Part II Converter Dynamics and Control

Part II Converter Dynamics and Control 7. AC equivalent circuit modeling 8. Converter transfer functions 9. Controller design 10. Ac and dc equivalent circuit modeling of the discontinuous conduction mode 11. Current programmed control 1

Chapter 7. AC Equivalent Circuit Modeling 7.1. Introduction 7.2. The basic ac modeling approach 7.3. Example: A nonideal flyback converter 7.4. State-space averaging 7.5. Circuit averaging and averaged switch modeling 7.6. The canonical circuit model 7.7. Modeling the pulse-wih modulator 7.8. Summary of key points 2

7.1. Introduction Objective: maintain v equal to an accurate, constant value V. There are disturbances: in v g in There are uncertainties: in element values in V g in Power input v g A simple dc-dc regulator system, employing a buck converter transistor gate driver δ δ Switching converter pulse-wih modulator v c compensator v c dt s T s t t Controller G c (s) v voltage reference Load v ref v feedback connection 3

Applications of control in power electronics Dc-dc converters egulate dc output voltage. Control the duty cycle d such that v accurately follows a reference signal v ref. Dc-ac inverters egulate an ac output voltage. Control the duty cycle d such that v accurately follows a reference signal v ref. Ac-dc rectifiers egulate the dc output voltage. egulate the ac input current waveform. Control the duty cycle d such that i g accurately follows a reference signal i ref, and v accurately follows a reference signal v ref. 4

Objective of Part II Develop tools for modeling, analysis, and design of converter control systems Need dynamic models of converters: How do ac variations in v g,, or d affect the output voltage v? What are the small-signal transfer functions of the converter? Extend the steady-state converter models of Chapters 2 and 3, to include CCM converter dynamics (Chapter 7) Construct converter small-signal transfer functions (Chapter 8) Design converter control systems (Chapter 9) Model converters operating in DCM (Chapter 10) Current-programmed control of converters (Chapter 11) 5

Modeling epresentation of physical behavior by mathematical means Model dominant behavior of system, ignore other insignificant phenomena Simplified model yields physical insight, allowing engineer to design system to operate in specified manner Approximations neglect small but complicating phenomena After basic insight has been gained, model can be refined (if it is judged worthwhile to expend the engineering effort to do so), to account for some of the previously neglected phenomena 6

Neglecting the switching ripple Suppose the duty cycle is modulated sinusoidally: d=dd m cos ω m t The resulting variations in transistor gate drive signal and converter output voltage: gate drive where D and D m are constants, D m << D, and the modulation frequency ω m is much smaller than the converter switching frequency ω s = 2πf s. actual waveform v including ripple averaged waveform <v> Ts with ripple neglected t t 7

Output voltage spectrum with sinusoidal modulation of duty cycle spectrum of v modulation frequency and its harmonics switching frequency and sidebands switching harmonics { { { ω m ω s ω Contains frequency components at: Modulation frequency and its harmonics Switching frequency and its harmonics Sidebands of switching frequency 8 With small switching ripple, highfrequency components (switching harmonics and sidebands) are small. If ripple is neglected, then only lowfrequency components (modulation frequency and harmonics) remain.

Objective of ac converter modeling Predict how low-frequency variations in duty cycle induce lowfrequency variations in the converter voltages and currents Ignore the switching ripple Ignore complicated switching harmonics and sidebands Approach: emove switching harmonics by averaging all waveforms over one switching period 9

Averaging to remove switching ripple Average over one switching period to remove switching ripple: L di L Ts = v L Ts Note that, in steady-state, v L Ts =0 i C Ts =0 C dv C Ts where = i C Ts by inductor volt-second balance and capacitor charge balance. x L Ts = 1 T s t t T s x(τ) dτ 10

Nonlinear averaged equations The averaged voltages and currents are, in general, nonlinear functions of the converter duty cycle, voltages, and currents. Hence, the averaged equations L di L Ts C dv C Ts = v L Ts = i C Ts constitute a system of nonlinear differential equations. Hence, must linearize by constructing a small-signal converter model. 11

Small-signal modeling of the BJT Nonlinear Ebers-Moll model C Linearized small-signal model, active region C i B β F i B i B β F i B B β i B B r E E E 12

Buck-boost converter: nonlinear static control-to-output characteristic 0 0 0.5 1 D V g quiescent operating point linearized function Example: linearization at the quiescent operating point D = 0.5 V actual nonlinear characteristic 13

esult of averaged small-signal ac modeling Small-signal ac equivalent circuit model V L g V d 1 : D D' : 1 v g Id Id C v buck-boost example 14

7.2. The basic ac modeling approach Buck-boost converter example 1 2 v g L i C v 15

Switch in position 1 Inductor voltage and capacitor current are: v L =L di = v g v g i L C v i C =C dv = v Small ripple approximation: replace waveforms with their low-frequency averaged values: v L =L di v g Ts i C =C dv v T s 16

Switch in position 2 Inductor voltage and capacitor current are: v L =L di = v i C =C dv =i v v g i L C v Small ripple approximation: replace waveforms with their low-frequency averaged values: v L =L di i C =C dv v Ts i Ts v T s 17

7.2.1 Averaging the inductor waveforms Inductor voltage waveform v L v g Ts Low-frequency average is found by evaluation of x L Ts = 1 T s t t T s x(τ)dτ 0 v L Ts = d v g Ts d' v Ts dt s T s t Average the inductor voltage in this manner: v Ts v L Ts = 1 T s t t T s v L (τ)dτ d v g Ts d' v Ts Insert into Eq. (7.2): L di T s = d v g Ts d' v Ts This equation describes how the low-frequency components of the inductor waveforms evolve in time. 18

7.2.2 Discussion of the averaging approximation Use of the average inductor voltage allows us to determine the net change in inductor current over one switching period, while neglecting the switching ripple. v L 0 v g Ts v L Ts = d v g Ts d' v Ts dt s T s t In steady-state, the average inductor voltage is zero (volt-second balance), and hence the inductor current waveform is periodic: i(t T s ) = i. There is no net change in inductor current over one switching period. i i(0) v g L Ts v Ts i(dt s ) v Ts L i(t s ) During transients or ac variations, the average inductor voltage is not zero in general, and this leads to net variations in inductor current. 0 dt s T s Inductor voltage and current waveforms t 19

Net change in inductor current is correctly predicted by the average inductor voltage Inductor equation: L di = v L Divide by L and integrate over one switching period: t t T s di = 1 L t t T s v L (τ)dτ Left-hand side is the change in inductor current. ight-hand side can be related to average inductor voltage by multiplying and dividing by T s as follows: i(t T s )i= 1 L T s v L Ts So the net change in inductor current over one switching period is exactly equal to the period T s multiplied by the average slope v L T s /L. 20

Average inductor voltage correctly predicts average slope of i L i Actual waveform, including ripple v g L v L Averaged waveform i i(0) i(t s ) T s 0 d v g Ts d' v Ts dt s L T s t The net change in inductor current over one switching period is exactly equal to the period T s multiplied by the average slope v L T s /L. 21

di Ts We have i(t T s )i= 1 L T s v L Ts earrange: L i(t T s) i T s = v L Ts Define the derivative of i T s as (Euler formula): di Ts = i(t T s)i T s Hence, L di T s = v L Ts 22

Computing how the inductor current changes over one switching period i Let s compute the actual inductor current waveform, using the linear ripple approximation. i(0) v g L Ts i(dt s ) v Ts L i(t s ) 0 dt s T s t With switch in position 1: i(dt s ) = i(0) dt s v g Ts L (final value) = (initial value) (length of interval) (average slope) With switch in position 2: i(t s ) = i(dt s ) d't s v Ts L (final value) = (initial value) (length of interval) (average slope) 23

Net change in inductor current over one switching period Eliminate i(dt s ), to express i(t s ) directly as a function of i(0): i(t s )=i(0) T s L d v g Ts d' v Ts v L Ts The intermediate step of computing i(dt s ) is eliminated. The final value i(t s ) is equal to the initial value i(0), plus the switching period Ts multiplied by the average slope v L T s /L. i v g L v L i(0) i(t s ) 0 Actual waveform, including ripple d v g Ts d' v Ts dt s L Averaged waveform T s i T s t 24

7.2.3 Averaging the capacitor waveforms Average capacitor current: i C v T s i C Ts = d v T s d' i Ts v T s 0 dt s i C Ts T s t Collect terms, and equate to C d v T s /: C dv T s =d' i Ts v T s v 0 v T s dt s i Ts T s t v(dt s ) v Ts v(0) v(t s ) v T s C v T s C i T s C Capacitor voltage and current waveforms 25

7.2.4 The average input current We found in Chapter 3 that it was sometimes necessary to write an equation for the average converter input current, to derive a complete dc equivalent circuit model. It is likewise necessary to do this for the ac model. Buck-boost input current waveform is i g = i Ts during subinterval 1 0 during subinterval 2 i g 0 0 i Ts dt s i g Ts Converter input current waveform 0 T s t Average value: i g Ts = d i Ts 26

7.2.5. Perturbation and linearization Converter averaged equations: L di T s C dv T s = d v g Ts d' v Ts =d' i Ts v T s i g Ts =d i Ts nonlinear because of multiplication of the time-varying quantity d with other time-varying quantities such as i and v. 27

Construct small-signal model: Linearize about quiescent operating point If the converter is driven with some steady-state, or quiescent, inputs d=d v g Ts =V g then, from the analysis of Chapter 2, after transients have subsided the inductor current, capacitor voltage, and input current i Ts, v Ts, i g Ts reach the quiescent values I, V, and I g, given by the steady-state analysis as V = D D' V g I= D' V I g =DI 28

Perturbation So let us assume that the input voltage and duty cycle are equal to some given (dc) quiescent values, plus superimposed small ac variations: v g Ts = V g v g d=dd In response, and after any transients have subsided, the converter dependent voltages and currents will be equal to the corresponding quiescent values, plus small ac variations: i Ts = I i v Ts = V v i g Ts = I g i g 29

The small-signal assumption If the ac variations are much smaller in magnitude than the respective quiescent values, v g << V g d << D i << I v << V i g << I g then the nonlinear converter equations can be linearized. 30

Perturbation of inductor equation Insert the perturbed expressions into the inductor differential equation: L dii note that d is given by = D d V g v g D' d V v d'= 1d =1 Dd =D'd with D = 1 D Multiply out and collect terms: L di 0 di = DV g D'V Dv g D'v V g V d d v g v Dc terms 1 st order ac terms 2 nd order ac terms (linear) (nonlinear) 31

The perturbed inductor equation L di 0 di = DV g D'V Dv g D'v V g V d d v g v Dc terms 1 st order ac terms 2 nd order ac terms (linear) (nonlinear) Since I is a constant (dc) term, its derivative is zero The right-hand side contains three types of terms: Dc terms, containing only dc quantities First-order ac terms, containing a single ac quantity, usually multiplied by a constant coefficient such as a dc term. These are linear functions of the ac variations Second-order ac terms, containing products of ac quantities. These are nonlinear, because they involve multiplication of ac quantities 32

Neglect of second-order terms L di 0 di = DV g D'V Dv g D'v V g V d d v g v Dc terms 1 st order ac terms 2 nd order ac terms (linear) (nonlinear) Provided v g << V g d << D i << I v << V i g << I g then the second-order ac terms are much smaller than the first-order terms. For example, d v g << Dv g when d << D So neglect second-order terms. Also, dc terms on each side of equation are equal. 33

Linearized inductor equation Upon discarding second-order terms, and removing dc terms (which add to zero), we are left with L di = Dv g D'v V g V d This is the desired result: a linearized equation which describes smallsignal ac variations. Note that the quiescent values D, D, V, V g, are treated as given constants in the equation. 34

Capacitor equation Perturbation leads to C dvv Collect terms: = D'd Ii Vv C dv 0 dv = D'I V D'iv Id di Dc terms 1 st order ac terms 2 nd order ac term (linear) (nonlinear) Neglect second-order terms. Dc terms on both sides of equation are equal. The following terms remain: C dv =D'i v Id This is the desired small-signal linearized capacitor equation. 35

Average input current Perturbation leads to I g i g = Dd Ii Collect terms: I g i g = DI DiId di Dc term 1 st order ac term Dc term 1 st order ac terms 2 nd order ac term (linear) (nonlinear) Neglect second-order terms. Dc terms on both sides of equation are equal. The following first-order terms remain: i g =DiId This is the linearized small-signal equation which described the converter input port. 36

7.2.6. Construction of small-signal equivalent circuit model The linearized small-signal converter equations: L di C dv = Dv g D'v V g V d =D'i v Id i g =DiId econstruct equivalent circuit corresponding to these equations, in manner similar to the process used in Chapter 3. 37

Inductor loop equation L di = Dv g D'v V g V d L V g V d D v g L di i D'v 38

Capacitor node equation C dv =D'i v Id C dv v D' i Id C v 39

Input port node equation i g =DiId i g v g Id D i 40

Complete equivalent circuit Collect the three circuits: i L V g V d v g Id D i Dv g D'v D'i Id C v eplace dependent sources with ideal dc transformers: V L g V d 1 : D D' : 1 v g Id Id C v Small-signal ac equivalent circuit model of the buck-boost converter 41

7.2.7. esults for several basic converters buck 1 : D V g d L i v g Id C v boost Vd L D' : 1 i v g Id C v 42

esults for several basic converters buck-boost V L g V d 1 : D D' : 1 i v g Id Id C v 43

7.3. Example: a nonideal flyback converter Flyback converter example v g i g L 1 : n D 1 C v MOSFET has onresistance on Flyback transformer has magnetizing inductance L, referred to primary Q 1 44

Circuits during subintervals 1 and 2 Flyback converter, with transformer equivalent circuit i g i L v L 1 : n D 1 i C C v Subinterval 1 v g i g L transformer model i 1:n v L C i C v v g ideal on Subinterval 2 Q 1 v g i g =0 transformer model i v L v/n 1:n i/n C i C v 45

Subinterval 1 Circuit equations: transformer model v L =v g i on i C = v i g =i v g i g L i v L 1:n C i C v Small ripple approximation: v L = v g Ts i Ts on i C = v T s i g = i Ts on MOSFET conducts, diode is reverse-biased 46

Subinterval 2 Circuit equations: v L = v n i C = i n v i g =0 Small ripple approximation: v g i g =0 transformer model i v L v/n 1:n i/n C i C v v L = v T s n i C = i T s n i g =0 v T s MOSFET is off, diode conducts 47

Inductor waveforms v L v g i on i v g Ts on i Ts L i Ts 0 dt s v L Ts T s t v Ts nl v/n Average inductor voltage: 0 dt s T s t v L Ts = d v g Ts i Ts on d' v Ts n Hence, we can write: L di T s = d v g Ts d i Ts on d' v Ts n 48

Capacitor waveforms i C i n v v i Ts nc v T s C i C Ts 0 dt s T s t v T s C v Ts v/ 0 dt s T s t Average capacitor current: i C Ts = d v Ts d' i Ts n v T s Hence, we can write: C dv T s = d' i Ts n v T s 49

Input current waveform i g i Ts i g Ts 0 0 dt s 0 T s t Average input current: i g Ts = d i Ts 50

The averaged converter equations L di T s = d v g Ts d i Ts on d' v Ts n C dv T s = d' i Ts n v T s i g Ts = d i Ts a system of nonlinear differential equations Next step: perturbation and linearization. Let v g Ts = V g v g d=dd i Ts =Ii v Ts =Vv i g Ts =I g i g 51

Perturbation of the averaged inductor equation L di T s = d v g Ts d i Ts on d' v Ts n L dii = D d V g v g D' d V v n D d I i on L di 0 di = DV g D' V n D oni Dv g D' v n V g V n I on dd on i Dc terms 1 st order ac terms (linear) dv g d v n di on 2 nd order ac terms (nonlinear) 52

Linearization of averaged inductor equation Dc terms: 0=DV g D' V n D oni Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are: L di = Dv g D' v n V g V n I on dd on i This is the desired linearized inductor equation. 53

Perturbation of averaged capacitor equation Original averaged equation: C dv T s = d' i Ts n v T s Perturb about quiescent operating point: C dvv = D' d I i n V v Collect terms: C dv 0 dv = D'I n V D'i n v Id n di n Dc terms 1 st order ac terms 2 nd order ac term (linear) (nonlinear) 54

Linearization of averaged capacitor equation Dc terms: 0= D'I n V Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are: C dv = D'i n v Id n This is the desired linearized capacitor equation. 55

Perturbation of averaged input current equation Original averaged equation: i g Ts = d i Ts Perturb about quiescent operating point: I g i g = Dd Ii Collect terms: I g i g = DI DiId di Dc term 1 st order ac term Dc term 1 st order ac terms 2 nd order ac term (linear) (nonlinear) 56

Linearization of averaged input current equation Dc terms: I g = DI Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are: i g =DiId This is the desired linearized input current equation. 57

Summary: dc and small-signal ac converter equations Dc equations: 0=DV g D' V n D oni 0= D'I n V I g =DI Small-signal ac equations: L di C dv = Dv g D' v n V g V n I on dd on i = D'i n v Id n i g =DiId Next step: construct equivalent circuit models. 58

Small-signal ac equivalent circuit: inductor loop L di = Dv g D' v n V g V n I on dd on i L D on d V g I on V n D v g L di i D' v n 59

Small-signal ac equivalent circuit: capacitor node C dv = D'i n v Id n D' i n Id n C dv C v v 60

Small-signal ac equivalent circuit: converter input node i g =DiId i g v g Id D i 61

Small-signal ac model, nonideal flyback converter example Combine circuits: i g L D on d V g I on V n i v g Id D i D v g D'v n D' i n Id n C v eplace dependent sources with ideal transformers: i g 1 : D L d V g I on V n D' : n i D on v g Id Id n C v 62

7.4. State Space Averaging A formal method for deriving the small-signal ac equations of a switching converter Equivalent to the modeling method of the previous sections Uses the state-space matrix description of linear circuits Often cited in the literature A general approach: if the state equations of the converter can be written for each subinterval, then the small-signal averaged model can always be derived Computer programs exist which utilize the state-space averaging method 63

7.4.1. The state equations of a network A canonical form for writing the differential equations of a system If the system is linear, then the derivatives of the state variables are expressed as linear combinations of the system independent inputs and state variables themselves The physical state variables of a system are usually associated with the storage of energy For a typical converter circuit, the physical state variables are the inductor currents and capacitor voltages Other typical physical state variables: position and velocity of a motor shaft At a given point in time, the values of the state variables depend on the previous history of the system, rather than the present values of the system inputs To solve the differential equations of a system, the initial values of the state variables must be specified 64

State equations of a linear system, in matrix form A canonical matrix form: State vector x contains inductor currents, capacitor voltages, etc.: K dx = AxBu y=cxeu x= x 1 x 2, dx = dx 1 dx 2 Input vector u contains independent sources such as v g Output vector y contains other dependent quantities to be computed, such as i g Matrix K contains values of capacitance, inductance, and mutual inductance, so that K dx/ is a vector containing capacitor currents and inductor winding voltages. These quantities are expressed as linear combinations of the independent inputs and state variables. The matrices A, B, C, and E contain the constants of proportionality. 65

Example State vector x= Matrix K K = v 1 v 2 i C 1 0 0 0 C 2 0 0 0 L i v L 1 i C1 i C2 i in 1 C 1 Input vector u= i in i v 1 L C 2 v 2 Choose output vector as y= v out i 1 2 3 v out To write the state equations of this circuit, we must express the inductor voltages and capacitor currents as linear combinations of the elements of the x and u( t) vectors. 66

Circuit equations i L i v L 1 i C1 i C2 2 i in 1 C 1 v 1 C 2 v 2 3 v out Find i C1 via node equation: i C1 =C 1 dv 1 = i in v 1 i Find i C2 via node equation: i C2 =C 2 dv 2 = i v 2 2 3 Find v L via loop equation: v L =L di = v 1 v 2 67

Equations in matrix form The same equations: Express in matrix form: i C1 =C 1 dv 1 i C2 =C 2 dv 2 v L =L di = i in v 1 i = i v 2 2 3 = v 1 v 2 C 1 0 0 0 C 2 0 0 0 L dv 1 dv 2 di = 1 1 0 1 0 1 1 2 3 1 1 0 v 1 v 2 i 1 0 0 i in K dx = A x B u 68

Output (dependent signal) equations y= v out i 1 i L i v L 1 i C1 i C2 2 i in 1 C 1 v 1 C 2 v 2 3 v out Express elements of the vector y as linear combinations of elements of x and u: v out =v 2 3 2 3 i 1 = v 1 1 69

Express in matrix form The same equations: Express in matrix form: v out =v 2 i 1 = v 1 1 3 2 3 v out i 1 = 0 3 2 3 0 1 1 0 0 v 1 v 2 i 0 0 i in y = C x E u 70

7.4.2. The basic state-space averaged model Given: a PWM converter, operating in continuous conduction mode, with two subintervals during each switching period. During subinterval 1, when the switches are in position 1, the converter reduces to a linear circuit that can be described by the following state equations: K dx = A 1 xb 1 u y=c 1 xe 1 u During subinterval 2, when the switches are in position 2, the converter reduces to another linear circuit, that can be described by the following state equations: K dx = A 2 xb 2 u y=c 2 xe 2 u 71

Equilibrium (dc) state-space averaged model Provided that the natural frequencies of the converter, as well as the frequencies of variations of the converter inputs, are much slower than the switching frequency, then the state-space averaged model that describes the converter in equilibrium is 0 = AXBU Y=CXEU where the averaged matrices are A = D A 1 D' A 2 B = D B 1 D' B 2 C = D C 1 D' C 2 E = D E 1 D' E 2 and the equilibrium dc components are X = equilibrium (dc) state vector U = equilibrium (dc) input vector Y = equilibrium (dc) output vector D = equilibrium (dc) duty cycle 72

Solution of equilibrium averaged model Equilibrium state-space averaged model: 0=AXBU Y=CXEU Solution for X and Y: X =A 1 BU Y= CA 1 BE U 73

Small-signal ac state-space averaged model K dx = AxBu A 1 A 2 X B 1 B 2 U d y=cxeu C 1 C 2 X E 1 E 2 U d where x=small signal (ac) perturbation in state vector u=small signal (ac) perturbation in input vector y=small signal (ac) perturbation in output vector d=small signal (ac) perturbation in duty cycle So if we can write the converter state equations during subintervals 1 and 2, then we can always find the averaged dc and small-signal ac models 74

7.4.3. Discussion of the state-space averaging result As in Sections 7.1 and 7.2, the low-frequency components of the inductor currents and capacitor voltages are modeled by averaging over an interval of length T s. Hence, we define the average of the state vector as: x Ts = 1 T s t t T s x(τ) dτ The low-frequency components of the input and output vectors are modeled in a similar manner. By averaging the inductor voltages and capacitor currents, one obtains: K d x T s = d A 1 d' A 2 x Ts d B 1 d' B 2 u Ts 75

Change in state vector during first subinterval During subinterval 1, we have K dx = A 1 xb 1 u y=c 1 xe 1 u So the elements of x change with the slope dx = K 1 A 1 xb 1 u Small ripple assumption: the elements of x and u do not change significantly during the subinterval. Hence the slopes are essentially constant and are equal to dx = K 1 A 1 x Ts B 1 u Ts 76

Change in state vector during first subinterval dx = K 1 A 1 x Ts B 1 u Ts x K 1 A 1 x Ts B 1 u Ts x(0) K 1 da 1 d'a 2 x Ts db Net change in state vector over first subinterval: 0 dt s x(dt s ) = x(0) dt s K 1 A 1 x Ts B 1 u Ts final initial interval slope value value length 77

Change in state vector during second subinterval Use similar arguments. State vector now changes with the essentially constant slope dx = K 1 A 2 x Ts B 2 u Ts The value of the state vector at the end of the second subinterval is therefore x(t s ) = x(dt s ) d't s K 1 A 2 x Ts B 2 u Ts final initial interval slope value value length 78

Net change in state vector over one switching period We have: x(dt s )=x(0) dt s K 1 A 1 x Ts B 1 u Ts x(t s )=x(dt s ) d't s K 1 A 2 x Ts B 2 u Ts Eliminate x(dt s ), to express x(t s ) directly in terms of x(0) : x(t s )=x(0) dt s K 1 A 1 x Ts B 1 u Ts d't s K 1 A 2 x Ts B 2 u Ts Collect terms: x(t s )=x(0) T s K 1 da 1 d'a 2 x Ts T s K 1 db 1 d'b 2 u Ts 79

Approximate derivative of state vector x K 1 A 1 x Ts B 1 u Ts K 1 A 2 x Ts B 2 u Ts x(0) x Ts x(t s ) K 1 da 1 d'a 2 x Ts db 1 d'b 2 u Ts 0 dt s T s t Use Euler approximation: d x Ts We obtain: x(t s)x(0) T s K d x T s = d A 1 d' A 2 x Ts d B 1 d' B 2 u Ts 80

Low-frequency components of output vector y C 1 x Ts E 1 u Ts y Ts C 2 x Ts E 2 u Ts 0 0 dt s T s t emove switching harmonics by averaging over one switching period: y Ts = d C 1 x Ts E 1 u Ts d' C 2 x Ts E 2 u Ts Collect terms: y Ts = d C 1 d' C 2 x Ts d E 1 d' E 2 u Ts 81

Averaged state equations: quiescent operating point The averaged (nonlinear) state equations: K d x T s = d A 1 d' A 2 x Ts d B 1 d' B 2 u Ts y Ts = d C 1 d' C 2 x Ts d E 1 d' E 2 u Ts The converter operates in equilibrium when the derivatives of all elements of < x > T are zero. Hence, the converter quiescent s operating point is the solution of 0 = AXBU Y=CXEU where A = D A 1 D' A 2 B = D B 1 D' B 2 C = D C 1 D' C 2 E = D E 1 D' E 2 and 82 X = equilibrium (dc) state vector U = equilibrium (dc) input vector Y = equilibrium (dc) output vector D = equilibrium (dc) duty cycle

Averaged state equations: perturbation and linearization Let x Ts = X x with U >> u u Ts = U u D >> d y Ts = Y y d=dd d'=d'd X >> x Y >> y Substitute into averaged state equations: K d Xx = Dd A 1 D'd A 2 Xx Dd B 1 D'd B 2 Uu Yy = Dd C 1 D'd C 2 Xx Dd E 1 D'd E 2 Uu 83

Averaged state equations: perturbation and linearization K dx = AX BU AxBu A 1 A 2 X B 1 B 2 U d firstorder ac dc terms firstorder ac terms A 1 A 2 xd B 1 B 2 ud secondorder nonlinear terms Yy = CX EU CxEu C 1 C 2 X E 1 E 2 U d dc 1st order ac dc terms firstorder ac terms C 1 C 2 xd E 1 E 2 ud secondorder nonlinear terms 84

Linearized small-signal state equations Dc terms drop out of equations. Second-order (nonlinear) terms are small when the small-signal assumption is satisfied. We are left with: K dx = AxBu A 1 A 2 X B 1 B 2 U d y=cxeu C 1 C 2 X E 1 E 2 U d This is the desired result. 85

7.4.4. Example: State-space averaging of a nonideal buck-boost converter v g Q 1 D i 1 g i L C v Model nonidealities: MOSFET onresistance on Diode forward voltage drop V D state vector input vector output vector x= i v u= v g V D y= i g 86

Subinterval 1 L di = v g i on C dv = v i g =i v g i g on i L C v L 0 0C d i v = on 0 0 1 i v 10 00 v g V D K dx A 1 x B 1 u i g = 1 0 i v 0 0 v g V D y C 1 x E 1 u 87

Subinterval 2 L di = vv D C dv = v i i g =0 v g i g V D L C i v L 0 0C d i v = 0 1 1 1 i v 0 1 0 0 v g V D K dx A 2 x B 2 u i g = 0 0 i v 0 0 v g V D y C 2 x E 2 u 88

Evaluate averaged matrices A = DA 1 D'A 2 = D on 0 0 1 D' 0 1 1 1 = D on D' D' 1 In a similar manner, B = DB 1 D'B 2 = D D' 0 0 C=DC 1 D'C 2 = D 0 E=DE 1 D'E 2 = 0 0 89

DC state equations 0=AXBU Y=CXEU or, 0 0 = D on D' D' 1 I V D D' 0 0 V g V D I g = D 0 I V 0 0 V g V D DC solution: I V = 1 1 D D' 2 on D D' 2 D' D 1 D' 1 V g V D I g = 1 1 D D' 2 on D 2 D' 2 D D' V g V D 90

Steady-state equivalent circuit DC state equations: 0 0 = D on D' D' 1 I V D D' 0 0 V g V D I g = D 0 I V 0 0 V g V D Corresponding equivalent circuit: I g D on D'V D 1 : D D' : 1 I V g V 91

Small-signal ac model Evaluate matrices in small-signal model: A 1 A 2 X B 1 B 2 U = V I C 1 C 2 X E 1 E 2 U = I V g I on V D 0 = V g V I on V D I Small-signal ac state equations: L 0 0 C d i v = D on D' D' 1 i v D D' 0 0 v g v D 0 V g V I on V D I d i g = D 0 i v 00 00 v g v D 0 0 I d 92

Construction of ac equivalent circuit Small-signal ac equations, in scalar form: L di C dv = D' vd on idv g V g VI on V D d =D'i v Id i g =DiId Corresponding equivalent circuits: inductor equation L D on d V g V V D I on input eqn v g i g Id Di D v g L di i D' v capacitor eqn D' i Id C dv C v v 93

Complete small-signal ac equivalent circuit Combine individual circuits to obtain i g 1 : D L d V g VV D I on D' : 1 i D on v g Id Id C v 94

7.5. Circuit Averaging and Averaged Switch Modeling Historically, circuit averaging was the first method known for modeling the small-signal ac behavior of CCM PWM converters It was originally thought to be difficult to apply in some cases There has been renewed interest in circuit averaging and its corrolary, averaged switch modeling, in the last decade Can be applied to a wide variety of converters We will use it to model DCM, CPM, and resonant converters Also useful for incorporating switching loss into ac model of CCM converters Applicable to 3ø PWM inverters and rectifiers Can be applied to phase-controlled rectifiers ather than averaging and linearizing the converter state equations, the averaging and linearization operations are performed directly on the converter circuit 95

Separate switch network from remainder of converter Power input Time-invariant network containing converter reactive elements Load v g C L v v C i L i 1 i 2 v 1 port 1 Switch network port 2 v 2 Control input d 96

Boost converter example Ideal boost converter example i L v g C v Two ways to define the switch network (a) i 1 i 2 (b) i 1 i 2 v 1 v 2 v 1 v 2 97

Discussion The number of ports in the switch network is less than or equal to the number of SPST switches 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. Definition of the switch network terminal quantities is not unique. Different definitions lead equivalent results having different forms 98

Boost converter example Let s use definition (a): i 1 i 2 L i v 1 v 2 v g C v Since i 1 and v 2 coincide with the converter inductor current and output voltage, it is convenient to define these waveforms as the independent inputs to the switch network. The switch network dependent outputs are then v 1 and i 2. 99

Obtaining a time-invariant network: Modeling the terminal behavior of the switch network eplace the switch network with dependent sources, which correctly represent the dependent output waveforms of the switch network L i i 1 v g v 1 i 2 v 2 C v Switch network Boost converter example 100

Definition of dependent generator waveforms v 1 v 2 i L i 1 v 1 Ts v g v 1 i 2 v 2 C v 0 0 0 i 2 i 2 Ts dt s i 1 T s t Switch network The waveforms of the dependent generators are defined to be identical to the actual terminal waveforms of the switch network. 0 0 0 dt s T s t The circuit is therefore electrical identical to the original converter. So far, no approximations have been made. 101

The circuit averaging step Now average all waveforms over one switching period: Power input Averaged time-invariant network containing converter reactive elements Load v g Ts C L v Ts v C Ts i L Ts i 1 Ts i 2 Ts v 1 Ts port 1 Averaged switch network port 2 v 2 Ts Control input d 102

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. In practice, the only work needed for this step is to average the switch dependent waveforms. 103

Averaging step: boost converter example L i i 1 v g v 1 i 2 v 2 C v Switch network L i Ts i 1 Ts v g Ts v 1 Ts i 2 Ts v 2 Ts C v Ts Averaged switch network 104

Compute average values of dependent sources v 1 v 1 Ts v 2 Average the waveforms of the dependent sources: i 2 0 0 0 dt s i 1 T s t v 1 Ts = d' v 2 Ts i 2 Ts = d' i 1 Ts i 2 Ts 0 0 0 dt s L T s t i Ts i 1 Ts v g Ts d' v 2 Ts d' i 1 Ts v 2 Ts C v Ts Averaged switch model 105

Perturb and linearize As usual, let: The circuit becomes: L I i v g Ts = V g v g d=dd d'=d'd i Ts = i 1 Ts =Ii v Ts = v 2 Ts =Vv v 1 Ts =V 1 v 1 i 2 Ts =I 2 i 2 V g v g D'd V v D'd I i C V v 106

Dependent voltage source D'd Vv =D' Vv Vdvd nonlinear, 2nd order Vd D' Vv 107

Dependent current source D' d I i = D' I i Idid nonlinear, 2nd order D' I i Id 108

Linearized circuit-averaged model L Vd I i V g v g D' Vv D' Ii Id C V v I i L Vd D' : 1 V g v g Id C V v 109

Summary: Circuit averaging method Model the switch network with equivalent voltage and current sources, such that an equivalent time-invariant network is obtained Average converter waveforms over one switching period, to remove the switching harmonics Perturb and linearize the resulting low-frequency model, to obtain a small-signal equivalent circuit 110

Averaged switch modeling: CCM Circuit averaging of the boost converter: in essence, the switch network was replaced with an effective ideal transformer and generators: i 1 2 v I i Vd D' : 1 Id V v Switch network 111

Basic functions performed by switch network i 1 2 v I i Vd D' : 1 Id V v Switch network For the boost example, we can conclude that the switch network performs two basic functions: Transformation of dc and small-signal ac voltage and current levels, according to the D :1 conversion ratio Introduction of ac voltage and current variations, drive by the control input duty cycle variations Circuit averaging modifies only the switch network. Hence, to obtain a smallsignal converter model, we need only replace the switch network with its averaged model. Such a procedure is called averaged switch modeling. 112

Averaged switch modeling: Procedure 1. Define a switch network and its terminal waveforms. For a simple transistor-diode switch network as in the buck, boost, etc., there are two ports and four terminal quantities: v 1, i 1, v 2, i 2.The switch network also contains a control input d. Buck example: i 1 v 1 i 2 v 2 2. To derive an averaged switch model, express the average values of two of the terminal quantities, for example v 2 and i T s 1, as Ts functions of the other average terminal quantities v 1 and i T s 1. Ts v 2 and i T s 1 may also be functions of the control input d, but they Ts should not be expressed in terms of other converter signals. 113

The basic buck-type CCM switch cell i 1 v CE i C i 2 L i i 1 i 2 T2 v g v 1 v 2 C v i 2 i 1 T2 Switch network 0 0 dt s 0 T s t v 2 v 1 i1 Ts = d i 2 Ts v 2 Ts = d v 1 Ts v 2 T2 0 0 dt s T s t 0 114

eplacement of switch network by dependent sources, CCM buck example v g Circuit-averaged model v 1 i 1 i 2 v 2 Switch network L i C v Perturbation and linearization of switch network: I 1 i 1 =D I 2 i 2 I 2 d V 2 v 2 =D V 1 v 1 V 1 d I 1 i 1 1 : D I 2 i 2 V 1 d V 1 v 1 I 2 d V 2 v 2 esulting averaged switch model: CCM buck converter V g v g I 1 i 1 1 : D I 2 i 2 V 1 d V 1 v 1 I 2 d V 2 v 2 L I i C V v Switch network 115

Three basic switch networks, and their CCM dc and small-signal ac averaged switch models i 1 v 1 i 1 i 2 v 2 i 2 I 1 i 1 1 : D I 2 i 2 V 1 d V 1 v 1 I 2 d V 2 v 2 I 1 i 1 D' : 1 I 2 i 2 V 2 d see also Appendix 3 Averaged switch modeling of a CCM SEPIC v 1 v 2 V 1 v 1 I 1 d V 2 v 2 i 1 v 1 i 2 v 2 I 1 i 1 D' : D I 2 i 2 V 1 v 1 V 1 DD' d I 2 V 2 v 2 DD' d 116

Example: Averaged switch modeling of CCM buck converter, including switching loss v g i 1 v 1 v CE i C i 2 v 2 L i C v i 1 =i C v 2 =v 1 v CE Switch network v CE i C Switch network terminal waveforms: v 1, i 1, v 2, i 2. To derive averaged switch v 1 model, express v 2 T s and i 1 T as functions of s 0 i 2 0 t 1 t 2 tir t vf t vr t if T s t v 1 and i T s 1. v Ts 2 Ts and i 1 T may also be s functions of the control input d, but they should not be expressed in terms of other converter signals. 117

Averaging i 1 v CE i C v 1 i 2 0 0 t 1 t 2 tir t vf t vr t if t T s i 1 Ts = 1 T s 0 T s i 1 = i 2 Ts t 1 t vf t vr 1 2 t ir 1 2 t if T s 118

Expression for i 1 Given i 1 Ts = 1 T s 0 T s i 1 = i 2 Ts t 1 t vf t vr 1 2 t ir 1 2 t if T s Let Then we can write d = t 1 1 2 t vf 1 2 t vr 1 2 t ir 1 2 t if T s i 1 Ts = i 2 Ts d 1 2 d v d v = t vf t vr T s d i = t ir t if T s 119

Averaging the switch network output voltage v 2 v CE i C v 1 i 2 0 0 t 1 t 2 tir t vf t vr t if t T s v 2 Ts = v 1 v CE Ts = 1 T s 0 T s v CE v 1 Ts t 1 1 2 t vf 1 2 t vr v 2 Ts = v 1 Ts T s v 2 Ts = v 1 Ts d 1 2 d i 120

Construction of large-signal averaged-switch model i 1 Ts = i 2 Ts d 1 2 d v v 2 Ts = v 1 Ts d 1 2 d i i 1 Ts 1 2 d i v 1 Ts i 2 Ts v 1 Ts 1 2 d v i 2 Ts d i 2 Ts d v 1 Ts v 2 Ts i 1 Ts 1 : d 1 2 d i v 1 Ts i 2 Ts v 1 Ts 1 2 d v i 2 Ts v 2 Ts 121

Switching loss predicted by averaged switch model i 1 Ts 1 : d 1 2 d i v 1 Ts i 2 Ts v 1 Ts 1 2 d v i 2 Ts v 2 Ts P sw = 1 2 d v d i i 2 Ts v 1 Ts 122

Solution of averaged converter model in steady state I 1 1 : D 1 2 D i V 1 I 2 L I V g V 1 1 2 D v I 2 V 2 C V Averaged switch network model Output voltage: Efficiency calcuation: V = D 1 2 D i V g = DV g 1 D i 2D P in = V g I 1 = V 1 I 2 P out = VI 2 = V 1 I 2 D 1 2 D v D 1 2 D i 123 η = P out P in = D 1 2 D i D 1 2 D v = 1 D i 2D 1 D v 2D

7.6. The canonical circuit model All PWM CCM dc-dc converters perform the same basic functions: Transformation of voltage and current levels, ideally with 100% efficiency Low-pass filtering of waveforms Control of waveforms by variation of duty cycle Hence, we expect their equivalent circuit models to be qualitatively similar. Canonical model: A standard form of equivalent circuit model, which represents the above physical properties Plug in parameter values for a given specific converter 124

7.6.1. Development of the canonical circuit model 1. Transformation of dc voltage and current levels modeled as in Chapter 3 with ideal dc transformer V g Converter model 1 : M(D) V effective turns ratio M(D) can refine dc model by addition of effective loss elements, as in Chapter 3 Power input D Control input Load 125

Steps in the development of the canonical circuit model 2. Ac variations in v g induce ac variations in v these variations are also transformed by the conversion ratio M(D) V g v g (s) 1 : M(D) V v(s) Power input D Control input Load 126

Steps in the development of the canonical circuit model 3. Converter must contain an effective lowpass filter characteristic necessary to filter switching ripple V g v g (s) 1 : M(D) Z ei (s) H e (s) Effective low-pass filter Z eo (s) V v(s) also filters ac variations effective filter elements may Power input not coincide with actual element values, but can also depend on operating point D Control input Load 127

Steps in the development of the canonical circuit model e(s) d(s) 1 : M(D) H e (s) V g v g (s) j(s) d(s) Z ei (s) Effective low-pass Z eo (s) V v(s) filter Power input D d(s) Control input Load 4. Control input variations also induce ac variations in converter waveforms Independent sources represent effects of variations in duty cycle Can push all sources to input side as shown. Sources may then become frequency-dependent 128

Transfer functions predicted by canonical model e(s) d(s) 1 : M(D) H e (s) V g v g (s) j(s) d(s) Z ei (s) Effective low-pass Z eo (s) V v(s) filter Power input D d(s) Control input Load Line-to-output transfer function: Control-to-output transfer function: G vg (s)= v(s) v g (s) =M(D)H e(s) G vd (s)= v(s) d(s) =e(s)m(d)h e(s) 129

7.6.2. Example: manipulation of the buck-boost converter model into canonical form Small-signal ac model of the buck-boost converter V L g V d 1 : D D' : 1 V g v g (s) Id Id C V v(s) Push independent sources to input side of transformers Push inductor to output side of transformers Combine transformers 130

Step 1 Push voltage source through 1:D transformer Move current source through D :1 transformer V g V D d L 1 : D D' : 1 V g v g (s) Id I D' d C V v(s) 131

Step 2 How to move the current source past the inductor: Break ground connection of current source, and connect to node A instead. Connect an identical current source from node A to ground, so that the node equations are unchanged. V g V D d node A L 1 : D D' : 1 V g v g (s) Id I D' d I D' d C V v(s) 132

Step 3 The parallel-connected current source and inductor can now be replaced by a Thevenin-equivalent network: V g V D d sli d D' L 1 : D D' : 1 V g v g (s) Id I D' d C V v(s) 133

Step 4 Now push current source through 1:D transformer. Push current source past voltage source, again by: Breaking ground connection of current source, and connecting to node B instead. Connecting an identical current source from node B to ground, so that the node equations are unchanged. Note that the resulting parallel-connected voltage and current sources are equivalent to a single voltage source. node B V g V D d sli d D' L 1 : D D' : 1 V g v g (s) Id DI D' d DI D' d C V v(s) 134

Step 5: final result Push voltage source through 1:D transformer, and combine with existing input-side transformer. Combine series-connected transformers. V g V D s LI DD' d(s) D' : D L D' 2 V g v g (s) I D' d(s) C V v(s) Effective low-pass filter 135

Coefficient of control-input voltage generator Voltage source coefficient is: e(s)= V gv D sli DD' Simplification, using dc relations, leads to e(s)= V D 2 1 sdl D' 2 Pushing the sources past the inductor causes the generator to become frequency-dependent. 136

7.6.3. Canonical circuit parameters for some common converters e(s) d(s) 1 : M(D) L e V g v g (s) j(s) d(s) C V v(s) Table 7.1. Canonical model parameters for the ideal buck, boost, and buck-boost converters Converter M(D) L e e(s) j(s) Buck D L V D 2 V Boost Buck-boost 1 D' D D' L D' 2 V 1 sl D' 2 L D' 2 V D 2 1 sdl D' 2 V D' 2 V D' 2 137

7.7. Modeling the pulse-wih modulator Pulse-wih modulator converts voltage signal v c into duty cycle signal d. What is the relation between v c and d? Power input v g transistor gate driver δ Switching converter pulse-wih modulator compensator v c G c (s) v Load v feedback connection δ v c voltage reference v ref dt s T s t t Controller 138

A simple pulse-wih modulator V M v saw Sawtooth wave v saw v c generator analog input v c comparator δ PWM waveform 0 δ t T s 0 dt s 2T s 139

Equation of pulse-wih modulator For a linear sawtooth waveform: d= v c V M for 0 v c V M V M v c v saw So d is a linear function of v c. 0 δ t T s 0 dt s 2T s 140

Perturbed equation of pulse-wih modulator PWM equation: Block diagram: d= v c V M Perturb: for 0 v c V M V c v c (s) 1 V M D d(s) v c =V c v c d=dd esult: D d= V cv c V M D = V c V M d= v c V M pulse-wih modulator Dc and ac relations: 141

Sampling in the pulse-wih modulator The input voltage is a continuous function of time, but there can be only one discrete value of the duty cycle for each switching period. v c 1 V M sampler f s d Therefore, the pulsewih modulator pulse-wih modulator samples the control waveform, with sampling rate equal to the switching frequency. In practice, this limits the useful frequencies of ac variations to values much less than the switching frequency. Control system bandwih must be sufficiently less than the Nyquist rate f s /2. Models that do not account for sampling are accurate only at frequencies much less than f s /2. 142

7.8. Summary of key points 1. The CCM converter analytical techniques of Chapters 2 and 3 can be extended to predict converter ac behavior. The key step is to average the converter waveforms over one switching period. This removes the switching harmonics, thereby exposing directly the desired dc and low-frequency ac components of the waveforms. In particular, expressions for the averaged inductor voltages, capacitor currents, and converter input current are usually found. 2. Since switching converters are nonlinear systems, it is desirable to construct small-signal linearized models. This is accomplished by perturbing and linearizing the averaged model about a quiescent operating point. 3. Ac equivalent circuits can be constructed, in the same manner used in Chapter 3 to construct dc equivalent circuits. If desired, the ac equivalent circuits may be refined to account for the effects of converter losses and other nonidealities. 143

Summary of key points 4. The state-space averaging method of section 7.4 is essentially the same as the basic approach of section 7.2, except that the formality of the state-space network description is used. The general results are listed in section 7.4.2. 5. The circuit averaging technique also yields equivalent results, but the derivation involves manipulation of circuits rather than equations. Switching elements are replaced by dependent voltage and current sources, whose waveforms are defined to be identical to the switch waveforms of the actual circuit. This leads to a circuit having a time-invariant topology. The waveforms are then averaged to remove the switching ripple, and perturbed and linearized about a quiescent operating point to obtain a small-signal model. 144

Summary of key points 6. When the switches are the only time-varying elements in the converter, then circuit averaging affects only the switch network. The converter model can then be derived by simply replacing the switch network with its averaged model. Dc and small-signal ac models of several common CCM switch networks are listed in section 7.5.4. Switching losses can also be modeled using this approach. 7. The canonical circuit describes the basic properties shared by all dc-dc PWM converters operating in the continuous conduction mode. At the heart of the model is the ideal 1:M(D) transformer, introduced in Chapter 3 to represent the basic dc-dc conversion function, and generalized here to include ac variations. The converter reactive elements introduce an effective low-pass filter into the network. The model also includes independent sources which represent the effect of duty cycle variations. The parameter values in the canonical models of several basic converters are tabulated for easy reference. 145

Summary of key points 8. The conventional pulse-wih modulator circuit has linear gain, dependent on the slope of the sawtooth waveform, or equivalently on its peak-to-peak magnitude. 146