure 1.5: Operation principle of electromagnetic shielding

1.3.3 Soft Switching

The technique of soft switching was first presented [15] in 1990 and has rapidly developed inrecent years [20, 21]. The main goal of soft switching is to reduce the switching loss whenconverters operate in high frequencies by switching on and off at zero current or zero voltage.Consequently, the high rates of changes in voltage and current are alleviated, thus EMI can bereduced. The operation principle and the effectiveness of soft switching are shown in Figures 1.6and 1.7, respectively.Meanwhile, soft switching has its own limitations in improving EMC: the effect to reduce EMIfocuses on the frequency band 150kHz – 30MHz, but it almost does not work on the frequencyband 10kHz – 150KHz; and more components are needed, such as resonant inductors, resonantcapacitors, auxiliary diodes, and even auxiliary switches, which increases the power loss on theother side and makes the design of switched mode converters more complicated.

1.3.4 Random Modulation

Random modulation is a new method proposed in the last two decades [29] to reduce EMI.Random modulation means that the switch frequency is varied according to a given randomsignal, thus the total energy is spread over a wider frequency band, which can be illustrated asin Figure 1.8.The peaks appearing in the frequency band when converters operate in periodic mode can bereduced and eliminated. In this way, EMI can be suppressed. For random modulation, thereare two main limitations: one is that in practice real random signals are difficult to generate,and the other is that the design of converter parameters becomes difficult, since it is based on

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(a) Turn-on process of hard switching (b) Turn-off process of hard switching

(c) Turn-on process of soft switching (d) Turn-off process of soft switching

Figure 1.6: The turn-on and turn-off processes of a hard-switching and a soft-switching MOS-FET

(a) Hard switching (b) Soft switching

Figure 1.7: The power-loss waveforms for a power MOSFET used in a DC-DC converter withhard- or soft-switching topologies

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Figure 1.8: Spectrum of a frequency-modulated sine signal following a sine modulation profilein time (Initial frequency fC , peak deviation ∆fC)

the random frequency, for example, when a converter operates in frequency f1, the equivalentinductance is 2πf1L. Due to the difficulty of obtaining a real random signal, a pseudo-randomsignal is used, which is called pseudo-random modulation. Chaotic modulation is one kind ofcommon and important pseudo-random modulation, since chaos is characterized by pseudo-randomness and continuous spectra, and can be generated by deterministic equations [27].

1.4 Motivation

Using chaos theory in engineering applications has emerged as an attractive new concept.Chaos as a special dynamical phenomenon has extensively been studied for more than fourdecades, but only recently it has been put forward for scientific and engineering applications.The continuous-spectrum feature of chaos is perfectly fitting to fight EMI by spreading thespectra of output signals over the entire frequency band and, thus, the peaks, which appear atthe multiples of the fundamental frequency and lead to EMI, can be suppressed, implying thereduction of EMI.

Having this feature in mind, we focus on DC-DC converter circuits themselves by integratingchaotic carriers with some conventional control methods for DC-DC converters, such as PWMcontrol, to propose some novel chaos-based control methods, which cannot only overcome thedisadvantages of conventional EMI filters and electromagnetic shielding, but also solve someproblems like big ripples of output current resulting from using chaos control. Therefore, theproposed methods will be a perfect solution for EMI suppression. Simulations and experimentswill be carried out to verify the effectiveness of the methods, which lays a foundation for futuremarketing.

In addition, some theoretical problems, such as stability, parameter design, and ripple estima-tion for DC-DC converters with chaos controls will be addressed to facilitate system design.

1.5 About this Dissertation

This thesis aims to propose approaches to fight EMI in the widely applied DC-DC convertersby employing chaos control, to carry out simulations and hardware implementations, and to

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provide theoretical analyses on some important issues, like stability and ripple estimation. Itis organised in the following way.Chapter 2 is to give an overview to the chaos control of DC-DC converters, which is classifiedinto two categories, parameter modulation and chaotic PWM control.Chapter 3 focuses on improving chaos control via parameter modulation in terms of ripples.Although this kind of chaos control applied to DC-DC converters has the advantage of EMIreduction, there is a big problem that the output ripples of DC-DC converters are too big tobe useful in practice. To cope with it, a novel chaos control method for ripple suppression isproposed and analysed. The chaotic mapping of a peak current boost converter with this novelchaos control is derived, which can facilitate further theoretical analysis.Chapter 4 introduces the concept of chaos into traditional PWM control. Unlike chaos controlvia parameter modulation, chaotic PWM control drives DC-DC converters to operate in chaoticmode by adding external chaotic signals, which renders the design of DC-DC converters moreflexible. Since the external chaotic signals, i.e., chaotic carriers, can be generated by digitalprocessors, accordingly the magnitudes of ripples can also be controlled by computer programs.Simulation and experimental results illustrate the effectiveness of this novel chaos control forEMI reduction. Moreover, to realise chaotic PWM control, control circuits more complicatedthan those for traditional PWM control need be implemented. Fortunately, these controlcircuits can be integrated on printed circuit boards or even in small chips.Chapter 5 deals with further improvements of chaotic PWM control. Considering the rela-tively high costs and speed limitations of digital processors, the chaotic carrier generated bya digital processor will be re-designed and replaced by a novel analogue chaotic carrier suit-ing high-frequency DC-DC converters. The design of the analogue chaotic carrier is detailed,and eventually, the evident EMI reduction can be observed at and proved by a DC-DC con-verter using the analogue chaotic carrier with the help of both simulation and experiments incomparison with the EMI of a DC-DC converter controlled by traditional PWM.Chapter 6 notices the different principles of reducing EMI by the popular soft switching tech-nique and chaos control. It is well known that soft switching can reduce EMI for DC-DCconverters, by turning the switchs on or off at zero current or zero voltage to alleviate the highrates of changes of voltage and current, thus reducing both switching loss and EMI; while chaoscontrol reduces EMI by spreading the spectra of signals or time series over the whole frequencyband. Obviously, soft switching and chaos control provide different ways to suppress EMI. InChapter 6, these two methods are combined, named chaotic soft switching PWM control, formore pronounced improvement of EMC for DC-DC converters.Chapters 7 and 8 address some theoretical considerations on chaotically controlled DC-DCconverters. Firstly, the chaotic features of DC-DC converters using chaos control via parametermodulation are deduced and analysed, and some applications based on these analytical resultsare given in Chapter 7. The analysis is carried out further for DC-DC converters using chaoticPWM control in Chapter 8, where stability and estimations of ripples and outputs for this kindof chaotic DC-DC converters are investigated, too.Chapter 9 attempts to find an appropriate spectral estimation method for chaotic signals. It isknown that EMI is conventionally estimated by measuring its spectrum which is then subjectedto fast Fourier transform (FFT). However, due to the special characteristics of chaotic signals,such as inner harmonics, FFT has evident drawbacks in analysing chaotic spectra. Here, a newspectral estimation method, the Prony method, is employed to analyse chaotic spectra in orderto improve spectral resolution.Chapter 10 summarises this dissertation, outlines the contributions made, and points out di-rections for further research.

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10 2 Chaos Control of EMI

Chapter 2

Chaos Control of EMI

Chaotic phenomena exist ubiquitously in nature. As non-linear systems, DC-DC converters canexhibit chaotic behaviour. The chaotic behaviour of DC-DC converters as well as chaos controlapproaches to suppress EMI in DC-DC converters are introduced in this chapter. Further,analytical tools for chaos, such as bifurcation diagram, Poincare section and spectrum, areillustrated. The advantages and disadvantages of these chaos control approaches are described,showing the research direction to follow in this dissertation.

2.1 Chaos in DC-DC Converters

Since E. Lorenz discovered in 1963 the first physical chaotic system, viz., the Lorenz attractor,chaos has matured as a science, and is considered as one of the three seminal scientific discoveriesof the twentieth century, together with relativity and quantum mechanics. Chaos typically refersto unpredictability. Mathematically, chaos means a deterministic aperiodic behaviour, whichis very sensitive to its initial conditions, known as “butterfly effect”, saying that a butterflyflapping its wings in Kansas can cause a tornado in Oklahoma a few days later [13]. Chaostheory describes the behaviour of certain non-linear dynamical systems that under certainconditions exhibiting chaos.Since chaotic phenomena in DC-DC converters were first reported in [26], great efforts havebeen devoted to study chaotic phenomena in various converters, such as boost, buck, boost-buck, and Cuk converters [1, 27, 59]. DC-DC converters are strongly non-linear systems andcan, thus, exhibit rich chaotic behaviour. As an example, periodic and chaotic behaviour canbe observed in a current mode boost converter under certain parameter conditions.

2.1.1 System Description

Typical DC-DC converters include buck, boost, buck-boost converters, and some other varia-tions. Due to its simple model, the boost converter is taken here as an example and describedas follows [25],

xn+1 = f(xn) = α(1− xn) mod 1, (2.1)

where xn =tnTC

, α =V0

VI

− 1, tn =(Iref − in)L

VI

, tn is the switching-on time length at the nth

switch, in the inductor current at the instant of switching on, TC the clock period, Iref thereference current, VI the given input voltage, and VO the average output voltage. The circuitdiagram of the peak current mode controlled boost converter is depicted in Figure 2.1 (a) andthe current waveform i is shown in Figure 2.1 (b). It is obvious that α > 0 if VO > VI . Based on

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2 Chaos Control of EMI 11

the criterion for the Lyapunov exponent, when α > 1, the sequence x0, x1, x2, . . . is chaoticwithin [0, α] [44].

(a) (b)

Figure 2.1: (a) Peak current mode controlled boost converter, (b) current waveform i(t)

2.1.2 Experimental Observations

The circuit parameters are set as follows: VI = 10V , L = 1mH, C = 92µF , Tc = 100µs,A = 8.4, and Iref = 1.8A. Here, A is the amplifier’s gain, and the load resistance RL serves asthe control parameter.The MOSFET IRF530 is selected here as the power switch, whose drain-to-source breakdownvoltage and continuous drain current are 100V and 14A, respectively. Since the maximumreverse voltage of the fly-wheel diode is about 16V when the MOSFET is on, and the maximumcurrent is about 4A, the diode MBR20100CT is selected, whose withstand voltage is 63V andrating current is 10A.Setting the value of RL to 8Ω, 12Ω, 14Ω, 15Ω, or 16.5Ω, the boost converter can operate in fourperiodic or chaotic modes, respectively, as shown in Figure 2.2 (the x-axis represents time, they-axis inductor current (upper) and output voltage (lower)) and Figure 2.3 (inductor currentgiven on the x-axis and output voltage on the y-axis).It is seen that the boost converter exhibits periodic or chaotic behaviour under certain parameterconditions, the ripples of the current and voltage become very big in chaotic mode, and theaverage values of current and voltage vary as parameters are changed, which is not allowed forDC-DC converters in most cases in practice.

2.1.3 Chaos Control

Today, it is well known that most conventional control methods and many special techniquescan be used for chaos control, regardless whether the purpose is to reduce “bad” chaos orto introduce “good” chaos. Numerous control methodologies have been proposed, developed,tested, and applied. Similar to conventional systems control, the concept of “controlling chaos”is first to mean ordering or suppressing chaos in the sense of stabilising chaotic system responses.In this pursuit, numerical and experimental simulations have convincingly demonstrated thatchaotic systems respond well to these control strategies. These methods of ordering chaosinclude the now familiar OGY method [58], feedback controls, and fuzzy control, to list just afew.However, controlling chaos has also encompassed many non-traditional tasks, particularly thoseof enhancing or generating chaos when it is beneficial. The process of chaos control is nowunderstood as a transition between chaos and order, and sometimes from order to chaos, de-pending on the application of interest. The task of purposely creating chaos, sometimes called

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(a) Period-1 (b) Period-2

(c) Period-3 (d) Period-4

(e) Chaos

Figure 2.2: Waveforms of inductor cur-rent (A) (upper) and capacitor voltage(V) (lower) for different modes

chaotification or anticontrol of chaos, has attracted increasing attention in recent years dueto its great potential in non-traditional applications such as those found within the contextof physical, chemical, mechanical, electrical, optical, and particularly biological and medicalsystems.

It was shown in the last subsection that a DC-DC converter running in chaotic mode has largecurrent and voltage ripples, and that it is difficult to design circuitry parameters. This is notacceptable in practice. Therefore, it seems that chaos should be avoided in DC-DC converters.On the other hand, chaos has the prominent feature of a continuous power spectrum, whichcan be used to spread the spectra of the output signals over the whole frequency band, andthus allows the peaks can be suppressed, which appear at the multiples of the fundamentalfrequency and lead to EMI, implying the reduction of EM [27]. Here, a question is if thereis an approach, which can utilise the beneficial feature of chaos, but overcome the drawbacksresulting from the use of chaos control? As we shall show, the answer is positive.

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(a) Period-1 (b) Period-2

(c) Period-3 (d) Period-4

(e) Chaos

Figure 2.3: Phase portraits (V −A) fordifferent modes

2.2 Approaches of Chaos Control for EMI Suppression

Fundamentally, chaos control methodologies can be divided into two categories: one is tomodulate circuitry parameters without any auxiliary circuits, while the other one is to appendexternal chaotic circuits to the main control parts to drive entire systems chaotic. The secondmethodology is mainly involved with the widely used PWM control, thus it is called chaoticPWM control.

2.2.1 Chaos Control via Parameter Modulation

To illustrate the chaos control method by parameter modulation, the voltage-controlled buckconverter shown in Figure 2.4 is used here.

The output voltage v of the converter is the non-inverting input to the amplifier, and thereference voltage Vref is the inverting input to the amplifier. The gain of the amplifier is A.The controlled output voltage vco can be expressed as

vco = A(vo − Vref ). (2.2)

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i

E

S L

C RD v

vramp

t

VU

VL

AC1 vco

Figure 2.4: Voltage-controlled buck converter

This controlled voltage vco is the inverting input of the comparator and the non-inverting oneis the saw-tooth carrier vramp, which has the period T , the lower limit VL and the upper limitVU , and satisfies the relationship,

vramp = VL + (VU − VL)[t mod T ], (2.3)

where mod refers to the modulo operation.The switch S is controlled by the pulse signal from the output of the comparator C1. Assumethat the converter operates in continuous current mode (CCM). As vco < vramp, the outputof the comparator is at high level, S is on and diode D is off, which corresponds to Mode I;and as vco > vramp, the output of the comparator is at low level, S is off and D is conducting,which corresponds to Mode II. According to circuitry theory, the state equations of the buckconverter can be written as

x = A1x + B1E for Mode I, (2.4)

x = A2x + B2E for Mode II, (2.5)

where, x =[v i

]T, and A1 = A2 =

[− 1

RC1C

− 1L

0

], B1 =

[01L

], and B2 =

[00

]are state matrices.

Chaos control by parameter modulation means that the system can exhibit chaos by only tuningone or more system parameters. Now some examples will be shown. First, the parameters ofthe buck converter which operates in periodic mode are: L = 20mH, C = 47µF , A = 8.4,VL = 3.8V , VU = 8.2V , TC = 400µs, R = 22Ω, Vref = 11.3V , and E = 20V .To illustrate this method, the input voltage E is used as the control parameter, and the bifur-cation diagram of E vs. i is depicted in Figure 2.5. From the figure it is seen that, when E islarger than about 32.3, the buck converter begins to operate chaotically. It is remarked thatthe values of the control variable, such as E here, with which the DC-DC converter exhibitschaotic behaviour, can be derived by solving the Lyapunov exponents of the Jacobian matrixof the state equations [9].The Poincare section provides another means to visualise an otherwise messy, possibly aperi-odic, attractor. A Poincare map is the intersection of a periodic orbit in the state space of acontinuous dynamical system with a certain lower-dimensional subspace, called the Poincaresection, transversal to the flow of the system, as shown in Figure 2.6. It can be interpreted as adiscrete dynamical system within a state space that is one dimension smaller than the originalcontinuous dynamical system. Since it preserves many properties of periodic and quasi-periodic

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Figure 2.5: Bifurcation diagram (E ∼ i)

orbits of the original system and has a lower-dimensional state space, it is often used to analysethe original system.

Figure 2.6: Illustration of Poincare section

In terms of power spectra, there are three types of flows, viz., periodic, quasi-periodic, andaperiodic. A fixed point, a closed curve, and a point cloud on the Poincare section correspondto a closed orbit, a quasi-periodic flow, and an aperiodic flow or chaos in the original statespace, respectively.Similarly, to illustrate the chaotic behaviour in the voltage-controlled buck converter, thePoincare section can be selected in the way shown in Figure 2.7, where the planes S = 1and S = 0 are called “switching planes”. Passing through the planes, the switch will change itsstate from turned-on to turned-off (S = 0), or from turned-off to turned-on (S = 1) [28, 52].Here, plane S = 1 is selected as the Poincare section of the buck converter, and the correspond-ing Poincare map is shown in Figure 2.8, where vn and in mean the values of output voltageand input current at the instant of the switch being on, respectively. It is seen that the DC-DCbuck converter operates in chaotic mode when E = 37V .It is remarked that some other parameters, such as Vref , can also be used as control parameter,for instance, as shown by the bifurcation diagram of Vref vs. i with E = 30V in Figure 2.9(a).Similarly, the Poincare section of the buck converter at Vref = 25V and E = 30V is shown in

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S=1

S=0

Mode IMode II

(in, vn)

Poincare Section

Figure 2.7: Selection of Poincare section for a DC-DC converter

Figure 2.9(b).

Moreover, the bifurcation diagram of 1/R vs. i with Vref = 11.3V and E = 35V is shownin Figure 2.10(a), and the corresponding Poincare cross section of the buck converter, whenR = 12.2Ω, is shown in Figure 2.10(b), respectively.

It is remarked that DC-DC converters can exhibit rich chaotic behaviour by tuning circuitryparameters. For comparison, the spectra of the buck converter operating in periodic modeand in chaotic mode are given in Figures 2.11 and 2.12, respectively. It is seen that the peak

Figure 2.8: Poincare section

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(a) Bifurcation of Vref vs. i (b) Poincare section

Figure 2.9: Bifurcation and Poincare section

(a) Bifurcation of 1/R vs. i (b) Poincare section

Figure 2.10: Bifurcation and Poincare section

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values of the spectrum are greatly reduced when the buck converter operates in chaotic mode,as compared with those when it runs in periodic mode.

0 10 20 30 40 50 60 70 80 90 100-100

-80

-60

-40

-20

0

20

40

Frequency (kHz)

Am

plitu

de

Frame: 63

Figure 2.11: Spectrum of the buck converter when E=31V

0 10 20 30 40 50 60 70 80 90 100-100

-80

-60

-40

-20

0

20

40

Frequency (kHz)

Am

plitu

de

Frame: 105

Figure 2.12: Spectrum of the buck converter when E=34V

Remarks

It is seen that DC-DC converters can exhibit rich chaotic behaviour by parameter modulation,which is used to reduce EMI as shown in Figures 2.11 and 2.12. Meanwhile, it is also observedthat the output ripples of the DC-DC converter with chaotic parameter modulation control areobviously increased. As shown in Figure 2.1, the ripple of the boost converter’s input currentis 0.38A with periodic control, while it increases to more than 0.7A under chaotic parametermodulation control. Since the main function of DC-DC converters is to provide stable andsmooth power supply, large ripple is not allowed for DC-DC converters in practice.On the other hand, the chaotic parameter modulation control approach makes system designdifficult, because the operation frequency of a chaotic system is uncertain. Furthermore, DC-DC converters with chaotic parameter modulation control may run out of chaotic regions whentheir power supplies or loads fluctuate. These fluctuations are normally unpredictable, becausethe input voltages (or loads) of DC-DC converters, such as E, are supplied by other DC sourcesor batteries, and changes of these DC voltages can influence the operation modes (chaotic orperiodic mode) of DC-DC converters according to the bifurcation diagram. Finally, there is alack of theory, such as to estimate the mean switching frequency of chaotic DC-DC converters,so that system design becomes difficult.

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(a) periodic waveforms (b) chaotic waveforms

Figure 2.13: Periodic and chaotic input current waveforms of a buck converter

2.2.2 Chaotic PWM Control

Due to the above mentioned disadvantages of chaotic parameter modulation, merging chaoscontrol with the most popular and successful control method for DC-DC converters, viz., PWM,in order to reduce EMI constitutes the main concern of this dissertation, which is to be detailedin Chapters 4 – 6.

2.3 Summary

In this chapter, it has been shown that DC-DC converters can exhibit chaotic behaviour undercertain parameter conditions. Therefore, the use of chaos control is possible. This chapterintroduced chaotic parameter modulation and its drawbacks, and pointed out a potential chaoticPWM control for EMI suppression to be detailed in this dissertation.

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20 3 Chaotic Peak Current Mode Boost Converters

Chapter 3

Chaotic Peak Current Mode BoostConverters

A by-product of applying chaos control in reducing EMI are the increased output ripples ofDC-DC converters, which are not acceptable in practice. In this chapter, a novel chaotic peakcurrent mode boost converter is proposed, which is based on parameter modulation and caneffectively restrain the ripples. A current mapping function is derived, and its chaotic behaviouris analysed. Further, simulations and experiments are carried out to illustrate the effectivenessof the proposed design in reducing EMI and restraining the output ripples of the converter.

3.1 Introduction

Over the last two decades, chaotic parameter modulation control to the end of reducing EMIin DC-DC converters has attracted great interest [3, 4, 6, 25, 27, 32, 33, 57, 75, 76]. Since thepioneering work of Deane and Hamill [27], who used chaotic parameter modulation control todesign a peak current mode controlled boost converter, some variations have been proposedand tested [32, 33], showing that in power converters EMC can effectively be improved by theintroduction of chaos via current mode control.

A detailed study on a chaotic DC-DC converter has also been carried out by computing itsperiodic spectral components [25]. For the same purpose of improving EMC, the switchingoperation of a boost converter controlled by a chaotic return map was proposed in [6], andthe spectral analysis of the converter’s input current demonstrates how a return map affectsthe power density spectrum of the input current, which provides an approach to design thereturn map to satisfy EMC standards. Further experimental research of a chaos-based current-programmed boost converter was reported in [3].

Despite of the success of applying chaos control in EMI suppression, there remain two prominentproblems unsolved, viz., the ripples of the outputs are much greater than those of periodicallyrunning DC-DC converters [4], and the power of the background spectra has been increased inmost designs of chaos control by parameter modulation, resulting in larger power consumption,although the peak values of the power spectrum are reduced. Since the basic purpose of DC-DCconverters is supplying power, large ripples simply imply a degradation of performance. Thisproblem has previously been pointed out, and an explicit expression between the ripples andthe spectral spread of the current was given in [5]. Anyway, it is a difficult task to design asuitable control suppressing the ripples to a desired level.

These two disadvantages do not only exist in the peak current mode controlled boost convert-ers, but also in other chaotic power converters [57, 75], which have seriously impeded theirpopularity.

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3 Chaotic Peak Current Mode Boost Converters 21

Hence, answering the questions of how to improve the control method for chaotic DC-DCconverters so that both low EMI and small output ripples can be achieved simultaneously, andhow to verify the relationship between the ripples and the background spectrum constitute theconcern of this chapter.This chapter proposes a novel chaotic peak current mode control by setting a lower limit forthe controlled current, by which the ripple can easily be restrained between the peak value,i.e., the upper limit, and the lower limit. Meanwhile, the chaotic characteristics of the DC-DCconverter are well maintained.Compared with other peak current mode controls, where there is only one control input, thepeak current (upper limit), the proposed chaotic peak current mode control leads to morecomplex and richer chaotic behaviour in the DC-DC converters.This chapter is organised as follows. In Section 3.2, a novel peak current mode boost converteris presented and its corresponding chaotic mapping function is derived. The characteristicsof the mapping are then analysed in Section 3.3 with focus on its spectrum, and bifurcatingand chaotic behaviour. Its effects on EMI reduction and ripple suppression are studied andillustrated with simulations. To further verify this approach, the entire system is built andexperimental results are presented in Section 3.4.

3.2 Chaotic Current Mode Boost Converter Model

Inspired by [25], a novel chaotic current mode boost converter is proposed and depicted inFigure 3.1.

Figure 3.1: A chaotic current mode boost converter

Unlike the design in [25], the switch S is now controlled by a clock with period TC , a lowerreference current signal and an upper one, denoted by Ilow and Iupp, respectively. Differentinductor current waveforms can be obtained as shown in Figures 3.2 (a)–(c), corresponding tothe following three cases, respectively:

1. Case I: t2 ≥ TC ,

2. Case II: TA ≥ TC > t2, and

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(a) Case I: t2 ≥ TC

(b) Case II: TA ≥ TC > t2

(c) Case III: TC ≥ TA

Figure 3.2: Different current waveforms i(t) obtained from the boost converter

3. Case III: TC ≥ TA,

where t1 is the time for i(t) to rise from Ilow to Iupp, t2 is the time for i(t) to fall from Iupp toIlow, and TA = t1 + t2.In order to facilitate the analysis of the proposed converter, the discrete-time mapping of i(t)is derived.Referring to Figure 3.2, the time interval of variant length [in, in+1) is focused, in which i(t)changes from in to in+1, with in defined as the inductor current sampled at the instants of theclock pulses as i(t) is decreasing (e.g., in in Figures 3.2 (a)–(c)) and the instants of the clockpulses as i(t) is increasing with switch S activated twice or more within a single clock cycle(e.g. in+2 in Figures 3.2 (b) and (c)). For clarity, a time mapping is also assumed, such that

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i(τn) = in when τn = 0.Referring to Figure 3.2, S is closed at τn = 0, and hence

didτn

= VI

L,

i(τn) = in + VI

Lτn,

(3.1)

where VI is input voltage and L the inductance.Let tn be the time required for the current to rise from in to Iupp. Based on (3.1), one has

tn =(Iupp − in)L

VI

. (3.2)

The switch S is then opened and i(τn) is governed by

di

dτn

=(VI − V O)

L, (3.3)

where V O is the mean output voltage. Therefore,

i(τn) = Iupp +VI − V O

L(τn − tn) (3.4)

until the next clock pulse arrives or i(τn) = Ilow.As explained in [25], it is possible to estimate the mean output voltage V O by equating themean of the aperiodic inductor current to a periodic one. It is derived that V O is governed bythe input-output relationship,

V3

O + V O(VITp/2L− Iupp)RVI −RTpV3I /2L = 0, (3.5)

where Tp = TC is based on the design given in [25].Here, a similar approximation is performed, and (3.5) is still applicable, except that Tp does notonly depend on TC , but also on the values of Iupp and Ilow for the Cases II and III — which areour main concern. It is also observed that Tp is proportional to Iupp but inversely proportionalto Ilow. Using the first-order approximation, Tp can be expressed as

Tp =

(aIupp

Ilow

+ b

)TC , (3.6)

where a and b are constants to be determined. Based on extensive experimental results, it isfound that a = 2.0499 and b = 1.5455 and, hence, V O can be obtained by solving (3.5). Basedon circuit simulation, the relative errors of V O are well within 2%, which is much better thanthe ones obtained by [25].Now, let t

′n be the time interval from the last action of S within a clock period to the next

clock pulse, which can be given as

t′

n =

ε if ε ≤ t2,

ε− t2 otherwise,(3.7)

where ε = [TC − (tn mod TC)] mod TA.Referring to Figure 3.2, we obtain

in+1 =

Iupp + (VI−V O)

Lε if ε ≤ t2,

Ilow + VI

L(ε− t2) otherwise.

(3.8)

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Defining

xn =tnTC

=(Iupp − in)L

VITC

and α =V O

VI

− 1,

based on (3.8), a chaotic mapping can be constructed as

xn+1 =

αx′n, if x′n ≤ γ,ρ + γ − x′n, otherwise,

(3.9)

where

x′n = β[ 1β

(1− (xn mod 1))] mod 1,

γ =t2TC

, β =TA

TC

and ρ =(Iupp − Ilow)L

VITC

.

It should be noticed that, for Case I or t2 > TC , (3.9) can be simplified as

xn+1 = α [1− (xn mod 1)] ,

which is equivalent to the chaotic mapping obtained in [25]. Therefore, the situation in [25]can be considered as a special case of the one studied in this chapter.

3.3 Characteristics of the Chaotic Mapping

In this section, the characteristics of the chaotic mapping (3.9) are studied. Although thesecharacteristics depend on the all related parameters, such as VI and R, the study here will onlyfocus on their dependence on Ilow. Hence, referring to Figure 3.1, the following parameters areassumed fixed as VI = 10V , L = 1mH, C = 12µF , TC = 100µs, and R = 30Ω.

3.3.1 Spectrum Analysis

As explained in Section 3.2, there are three possible cases associated with the reference currents.Throughout this chapter, it is assumed that Iupp = 4A while Ilow takes values of 0A, 3A and3.5A, for Cases I, II, and III, respectively.Figure 3.3 shows the time evolutions of the inductor currents i(t) and the corresponding spectrafor the three cases. Comparing the waveforms in Figures 3.3 (a), (c) and (e), it can be observedthat the ripples of i(t) are greatly reduced when a larger Ilow is applied. Moreover, it is shownby the spectra in Figures 3.3 (b), (d), and (f) that power is well spread over the entire frequencyband. It is also interesting to notice that, instead of having a maximum peak of a magnitudeclose to the clock frequency TC as in Cases I and II, in Case III the peak is shifted to a frequencyclose to 1

TA= 13.9kHz.

Since the low-frequency components are suppressed, a better spectrum distribution is obtainedin all cases. However, it is also noticed that the background spectrum is not significantlyimproved with the reduction of current ripples.

3.3.2 Bifurcation and Lyapunov Exponents

The broadband spectrum discussed in the previous section suggests the chaotic nature of theboost converter expressed in (3.9). In the sequel, this nature is further investigated with theuse of bifurcation diagram and Lyapunov exponents.

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(a) i(t) for Case I (b) Spectrum of (a)

(c) i(t) for Case II (d) Spectrum of (c)

(e) i(t) for Case III (f) Spectrum of (e)

Figure 3.3: (a, b) Case I: t2 ≥ TC ; (c, d) Case II: TA ≥ TC > t2; (e, f) Case III: TC ≥ TA

Figure 3.4 depicts the bifurcation diagram of xn vs. Ilow and the corresponding maximumLyapunov exponent spectrum (LEs). The chaotic nature is confirmed with the existence of apositive LEs, while some periodic windows are observed in between. According to (3.9), periodicwindows exist when ρ + γ = β, and (3.9) can be written as xn+1 = β(1 − 1

βx′n) corresponding

to LE = 0.

Similarly, the bifurcation diagrams of xn vs. VI and xn vs. TC are obtained and shown inFigures 3.5 and 3.6. In Figure 3.5, a route from periodicity to chaos is clearly observed whenthe input voltage VI is decreased although some periodic windows exist. A similar conclusioncan be drawn from the bifurcation diagram given in Figure 3.6. Therefore, the mapping (3.9)can generate rich dynamical behaviour like bifurcation and chaos, which constitutes the cornerstone of the proposed approach to reduce EMI and improve EMC.

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(a) Bifurcation of xn vs. Ilow (b) Maximum LE

Figure 3.4: Bifurcation of xn vs. Ilow and corresponding maximum LE

(a) Bifurcation of xn vs. VI (b) Maximum LE

Figure 3.5: Bifurcation of xn vs. VI and corresponding maximum LE

(a) Bifurcation of xn vs. TC (b) Maximum LE

Figure 3.6: Bifurcation of xn vs. TC and corresponding maximum LE

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3.3.3 EMC Performance

In this subsection, the EMC performance of the proposed chaotic peak current mode boostconverter is studied. As shown in the bifurcation diagram, the boost converter can operateeither in chaotic or periodic mode. Therefore, simulations are to be conducted to comparewhich mode provides better EMI suppression performance.

(a) Ilow = 1.979A (b) Ilow = 2.62A

(c) Ilow = 2.958A (d) Ilow = 0A

(e) Ilow = 2.4A (f) Ilow = 3A

Figure 3.7: Spectra for different Ilow: (a)–(c) periodic mode and (d)–(e) chaotic mode

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It can be observed in Figure 3.4 (a) that the boost converter operates in periodic mode at, e.g.,Ilow = 1.979A, 2.62A, and 2.958A (Iupp = 4A) among many other options, while the powerspectra of the corresponding inductor currents are depicted in Figure 3.7 (a)–(c). It revealsthat the peak amplitude remains almost the same with the fundamental frequency shifting toa higher frequency as Ilow increases, which implies that the EMI is not increased, while theincrease of Ilow means a decrease of ripple amplitudes.On the contrary, Figures 3.7 (d)–(f) depict the spectra when the boost converter operates inchaotic mode with Ilow = 0A, 2.4A, and 3A (Iupp = 4A) for the three specific cases. A smallermaximum peak value is obtained when Ilow = 3A, as compared with the case of Ilow = 0A,corresponding to the original design given in [25], which means that the EMC of the boostconverter is improved, and a slight shift of the fundamental frequency is also observed.It is remarked that, theoretically, Ilow can be infinitesimally close to Iupp to restrain the currentripple to very small values. Due to the limited operation frequency of real switches, implementedwith MOSFETs, IGBTs etc., however, Ilow is dependent on the combination of the switches’operation frequency, ripple requirement, and EMC standards.Therefore, it can be concluded that, by controlling the boost converter to run in chaotic mode,the switch control strategy proposed in Figure 3.1 cannot only suppress the ripples, but alsoimprove the EMC at the same time.

3.4 Experimental Verification

The design shown in Figure 3.1 is realised with discrete components, the major ones of whichare tabulated in Table 3.1. Assume that VI = 10V , TC = 100µs, L = 0.56mH, C = 47µF , andR = 30Ω.

Table 3.1: List of main components

Component Devicediode MBR2045CTswitch IRFZ234Ncurrent sensor LA-55-Pflip-flop 74HC74Ncomparator LM393driver 34152P

Figure 3.8: Operation principle of LA 58-P mutual inductor

Current sampling is important in circuit implementation. In an experiment carried out forverification purposes, a type LA 58-P mutual inductor is used to detect the input current. Itsoperation principle is introduced in Figure 3.8, and its main characteristics are

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+ : DC source +12V .. 15V

- : DC source -12V .. 15V

RM : Measurement

χ : Accuracy 0.5 %

f : Frequency band DC .. 200 kHz

KN : Conversion rate 1:1000

Since the conversion rate, i.e., IS:Ip, is equal to 1:1000, to obtain the real value of the measuredcurrent, R should be 1000Ω in the experiment. Finally, the circuit is implemented as shown inthe circuit diagram Figure 3.9 as the circuit board shown in Figure 3.10.

Figure 3.9: Circuit diagram of the chaotic peak current mode boost converter

The current waveforms of the three cases with the boost converter operating in periodic modeare depicted in Figures 3.11 (a), (c) and (e), while the corresponding spectra are given inFigures 3.11 (b), (d) and (f). The experimental results are well matched by the simulationspresented in Section 3.3.3. It is also noticed that the maximum peaks of the spectra remainunchanged, even though the ripples, which haves the sizes 2.4A, 1.4A, and 0.9, respectively,have been reduced greatly.

Figure 3.12 shows the cases when the boost converter operates in chaotic mode. It is worth toemphasize that the case presented in [25] is equivalent to that with Ilow = 0A. By comparingthe results depicted in Figure 3.12, an improvement of EMI suppression is clearly demonstratedwith an increase of Ilow, while a large reduction of the ripples can be achieved at the same time.This is also consistent to the observations in Section 3.3.1 that there is no obvious relationshipbetween ripple magnitude and background spectrum.

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Figure 3.10: Circuit board of the chaotic peak current mode boost converter

3.5 Summary

This chapter proposed a chaotic parameter modulation, i.e., a novel chaotic peak current modeboost converter. This method cannot only reduce EMI but can also effectively restrain theripples. A current mapping function has been derived, with which its chaotic behaviour hasbeen analysed. Further, simulations and experiments have been carried out to illustrate theeffectiveness of the proposed design in reducing EMI and restraining the converter’s outputripples. Diss

erta

tion

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(a) i(t) with Ilow = 1.6A (b) spectrum of (a)

(c) i(t) with Ilow = 2.6A (d) spectrum of (c)

(e) i(t) with Ilow = 3.1A (f) spectrum of (e)

Figure 3.11: Current waveforms and corresponding spectra in periodic mode for three differentcases

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(a) i(t) with Ilow = 0A (b) spectrum of (a)

(c) i(t) with Ilow = 3A (d) spectrum of (c)

(e) i(t) with Ilow = 3.2A (f) spectrum of (e)

Figure 3.12: Current waveforms and corresponding spectra in chaotic mode for three differentcases

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4 Chaotic Pulse Width Modulation 33

Chapter 4

Chaotic Pulse Width Modulation

Since pulse width modulation (PWM) control is the most common and important controlmethod for DC-DC converters, combining chaos control and PWM can distribute the harmonicsof DC-DC converters continuously and evenly over a wide frequency range, thereby reducingthe EMI. Simulation and experimental results are given to illustrate the effectiveness of theproposed chaotic pulse width modulation (CPWM), which provides a good example of applyingchaos theory in engineering practice.

4.1 Introduction

It has been suggested in Chapter 3 and the literature [27, 34] that in a DC-DC converter chaoscontrol by parameter modulation can be used to reduce EMI. Although chaos is very desirablein this case, there exist some by-products that need to be eliminated. The most prominentone is the difficulty of design, because the circuit may run out of chaos when its power supplyor load fluctuate. As these fluctuations are normally unpredictable, this kind of chaos controlonly suits DC-DC converters running under stable working condition. The second one are largeoutput ripples. Although in Chapter 3 some efforts have been devoted to this problem, thecontrol method proposed in Chapter 3 is only available for the controls with current referenceor voltage reference.

PWM control is the most popular and widely used control method for DC-DC converters,and it can mainly be divided into three parts, sampling and error amplifying, PWM carrier,and PWM signal output. Due to the cluster harmonics around the multiples of the carrierfrequency in output waveforms, for a DC-DC converter with PWM control is difficult to satisfythe more and more strict international EMC standards. EMI filters are always needed asauxiliary circuits together with DC-DC converters, which largely increase the products’ costand weight. Chaos provides a new way to reduce EMI for DC-DC converters. Therefore,in this chapter, combining chaos with PWM, named chaotic PWM control, is proposed byreplacing the periodic PWM carrier by a chaotic one. The harmonics of DC-DC converters willthen be distributed continuously and evenly over a wide frequency range. Consequently, theEMI can be controlled and reduced, and the EMC can be improved. Furthermore, the outputwaveforms and spectral properties of the EMI will be analysed in Section 4.3 as the carrierfrequency changes with different chaotic maps, and an analysis of the chaotic PWM converteras the carrier amplitudes change is conducted in Section 4.4. Both simulation and experimentalresults are given to illustrate the effectiveness of the proposed CPWM. This provides a goodexample of applying chaos theory in engineering practice.

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34 4 Chaotic Pulse Width Modulation

4.2 Design Considerations

The output waveform of a DC-DC converter controlled by traditional PWM, as introduced in[40], is constituted of many harmonic components. The distribution of harmonics is influencedby the carrier. Carrier frequency f∆ and carrier amplitude A∆ are invariant under traditionalPWM, thus the spectrum has biggish peaks close to the carrier frequency and its multiples.This makes it difficult for the DC-DC converter to satisfy the international EMC standards.Conventionally, filters are used to reduce EMI of DC-DC converters. However, due to therelationship between harmonics and signals, filters do not only restrain the harmonics but alsothe effective current signals. Moreover, each filter can only restrain EMI in a certain, relativelynarrow frequency band. The existence of a number of biggish peaks of the spectrum undertraditional PWM makes it difficult to design filters for DC-DC converters. It is remarked thatthe pulse width generated by traditional PWM is determined by the intersection of the carrierand modulation waves. The carrier wave can have triangular or sawtooth shape.

Figure 4.1: Chaotic PWM boost converter

It is desirable for DC-DC converters to eliminate EMI without using filters. Since the distrib-ution of harmonics is influenced by the carrier and the chaotic behaviour of DC-DC converterscan be used to reduce EMI, chaotic f∆ or chaotic A∆ are used to distribute the harmonics con-tinuously and evenly over a wide frequency range. Although the total energy is not changed,the peaks of the harmonics are reduced, thus EMI is restrained. Therefore, in order to ob-tain chaotic f∆ or chaotic A∆, chaotic PWM (CPWM), as shown in Figure 4.1, is proposed,analysed, and tested.

4.3 CPWM with Varying Carrier Frequencies

CPWM adopts triangular or sawtooth waves to modulate, but its carrier period T ′∆ changes

according to

T ′∆ =

xi

Mean(x)T∆ (4.1)

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(a) µ = 0.7 (b) µ = 0.8 (c) µ = 0.9

Figure 4.2: Chaotic sequences generated by the tent map

where T∆ is the invariant period, xi, i = 1, 2, . . . , N , a chaotic sequence is denoted by x =x1, x2, . . . , N, and Mean(x), the average of the sequence, is defined as

Mean(x) =

N∑i=1

xi

N.

For simplicity, the tent map is employed here to generate chaotic sequences [35], which isdescribed as

f(xn) =

2µxn if xn 6 0.5,2µ(1− xn) if xn > 0.5,

(4.2)

with xn ∈ [0, 1]. Note that when 0.5 < µ < 1, |f ′(xi)| > 1. Its Lyapunov exponent is

λ = limn→∞

1

n

n∑i=1

ln |f ′(xi)| = ln (2µ) > 0. (4.3)

The positive Lyapunov exponent implies that the system is chaotic. Figure 4.2 shows thechaotic sequences of the map at µ = 0.7, µ = 0.8 and µ = 0.9, respectively. Therefore, chaoticPWM is realised by properly tuning the period length of the carrier.

4.3.1 Simulations

For practical evaluation of CPWM, here a boost converter is taken as test-bed and is describedas the main circuit in Figure 4.1. The values of its parameters are chosen as VI = 10V ,L = 1mH, C = 330µF , RL = 15Ω, Iref = 2A, and T∆ = 0.0001s. Then, the modulationwaves, carrier and PWM waves of the boost converters controlled by traditional PWM and byCPWM at µ = 0.7, 0.8, and 0.9 are simulated as shown in Figures 4.3 and 4.4, respectively.The corresponding spectra are shown in Figure 4.5.It is seen in Figure 4.5 that the peak values of the spectrum generated by traditional PWM(Figure 4.5 (a)) may lead to exceed the limits set in EMC standards, while the spectrumgenerated by CPWM distributes continuously and evenly over a wide frequency range (Fig-ures 4.5 (b)–(d)), which satisfies the international EMC standards. Furthermore, by CPWMthe average switching frequency has been greatly reduced (Figure 4.4 (a)(c)(e)) as comparedwith that by traditional PWM (Figure 4.3 (a)). This reduces the dissipation of DC-DC con-verters and enhances their stability. Meanwhile, it can be seen that increasing µ results in someslightly larger ripples of the output waveforms and smoother spectra under CPWM. Therefore,an appropriate µ needs to be determined to reach a good trade-off between ripples and spectrain practice.

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(a) PWM control signals (b) Current wave (upper), andoutput voltage wave (lower)

Figure 4.3: PWM control signals and output waveforms of the boost converter controlled bytraditional PWM

Spectral Characteristics

Now, the logistic map and the shift map are employed to generate chaotic sequences, and thentheir spectral characteristics are compared to that of the boost converter controlled by CPWMwith the tent map. The logistic map is defined as

f(xn) = 1− µx2n, (4.4)

where x ∈ [−1, 1] and µ = 2.0, and the shift map as

f(xn) =

µ

(xn − 1

2

)+ 1, if 0 6 xn 6 1

2,

µ(xn − 1

2

), if 1

2< xn 6 1,

(4.5)

where x ∈ [0, 1] and µ = 1.8.

The output waveforms and spectra of the currents in the DC-DC converter controlled by CPWMemploying the logistic map and the shift map are shown in Figure 4.6. Comparing the spectrain Figure 4.6 with that in Figure 4.5(d), it is seen that the current spectra with the logisticand shift maps are better than that of the tent map. Comparing the output waveforms showsthat using the tent map leads to the least ripple. This means that various chaotic maps can beused to design CPWM just dependent on the application of interest in practice.

4.3.2 Experiments

To verify the simulation results, an experiment is conducted. The block diagram of the exper-imental configuration is drawn in Figure 4.7.

The experimental results of using the logistic map are shown in Figures 4.8 – 4.10, which appearto be consistent with the simulation results. Furthermore, in Figures 4.8 and 4.9 it is seen thatthe peak values of the spectra in the low frequency band obtained by CPWM are reduced by10% in comparison with those yielded by traditional PWM.

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(a) µ = 0.7 (b) µ = 0.7

(c) µ = 0.8 (d) µ = 0.8

(e) µ = 0.9 (f) µ = 0.9

Figure 4.4: Control signals (left column) and current and output voltage waveforms (rightcolumn) of the boost converter controlled by CPWM

4.4 CPWM with Varying Carrier Amplitudes

CPWM also adopts triangular or sawtooth waves to modulate, but its carrier amplitude A′∆

changes according to

A′∆ = 1 + λ

xi

Mean(x)A∆, (4.6)

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(a) By traditional PWM (b) By CPWM at µ = 0.7

(c) By CPWM at µ = 0.8 (d) By CPWM at µ = 0.9

Figure 4.5: Spectra of the current in the boost converter controlled by traditional PWM andCPWM, respectively

where A∆ is the invariant amplitude, xi, i = 1, 2, . . ., a chaotic sequence, x = x1, x2, . . ., λ themodulation factor of the amplitude, which is determined as required in practice, and Mean(x)the average of the sequence as defined in Section 4.3.

4.4.1 Simulations

The same converter with the same circuitry parameters as used in Section 4.3 is employed (seeFigure 4.1). Here, when A∆ = 1.5V , the same output voltage of the boost converter withvarying carrier frequency can be obtained. The logistic map is adopted to generate chaoticsequences. Now, the output characteristics and spectra of the boost converter at λ = 0,λ = 0.4, and λ = 0.8 are to be simulated.At λ = 0, the output waveforms and PWM control signals are the same as the ones in Figure 4.3;therefore, only the output waveforms and spectra at λ = 0.4 and λ = 0.8 are given here. Theoutput waveforms of the boost converter controlled by CPWM at λ = 0.4 and λ = 0.8 areshown in Figure 4.11 (a) and (b). Figure 4.11 (c) and (d) show the inductor current spectra ofthe boost converter at λ = 0.4 and λ = 0.8.It is seen in Figure 4.11 that under CPWM control with varying amplitudes the ripples ofthe output waveforms are relatively larger than under CPWM control with varying carrierfrequencies. However, their spectra are similar. It is also seen that as λ increases, the ripples ofthe output waveforms increase, but the spectra remain unchanged. Thus, if the spectra alreadysatisfy the EMC standards, λ should be as small as possible in practice.

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(a) Output waveformwith logistic map

(b) Output waveformwith shift map

(c) Spectrum of currentwith logistic map

(d) Spectrum of currentwith shift map

Figure 4.6: Output waveforms and spectra of currents in the boost converter controlled byCPWM

Figure 4.7: Block diagram of experimental set-up

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(a) (b)

Figure 4.8: Output waveforms and spectra of input current (a) and output voltage (b) of theboost converter controlled by traditional PWM

(a) (b)

Figure 4.9: Output waveforms and spectra of input current (a) and output voltage (b) of theboost converter controlled by CPWM

(a) Periodic carrier wave (b) Chaotic carrier wave

(c) Periodic drive wave (d) Chaotic drive wave

Figure 4.10: Comparison of two kinds of carrier waves and drive waves

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(a) Output waveform at λ = 0.4 (b) Outt waveform at λ = 0.8

(c) Current spectrum at λ = 0.4 (d) Current spectrum at λ = 0.8

Figure 4.11: Output waveforms and current spectra of the boost converter controlled by CPWMat λ = 0.4 and λ = 0.8

4.4.2 Experiments

Likewise, experimental results obtained by using the logistic map at λ = 0.4 are given to testifythe simulation results. It is shown in Figure 4.12 that they are consistent.

4.5 Summary

Chaotic PWM control has been proposed in this chapter. According to the results of simula-tions and experiments, it can be observed that the output spectra of DC-DC converters withCPWM control can be distributed evenly over a wide frequency band, thus reducing EMI.Some important problems, such as long-time stability or average value estimations of inputand output variables of DC-DC converters controlled by CPWM, remain to be answered inChapter 8.

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(a) Periodic carrier (b) Chaotic carrier

(c) Chaotic drive waveform (d) Output waveform and its spectrum

Figure 4.12: Experimental waveforms of the boost converter with varying carrier amplitudesDisser

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5 Analogue Chaotic PWM 43

Chapter 5

Analogue Chaotic PWM

CPWM control can widely be applied in DC-DC converters and is very effective to suppressEMI. However, the high cost of digitally generated chaotic carriers used in Chapter 4 greatlyimpedes the applicability of this control. Thus, a novel method to generate a chaotic carrier inanalogue form using chaotic oscillators is to be proposed, analysed, simulated, and experimen-tally validated in this chapter.

5.1 Introduction

Generally, chaotic carriers can be generated in digital or analogue ways. The advantages of digi-tally generated chaotic carriers are that digital chaotic signals are accurate, and that frequencyand amplitude of the carriers can easily be adjusted by programming the digital processorswithout changing their external interface circuits; while the disadvantages are also obvious,namely, that the regulable frequency range of chaotic carriers generated by digital processorsis dependent on the speed of Digital Signal Processors (DSP) or other digital computers suchas single-chip ones, that sometimes external interface circuits are necessary, and that the costsof digital chaotic carriers are high. On the other hand, the costs of analogue chaotic carrier aremuch lower; and the regulable frequency range can be much broader by changing resistance andcapacitance of the analogue chaotic carrier circuits suitable to function in high-frequency DC-DC converters. Furthermore, numerous existing chaotic oscillators can be employed to designanalogue chaotic carriers. However, analogue chaotic carriers cannot be adjusted as accuratelyas digital ones due to the non-ideal performance characteristics of the components, and theirhardware implementation is a little more complex, since chaotic carriers are not realised byprogramming, but by components.

It is known that DC-DC converters always operate with high frequencies, and that the frequencyof chaotic carriers must as high as of the DC-DC converters. Therefore, if a digital chaotic carrierwere used, the speed of the generating DSP, single-chip or or other computer would be requiredto be correspondingly high, resulting in very high cost. Even so, existing processors can hardlysatisfy the practical requirements. Instead, analogue chaotic carriers can be employed, leavingthe problem of how to design them.

Actually, in [57] a design method is proposed using three switches (a main switch and twoauxiliary ones), leading to large switching loss. Moreover, the chaotic generator circuit describedin [57] can generate one kind of chaotic signals, only.

In this chapter, only one switch is adopted in generating a chaotic carrier by porting one ofthe numerous existing chaotic oscillator circuits, i.e., Chua’s chaotic oscillator, which rendersthe circuit design more flexible. Another contribution of this chapter is to propose a transformto increase the frequency of the chaotic oscillator to a value required. Then, simulations and

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44 5 Analogue Chaotic PWM

experiments will be conducted to verify the effectiveness of the novel analogue chaotic carrierin suppressing EMI, which refers to conducted EMI here and throughout the dissertation.

5.2 Analogue Chaotic Carrier

Analogue carriers used for DC-DC converters, such as triangle waves and sawtooth waves, aregenerated by charging and discharging a capacitor. The proposed chaotic analogue carrier usesthe same principle, and employs a chaotic signal v′chaos generated by a chaotic oscillator asshown in Figure 5.1.

VCC

R5

C6S7

vc

Drivercircuitvc

VCC

R1

R2

Vlow

Vupp

Chaotic oscillatorcircuit

Comparator

Comparator

Control part Main circuit

VCC

R3

R4

Sum circuit

Proportional circuit

R5’

Vu

vchaos

v'chaos

S

R Q

Q

SET

CLR

Figure 5.1: A chaotic sawtooth carrier generator

5.2.1 Circuit Design

The circuit diagram of the analogue chaotic carrier is drawn in Figure 5.1, which can generateboth chaotic sawtooth and chaotic triangle waveforms. It is shown in Figure 5.1 that the lowerlimit of the chaotic carrier, Vlow, is determined by R1 and R2, while its upper limit, Vupp, byVu and vchaos. The latter is obtained from the output voltage v′chaos of the chaotic oscillatorcircuit via a proportional modulation. According to the characteristic table of R-S flip-flopin Table 5.1, the chaotic carrier circuit operates in the following way. Initially, vc is zero andvc < Vlow < Vupp. Then, R = 1 and S = 0, which result in Qn+1 = 1, the switch S7 turnson, and C6 will be charged through R5 and R′

5 by V CC. When vc > Vlow and vc < Vupp, onehas that R = 1 and S = 1. In terms of Table 5.1, it holds Qn+1 = Qn, which means that theswitch remains “on” until vc arrives or exceeds Vupp. When S = 1, R = 0 and Qn+1 = 0, theswitch turns off, and C6 begins to discharge through R′

5 until vc ≤ Vlow. Thereafter, a newcircle begins.When R′

5 is very small or close to zero, C6 discharges very fast, and the output voltage of C6 isclose to be a sawtooth waveform. If R′ is equal to or larger than R, then a triangle waveformappears. Based on circuit theory, the frequency of the chaotic carrier can be calculated by thefollowing expression

fcn =1

tncharge + tndischarge

, (5.1)

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Table 5.1: Characteristic table of RS flip-flop

R S Qn+1

0 1 01 0 11 1 Qn

0 0 unstable

where tncharge = −(R5 + R′5)C6 ln(1− Vupp−Vlow

V CC−Vlow) and tndischarge = −R′

5C6 ln(Vlow

Vupp).

In practice, a reference frequency fC always needs to be defined, since the design of inductorand capacitor in DC-DC converters is based on a certain frequency. In this chapter, fC isdefined as the frequency when Vupp = Vu. Normally, vchaos ∈ (−M, M), where M is a positivereal number, so that fcn will fluctuate around fC , and the fluctuating range is dependent onv′chaos and the proportional circuit.

Due to the chaotic characteristics of Vupp, fcn = 1Tn

varies chaotically, as shown in Figure 5.2.Therefore, it is called chaotic carrier.

Vu

Vlow

Vupp

vchaos

Vupp=Vu+vchaos

vc

t / s

vc / V

Tn

Figure 5.2: Chaotic carrier

5.2.2 Chaotic Oscillator

In recent decades, chaotic oscillators have widely been investigated [8, 14, 17], and are exten-sively applied in many fields, such as communication security and industrial mixing. Here,chaotic oscillators are used for the first time in PWM control of DC-DC converters to reduceEMI.

Among the existing chaotic oscillators, Chua’s, Lorentz’s, and Chen’s oscillators are most wellknown. In this section, Chua’s oscillator is adopted due to its simplicity and maturity. Fig-ure 5.3 shows Chua’s oscillator, where NR is Chua’s diode (cp. Figure 5.4), and VR and iRsatisfy the relationship,

iR = f(VR) = GbVR +1

2(Ga −Gb)(|VR + E| − |VR − E|). (5.2)

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R0

L1

C2 C1

+

-

+

-V2 V1

R iR

i3

NR

-

VR

+

Figure 5.3: Chua’ oscillator circuit

-EE

iR

VR

Gb

Ga

Figure 5.4: Typical iR-VR characteristic of Chua’s diode

Chua’s oscillator can be described by the following differential equations;dV1

dt= 1

C1[(V2 − V1)G− f(V1)],

dV2

dt= 1

C2[(V1 − V2)G + i3],

di3dt

= − 1L1

(V2 + R0i3),

(5.3)

where G stands for the reciprocal of Ohm.For the case R = 1858Ω, R0 = 0Ω, L1 = 18mH, C1 = 10nF , C2 = 100nF , E = 1.075V ,Ga = −757.58µS, and Gb = −409.09µS the phase portraits of the chaotic oscillator are shownin Figure 5.5.It is noted here that when a chaotic oscillator is used for a chaotic carrier, but the frequency ofthe existing chaotic oscillators cannot follow the required switching frequency, these oscillators’frequencies should be increased by adjusting their circuits’ parameters. To maintain the samechaotic characteristics of these oscillators, the relationship between the parameters and thefrequencies should be found. For Chua’s chaotic oscillator, to increase the frequency of vchua

from fv to Nfv, one just needs to apply the transform t = Nτ . To this end, the differentialequations (5.4) can be re-written an

dV1

dτ= 1

C1/N[(V2 − V1)G− f(V1)],

dV2

dτ= 1

C2/N[(V1 − V2)G + i3, ]

di3dτ

= − 1L1/N

(V2 + R0i3).

(5.4)

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(a) Phase portrait of V1 - V2 (b) Phase portrait of V1 - i3

Figure 5.5: Phase portraits of Chua’s oscillator

Consequently, the frequencies of outputs, such as vchaos, will be increased N times when theparameters C1, C2 and L1 are replaced by C1/N , C2/N and L1/N . The approach is alsoapplicable to other chaotic oscillators. However, it is remarked that, in practice, the transformedparameters should be adjusted by trial and error, because circuit components are normally notideal.

5.3 Analogue Chaotic PWM

5.3.1 A Boost Converter

Here, an analogue chaotic carrier is to be embedded in a PWM boost converter as shown inFigure 5.6, because it is one of the basic topologies of DC-DC converters and very popular inmany practical circuits, such as power factor correction (PFC) circuits, power inverters, and soon. The switch S, the input inductor L, the freewheel diode D, and the output filter capacitorC constitute the main circuit of the boost converter; while RL representing a resistive load, thesampling circuit for iL, the reference circuit for Iref , a operational amplifier, a comparator, anda carrier (periodic carrier or chaotic carrier) form the PWM control part as shown in Figure 5.6.

5.3.2 Simulations

Two different control methods, including traditional PWM, i.e., PWM with periodic carrier,and chaotic PWM, i.e., PWM with chaotic carrier, are now simulated and compared in termsof their performance on suppressing ripple and EMI, and improving efficiency.

The circuit diagram of the boost converter is shown in Figure 5.6, where VI = 10V , L = 1mH,C = 10µF , R = 200Ω and fC = 10KHz. For the control part, Vlow = 0V , Vu = 2V andIref = 1A are set.

The periodic carrier can easily be generated as Vupp = Vu = 2V (see Figure 5.1). In orderto generate the chaotic carrier, just assume that the parameters of the embedded chaoticoscillator assume the values as given in Section 5.2.2, and V2 = v′chaos. If v′chaos is proportionallymodulated within (−0.3, 0.3), then one has Vupp ∈ (1.7, 2.3). The periodic and the chaoticcarriers generated are shown in Figures 5.7(a) and 5.7(b), respectively.

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Figure 5.6: A PWM boost converter

(a) (b)

Figure 5.7: Periodic carrier (a) and chaotic carrier (b) with Chua’s oscillator

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It is remarked that due to the chaotic carrier’s frequency being around 10kHz, the frequencyof Chua’s chaotic oscillator with the above selected parameters should be accelerated 104 timesbased on its original frequency, which can be estimated by observing the frequency with biggestamplitude in its spectrum revealed by fast Fourier transform (FFT). With the transformationt = 104τ , the FFT spectrum of Vupp is shown in Figure 5.8. It is obvious that the frequency ofChua’s oscillator can now catch up with the switching frequency of the boost converter.

0 10 20 30 40 50 60 70 80 90 100

-20

-10

0

10

20

Frame: 24 Frequency (kHz)

Am

plitu

de

Figure 5.8: FFT spectrum of Vupp

Comparison results for the output waveforms, the phase portraits, and the input current spectraof the boost converters under PWM control using the chaotic carrier (Figure 5.7(b)) and theperiodic carrier (Figure 5.7(a)), respectively, are shown in Figures 5.9 – 5.11 and in Table 5.2.It is remarked that the current and voltage overshoots are almost the same, the current andvoltage ripples increase slightly, the efficiency is improved, and EMI is greatly reduced, whenthe periodic carrier is replaced by the chaotic one in the PWM control. In summary, thechaotic carrier does not change the DC-DC converters’ characteristics, such as the basic outputwaveforms and stability, however, it improves EMC considerably according to Figure 5.11,especially in the low frequency band.

Table 5.2: Performance comparison of the boost converter with different control methods

Parameters for comparison Traditional PWM Chaotic PWMcurrent

overshoot(A) 1.064 1.053voltage

overshoot(V) 16.70 16.75current

ripple(A) 0.2607 0.3404voltage

ripple(V) 0.7326 1.0592efficiency(%) 91.78 93.45

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(a) Waveforms of the traditional boost converter (b) Waveforms of the chaotic boost converter

Figure 5.9: Output waveforms of the boost converter

(a) Periodic phase (b) Chaotic phase

Figure 5.10: Phase portraits of the input current and output voltage when the boost converteroperates in periodic and chaotic modes

5.4 Experiments

To further verify the effectiveness of the analogue chaotic PWM, also an experiment and hard-ware were designed. First, as the chaotic oscillator’s core, the circuit design of Chua’s diode isintroduced.

5.4.1 Chua’s Diode

So far, many methods have been reported to build Chua’s diode [18], among which the mostpopular one is shown in Figure 5.12, and its parameter design is given in [31]. Here, theparameters for Chua’s diode are chosen as Rd1 = 2.4KΩ, Rd2 = 3.3KΩ, Rd3 = Rd4 = 220Ω,and Rd5 = Rd6 = 20KΩ.The other parameters of Chua’s oscillator in the experiment are L1 = 2.2mH, C1 = 4.7nF ,C2 = 500pF , and R = 1.75KΩ. The parameters for the main circuit of a chaotic sawtoothgenerator are Rs = 1KΩ, R′

s = 3.9Ω, Cs = 22nF , and V CC = 5V . For the main circuit of the

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(b) Spectrum of inductor current of chaotic hard switching boost converter

Figure 5.11: Spectra of inductor current of the boost converter

boost converter and the PWM control part, assume that VI = 10V , L = 680mH, C = 10µFand RL = 200Ω; Vlow = 0V , Vu = 2.5V , Iref = 1A, and fC ≈ 60KHz. With these parametersettings, the boost converter will operate in current continuous mode (CCM) with a duty cycleof around 40%.Circuit diagram, printed circuit board, and an experimental board of the PWM boost converterare shown in Figure 5.15. The boost converter can be induced to operate in periodic or chaoticmode through jumpers J7 and J8, which have been marked on Figures 5.13 – 5.15.

5.4.2 Experimental Results

The waveforms of periodic and chaotic carrier are given in Figure 5.16, and the output voltageswith ripple measurements of the PWM boost converter with two kinds of carriers are providedin Figure 5.17.It is seen from Figure 5.17 that the ripple increases by 120mV as the periodic carrier is replacedby a chaotic one, while the efficiency of the boost converter is improved from 86.40% to 89.43%.In this experiment, the EMC standard GB9254-1998 CE (AV class A and QP class A) is applied,the measurement bandwidth is 9kHz, the frequency step 5kHz, the attenuation 10dB, and thefrequency range 0.15–30MHz.The measurement results of the boost converter’s EMI with the periodic and chaotic carriers

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+-

+-

A1

A2

Rd1 Rd2

Rd3

Rd4

Rd5

Rd6

Chua’s Diode

iR

Figure 5.12: Chua’ Diode

Figure 5.13: Circuit diagram of the boost converter

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Figure 5.14: Printed circuit board of the boost converter

Figure 5.15: Experimentation board of the boost converter

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(a) (b)

Figure 5.16: Periodic carrier (a), and chaotic carrier (b)

(a) (b)

Figure 5.17: Ripples of the output voltage as the boost converter operates: (a) in periodicmode and, (b) in chaotic mode

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Figure 5.18: EMI of the periodic PWM boost converter

are given in Figures 5.18 and 5.19, respectively, which show that applying the chaotic carrierin reducing EMI is much more effective in the low frequency band, which is consistent with thesimulation results.

Figure 5.19: EMI of the chaotic PWM boost converter

5.5 Summary

This chapter is concerned with analogue chaotic PWM, where the key is to design an analoguechaotic carrier using chaotic oscillators. According to the simulation and experimental results,although the ripple in the output voltage is slightly increased by adopting the chaotic carrierinstead the periodic one, the efficiency of the boost converter is much improved and the EMI isdistributed much smoother on the frequency band, which allows the boost converter to bettersatisfy the EMC standards.

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56 6 A Chaotic Soft Switching PWM Boost Converter

Chapter 6

A Chaotic Soft Switching PWM BoostConverter

So far, we have shown that CPWM can suppress EMI significantly by spreading the spectraover a wide frequency band. Moreover, EMI is mainly caused by rapid di/dt and dv/dt, whichcan be reduced by the soft switching technique. Therefore, in this chapter, a novel methodbased on CPWM and soft switching control is proposed for the reduction of the EMI in DC-DC converters. Here, a digital generator of the chaotic carrier is proposed based on a chaoticmapping and a sawtooth wave generator, which convert the periodic sawtooth wave into achaotic one. Simulation results show that the EMI of the DC-DC boost converter is muchreduced due to the total energy more evenly spreaded over the frequency band and reducedenergy loss. It is also found that the efficiency of the DC-DC boost converter is improved ascompared with the hard and soft switching PWM controls.

6.1 Introduction

Since CPWM control cannot directly reduce the rapid change rate of voltage and current,another earlier proposed, more popular and practical technique, i.e., soft switching, will beintroduced. The technique of soft switching was first presented in [15] and was rapidly developedin recent years [19, 21, 65]. The concept is to open and close the switch at zero current or zerovoltage to alleviate the high rates of changes in voltage and current so that EMI can be reduced.Thus, the switching loss is reduced, which implies that the energy loss is also reduced, resultingin improved efficiency.

CPWM has been proposed and simulated [7, 37, 50, 69, 70, 72, 75], but there are no hardwareimplementations. In addition to the hardware implementation of the analogue chaotic carriergiven in Chapter 5, an implementation of a digital chaotic carrier generated by a sawtoothgenerator, whose period length is governed by a chaotic mapping, will be detailed in thischapter.

Further, this chapter is concerned with combining CPWM with soft switching in order not tospread the energy distribution over the whole frequency band (thus reducing the peaks in thespectrum), only, but also to reduce the switching loss or energy loss, such that EMI cannotonly be greatly reduced, but that the efficiency is improved, too.

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6 A Chaotic Soft Switching PWM Boost Converter 57

6.2 Circuitry and Control

6.2.1 Circuit Description

The chaotic soft switching PWM boost converter is depicted in Figure 6.1, where the switchS1, the inductor L1, the diode D3, and the capacitor C2 form the main circuit of the boostconverter, and R represents a resistive load. The soft switching of S1, which was proposed in[2], is governed by the auxiliary circuit consisting of inductors L2 and L3, diodes D1 and D2,and capacitor C1. Usually, the inductances of L2 and L3 are much smaller than that of L1, andthe capacitance of C1 is much smaller than that of C2.

Figure 6.1: Chaotic soft switching PWM boost converter

It is possible to classify the operations of the boost converter into seven different modes basedon the principle of soft switching. They are described briefly as follows (cf. [2] for details).According to [2], Vout and iL1 are assumed as constants V1 and I1 for Modes 1 and 2, and V2

and I2 for Modes 5 and 6, respectively, since iL1 is quite small in Modes 1, 2, 5 and 6.

Mode 1 (t ∈ [t0, t1))

Let the initial values of L2 and L3 be zeros, and C1 previously be charged to a value VC1(t0).Assume that the switch S1 is turned on when the current is zero at time t0, while the currentiL2(t) will then gradually rise and become I1 + iL3(t) at t1 when D3 turns off.

The equivalent circuit is shown in Figure 6.2(a), and the expressions for iL2(t), iL3(t) and VC1(t)can be derived as

iL2(t) =V1

L2

t,

VC1(t) = [V1 − VC1(t0)][1− cos ω1t]] + VC1(t0),

iL3(t) = [VC1(t0)− V1]sin ω1t

ω1L3

, (6.1)

where ω1 = 1√L3C1

.

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(a) mode 1

(b) mode 2 (c) mode 3

(d) mode 4 (e) mode 5

(f) mode 6 (g) mode 7

Figure 6.2: Circuits equivalent to the soft switching boost converter in different modes

Mode 2 (t ∈ [t1, t2))

Since D1 is off, the operations of this mode can be represented by the equivalent circuit shownin Figure 6.2(b). The capacitor C1 is to be completely discharged and VC1 eventually reacheszero at t2. Assuming that the initial values of L3, L2, and C1 are equal to iL3(t1), iL2(t1) + I1,and VC1(t1), respectively, evaluated at the end of Mode 1, one has

VC1(t) = −VC1(t1)(2− cos ω2t)−iL3(t1)

ω2C1

sin ω2t,

iL3(t) =VC1(t1)

ω2(L2 + L3)sin ω2t + iL3(t1) cos ω2t,

iL2(t) =VC1(t1)

ω2(L2 + L3)sin ω2t + iL3(t1) cos ω2t + I1, (6.2)

where ω2 = 1√(L2+L3)C1

.

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Mode 3 (t ∈ [t2, t3))

The equivalent circuit for this mode is shown in Figure 6.2(c), where the initial conditions ofiL2, iL3, and VC1 are iL2(t2), iL3(t2), and zero, respectively. From t2 to t3, the current iL3(t)drops and becomes zero at t3. The expression for iL3 is given by

iL3(t) =−VSL2

L1L2 + L2L3 + L3L1

t + iL3(t2). (6.3)

Mode 4 ( t ∈ [t3, t4))

The equivalent circuit for this mode is shown in Figure 6.2(d). At t4, the end of this operationalmode, iL1(t) and Vout(t) attain the values I2 and V2, respectively, and the switch S1 is turnedoff. Hence, one has

iL1(t) = iL2(t) =VS

(L1 + L2)t + I1,

Vout(t) = V1e− 1

RC2t. (6.4)

Mode 5 (t ∈ [t4, t5))

For this mode, after S1 turns off, the current iL2(t) drops and reaches zero at t5. The equivalentcircuit is shown in Figure 6.2(e), where the initial condition of iL2 is I2. The expressions foriL2, iL3, and VC1 are then obtained as

VC1(t) = V2(1− cos ω3t) +I2

ω3C1

sin ω3t,

iL2(t) =L2

L2 + L3

[V2C1ω3 sin ω3t− I2(1− cos ω3t)] + I2,

iL3(t) =L2

L2 + L3

[V2C1ω3 sin ω3t− I2(1− cos ω3t)], (6.5)

where ω3 = 1qL2L3

(L2+L3)C1

.

Mode 6 (t ∈ [t5, t6))

In this mode, the current iL3 decreases and becomes zero at t6, in terms of the equivalent circuitgiven in Figure 6.2(f). The expressions for iL3 and VC1 can be derived as

iL3(t) =VC1(t5)− V2

L3ω1

sin ω1t + iL3(t5) cos ω1t,

VC1(t) =[VC1(t5)− V2][cos ω1t− 1]− iL3(t5)

ω1C1

sin ω1t. (6.6)

Mode 7 (t ∈ [t6, t7))

The last mode is under the conditions of having zero iL2 and zero iL3. Figure 6.2(g) depictsits equivalent circuit, which is also the normal mode of the boost converter. At the end of thismode or at t7, S1 is turned on at zero current, the inductor current iL1 will reach I1 and Vout

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will reach V1. Therefore, one has

Vout(t) = e−αt[A sin ω4t + B sin ω4t] + VS,

iL1(t) =Vout(t)

R+ e−αt

[(−BC2α + AC2ω4) cos ω4t

−(AC2α + BC2ω4) sin ω4t], (6.7)

where α = 12RC2

, ω4 = 1√L1C2

, A = I2ω4C2

− V2

Rω4C2+ α(V2−VS)

ω4and B = V2 − VS.

6.2.2 Chaotic Soft Switching PWM Control

With traditional PWM control, the carrier frequency fC is invariant and has biggish peaksclose to the carrier frequency or its multiples in the spectrum, making it difficult for the DC-DC converters to satisfy the international standards for Electromagnetic Compatibility (EMC).The problem can be solved by using CPWM control [7, 37, 50, 69, 70, 72], in which a chaoticcarrier is integrated. The reason is that the chaotic carrier can distribute the spectrum con-tinuously and evenly over a wide range of frequencies. Although the total energy may not bealtered, the magnitudes of the peaks are reduced, thus EMI is restrained.

A Digital Chaotic Carrier

The chaotic carrier to be combined with soft switching can be analogue or digital dependingon the application of interest. The design of an analogue chaotic carrier has been introduced inChapter 5 Therefore, this subsection just introduces the design of an applicable digital chaoticcarrier.

Chaotic mapping

xn=u u.ß.TC + TCXn-1

Sawtooth generator

T’nC

Pulse signal

, the given samplesin each period length

T’nC /N

Vlow+(Vupp-Vlow)(N-1)/(N-1)=Vupp

Vlow+(Vupp-Vlow)(N-2)/(N-1)

n=N?

Vlow+(Vupp-Vlow)(N-n)/(N-1)

Yes

No

Vlow+(Vupp-Vlow)/(N-1)Vlow

Figure 6.3: Generation of chaotic carrier

The diagram of the proposed design is depicted in Figure 6.3, based on a chaotic mapping anda sawtooth generator. It is remarked that the chaotic carrier is being generated as the DC-DCconverter is running. The period length of the n-th sawtooth signal can be determined by thefollowing mapping:

T ′nC = xnβTC + TC , xn ∈ [−1, 1], β ∈ [0, 1), (6.8)

where TC is the fundamental frequency of the switch, which is a constant, xn is the n-th outputof the chaotic mapping, and β is a modulation factor, which can slightly modulate the trade-offbetween ripple and EMI.

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Here, the chaotic sequence xn is generated by the logistic mapping, which is described as

f(xn) = 1− µx2n, x ∈ [−1, 1]. (6.9)

where µ = 2 (at which the largest chaoticity is reached).Let TC = 10µs, the corresponding periodic and chaotic sawtooth carriers are shown in Figure 6.4for β = 0.05 and 0.2. It should be emphasised that some other chaotic mappings, such as theshift mapping or tent mapping, can also be applied.

(a) Periodic carrier (b) Chaotic carrier at β = 0.05

(c) Chaotic carrier at β = 0.2

Figure 6.4: Different carrier waveforms generated according to Figure 6.3

Experiment

The generation process of the chaotic carrier is shown in Figure 6.3 in form of a flow diagram. Anexperiment is conducted using a single-chip computer of type C8051F410, which can downloadprograms from a PC through a USB debug adaptor, as shown in Figure 6.5.

Figure 6.5: Illustration of the hardware connection

Let TC = 0.001s, Vupp = 1.5V , Vlow = 0V , and β = 0.2 and β = 0.5, respectively. Afterprogramming the single-chip computer with the method introduced in Section 6.2.2, the digitalchaotic carrier is obtained as shown in Figure 6.6.

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(a) β = 0.2 (b) β = 0.5

Figure 6.6: Digital chaotic carriers for different β

6.3 Simulations and Performance Evaluation

In this section, the proposed chaotic soft switching PWM boost converter is first simulated. Inorder to highlight its merits, then comparisons with hard switching PWM and soft switchingPWM are carried out focusing on their performance in ripple suppression and the improvementof EMC and efficiency.The chaotic soft switching PWM boost converter is shown in Figure 6.1, where VS = 10V ,L1 = 0.6mH, C2 = 10µF , R = 200Ω, Iref = 1A, and TC = 10µs. For the soft switchingcontrol, assume that L2 = L3 = 10µH and C1 = 10nF , while the components L1, L2, C1, D1,and D2 are not necessary for hard switching PWM control.

(a) Inductor current (b) Output voltage

Figure 6.7: Output waveforms of the boost converter with hard switching PWM

The inductor currents iL1 and output voltages Vout obtained for the three control methods areshown in Figures 6.7 – 6.10, respectively, and the corresponding power spectral densities (PSD)of the inductor currents are depicted in Figure 6.11. From this figure it is obvious that even avery small chaotic disturbance to a sawtooth carrier frequency can greatly improve the EMC.For ease of comparison, in Table 6.1 the results are compiled. It is observed that the ripples aresimilar; however, significant improvements of EMC and efficiency are observed, as comparedwith the results for hard and soft switching PWM.It is also observed that the overshoot of the inductor current is largest for hard switching PWM,but that its voltage overshoot is smallest. By comparing the results with soft switching andchaotic soft switching, the current overshoot and the voltage overshoot are found to be almostthe same.

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Figure 6.8: Output waveforms of the boost converter with soft switching PWM


Figure 6.9: Output waveforms of the boost converter with chaotic soft switching PWM atβ=0.05


Figure 6.10: Output waveforms of the boost converter with chaotic soft switching PWM atβ=0.2

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Figure 6.11: PSDs of inductor currents based on different control methods

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It should be noticed that the proposed chaotic soft switching PWM control can be tunedeasily. The modulation factor β can be tuned to reach any trade-off performance betweenripple magnitude and EMC. In addition, since a constant TC is used and the chaotic carrierfrequency is close to TC , the system parameters of the DC-DC converter can easily be obtainedaccording to the standard design procedures for the periodic mode, depending on the switchingfrequency. This is particularly obvious for the case that β is very small.

Table 6.1: Performance comparison of the boost converter with three different control methods

chaotic softParameters hard switching soft switching switching PWM

for comparison PWM PWM β=0.05 β=0.2

currentovershoot(A) 1.2765 -0.216 -0.216 -0.216

voltageovershoot(V) 18.9355 28.05 28.045 28.0532

currentripple(A) 0.0679 0.0669 0.0783 0.0840voltage

ripple(V) 0.0410 0.0481 0.0507 0.1105efficiency(%) 78.92 87.52 91.56 91.32

6.4 Summary

A chaotic switching PWM has been proposed in this chapter. It can improve EMI and efficiencyas compared with both hard and soft switching PWM, at the price of a small increase in ripplemagnitude. However, it is noted that this approach leads to a relatively complicated circuit,increasing cost and size of the final circuit. Fortunately, this problem can be alleviated by therapid development of large scale integration.

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66 7 Invariant Densities of Chaotic Mappings

Chapter 7

Invariant Densities of ChaoticMappings

This chapter is concerned with applying probability analysis to the chaotic mappings employedin the control of DC-DC converters. A computation method for the invariant density of a chaoticmapping is proposed by using the eigenvector method, which is to facilitate the accurate designof the DC-DC converter parameters. Moreover, the power spectral density of the input to aDC-DC converter and the average frequency of switching are deduced. Finally, some applicationexamples are given to illustrate the effectiveness of the method proposed.

7.1 Introduction

It is known that chaotic motion is an unstable, aperiodic behaviour within a bounded area,and that its long-term behaviour shows random-like characteristics, which can be studied usingprobability theory.

The invariant density is a basic and important characteristic of chaos. For a DC-DC converter,a one-dimensional mapping can be derived under some reasonable assumptions, which canthen be used to analyse the chaotic behaviour of the DC-DC converter. Several methods wereproposed to calculate the invariant densities of chaotic mappings used for DC-DC converters.However, these methods have their own drawbacks. For instance, the method presented in [25]is difficult to realise by computer due to the immense increase of calculation complexity asthe iteration of the mapping advances just slightly. Moreover, this method does not requirethe mapping to have a finite number of Markov partitions [38]. The method described in [71]uses the Frobenius-Perron operator equation to calculate invariant densities. Since it is wellknown that very few Frobenius-Perron operator equations of chaotic mappings can be solvedanalytically, this method can be applied in a few special cases, only.

In this chapter, a boost converter operating in a chaotic mode is described by a one-dimensionalmapping, based on which the chaotic mapping’s invariant density is then calculated usingthe eigenvector method. Comparing the invariant density of the chaotic mapping with itsphase portrait and its bifurcation diagram shows that the method is appropriate to calculateinvariant densities of the chaotic mappings used to control DC-DC converters. Furthermore,The calculation results can also be used to estimate the power spectral densities of the inputs,calculate the average switching frequencies of DC-DC converters, and accurately design thesystem parameters. Finally, simulation examples will be given to illustrate the effectiveness ofthe method.

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7 Invariant Densities of Chaotic Mappings 67

7.2 1-D Mapping for a Boost Converter

A one-dimensional mapping describing the behaviour of the boost converter in Figure 7.1 wasgiven in [25], and has the form of (7.1), with the inductor current i(t) sketched in Figure 7.2:

xn+1 = α(1− (xn mod 1)), (7.1)

where xn = tnTc

, α = VO

VI− 1, and tn =

(Iref−in)L

VI.

Figure 7.1: Peak current mode controlled boost converter

Figure 7.2: Current waveform iL(t) in a boost converter

For a boost converter, one has α > 0 due to VO > VI . It is easy to see by the Lyapunov exponentthat for α > 1 the sequence x0, x1, . . . , xn, . . . is chaotic within the range [0, α] [35]. Themapping (7.1) or its normalisation has extensively been studied, most notably by Renyi [41, 60].It is shown there by the Renyi transformation that the Frobenius-Perron equation — to bedefined in Section 7.3 — has an invariant density ρ(x), which is (1) absolutely continuous withrespect to the Lebesgue measure on the interval [0, α], as well as (2) ergodic and asymptoticallystable [41]. Due to the random-like characteristic of chaos, the eigenvector method derivedfrom probability theory and to be introduced in Section 7.4 is employed here to calculate theinvariant density of a chaotic mapping.

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7.3 Invariant Density of a Chaotic Mapping

Chaos is a kind of unstable behaviour in a bounded area. Its long-term behaviour showsrandom-like characteristics. Thus, it is possible to characterise it with probability theory, usingthe invariant densities ρ(x) of chaotic mappings. The term “invariant” means that the numberof orbit points of a chaotic mapping is invariant under the iterations of the mapping [35].For some simple cases, such as the parabola mapping, it is possible to represent the invariantdensities analytically. But for general cases, calculating ρ(x) requires to employ the Perron-Frobenious equation to obtain numerical solutions. The Perron-Frobenious equation is basedon “conservation of quantity” [35]. Figure 7.3 shows a non-linear function, where y has twoinverse images x1 and x2, namely, y = f(x1) = f(x2).

1x

y

2x

( )f x

1∆ 2∆

∆

x

Figure 7.3: Mapping of a non-linear function

Denote the small neighbourhoods of x1, x2, and y as ∆1, ∆2, and ∆, respectively, and thecorresponding probability densities as ρ(x1), ρ(x2), and ρ(y). According to the law of conservingquantity [35], one has

ρ(y)∆ = ρ(x1)∆1 + ρ(x2)∆2. (7.2)

When ∆1, ∆2, and ∆ are small enough, (7.2) can be recast as,

ρ(y) =ρ(x1)

|f ′(x1)|+

ρ(x2)

|f ′(x2)|, (7.3)

where f ′(x1) = ∆∆1

and f ′(x2) = ∆∆2

. If f(x) has more than 2 inverse images, there exist

xi = f−1(y), i > 2, and (7.3) can be denoted as

ρ(y) =∑

xi=f−1(y)

ρ(xi)

|f ′(xi)|. (7.4)

This is the so-called Perron-Frobenious equation, on which the calculation of invariant densitiesusing the eigenvector method can be based.

7.4 Eigenvector Method

For a non-linear function f(x), f : I → I, the interval I can equally be divided into Msegments. If M is large enough, ρ(x) can be regarded as “invariant” in each small interval.

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Figure 7.4: Partial sketch of a chaotic mapping

Then, ρ(x) can be expressed as M discrete values ρ(x1), ρ(x2), . . . , ρ(xM) or in vector formR = [ρ(x1), ρ(x2), . . . , ρ(xM)] [24].In Figure 7.4, pi,j is the transition probability of the j-th interval, and the transition probabilitymatrix is denoted by

P =

∣∣∣∣∣∣∣∣p1,1 p1,2 · · · p1,M

p2,1 p2,2 · · · p2,M

. . . . . . . . . . . . . . . . . . . . .

M,1 pM,2 · · · pM,M

∣∣∣∣∣∣∣∣ , (7.5)

in which the entries can be derived bypm,j = (xn − xm)/L,pm+1,j = (xs − xn)/L,pm+2,j = (xc − xs)/L,pi,j = 0 (1 ≤ i ≤ M, i 6= m, m + 1, m + 2).

(7.6)

Thus, it is easy to see that the calculation of the transition probability matrix P is easy as longas f(x) and M are known. From the definitions of P , R, and the Perron-Frobenious equation,P and R satisfy the following equality,

PR = R. (7.7)

It is concluded from (7.7) that R is the eigenvector of P with eigenvalue 1. Thus, the calculationof the invariant density is reduced to a calculation of the eigenvector of the transition probabilitymatrix P .

7.5 Invariant Density of the Boost Converter’s Chaotic

Mapping

For the above mentioned boost converter, according to Eqs. (7.1)–7.7, and by dividing theinterval [0, α] into M equal segments, the eigenvector R = [ρ(x1), ρ(x2), . . . , ρ(xM)] of P , i.e.,the invariant density of the chaotic mapping, can be calculated. For α assuming different values,simulation results are presented below.For α = 1.30, the phase portrait of the mapping is shown in Figure 7.5(a), and the correspondingbifurcation diagram and invariant density are given in Figures 7.5(b) and 7.5(c). From these

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(a) (b)

(c)

Figure 7.5: Chaotic mapping (a), bifurcation diagram (b), and invariant density (c) at α = 1.30

figures, it is obvious to see that they inosculate quite well. It is remarked that the invariantdensity reflects the operating status of the boost converter from a special perspective.It is seen from Figure 7.5(a) that there are no orbit points in the intervals [0.13, 0.91] and[1.10, 1.15], corresponding to the zero invariant density in these intervals. Similarly, for thecases α = 1.52 and α = 2.65, the simulation results are illustrated in Figures 7.6 and 7.7. Thesimulation results illustrate the accuracy of the eigenvector method in calculating the invariantdensity.

7.6 Examples of Applying Invariant Densities

The invariant density of a DC-DC converter can be used to calculate the power spectral densityof its input, to estimate its average switching frequency, and to accurately design its parameters.Two examples are given in the following for illustration.

7.6.1 Power Spectral Density of a DC-DC Converter’s Input Cur-rent

Consider the boost converter introduced above. The quadratic derivative of its inductor currentdepicted in Figure 7.2 is shown in Figure 7.8. According to [25], the inductor current can beexpressed by

d2i

dt2= − VO

Lδ(t)− δ(t− TCx1) + δ[t− TC(1 + bx1c)]− δ[t− TC(1 + bx1c+ x2)]

+ · · · − δ[t− TC(N − 1 +N−1∑k=1

bxkc) + xN ] + δ[t− TC(N +N∑

k=1

bxkc)], (7.8)

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where bxc means the round-off number. Employing the following Fourier transformation,

g(t) G(ω) ⇒∫ t

−∞g(u)du

1

jωG(ω),

(a) (b)

(c)

Figure 7.6: Chaotic mapping (a), bifurcation diagram (b), and invariant density (c) at α = 1.52

(a) (b)

(c)

Figure 7.7: ´Chaotic mapping (a), bifurcation diagram (b), and invariant density (c) at α = 2.65

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Figure 7.8: Quadratic derivative of inductor current

g(t) G(ω) ⇒ g(t− τ) e−jωtG(ω),

and

δ(t) 1,

results in the Fourier transform of the inductor current to be

A(ω) =− VO

ω2Llim

N→∞

1

TN

[1− exp(−jωTCx1)+ exp(−jωTC [1 + bx1c])1− exp(jωTCx2)

+ · · ·+ exp(−jωTC [N − 1 +N−1∑k=1

bxkc])(1− exp(−jωTCxN)]. (7.9)

With the denotations

Jn =

0 for n = 1∑N−1

k=1 1 + bxkc = n− 1 +∑N−1

k=1 bxkc for n > 1, (7.10)

and

Tn =N∑

n=1

1 + bxnc, (7.11)

Eq. (7.9) can be re-written as

A(ω) = − VO

ω2Llim

N→∞

1

TN

N∑n=1

e−jωTCJn1− e−jωTCxn. (7.12)

The power spectral density of the inductor current is defined as |A(ω)|2.When ω = mωc, with ωc the clock angular frequency, one has

Am = − VO

ω2Llim

N→∞

1

TN

N∑n=1

1− e−2jπmxn , (7.13)

where Am stands for the peak values.According to Birkhoff’s ergodic theory [30], a mapping f , which is an invariant density, satisfiesthe relationship,

limN→∞

1

N

N∑n=1

φ(fn−1(x)) =

∫ α

0

φ(y)ρ(y)dy. (7.14)

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Thus, Am , |Am|2 can be expressed by the invariant density ρ(x) as

Am = |Am|2 = [− VO

m2ω2cL〈T 〉

]2 × [(

∫ α

0

cos 2πmxdx− 1)2 + (

∫ α

0

sin2πmxdx)2], (7.15)

where

〈T 〉 = limN→∞

TN

N= TC(1 + lim

N→∞

1

Nbxnc).

A comparison of the power spectral densities calculated by (7.13) without using the invariantdensity, and by (7.15) using the invariant density is illustrated in Figure 7.9, and shows thatboth have almost the same accuracy, but that the calculation with the invariant density takesmuch shorter time, because (7.13) includes exponential operations to be calculated N timeswith N → ∞; whereas (7.15) just needs a single calculation, since the invariant density isknown.

(a) (b) Enlargement of (a)

Figure 7.9: Comparison of (7.13) shown as “+”, and (7.15) shown as “x”

7.6.2 Average Switching Frequency

Chaos control of DC-DC converters cannot only reduce electromagnetic interference of the con-verters [27, 34, 74], but also reduce their average switching frequencies, which is very importantfor reducing switching loss and increasing stability. The average switching frequency can becalculated with the invariant density.If the boost converter shown in Figure 7.1 operates properly, one can assume the total incrementof the inductor current ∆i+(total) to be equal to the total decrement of the inductor current∆i−(total) for a relatively long time, namely, ∆i+(total) = ∆i−(total) as shown in Figure 7.10.From Figure 7.10, the total time corresponding to the increasing inductor current is (t0 + t1 +· · · + tN−1). Then, the total time corresponding to the decreasing inductor current, tdown, canbe obtained by,

(t0 + t1 + · · ·+ tN−1)m1 = tdownm2, (7.16)

where m1 = VI

Land m2 = VO−VI

Lare the rates of increment and decrement of the inductor

current, respectively. Then, tdown can be obtained from (7.16) as

tdown =m1

m2

(t0 + t1 + · · ·+ tN−1) =1

α(t0 + t1 + · · ·+ tN−1), (7.17)

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Figure 7.10: Times of rising and falling inductor current

and the total duration of switching N times is

TN = (1 +1

α)(t0 + t1 + · · ·+ tN−1) = (1 +

1

α)(x0 + x1 + · · ·+ xN−1)TC . (7.18)

Thus, the total number of clock cycles, denoted by L, is

L =TN

TC

= (1 +1

α)(x0 + x1 + · · ·+ xN−1), (7.19)

and the total number of switchings is N .The average switching frequency is defined [22] as,

〈s〉 = limN→∞

N

L= lim

N→∞

N

(1 + 1α)(x0 + x1 + · · ·+ xN−1)

=1

(1 + 1α)(

∫ α

0ρ(x)xdx)

. (7.20)

To simplify the analysis, let α be an integer larger than 1. By the chaotic mapping, it is easy tofind that ρ(x) = 1

αfor integers α ≥ 1. Then, the average switching frequency can be obtained

as

〈s〉 =2

1 + α. (7.21)

From (7.21), it is obvious that 〈s〉 = 1 when α = 1, implying that the boost converter runsperiodically; and 〈s〉 < 1 when α > 1. The boost converter will operate in a chaotic mode whenα > 1, by which the boost converter has a low average switching frequency and low switchingloss. Further, as α increases, the average switching frequency decreases.

7.6.3 Parameter Design with Invariant Density

In designing a DC-DC converter, e.g., the boost converter shown in Figure 7.1, one needs knowthe value of the reference current Iref . Generally speaking, the values of input and outputvoltage are known conditions. According to [25], Iref can be calculated from the formula,

VO3+ VO(

VITc

2L− Iref )RVI −

RTcV3I

2L= 0. (7.22)

Employing the invariant density, one can accurately design the parameters for a chaotic DC-DCconverter. To simplify the calculation, α is restricted to integers between 2 to 10, because theinvariant density is 1

αwhen α takes on integer values larger than 1.

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Denote the quantity of electric charge through the diode D at the n-th time as Q(xn). Referringto the Figure 7.2 and using the physical definition of quantity of electric charge, one has

Q(xn) = (Iref −m2(1 + bxnc − xn)Tc

2)(1 + bxnc − xn)Tc). (7.23)

Using Birkhoff’s ergodic theory and the invariant density, one can obtain

〈T 〉 = limN→∞

1

N

N−1∑n=0

Tc(1 + bxnc) = Tc

∫ α

0

(1 + bxc)ρ(x)dx =α + 1

2Tc, (7.24)

and

〈Q〉 = limN→∞

1

N

N−1∑n=0

Q(xn) =

∫ α

0

Q(x)f(x)dx =1

2IrefTc −

m2Tc2

6. (7.25)

Because of

ID =Q

T, Q = 〈Q〉, T = 〈T 〉, ID =

VO

R, and VO = (1 + α)VI , (7.26)

the reference current Iref can be expressed as

Iref =(1 + α)2VI

R+

αVITc

3L. (7.27)

A comparison of the Irefs calculated by (7.22) and (7.27) using the invariant density, anddetermined experimentally, as shown in Figure 7.11, reveals that the estimation of Iref withthe invariant density is much more accurate.

Figure 7.11: Comparison of Irefs obtained by (7.22) (“*”), (7.27) (“x”), and experimentally(“.”)

7.7 Summary

The invariant density of a one-dimensional chaotic mapping used in the control of DC-DCconverters has been calculated in terms of the eigenvector method in this chapter. Further,applications of the invariant density have been introduced.

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76 8 Stability of a Chaotic PWM Boost Converter

Chapter 8

Stability of a Chaotic PWM BoostConverter

In the previous chapters, a chaotic pulse width modulation (PWM) boost converter has beenproposed to reduce EMI in DC-DC converters, circuit design and simulations have been con-ducted. Remaining problems such as mean value estimation of state variables for circuit pa-rameter design, ripple estimation of the input current, and stability analysis are addressed inthis chapter. First, a mean value estimation method is proposed, which is used to estimatethe mean values of state variables of chaotic PWM boost converters to facilitate the designof circuit parameters and the selection of circuit components. Although ripples are slightlyincreased when adopting chaotic carriers, DC-DC converters with reduced EMI are still stableunder chaotic PWM control. This chapter provides a theoretic verification of the effectivenessand practicability of the chaotic PWM DC-DC converters proposed.

8.1 Introduction

Chaotic PWM control has recently been recognised as an effective technology to suppresselectromagnetic interference (EMI) [23, 44, 45, 69, 70, 75], and is used in switched-mode powersupply (SMPS) converters [44, 69, 75] and in motor drives [23, 70]. Literature shows thatprevious research was focused on analysing the introduced chaotic signals and the improvedspectra, but ignored some basic problems such as the mean values of inputs and outputs usedfor system design and ripple estimation, as well as system stability under chaotic PWM control.

In Chapters 4 and 6, and in [44, 45] chaotic PWM has been proposed to control a boost DC-DC converter in order to suppress EMI by applying the continuous power spectrum featureof chaos to spread the harmonics of DC-DC converters continuously and evenly over widefrequency ranges. Therein, a chaotic carrier plays a key role in generating chaotic signals,whose circuit was designed. Simulation results have shown the effectiveness of the technologyproposed. However, the problems of how to estimate the mean value of the input current tofacilitate circuit parameter design and selection of circuit components, of how to calculate theripple increment, and of how to analyse stability of chaotic PWM DC-DC converters remainopen. To the best of our knowledge, these problems are addressed here first.

This chapter is organised as follows: Section 8.2 describes the circuit of the chaotic PWMboost converter; Section 8.3 proposes an estimation method for the mean values of the statevariables, i.e., input current and output voltage, to facilitate parameter design of the controlpart; in Section 8.4 only the ripple of the input current is estimated, since this chaotic PWMcontrol is a kind of current mode control; finally, in Section 8.5.2, the stability of the chaoticPWM boost converter is analysed.

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8 Stability of a Chaotic PWM Boost Converter 77

(a) Main circuit of boost converter

(b) Chaotic PWM control

Figure 8.1: Chaotic PWM boost converter

8.2 Chaotic PWM Boost Converters

The chaotic PWM control proposed in Chapter 4 can be used for many kinds of SMPS con-verters. Here, a boost converter with chaotic PWM control is adopted as test-bed due to itssimplicity and wide application. The main circuit and control part of the boost converter areshown in Figure 8.1. The difference to traditional PWM lies in the fact that the periodic carrieris replaced by a chaotic one, whose frequency is determined by a chaos mapping.

Here, the logistic mapping is employed to generate chaotic sequences, which is described by

f(xn) = 1− µx2n, x ∈ [−1, 1], (8.1)

with µ = 2, where the largest chaoticity is reached.

The circuit parameters are the same as the ones used in Chapter 6, i.e., VI = 10V , L = 6e−4H,C = 1e− 5F , R = 200Ω, Iref = 1A, and TC = 1e− 5s.

8.3 Estimation of the Mean State Variables

Estimation of the mean state variables is of significance to facilitate proper design of the systemparameters.

To obtain the mean values of input current iL and output voltage uC , first, the output voltageis assumed to be a constant VO for the big output filter capacitance C, and the input current’schaotic waveform is regarded to be equivalent to periodic waveforms in terms of the same meanvalue, as shown in Figure 8.2.

Denote the mean clock cycle as T , then VO can be estimated by assuming that S is switched onfor a time DT within the mean clock cycle T , where D is the mean duty cycle of S. Therefore,

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Chaotic input current waveform Equivalent periodic input currentwaveform

Same mean values

Rising slope = VI/L and falling slope = (VO-VI)/L Rising slope = VI/L and falling slope = (VO-VI)/L

IL

Imax

ΔIL

T

iL

D T

Figure 8.2: Sketch of equivalent input currents

VO is estimated by

VO = R(1− D)(Imax −∆i

2), (8.2)

where R is the load resistance, Imax is the maximum value of the equivalent periodic inputcurrent, and ∆IL is the ripple of the equivalent periodic input current (refer to Figure 8.2).The equation implies that the current through the diode is either zero (for the time DT ) orImax −∆IL/2 (for the time (1− D)T ). As S is switched on, iL rises at a rate of VI

Lfor a time

DT , while as S is switched off, iL falls at a rate VO−VI

Lfor a time (1− D)T [25].

Since the chaotic carrier can be equivalent to a periodic carrier with period T in terms ofequivalence of the mean input current IL, one has

A(Iref − Imax) = Vlow +(Vupp − Vlow)

TDT , (8.3)

where A is the amplification coefficient.Here, assuming A = 1 and Vlow = 0, one has

Iref − Imax = VuppD. (8.4)

In terms of the mean input current increment ¯∆iL+ and decrement ¯∆iL−, it is easy to obtainthat

¯∆iL+ =VI

LDT , ¯∆iL− =

VO − VI

L(1− D)T , (8.5)

and ¯∆iL+ = ¯∆iL− = ∆IL.

Eliminating ¯∆iL+, ¯∆iL−, ∆IL, Imax, and D from (8.2), (8.4), and (8.5) yields

VO3+ VIR(Vupp +

VI T

2L− Iref )VO − V 2

I R(Vupp +VI T

2L) = 0 (8.6)

It is obvious that the mean output voltage and mean input current can be obtained if T isknown, which is determined by the corresponding chaotic mapping.In chaotic PWM control, each period length of the chaotic carrier is determined by

Tn = xnβTC + TC , n = 1, 2, 3, ... (8.7)

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where xn is the output of the logistic mapping.Thus, the mean period of the carrier can be expressed as

T = limN→∞

1

N

N∑n=1

Tn = TC + βTC limN→∞

1

N

N∑n=1

xn. (8.8)

Now, the problem remaining is to derive (8.8), which can be addressed by using the ergodicityof the invariant density ρ of the mapping f in terms of Birkhoff’s ergodic theory [36]. Givenan expanding mapping f , which preservers the ergodic measure with density ρ(x) on (−1, 1),one has

limN→∞

1

N

N∑n=1

φ(f [n−1](x)) =

∫ 1

−1

φ(y)ρ(y)dy. (8.9)

For the logistic mapping the invariant density can easily be obtained analytically [36] as

ρ(x) =1

π√

1− x2. (8.10)

Substituting (8.9) and (8.10) into (8.8) yields

T = TC + βTC limN→∞

1

N

N∑n=1

xn (8.11)

= TC + βTC

∫ 1

−1

x1

π√

1− x2dx = TC .

Now, substituting T = TC into (8.6), one obtains VO. It follows that D = 1 − VI

VO, Imax =

Iref − VuppD, ∆i = VI

LDT , and the mean input current IL = Imax − ∆IL

2can thus be derived.

Table 8.1 shows the mean input currents and output voltages obtained by the estimationmethod outlined above and by circuit simulation based on Simulink with various input voltagesand values for β. The table indicates that β does not contribute to the mean values of thestate variables when the logistic mapping is employed to generate chaotic sequences, which isconsistent with the results of estimation and simulation. It is also remarked that the differencesbetween estimation and circuit simulation are caused by circuit components.

Table 8.1: Mean values of state variables obtained by estimation and simulation

Mean state variables obtained byParameters estimation method circuit simulation

for comparison IL VO IL VO

VI = 10V β = 0.05 0.1156A 15.2056V 0.1193A 15.4467VVI = 10V β = 0.2 0.1156A 15.2056V 0.1192A 15.4402VVI = 12V β = 0.05 0.1348A 17.9850V 0.1398A 18.3172VVI = 12V β = 0.2 0.1348A 17.9850V 0.1397A 18.3107V

The above proposed estimation method is also applicable to other chaotic mappings, althoughthe invariant densities of some chaotic mappings might not be obtained analytically. Fortu-nately, a numerical method to solve for invariant densities has been reported in [25, 46].The estimation results of the mean state variables are very helpful in practice to choose thecircuit components, because different currents are allowed for different components. Further,(8.6) appearing in the estimation method can be used to design the parameters in the controlpart: normally, VI , VO, R, and TC are given, therefore, Iref and Vupp can be obtained if any oneis given.

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8.4 Ripple Estimation of the Input Current

It is known from Chapters 4 and 6, that chaotic PWM control can greatly suppress EMI. Indoing so, it causes the ripples of input current and output voltage to increase. Since ripple isan important index of SMPS converters, it is of significance to know how much the ripples willbe increased.As current mode control is adopted in this chapter, the ripple of the current will be estimated.The output waveforms of chaotic PWM control are shown in Figure 8.3.

Traditional PWM control

Chaotic PWM control

Chaotic carrier

Feedback variable

Chaotic PWM waveform

Periodic PWM waveform

Periodic carrier

Feedback variable

Chaotic carrier

Iref - iL

In

I’n

In+1

I’n+1

In+2

I’n+2

In+3

I’n+3

Tn-1 Tn Tn+1 Tn+2 Tn+3tn tn+1 tn+2 tn+3

PWM signal

Figure 8.3: Output waveforms for chaotic PWM control

It is known that the rising slope of iL is VI/L, and the falling slope is VO−VI

L. Therefore, in

terms of Figure 8.3, one has

In = Iref − iLn, (8.12)

where iLn means the input current at any moment when S turns on, and

I ′n = In −VI

Ltn. (8.13)

In terms of the control part, one has

I ′n =Vupp

Tn

tn. (8.14)

Eliminating I ′n from (8.13) and (8.14) yields

tn =In

VI

L+ Vupp

Tn

. (8.15)

Substituting (8.15) into (8.14) results in

I ′n =Vupp

Tn

(In

VI

L+ Vupp

Tn

). (8.16)

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(a) by iteration method (b) by circuit simulation

Figure 8.4: Estimation of input current iL

Then, the current mapping can be derived from Figure 8.3 and (8.12) – (8.16),

In+1 = I ′n +VO − VI

L(Tn − tn) (8.17)

= (Vupp

Tn

− VO − VI

L)tn +

VO − VI

LTn

= (Vupp

Tn

− VO − VI

L)

In

VI

L+ Vupp

Tn

+VO − VI

LTn,

Due to the complexity of the chaotic mapping (8.17), it is impossible to obtain an analyticalrepresentation of the input current ripple. Instead, it has to be determined numerically.By observing the maximum of In and the minimum of I ′n within 1000 or more iterations, theripple can be obtained approximately by max(In)−min(I ′n).Here, let (8.17) and (8.16) iterate 1000 times with the initial values I0 = Iref − Imax andx0 = 0.625. Then, the input current iL can be drawn according to the iteration, as shown inFigure 8.4(a). Figure 8.4(b) shows the resulting input current when simulating the circuit withthe same parameters as used in the above iteration. It is seen in Figure 8.4 that their ripplesare very close.Table 8.2 shows the ripple of the input currents obtained by iteration and circuit simulationwith various selections of β and input voltage VI , and the corresponding increments of theripples in chaotic mode and in periodic mode (i.e., β = 0).Normally, the output voltage ripple is not allowed to exceed 1% of the output voltage. It isseen from Table 8.2 that, although the current ripple seems to increase somewhat, the ripple ofthe output voltage is still very small, which can be estimated by multiplying the input currentripple with the equivalent series resistance (ESR). For instance, as VI = 10V and β = 0.2, theripple of the output voltage is only 0.48% of the latter.Moreover, based on ripple estimation, the relationship between input current ripple and βis illustrated in Figure 8.5. It is obvious that as β grows, the input current increases asshown with the “black line” in Figure 8.5, which can be fitted with the polynomial ripple =−0.0469β2 + 0.1084 β + 0.0559 as shown with the “red line” in Figure 8.5.Therefore, the ripple can be calculated from β directly, which is an easy way to estimate ripplein practice.

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Table 8.2: Comparison of input current ripples obtained by iteration method and circuit sim-ulation, and ripple increments

Ripple obtained by Ripple with RippleParameters iteration simulation β = 0 increment

VI = 10V β = 0.05 0.0618A 0.0618A 0.0571A 0.0047AVI = 10V β = 0.2 0.0753A 0.0755A 0.0571A 0.0184AVI = 12V β = 0.05 0.0722A 0.0721A 0.0664A 0.0057AVI = 12V β = 0.2 0.0884A 0.0884A 0.0664A 0.022A

Figure 8.5: Relationship between β and ripple

8.5 Stability

8.5.1 Two Operation Modes of the Boost Converter

A boost converter has two operation phases or two switching modes: when the switch S isturned on, the state equation refers to Mode I, described by (8.18) and shown in the upperpart of Figure 8.6, and when the switch S is turned off, the state equation refers to Mode II,described by (8.19) and shown in the lower part of Figure 8.6.

Mode I

duC

dt= − 1

RCuC

diLdt

= − 1LVI

(8.18)

Mode II

duC

dt= − 1

RCuC + 1

CiL

diLdt

= − 1LuC + 1

LVI

(8.19)

Assume that the mean duty cycle of S is D, the mean state equations can be obtained byapplying state space averaging [51, 64] to (8.18) and (8.19),

duC

dt= − 1

RCuC + 1−D

CiL

diLdt

= −1−DL

uC + 1LVI

(8.20)

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Figure 8.6: The two operation modes of the boost converter

From (8.20), the output voltage uC and the input current iL in the steady state can be obtainedas follows,

VO = VI/(1− D)

IL = VO

(1−D)R

(8.21)

8.5.2 Stability

First, it is assumed that every transition state in starting up the converter is supposed to be a“quasi-steady state”. The state variables slowly increase in the start-up transition and, finally,reach their own values of the “complete steady state”. So, it seems reasonable to assume the“quasi-steady state” in the start-up transition. Secondly, suppose that the duty ratio changesfrom cycle to cycle, i.e., D(t) = D(t)+∆D, where D is the duty cycle of the “quasi-steady state”and ∆D is a super-imposed variation. With the corresponding disturbance, the load resistanceR(t) = R + ∆R, the input voltage VI(t) = VI + ∆VI , the input current IL(t) = IL + ∆IL, andthe output voltage VO(t) = VO + ∆VO, the basic equations become

d∆VO

dt= −R∆VO−VO∆R

RC(R+∆R)+ (1−D)∆IL−IL∆D

Cd∆IL

dt= VO∆D−(1−D)∆VO

L+ ∆VI

L

(8.22)

in which the second-order terms of (8.22) have been neglected. Since chaotic PWM control is acurrent mode control, the Laplace transform of (8.22) leads to an expression for the disturbanceof the input current of the form,

(R2LCs2 + RLs + R2(1− D)2)∆IL(s) (8.23)

= (R2VOCs + RVO + R2IL(1− D))∆D(s)

− VO(1− D)∆R(s) + (R2Cs + R)∆VI(s)

Similarly, according to the control part, there are Iref − (Imax + ∆Imax) = Vupp(D + ∆D),IL + ∆IL = (Imax + ∆Imax)− (IL∆ + ∆IL∆)/2), and i∆ + ∆i∆/2 = (VI + ∆VI)(D + ∆D)T /L,

thus ∆IL = −Vupp+VI T

L∆D − DT

L∆VI , and ∆D(s) = −k1∆IL(s) − k2∆VI(s), k1, k2 ∈ (0, +∞),

where k1 and k2 are the feedback gains of the control circuit. Then, the disturbance of theinput current can be re-written as:

∆IL(s) =GR(s)

1 + k1G(s)∆R(s)− GV (s)

1 + k1G(s)∆VI , (8.24)

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where

GR(s) =(1− D)2IL

RLCs2 + Ls + R(1− D)2,

GV (s) =(RC − k2RCVO)s + 1− 2k2VO

RLCs2 + Ls + R(1− D)2,

and

G(s) =RCVOs + 2VO

RLCs2 + Ls + R(1− D)2.

Therefore, the characteristic equation can be obtained as

1 + k1G(s) = 0 (8.25)

and (8.25) can be further written as

RLCs2 + (L + k1RCVO)s + R(1− D)2 + 2VO = 0. (8.26)

It is well known that the root locus of the characteristic equation can be used to judge thestability of a system [61]. If all roots, obtained when k1 increases from 0 to infinity, distributeon the left plane, then the system will be stable. The root locus of (8.26) is shown as Figure 8.7.

Figure 8.7: Root locus of characteristic equation (8.26)), k1 ∈ [0, +∞)

According to the root locus of characteristic equation (8.26), the boost converter is stablefor k1 > 0. Furthermore, according to the control part, one has k1 = L

Vupp+VI T. Therefore, the

difference between chaotic PWM control and traditional PWM control lies in T . For traditionalPWM control as well as for chaotic PWM control using the logistic mapping it holds T = TC .If other chaotic mappings are employed, it holds always T > 0, implying that k1 > 0. Insummary, the boost converter is stable under this kind of chaotic PWM control.

8.6 Summary

The chapter has addressed estimating the mean values of state variables and the ripples forchaotic PWM DC-DC converters, which are significant for their design. Finally, the stabilityof DC-DC converters under CPWM control has been verified.

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9 Chaotic Spectra Analysis Using the Prony Method 85

Chapter 9

Chaotic Spectra Analysis Using theProny Method

It is well known that chaotic DC-DC converters are mainly used to reduce EMI, which isestimated by its spectrum. Conventionally, the Fast Fourier Transform (FFT) is used to analysethe spectra. However, it is not applicable to the inner-harmonics, i.e., the non-integral multiplesof the fundamental frequency, which is a prominent feature of chaotic signals. In this chapter,the Prony method is suggested for spectral estimation of chaos-controlled DC-DC converters.Numerical simulations show its advantages over the traditional FFT.

9.1 Introduction

Traditionally, the strength of EMI is measured by estimating the system harmonics, namely, byderiving the power spectral density (PSD) based on FFT [49]. This spectral analysis approach iscomputationally efficient and, in most cases, can provide reasonable results for signal processes.It has, however, some drawbacks. The most prominent one is that of frequency resolution, i.e.,the ability to distinguish the spectral responses of two or more signals. The frequency resolutionmeasured in Hertz is roughly the reciprocal of the time interval in seconds, over which sampleddata are available. The second shortcoming is due to the implicit windowing of the datathat occurs when processing by FFT. Windowing manifests itself as “leakage” in the spectraldomain, i.e., energy in the main lobe of a spectral response “leaks” into the side-lobes, obscuringand distorting other nearby spectral responses being present [54]. These two drawbacks limitthe application of FFT in analysing short sampled data sequences, which occur frequently inpractice, because many process measurements are short in duration or have slowly time-varyingspectra that are often considered as constant in short sampling intervals. Further, FFT cannotefficiently estimate inner-harmonics, since it assumes the harmonics to be integral multiples ofthe fundamental frequency [47].

To alleviate the limitations of FFT, several new modern spectral estimation methods have beenproposed [39, 53, 56, 63, 68]. In this chapter, one of the available spectral estimation methods,the Prony method, is employed to investigate and analyse chaotic signals [43, 68]. The Pronymethod improves the frequency resolution and is not affected by windowing. Thus, the Pronymethod cannot only be applied to spectral estimation, but also to obtaining information aboutamplitudes, phases, frequencies, and damping factors of harmonics. Furthermore, it is shownthat the Prony method can be used to reconstruct or to fit sampled data. Finally, somesimulation results are presented for illustration.

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86 9 Chaotic Spectra Analysis Using the Prony Method

9.2 Prony Method

Consider N complex sampled data, x(0), x(1), . . . , x(N − 1), which can be fitted by using Ppolynomial exponential functions:

x(n) =P∑

k=1

bkznk , n = 0, 1, . . . , N − 1, (9.1)

bk = Akejθk , (9.2)

zk = e(αk+j2πfk)∆t, (9.3)

where x(n), n = 0, 1, . . . , N − 1, are the fitted data, θk the phase, ∆t the sampling period, Ak

the amplitude, αk the damping factor, and fk the frequency. Traditionally, the fitting problemis based on minimising the sum of squared errors between measured data x(n) and fitted valuesx(n):

ε =N−1∑n=0

|x(n)− x(n)|2. (9.4)

However, it is very difficult, if not impossible, to derive the coefficients Ak, αk, fk, θk due to theexistence of the exponential terms, which require to solve a complicated non-linear problem.Thanks to the Prony method, one can convert this problem to deriving the homogeneoussolution of a constant-coefficient linear difference equation of the form [66]:

x(n) = −P∑

k=1

akx(n− k), (9.5)

by defining the polynomial that has the exponents zk as its roots

F (z) =P∏

k=1

(z − zk) = (z − z1)(z − z2)...(z − zP )

=P∑

k=0

akzP−k, a0 = 1. (9.6)

Denote e(n) = x(n)− x(n). Then (9.5) can be written as

x(n) = −P∑

k=1

akx(n− k) +P∑

k=0

ak · e(n− k), a0 = 1. (9.7)

Define

u(n) =P∑

k=0

ak · e(n− k), a0 = 1, (9.8)

then, (9.8) can be recast as

x(n) = −P∑

k=1

akx(n− k) + u(n). (9.9)

Here, x(n) is regarded as the output of the P -th order autoregressive (AR) model driven bynoise u(n). Minimising the quadratic sum of u(n) results in the parameter ak (k = 1, 2, . . . , P )

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[43]. Substituting ak (k = 1, 2, . . . , P ) into (9.5), one can obtain the polynomial equation (9.10),whose roots are zk (k = 1, 2, . . . , P ), which can easily be calculated by using Matlab,

P∑k=0

akzP−kk = 0. (9.10)

Further substituting zk (k = 1, 2, . . . , P ) into (9.3) yields the frequency fk and the dampingfactor αk,

fk = arctan[Im(zk)/Re(zk)]/2π∆t,αk = ln|zk|/∆t,

k = 1, 2, . . . , P, (9.11)

where Im(∗) and Re(∗) denote the imaginary part and the real part of complex numbers.Replacing the fitted data x(n) by the sampled data x(n) in (9.1)) results in the matrix equation,

V b = x, (9.12)

where V =

1 1 . . . 1z1 z2 . . . zP...

......

...zN−11 zN−1

2 . . . zN−1P

, b =

b1

b2...

bP

, and x =

x(0)x(1)

...x(N − 1)

. Solving the

least-square equation (9.12) gives

b = (V HV )−1V Hx, (9.13)

in which V H stands for the conjugate transpose matrix of V .Finally, in terms of (9.2), the amplitudes Ak and the phases θk are obtained as

Ak = |bk|,θk = arctan[Im(bk)/Re(bk)],

k = 1, 2, . . . , P. (9.14)

Thus, x(n) (n = 0, 1, . . . , N − 1) are obtained and denoted in vector form as x.Denote the Fourier transform of x by X(f). Then, the PSD of the N sampled data (PProny(f))can be expressed as

PProny(f) = |X(f)|2, (9.15)

where

X(f) =P∑

k=1

Akejθ 2αk

|αk|2 + (2π(f − fk))2.

9.3 Deriving the Power Spectral Density

It is known that the frequency resolution of FFT is proportional to 1/N∆t, where ∆t is thesampling period. DC-DC converters always work at high frequencies, so that the samplingperiod must be very small. Thus, the resolution of FFT is not satisfactory in practice. Inaddition, for the case of short data sequences, e.g., for data obtained in failure diagnosis, whereN is small, the resolution of FFT is also very low.The Prony method overcomes these drawbacks, at the price that its computation is a little bitmore complex than that of FFT. It can be used to estimate the PSD of DC-DC converters,especially when converters work in chaotic mode.

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To show the effectiveness of the Prony method in improving the frequency resolution of thePSD as compared with that of the FFT, the periodic signal equation below is taken as anexample to derive its PSD with the two methods, respectively:

y(t) = sin(2πf1t) + 0.6 cos(2πf2t) + 2 sin(2πf3t), (9.16)

where f1 = 100Hz, f2 = 98Hz, and f3 = 25Hz. Let N = 128 be the number of data sampled,and fs = 1000Hz the sampling frequency.

Figure 9.1: PSD obtained by using the Prony method

Figure 9.1 shows the PSD plot, and the related coefficients derived for P = 10 are given inTable 9.1.

Table 9.1: Coefficients derived for P = 10

Ak fk αk θk

5.97E-38 500 654.48 -3.14162.58E-12 467.34 26.938 0.68762.58E-12 -467.3 26.938 -0.68759

1 25 -2.09E-10 -1.39821 -25 -2.09E-10 1.3982

0.5 100 3.09E-06 -0.880270.5 -100 3.09E-06 0.880270.3 98 2.46E-06 0.676710.3 -98 2.46E-06 -0.67671

4.21E-09 0 -1106.9 -3.1416

Investigating Ak and fk in Table 9.1, it can be seen that by discarding the negative frequenciesand those corresponding to small values of Ak, only three positive frequencies, i.e., f = 25Hz,f = 100Hz, and f = 98Hz, remain. This is consistent with Eq. (9.16). Further investigatingthe damping factors αk, it is noted that the three damping factors corresponding to the threepositive frequencies are very small, implying that the corresponding signals in the polynomialexponential function (9.1) are periodic, while the others with big damping factors are constant.In addition, the Prony method can be employed to reconstruct the sampled data of the signaly using the obtained parameters Ak, fk, αk, and θk as shown in Figure 9.2. It can be seen inFigure 9.3 that the error between the real signal and the reconstructed one is very small, viz.,of the order of 10−9.

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Figure 9.2: Reconstructed signal Figure 9.3: Error signal

For comparison purposes, FFT is adopted, whose frequency resolution is ∆f = fs/N =7.8125Hz [48]. This means, if |f1−f2| ≤ ∆f , FFT is not able to distinguish these two frequen-cies. It is shown in Figure 9.4 that using FFT only two peaks are identified, at f1 = 23.4375Hzand f2 = 101.5625Hz, respectively.

Figure 9.4: PSD obtained by using FFT

It is remarked that although the FFT method is simple and computationally effective, itsfrequency resolution is low, especially for short sampled data sequences. In contrast, the Pronymethod has its merits in improving frequency resolution and data reconstruction. In particular,due to the existence of rich inner-harmonics and random-like behaviour in chaotic systems, theProny method is more powerful and effective than the FFT method.

9.4 Chaotic Spectral Estimation of DC-DC Converters

It is known that DC-DC converters produce electromagnetic interferences and, thus, electro-magnetic pollution. With the increasing use of electronic equipment, the problem of EMI hasattracted increasing attention from engineers [4, 10, 16]. Recently, studies have shown that DC-DC converters have broadband spectra when they operate in chaotic modes, and the energyof EMI is more evenly distributed on the frequency band [27]. Thus, the peak values of EMIcan be decreased, but rich inner-harmonics are generated. The inner-harmonics may result inquality degradation of the transmission energy, increase of power loss, reliability degradationof the converter systems, etc. [12, 42]. Thus, it is of significance to detect the inner-harmonicsin the control systems.

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Figure 9.5: Ccurrent-controlled boost converter

As traditional FFT can only detect the fundamental frequency and its integral multiples, it isnot applicable for this case of inner-harmonics. Instead, the Prony method is employed herefor the spectral estimation of the inductor current of a basic DC-DC converter, viz., the boostconverter, whose circuitry is shown in Figure 9.5. Therein, the reference current Iref servesas the control parameter. By adjusting the reference current, the boost converter can exhibitperiod-1, period-2 and chaotic oscillations. In the sequel, the Prony Method is used for spectralestimation of the inductor current corresponding to the three operating modes.Assume the circuit parameters to be Vin = 10V , L = 1mH, C = 12µF , R = 20Ω, andfc = 10kHz, where Vin is the input voltage, L the input inductance, C the output capacity, Rthe load resistance, and fc the clock frequency, which lead the converter to operate in continuouscurrent mode (CCM).In the simulation, 128 sampled data, in (n = 0, 1, . . . , 127) are taken from the input endof inductor current. It is shown that the system exhibits period-1 behaviour for Iref = 1A(Figure 9.6), period-2 behaviour for Iref = 1.8A (Figure 9.7), and chaotic behaviour for Iref =4A (Figure 9.8).

Figure 9.6: Sampled currentwaveform for Iref = 1A

Figure 9.7: Sampled currentwaveform for Iref = 1.8A

Figure 9.8: Sampled currentwaveform for Iref = 4A

In order to carry out the spectral estimation, we assume that Iref = 1A corresponding to theperiod-1 mode and P = 40, which is an empirical value. Using the Prony method introducedin Section 9.2, the coefficients can be derived as given in Table 9.2 by omitting the negativefrequencies.It is seen from Table 9.2 that the direct current (DC) component 0.88179A with zero valuesof fk and the alternating current (AC) components with non-zero values of fk can be decom-posed. That is, by investigating fk, one cannot only distinguish the fundamental frequency andits integral multiples but also the inner-harmonics. By observing Ak it is known that the am-plitude of the fundamental frequency component is the largest one among all AC components.

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Table 9.2: Parameter values derived for P = 40

Ak fk αk θk

0.0015954 1.00E+05 23.547 3.95E-171.35E-05 95246 -5102.4 -0.219880.0011592 90000 19.932 0.393812.38E-05 85613 -4704.9 0.34639

0.00013856 79998 -358.21 -0.683133.13E-05 75738 -4605.4 1.51960.0014441 69998 14.556 -1.65534.96E-05 65823 -5040 2.63540.0024704 60000 10.724 -1.25496.79E-05 55824 -5306.1 -2.49410.0022907 50002 5.6684 -0.84738.80E-05 45704 -5582 -1.23010.0002811 40000 -92.27 -1.94390.00012273 35518 -6200.1 0.122540.0056313 30000 3.0246 -2.93120.00019746 25385 -6989.8 1.41450.017208 20000 1.138 -2.52080.88179 0 0.060618 1.24E-19

0.0010786 5085.9 -6929.6 -2.6090.048575 10000 0.096451 -2.1129

0.00034827 15281 -7176.4 2.6193

(a) (b)

Figure 9.9: PSD obtained by using (a) Prony (b) FFT method for Iref = 1A

Figures 9.9(a) and 9.9(b) show that the two Ak corresponding to fk = 40kHz and fk = 80kHzare much smaller than that corresponding to the fundamental frequency and its integral mul-tiples. For the cases Iref = 1.8A and Iref = 4A corresponding to period-2 and chaotic modes,respectively, similar results can be obtained.Figures 9.10(a), 9.10(b), and 9.10(c) show the errors between the real signals and the recon-structed ones in the three respective cases considered here.The simulation results of the spectral estimation using the Prony method are illustrated inFigures 9.9(a), 9.11(a), and 9.12(a). For comparison, a similar simulation using the FFTmethod was also carried out and its results are shown in Figures 9.9(b), 9.11(b), and 9.12(b).It is obvious that the Prony method can much more accurately locate the frequencies of theharmonics corresponding to peaks for all cases.

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(a) (b) (c)

Figure 9.10: Error signals obtained with Prony method for (a) Iref = 1A (b) Iref = 1.8A and(c) Iref = 4A

(a) (b)

Figure 9.11: PSD obtained by using (a) Prony (b) FFT method for Iref = 1.8A

Figure 9.12(a) shows that there exist two inner-harmonics, 5kHz and 17kHz, correspondingto two peaks of the PSD, which are not made visible by the FFT method (see Figure 9.12(b)).Therefore, for chaotic signals, the Prony method is more accurate and effective than FFT.

(a) (b)

Figure 9.12: PSD obtained by using (a) Prony (b) FFT method for Iref = 4A

9.5 Summary

This chapter put effort into finding a more accurate algorithm to calculate the spectra of chaoticsignals. Simulation results reveal that the proposed Prony method is more effective than theconventional FFT method in estimating chaotic spectra accurately.

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10 Conclusion 93

Chapter 10

Conclusion

This dissertation has contributed to the application of chaos control in DC-DC converters to theend of reducing EMI, but also to system design, dynamics analysis, simulation, and hardwareimplementation of chaos-controlled DC-DC converters. In particular, the contributions of thedissertation are the following.

1. The rapid development and application of electronic devices and products have causedserious EMI problem. The EMI standards and the international EMC standards requiredto be satisfied by converters have been introduced. After surveying the conventionalEMI suppression techniques for DC-DC converters, it has been pointed out that a newtheory, i.e., chaos theory, has great potential to provide a new means for coping with EMIproblems.

2. The periodic and chaotic behaviour of DC-DC converters under different parametric con-ditions has experimentally been exhibited. Since chaos control has been proposed toimprove EMC of DC-DC converters for several decades, the conventional chaos controlmethods and their advantages and disadvantages have been discussed. Some examples ofchaos control in DC-DC converters have been considered to verify their good performancein reducing EMI.

3. Based on the conventional chaos control methods, a novel chaotic peak current modeboost converter has been proposed. By the use of upper and lower reference currents,the chaos control proposed can adjust the magnitudes of the output ripples easily, aswell as reduce EMI. A chaotic mapping corresponding to this boost converter has alsobeen derived, showing more complex bifurcation and chaotic phenomena. It has also beennoticed that the introduction of Ilow can facilitate bifurcation and drive the system intochaotic mode more easily. The novel chaos control has been verified both by simulationsand experiments with simple circuitry design. It has been confirmed that not only EMIcan be suppressed, but that also the output ripples can be duly reduced by the control ofthe reference current Ilow as compared with [25]. From both simulation and experimentalresults, a shift of the dominant frequencies has been observed in the power spectrum whenIlow is increased. Some further studies will be carried out in the future, so as to identifythe factors influencing the energy distribution in the chaotic power converter proposed.

4. A method for CPWM control by varying carrier frequencies or varying carrier amplitudeshas been proposed. It can distribute spectra continuously and evenly over wide frequencyranges, thus improving the EMC of DC-DC converters. In addition, the average switch-ing frequencies and switching dissipation of DC-DC converters are accordingly reduced,and stability is enhanced. Analyses of the output waves and EMI properties of DC-DC

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94 10 Conclusion

converters under CPWM control have been carried out. This approach provides a goodexample of applying chaos control in engineering practice.

5. For implementing CPWM in practice, a novel analogue chaotic carrier has been pro-posed and applied in a boost converter. To generate the analogue chaotic carrier, chaoticoscillator circuits have been introduced. The generation of analogue chaotic carriers issimpler and cheaper than of digital ones. The simulation and experimental results showthat CPWM control with analogue chaotic carriers can greatly suppress EMI of boostconverters while the other characteristics of operation performance are well maintained.

6. A novel approach combining the technique of soft switching and chaos control has beenproposed for EMI reduction. Further, the digital design of chaotic carriers has beenaddressed, too. Chaotic soft switching PWM has been applied in a boost converter, andthe results obtained show that EMI and efficiency of the boost converter can be improvedby chaotic soft switching PWM as compared with hard switching PWM and conventionalsoft switching PWM control. This chaotic soft switching PWM control can easily be usedin different kinds of DC-DC converters. In the future, a hardware implementation andexperimental verifications will be carried further.

7. A one-dimensional chaotic mapping for DC-DC converters has been derived, and use of theeigenvector method from probability theory has been proposed to calculate the invariantdensity of the chaotic mapping, since chaos has random-like characteristic. Further, theinvariant density has been used to calculate the PSD and the average switching frequencyof a DC-DC converter. When a DC-DC converter works in a chaotic mode, its averageswitching frequency is lower than when it works in a periodic mode. Consequently, theswitching loss of the DC-DC converter can be reduced. Moreover, the invariant densitycan be used to accurately design the parameters of DC-DC converters. Simulation resultshave illustrated the effectiveness of the eigenvector method.

8. The mean values of state variables and the size of the ripples in the input current of aCPWM controlled DC-DC converter have been estimated. Comparing these estimationresults with ones obtained by circuit simulation, it has been found that the estimationmethods proposed are very accurate. Finally, stability, not only for the steady state butalso the dynamic state, has been proven based on the state space averaging method.According to the above mentioned analysis, it can be concluded that CPWM control canbe applied in practice, since it is effective in suppressing EMI, stable, and causes a littleripple increment, only.

9. The Prony method has been employed for spectral estimation of the inputs (or outputs)of chaotic DC-DC converters. As compared with FFT, the Prony method has shown itsmerits, such as improving the frequency resolution and accuracy in locating harmonics.Thus, for analysing chaotic signals it is a better tool. In addition, the frequencies, phases,amplitudes, and damping factors of the harmonics of currents or voltages of DC-DCconverters can also be obtained with the Prony method. The Prony method can alsodistinguish between the DC and AC components of a signal. Therefore, it is recommendedto employ the Prony method of the popular FFT in such applications as the spectralanalysis of converters involving chaotic signals.

Although great effort has been practically and theoretically made in this dissertation to makechaos control more suitable for practical applications, there are still some issues to be furtheraddressed in the future.

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10 Conclusion 95

1. The theoretical analysis of chaotic DC-DC converters is still not self-contained, althoughsome analyses have been given in Chapters 7 – 9. Further issues, such as lifetime analysisof the components in chaotic DC-DC converters, or the factors influencing the backgroundspectrum, are worth being investigated.

2. For CPWM control, the control circuits are to be further integrated. New applicationfields for chaos control in power electronics should be explored.

3. In this dissertation, chaos control has been combined with peak current mode control,PWM control, and soft switching PWM control. Similarly, chaos control could be com-bined with other control schemes, such as PID or sliding mode control, to realise morefunctions desirable for certain purposes.

4. Chaos control should be tested in real products, such as adaptors of laptop computers ormobile charger, to further prove the good characteristic of suppressing EMI.

5. The toolbox for spectral estimation toolbox of chaotic signals will continuously be devel-oped and improved.

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96 References

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