
Digital Control of High-Frequency Switched-Mode Power Converters
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Preface ix
Introduction 1
Chapter 1 Continuous-Time Averaged Modeling of DC-DC Converters 13
1.1 Pulse Width Modulated Converters 14
1.2 Converters in Steady State 16
1.2.1 Boost Converter Example 17
1.2.2 Estimation of the Switching Ripple 19
1.2.3 Voltage Conversion Ratios of Basic Converters 20
1.3 Converter Dynamics and Control 21
1.3.1 Converter Averaging and Linearization 22
1.3.2 Modeling of the Pulse Width Modulator 24
1.3.3 The System Loop Gain 25
1.3.4 Averaged Small-Signal Models of Basic Converters 26
1.4 State-Space Averaging 28
1.4.1 Converter Steady-State Operating Point 28
1.4.2 Averaged Small-Signal State-Space Model 29
1.4.3 Boost Converter Example 30
1.5 Design Examples 32
1.5.1 Voltage-Mode Control of a Synchronous Buck Converter 32
1.5.2 Average Current-Mode Control of a Boost Converter 42
1.6 Duty Ratio d[k] Versus d(t) 48
1.7 Summary of Key Points 50
Chapter 2 The Digital Control Loop 51
2.1 Case Study: Digital Voltage-Mode Control 52
2.2 A/D Conversion 53
2.2.1 Sampling Rate 53
2.2.2 Amplitude Quantization 56
2.3 The Digital Compensator 58
2.4 Digital Pulse Width Modulation 63
2.5 Loop Delays 65
2.5.1 Control Delays 65
2.5.2 Modulation Delay 66
2.5.3 Total Loop Delay 70
2.6 Use of Averaged Models in Digital Control Design 71
2.6.1 Limitations of Averaged Modeling 71
2.6.2 Averaged Modeling of a Digitally Controlled Converter 74
2.7 Summary of Key Points 78
Chapter 3 Discrete-Time Modeling 79
3.1 Discrete-Time Small-Signal Modeling 80
3.1.1 A Preliminary Example: A Switched Inductor 82
3.1.2 The General Case 85
3.1.3 Discrete-Time Models for Basic Types of PWM Modulation 87
3.2 Discrete-Time Modeling Examples 88
3.2.1 Synchronous Buck Converter 90
3.2.2 Boost Converter 97
3.3 Discrete-Time Modeling of Time-Invariant Topologies 102
3.3.1 Equivalence to Discrete-Time Modeling 106
3.3.2 Relationship with the Modified Z-Transform 108
3.3.3 Calculation of Tu(z) 108
3.3.4 Buck Converter Example Revisited 112
3.4 Matlab® Discrete-Time Modeling of Basic Converters 112
3.5 Summary of Key Points 117
Chapter 4 Digital Control 119
4.1 System-Level Compensator Design 119
4.1.1 Direct-Digital Design Using the Bilinear Transform Method 120
4.1.2 Digital PID Compensators in the z- and the p-Domains 123
4.2 Design Examples 126
4.2.1 Digital Voltage-Mode Control of a Synchronous Buck Converter 126
4.2.2 Digital Current-Mode Control of a Boost Converter 134
4.2.3 Multiloop Control of a Synchronous Buck Converter 136
4.2.4 Boost Power Factor Corrector 141
4.3 Other Converter Transfer Functions 154
4.4 Actuator Saturation and Integral Anti-Windup Provisions 160
4.5 Summary of Key Points 165
Chapter 5 Amplitude Quantization 167
5.1 System Quantizations 167
5.1.1 A/D Converter 167
5.1.2 DPWM Quantization 169
5.2 Steady-State Solution 172
5.3 No-Limit-Cycling Conditions 175
5.3.1 DPWM versus A/D Resolution 175
5.3.2 Integral Gain 178
5.3.3 Dynamic Quantization Effects 181
5.4 DPWM and A/D Implementation Techniques 182
5.4.1 DPWM Hardware Implementation Techniques 182
5.4.2 Effective DPWM Resolution Improvements via S¿ Modulation 186
5.4.3 A/D Converters 187
5.5 Summary of Key Points 190
Chapter 6 Compensator Implementation 191
6.1 PID Compensator Realizations 194
6.2 Coefficient Scaling and Quantization 197
6.2.1 Coefficients Scaling 198
6.2.2 Coefficients Quantization 200
6.3 Voltage-Mode Control Example: Coefficients Quantization 203
6.3.1 Parallel Structure 204
6.3.2 Direct Structure 206
6.3.3 Cascade Structure 208
6.4 Fixed-Point Controller Implementation 213
6.4.1 Effective Dynamic Range and Hardware Dynamic Range 214
6.4.2 Upper Bound of a Signal and the L1-Norm 216
6.5 Voltage-Mode Converter Example: Fixed-Point Implementation 218
6.5.1 Parallel Realization 220
6.5.2 Direct Realization 225
6.5.3 Cascade Realization 229
6.5.4 Linear versus Quantized System Response 233
6.6 HDL Implementation of the Controller 234
6.6.1 VHDL Example 235
6.6.2 Verilog Example 237
6.7 Summary of Key Points 239
Chapter 7 Digital Autotuning 241
7.1 Introduction to Digital Autotuning 242
7.2 Programmable PID Structures 243
7.3 Autotuning VIA Injection of a Digital Perturbation 247
7.3.1 Theory of Operation 249
7.3.2 Implementation of a PD Autotuner 253
7.3.3 Simulation Example 255
7.3.4 Small-Signal Analysis of the PD Autotuning Loop 261
7.4 Digital Autotuning Based on Relay Feedback 265
7.4.1 Theory of Operation 266
7.4.2 Implementation of a Digital Relay Feedback Autotuner 267
7.4.3 Simulation Example 271
7.5 Implementation Issues 272
7.6 Summary of Key Points 275
Appendix A Discrete-Time Linear Systems and The Z-Transform 277
A.1 Difference Equations 277
A.1.1 Forced Response 278
A.1.2 Free Response 279
A.1.3 Impulse Response and System Modes 281
A.1.4 Asymptotic Behavior of the Modes 282
A.1.5 Further Examples 283
A.2 Z-Transform 284
A.2.1 Definition 284
A.2.2 Properties 285
A.3 The Transfer Function 287
A.3.1 Stability 287
A.3.2 Frequency Response 288
A.4 State-Space Representation 288
Appendix B Fixed-Point Arithmetic and HDL Coding 291
B.1 Rounding Operation and Round-Off Error 291
B.2 Floating-Point versus Fixed-Point Arithmetic Systems 293
B.3 Binary Two's Complement (B2C) Fixed-Point Representation 294
B.4 Signal Notation 296
B.5 Manipulation of B2C Quantities and HDL Examples 297
B.5.1 Sign Extension 298
B.5.2 Alignment 299
B.5.3 Sign Reversal 301
B.5.4 LSB and MSB Truncation 302
B.5.5 Addition and Subtraction 304
B.5.6 Multiplication 305
B.5.7 Overflow Detection and Saturated Arithmetic 307
Appendix C Small-Signal Phase Lag of Uniformly Sampled Pulse Width Modulators 313
C.1 Trailing-Edge Modulators 313
C.2 Leading-Edge Modulators 317
C.3 Symmetrical Modulators 318
References 321
Index 335
Introduction
Efficient processing and control of electric power is required in applications ranging from submilliwatt on-chip power management to hundreds of kilowatt and megawatt power levels in motor drives and utility applications. The objectives of high efficiency, as well as static and dynamic control of inputs or outputs under a range of operating conditions, are accomplished using power electronics, that is, switched-mode power converters consisting of passive (capacitive and inductive) components, and power semiconductor devices operated as switches. In high-power applications, control and monitoring tasks are often more complex, while the power semiconductor devices are operated at relatively low switching frequencies, for example, up to tens of kilohertz. The controller cost and power consumption are relatively low compared to the overall system cost and power rating. In these applications, digital control offers clear technical and economic advantages in addressing complex control, management, and monitoring tasks. As a result, for many years now, digital control methods and digital controllers based on general-purpose or dedicated microprocessors, digital signal processors (DSPs), or programmable logic devices have been widely adopted in power electronics applications at relatively high power levels.
In ubiquitous low-to-medium power switched-mode power supply (SMPS) applications, including point-of-load (POL) regulators, nonisolated and isolated dc-dc converters, single-phase power factor correction (PFC) rectifiers, single-phase inverters, and lighting applications, adoption of digital power management and digital control has been slower. In these applications, switching frequencies are often in the range from hundreds of kilohertz to multiple megahertz, and much faster dynamic responses are required. The controller cost and the controller power consumption can easily present significant portions of the system cost and power dissipation. Furthermore, in many applications, control challenges have been successfully met by continuous advances of readily available analog controllers, using well-established analog analysis, modeling, and design techniques [1-5]. Nevertheless, practical digital control of high-frequency switched-mode power converters has moved from proof-of-concept demonstrations to digital pulse width modulation (DPWM) controller chips commercially available from multiple vendors, with growing adoption rates in many applications. Several factors have contributed to the increasing penetration of the concept of "digital power" in high-frequency power electronics applications:
- Ongoing advances in digital integrated-circuit processes have continued to increase processing capabilities while bringing the cost down.
- The needs for improved system integration and increasingly complex power management and monitoring functions have translated into the needs for digital interfaces and programmability in switched-mode power conversion applications [9-11].
- Practical high-performance digital control techniques have been introduced and demonstrated, together with innovative approaches offering performance gains or entirely new capabilities that would be difficult or impractical to realize using traditional analog techniques [12-14].
The "digital power" concept encompasses several aspects:
- Digital power management, which refers to system-level control and monitoring of power conversion and distribution, usually over a serial communication bus [9-11]. Power management functions include turning on and off or sequencing system power rails, adjusting setpoints for converter control loops, programming control loop parameters, monitoring and reporting of measured status or variables, and so on [15, 16]. These functions are typically performed at timescales that are relatively long compared to a switching period.
- Digital control, which includes time-domain and frequency-domain converter modeling and control techniques, with control actions performed at timescales comparable to a switching period.
- Digital implementation techniques, which can be classified into two main groups:
- Software-based controllers, where control algorithms are designed and implemented in code executed on general-purpose or specialized microcontrollers or DSP chips. An early example of application of microprogrammed digital control to power factor preregulators is presented in [17].
- Hardware-based controllers, based on custom-integrated circuits or programmable logic devices such as field-programmable gate arrays (FPGAs) [18, 19]. Early examples of such hardware-based digital controllers can be found in [20-22].
This book is focused on the fundamental aspects of analysis, modeling, and design of digital control loops around high-frequency switched-mode power converters in a systematic and rigorous manner. The objectives are to enable the reader to understand, analyze, model, design, and implement digital feedback loops around power converters, from system-level transfer function formulations to practical implementation details. The purpose of this chapter is to introduce the topics covered in the book and to motivate the reader to pursue the theoretical and practical concepts covered in the remaining chapters of this book. Furthermore, this introductory chapter points to some of the more advanced digital techniques reported in the literature, including approaches to dynamic response improvements, system identification, autotuning of digital control loops, and on-line efficiency optimization.
Digitally Controlled Switched-Mode Converters
A number of DPWM controller architectures and implementation strategies have been investigated and realized in practice. Many standard microcontrollers and DSP chips are now available, featuring multiple high-resolution PWM and analog-to-digital (A/D) channels, which allow software-based implementation of control and management functions. While advances in this area have been rapid, the software-based approaches are still better suited for applications where switching converters operate at relatively low switching frequencies. On the other hand, at switching frequencies in the hundreds of kilohertz to megahertz range, specialized hardware-based control loops are often preferred. This approach is illustrated in the architecture shown in Fig. 1 [12, 13]. The control loop is digital, using specialized, programmable A/D, DPWM, and compensator blocks to achieve high-performance closed-loop dynamic responses, while programmability, power management, and system interface functions are delegated to a microcontroller core. Similar combinations of programmable hardware peripherals specialized for switched-mode power converter applications, with software-based realizations of higher-level management and communication functions are often found in commercially available DPWM controllers.
Figure 1 Digital controller architecture for high-frequency switched-mode power converters [13].
Controllers of the type shown in Fig. 1 can be developed, realized, and tested using standard digital VLSI design flow starting from logic functions described using hardware description language (VHDL or Verilog), followed by prototyping and experimental verifications using FPGA development platforms, ultimately targeting relatively small, relatively low-gate-count integrated circuits capable of matching or surpassing the state-of-the-art analog solutions in terms of dynamic performance, power consumption, and cost. At the same time, digital PWM controllers offer digital system interface, programmability and flexibility, power management functions, reductions in the number of passive components, reduced sensitivity to process and temperature variations, and potentials for practical realizations of more advanced features.
Figure 2 shows a more detailed block diagram of a hardware-based digital controller around a POL synchronous Buck converter. Output voltage is sampled by an A/D converter and compared to a setpoint reference to produce a digital voltage error signal . The error signal is processed by a discrete-time digital proportional-integral-derivative (PID) compensator to generate a duty cycle command . In the basic version of the controller, the compensator gains , , and are found by design to meet control loop specifications, such as the crossover frequency and phase margin, as detailed in Chapters 1-6 of this book. Once a compensator is designed, the gains can be realized using digital multipliers, as shown in Fig. 2. As only a few bits are sufficient to represent the error signal , the entire compensator can also be implemented as a lookup table [21-24]. In a more advanced case, as illustrated by the digital autotuner block in Fig. 2, the compensator gains can be tuned in response to the actual system dynamics to meet the desired specifications. Autotuning is addressed in Chapter 7. Finally, a DPWM block generates the complementary gate-drive control signals and with duty cycle set by the digital command and with appropriate dead times. Together with various enhancements, such a controller can be realized in about 10,000 equivalent logic gates, which translates to about one-third of a square millimeter in a standard 0.35 µ m CMOS process. Furthermore, higher-density CMOS processes with high-voltage extensions suitable for power electronics applications are now readily available, making power and cost-effective digital controllers for high-frequency switched-mode power converters a reality. Examples of...
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