8/21/2019 00048525

1/8

EXTENSION OF STATE SPACE AVERAGING TO

RESONANT SWITCHES

=

A N D B E Y O N D

ArthurF.

Witulski and RobertW. nckson

Department of Electrical and Computer Engineering

University

of

Colorado at Boulder, 80309-0425

Abstrac t

In this pap er it is shown that the state space averaging melhod can be

extended

by

linear network theory from the domain of pulse w idth modulated

converters to a much larger class of converters, including resonant switch

converters, current programmed mode, and others. The canonical model

concept

is

also extend ed. and it is shown that the effectof resonant switching is

to

introduce a feedback block into the generalized canonical model. These

results are applied to linear zero current and zero voltage resonant swilches, a

new classof nonlinear resonant switch converters. nd the current programmed

mode. Equivalent circuit models are developed for both full and half wave

operation. and experimental verjficatwn

s

presented.

1. Introduction

Equivalent circuit modeling is well e stablishedas a useful tool for small

signal ac analysis of switching converters [l ]. Small signal analysis is necessary

because of the nonlinear and time varying nature of the converter caused by the

switching element. The most systematic modeling technique has been that of

state space averaging, which can be applied to

all

pulse width modulated (PWM)

converters provided the linear ripple assumption is satisfied. State space

averaging is valuable because it provides a method of analyzing both the dc and

ac behavior of a large number of converters in a systematic manner.

Furthermore, it has led to development of the canonical circuit model, which

permits the representation of the dynamics of a number of different converten by

a single equiva lent circuit model [l ]. Recently a large number of quasi-resonant

switch (QRS) converters have been introduced both of the linear [2-51 and

nonlinear variety [6]. These. converters operate n a different manner than PWM

converters, yet present the same kind of analytical difficulty, namely, nonlinear

nd

time varying behavior. It is desirable to apply the same analytical techniques

to the QRS

as

to the PWM converters, but state space averaging does not appear

applicable to QRS converters because of the presence of resonant elements for

which the linear ripple approximationisnot satisfied.

Consequently the analysis done

to

date on QRS conveners has been done

via circuit oriented techniques based on circuit averaging [71, both for the steady

state analysis [2-6] and for small signal ac analysis [8,9]. The ac modeling is

done by finding the average value of the switch waveforms, then deriving a two

port small signal model for the low frequency ac behavior of the switch. The

two port model is rotated as necessary and installed in the position the switch

occupies n the original converter [8,9]. Althoughthis circuit oriented approach

yields a great deal of physical insight into conver ter performance in a direct and

intuit ive manner, it does not necessar ily build on the body of knowledge about

PWM converters obtained from state space averaging. In addition, the form of

the circuit model is quite different fromthatof the PWM state space model, and

for more complicated switches and converters (e.g.. a

half

wave zero current

switched buck boost converter) reduction of the model to canonical form by

circuit manipulations is not straightfonvard. However, the end results of the

circuit oriented analysis of QRS converters and the state space averag ing of

PWM converters

are

similar and suggest that both classes of converters may

yield to the same kind of analysis. It is shown in this paper that state space

averaging can be extended to a much wider class of converters

th n

previously

thought, and that by doing

so

the analysis of PWM and QRS converters

can be

unified.

Consider first the primary results of state space averaging. The state

space averaging result states that i fa PWM converter (Fig. 1) is characterized by

state matrix Ai, input matrix B1, output matrix Ci, and feedforward matrix El

during the first s witching interval, (i.e., while the transistor is on), and by the

matrices A2, B2, Cz. and E2

n

the second interval (the transistor is off), then

the two separate topologies can be replaced by a single averaged equivalent

system given by

X F = (DA~+(I-D)A~)XF(DBl+(l-D)&)u

Y

=

(DC~+(~-D)CZNF+DEi+(l-D)EL)u (1)

The quantity D

is

the switch duty cycle,

XF

is the vector of the converter filter

states, and U is the vector of power inputs. This approximation s valid as long

as

the averaging is done over an interval

T,

short with respect to the natural time

constants Ton of the filter elements or

f,>>f, (2)

This work was supported in part by the Delco-Remy Division of the General

Motors Corporation, Anderson, IN and by the General Electric Foundation.

for each converter natural frequency f h . where fs is the switching frequency, a

condition refemd to

as the

linear ripple

assumption.

Application of these results

to the converters of Fig. 1 yields the following conversion ratios:

M = D (buck) M=4 boost) M= (buck-boost)

V8 D D*

(3)

The quantityD is equal to 1-D.

A small signal dynamic model can also be obtained from the state space

averaged result when each quantity

in Eq.

(1) is replaced by a steady state

component (upper case, zero subscript)plus a small time varying component

(lower case, with a caret). The small signal state space averaged model is given

by:

& = ( D A ~ + ( I - D ) A ~ ) ~ ~ ~( D B ~ + ( I - D ) B ~ ) ~ ~

+ [(AI-AZFF + (B1-Bz)Uol~

?=

DCI+(~-D)CZ)XF(DEi+(l-D_)&)ii

+ [(Cl-Cz)Xm + (El-&)Uold

4)

Equivalent small signal circuit models can then be constructed from Eqs.(4).

The buck, boost, and buck boost converters can all be modeled by a single

canonical model [l], in which the circuit parameters depend on the converter

topology. The canonical model shows that each PWM converter equivalent

circuit model possesses hree basic properties: independent conuol voltage and

current sources, dc to dc power conversion , and

an

equivalent ow pass filtering

network [l ]. The canonical model, therefore, provides a useful tool for

understanding he physical mechanisms of the power conversion process, and a

systematic means of evaluating and comparing the dynamic behavior of different

converters.

The operation of the quasi-resonantconverter is quite different from that

of the PWM converters. In a QRS converter a resonant inductor and capacitor

are added to the s witch network (Fig. 2b). When the transistor is turned on at

zero current

or

zero voltage, the inductor and capacitor resonate to form a

piecewise sinusoidal waveform which transfers the energy from the source to the

output filter. Because the switching occurs at zero current (ZCS) [2]or zero

voltage (ZVS) [3], the switching loss is greatly reduced from that of a PWM

converter. To reduce conduction losses, a nonlinearity can be added to the

resonant inductor (Fig.

2c)

which limits the peak current in the switch [61 and

forms a nonlinear resonant switch (NRS). In any case, the resonant switch

converter, in contrast to the PWM converter, has high frequency energy storage

elements, whose natural time constants Tor are the same or smaller

th n

the

switching period,hence

which apparently violates

the

linear ripple assumption. Therefore it appean that

state space averaging is not viable for these converters.

The steady state s olution for the QRS onveners [2-61 has been obtained

f,5 f, 5 )

Fi g . 1 . Three basic converler topologies: (a ) buck ( b) boost and ( c) buck-

boost .

476

CH2721-9/89/0000-0476 1.00'

989

IEEE

8/21/2019 00048525

2/8

by means of circuit averaging

[71.

The

use

of circuit averaging is justifiable

since the switch waveform is still applied to a filter network, whose average

output is equal to the average value of the switch waveform. Circuit averaging

of the resonant switch converters demonsaates that the average output voltage

=

KvT

M =

p

(buck) M= boost) M= (buck-boost)

v,

P F 7)

where p' s equal to 1-p. The conversion ratio for the full wave zero current

switch boost converter. for example , is

where F is the normalized switching frequency f a 0[8]. The conversion ratios

in

Eqs.

7)

are intriguingly similar to those of thePWM converters,

nd

lead one

to suspectthat the averaged state waveformsof t heresonant switch converter

can

be expressed in a form similarto that of

Eq.

(1):

M , 1

1

-F (8)

XF

= (CIA~+(~-IL)AZ)XFW I + ( ~ - I ~ & ) U

Y =

@cI+(1-p)c2)xF

+

O I E I + ( ~ - M W ~

9)

in which p de ds suictly

on

he variety of the switch

and

he matricesAi, A2.

B1. Bz,

Ci. K E i ,

and EZ

depend strictly on the topology of the PWM

parent converter. A result of the form of Eq. (9) is desirable

because

it is

systematic ndbecause it

unifies

the analysisof resonant switch mve rte rs with

the large body

of

existing knowledge concerning PWM converters.

Furthermore, the extensionof the small signal ac state space averaging result of

J?q.

(4)

permits the

use

of the canonical modeling concept in the small signal

dynamic analysis of resonant switch converters.

I nthis

paper

the

validity of extending

state

sp ceaveraging in the formof

Eq. (9) to a wider class of converters th n the PWM continuous conduction

mode class is demonstrated, and the results areapplied in particular

to

nonlinear

resonant switch converters. In Section

2

a more in depth discussion of

the

switch parameter

p

s given, along with a brief explanation and summary of

NRS

converter dc analysis. In Section

3 a

derivation is given in which state

spaceaveraging is extended by

linear

network

theory

from the domain of

PWM

converters

to

a much larger class of converters. A detailed example is given in

Section4

of

the state space averaged model of a nonliiear resonant switch boost

converter, for both full and half wave operation. The full wave converter is

found

to have only output voltage fe ed kk , but the

half

wave converter

has

both

filter capacitor voltage and filter inductorcurrent feedback. Thecanonicalmodel

concept is expanded

o

include linear

ad

nonlinear resonant switches n

Section

5 . It is shown

th t

the primary effect of resonant switching is to inaoduce a

feedback block of the filte r states and power inputs to the basic functions of the

PWM

canonical

model.

In

Section

6

the significance of the feedback loops

aroundthe

canonical

model is explored for the

resonant

switch boost converter,

a general table of W e r unctions

is

given for the buck, boost,and buck-boost

converters, and

the

switch average coefficient

)I

s

given together with the values

of the feedback gains for zero voltage switched converters and the current

programmed mode. Section7 contains a description of an experimental N R S

boost converter. The control to output transfer function of the prototype was

measured

ad

foundto

agree

well with that

predicted

by the state.

sp ce

veraged

model. A

summary

of the discussionis given inSection 8.

K r

-

Vout

Vout

L1

Fig. 2 .

Three diferent power switches

and

their input current waveforms: ( a)

pulse width modulated (PWM) b) linear zero current sw itch

(ZCS)

(e ) nonlinear resonant switch

(NRS).

r 1

r

I

L - - - J

Fig. 3.

A

general pulse width modulated two-port switch.

2 . The Switch Average Parameter p

A

two port power switch is shown in Fig.

3.  

As with any two port

network, it

has

two independent

inputs

aud twodependent outputs. In gene it

is possible fora converter to have an n-pon s witch. The independent quanuUes

aredemmimdby

the

converter

and

in

the

wo

pnt

caseare most often chosento

be the input voltage VT and the output current IT. The resultant dependent

quantities are therefore most often the nput current ih (t ) and output voltage

VmI(t) (the tilde denotes an

instantarmus

quantity,

as

n

[9]). The

variety of the

switch, e.g., ZCS. VS,

NRS,

etc., determines the exact value of the dependent

quantities n a given converter. i he dependent quantities i d t ) andVO&

always occur in conjunction with large filter elements in the converter. tt is

appropriate to consider the average values iin(t) and v,,t(t) and their

relationship to switch inputs IT

and

VT via the witch average parameter p. The

natureof this elationship s

critical

o the extension of state sp ceaveraging, and

isdiscussed n detail inSection 2.1. In Section22 he behaviorof the nonlinear

resonant switch is

examined

and he value of the switch average parameter for

N R S converters

is

reviewed.

2.1

The Averaged Waveforms of

a

Switch

The class of switches to which state space averaging is extended are

ideal, lossless switches whichcontain

no

energy storage elements. It

is

required

that the average values of the switch dependent waveformsbe related to the

instantaneous waveforms of the

PWM

parent converter in a particular form,

namely:

where p s the switch average parameter

and p'

is equal to 1-p. The vector xs

contains the averaged switch dependent quantities, and

the

vectors

Xsl

and XSZ

are

the

values of theswitch waveforms of the PWM converter during the first

and

seam3

switching nterval s respectively.

The PWM switch

is

the most obvious example of a switch meeting the

above criteria. The ideal

PWM

switch contains no lossy or energy storing

elements. The average value of the switch input current over one switching

cycle is given by:

xs = px't

+

p'X.2

(10)

(11)

which

can

be expressedas

(12)

p =DLt

+

DIS2

where D

is

equal to

p

and is

the

switch average parameter. Likewise

the

average

output voltagecan be expressedas

V DV,,

+

DV,,

(13)

The meaning of these expressions is made more clear by reference to Fig.

4,

which shows a buck converter and its dependent switch waveforms. For a buck

converter the switch independent quantities are IFIF, the inductor current, and

VFV,. the input voltage. During the f mt interval the switch is closed and

the

dependent switch waveforms i(t)in and J(t),,, are equal to IF and V

.

respectively. During the

second

interval the switch is

open

and both dependekt

waveformsare

?era.

Hence the average switch Waveformsare expressed

as

xg

= [

lm ] = D f ] + D [ :]

van (14)

- - - - - .

k)

(4

j;

r,

/ I

r,

s2

=

0 &2=0

The switching wa vefo rm of a PWM buck converter (a)Definition of

terminal quantities

b)

Switch output voltage waveform (c ) Switch

itput current waveform.

Vsl

=v*

41=IF

t

Fig.

4 .

477

8/21/2019 00048525

3/8

which is in the form of

Eq.

(10) with p=D. For many switches,XQ s zero and

the averaged switch waveforms

can

be expressed as

X s

=

)LxT (15)

where the vector

XT

is the vector of independent switch inputs (IT,VT).

Another example of a switch that meets the criteria for extended state

space averaging is the linear zero current resonant switch. Again the average

value of the input current waveform is obtained by averaging the input current

over one switching nterval and expressing it in the form of Eq. 10).

TS

i, =kL Tj,,(t)dt = pL1 + p'I,z

(16)

For the buck converter example, Isl and

I,z

are the same as they were in the

PWM case nEq. (14), but the switch average parameterp s given by

where F is the normalized switching frequency fs /fo and P is a function

dependent on the variety of switch [21.

(17)

=FP

P =1 1. x

sin-'(J~)

$ l + G ) )

2x 2

half wave

P = k+% sin-'(JT) 4 1 - G ) )

2x 2

JT full

wave (18)

The normalization conventions are:

vr

& (19)

JT=A

Ibse

in which L and C are the values of the resonant elements in the switch. The h f

wave value of P is highly dependenton the switch output current

JT,

but the full

wave value of P is almost independent of JT and is approximatelyequal to one

[Z].

Hence for the full wave switch

The parameter F plays the same role in analysis of full wave ZCS convertersas

the parameter D does in the analysis of PWM converters[8,9].

2 .2

Nonlinear Resonant Switch Average Parameters

A full wave nonlinear resonant switch boost converter is obtained by

substitution of the switch of Fig. 2c into the converter of Fig. lb.

The

normalized switch input current waveform is shown inFig. 5 [6]. Note that the

nonlinear inductor has both a bias winding and a winding in series with the

output of the switch. The turns ratio of the nonlinear inductor is I:NB:NT where

NB s the the

turns

ratio of the bias current winding and NT is the turns ratio of

the output current winding. A common choice for NT

in

the full wave converter

is N y l . When the nansistor is first turned on the nonlinear inductor is biased

into saturation by the current in the bias winding ibias and the load current iT.

Hence the inductance present at the primary winding in series with the aansistor

is L1, a small value, and the resonant currentrings at a high frequency. At some

transistor current given approximately (because of the shape of the saturation

(20)

= F P = F

curve) by

lj,,=iT+&[ =Nbibi.r+NTiT (21)

the amp-tums of the primary winding equals that of the two secondary windings

and the inductor unsaturates. Consequently a large inductance Lz is present at

the primary winding, and the switch current is limited. After some time the

current rings down to the critical value again, the inductor saturates, and the

current

ings

at the high resonant frequencyuntil the transistor is turned off. The

ratio of saturated inductance

to

unsaturated inductance is

Mined

as

the parameter

k

122)

-~,

which is very small, k

8/21/2019 00048525

4/8

Linear, Time-invariant

Network

Inputs outputs

U

zcs

k

4

w

Fig.

7. The

instan aneous switch waveforms can be replaced by equivalent

averaged voltage and urrent sources.

and the general

state

space veraging result

is

obtained. In the last section the

small signal form of the generalized state space result

is

developed,

and

the

diffexences

between he g e n d model and hePWMormare discussed

3.1 State Space Representation of the Switching Waveforms

Figure 6 shows a mv er te r in which the nonlinear element,

the

switch, is

separated from the linear time nvariant convener network. Consequently, by

superposition. the state equations of the converter

can

be written as a linear

combination of the ilter

states

XF he power inputsU,

and

he switch dependent

waveforms 4.

F

= AF& BFG A

7

CFCF EFL

+

C.Z.

(31)

where the tilde demotes the instantaneous values. The matrices AF and CF

describe the connection of the filter states, and B F and E F describe the

connection of the power inputs. The matricesA, and Cs indicate he connection

of the switch

sources

to theconverter, and the vector x ,

contains

he the switch

waveforms (here chosen to be i b and vat for a PWM ZCS, orZVS wo port

switch). Note that these matrices AF B F. CF EF, snd Cs are now time

invariantand independent of the

s w i t c k

actio;. N

&

nformationabout the

switching is contained in thevectorf,.

With reference to Figs. 6 and 7. it

can

be

seen

that an equivalent

representationof the convener can be obtained by averaging of all convener and

switch waveforms. The averagingstep

is

valid as

ong as

the averaging is done

over an inte nd

shon

with respect

to

the

m u d

i meconstants

of

the ilter srates.

The result is:

XF =

AFXF

+

BFU Aaxg

y

= CFXF

+

EFU

C.X.

(32)

The filter state vector XF. the power input vector

U.

and the switch vector xs

now contain information only about the slowly varying components of the

converter waveforms.

Equations (31)

and

(32) are significant because they are completely

generalrepresemations of

the.

converter

state

equations. These equations indicate

that the

switching

waveform s a function of the states

and

inputs of thelinear

time invariant systa n in

addition

o the control inputs.

3.2 Relationship of the Switching Vector to the PWM Case:

The General State Space Averaging Result

The next step in extension of sta te

space

veraging is to relate

the

general

averaged

Eq.(32)

to the matrices of the

FWM

conveners of Eq.

(30).

Therefore

it is necessary to express the averaged waveforms x, in term of the PWM

switching inte rval waveformsX.1 and X d

and

he switch average parameterF.

The vector X.1 contains the value of the PWM switch waveforms when the

switch is in position 1. and X.2 contains the values of the switch waveforms

when the switch is in position

2

(Fig.

4).

Equation

(32)

is the essential

requirement for application of state space averaging to a given converter and

switch.

As

discussed in Section 2.1, this expression is valid for a wide variety

of lossless switches which have zero energy storage, including PWM, ZCS,

ZVS,

N R S

nd the PWM current programmed mode.

(33)

.

=

l X . 1

+

P'X.2

Equation

(33)

can

be

used

to e l i from Eq.

(32).

The result is:

XF =

AFXF

+BFU

+ pA.X.1 + p'A.X.2

y = CFXF+EFU+ pCgX.i+ F'C.X.2

(34)

The next step is to relate the terms containing X s l and Xs2 to the PWM

matrices. Consider the inear networlc obtained wiih thePWMswitch in position

1. The stateequationsof thisnetwork are:

X F = A ~ X F + B ~ U

y

=

C I X F + E I U

(35)

The

lmear

filter network of

Fig. 7

describes the same system, provided that the

switch signal xs is chosen to be

X,1.

Consequently the tworepresentations anbe equated.

Similar quatiom

canbe

written for the

PWM

converter when the switch is in

position 2:

XF = AFXF+ BFU AaX.l

y

= CFXF+

EFU

+ C.X.1

XF'ALXF +B iu

=

AFXF

+

B F U+ A.X.1

Y

= C iX F + E i u = CFXF

EFU

+ C.X.1

(36)

(37)

XF

= AZXF B ~ u AFXF

+

BFU 4 X . z

Y = CZXF

+

E Z U= CFXF+ E F U

+

C.X.2

Equations (35)-(37)

can e

solved for the switching

tam:

A&i

=

( A I - A F ~ 0 3 1 - B ~ ) ~

CaX.1

=

( c 1 - c ~ ) ~ ~( E ~ - E F ) u

A&z

=

(Az-AFIxF

+ f 3 2 - B ~ ) ~

c ~ x a z ( CZ -C F) XF+ ( E ~ - E F ) u

(38)

(39)

The switching terms can now be e l i t e d rom Eqs. (34) by substitution of

Eqs. (39).

The

defmition

w+p'=1

is employed

and

the generalized state space

result is

obtained:

XF =

( P A I + ~ ' A Z ) X F(pBi+P'Bz)u

Y = @cl+P'CZ)xF + (PEi+p'Ez)u

(40)

Equation

(40)

is

the

desired form of the averaged model and demonsuates that

the state

space

averaging method isvalid not only for PWM converters but also

for any converter whose averaged switch waveforms can be expressed in the

3.3

The Small Signal State Space Averaged Model

The extended state spaceaveraged model can be applied to small signal

variations

via

the usual method of perturbation and linearization, wherein each

quantity in Eq. (40) is replaced by a steady statepart plus a time varying small

signal

part:

The re$ is the small signal

state space

averaged model:

+

[ ( A l - AdXm

+

f 3 i - B z N J o l ~

form of Eq. (33).

X F = X F O + % F U = u O + G

P = b o + c

(41)

xF = ( ~ A ~ + ) I O ' A Z ) G FWOBI%LBZ)G

? =

( b c l + b ' c Z ) % F

+

(@l+b'EZ)ii

+ [(C i - W X m + (El -&)Uol%

42)

where w designates the steady state value of the switch average parameter.

Note th t there is an important difference between the PWM result and the

generalized state

space

result. In the PWM model the control parameter

a

is an

independent parameter, but in the generalized result the control input is the

switch avera$e parametera given by

where the vector derivatives are evaluated at the steady state

Operating

pomt, and

i,

s the vector of variation in

the

independent control inputs. Thus for a given

topology the generalized state space model is the sameas the PWM model, but

contains eedback of the filter state variations . and feedforward of the power

input variations6 in addition to the switch control variablesis This result is

found to be the principal effect of resonant switching and current programming

mode

[IO]

on switching conveners. and is indicated schematically n

Fig.8,

in

which the generalized result is shown to be a feedback loop around the

PWM

canonical model.

1 z F

%

+k .

aXF

au

aU.

(43)

I i

~ fa, Co;ye;;riables,

Fig. 8 .

The effect

of

resonant switching b to introduce a eedback around the

original PWM canonical model.

479

8/21/2019 00048525

5/8

In summary,a new method of writing the state equations of a converter

in a manner that separates the fixed topological elements and the switching

waveforms of the converter has been presented.This representation is used to

demonstrate that the generalized state space averaging result ofa s .

(40)

nd

(42)

s valid for a given converter and switch

if

the switching period of the

converter is small in comparison with the natural period of the the filter states,

and if the average switch waveforms can

be

written as functions of the PWM

waveforms as expressed in Eq.

(33).

Furthermore, he form of the small signal

model is the same as that of the PWM converter, but the effect of resonant

switching is to introduce feedback of the filter states and feedforward of the

power inputs.

4 .

An Example

of

Small Signal Modeling of an NRS Boost

Converter

A nonlinear resonant switch boost converter is obtained by substitu tion

of

the nonlinear resonant switch of Fig. 2c into the boost converterof Fig. Ib.

It is possible to obtain either a h f wave switch by insertion of a diode in series

with the transistor,

or

a full wave switch by means of a diode in parallel with the

transistor. As mentioned in Section 2.2, it is common for the half wave

nonlinear inductor bias current to be derived entirely from the filter inductor

current, with NT > ~ ,nd for the full wave bias current to be derived both from

the inductor current, N F ~ , and from an independent bias current ibiaswith turns

ratio 1:Ng. The small signal model of the NRS half wave

boost

converter is

discussed in Section

4.1,

nd the small signal model of the full wave

M1s

boost

converter in Section 4.2.

4 . 1

The Half Wave NRS Boost Converter

The first t sk in analysis of the nonlinear resonant switch converter is

to

find the matrices for the PWM

boost

converter during each of its two switching

intervals. Substitution of these

ma ices into

thegeneralized state space averaged

small

signal ;esut of Eq. (422 yields:

, ,

.

Small signal circuit eq&ons in the Laplace dom&can-be written from>& state

space description of Eq. (44):

SL& =

-&?

+ vo;

A

(45)

SCFG = I ~ K

R

An equivalent circuit model

can

be conshucted from

Eqs.

(45)

nd is shown in

Fig. 

9.

The form of the model is

seen to be

the same as hatobtained for a PWM

converter. As with PWM converters, the relation for a transformer with turns

ratio

I

is obtained

[I].

The remaining task in the modeling of the NRS converter is the

evaluation of 5 In the equation for ).t of a half wave NRSswitch, p s equal to

FP, where P is given by

Eqs.

(24-26). and the independent variables are fa, iT,

and VT. Substitutionof the half wave expression for

p

into

Eq.

(43)

ields:

~ J TIT ~ J T

V T af

(46)

-

~ J T : all ~ J T all*

=

-?T - s

The uartial derivative of

U

with resuect

to

JT

s:

where Jcrit is here defined as Jcrit

=

(NF-I)JT.

In

the special case when

k

8/21/2019 00048525

6/8

would

be

present. across the filter capacitor, but two independent voltage and

two independent current sources would be present to represent the effect of

2,

and ?bias. sin ce it is common to control the converter by switching frequency

alone, in many applicatiOns ?bias iS q u i d to zero, and a fairly simple model

results for the full wave converter.

5 .

A

Generalized Canonical Model

Since the form of the state spaceaveraged model is the same regardless

of the switch employed, the canonical model concept derived for PWM

converters

can be

generalized

to

include

other

converters such

as

QRS

conveners

or the current programmed mode. The only difference in the derivation of the

generalized model is that the average parameter0 is no longer an independent

quantity, but instead depends on the control signals, the filter states, and the

power inputs. Therefore the variations in the switch average parameter

fi are

WTitIen?

~

The control signals

are

assigned

tob,

nd the filter states and power inputs are

assignet0

b,

F = k + P €

57)

14

=

K1G1 +K2G2 +...+

K.6.

= -K~KK"GT (58)

which allows for switching frequency, bias current, duty cycle control,

transistor current control, and other possible control schemes, and for filter

capacitor voltage, input voltage, and filter inductor feedback.

Consider

the

deriva tion of the canonical model via

the

boost converter

as

an example. The statespaceaveraged small signal model of the boost converter

is shown in Fig. 9a. This circuit model is the same regardless of which type of

switch is employed in the boost converter. When the switch variationsfi consist

of only one source,

as

in the PWM case, the canonical model is derived by

reflecting the current source representing the switch to the converter input, and

reflecting the filter inductor

to

the

output

[

11.

In

the process a new equivalent

voltage source

is

created, whose value is:

e s)Z= VO 1

-

* i

59)

When the canonical model is derived in the general

case

the voltage and current

sources arising from the switching action consist of both independent control

source variations and dependent feedback sources, as shown in Fig. 12a.

However, since the system is linear. the same circuit manipulations can be

performed by superposition to move the s econd currentsource

to

the converter

input. The coefficients of the canonical model obtained in the generalized case

are the same

as hose

of thePWM canmicalmodel, except that is replaced by

LO.

The canonical model for the generalized state space averaging result is

shown in Fig. 12b. The coefficients of the equivalent circuit components are

given in Table 1.  Figure 12b demonstrates that in general resonant switchor

current programmed converters contain the sa me functional blocks as PWM

converters, but in addition contain elements of feedback arising from the

resonant switching action. 'he basic elements of the generalized canonic al

model are as follows: First, the generalized converter model

possesses

independent voltage and current control sources, which may be switching

frequency, bias current, duty cycle, or transistor current control. Second. the

general converter model possesses either filter inductor current

or

capacitor

voltage feedback, or both. In some converters there is a feedforward term as

well that changes the line to output transfer function. Third, the general

converter model has an equivalent low

pass

filtering network

that

models the

attenuation of signals by the converter filter states. The lone exception to

this

model is the

h f

wave buck-boost, which is similar

to

the canonical model for

current programmed conuol in that to retain the physical interpretation of the

branch containing current

pot^

the two fi current sources in the state space

averaged model cannot be completely combined [lo]. Altogether, the basic

p o :

1

b )

r

-

ir

- -

ir - ir

-

- -lr -

-

1

I ICtrl IIFeedhacklIVCtrllI

Ddtbk

11

L -

J L

-

1 -

- -

- 11- -

- 1

Fig . 12 . The derivation of

the

generalized canonical model via the boost

conver ter . (a)Separation of

the

switch voltage and current sources

info confrol and feedback

quanti i ies. b)The

generalized canonical

model.

ILConversion Filtering

elements of the canonical circuit model compose a general circuit model that

utilizes the existing body of knowledge concerning PWM converters, but also

highlights he effect of alternativeswitching schemes.

Table 1. Coefficients

of

the generalized canonical N circuit model (a fter [ l ] .

Note that eB)= EfI( s) nd s ) =Jfz(s).

E

6 . Discussion

of

the Generalized Canonical Model

An alternative epresentationof the generalized canonical model is sh o w

in Fig. 13.   The feedback loops are shown external

to

the basic state space

averaged model in a configuration specific

to

boost resonant switch converters.

Both inductor current and capacitor voltage feedback loops are present whenever

half wave resonant switches are used.

In

linear full wave ZCS and ZVS

configurations neither current or voltage feedback is present and the model

reduces

to

the PWM canonical form. The

NRS

full wave converter is

unique

in

th t it has voltage feedback but no current feedback loop.

with respect to its normalized

currents, and the feedback gains of Fig.

13 

and Eqs. (57) and (58) are given in

Table 

2.

The gains or the linear half wave ZCS converters are very similar to

those of the nonlinear resonant switch differing only in the value of p and its

derivative. It

has

been shown that the switch average parameter of the zero

voltage switch

b,

is related to that of the

zero

m t

witch pi:

which is true for both half and full wave zero voltage switch converters.

Therefore the parameters of the ZVS can

be

ound from those of the ZCS, and

they share the same general dynamic characteristics. Hence, given the resonant

switch canonical model of Fig. 13, the equivalent circuit model of any of the

resonant switch convemrs in Table 2 can

be

found by evaluating

he

coefficients

of the model with the correct p and feedback gains for the desired converter.

The line to

output

and control to output ransfer functions can

be

olved

in general for the resonant switch buck, boost and buck-boost converters either

by direct solution of the circuit equations or by solution of the feedback loops

around the basic canonical model. These transfer functions are tabulated in

standard form in Table

3 

for thecase when both filter capacitor voltage and filter

inductor current feedback are present, i.e.. the half wave case. Note that a zero

is obtained for the line

transfer

function in the buck-boost converter which is not

present in the PWM converter, introduced by

the

fd on va rd of the resonant

switch. When capacitor voltage feedback but no inductor current feedback is

present, as in the NRS full wave converter, the coefficients of the transfer

functions may

be

obtained from Table 3 by setting Ki equal to zero. When

neither voltage or current feedback is present. as in the PWM

or

full wave ZCS

and ZVS converters, the transfer function coefficients are obtained from Table3 

by setting to

zeroboth

Ki and K,.

The switch averageparameter of the current program mode is found [101

from the

equation:

FTS

iT =i-m+Ts -ml T

where

L

s the control current,m,

is

the slope of the anif cial ramp, and ml is the

slope of inductor current in the

f is t

switching period. Consequently he switch

averagepram-p

is

The values of

p,,

the derivatives of

=l- )

JT

60)

(61)

=

k- iT

(62)

m,

+Y ) T s

Application of Eq.

(43)

yields the following express ion of the feedback equation:

e

-Im a

IC0

iT+ -~Im-Im) Gl

& e ) T a '+&+)Ta 2(&*)2Ta

(63)

which is of the form of Eqs. (57)and (58). but the feedback voltage is different

from that of the resonant switch conveners. The feedback voltage is

the

inductor

voltage during the fmt switching interval, which is v v for the buck converter

and v for both the boost and buck-boost converters?Clearly the generalized

canon cal model provides a unified interpretation of the resonant switch and

current mode equivalent circuit models thatwas not previously possible to attain.

7 . Experimental Measurement of an

NRS

Boost Converter

A nonlinear resonant switch half wave boost converter has been

constructed for experimental verification of the state space averaging analysis.

The nonlinear inductorconsisted of a femte

3C8

1408

pot

core with 12

turns

on

the primary and 15

turn

on the secondary, for a tum ratio of

NT

1.25. There

was no independent bias winding. The switch controller was an

SG3523

pulse

width modulator modified for variable frequency, variable duty cycle operation.

The

other ca n p e n t values were:

L ~ z 6 8 3 t 1 - H cF=11 . 1@ QI=lRF640 R z 4 3 . 6 R

481

8/21/2019 00048525

7/8

Table

2 .

The switch average parameter

p ,

its derivative, and the feedback and conlrol gains for zero

current, zero voltone,

nd

nonlinear resonan1 switch converlers

Switch

PWM

LZCS

m

LZCS Fw

LZVS HU

LZVS Fw

NRS

HW

NT>

1

NB=O

NRS FW

N T = ~

NB O

PO

T)

Table 3 . Transferfunclions

for

halfw ave resonant switch buck, boost, and b

Ki=U and K v d , he

fu ll

wave NR S transfer functions are obtainel

full wave linear zero current

and

ero voltage and PWM framfer

wu

Ki

0

0

0

0

0

-boost converters. When

vhen K i d and Kv=U fhe

n are obtained.

Kc

& = l

Kf =

,n

I

Buck Boost

I

Buck-BWst

482

8/21/2019 00048525

8/8

f.

Fig. 13 . Ap plication of the generalized canonical model to the boost conve rter

resonant switching co&yation.

38 115

SB U

B I

U U

-

?

m -1s

-.5

2

-

-

U -zm -m

Y

s

-

c -38 - 135 f

E -.n - , I n

- se - 2 2 5

-60

->,a

-3 , s

10 I ea I K I. I ,m

H

-,a

Frequency (Ha)

Fig.

14.

Theoretical

(T)and

experimental

(E)plot

qf he magnitude

solid)

and

phase (dashe d) of the NRS open loop control to output tranrfer

funclion.

Ll

= 17.1p.H C-= 49 nF

'

18.7 R Di,D2=MUR810

fs

=

37.2 kHz Vgo

=

29 V V,

=

34.1 V k = 0.075

The experimentally

meawed

operating point was:

The theoretical operamg point was obtained by choosing Vgo. fso, fo,

and Ro, hen numerically solving the steady state expressions for the

h f wave converter:

M , v O = 1

I m = + h

V@ LbR 64)

to obtain Im and Vo.

The

resultant theoretical operating point was:

which yielded a theoretical output voltage of 36.6 V. The gains in the

feedback exuression of Ea. 50) were therefore:

= 0.45 10 0.55 M=1.83

F

= 0.214

J T =

JF =0.783

P

=

2.12

Kv=

oodo

V-1 Ki= 022 2 A-1 Kf

=

12.2 p e c

The control to output uansfer function

can

be found from Table

3 

and

evaluated:

1-

G(s)

=

6.17

*)

+(* (65)

where the gain of the modulator, 12.4 m o l t , has been included in

the

dc

gain. A plot of the theoretical and experimental ransfer functions is shown in

Fig. 14 to half the switching frequency. Although some deviation (?ccurs at

higher frequencies because sampling is neglected, the agreement between

experiment and theory at low perturbation frequencies s quite good.

8 . Conclusion

State space averaging is shown in this paper to extend beyond the

domain of pulse width modulated convertersto the domain of converters whose

switches are lossless, have no energy storage,

and

whose average dependent

switch waveforms

x

can be expressedas

where X,1 and

Xs2

are the vectors of the switch waveforms during the switching

intervals of the PWM parent converter. The quantity1 s the switch average

parameter, and characterizes

the

average values of the switch waveforms over

one switching

period.

A large n u m k of switching schemes fall into this class,

including linear and nonlinear resonant switches and the current programmed

mode. The steady state values of

p

for linear zero current switches and

nonlinear resonant switchesare reviewed in Section 2.

A derivation by linear network theory is given in Section 3 which

demonstrates that conveners whose switching elements satisfy

Eq. 64)

may be

expressed n generalized state

space

averaged form:

XF = (PA~+(~-IL)Az)XF(@31+(1-1U32)~

Y

= (1c1+(1-1)cZ)xF

+

W1+(1-II)Ez)U

x.

=

1X.I + 1'X.Z

66)

(67)

where XF contains the filter states of the converter and AI, Ai, B1, Bz, C1,

C7. El. and E?.

are

the matrices correspondinn to the first and second

switching intervals of the parent PWM concerter. Furthermore, the state space

averaged form of the snal l signal model,

CF = (POAIYIO'AZ)~F (p oB i+ k~ B2 )~

+ [(Ai -Az)Xm + (Bi -Bz)Uolp

+ [(Ci-Cz)Xm + (Ei-fi)Uol~~ (68)

?=

bCl+bJ'cZ)%F

+ POEl+b~EZ)ii

is also valid for the extended domain of converters. The perturbation of the

switch a v m e uarameter.

is found to co & u t e a feedback loop around the state space averag& PWM

equivalent circuit model of a given circuit model.

These results are illustrated by application to both

half

and full wave

nonlinear resonant switch boost converters in Section 4. The

h f

wave NRS

converter s found to have both filter inductor current and filter capacitor voltage

feedback, while the full wave NRS converter has only capacitor voltage

feedback. The control

to

output uansfer functions of the full wave

NRS

converters

are.

the same for both switching frequency and bias current control,

differingonly in their dcgain.

The extended state space averaging result is used to generalize the

canonical model concept [11 in Section 5. The variations in the switch average

parameter are found in general to be a l i ombimtion of feedback of

the.

ilter

states and power inputs and the ndependent control inputsb. Consequently

the generalized canonical model

has

the same functional blocks as the PWM

canonical model. namely independent current and voltage control sources, dc to

dc power conversion. and low pass filter elements, but also has a block that

models the feedback &introduced by alternative switching schemes. The

values of the circuit elements in the generalized canonical model are given in

Table 1.

In Section 6 the generalized canonical model is shown for a half wave

resonant switch converter with the feedback loops external to the model

aw rd ii g to the equation

p =

-Ki& K v ? ~ KIC1 +K2Cz ...+KnCn

where

?T

and C-r are the feedback quantities and &i is a control input. The

switch average

parameter

p, the derivative of p, and the feedback and control

gains of a number of resonant switching schemes, including zero current, zero

voltage, and nonlinear resonant switches are given in Table 2.  Line to output

and control to output transfer functions for the buck, boost and buck-boost

transfer functions are given in standard form in Table 3.  The switch average

parameter of the PWM current

prognUnming

mode is also found. Current mode

control differs from resonant switch converters in that it is the inductor voltage

during the fm t switching interval, not the switch input voltage, that constitutes

the inner voltage feedback loop.

This

conclusion agrees with previous current

program mode models [lo].

Experimental verifcation of the

state

space averaged model is presented

in Section 7. The control to output transfer function of a h f wave nonlinear

resonant switch boost converter is measured and compared to the transfer

function predicted by the theoretical model. Good agreement between

experiment and theory is obtained.

(70)

References

R.D.Middlebrook and

S.

Cuk, "A General Unified Approach to

Modelling SwitchingConvener Power Stages." IEEEPower Electronics

specialists Conference, 1976 Record, pp. 18-3 1.

K. Liu, R. Oruganti, and F. C. Lee, "Resonant Switches

-

Topologies

and Characteristics,"

IEEE

Power Electronics Specialists Conference,

1985 Record, pp. 106-1 16

(IEEE

Publication 85-1 17-0).

K. H. Liu and F.C.

Lee,

ZemVoltage Switching Technique n DCDC

Converters,'' IEEE Power Electronics Specialists Conference, 1986

Record, pp.58-70

IEEE

Publication 86CH2310-1).

T.

Zeng,

D. Y,Chm, and F. C. Lee, "Variations of Quasi-Resonant

DC-DC Converter Topologies," IEEEPower Electronics Specialists

Conference, 1986 Record, pp.381-392 (IEEE ublication 86CH23 10-

1).

K.D.T.

Ngo

"Generalization of Resonant Switches and Quasi-Resonant

DC-DC onverters," IEEEPower Electronics Specialists Conference,

1987 Record, pp.395403

(IEEE

Publication 87-59-6).

R. W. rickson,A. F. Hernandez, A. F. Witulski, R.

Xu

"A Nonlinear

Resonant Switch, JEEE Power Electronics Specialists Conference,

1989 Record, also acceuted for the IEEE Transactions in Power

Electronics.

G. W. Wester and R.D.Middlebrook, "Low Frequency Characterization

of Switched DC-DC Converters," Third IEEE Power Processing and

Elecuonics SpecialistsConferen=, Atlantic City, May 1972.

S. Freeland and R.D. Middlebrook,

A

Unified Analysis of Converters

with Resonant Switches," IEEE Power Electronics Specialists

Conference, 1987 Record,pp. 20-30

(IEEE

Publication 87CHZ459-6).

V. Vorperian, "Equivalent Circuit Models for Resonant and PWM

Switches," IEEE International Symposium on Circuits and Systems,

1987 Record,Vol. 3, pp.1080-1087.

R. D. Middlebrook, "Topics in Multiple Loop Regulators and Current

Mode Programming,'' IEEE Power Electronics Specialists Conference,

1985 Record, pp. 716-732.

483