Fundamentals of Power Electronics


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1 Fundamentals of Power Electronics SECOND EDITION Robert W. Erickson Dragan Maksimovic University of Colorado Boulder, Colorado
2 Preface 1 Introduction Introduction to Power Processing Several Applications of Power Electronics Elements of Power Electronics 9 References 1 Converters in Equilibrium 11 2 Principles of Steady State Converter Analysis Introduction Inductor VoltSecond Balance, Capacitor Charge Balance, and the SmallRipple Approximation Boost Converter Example Cuk Converter Example Estimating the Output Voltage Ripple in Converters Containing TwoPole LowPass Filters Summary of Key Points 34 References 34 Problems 35 3 SteadyState Equivalent Circuit Modeling, Losses, and Efficiency The DC Transformer Model Inclusion of Inductor Copper Loss Construction of Equivalent Circuit Model 45 xix
3 viii Inductor Voltage Equation Capacitor Current Equation Complete Circuit Model Efficiency How to Obtain the Input Port of the Model Example: Inclusion of Semiconductor Conduction Losses in the Boost Converter Model Summary of Key Points 56 References 56 Problems 57 4 Switch Realization Switch Applications SingleQuadrant Switches CurrentBidirectional TwoQuadrant Switches VoltageBidirectional TwoQuadrant Switches FourQuadrant Switches Synchronous Rectifiers A Brief Survey of Power Semiconductor Devices Power Diodes MetalOxideSemiconductor FieldEffect Transistor (MOSFET) Bipolar Junction Transistor (BJT) Insulated Gate Bipolar Transistor (IGBT) Thyristors (SCR, GTO, MCT) Switching Loss Transistor Switching with Clamped Inductive Load Diode Recovered Charge Device Capacitances, and Leakage, Package, and Stray Inductances Efficiency vs. Switching Frequency Summary of Key Points 101 References 102 Problems The Discontinuous Conduction Mode Origin of the Discontinuous Conduction Mode, and Mode Boundary Analysis of the Conversion Ratio Мф,К) Boost Converter Example Summary of Results and Key Points 124 Problems Converter Circuits Circuit Manipulations Inversion of Source and Load Cascade Connection of Converters Rotation of ThreeTerminal Cell 137
4 ix Differential Connection of the Load A Short List of Converters Transformer Isolation FullBridge and HalfBridge Isolated Buck Converters Forward Converter PushPull Isolated Buck Converter Flyback Converter BoostDerived Isolated Converters Isolated Versions of the SEPIC and the Cuk Converter Converter Evaluation and Design Switch Stress and Utilization Design Using Computer Spreadsheet Summary of Key Points 177 References 177 Problems 179 Converter Dynamics and Control 185 AC Equivalent Circuit Modeling Introduction The Basic AC Modeling Approach Averaging the Inductor Waveforms Discussion of the Averaging Approximation Averaging the Capacitor Waveforms The Average Input Current Perturbation and Linearization Construction of the SmallSignal Equivalent Circuit Model Discussion of the Perturbation and Linearization Step Results for Several Basic Converters Example: A Nonideal Flyback Converter StateSpace Averaging The State Equations of a Network The Basic StateSpace Averaged Model Discussion of the StateSpace Averaging Result Example: StateSpace Averaging of a Nonideal BuckBoost Converter Circuit Averaging and Averaged Switch Modeling Obtaining a TimeInvariant Circuit Circuit Averaging Perturbation and Linearization Switch Networks Example: Averaged Switch Modeling of Conduction Losses Example: Averaged Switch Modeling of Switching Losses The Canonical Circuit Model Development of the Canonical Circuit Model 248
5 x Example: Manipulation of the BuckBoost Converter Model into Canonical Form Canonical Circuit Parameter Values for Some Common Converters Modeling the PulseWidth Modulator Summary of Key Points 256 References 257 Problems Converter Transfer Functions Review of Bode Plots Single Pole Response Single Zero Response Right HalfPlane Zero Frequency Inversion Combinations Quadratic Pole Response: Resonance The LowQ Approximation Approximate Roots of an ArbitraryDegree Polynomial Analysis of Converter Transfer Functions Example: Transfer Functions of the BuckBoost Converter Transfer Functions of Some Basic CCM Converters Physical Origins of the RHP Zero in Converters Graphical Construction of Impedances and Transfer Functions Series Impedances: Addition of Asymptotes Series Resonant Circuit Example Parallel Impedances: Inverse Addition of Asymptotes Parallel Resonant Circuit Example Voltage Divider Transfer Functions: Division of Asymptotes Graphical Construction of Converter Transfer Functions Measurement of AC Transfer Functions and Impedances Summary of Key Points 321 References 322 Problems Controller Design Introduction Effect of Negative Feedback on the Network Transfer Functions Feedback Reduces the Transfer Functions from Disturbances to the Output Feedback Causes the Transfer Function from the Reference Input to the Output to be Insensitive to Variations in the Gains in the Forward Path of the Loop Construction of the Important Quantities 1/(1 + T) and 77(1 + T) and the ClosedLoop Transfer Functions Stability 340
6 xi The Phase Margin Test The Relationship Between Phase Margin and ClosedLoop Damping Factor Transient Response vs. Damping Factor Regulator Design Lead (PD) Compensator Lag (PI) Compensator Combined (PID) Compensator Design Example Measurement of Loop Gains Voltage Injection Current Injection Measurement of Unstable Systems Summary of Key Points 369 References 369 Problems 369 Input Filter Design Introduction Conducted EMI The Input Filter Design Problem Effect of an Input Filter on Converter Transfer Functions Discussion Impedance Inequalities Buck Converter Example Effect of Undamped Input Filter Damping the Input Filter Design of a Damped Input Filter R f C b Parallel Damping R f L b Parallel Damping R f L b Series Damping Cascading Filter Sections Example: Two Stage Input Filter Summary of Key Points 403 References 405 Problems 406 AC and DC Equivalent Circuit Modeling of the Discontinuous Conduction Mode DCM Averaged Switch Model SmallSignal AC Modeling of the DCM Switch Network Example: ControltoOutput Frequency Response of a DCM Boost Converter Example: ControltoOutput Frequency Responses ofaccm/dcmsepic 429
7 xü 11.3 HighFrequency Dynamics of Converters in DCM Summary of Key Points 434 References 434 Problems Current Programmed Control Oscillation for D> A Simple FirstOrder Model Simple Model via Algebraic Approach: BuckBoost Example Averaged Switch Modeling A More Accurate Model CurrentProgrammed Controller Model Solution of the CPM Transfer Functions Discussion CurrentProgrammed Transfer Functions of the CCM Buck Converter Results for Basic Converters Quantitative Effects of CurrentProgrammed Control on the Converter Transfer Functions Discontinuous Conduction Mode Summary of Key Points 480 References 481 Problems 482 III Magnetics Basic Magnetics Theory Review of Basic Magnetics Basic Relationships Magnetic Circuits Transformer Modeling The Ideal Transformer The Magnetizing Inductance Leakage Inductances Loss Mechanisms in Magnetic Devices Core Loss LowFrequency Copper Loss Eddy Currents in Winding Conductors Introduction to the Skin and Proximity Effects Leakage Flux in Windings Foil Windings and Layers Power Loss in a Layer Example: Power Loss in a Transformer Winding Interleaving the Windings PWM Waveform Harmonics 522
8 xiii 13.5 Several Types of Magnetic Devices, Their BH Loops, and Core vs. Copper Loss Filter Inductor AC Inductor Transformer Coupled Inductor Flyback Transformer 13.6 Summary of Key Points References Problems Inductor Design Filter Inductor Design Constraints Maximum Flux Density Inductance Winding Area Winding Resistance The Core Geometrical Constant К A StepbyStep Procedure MultipleWinding Magnetics Design via the К Method Window Area Allocation Coupled Inductor Design Constraints Design Procedure 14.4 Examples Coupled Inductor for a TwoOutput Forward Converter CCM Flyback Transformer 14.5 Summary of Key Points References Problems Transformer Design Transformer Design: Basic Constraints Core Loss Flux Density Copper Loss Total Power Loss vs. ДВ Optimum Flux Density A StepbyStep Transformer Design Procedure Examples Example 1: SingleOutput Isolated Cuk Converter Example 2: MultipleOutput FullBridge Buck Converter AC Inductor Design Outline of Derivation StepbyStep AC Inductor Design Procedure
9 xiv 15.5 Summary 583 References 583 Problems 584 IV Modern Rectifiers and Power System Harmonics Power and Harmonics in Nonsinusoidal Systems Average Power RootMeanSquare (RMS) Value of a Waveform Power Factor Linear Resistive Load, Nonsinusoidal Voltage Nonlinear Dynamic Load, Sinusoidal Voltage Power Phasors in Sinusoidal Systems Harmonic Currents in ThreePhase Systems Harmonic Currents in ThreePhase FourWire Networks Harmonic Currents in ThreePhase ThreeWire Networks Harmonic Current Flow in Power Factor Correction Capacitors AC Line Current Harmonic Standards International Electrotechnical Commission Standard IEEE/ANSI Standard Bibliography 605 Problems LineCommutated Rectifiers The SinglePhase FullWave Rectifier Continuous Conduction Mode Discontinuous Conduction Mode Behavior when С is Large Minimizing THD when С is Small The ThreePhase Bridge Rectifier Continuous Conduction Mode Discontinuous Conduction Mode Phase Control Inverter Mode Harmonics and Power Factor Commutation Harmonic Trap Filters Transformer Connections Summary 630 References 631 Problems PulseWidth Modulated Rectifiers Properties of the Ideal Rectifier 638
10 18.2 Realization of a NearIdeal Rectifier CCM Boost Converter DCM Flyback Converter Control of the Current Waveform Average Current Control Current Programmed Control Critical Conduction Mode and Hysteretic Control Nonlinear Carrier Control SinglePhase Converter Systems Incorporating Ideal Rectifiers Energy Storage Modeling the Outer LowBandwidth Control System RMS Values of Rectifier Waveforms Boost Rectifier Example Comparison of SinglePhase Rectifier Topologies Modeling Losses and Efficiency in CCM HighQuality Rectifiers Expression for Controller Duty Cycle d(t) Expression for the DC Load Current Solution for Converter Efficiency Ti Design Example Ideal ThreePhase Rectifiers Summary of Key Points 691 References 692 Problems 696 V Resonant Converters Resonant Conversion Sinusoidal Analysis of Resonant Converters Controlled Switch Network Model Modeling the Rectifier and Capacitive Filter Networks Resonant Tank Network Solution of Converter Voltage Conversion Ratio M = V/V Examples Series Resonant DCDC Converter Example Subharmonic Modes of the Series Resonant Converter Parallel Resonant DCDC Converter Example Soft Switching Operation of the Full Bridge Below Resonance: ZeroCurrent Switching Operation of the Full Bridge Above Resonance: ZeroVoltage Switching LoadDependent Properties of Resonant Converters Inverter Output Characteristics Dependence of Transistor Current on Load Dependence of the ZVS/ZCS Boundary on Load Resistance 734
11 xvi Another Example Exact Characteristics of the Series and Parallel Resonant Converters Series Resonant Converter Parallel Resonant Converter Summary of Key Points 752 References 752 Problems Soft Switching SoftSwitching Mechanisms of Semiconductor Devices Diode Switching MOSFET Switching IGBT Switching The ZeroCurrentSwitching QuasiResonant Switch Cell Waveforms of the HalfWave ZCS QuasiResonant Switch Cell The Average Terminal Waveforms The FullWave ZCS QuasiResonant Switch Cell Resonant Switch Topologies The ZeroVoltageSwitching QuasiResonant Switch The ZeroVoltageSwitching MultiResonant Switch QuasiSquareWave Resonant Switches Soft Switching in PWM Converters The ZeroVoltage Transition FullBridge Converter The Auxiliary Switch Approach Auxiliary Resonant Commutated Pole Summary of Key Points 797 References 798 Problems 800 Appendices 803 Appendix A RMS Values of CommonlyObserved Converter Waveforms 805 A.l Some Common Waveforms 805 A.2 General Piecewise Waveform 809 Appendix В Simulation of Converters 813 B. 1 Averaged Switch Models for Continuous Conduction Mode 815 B.l. 1 Basic CCM Averaged Switch Model 815 B.1.2 CCM Subcircuit Model that Includes Switch Conduction Losses 816 B. 1.3 Example: SEPIC DC Conversion Ratio and Efficiency 818 B.1.4 Example: Transient Response of a BuckBoost Converter 819 B.2 Combined CCM/DCM Averaged Switch Model 822 B.2.1 Example: SEPIC Frequency Responses 825 B.2.2 Example: Loop Gain and ClosedLoop Responses of a Buck Voltage Regulator 827
12 xvii B.2.3 Example: DCM Boost Rectifier 832 B.3 Current Programmed Control 834 B.3.1 Current Programmed Mode Model for Simulation 834 B.3.2 Example: Frequency Responses of a Buck Converter with Current Programmed Control 837 References 840 Appendix С Middlebrook's Extra Element Theorem 843 C.l Basic Result 843 C.2 Derivation 846 C.3 Discussion 849 C.4 Examples 850 C.4.1 A Simple Transfer Function 850 C.4.2 An Unmodeled Element 855 C.4.3 Addition of an Input Filter to a Converter 857 C.4.4 Dependence of Transistor Current on Load in a Resonant Inverter 859 References 861 Appendix D Magnetics Design Tables 863 D.l Pot Core Data 864 D.2 ЕЕ Core Data 865 D.3 EC Core Data 866 D.4 ETD Core Data 866 D.5 PQ Core Data 867 D.6 American Wire Gauge Data 868 References 869 Index 871
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