Department of Electrical, Computer, and Energy Engineering...

R.W.EricksonDepartmentofElectrical,Computer,andEnergyEngineering

UniversityofColorado,Boulder

Fundamentals of Power Electronics! Chapter 7: AC equivalent circuit modeling!1!

Part II"Converter Dynamics and Control!

7. !AC equivalent circuit modeling!8. !Converter transfer functions!9.  Controller design!10.  Input filter design!11. !AC and DC equivalent circuit modeling of the

discontinuous conduction mode!12. !Current programmed control!


Chapter 7. AC Equivalent Circuit Modeling!

7.1 !Introduction!7.2 !The basic AC modeling approach!7.3 !State-space averaging!7.4 !Circuit averaging and averaged switch modeling!7.5 !The canonical circuit model!7.6 !Modeling the pulse-width modulator!7.7 !Summary of key points!


7.1. !Introduction!

A simple dc-dc regulator system, employing a buck converter!

Objective: maintain v(t) equal to an accurate, constant value V.!There are disturbances:!

•  in vg(t)!

•  in R!There are uncertainties:!

•  in element values!

•  in Vg!

•  in R!

+–

+

v(t)

–

vg(t)

Switching converterPowerinput

Load

–+

R

Compensator

Gc(s)

vrefVoltage

reference

v

Feedbackconnection

Pulse-widthmodulator

vc

Transistorgate driver

c(t)

c(t)

TsdTs t t

vc(t)

Controller


Applications of control in power electronics!

DC-DC converters!Regulate dc output voltage.!Control the duty cycle d(t) such that v(t) accurately follows a reference signal vref.!

DC-AC inverters!Regulate an ac output voltage. !Control the duty cycle d(t) such that v(t) accurately follows a reference signal vref (t).!

AC-DC rectifiers!Regulate the dc output voltage.!Regulate the ac input current waveform.!Control the duty cycle d(t) such that ig (t) accurately follows a reference signal iref (t), and v(t) accurately follows a reference signal vref.!


Converter Modeling!

Applications!Aerospace worst-case analysis!Commercial high-volume production: design for reliability and yield!

High quality design!Ensure that the converter works well under worst-case conditions!–  Steady state (losses, efficiency, voltage regulation)!–  Small-signal ac (controller stability and transient response)!

Engineering methodology!Simulate model during preliminary design (design verification)!Construct laboratory prototype converter system and make it work under

nominal conditions!Develop a converter model. Refine model until it predicts behavior of

nominal laboratory prototype!Use model to predict behavior under worst-case conditions!Improve design until worst-case behavior meets specifications (or until

reliability and production yield are acceptable)!!


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 vg(t), R, or d(t) affect the output voltage v(t)?!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)!•  Design input EMI filters that do not disrupt control system operation

(Chapter 10)!•  Model converters operating in DCM (Chapter 11)!•  Current-programmed control of converters (Chapter 12)!


Modeling!

•  Representation 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!


Finding the response to ac variations"Neglecting the switching ripple!

This causes the duty cycle to be modulated sinusoidally:!

Assume D and Dm are constants, !Dm! < D, and the modulation frequency ωm is much smaller than the converter switching frequency ωs = 2πfs.!

d(t) = D + Dmcostmt

0 5 10 15 200

0.5

1

0 5 10 15 200

2

4

6

0 5 10 15 20−17

−16

−15

−14

−13

hiL(t)iTs

iL(t)

vc(t)

Gate drive!(a)!

(b)!

(c)!

tTs

tTs

tTs

hv(t)iTs

v(t)

228 7 AC Equivalent Circuit Modeling

+–

+

v(t)

–

vg(t)

Switching converterPowerinput

Load

–+

R

Compensator

Gc(s)

vrefVoltage

reference

v

Feedbackconnection

Pulse-widthmodulator

vc

Transistorgate driver

c(t)

c(t)

TsdTs t t

vc(t)

Controller

Fig. 7.1. A simple dc–dc regulator system, including a buck converter power stage and a feedback network.

eling process involves use of approximations to neglect small but complicating phenomena, inan attempt to understand what is most important. Once this basic insight is gained, it may bedesirable to carefully refine the model, by accounting for some of the previously ignored phe-nomena. It is a fact of life that real, physical systems are complex, and their detailed analysiscan easily lead to an intractable and useless mathematical mess. Approximate models are animportant tool for gaining understanding and physical insight.

As discussed in Chapter 2, the switching ripple is small in a well-designed converter operat-ing in continuous conduction mode (CCM). Hence, we should ignore the switching ripple, andmodel only the underlying ac variations in the converter waveforms. For example, suppose thatsome ac variation is introduced into the control signal vc(t), such that

vc(t) = Vc + Vcm cos!mt (7.1)

where Vc and Vcm are constants, |Vcm| ⌧ Vc, and the modulation frequency !m is much smallerthan the converter switching frequency !S = 2⇡ fs. This control signal is fed into a pulse-widthmodulator (PWM) that generates a gate drive signal having switching frequency fs = 1/Tsand whose duty cycle during each switching period depends on the control signal vc(t) appliedduring that period. The resulting transistor gate drive signal is illustrated in Fig. 7.2(a), and atypical converter inductor current and output voltage waveforms iL(t) and v(t) are illustrated inFig. 7.2(b). The spectrum of v(t) is illustrated in Fig. 7.3. This spectrum contains components atthe switching frequency as well as its harmonics and sidebands; these components are small inmagnitude if the switching ripple is small. In addition, the spectrum contains a low-frequencycomponent at the modulation frequency !m. The magnitude and phase of this component de-pend not only on the control signal and duty cycle variation, but also on the frequency responseof the converter. If we neglect the switching ripple, then this low-frequency component remains

Suppose the control signal varies sinusoidally:!


Output voltage spectrum"with sinusoidal modulation of duty cycle!

Contains frequency components at:!•  Modulation frequency and its

harmonics!•  Switching frequency and its

harmonics!•  Sidebands of switching frequency!

With small switching ripple, high-frequency components (switching harmonics and sidebands) are small.!If ripple is neglected, then only low-frequency components (modulation frequency and harmonics) remain.!

Spectrumof v(t)

tm ts t

{Modulationfrequency and its

harmonics {Switchingfrequency and

sidebands {Switchingharmonics


Objective of ac converter modeling!

•  Predict how low-frequency variations in duty cycle induce low-frequency variations in the converter voltages and currents!

•  Ignore the switching ripple!•  Ignore complicated switching harmonics and sidebands!Approach:!•  Remove switching harmonics by averaging all waveforms

over one switching period!


Averaging to remove switching ripple!

where!

Average over one switching period to remove switching ripple:!

Note that, in steady-state,!

vL(t) Ts= 0

iC(t) Ts= 0

by inductor volt-second balance and capacitor charge balance.!

Ld iL(t) Ts

dt = vL(t) Ts

Cd vC(t) Ts

dt = iC(t) Ts


LdhiL(t)iTs

dt= hvL(t)iTs

CdhvC(t)iTs

dt= hiC(t)iTs (7.2)

where hx(t)iTs denotes the average of x(t) over an interval of length Ts:

hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.3)

So we will employ the basic approximation of removing the high-frequency switching rippleby averaging over one switching period. Yet the average value is allowed to vary from oneswitching period to the next, such that low-frequency variations are modeled. In e↵ect, the“moving average” of Eq. (7.3) constitutes low-pass filtering of the waveform. A few of thenumerous references on averaged modeling of switching converters are listed at the end of thischapter [69–88].

Note that the principles of inductor volt-second balance and capacitor charge balance predictthat the right-hand sides of Eqs. (7.2) are zero when the converter operates in equilibrium.Equations (7.2) describe how the inductor currents and capacitor voltages change when nonzeroaverage inductor voltage and capacitor current are applied over a switching period.

The averaged inductor voltage and capacitor currents of Eq. (7.2) are, in general, nonlin-ear functions of the signals in the converter, and hence Eqs. (7.2) constitute a set of nonlineardi↵erential equations. Indeed, the spectrum in Fig. 7.3 also contains harmonics of the modu-lation frequency !m. In most converters, these harmonics become significant in magnitude asthe modulation frequency !m approaches the switching frequency !s, or as the modulationamplitude Dm approaches the quiescent duty cycle D. Nonlinear elements are not uncommon inelectrical engineering; indeed, all semiconductor devices exhibit nonlinear behavior. To obtaina linear model that is easier to analyze, we usually construct a small-signal model that has beenlinearized about a quiescent operating point, in which the harmonics of the modulation or ex-citation frequency are neglected. As an example, Fig. 7.4 illustrates linearization of the familiardiode i–v characteristic shown in Fig. 7.4(b). Suppose that the diode current i(t) has a quiescent(dc) value I and a signal component i(t). As a result, the voltage v(t) across the diode has aquiescent value V and a signal component v(t). If the signal components are small compared tothe quiescent values,

|v| ⌧ |V |, |i| ⌧ |I| (7.4)

then the relationship between v(t) and i(t) is approximately linear, v(t) = rDi(t). The conductance1/rD represents the slope of the diode characteristic, evaluated at the quiescent operating point.The small-signal equivalent circuit model of Fig. 7.4(c) describes the diode behavior for smallvariations around the quiescent operating point.

An example of a nonlinear converter characteristic is the dependence of the steady-stateoutput voltage V of the buck-boost converter on the duty cycle D, illustrated in Fig. 7.5. Supposethat the converter operates with some dc output voltage, say, V = �Vg, corresponding to aquiescent duty cycle of D = 0.5. Duty cycle variations d about this quiescent value will excitevariations v in the output voltage. If the magnitude of the duty cycle variation is su�cientlysmall, then we can compute the resulting output voltage variations by linearizing the curve. Theslope of the linearized characteristic in Fig. 7.5 is chosen to be equal to the slope of the actualnonlinear characteristic at the quiescent operating point; this slope is the dc control-to-output


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!

constitute a system of nonlinear differential equations.!Hence, must linearize by constructing a small-signal converter model.!

Ld iL(t) Ts

dt = vL(t) Ts

Cd vC(t) Ts

dt = iC(t) Ts


Small-signal modeling of the diode!

Nonlinear diode, driven by current source having a DC and small AC component!

Linearization of the diode i-v characteristic about a quiescent operating point!

+

v = V+v

–

i = I+i

0 1 V

5 A

4 A

3 A

2 A

1 A

0

v

i

Quiescentoperatingpoint

Actualnonlinear

characteristicLinearizedfunction

i(t)I

v(t)

V

0.5 V

+

v

–

i

rD

Small-signal AC model!


Buck-boost converter: "nonlinear static control-to-output characteristic!

Example: linearization at the quiescent operating point!

!D = 0.5"

D

V

–Vg

0.5 100

Actualnonlinear

characteristic

Linearizedfunction

Quiescentoperatingpoint

V = Vg D/(1 – D)!


Result of averaged small-signal ac modeling!

Small-signal ac equivalent circuit model!

buck-boost example!

+–

+–

L

RC

1 : D Dv : 1

vg(t) Id (t)

Vg – V d (t)

Id (t) v(t)

+

–

Fundamentals of Power Electronics Chapter 7: AC equivalent circuit modeling15

7.2. The basic ac modeling approach

+–

LC R

+

v(t)

–

1 2

i(t)vg(t)

Buck-boost converter example


Switch in position 1

vL(t) = Ldi(t)dt = vg(t)

iC(t) = Cdv(t)dt = – v(t)R

iC(t) = Cdv(t)dt ≈ –

v(t) Ts

R

vL(t) = Ldi(t)dt ≈ vg(t) Ts

Inductor voltage and capacitor current are:

Small ripple approximation: replace waveforms with their low-frequency averaged values:

+– L C R

+

v(t)

–

i(t)

vg(t)


Switch in position 2

Inductor voltage and capacitor current are:

Small ripple approximation: replace waveforms with their low-frequency averaged values:

+– L C R

+

v(t)

–

i(t)

vg(t)vL(t) = L

di(t)dt = v(t)

iC(t) = Cdv(t)dt = – i(t) – v(t)R

vL(t) = Ldi(t)dt ≈ v(t) Ts

iC(t) = Cdv(t)dt ≈ – i(t) Ts

–v(t) TsR


7.2.1 Averaging the inductor waveforms!

Inductor voltage waveform!For the instantaneous inductor waveforms, we have:!

Can we write a similar equation for the averaged components? Let us investigate the derivative of the average inductor current:!

On the right hand side, we are allowed to interchange the order of integration and differentiation because the inductor current is continuous and its derivative v/L has a finite number of discontinuities in the period of integration. Hence: !


(a)

+– L C R

+

v(t)

–

i(t)

vg(t)

(b)

+– L C R

+

v(t)

–

i(t)

vg(t)

Fig. 7.8. Buck–boost converter circuit: (a) when the switch is in position 1, (b) when the switch is inposition 2.

Ldi(t)dt= vL(t) (7.9)

Is there a similar relationship between the averages of the inductor voltage and current? Let uscompute the derivative of the average inductor current:

dhi(t)iTs

dt=

ddt

"1Ts

Z t+Ts/2

t�Ts/2i(⌧)d⌧

#(7.10)

On the right side of this equation, we can interchange the order of di↵erentiation and integrationbecause the inductor current is continuous, and because its derivative vL(t)/L has a finite numberof discontinuities over the period of integration. Hence, the above equation becomes

dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

d i(⌧)d⌧

d⌧ (7.11)

Finally, we can use Eq. (7.9) to replace di(⌧)/d⌧ with vL(⌧):

dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

vL(⌧)L

d⌧ (7.12)

This can be rearranged to obtain

Ldhi(t)iTs

dt= hvL(t)iTs (7.13)

This result shows that average components of the inductor voltage and current follow the samedefining equation (7.9), with no change in L and no additional terms.


(a)

+– L C R

+

v(t)

–

i(t)

vg(t)

(b)

+– L C R

+

v(t)

–

i(t)

vg(t)




dhi(t)iTs

dt=

ddt

"1Ts

Z t+Ts/2

t�Ts/2i(⌧)d⌧

#(7.10)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

d i(⌧)d⌧

d⌧ (7.11)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

vL(⌧)L

d⌧ (7.12)


Ldhi(t)iTs




(a)

+– L C R

+

v(t)

–

i(t)

vg(t)

(b)

+– L C R

+

v(t)

–

i(t)

vg(t)




dhi(t)iTs

dt=

ddt

"1Ts

Z t+Ts/2

t�Ts/2i(⌧)d⌧

#(7.10)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

d i(⌧)d⌧

d⌧ (7.11)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

vL(⌧)L

d⌧ (7.12)


Ldhi(t)iTs




Averaging the inductor waveforms, p. 2!

Now replace the derivative of the inductor current with v/L:!


(a)

+– L C R

+

v(t)

–

i(t)

vg(t)

(b)

+– L C R

+

v(t)

–

i(t)

vg(t)




dhi(t)iTs

dt=

ddt

"1Ts

Z t+Ts/2

t�Ts/2i(⌧)d⌧

#(7.10)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

d i(⌧)d⌧

d⌧ (7.11)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

vL(⌧)L

d⌧ (7.12)


Ldhi(t)iTs



Finally, rearrange to obtain:!


(a)

+– L C R

+

v(t)

–

i(t)

vg(t)

(b)

+– L C R

+

v(t)

–

i(t)

vg(t)




dhi(t)iTs

dt=

ddt

"1Ts

Z t+Ts/2

t�Ts/2i(⌧)d⌧

#(7.10)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

d i(⌧)d⌧

d⌧ (7.11)


dhi(t)iTs

dt=

1Ts

Z t+Ts/2

t�Ts/2

vL(⌧)L

d⌧ (7.12)


Ldhi(t)iTs



So the averaged inductor current and voltage follow the same defining equation, with no additional terms and with the same value of L. A similar derivation for the capacitor leads to:!

7.2 The Basic AC Modeling Approach 235

(a)

o

vL(o)

dTs

v(t) Ts

vg(t) Ts

Averaging intervalTotal timedvTs

t + Ts/2t – Ts/2 t

(b)

v TsL

i(o)

t

dTs

t + Ts/2t – Ts/2

Averaging interval

vg TsL Total time

dvTs

v TsL

o

Fig. 7.9. Mechanics of evaluating the average inductor waveforms at some arbitrary time t: (a) averagingthe inductor voltage vL(t), (b) averaging the inductor current i(t).

We can employ a similar analysis to find the relationship between the average componentsof a capacitor voltage and current, with the following result:

Cdhv(t)iTs

dt= hiC(t)iTs (7.14)

We next need to evaluate the right sides of the above two equations, by averaging the inductorvoltage and capacitor current waveforms.

7.2.2 The Average Inductor Voltage and the Small-Ripple Approximation

The inductor voltage and current waveforms for the buck–boost converter example are illus-trated in Fig. 7.9. It is desired to compute the average inductor voltage hvL(t)iTs at some arbi-trary time t. As illustrated in Fig. 7.9, the averaging interval extends over the interval beginningat t� Ts/2 and ending at t+ Ts/2. For the example time illustrated, there is an interval of lengthdTs in which the inductor voltage is vL = vg, and there are two intervals of total length d0Ts inwhich the inductor voltage is vL = v.

We now make the small-ripple approximation. But rather than replacing vg(t), v(t), and i(t)with their dc components Vg, V and I as in Chapter 2, we now replace them with their low-frequency averaged values hvg(t)iTs , hv(t)iTs , and hi(t)iTs , defined by Eq. (7.3). It is important tonote that it is valid to apply the small-ripple approximation only to quantities that actually havesmall ripple and are nonpulsating; hence, we apply this approximation to the inductor currents,capacitor voltages, and independent sources that indeed have small ripple and are continuousfunctions of time.


7.2.2 The average inductor voltage "and small ripple approximation! 7.2 The Basic AC Modeling Approach 235

(a)

o

vL(o)

dTs

v(t) Ts

vg(t) Ts


t + Ts/2t – Ts/2 t

(b)

v TsL

i(o)

t

dTs

t + Ts/2t – Ts/2

Averaging interval

vg TsL Total time

dvTs

v TsL

o



Cdhv(t)iTs






The actual inductor voltage waveform, sketched at some arbitrary time t. Let’s compute the average over the averaging interval (t – Ts/2, t + Ts/2)!

While the transistor conducts, the inductor voltage is!


The usefulness of the small-ripple approximation here is that we ignore the changes in thesequantites during one switching period or during the averaging interval (t�Ts/2, t+Ts/2). As inthe steady-state case, the small ripple approximation considerably simplifies the mathematics.This approximation is valid provided that the natural frequencies of the circuit are su�cienctlyslower than the switching frequency, so that the ripples in the actual inductor current and capac-itor voltage waveforms are indeed small.

With the small-ripple approximation, we can express the inductor voltage for the subintervalof length dTs [Eq. (7.5)] as

vL(t) = Ldi(t)dt⇡ hvg(t)iTs (7.15)

In a similar manner, for the remaining subintervals of total length d0Ts [Eq. (7.7)], we canexpress the inductor voltage as

vL(t) = Ldi(t)dt⇡ hv(t)iTs (7.16)

The average inductor voltage is therefore

hvL(t)iTs =1Ts

Z t+Ts/2

t�Ts/2vL(⌧) d⌧ ⇡ d(t)hvg(t)iTs + d0(t)hv(t)iTs (7.17)

Insertion of this expression into Eq. (7.13) leads to

Ldhi(t)iTs

dt= d(t)hvg(t)iTs + d0(t)hv(t)iTs (7.18)

This equation describes how the low-frequency components of the inductor current vary withtime, and is the desired result.

7.2.3 Discussion of the Averaging Approximation

The averaging operator, Eq. (7.3), is repeated below:

hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.19)

Averaging is an artifice that facilitates the derivation of tractable equations describing the lowfrequency dynamics of the switching converter. It removes the waveform components at theswitching frequency and its harmonics, while preserving the magnitude and phase of the wave-form low-frequency components. In this chapter, we replace the converter waveforms by theiraverages, to find models that describe the dynamic properties of switching converters operatingin the continuous conduction mode. In later chapters of this text, this averaging operator is em-ployed in other situations such as the discontinuous conduction mode or current-programmedcontrol.

Figure ?? illustrates the inductor current and voltage waveforms of a buck–boost converterin which the duty cycle is varied sinusoidally. The waveform averages as computed by Eq. (7.19)are superimposed. It can be seen that the hi(t)iTs waveform indeed passes through the center ofthe actual i(t) waveform. Additionally, an increase in hvL(t)iTs causes an increase in the slope ofhi(t)iTs , as predicted by Eq. (7.13).

Small ripple approximation: assume that vg has small ripple, and does not change significantly over the averaging interval.!

While the diode conducts, the inductor voltage is!

Small ripple approximation: assume that v has small ripple, and does not change significantly over the averaging interval.!








hvL(t)iTs =1Ts

Z t+Ts/2



Ldhi(t)iTs





hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.19)




Average inductor voltage, p. 2!7.2 The Basic AC Modeling Approach 235

(a)

o

vL(o)

dTs

v(t) Ts

vg(t) Ts


t + Ts/2t – Ts/2 t

(b)

v TsL

i(o)

t

dTs

t + Ts/2t – Ts/2

Averaging interval

vg TsL Total time

dvTs

v TsL

o



Cdhv(t)iTs













hvL(t)iTs =1Ts

Z t+Ts/2



Ldhi(t)iTs





hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.19)



The average inductor voltage is therefore:!








hvL(t)iTs =1Ts

Z t+Ts/2



Ldhi(t)iTs





hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.19)



Hence:!


7.2.3 Discussion of the averaging approximation!








hvL(t)iTs =1Ts

Z t+Ts/2



Ldhi(t)iTs





hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.19)



The averaging operator!

•  Averaging facilitates derivation of tractable equations describing the dynamics of the converter!

•  Averaging removes the waveform components at the switching frequency and its harmonics, while preserving the magnitude and phase of the waveform low-frequency components!

•  Averaging can be viewed as a form of low-pass filtering!


Simulation of converter waveforms "and their averages!

0 5 10 15 200

0.5

1

0 5 10 15 200

2

4

6

0 5 10 15 20−17

−16

−15

−14

−13

hiL(t)iTs

iL(t)

vc(t)

Gate drive!(a)!

(b)!

(c)!

tTs

tTs

tTs

hv(t)iTs

v(t)

Computer-generated plots of inductor current and voltage, and comparison with their averages, for sinusoidal modulation of duty cycle!


The averaging operator as a low-pass filter!








hvL(t)iTs =1Ts

Z t+Ts/2



Ldhi(t)iTs





hx(t)iTs =1Ts

Z t+Ts/2

t�Ts/2x(⌧)d⌧ (7.19)



The averaging operator:!


!"# !"$ !"% !"& !"' !"( # $ % &�&!

�%'

�%!

�$'

�$!

�#'

�#!

�'

!

f /fs

)*+,-./012304

5617/1,893:1;<=,;13=>3?@16*+-,+3A<16*.=6

Fig. 7.10. Frequency response of the averaging operator: kGav( j!) k given by Eq. (7.22).

The averaging operator of Eq. (7.19) is a transformation that e↵ectively performs a low-passfunction to remove the switching ripple. Indeed, we can take the Laplace transformation of Eq.(7.19) to obtain:

hx(s)iTs = Gav(s)x(s) (7.20)

It can be shown that Gav(s) is given by

Gav(s) =esTs/2 � e�sTs/2

sTs(7.21)

We can compute the e↵ect of the averaging operator on a sinusoid of angular frequency ! byletting s = j! in the above equation. The transfer function Gav then becomes:

Gav( j!) =e j!Ts/2 � e� j!Ts/2

j!Ts=

sin(!Ts/2)!Ts/2

(7.22)

Figure 7.10 contains a plot of the magnitude (expressed in decibels) of Eq. 7.22 vs. frequency(for more information on frequency response plots, see Section 8.1). The averaging operatorexhibits a low frequency gain of 1 (or 0 dB), and a gain of zero (or �1 dB) at the switch-ing frequency fs and its harmonics. Equation 7.22 is purely real, and exhibits zero phase shiftfor frequencies less than the switching frequency. Thus, the averaging operator preserves themagnitude and phase of the low-frequency components of the waveform, while removing com-ponents at the switching frequency and its harmonics.

For frequencies f greater than approximately fs/3, Fig. 7.10 exhibits substantial attenuation.This suggests that averaged models may not accurately predict transient responses at higherfrequencies. The high-frequency dynamics of the discontinuous conduction mode is an exampleof this behavior, and is discussed further in Section 16.5.

Unlike the steady-state analyses of Chapters 2 and 3, Fig. 7.9 is sketched at an arbitrarytime t, with an averaging interval that does not necessarily begin when the transistor is switched


!"# !"$ !"% !"& !"' !"( # $ % &�&!

�%'

�%!

�$'

�$!

�#'

�#!

�'

!

f /fs

)*+,-./012304

5617/1,893:1;<=,;13=>3?@16*+-,+3A<16*.=6






sTs(7.21)



j!Ts=

sin(!Ts/2)!Ts/2

(7.22)





!"# !"$ !"% !"& !"' !"( # $ % &�&!

�%'

�%!

�$'

�$!

�#'

�#!

�'

!

f /fs

)*+,-./012304

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sTs(7.21)



j!Ts=

sin(!Ts/2)!Ts/2

(7.22)





!"# !"$ !"% !"& !"' !"( # $ % &�&!

�%'

�%!

�$'

�$!

�#'

�#!

�'

!

f /fs

)*+,-./012304

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sTs(7.21)



j!Ts=

sin(!Ts/2)!Ts/2

(7.22)




Take Laplace transform:!

•  Notches at the switching frequency and its harmonics!

•  Zero phase shift!•  Magnitude of – 3 dB and

half of the switching frequency!


Averaging the capacitor waveforms!238 7 AC Equivalent Circuit Modeling

(a)

t

iC(t)

dTs Ts0

–v(t) TsR – i(t) Ts

–v(t) TsR

iC(t) Ts

(b)tv(t)

dTs Ts0

v(0)

v(dTs)

v(Ts)

v(t) Ts

–v(t) TsRC –

v(t) TsRC –

i(t) TsC

Fig. 7.11. Buck–boost converter waveforms: (a) capacitor current, (b) capacitor voltage.

on. This rigorous definition of averaging is important when modeling high-bandwidth controlschemes such as the current-programmed mode of Chapter 17. The choice of averaging intervalextending from (t � Ts/2) to (t + Ts/2) preserves the phase of the waveform, and thereforecorrectly predicts the behavior of current-programmed converters. It should also be noted thatcomputing the average by integrating one half-cycle into the future [i.e., to (t + Ts/2)] does notviolate any physical causality constraint, because this is merely a modeling artifice that is notimplemented in a physical circuit.

We may also note that the result of Eq. (7.18) can be derived without such rigor. For deriva-tion of continuous-time models of the continuous conduction mode, the same result is obtainedregardless of whether the averaging interval begins at (t�Ts/2) or at the instant when the transis-tor is switched on. For the remainder of this textbook, we will continue to employ the simplerarguments begun in Chapter 2, in which the averaging interval begins when the transistor isswitched on. In later chapters, the more rigorous treatment will be employed when necessary,such as when modeling the high-frequency dynamics of current-programmed control.

7.2.4 Averaging the Capacitor Waveforms

A similar procedure leads to the capacitor dynamic equation. The capacitor voltage and currentwaveforms are sketched in Fig. 7.11. When the switch is in position 1, the capacitor current isgiven by

iC(t) = Cdv(t)

dt= �v(t)

R⇡ �hv(t)iTs

R(7.23)

With the switch in position 2, the capacitor current is


(a)

t

iC(t)

dTs Ts0

–v(t) TsR – i(t) Ts

–v(t) TsR

iC(t) Ts

(b)tv(t)

dTs Ts0

v(0)

v(dTs)

v(Ts)

v(t) Ts

–v(t) TsRC –

v(t) TsRC –

i(t) TsC

Fig. 7.11. Buck–boost converter waveforms: (a) capacitor current, (b) capacitor voltage.

on. This rigorous definition of averaging is important when modeling high-bandwidth controlschemes such as the current-programmed mode of Chapter 17. The choice of averaging intervalextending from (t � Ts/2) to (t + Ts/2) preserves the phase of the waveform, and thereforecorrectly predicts the behavior of current-programmed converters. It should also be noted thatcomputing the average by integrating one half-cycle into the future [i.e., to (t + Ts/2)] does notviolate any physical causality constraint, because this is merely a modeling artifice that is notimplemented in a physical circuit.

We may also note that the result of Eq. (7.18) can be derived without such rigor. For deriva-tion of continuous-time models of the continuous conduction mode, the same result is obtainedregardless of whether the averaging interval begins at (t�Ts/2) or at the instant when the transis-tor is switched on. For the remainder of this textbook, we will continue to employ the simplerarguments begun in Chapter 2, in which the averaging interval begins when the transistor isswitched on. In later chapters, the more rigorous treatment will be employed when necessary,such as when modeling the high-frequency dynamics of current-programmed control.

7.2.4 Averaging the Capacitor Waveforms

A similar procedure leads to the capacitor dynamic equation. The capacitor voltage and currentwaveforms are sketched in Fig. 7.11. When the switch is in position 1, the capacitor current isgiven by

iC(t) = Cdv(t)

dt= �v(t)

R⇡ �hv(t)iTs

R(7.23)

With the switch in position 2, the capacitor current is


t

ig(t)

dTs Ts00

i(t) Ts

ig(t) Ts

0

Fig. 7.12. Buck–boost converter waveforms: input source current ig(t).

iC(t) = Cdv(t)

dt= �i(t) � v(t)

R⇡ � hi(t)iTs

�hv(t)iTs

R(7.24)

The average capacitor current can be found by averaging Eqs. (7.23) and (7.24); the result is

hiC(t)iTs = d(t) �hv(t)iTs

R

!+ d0(t)

�hi(t)iTs �

hv(t)iTs

R

!(7.25)

Upon inserting this equation into Eq. (7.2) and collecting terms, one obtains

Cdhv(t)iTs

dt= �d0(t) hi(t)iTs �

hv(t)iTs

R(7.26)

This is the basic averaged equation which describes dc and low-frequency ac variations in thecapacitor voltage.

7.2.5 The Average Input Current

In Chapter 3, it was found to be necessary to write an additional equation that models the dccomponent of the converter input current. This allowed the input port of the converter to bemodeled by the dc equivalent circuit. A similar procedure must be followed here, so that low-frequency variations at the converter input port are modeled by the ac equivalent circuit.

For the buck-boost converter example, the current ig(t) drawn by the converter from theinput source is equal to the inductor current i(t) during the first subinterval, and zero during thesecond subinterval. By neglecting the inductor current ripple and replacing i(t) with its averagedvalue hi(t)iTS , we can express the input current as follows:

ig(t) =(hi(t)iTs during subinterval 1

0 during subinterval 2 (7.27)

The input current waveform is illustrated in Fig. 7.12. Upon averaging over one switching pe-riod, one obtains

hig(t)iTs = d(t)hi(t)iTs (7.28)

This is the basic averaged equation which describes dc and low-frequency ac variations in theconverter input current.

Switch position 1:!

Average:!


t

ig(t)

dTs Ts00

i(t) Ts

ig(t) Ts

0


iC(t) = Cdv(t)

dt= �i(t) � v(t)

R⇡ � hi(t)iTs

�hv(t)iTs

R(7.24)



R

!+ d0(t)

�hi(t)iTs �

hv(t)iTs

R

!(7.25)


Cdhv(t)iTs


hv(t)iTs

R(7.26)










Switch position 2:!


t

ig(t)

dTs Ts00

i(t) Ts

ig(t) Ts

0


iC(t) = Cdv(t)

dt= �i(t) � v(t)

R⇡ � hi(t)iTs

�hv(t)iTs

R(7.24)



R

!+ d0(t)

�hi(t)iTs �

hv(t)iTs

R

!(7.25)


Cdhv(t)iTs


hv(t)iTs

R(7.26)










Hence:! For the inductor voltage, the procedure was explained in most correct in general way for arbitrary t. But result is the same when t coincides with beginning of switching period, as illustrated here.!


7.2.4 The average input current

t

ig(t)

dTs Ts0

0

i(t) Ts

ig(t) Ts

0

Converter input current waveform

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

ig(t) =i(t) Ts

during subinterval 1

0 during subinterval 2

Average value:

ig(t) Ts= d(t) i(t) Ts


7.2.5. Perturbation and linearization

Ld i(t) Ts

dt = d(t) vg(t) Ts+ d'(t) v(t) Ts

Cd v(t) Ts

dt = – d'(t) i(t) Ts–

v(t) Ts

R


Converter averaged equations:

—nonlinear because of multiplication of the time-varying quantity d(t) with other time-varying quantities such as i(t) and v(t).


Construct small-signal model:Linearize about quiescent operating point

If the converter is driven with some steady-state, or quiescent, inputsd(t) = Dvg(t) Ts

= Vg

then, from the analysis of Chapter 2, after transients have subsided the inductor current, capacitor voltage, and input current

i(t) Ts, v(t) Ts

, ig(t) Ts

reach the quiescent values I, V, and Ig, given by the steady-state analysis as

V = – DD' Vg

I = – VD' R

Ig = D I


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:

vg(t) Ts= Vg + vg(t)

d(t) = D + d(t)

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(t) Ts= I + i(t)

v(t) Ts= V + v(t)

ig(t) Ts= Ig + ig(t)


The small-signal assumption

vg(t) << Vg

d(t) << D

i(t) << Iv(t) << V

ig(t) << Ig

If the ac variations are much smaller in magnitude than the respective quiescent values,

then the nonlinear converter equations can be linearized.


Perturbation of inductor equation

Insert the perturbed expressions into the inductor differential equation:

Ld I + i(t)

dt = D + d(t) Vg + vg(t) + D' – d(t) V + v(t)

note that d’(t) is given by

d'(t) = 1 – d(t) = 1 – D + d(t) = D' – d(t) with D’ = 1 – D

Multiply out and collect terms:

L dIdt⁄0+d i(t)dt = DVg+ D'V + Dvg(t) + D'v(t) + Vg – V d(t) + d(t) vg(t) – v(t)

Dc terms 1 st order ac terms 2nd order ac terms(linear) (nonlinear)


The perturbed inductor equation



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


Neglect of second-order terms



vg(t) << Vg

d(t) << D

i(t) << Iv(t) << V

ig(t) << Ig

Provided then the second-order ac terms are much smaller than the first-order terms. For example,d(t) vg(t) << D vg(t) d(t) << Dwhen

So neglect second-order terms.Also, dc terms on each side of equation

are equal.


Linearized inductor equation

Upon discarding second-order terms, and removing dc terms (which add to zero), we are left with

Ld i(t)dt = Dvg(t) + D'v(t) + Vg – V d(t)

This is the desired result: a linearized equation which describes small-signal ac variations.

Note that the quiescent values D, D’, V, Vg, are treated as given constants in the equation.


Capacitor equation

Perturbation leads to

C dVdt⁄0+dv(t)dt = – D'I – VR + – D'i(t) – v(t)R + Id(t) + d(t)i(t)

Dc terms 1 st order ac terms 2nd order ac term(linear) (nonlinear)

Neglect second-order terms. Dc terms on both sides of equation are equal. The following terms remain:

Cdv(t)dt = – D'i(t) – v(t)R + Id(t)

This is the desired small-signal linearized capacitor equation.

Cd V + v(t)

dt = – D' – d(t) I + i(t) –V + v(t)

RCollect terms:


Average input current

Perturbation leads to

Ig + ig(t) = D + d(t) I + i(t)

Collect terms:

Ig + ig(t) = DI + Di(t) + Id(t) + d(t)i(t)

Dc term 1 st order ac term Dc term 1 st order ac terms 2nd order ac term(linear) (nonlinear)

Neglect second-order terms. Dc terms on both sides of equation are equal. The following first-order terms remain:

ig(t) = Di(t) + Id(t)

This is the linearized small-signal equation which described the converter input port.


7.2.6. Construction of small-signalequivalent circuit model

The linearized small-signal converter equations:




Reconstruct equivalent circuit corresponding to these equations, in manner similar to the process used in Chapter 3.


Inductor loop equation


+–

+–

+–

L

D' v(t)

Vg – V d(t)

L d i(t)dtD vg(t)

i(t)

+ –


Capacitor node equation

+

v(t)

–

RC

C dv(t)dt

D' i(t) I d(t)

v(t)R



Input port node equation


+– D i(t)I d(t)

i g(t)

vg(t)


Complete equivalent circuit

Collect the three circuits:

+– D i(t)I d(t)vg(t) +

–

+–

+–

L

D' v(t)

Vg – V d(t)

D vg(t)

i(t)

+

v(t)

–

RCD' i(t) I d(t)

Replace dependent sources with ideal dc transformers:

+– I d(t)vg(t)

+–

L Vg – V d(t)

+

v(t)

–

RCI d(t)

1 : D D' : 1

Small-signal ac equivalent circuit model of the buck-boost converter


7.2.7. Results for several basic converters

+– I d(t)vg(t)

+–

LVg d(t)

+

v(t)

–

RC

1 : D

i(t)

+–

L

C Rvg(t)

i(t) +

v(t)

–

+–

V d(t)

I d(t)

D' : 1

buck

boost


Results for several basic converters

+– I d(t)vg(t)

+–

L Vg – V d(t)

+

v(t)

–

RCI d(t)

1 : D D' : 1

i(t)

buck-boost


7.3. Example: a nonideal flyback converter

+–

D1

Q1

C R

+

v(t)

–

1 : n

vg(t)

ig(t)

L

Flyback converter example

• MOSFET has on-resistance Ron

• Flyback transformer has magnetizing inductance L, referred to primary


Circuits during subintervals 1 and 2

+–

D1

Q1

C R

+

v(t)

–vg(t)

ig(t)

L

i(t) iC(t)+

vL(t)

–

1 : n

ideal

+–

L

+

v

–

vg

1:n

C

transformer model

iig

R

iC+

vL

–

Ron

+–

+

v

–

vg

1:n

C

transformer model

i

R

iC

i/n

–v/n

+

+

vL

–

ig=0

Flyback converter, with transformer equivalent circuit

Subinterval 1

Subinterval 2


Subinterval 1

+–

L

+

v

–

vg

1:n

C

transformer model

iig

R

iC+

vL

–

Ron

vL(t) = vg(t) – i(t) Ron

iC(t) = –v(t)R

ig(t) = i(t)

Circuit equations:

Small ripple approximation:

vL(t) = vg(t) Ts– i(t) Ts

Ron

iC(t) = –v(t) Ts

Rig(t) = i(t) Ts

MOSFET conducts, diode is reverse-biased


Subinterval 2

Circuit equations:

Small ripple approximation:

MOSFET is off, diode conducts

vL(t) = –v(t)n

iC(t) = –i(t)n – v(t)R

ig(t) = 0

vL(t) = –v(t) Tsn

iC(t) = –i(t) Tsn –

v(t) Ts

Rig(t) = 0

+–

+

v

–

vg

1:n

C

transformer model

i

R

iC

i/n

–v/n

+

+

vL

–

ig=0


Inductor waveforms

t

vL(t)

dTs Ts0

vg – iRon

– v/n

vL(t) Ts

t

i(t)

dTs Ts0

i(t) Ts

– v(t) TsnL

vg(t) Ts– Ron i(t) TsL

vL(t) Ts= d(t) vg(t) Ts

– i(t) TsRon + d'(t)

– v(t) Tsn

Ld i(t) Ts

dt = d(t) vg(t) Ts– d(t) i(t) Ts

Ron – d'(t)v(t) Tsn

Average inductor voltage:

Hence, we can write:


Capacitor waveforms

Average capacitor current:

Hence, we can write:

t

iC(t)

dTs Ts0

– v/R

iC(t) Ts

in –

vR

t

v(t)

dTs Ts0

v(t) Ts

–v(t) TsRC

i(t) TsnC –

v(t) TsRC

iC(t) Ts= d(t)

– v(t) Ts

R + d'(t)i(t) Tsn –

v(t) Ts

R

Cd v(t) Ts

dt = d'(t)i(t) Tsn –

v(t) Ts

R


Input current waveform

Average input current:

t

ig(t)

dTs Ts0

0

i(t) Ts

ig(t) Ts

0



The averaged converter equations

Ld i(t) Ts



Cd v(t) Ts


v(t) Ts

R


— a system of nonlinear differential equations

Next step: perturbation and linearization. Let

vg(t) Ts= Vg + vg(t)

d(t) = D + d(t)

i(t) Ts= I + i(t)

v(t) Ts= V + v(t)

ig(t) Ts= Ig + ig(t)


L dIdt⁄0+d i(t)dt = DVg– D'Vn – DRonI + Dvg(t) – D'

v(t)n + Vg + Vn – IRon d(t) – DRoni(t)

Dc terms 1 st order ac terms (linear)

+ d(t)vg(t) + d(t)v(t)n – d(t)i(t)Ron

2nd order ac terms (nonlinear)

Perturbation of the averaged inductor equation

Ld i(t) Ts



Ld I + i(t)

dt = D + d(t) Vg + vg(t) – D' – d(t)V + v(t)

n – D + d(t) I + i(t) Ron


Linearization of averaged inductor equation

Dc terms:

Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are:

0 = DVg– D'Vn – DRonI

Ld i(t)dt = Dvg(t) – D'


This is the desired linearized inductor equation.


Perturbation of averaged capacitor equation

Cd v(t) Ts


v(t) Ts

R

Cd V + v(t)

dt = D' – d(t)I + i(t)n –

V + v(t)R

C dVdt⁄0+dv(t)dt = D'I

n – VR + D'i(t)n – v(t)R – Id(t)n – d(t)i(t)

n

Dc terms 1 st order ac terms 2nd order ac term(linear) (nonlinear)

Original averaged equation:

Perturb about quiescent operating point:

Collect terms:


Linearization of averaged capacitor equation

0 = D'In – VR

Cdv(t)dt = D'i(t)n – v(t)R – Id(t)n

Dc terms:


This is the desired linearized capacitor equation.


Perturbation of averaged input current equation

Original averaged equation:

Perturb about quiescent operating point:

Collect terms:


Ig + ig(t) = D + d(t) I + i(t)

Ig + ig(t) = DI + Di(t) + Id(t) + d(t)i(t)

Dc term 1 st order ac term Dc term 1 st order ac terms 2nd order ac term(linear) (nonlinear)


Linearization of averaged input current equation

Dc terms:


This is the desired linearized input current equation.

Ig = DI



Summary: dc and small-signal acconverter equations

0 = DVg– D'Vn – DRonI

0 = D'In – VR

Ig = DI





Dc equations:

Small-signal ac equations:

Next step: construct equivalent circuit models.


Small-signal ac equivalent circuit:inductor loop



+–

+–

+–

L

D' v(t)n

d(t) Vg – IRon + Vn

L d i(t)dtD vg(t)

i(t)

+ –

DRon


Small-signal ac equivalent circuit:capacitor node


+

v(t)

–

RC

C dv(t)dt

D' i(t)n

I d(t)n

v(t)R


Small-signal ac equivalent circuit:converter input node


+– D i(t)I d(t)

i g(t)

vg(t)


Small-signal ac model,nonideal flyback converter example

Combine circuits:

Replace dependent sources with ideal transformers:

+– D i(t)I d(t)

i g(t)

vg(t) +–

+–

+–

L

D' v(t)n

d(t) Vg – IRon + Vn

D vg(t)

i(t)

DRon

+

v(t)

–

RCD' i(t)n

I d(t)n

+– I d(t)

i g(t)

vg(t)

L d(t) Vg – IRon + Vn

i(t) DRon+

v(t)

–

RCI d(t)n

1 : D +– D' : n

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