Advances in Averaged Switch Modeling

Advances in Averaged Switch Modeling Robert W. Erickson Power Electronics Group University of Colorado Boulder, Colorado USA 80309-0425 rwe@boulder.colorado.edu http://ece-www.colorado.edu/~pwrelect 1

Outline of tutorial presentation 1. Introduction: Objectives of converter modeling via averaged switch approach 2. Averaged switch modeling of PWM converters operating in continuous conduction mode 2.1 Basics of averaged switch modeling 2.2 Modeling switching loss 2.3 Modeling converter dynamics 3. Discontinuous conduction mode in PWM converters 3.1 Averaged switch model 3.2 Properties of the dependent power source 3.3 Solution of converter characteristics 3.4 Modeling converter dynamics 2

Outline of tutorial presentation continued 4. Current programmed control of PWM converters 4.1 Averaged switch model, CCM 4.2 Ac model, CCM 4.3 DCM models 5. Single-phase low-harmonic rectifiers 5.1 The ideal rectifier 5.2 Integrated High-Quality Rectifier-Regulators 6. Single-switch three-phase low-harmonic rectifiers 6.1 The ideal 3ø rectifier 6.2 Single-switch approaches to three-phase rectification 7. Summary 3

1. Introduction Objectives of converter modeling via the averaged switch approach 4

Objectives of converter modeling Dc-dc converter system: modeling of efficiency and losses Boost converter example i L i C Develop loss model Solve for steady-state losses and efficiency V g DT s T s C R v Specify component requirements such that efficiency specifications are met Steady-state model R L DRon D'V D D'R D D' : 1 V g I V R 5

Objectives of converter modeling Dc-dc converter system: modeling of dynamics Develop block diagram of system Determine the relevant small-signal transfer functions Control-to-output Disturbance-to-output Design system to meet specifications Reference input Error signal Compensator G c (s) Pulse-width modulator ac line variation Duty cycle variation G vg (s) G vd (s) Z out (s) Load current variation Converter power stage Output voltage variation H(s) Sensor gain 6

Objectives of converter modeling Ac-dc low-harmonic rectifier system: efficiency and losses Boost rectifier example i g i ac v ac v g R L L Controller Q 1 D 1 i d C i v R Develop loss model Solve for steady-state losses and efficiency Specify component requirements such that efficiency specifications are met Low-frequency averaged model η 1 0.95 i g R L dr on d' : 1 V F i d i = I 0.9 v g C R (Large) v = V 0.85 0.8 0.75 0.0 0.2 0.4 0.6 0.8 1.0 V M /V 7

Objectives of converter modeling Ac-dc low-harmonic rectifier system: dynamic modeling v ac i g i ac v g Boost converter L D 1 Q 1 i 2 v C C DCDC Converter i Load v v control Multiplier X v g R s i g v a PWM d v v ref1 = k x v g v control v err G c (s) Compensator Compensator and modulator v ref3 Wide-bandwidth input current controller Wide-bandwidth output voltage controller Model inner and outer loops of rectifier control system Design to meet specifications v C Compensator v ref2 Low-bandwidth energy-storage capacitor voltage controller 8

Averaged switch modeling Switch network is replaced by averaged circuit model. Switching harmonics are removed, and low-frequency components of waveforms are modeled in a simple way. A very general approach to modeling converter losses, efficiency, and dynamics. Easily taught to students. Yields an intuitive understanding of converter behavior in CCM, DCM, current-programmed mode, etc. Results are easy to generalize to other converters. Well-suited to simulation using PSPICE. Applicable to dc-dc converters, as well as dc-ac inverters, ac-dc low-harmonic rectifiers, ac-ac matrix converters. 9

2. Averaged switch modeling of PWM converters operating in the continuous conduction mode 2.1 Basics of averaged switch modeling 2.2 Modeling switching loss 2.3 Modeling converter dynamics 10

2.1. Averaged switch modeling Basic approach Given a PWM converter operating in continuous conduction mode: L 1 C 1 D 1 V g L 2 C 2 R v SEPIC example Q 1 Separate the switching elements from the remainder of the converter... 11

Definition of switch network, SEPIC example Define a switch network, containing all of the converter switching elements. The remainder of the converter is linear and timeinvariant. The terminal voltages and currents of the switch network can be arbitrarily defined. v g i L1 L 1 i 1 v 1 C 1 v C1 Q 1 L 2 i L2 Switch network Duty cycle D 1 d v 2 i 2 C 2 v C2 R 12

Switching converter system with switch network explicitly defined Power input Time-invariant network containing converter reactive elements Load v g C L R v v C i L i 1 v 1 Switch network i 2 v 2 Control input d 13

Discussion The number of ports in the switch network is less than or equal to the number of SPST switches in the converter Simple dc-dc case, in which converter contains two SPST switches: switch network contains two ports The switch network terminal waveforms are then the port voltages and currents: v 1, i 1, v 2, and i 2. Two of these waveforms can be taken as independent inputs to the switch network; the remaining two waveforms are then viewed as dependent outputs of the switch network. Switch network also includes control input d Definition of the switch network terminal quantities is not unique. Different definitions lead equivalent results having different forms 14

A few points regarding averaged switch modeling The switch network can be defined arbitrarily, as long as its terminal voltages and currents are independent, and the switch network contains no reactive elements. It is not necessary that some of the switch network terminal quantities coincide with inductor currents or capacitor voltages of the converter, or be nonpulsating. The object is simply to write the averaged equations of the switch network; i.e., to express the average values of half of the switch network terminal waveforms as functions of the average values of the remaining switch network terminal waveforms, and the control input. 15

Terminal waveforms of the switch network v 1 v C1 v C2 L 1 C 1 0 0 0 dt s T s t v g i L1 v C1 L 2 C 2 v C2 R i 1 i L1 i L2 i L2 0 0 0 dt s v 2 v C1 v C2 T s t i 1 v 1 Switch network Q 1 D 1 v 2 i 2 0 0 0 dt s T s t Duty cycle d i 2 i L1 i L2 0 0 0 dt s T s t 16

The averaging step x Ts = 1 T s t t Ts xdt Now average all waveforms over one switching period: Power input Averaged time-invariant network containing converter reactive elements Load v g Ts C L R v Ts v C Ts i L Ts i 1 Ts i 2 Ts v 1 Ts Averaged switch network v 2 Ts Control input d 17

The averaging step The basic assumption is made that the natural time constants of the converter are much longer than the switching period, so that the converter contains low-pass filtering of the switching harmonics: One may average the waveforms over an interval that is short compared to the system natural time constants, without significantly altering the system response. In particular, averaging over the switching period T s removes the switching harmonics, while preserving the low-frequency components of the waveforms. This step removes the small but mathematically-complicated switching harmonics, leading to a relatively simple and tractable converter model. In practice, the only work needed for this step is to average the switch dependent waveforms. 18

Averaged terminal equations of the switch network (small switching ripple is neglected) v 1 v C1 v C2 v 2 v C1 v C2 0 0 0 dt s T s t 0 0 0 dt s T s t i 1 i L1 i L2 i 2 i L1 i L2 0 0 0 dt s T s t 0 0 0 dt s T s t 19

Derivation of switch network equations (Algebra steps) We can write Result i 1 Ts v 1 Ts v 2 Ts Hence i 2 Ts Averaged switch network Modeling the switch network via averaged dependent sources 20

Steady-state switch model: Dc transformer model Original switch network i 1 Switch network i 2 v 1 Q 1 D 1 v 2 Averaged steady-state model: DC transformer Correctly represents the relationships between the dc and low-frequency components of the terminal waveforms of the switch network V 1 I 1 Duty cycle d D' : D I 2 V 2 21

Steady-state CCM SEPIC model Replace switch network with dc transformer model L 1 C 1 I L1 V C1 V g L 2 C 2 V C2 R I L2 I 1 V 1 D' : D V 2 Can now let inductors become short circuits, capacitors become open circuits, and solve for dc conditions. I 2 Can simulate this model using PSPICE, to find transient waveforms 22

The ideal dc transformer model Switch network Averaged switch model i 1 i 2 1 : d CCM v 1 v 2 Correctly represents the basic properties of the ideal CCM PWM switch network: 100% efficiency Voltage and current conversion ratios, controllable by duty cycle d 23

Several ways to define the PWM switch network, and the corresponding CCM models i 1 i 2 i 1 Ts 1 : D i 2 Ts v 1 v 2 v 1 Ts v 2 Ts i 1 i 2 i 1 Ts D' : 1 i 2 Ts v 1 v 2 v 1 Ts v 2 Ts i 1 i 2 i 1 Ts D' : D i 2 Ts v 1 v 2 v 1 Ts v 2 Ts 24

PSPICE simulation of the dc transformer model Use dependent sources: Switch network Averaged switch model PSPICE switch model i 1 i 2 1 : d v 1 v 2 d i 2 Ts d v 1 T s Polynomial dependent sources can be used to simulate the terminal equations of the ideal dc transformer model 25

2.2. Modeling switching loss Example: diode stored charge in boost converter i L L i 1 i 2 v g v 1 v 2 C R v v 1 Waveforms: v 2 v 2 i 2 t r 0 dt s T s 0 t Other switching loss mechanisms are ignored in this example; one can include other losses if desired, using a similar procedure i 1 i 1 Determine averaged terminal waveforms of switch network 0 0 t Construct averaged equivalent circuit model Area Q r 26

Expressions for average terminal waveforms Boost converter, switching loss example v 1 v 1 T s = 1 T s t r (1 d)t s v 2 Ts v 2 v 2 i 2 Ts = 1 T s Q r i 1 Ts t r (1 d)t s t r = diode reverse recovery time Q r = diode recovered charge i 2 i 1 t r 0 dt s T s 0 i 1 t 0 0 t Area Q r 27

Averaged equivalent circuit of switch network v 1 Ts = t r T s (1 d) v 2 Ts i 2 Ts = Q r T s t r T s (1 d) i 1 T s i 1 i 2 i 1 Ts i 2 Ts v 1 v 2 v 1 Ts v 2 Ts Diode reverse recovery time affects conversion ratio Stored charge leads to power loss, modeled by current sink 28

Insert averaged switch model into converter circuit Original converter v g i L L i 1 v 1 i 2 v 2 C R v Averaged model v g Ts i L Ts L i 1 Ts i 2 Ts v 1 Ts v 2 Ts C R v Ts Steadystate solution: 29

2.3. Modeling converter dynamics: Small-signal linearization of model Perturb and linearize the switch network averaged waveforms about a quiescent operating point. Let: Voltage equation becomes Eliminate nonlinear terms and solve for v 1 terms: 30

Linearization, continued Current equation becomes Eliminate nonlinear terms and solve for i 2 terms: 31

Switch network: Small-signal ac model Reconstruct an equivalent circuit that corresponds to these smallsignal equations: D' : D Transistor port Diode port A general small-signal ac model for the PWM switch network operating in CCM. 32

Small-signal ac model of the CCM SEPIC Replace switch network with small-signal ac model: L 1 C 1 L 2 C 2 R D' : D Can now solve this model to determine ac transfer functions 33

Small-signal models of several basic switch networks i 1 i 2 1 : D v 1 v 2 i 1 i 2 D' : 1 v 1 v 2 i 1 i 2 D' : D v 1 v 2 34

Table of results Transfer functions of the basic buck, boost, and buck-boost converters Control-to-output and line-to-output transfer functions G vd (s) and G vg (s) Converter G g0 G d0 ω 0 Q ω z V 1 buck D D R C LC L 1 V D' boost D' D' D'R C D' 2 R LC L L buck-boost D' D V D' D D' 2 D'R C D' 2 R LC L D L where the transfer functions are written in the standard forms 35

3. Discontinuous conduction mode in PWM converters 3.1 Averaged switch model 3.2 Properties of the dependent power source 3.3 Solution of averaged model 3.4 Modeling converter dynamics 36

Equivalent circuit modeling of discontinuous conduction mode converters DC AC 1 : M(D) 1 : M(D) L e CCM V g R V C R DCM V g? R V? R 37

Change in characteristics at the CCM/DCM boundary Steady-state output voltage becomes strongly load-dependent Simpler dynamics: one pole and the RHP zero are moved to very high frequency, and can normally be ignored Traditionally, boost and buck-boost converters are designed to operate in DCM at full load All converters may operate in DCM at light load So we need equivalent circuits that model the steady-state and smallsignal ac models of converters operating in DCM The averaged switch approach yields an intuitive result that is relatively easy to solve 38

3.1. Derivation of DCM averaged switch model Buck-boost example Define switch terminal quantities v 1, i 1, v 2, i 2, as shown Let us find the averaged quantities v 1, i 1, v 2, i 2, for operation in DCM, and determine the relations between them v g Switch network i 1 i 2 v 1 v L L i L v 2 C R v 39

DCM waveforms i L i pk i 1 Area q 1 i pk 0 v L v g t v 1 v g v 0 0 v g v i 2 i pk Area q 2 Switch network i 1 i 2 v g v 1 v L L i L v 2 C R v v 2 T s v g v v 0 d 1 T s d 2 T s d 3 T s t 40

Basic DCM equations Peak inductor current: i 1 Area q 1 i pk Average inductor voltage: v 1 v g v v g In DCM, the diode switches off when the inductor current reaches zero. Hence, i(0) = i(t s ) = 0, and the average inductor voltage is zero. This is true even during transients. i 2 0 i pk Area q 2 v 2 v g v Solve for d 2 : T s v 0 d 1 T s d 2 T s d 3 T s t 41

Average switch network terminal voltages Average the v 1 waveform: i 1 Area q 1 i pk Eliminate d 2 and d 3 : v 1 v g v v g Similar analysis for v 2 waveform leads to i 2 0 i pk Area q 2 v 2 v g v T s v 0 d 1 T s d 2 T s d 3 T s t 42

Average switch network terminal currents Average the i 1 waveform: i 1 Area q 1 i pk The integral q 1 is the area under the i 1 waveform during first subinterval. Use triangle area formula: v 1 v g v v g 0 i 2 Eliminate i pk : i pk Area q 2 Note i 1 Ts is not equal to d i L Ts! Similar analysis for i 2 waveform leads to v 2 T s v g v v 0 d 1 T s d 2 T s d 3 T s t 43

Input port: Averaged equivalent circuit R e (d 1 ) Low-frequency components of input port waveforms obey Ohm s law 44

Output port: Averaged equivalent circuit i p v Output port is a source of power p Power p is independent of load characteristics Power p is dependent on (equal to) the power apparently consumed by the switch network input port 45

3.2. The dependent power source i i vi = p p v v Must avoid open- and short-circuit connections of power sources Power sink: negative p 46

How the power source arises in lossless two-port networks In a lossless two-port network without internal energy storage: instantaneous input power is equal to instantaneous output power In all but a small number of special cases, the instantaneous power throughput is dependent on the applied external source and load If the instantaneous power depends only on the external elements connected to one port, then the power is not dependent on the characteristics of the elements connected to the other port. The other port becomes a source of power, equal to the power flowing through the first port A power source (or power sink) element is obtained 47

Properties of power sources Series and parallel connection of power sources P 1 P 1 P 2 P 3 P 2 P 3 Reflection of power source through a transformer P 1 n 1 : n 2 P 1 48

The loss-free resistor (LFR) R e (d 1 ) A two-port lossless network Input port obeys Ohm s Law Power entering input port is transferred to output port 49

Averaged modeling of CCM and DCM switch networks Switch network Averaged switch model i 1 i 2 1 : d CCM v 1 v 2 i 1 i 2 DCM v 1 v 2 R e (d 1 ) 50

Averaged switch model: buck-boost example Original circuit Switch network i 1 i 2 v 1 v 2 v g v L L i L C R v Averaged model R e (d) C R L 51

3.3. Solution of averaged model: steady state Let L short circuit I 1 P C open circuit V g R e (D) R V Converter input power: Equate and solve: Converter output power: 52

Steady-state LFR solution is a general result, for any system that can be modeled as an LFR. For the buck-boost converter, we have Eliminate R e : which agrees with the results of previous steady-state analyses. 53

Steady-state LFR solution with ac terminal waveforms v g i 1 i 2 p C R R e v Converter average input power: Converter average output power: Note that no average power flows into capacitor Equate and solve: 54

Averaged models of other DCM converters Determine averaged terminal waveforms of switch network In each case, averaged transistor waveforms obey Ohm s law, while averaged diode waveforms behave as dependent power source Can simply replace transistor and diode with the averaged model as follows: i 1 i 2 v 1 v 2 R e (d 1 ) 55

DCM buck, boost Buck R e (d) L C R Boost L R e (d) C R 56

DCM Cuk, SEPIC Cuk L 1 C 1 L 2 R e (d) C 2 R SEPIC L 1 C 1 R e (d) L 2 C 2 R 57

Steady-state solution: DCM buck, boost Let L short circuit C open circuit Buck V g R e (D) P R V Boost V g R e (D) P R V 58

Steady-state solution of DCM/LFR models Converter M, CCM M, DCM Buck D 2 1 1 4R e /R Boost Buck-boost, Cuk SEPIC 1 1 D D 1 D D 1 D 1 1 4R/R e 2 R R e R R e 59

3.4. Small-signal ac modeling of the DCM switch network Large-signal averaged model Perturb and linearize: let R e (d) d 60

Linearization via Taylor series Given the nonlinear equation Expand in three-dimensional Taylor series about the quiescent operating point: (for simple notation, drop angle brackets) 61

Equate dc and first-order ac terms AC DC 62

Output port same approach DC terms Small-signal ac linearization 63

Output resistance parameter r 2 Power source characteristic Quiescent operating point Load characteristic Linearized model 64

Small-signal DCM switch model parameters Switch type g 1 j 1 r 1 g 2 j 2 r 2 Buck, Fig. 10.16(a) 1 R e 2(1 M)V 1 DR e R e 2 M MR e 2(1 M)V 1 DMR e M 2 R e Boost, Fig. 10.16(b) 1 2 (M 1) 2 2MV (M 1) R 1 e M D(M 1)R e R e 2M 1 (M 1) 2 R e 2V 1 D(M 1)R e (M 1) 2 R e Buck-boost, Fig. 10.7(b) 0 2V 1 DR e R e 2M R e 2V 1 DMR e M 2 R e 65

Small-signal ac model, DCM buck-boost example Switch network small-signal ac model C R L 66

A more convenient way to model the buck and boost small-signal DCM switch networks i 1 i 2 i 1 i 2 v 1 v 2 v 1 v 2 In any event, a small-signal two-port model is used, of the form 67

Small-signal ac models of the DCM buck and boost converters (more convenient forms) DCM buck switch network small-signal ac model L C R L DCM boost switch network small-signal ac model C R 68

DCM small-signal transfer functions When expressed in terms of R, L, C, and M (not D), the smallsignal transfer functions are the same in DCM as in CCM Hence, DCM boost and buck-boost converters exhibit two poles and one RHP zero in control-to-output transfer functions But, value of L is small in DCM. Hence RHP zero appears at high frequency, usually greater than switching frequency Pole due to inductor dynamics appears at high frequency, near to or greater than switching frequency So DCM buck, boost, and buck-boost converters exhibit essentially a single-pole response A simple approximation: let L 0 69

The simple approximation L 0 Buck, boost, and buck-boost converter models all reduce to DCM switch network small-signal ac model C R Transfer functions control-to-output with line-to-output 70

Transfer function salient features Converter G d0 G g0 ω p Buck 2V D 1 M 2 M M 2 M (1 M)RC Boost 2V D M 1 2M 1 M 2M 1 (M 1)RC Buck-boost V D M 2 RC 71

DCM boost example Control-to-output transfer function G vd (s) R = 12 Ω i L v L D 1 i D i C L = 5 µh V g Q 1 C R v C = 470 µf f s = 100 khz The output voltage is regulated to be V = 36 V. It is desired to determine G vd (s) at the operating point where the load current is I = 3 A and the dc input voltage is V g = 24 V. 72

Evaluate simple model parameters 73

Control-to-output transfer function, boost example 60 dbv G vd G vd 40 dbv G d0 37 dbv 20 dbv 0 dbv 20 dbv 40 dbv G vd 0 G vd f p 112 Hz 20 db/decade 0 90 180 270 10 Hz 100 Hz 1 khz 10 khz 100 khz 74 f

4. Current programmed control of PWM converters 4.1. Averaged switch model, CCM 4.2. Ac model, CCM 4.3. DCM models 75

Current-programmed control Buck converter v g i s Q 1 D 1 L i L C v R The peak transistor current replaces the duty cycle as the converter control input. Measure switch current i s R f i c R f R f i s Clock Analog comparator 0 T s S Q R Latch m 1 Switch current i s Control signal i c Control input Current-programmed controller 0 dt s T s Transistor status: on off t Compensator v Clock turns transistor on Comparator turns transistor off v ref Conventional output voltage controller 76

A simple approximation Neglects switching ripple and artificial ramp (slope compensation) Yields physical insight and simple first-order model Accurate when converter operates well into CCM (so that switching ripple is small) and when the magnitude of the artificial ramp is not too large Well-accepted by practicing engineers Resulting small-signal relation: 77

4.1. Averaged switch modeling with the simple approximation Buck converter example i 1 i 2 L i L v g v 1 v 2 C R v Switch network Averaged terminal waveforms, CCM: The simple approximation: 78

CPM averaged switch equations Eliminate duty cycle: So: Output port is a current source Input port is a dependent current sink 79

CPM averaged switch model i 1 Ts i 2 Ts L i L Ts p Ts v g Ts v 1 Ts i c Ts v 2 Ts C R v Ts Averaged switch network 80

Results for other converters Boost i L Ts L v g Ts i c Ts p Ts C R v Ts Averaged switch network Averaged switch network Buck-boost p Ts i c Ts v g Ts C R v Ts L i L Ts 81

4.2. Perturbation and linearization to construct small-signal model, CCM Let Resulting input port equation: Small-signal result: Output port equation: î 2 = î c 82

Resulting small-signal model Buck example L C R Switch network small-signal ac model 83

Origin of input port negative incremental resistance i 1 Ts Power source characteristic v 1 Ts i 1 Ts = p Ts Quiescent operating point I 1 V 1 v 1 Ts 84

Expressing the equivalent circuit in terms of the converter input and output voltages L C R 85

Predicted transfer functions of the CPM buck converter L C R 86

Table of results basic converters r 1 r 2 C R Converter g 1 f 1 r 1 g 2 f 2 r 2 Buck D R D 1 sl R R D 2 0 1 Boost 0 1 1 D'R D' 1 sl D' 2 R R Buck-boost D R D 1 sl D'R D'R D 2 D2 D'R D' 1 sdl D' 2 R R D 87

4.3. Discontinuous conduction mode in current-programmed converters Again, use averaged switch modeling approach Result: simply replace Transistor by power sink Diode by power source Inductor dynamics appear at high frequency, near to or greater than the switching frequency Small-signal transfer functions contain a single low frequency pole DCM CPM boost and buck-boost are stable without artificial ramp DCM CPM buck without artificial ramp is stable for D < 2/3. A small artificial ramp m a 0.086m 2 leads to stability for all D. 88

DCM CPM buck-boost example i L i 1 Switch network i 2 v 1 v 2 i c m a i pk v g C R v L i L v L 0 t 0 89

Analysis i L i c m a i pk v L 0 t 0 90

Averaged switch input port equation i 1 Area q1 i pk i 2 i pk Area q 2 t d 1 T s d 2 T s d 3 T s T s 91

Discussion: switch network input port Averaged transistor waveforms obey a power sink characteristic During first subinterval, energy is transferred from input voltage source, through transistor, to inductor, equal to This energy transfer process accounts for power flow equal to which is equal to the power sink expression of the previous slide. 92

Averaged switch output port equation i 1 Area q1 i pk i 2 i pk Area q 2 t d 1 T s d 2 T s d 3 T s T s 93

Discussion: switch network output port Averaged diode waveforms obey a power sink characteristic During second subinterval, all stored energy in inductor is transferred, through diode, to load Hence, in averaged model, diode becomes a power source, having value equal to the power consumed by the transistor power sink element 94

Averaged equivalent circuit C R L 95

Steady state model: DCM CPM buck-boost V g P R V Solution for a resistive load 96

Models of buck and boost Buck L C R Boost L C R 97

Summary of steady-state DCM CPM characteristics Converter M I crit Stability range when m a = 0 Buck P load P P load 1 2 I c M m a T s 0 M < 2 3 Boost P load P load P I c M M 1 2 M m a T s 0 D 1 Buck-boost Depends on load characteristic: P load = P I c M m M 1 a T s 2 M 1 0 D 1 98

Buck converter: output characteristic with m a = 0 with a resistive load, there can be two operating points I I c CPM buck characteristic with m a = 0 resistive load line I = V/R the operating point having V > 0.67V g can be shown to be unstable A B CCM DCM V g V 99

Linearized small-signal models Buck L C R Boost L C R 100

Linearized small-signal models: Buck-boost C R L 101

DCM CPM small-signal parameters: input port Converter g 1 f 1 r 1 Buck 1 R M 2 1 M 1 m a m 1 1 m a m 1 2 I 1 I c R 1 M 1 M 2 m a m 1 1 m a m 1 Boost 1 R M M 1 2 I I c M 2 R M 2 M 2 1 m a m 1 1 m a m 1 Buck-boost 0 2 I 1 I c 1 R M 2 m a m 1 1 m a m 1 102

DCM CPM small-signal parameters: output port Converter g 2 f 2 r 2 Buck 1 R M 1 M m a m 1 2 M M 1 m a m 1 2 I I c R 1 M 1 m a m 1 1 2M m a m 1 Boost 1 R M M 1 2 I 2 R I M 1 c M Buck-boost 2M R m a m 1 2 I 2 I c R 1 m a m 1 103

Simplified DCM CPM model, with L = 0 Buck, boost, buck-boost all become C R 104

5. Single-phase low-harmonic rectification 5.1. The Ideal Rectifier 5.2. Integrated High-Quality Rectifier-Regulators 105

5.1 Properties of the Ideal Rectifier It is desired that the rectifier present a resistive load to the ac power system. This leads to unity power factor ac line current has same waveshape as voltage i ac R e is called the emulated resistance v ac R e 106

Control of power throughput i ac Power apparently consumed by R e is actually transferred to rectifier dc output port. To control the amount of output power, it must be possible to adjust the value of R e. v ac R e (v control ) v control 107

Output port model The ideal rectifier is lossless and contains no internal energy storage. Hence, the instantaneous input power equals the instantaneous output power. Since the instantaneous power is independent of the dc load characteristics, the output port obeys a power source characteristic. v ac i ac ac input Ideal rectifier (LFR) p = v 2 ac /R e R e (v control ) v control i v dc output 108

The dependent power source i i i vi = p p v v p v power source power sink i-v characteristic 109

Equations of the ideal rectifier / LFR Defining equations of the ideal rectifier: When connected to a resistive load of value R, the input and output rms voltages and currents are related as follows: A switch network that is capable of satisfying the above (averaged) equations can be employed in low-harmonic rectifier applications 110

Single-phase system with internal energy storage i g Ideal rectifier (LFR) i 2 p load = VI = P load v ac i ac v g R e p ac Ts C v C Dcdc converter v i load Energy storage capacitor voltage v C must be independent of input and output voltage waveforms, so that it can vary according to Energy storage capacitor This system is capable of Wide-bandwidth control of output voltage Wide-bandwidth control of input current waveform Internal independent energy storage 111

5.2. Integrated High-Quality Rectifier-Regulators The fact that converters can naturally exhibit the loss-free resistor characteristic suggests that one could create a single converter power stage, having a single active device, that simultaneously performs the functions of: Low-harmonic rectification / input resistor emulation Internal low-frequency energy storage Wide-bandwidth regulation of the dc output voltage High-frequency transformer isolation These properties can be achieved by: Beginning with a DCM or resonant converter that exhibits the properties of the loss-free resistor Adding a low-frequency energy storage capacitor Cascading a CCM converter, possibly having transformer isolation Integrating the cascaded converters, to share the switching element 112

BIBRED Boost Integrated with Buck: Rectifier / Energy storage / Dc-dc converter L 1 D 1 C 1 C 2 L2 Q 1 D 2 C 3 R v 1 : n Resembles a transformer-isolated Cuk converter, except that diode D 1 allows the input inductor to operate independently in DCM C 1 is the low-frequency energy-storage capacitor Automatic input power-factor and harmonic correction is obtained Output voltage is regulated via duty-cycle control, as in conventional dc-dc converter Variable switching frequency allows reduction of voltage variations in capacitor C 1 113

BIFRED Boost Integrated with Flyback: Rectifier / Energy storage / Dc-dc converter Converter power stage L 1 D 1 C 1 D 2 i 1 C 2 R v Q 1 Averaged model L 1 1 : n n : D L m (1D) : 1 R e p C 1 C 2 R Predicts converter waveforms Suitable for analytical solution or computer simulation 114 n : 1

Discussion Provided that the operating point variations are not too great, these converters allow reduction of the number of switches and control circuit complexity In a universal-input application in which the load power varies by a factor of 10:1 or more, the transistor voltage stress becomes too large The converter must be designed such that the input inductor operates in DCM while the output inductor operates in CCM PSPICE simulation of these circuits is not successful unless averaged models are used A number of other topologies are known 115

6. Three-phase low harmonic rectifiers 6.1. The ideal 3ø rectifier 6.2. Single-switch approaches to three-phase rectification 116

6.1. The ideal three-phase rectifier Ideal 3ø rectifier, modeled as three 1ø ideal rectifiers: 3øac input dc output ø a i a R e p a ø b i b R e p b R v ø c i c R e p c 117

Ideal 3ø rectifier model Combine parallel-connected power sources into a single source p tot : 3øac input dc output ø a i a R e ø b i b R e p tot = p a p b p c R v ø c i c R e 118

Value of p tot Ac input voltages: 3øac input ø a i a R e p a dc output ø b i b R e p b R v Instantaneous phase powers: ø c i c R e p c Total 3ø instantaneous power: 2 nd harmonic terms add to zero total 3ø power p tot is constant 119

Instantaneous power in ideal 3ø rectifier 3øac input dc output In a balanced system, the ideal 3ø rectifier supplies constant power to its dc output a constant power load can be supplied, without need for lowfrequency internal energy storage ø a ø b ø c i a i b i c R e R e R e p tot = p a p b p c R v 120

Summary The averaged switch modeling approach: replace switch network with an equivalent circuit that correctly predicts the low-frequency components of the switch network terminal waveforms PWM continuous conduction mode (CCM): switch network is modeled by a dc transformer PWM discontinuous conduction mode (DCM): transistor is replaced by effective resistor, and diode is replaced by dependent power source Current-programmed control, CCM: switch network is replaced by a current source and a dependent power sink Current-programmed control, DCM: switch network is replaced by power source and power sink Ideal rectifier: switch network is replaced by 1ø or 3ø loss-free resistor network, consisting of effective resistors and power source In each case, we can linearize to obtain the converter small-signal model. We can also easily implement these models in PSPICE 133