N

84 - 19 7 07

SCHOOL OF ENGINEERING

[UKE UNIVERSITY

DURHAM, NO 27706

CENTER FOR SOLID-STATE

POWER CONDITIONING

AND CONTROL

a a D

a a a a a

cnanaa a 0 o

e a

DO 1 0

Q DGS3Q E3 Q 1 1 I

U D O

i ODD

i D D D

i Q D E3 D

i D a OQQE3E3

i E3 a a ac3a

o Q a a C3 en

DERIVATION OF LINEARIZED TRANSFER FUNCTIONS

FOR SWITCHING-MODE REGULATORS

PHASE B — SINGLE-WINDING CURRENT-OR-VOLTAGE STEP-UP

AND TWO-WINDING CURRENT-OR-VOLTAGE STEP-UP CONVERTERS

Prepared by

Ronald C. Wong, Harry A. Owen, Jr., and Thomas G. Wilson

December 15, 1981

for

National Aeronautics and Space Administration

Goddard Space Flight Center

Order No. S-71440B
DERIVATION OF LINEARIZED TRANSFER FUNCTIONS

FOR SWITCHING-MODE REGULATORS

PHASE 6 — SINGLE-WINDING CURRENT-OR-VOLTAGE STEP-UP

AND TWO-WINDING CURRENT-OR-VOLTAGE STEP-UP CONVERTERS

Prepared for

National Aeronautics and Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

Order No. S-7H40B

Center for Solid-State Power Conditioning and Control

School of Engineering

Duke University

Durham, North Carolina 27706

Ronald C, Wong, Harry A. Owen, Jr., and Thomas G. Wilson

December 15, 1981
I. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

^ Title and Subtitle oerl vation of Linean zed Transfer

Functions for Switching-Mode Regulators, Phase

B — Single-Winding Current-or-Voltage Step-Up

and Two-Winding Current-or-Voltage Step-Up Converters

5. Report Dote

Dec 15, 1981

6. Performing Organization Cod*

7. Authors)

R.C. Worm. H.A. Owen. Jr.. and T.fi. Mil son

8. Performing Organisation Report No.

9. Performing Organization Name and Address

Department of Electrical Engineering

Duke University

Durham, N.C. 27706

10. Work Unit No.

11. Contract or Grant No.

Order No. S-71440B

13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address

G.E. Rodriguez, Code 711

Goddard Space Flight Center

Greenbelt, Maryland 20771

14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

Small-signal models are derived for the power stage of the

single-winding and two-winding current-or-voltage step-up (buck-boost)

converters. The modeling covers operation in both the continuous-mmf

mode and the discontinuous-mmf mode.

17. Key Words (Selected by Authors))

Small-Signal Models

DC-to-DC Converters

Continuous-Mmf Mode

Discontinuous-Mmf Mode

18. Distribution Statement

19. Security dossil, (of this report)

Unclassified

20. Security Clossif. (of this page)

Unclassified

21. No. of Pages

91

22. Price

•For sate by the National Technical Information Service, Springfield, Virginia 22151.
m

PREFACE

This Phase B Report documents thoroughly the derivation of a set of

small-signal models for the power stages in the single-winding and the twowinding current-or-vol tage step-up (buck-boost) energy-storage dc-to-dc

converters for operation in both the continuous-mmf mode and the

discontinuous-mmf mode. The derivation of the transfer functions describing

these power stages is shown in detail and the expressions for the coefficients

of these transfer functions are tabulated for ease of application.
IV

TABLE OF CONTENTS

PAGE

1. INTRODUCTION 1

2. DEVICE MODELS 7

2.1 Transistor Model 7

2.2 Diode Model 10

2.3 Energy-Storage Reactor Model 12

2.3.1 Energy-Storage Reactor Model for the Single-Winding

Current-or-Voltage Step-Up (SCVU) Converter 12

2.3.2 Energy-Storage Reactor Model for the Two-Winding

Current-or-Voltage Step-Up (TCVU) Converter 14

2.4 Capacitor Model 14

2.5 Load Model 16

3. SOLUTIONS OF STATE VARIABLES IN ONE SWITCHING CYCLE 17

3.1 Description of State Variables in General 19

3.2 Description of State Variables for both SCVU and TCVU 27

4. EQUILIBRIUM OPERATING POINT 32

4.1 Equilibrium Operating Point in the Continuous-Mmf Mode 32

4.2 Equilibrium Operating Point in the Discontinuous-Mmf Mode 38
PAGE

5. SMALL-SIGNAL TRANSFER FUNCTIONS CHARACTERIZING THE POWER STAGE 45

5.1 Small-Signal Transfer Functions in the Continuous-Mmf Mode 45

5.2 Small-Signal Transfer Functions in the Discontinuous-Mmf Mode .. 65

6. CLOSED-LOOP FUNCTIONS OF A REGULATED DC-TO-DC CONVERTER 78

6.1 Loop Gain of a Regulated DC-to-DC Converter 78

6.2 Input Impedance of a Regulated DC-to-DC Converter 80

6.3 Audio Susceptibility of a Regulated DC-to-DC Converter 81

6.4 Output Impedance of a Regulated DC-to-DC Converter 81

7 . CONCLUSIONS 82

8. REFERENCES 83
vi

APPENDICES

PAGE

A. DEFINITION OF SYMBOLS AND ABBREVIATIONS 84

B. EQUATIONS DEFINING THE CONSTANTS AND THE COEFFICIENTS

IN THE TRANSFER FUNCTIONS 86
vii

LIST OF FIGURES

\

FIGURES PAGE

1 Block diagram representing the

signal flow in a converter power stage 3

2 Steps in arriving at the small-signal

closed-loop functions of a regulated dc-to-dc converter 6

3 Piecewise-linear equivalent model of a BJT or FET switch 8

4 Piecewise-linear equivalent model of the diode switch 11

5(a) Model of the energy-storage reactor for the

single-winding current-or-voltage step-up converter 13

5(b) Model of the energy-storage reactor for the two-winding

current-or-voltage step-up converter with magnetizing

inductance L$ referred to the secondary circuit 13

6 Model of the output-filter capacitor 15

7 Model of the power-stage load 15

8(a) Circuit diagram of the power stage of the

single-winding current-or-voltage step-up converter 20

8(b) Equivalent model of the power stage shown in8(a) 20

8(c) Equivalent model of the power stage of the single-winding

current-or-voltage step-up converter as obtained from the

equivalent circuit shown in Fig. 8(b) 21

9(a) Circuit diagram of the power stage of the

two-winding current-or-voltage step-up converter 22

9(b) Equivalent model of the power stage shown in9(a) 22

9(c) Equivalent model of the power stage of the two-winding

current-or-voltage step-up converter as obtained from the

equivalent circuit shown in Fig. 9(b) 23

9(d) Equivalent model of the power stage of the two-winding

cur rent-or-voltage step-up converter after the primary circuit

has been reflected to the secondary circuit 23
vm

FIGURES PAGE

10 Equivalent circuit of the two-winding current- or- voltage stepup power stage for the three subintervals tQN, toFFl » and

 ^OFFZ ••• • 29

11 Interconnection of functional blocks to characterize

a converter power stage in the continuous- mmf mode ................. 46

12 Interconnection of functional blocks to characterize

a converter power stage in the discontinuous- mmf mode .............. 67

13 Block diagram of a regulated dc-to-dc

converter operated from a stiff voltage source ..................... 79
IX

LIST OF TABLES

TABLES PAGE

1 Six Small-Signal Transfer Functions for the Two State

Variables Under Continuous-Mmf-Mode Operation 51

2 Six Transfer Functions Describing the Load Voltage

and the Supply Current for Continuous-Mmf-Mode Operation 62

3 Three Transfer Functions for the Ideal Capacitor

Voltage Under Discontinuous-Mmf-Mode Operation 69

4 Five Transfer Functions Describing the Load Voltage and

the Supply Current for Discontinuous-Mmf-Mode Operation 73
1. INTRODUCTION

A brief survey of existing approaches to the modeling of switching-mode

power systems was presented in Section 1 of the Phase A Report [1] prepared as

part of this contract under NASA Order No. S-71440B. In that report, a set of

small-signal functions is derived to characterize the voltage step-up (boost)

and the current step-up (buck) converters operating under the constantfrequency control law. The basic modeling approach employed in that report

starts with the approximation of state transition matrices by second-order

Taylor series. From this approximation of the state transition matrices,

difference equations for the state variables are developed. Finally, transfer

functions are derived to describe the output variables.

Following the same modeling approach employed in the Phase A Report, this

Phase B Report documents thoroughly the derivation of a set of small-signal

functions for the single-winding and the two-winding current-or-voltage stepup (buck-boost) converters. The equation numbers in this report are assigned

the same sequence to match as closely as possible those in the Phase A Report

and thereby help the readers to follow the parallel treatment of material

common to the two reports. Equations in these two reports can be divided into

two groups, those associated with the general modeling approach and those

relating to particular converters. Equations relating to the general modeling

approach appear identically in these two reports because the same modeling

approach is shared between the two. An equation of this type is labeled as

(n) where n is the equation number. Equations relating to particular converters, however, differ from converter to converter, depending on the particular converter topology. As a result, an equation relating to a particular

converter is labeled as (n,#) where n is the equation number and # is an

acronym to identify the particular converter. For example, VU, CU, SCVU, and
TCVU are the acronyms used for the voltage step-up, current step-up, singlewinding current-or-voltage step-up, and the two-winding current-or-voltage

step-up converters, respectively. Also, there are equations which are common

to both the single-winding and the two-winding current-or-voltage step-up converters. Equations of this type are labeled as (n,CVU) where n is the equation number. The tabulation below illustrates the use of the acronyms and the

numbering of the equations :

VU voltage step-up converter

CU current step-up converter

SCVU single-winding current-or-voltage step-up converter

TCVU two-winding current-or-vol tage step-up converter

(14) equation 14, applies to all four converters

(30,VU) equation 30, applies to the voltage step-up converter

(30,CU) equation 30, applies to the current step-up converter

(30,CVU) equation 30, applies to both the single-winding and the

two-winding cur rent-or-vol tage step-up converters

The symbols used in this report are listed in Appendix A. As far as

signal or information flow is concerned, a regulated converter can be represented by the block diagram as shown in Fig. 1. It consists of a dc-to-dc

converter power stage with its load, a feedback network, an error detector,

and a pulse-width modulator. The power stage is symbolized as having three

inputs — the supply voltage vj, the load-disturbance current iy, and the duty

ratio ao — and two outputs — the load voltage VQ and the supply current 1j.

The feedback network is characterized as a network with the load voltage VQ as

its input and feedback voltage VF as its output. The error detector compares

the feedback voltage VF to the reference voltage VREF and provides an output
VREF PULSE-WIDTH

MODULATOR

FEEDBACK

NETWORK

VI • V i

POWER STAGE

AND LOAD V

0

Fig. 1. Block diagram representing the signal flow in a converter power stage,
V£

 3 vp - VRjrp. The pulse-width modulator has as its input the error voltage

VE and as its output the duty ratio OQ- The feedback network, the error

detector, and the pulse-width modulator often are grouped together into a

single functional block called the controller. The signal VREP

 is a

reference-voltage source that establishes the desired dc output voltage VQ.

The first and usually most involved step in characterizing a closed-loop

regulated converter under small-signal perturbation conditions is the derivation of the small-signal open-loop functions that describe the power stage

of the converter. In deriving these open-loop functions, we need to decide

how the various physical devices, such as the energy-storage reactor and the

semiconductor switches, are to be modeled in the analysis. We also need

information on how the state variables, such as reactor exciting current and

output filter-capacitor voltage, vary during one switching period. Determination of the values of the state variables at the beginning of a switching

period under equilibrium, i.e., steady-state operating conditions, also is an

essential piece of information for deriving these functions.

Section 2 of this report explains how the various physical devices are

modeled in this analysis. It also illustrates how the numerical parameters

characterizing these models can be obtained. Differential equations for the

state variables are set up and solved in Section 3 to describe how the state

variables vary during one switching period. Finally, in Section 4, the computation of the equilibrium operating point is illustrated. The various

small-signal open-loop functions of the power stage are then derived in

Section 5 and expressions for the coefficients of these transfer functions are

tabulated.
To completely characterize a closed-loop regulated converter, a model for

the controller must be developed. Although there has been some efforts to

standardize controller modules [2], the design of a controller is far from

unique. It is possible to design a great number of controllers, each achieving the goal of regulating the output voltage of the converter, with vastly

different circuitries. These controllers may, or may not, have the same

small-signal transfer function. This report does not cover the modeling of

controllers. It is assumed that the user can obtain the small-signal transfer

function of the controller, either analytically, or experimentally as shown

in [3].

Once the power stage and the controller are characterized by their respective small-signal transfer functions, functions such as input impedance,

output impedance, and audio susceptibility can be derived for the closed-loop

regulated converter. The derivations of such functions are elaborated on in

Section 6. The step-by-step procedures to obtain these closed-loop functions

are shown in the form of a flow chart in Fig. 2.
REPORT

SECTION ;:o.

fsTAR T J

5 '

LOOK UP DEVICE MODELS IN SECTION 2 AND '

OBTAIN VARIOUS PARAMETERS FOR THESE MODELS

COMPUTE THE SIX CONSTANTS IN EQS. (3) TO (8) IN SECTION 3

FOR SCVU, SET L$ = L, rp = rs = rx, AND Y = 1

COMPUTE THE APPROXIMATE DUTY RATIO A0 AND INDUCTOR

CURRENT IxUTs) FROM (30,#) and (31,#)

YES NO

CONTINUOUS-MMF MODE DISCONTINUOUS-MMF MODE

SUBSTITUTE EXISTING VALUE OF

IXUTS) INTO (32,#) TO SOLVE FOR

THEN SUBSTITUTE NEW VALUE OF An

INTO (28,#) TO SOLVE FOR IxUTS)

ITERATE ON THIS STEP

TIMES

SEVERAL

SUBSTITUTE EXISTING VALUE OF

AO INTO (38,#) TO SOLVE FOR TOFF1

THEN SUBSTITUTE NEW VALUE OF

Tom INTO (41,#) TO SOLVE FOR An

ITERATE

TIMES

ON THIS STEP SEVERAL

LOOK UP TABLES 1 AND 2 AND OBTAIN

THE 12 FUNCTIONS BY COMPUTING

THE CONSTANTS IN (46 a-j ,#) AND

(S7 a-e,#)

LOOK UP TABLES 3 AND 4 AND OBTAIN

THE 8 FUNCTIONS BY COMPUTING THE

CONSTANTS IN (71 ,#) , (72a-d,#)

(73,#), (62a-d,#), (74 ,f), AND

(69 a-d,#)

OBTAIN THE TRANSFER FUNCTION OF THE CONTROLLER

USE BLOCK DIAGRAM IN FIGURE 13 TO

DERIVE DESIRED CLOSED-LOOP FUNCTIONS

C

END

# = cvu

Equations that are mentioned in this flow chart are repeated in Appendix B for quick reference

Fig. 2. Steps In arriving at the small-signal closed-loop functions of a

regulated dc-to-dc converter.
2. DEVICE MODELS

In the modeling of the power stage of a dc-to-dc converter, the power

transistor, diode, energy-storage reactor, and filter capacitor are replaced

with equivalent-circuit model s consisting of ideal elements. These equivalent

models must be simple enough to make the analysis tractable, while accurate

enough to capture the essential features of the devices during the operation

of the converter. The device models are presented individually in the following subsections.

2 .1 Transi stor Model

The power switching transistor is modeled, as shown in Fig. 3, as a

series combination of an ideal switch Sq, a constant voltage YQ, and a resistor with equivalent resistance rq. The switch SQ is closed when the transistor is switched ON, and it is open when the transistor is switched OFF. The

quantity VQ is equal to the numerical value of the break-point voltage of the

transistor in the ic versus VQE plane for a bipolar junction transistor (BJT) ,

or in the I'D versus VQS plane for a field-effect transistor (FET). The value

of rq' is equal to the differential resistance of the transistor dig/dvq when

it is turned on. The equivalent resistance rq in the model, however, is

defined in terms of the equivalent resistance needed to account for all of the

resistive components of transistor loss, including switching loss, due to the

effective value of the transistor current Iqfrms- Defining

2 1 f 2

( iQ.rms) - Y I 1Q <it

'<; J Tc

where T$ is the switching period, then rq is the required value of resistance

such that the product ( Iq,rms )2 rQ corresponds to the total resistive con-
8

(C or D)

O

V

 1«

f 8 or C)

O

or

o:

O

(E or S)

(VCE or

Fig. 3. Piecewise-linear equivalent model of a BJT or an FET switch. For the

BJT switch, the base, emitter, and collector terminals are identified

by the letters B, E, and C, respectively. For the FET switch, the

gate, source, and drain terminals are identified by the letters G, S,

and D, respectively.
duction losses, as opposed to the loss due just to the constant component Vq

of the transistor saturation voltage. To find the value of Iq >rms and eventually the value of rq, the switching waveforms of the transistor must be

determined either by oscillographic measurement or by calculation.

Over one switching cycle of period TS, the average power loss in the

transistor switch is

1 f

PQ = — I VQ iq dt

T

s

 J Ts

This power loss Pq can be viewed as the sum of two components, the loss PV due

to the constant component of the transistor forward drop voltage Vq when it is

conducting, and the remainder of the power loss PR which can be considered as

due to an equivalent resistor rq [3]. .Therefore,

•W

and

]

PR = PO - PV = -

^s J TS

Equating PR to (iQ.rms)2 H}'

 we nave

2 1 f

(iQ.rms) rq =— I (vq - Vq) iq dt

c » Tc

and the equivalent resistance

f (vq - VQ) iq dt

J

 Ts

rq = (1)

f iq dt

J TS

PV --

 V d t

s

PR= PQ - PV -i- j (vq - vq) IQ dt

'

 J T
10

One possible way of evaluating the integrals shown in (1) is to sample

and digitize the actual waveforms of transistor voltage vg (VCE or VQ$) and

current iq HC or ig). The digitized results are stored in some storage

medium such as the memory registers of a computer or a disk in a computer

system. The integration is then performed through numerical summation of the

sampled values of (VQ - VQ) ig and ig2. This process can be accomplished

through the use of a digitizing recorder such as the Biomation 8100 in conjunction with a minicomputer, or through a digital processing oscilloscope

such as the Tektronix 7854. If such an instrument is not on hand, one can

approximate the switching waveforms with piecewise-linear waveforms. The

value of rq is then obtained according to (1), with the values of vg and ig

replaced by their pi ecewise-linear approximations [3].

Where the switching loss is negligible compared to ON-time conduction

loss, as for a converter operating at a relatively low switching frequency, rg

can be approximated by the dynamic resistance rg' of the transistor, where rg'

is defined graphically as shown in Fig. 3.

2.2 Diode Model

In a manner similar to that of the transistor, the diode is modeled as a

series combination of an ideal switch SQ, a constant voltage VQ, and a resistor with equivalent resistance re- The switch SQ is closed when there is forward current flowing in the physical diode, and it is open otherwise. VQ is

equal to the numerical value of the break-point voltage of the diode in the I'D

versus VQ plane as shown in Fig. 4. Similar to the definiton of rg, the

equivalent resistance of the diode rg is defined as
11

2 so

V

0

Ivo y

SLOPE = —

I)'

Fig. 4. Piecewise-llnear equivalent model of the diode switch.
12

"D -

 va' '„

(2)

2 i dt

where VD and ID are the Instantaneous voltage across the diode and the instantaneous current through the diode, respectively. The methods of evaluating

integrals outlined in Section 2.1 apply equally well in evaluating the

integrals in (2) .

In the case where the switching loss is negligible compared to the regular conduction loss, the resistance ro

 can De

 approximated by the dynamic

resistance ro of the diode, where ro' is defined graphically in Fig. 4.

2.3 Energy- Storage Reactor Model

2.3.1 Energy- Storage Reactor Model for the Single- Winding Current- or- Voltage

Step-Up (SCVU) converter

The energy- storage reactor for the single- wind ing current- or- voltage

step- up converter (SCVU) is modeled as an ideal inductor L in series with a

resistor rx as shown in Fig. 5{a). Assuming that the reactor core material

has a constant permeabilitiy, L is then the nominal inductance of the inductor. That is, L = yN2A/A, where u is the absolute permeability of the core

material, N is the number of turns, A is the effective cross- sectional area,

and i is the mean magnetic path length. The value of resistance for rx is the

winding resistance of the inductor, and it can be estimated when frequency

effects can be neglected by finding the wire size of the winding, the number

of turns in the winding, and the average length per turn of windings.
13

(a)

IDEAL

TRANSFORMER

Np NS

MS

(b)

Fig. 5. (a) Model of the energy-storage reactor for the single-winding

current-or-voltage step-up converter, (b) Model of the energy-storage

reactor for the two-winding current-or-voltage step-up converter with

magnetizing inductance L$ referred to the secondary circuit.
14

2.3.2 Energy-Storage Reactor Model for the Two-Winding Current-or-Voltage

Step-Up (TCVU) Converter

The energy-storage reactor for the two-winding current-or-voltage step-up

(TCVU) converter is modeled by a network of linear elements as shown in Fig.

5(b). Neglecting the leakage inductances associated with the primary and

secondary windings, the energy-storage reactor is modeled as an ideal transformer with the linear magnetizing inductor L$ connected across the secondary

winding. The secondary-to-primary turn ratio of the ideal transformer is

equal to Y = N$/Np, the secondary-to-primary turn ratio of the original twowinding reactor. Assuming the core material has a constant permeability, L$

is equal to the nominal inductance of the reactor referred to the secondary

winding. That is, L$ = yN^A/A, where u is the absolute permeability of the

core material, N$ is the number of turns in the secondary winding, A is the

effective cross-sectional area and i is the mean magnetic path length. The

-i .

value of rp and r$ are the primary and secondary winding resistances, respectively, and they can be estimated in the same way as the estimation of the

winding resistance rx in the single-winding reactor. It is justified to

neglect the leakage inductances because they usually result in circuit resonances of frequencies much higher than the switching frequency while the

small-signal analysis is valid only in the frequency spectrum below one-half

of the switching frequency.

2.4 Capacitor Model

The output filter capacitor of the power stage is modeled as an ideal

capacitor C with a voltage of VQ in series with a resistor re to represent the

equivalent series resistance (ESR) as shown in Fig. 6. The equivalent series

inductance (ESL) of the capacitor is ignored because the filter capacitor
15

VC

Fig. 6. Model of the output-filter capacitor.

LOAD VOLTAGE VQ

Fig. 7. Model of the power stage load.
16

employed in a dc-to-dc converter ususally is chosen so that its resonant

frequency is much higher than the switching frequency f$, while the smallsignal analysis is valid only in the frequency spectrum below one-half of the

switching frequency. The value of C is the nominal capacitance of the filter

capacitor and re can be found by measuring the small-signal impedance of the

capacitor at its resonant frequency or from the step jump in capacitor terminal voltage due to an injected step of current through the capacitor.

2.5 Load Model

The converter power-stage load is modeled as an ideal load resistor RL in

parallel with a current source iy as shown in Fig. 7. The principal purpose

of the source iy is to permit the injection of a disturbance signal at the

output port of the converter. Such a signal permits the measurement of the

output impedance of the converter and, in such cases, usually is a lowamplitude alternating current. It may, however, also be used to represent the

dc equilibrium value of a constant-current component of the total load

current.
17

3 . SOLUTIONS OF STATE VARIABLES IN ONE SWITCHING CYCLE

Before going into the description of state variables for the two converter power stages, it is appropriate to explain the convention for the

symbols used in this report. For any dynamic variable, such as voltage,

current, transistor conduction time, diode conduction time, and duty ratio,

the total instantaneous signals are represented by lower-case characters with

upper- case subscripts. For example,

Instantaneous values :

Supply voltage vj

Duty ratio ag

Transistor conduction time

Each instantaneous signal is assumed to be composed of a dc or equilibrium

term and a small- signal variational term. Equilibrium terms are represented

by upper-case characters with upper-case subscripts. For example,

Equilibrium values :

Supply voltage YI

Duty ratio AQ

Transistor conduction time TON

The small- signal variational terms are represented by lower-case characters

with lower-case subscripts. For example,

Variational values:

Supply voltage VT

duty ratio ad

transistor conduction time ton
18

Combining these terms, we have

vi = YI + v-j

oto = Ag + ad

tON =

 TON + ton

The frequency- domain transformed variables of the small- signal variational

terms are represented by upper-case characters with lower-case subscripts.

For example,

Frequency- domain variables:

Vi(s) = L[ vj(t) ]

) = L[

) = L[

where s represents the Laplace transform variable and L[ ] represents the

Laplace transformation of the enclosed time function.

Also let us define a set of constants, (3.CVU) to (8,CVU), which facilitate the derivations in the following subsections.

,3,cvu,

C( rC + R|J

(4.CVU)

NS

Y - — (5.CVU)

Np

oe= T— (6.CVU)

LS
19

Y

2

 ( rp + rg )

LI

r

S

 + H) + P rC

LS

(8.CVU)

3.1 Description of State Variables in General

Replacing the energy-storage reactor, the capacitor, and the load with

the models derived in Section 2, the power stage of a single-winding currentor-voltage step-up converter shown in Fig. 8(a) can be replaced by an equivalent model in Fig. 8(b) . Replacing the transistor and the diode switches with

their corresponding models, the circuit shown in Fig. 8(b) is reduced to an

equivalent circuit shown in Fig. 8(c). The switch Sq and SQ in the device

models have been combined into one single controllable switch S in Fig. 8(c) .

Similarly, the power stage of a two-winding current-or-voltage step-up converter shown in Fig. 9(a) is replaced step by step by its equivalent circuits

shown in Figs. 9(b) and 9(c). Reflecting the primary circuit of Fig. 9(c)

through the ideal transformer to the secondary circuit, the final equivalent

circuit shown in Fig. 9(d) is obtained. In Figs. 8(c) , 9(c) , and 9(d) , the

switch S assumes position 1 when the transistor is conducting and position 2

when the diode is conducting. If the converter operates in the discontinuousmmf mode, switch S assumes position 3 for part of the switching cycle when

neither the transistor nor the diode conducts.

In general, the converter power stage models in Figs. 8(c) and 9(d) have

three controllable input variables — the supply voltage vj, the loaddisturbance current iy, and the duty ratio ao» which is defined as the ratio

of the time that the transistor is in conduction in one switching cycle tQN to
20

o-*—>

VQ

\f

(a)

Fig. 8(a) Circuit diagram of the power stage of the single-winding currentor-voltage step-up converter.

x

(b)

Fig. 8(b) Equivalent model of the power stage shown in Fig. 8(a)
21

|—>/w^

r

o

VQ

(c)

Fig. 8(c) Equivalent model of the power stage of the single-winding currentor-voltage step-up converter as obtained from the equivalent

circuit shown in Fig. 8(b) .
22

(a)

Fig. 9(a) Circuit diagram of the power stage of the two-winding current-orvoltage step-up converter.

TI ig +

o > >

VQ vo

vi Np

fp

o ^WA-

-torc

C

/\

•VW^-

vc

•f

Mw

(b)

Fig. 9(b) Equivalent model of the power stage shown in Fig. 9(a)
23

°0 Q

Y= MS/NP

(c)

<|-^VWV-

rc|

VQ

Fig. 9(c) Equivalent model of the power stage of the two-winding current-orvoltage step- up converter as obtained from the equivalent

circuit shown in Fig. 9(b).

Y VI

>r ix

VX

(d)

WV

r

s

VQ

v

c

Fig. 9(d) Equivalent model of the power stage of the two-winding current-orvoltage step- up converter after the primary circuit has been

reflected to the secondary circuit.
24

the switching period T$. There are two state variables — the inductor

current i'x and the ideal capacitor voltage vr,. As far as signal flow is

concerned, the load voltage VQ and the supply current ij are looked upon as

the response, or the output variables, of the power stage.

The operation of a converter power stage can be classified into two categories: the continuous- mmf mode and the discontinuous-mmf mode. In the

continuous-mmf mode, the exciting current ix of the ideal inductance in the

equivalent circuit, L in Fig. 8(c) or L$ in Fig. 9(d) , is never equal to zero.

As a result, at any time in a switching cycle, either the transistor or the

diode is conducting. In the discontinuous-mmf mode, however, the exciting

current drops to zero during a certain portion of the switching cycle. Hence,

there is an interval during the switching cycle when neither the transistor

nor the diode is conducting. Define tQN

 as tne duration of time the transistor conducts in one switching cycle, tQFFl as the duration of time the diode

conducts, and tQFFZ

 as tne

 duration of time when neither the transistor nor

the diode is conducting. Then, in the continuous-mmf mode,

tON + tQFFl = TS (9)

where T$ is the switching period. In the discontinuous-mmf mode,

tQN

 + tQFFl + tQFFZ = T$ (10)

Section 3.2 is devoted to setting up the power- stage differential equations in terms of the state variables. The differential equations are then

solved, giving a picture of how the state variables vary in one switching

cycl e .
25

Whether the converter operates in the continuous-mmf mode or in the

discontinuous-mmf mode, in any one of the time intervals tQN, *OFF1'

 an(*

tOFF2»

 tne

 power stage state variables satisfy the piecewise-1 inear differential equations for that interval. This may be expressed for any time interval

in compact matrix notation as

d— _xs = A x$ + B ui

at

whereas is the state vector, juj is the input vector, and ^ and JJ are matrices

whose elements are topology dependent. Since the circuit topology of the

equivalent model of the power stage is altered according to the position of

switch S, the matrices^ and J3 during the time the transistor is conducting

usually are different from the ^ and JJ for the case when the diode is conducting. Using the device models established in Section 2, [ VQ ix 3^ is

chosen as the state vector, where

 T stands for the transpose of matrices and

vectors. Likewise, [ vj iy Vg Vn, ]T is a meaningful choice for the input

vector. In terms of these choices, we have

2LS = C vc ix 1T (ID

Jil - C vi iW VQ VD ]T (12)

To identify and distinguish relationships that are applicable during the

time interval when the transistor is conducting, we attach the subscript ON to

the corresponding^ and _B matrices. Therefore, during the ON-time in the kth

switching cycle for (kT$) <_ t£ (kT$ + ton), where k is an integer,

d— _xs » AON Jis

 + ION _yi (13)

Withjcs(kTs) as the initial solution for the vector jc$» the general solution

to (13) is
26

AoN(t-kTS) ft

is(t) = e~ IsUTs) +

J kT$

BQN UI(T) dr

~ ~

Assuming jji does not change in one switching cycle, then

AON<t-kTs)

jcs(t) = e

= e jcs(kTS) -

AQN( t- kTs)

p AON

. J kTS

 6

i r AW

^ON"1 e L

|(trt)

dr ION Jii

|(t-T) 1 t

BON ui

J

 kTS~ -

t

xS(kTS) + AQN'1 ( e" - _I ) BON Jil (14)

where ^QN"

 1S tne

 inverse of ^QN- ^ne assumption that_ui does not vary in

the switching period is seen to have a far reaching consequence in later

sections. This assumption is necessitated by the need to keep the analysis

tractable, otherwise the modeling becomes excessively complicated.

During the time interval ( kT$ + toh|) 1 1 1 ( kTS

 + tQN + tQFFl) > tne

 diode

is conducting. Similar to (13), the differential equation in this interval is

d

^ 2is = AOFFI is

 + IOFFI jii ( is)

where j^oFFl

 ana<

 J^OFFl

 are

 matrices whose elements are dependent on the

topology and elements in the circuit. Assuming the input vector _uj does not

change in one switching cycle, then, similar to (14), the solution for the

state variables in this interval is

AOFFI (t-kTS-toN)

+

 AQFF1'1 ( e - I } JJQFFI Jil (16)

where _xs(kT$+tON) is the initial condition for this time interval.
27

If the converter operates in the discontinuous-mmf mode, there will be a

third time interval during which neither the transistor nor the diode conducts. Similar to (13) and (15), the differential equation for this interval

is

d

•£ is = AOFFZ is

 + IOFFZ 21 ( 17)

where J\QFF2 and J5QFF2 are matrices whose elements are dependent on the

topology and elements in the circuit. Similar to (14) and (16) , the solution

for the state variables in this interval is

- e

AOFFZ ( tAOFF2'1

 ( e - 1 ) 10FF2 Jil

3.2 Description of State Variables for both SCVU and TCVU

An inspection of the equivalent circuits shown in Figs. 8(c) and 9(d)

shows that the equivalent circuits of SCVU and TCVU are very similar. In

fact, if Y = 1, LS = L, rp = rx, and r$ = rx, the equivalent circuit of TCVU

shown in Fig. 9(d) is reduced to the equivalent circuit of SCVU shown in Fig.

8(c) . Therefore, as far as the description in terms of the state variables is

concerned, SCVU is only a special case of TCVU where y = 1» L$ = L, rp = rx,

and r$= rx. Starting from this section, equations that are applicable to

both SCVU and TCVU are labeled with (n,CVU) where n is the equation number,

with the understanding that y =l » Ls=L,rp=rx , and r$ = rx for the case

of SCVU.

In the time interval in which the transistor is conducting, the equivalent model shown in Fig. 9(d) can be reduced further to the circuit shown in
28

Fig. 10(a). The power-stage differential equations written in terms of the

the capacitor voltage VQ and the inductor current i"x are :

ir = C — vr = — -

dt rc + RL rc + RL

vx = LS ^- ix = - Y2 ( rp + rq) ix + y vi - y VQ

Dividing by C throughout the first equation, and by L$ throughout the second

one, and substituting symbols defined in (4,CVU) (5.CVU), and (7 ,CVU)

— vc= -wa

 VC -

rt1*

Y vi Y VQ

— ' ~

These two differential equations can be combined to form a single matrix differential equation

0 U>g

0 0 0

Y_ 0 ^Y 0

Comparing the above equation with (13) gives

and

_§ON =

-ua 0

0 -uq * » •

0 -R|_ua 0 0 "

J- 0 11 0

I

 Ls Ls J

(19,CVU)

(20,CVU)

When the initial condition jc$(

 kT$) is known, the solution for the state

variables during this time interval can be obtained by substituting (19.CVU)

and (20.CVU) into (14).
29

Y Vl

iw

(a) kT$ £ t £ kTs

Y V J

>' 1\

-vwrs

RL

.vc

(b) kTs + tQN 1 t £ kTs

 + *OM + tQFFl

Y Vj

( I

\f

(c) kTs + tQN + tQFFl <. t <_ (k+l)Ts

Fig. 10. Equivalent circuit of the two-winding current-or-voltage step-up

power stage for the three subintervals tQN, tQFFl*

 and
30

When switch S in Fig. 9(d) is in position 2, corresponding to conduction

of the diode, the equivalent model is reduced to the one shown in Fig. 10(b) .

From Fig. 10 (b) , the differential equations are :

1C - -—

VX = LS — ; dt

RL ^

RL

rc

+ RL + RL

rc RL

RL

Dividing the first equation by C, and the second one by L$, and using the

constants defined in (3,CVU), (4,CVU), (6,CVU), and (8,CVU),

d

-. vc = - ua

 VC + RL<"a ix - RL"a ^W dt

d p

T l'X

 =

 ~ 7~

 V

C ' ut

dt LS

which can then be combined into

p •

-P -u

• o «

«s +

0 - RLU a

0 u e

0

0

0

L

s .

Comparing the above equation with (15) gives

.AOFFI =

and

(21.CVU)

iOFFl =

0

0

-RLu>a

(22.CVU)

When the initial condition j(s( kTs*-tQN)

 ls

 known, the solution for the state

variables in this time interval can be obtained by substituting (21,CVU) and

(22.CVU) into (16).
31

When switch S in Fig. 9(d) is in position 3, representing the condition

that both the transistor and the diode are not conducting, the equivalent

model is reduced to the one shown in Fig. 10(c) . From Fig. 10(c) , the differential equations are :

• - L -

 VG RL *H

1C

 dt

 VC

 rc + RL rc + RL

— iX = 0

Dividing by C throughout the first equation, and using the constant defined

in (4,CVU) ,

Jil

- ua

0

0 '

0

IS +

0 -R

0

L^a

0

0

0

0

0 •

Comparing the above equation with (17)

and

_§OFF2 =

• •

-u>a 0

0 0 ft a

0 -RLu>a 0 0

000 0

> •

(23,CVU)

(24,CVU)

When the initial condition is known, the solution for the state variables

during this time interval can then be obtained by substituting (23,CVU) and

(24.CVU) into (18).
32

4. EQUILIBRIUM OPERATING POINT

4.1 Equilibrium Operating Point in the Continuous- Mmf Mode

In the continuous-mmf mode, the switching period T$ is equal to the sum

of the transistor conduction time ton and the diode conduction time tQFFl • At

the end of the kth switching cycle, the time is t = (k+l)T$ = kT$ + tQN +•

' At this time, according to (16) ,

AOFFltQFFl .

2csC(k+l)T$] = e £s(kTs*toN) + ^OFFl"1 ( e - _I ) JJQFFl Ji

Substituting for _xs( kTjf tow) the value given by (14) evaluated at t =

_AOFF1*OFF1 A

_xsC(k+l)Tsl = e { e jcs( kTs) + Ao^1 ( e - _I ) _§QN

, ,

AQFFl"1 ( e - 1 ) lOFFl Jil

Analogous to the scalar exponential series, the matrix exponential e can be

expressed in the series form

M M M2 M3

e =i+ — + — + — +•• •

- 1! 2! 31

for a square matrix _M. Using the series expansion for the state transition

matrices e and e , and retaining terms only up to second order

in T$, the equation for j<sC( k+l)T$] reduces to

^ON2tQN2 + AOFFl2tQFFl2

jCsC(kfl)Ts] = ( 1 + AQNtQN

 + A0FFltQFFl + - ^ -

) *$<•

 kTs)

AQFFllOFFltOFFl2

(25).
33

The relationship between the transistor conduction time tQN and the duty

ratio OQ is tQN = <*DTS> where T$ is the switching period. Combining tQN

 =

aoT$ with (9) gives toppi = (l-an,)T$. Now substituting the respective matrices AON, ^OFFI » 10N»

 and

 JOFFl

 from

 U9.CVU) to (22.CVU) into (25), and carrying out the multiplication, the matrix equation can be decomposed into two

scalar equations.

vc(kTs) ( 1 - l

iw ( TS - — --- - - ) (26.CVU)

and

PvC(kTS) u)a(l-aD

2

)Ts2

(u qon+u h( 1-afl) ) 2T$2

-h ix(kTS) ( 1 - (a)gaD+a)h(l-ccD))Ts+ — = - -

VD a.h(l-

— ( (l-ao)Ts -- - - ) (27.CVU)
34

In steady state or equilibrium, all state variables are the same at the end of

an arbitrary switching cycle as at the beginning. As a result, using an

upper- case character with an upper- case subscript as discussed in the introduction to Sections to indicate an equilibrium value, we have vr,( kTs) =

+l)Ts3 = Vc (kTs ), ix(kTs) = 1XC( k+l)Ts] = Ix( kTs) , vj = Yj, iw = IW, and

AD. In equilibrium, (26,CVU) and (27.CVU) reduce to

0 = - o,a VC( kTs) ( 1 - —

Ix(kTS)

L^a ( 1 - 1 ) (28.CVU)

and

pVc(kTS)

o --- __ ( I-AO, ( i -

Y(VI-VQ)A D

iw

) (29.CVU)
35

Since the converters are designed to provide a constant voltage for the load,

the ripple on the ideal capacitor voltage VQ and the load voltage VQ are

usually very small with respect to their corresponding average values. Therefore, the ideal capacitor equilibrium voltage VQ( kT$) is very close to the

value of the specified load voltage VQ. Hence, VQ{ kT$) can be approximated by

the value of VQ without much loss of accuracy. Once Vj, Iy, VQ, Vq, and VQ

are specified, the two remaining unknowns Ix(kT$) and AQ can be computed from

(28,CVU) and (29.CVU). The terms TON and TQFFI are then calculated from the

values of Ag and TS. Equations (28,CVU) and (29.CVU) are linear in Ix( kT$) ,

but quadratic in AQ. As a result, a closed form solution is not directly

accessible. On the other hand, if an approximate solution to AQ and Ix(kT$)

can be found initially, then iteration on (28,CVU) and (29,CVU) can be carried

out to obtain more accurate solutions. As a first approximation, all parasitic dissipative elements rq, rg, rp, r$, and re are neglected by assuming

they are equal to zero. As a result, ua is equal to l/CRi_, while u>e, u>g, and

con are equal to zero, and p is equal to one. Equation (29,CVU) is then

approximated as

(i-Ao)2TS T(VI-VQ)AO

0 = - ( 1 -- ) — - + - - —

LS 2 } 2LSC LS

iwd-Ag2)TS VD (I-AD)

2LSC " ~~Ti

To maintain a low ripple voltage across the load, the values of L$ and C are

usually chosen so that u)aTs « 1, and Ts2/L$C « 1. Hence, the above equation

can be reduced further to
36

Y(VI-VQ)AD VD(I-AD)

o = - + ---

LS LS LS

= AD ( Y(VI-VQ) + v0 + VD ) - ( v0 + VD

which gives

VQ + vD AD = - (so.cvu)

Y(VI-VQ) + v0 + YD

With u)e> ug, u>h set equal to zero and p set equal to one, (28.CVU) can be

approximated as

a)a(l-AD)Ts

0 - -o,aV0 ( 1 - — + - - - ) + Ix(kTS)RL03a(l-AD) (

VD

Dividing byo>a and utilizing o>aT$ « 1,

o - - v0 ( i + - - - ] + ix(kTS) RL(I-AD)

Y(VI-VQ)RL(I-AD)ADTS VD

- iw RL

Dividing by R|_(!-AD) , the above equation becomes

i v0 {VO+VD )(I-AD)TS Y(VI-VQ)A D TS

0 • -TT ( + Iw ) -- r - +I x (kTs )n- L$

From (30), (VofVD)(l-AD) = Y(VI-VQ)AD, so that
37

i v0 Y(VI-VQ)AOTS

o = - - rL w — + iw

RL 2LS

i v0 Y(VI-VQ)ADTS kTs)

 • c +I«

 ]

 - —-

 (31>cvu)

Most dc-to-dc converters operate with very high efficiency, and the actual

values of rq, rg, rx, and re are so small that the equilibrium duty ratio AQ

given by(30,CVU) and the equilibrium inductor current Ix( kT$) given by

(31,CVU) are within one to two percent of the values obtained from experimental measurements. If these dissipative elements are large enough to affect

significantly the values of AQ and Ix( kT$) , the operating point can be located

by using iterative methods. Equation (29,CVU) can be rewritten as a quadratic

in AD :

0

 =

Y(Vl-YQ)Ug-2u,h)

AQ< (1-u

vpmi "I

37 /

hTS) (

LS

[2-o>hT$)

2

 ( " ~Ls

v0 - RL!X( kis)

(32.CVU)

First, substitute the specified constants Vj, VQ, Vq, VQ, and Iy, and the constants P , u)aj uej 0}gj and uh 1nto (32,cvu) and (28,CVU). Then substitute the
38

approximation for ty( kT$) from (31.CVU) into (32,CVU) and solve for a new

solution for AQ. Substituting the new Ag into (28,CVU), a better approximation for Ix( kl"s) can be computed. The new value for Ix(kT$) can then be

substituted into (32.CVU) to obtain a more accurate value for AQ. Working

back and forth between (32.CVU) and (28.CVU) once or twice will give highly

accurate solutions for the equilibrium duty ratio AQ and the equilibrium

inductor current Ix(kT$) •

4.2 Equilibrium Operating Point in the Discontinuous-Mmf Mode

In the discontinuous-mmf mode, the inductor current is always equal to

zero at the beginning of each switching cycle, that is, i'x( kT$) = ixC(k+l)T$]

= 0. Also, tQFF2

 = TS - tQN - tOFFl • Evaluating (18) at t =

= e

+ AQFF2"1 ( e - _I ) J3QFF2 Jil

Evaluating (14) at t = kT$*-toN and (16) at t =

AQNtON , % , A

jCS^kTs^-ton) = e £$( kTs) + Ao^1 ( e • 1 ) lOM Jil

and

A t

x$( kT^ t<jM)

, ,

+

 AQFFl"1 ( e - _! ) _B0FF1 Jil

Combining the three equations above gives
39

( A0FFltoFFl

JcsC(k+l)Ts] = e I e e

i

( e - i ) ION Jil I

.

AQFFl'1 ( e - _I ) J3QFF1

"I

Jil >

AQFF2"1 ( e - _I ) JJQFF2 J"!

Expanding the state transition matrices e , e , and

e Into a set of power series, and retaining terms only up

to second order in T$»

< 1 + AAQNtQN + AoFFltQFFl + A

+ AOFFZ AOFFI ( Ts- tor *OFFI ) toFFi \ *$(

 kTs)

( '

I l

AQFF210FF1 ( TS- tQM- tQFFl ) tflFFl > Jil ( 33 )
40

Substituting the respective matrices ^VQN, ^OFFl > j^OFF2» .§ON» lOFFl > and lOFFZ

from (19.CVU) to (24.CVU) into (33) and carrying out the multiplication, the

matrix equation can be decomposed into two scalar equations.

= VCUTS) ( 1 - »aTs + — --- — - )

Y (VI - VQ) RLo>a tQFFltQN VD RLua tQFFl2

2LS

(34,CVU)

L '£.

and

_ »/^i cT\i

0 vIA TM **J/

0 = — ( tOFFl - 2

OJgtOM2 VD

Ls ( ton - —- - .ntoFFitQN ) - - (

x ,

^^

 +

 - [ - ^ (35.CVU)

The identity i'x( kT$) = ixCd^DTs] = 0 has been used in deriving these two

equations. For (35.CVU), terms in powers of tQFFl

 can be

 grouped to obtain

<*)h(pvc(kTs)+VD)

o - ( - — - * - —

2L$
41

Equations (34, CVU) to (36, CVU) are equations based on the pi ecewise- linear

description of the instantaneous values of the state variables over one complete steady- state cycle. In steady state or equilibrium, all state variables

are the same at the end of an arbitrary switching cycle as at the beginning.

As a result, using an upper-case character with an upper-case subscript as

discussed in the introduction to Section 3 to indicate an equilibrium value,

we have VQ( kT$) = vcC(k+l)T$] = VQ( kT$) , vj = YI, and iy = Iy. Since a

stringent restriction on the ripple of the load voltage is usually placed on a

dc-to-dc converter, the ideal capacitor equilibrium voltage VG( kT$) is very

close to the load voltage VQ. Hence, VQ( kT$) can be approximated by the value

of VQ without much degradation of accuracy. Thus, under steady- state or

equilibrium, (34, CVU) and (36, CVU) reduce to

uaTS

2

 PRL

TOFF12

0 = — a>aV0 f Ts -- + a

2 2LS LS

weTOFFl2 ^a VD T0FF12

( TS - — --- - - ) -- — - (37 ,CVU)

and

0 - TOFF1 ( — - - - --- j— )

-h TQFFI (

Y(Vl-VQ)TON(2-a)gTON)

-r ( - - = - ) (38, CVU)
42

For specified supply voltage, load voltage, and load, the two unknowns in

the steady- state analysis of a converter operating in the continuous- mmf mode

are the duty ratio AQ and the inductor current Ix(kT$) • The diode conduction

time TQFFI is simply defined by TQFFI

 = TS(I - Ag) . On the other hand, for

specified supply voltage, load voltage, and load, the two unknowns in the

steady- state analysis of a converter operating in the discontinuous- mmf mode

are the duty ratio AQ and the diode conduction time TQFFI • The value of

Ix( kT$) , however, is always equal to zero in the case of the discontinuous- mmf

mode. The values of the two unknowns AD and TQFFI can be obtained from the

two simultaneous equations (37,CVU) and (38,CVU). First, approximations to AQ

and TQFFI are found by neglecting the parasitic dissipative elements rg, ro,

rp, r$, and rQ. As a consequence, with ua equal to 1/CRi., u>e , o>g, and un

equal to zero, and p equal to one, (38.CVU) is reduced to

VQ+RI.IW TO N(VO+RLIW) VQ+VQ Y(VI-VQ ) T O N T

°

FF1 (

 --- —

 ] + ( ~^i —

Since TQFFI ^s of tne order T$, and TQN is also of the order T$, the use of

the inequalities (i)aT$ « 1 and T$2/LC « 1 reduces the above equation to

Y(VI-VQ ) T O N o =

or

Y(VI-VQ)TON Y(VI-VQ)ADTS

Although (39,CVU) gives TQFFI as a function of AQ, Vj, VQ, VQ, and VQ, the

equilibrium duty ratio AQ is still unknown at this moment. With ue» ug, and

uh equal to zero and p equal to one, (37.CVU) reduces to
43

R

L

TOFF12

R

L»aTS

vOTOFFl2

0 . -

,

1- — )

Dividing the above equation by coal's, and noting thatu>aT$ is negligible when

compared to unity,

Y RL(

 v I-VQ) TOFFI AO 0

 « -

 vo ( 1

-YO2LSTS ' LS " 2LSTS

Y(Vi-VQ)RLTOFF1Ao (VofVD)RLT0FFl2

LS

which then leads to

V0 +

LS 2LSTS

Substituting the expression for TQFFI ^r°

m

 (39.CVU) into the above equation,

Y

2

(Vi-VQ)2RLAn2TS

VQ + RI.IW = —

2LS(Vo*-VD)

giving

2Ls(V0fRLIw)(V0*VD)

(40.CVU)

If the solutions for AQ and TQFFI given by (40,CVU) and (39.CVU) are not

accurate enough, more accurate solutions can be obtained through iterative

procedures. Dividing (37.CVU) by o>aTS» and rewriting it,
44

, Y(Vi-Vq)RLTOFF1

0 . Ao ( - ) + - V0 ( 1 - — *

}

- Ri iu f i ) > (4i,cvu)

2LSTS L w i. 2 2TS

 ; r

Now substitute the specified Vj, VQ, VQ, VQ, and Iy, and the constants p, toa,

coe> ug» and u>n into (38,CVU) and (41.CVU). Substitute TON = ADT$, where AD is

obtained from (40.CVU) into (38.CVU) and solve for a new value for TQFFI*

Then substitute the new TQFFI

 ln<to

 (41.CVU) and obtain a more accurate approximation on AD« Working back and forth between (38.CVU) and (41.CVU) once or

twice will give highly accurate solutions for the equilibrium point.
45

5. SMALL-SIGNAL TRANSFER FUNCTIONS CHARACTERIZING THE POWER STAGE

5.1 Small-Signal Transfer Functions in the Continuous-Mmf Mode

As outlined in Section 3.1, the signal flow of the power stage of a dcto-dc converter may be modeled as a network with three controllable input

variables, the duty ratio ao» the supply voltage vj, and the load-disturbance

current i^; two state variables, the capacitor voltage VQ and the inductor

current ix; and two output variables, the load voltage VQ and the supply current i'i. In order to model the power stage, including its load, in the smallsignal frequency domain, transfer functions relating the input variables, the

state variables, and the output variables have to be derived. Figure 11 shows

a block diagram representing the power stage and the twelve transfer functions

relating these variables. Six of these transfer functions involve the state

variables and the input variables while the remaining six involve the output

variables with the input variables and the state variables.

Equation (25) in Section 4.1 describes the state vector x.sH k+l)T$] at

the end of the switching cycle as a function of the state vector £$( kT$) at

the beginning of the switching cycle, the duty ratio ao, and the input vector

_uj. Rewriting (25) in the form of a difference equation, we have

_ AQFFl2tOFFl2

- _x$UTs)

.*$(

 kTS)

r

+ I lONtQN + lOFFltflFFl

 +

JJI
h

>-

vc(s)

\

>

•x.

1 ~ ^

G

Ii/Ad(s)

Gii/Vc(s)

GV0/Vc(s)

\-/'

V0 (s)

Fig. 11 Interconnection of functional blocks to characterize a converter

power stage in the continuous-mmf mode.
47

df

The derivative — of any function f(t) is defined as

dt

df _ 1 im f(tt-At) - f(t)

dt A1>0 It

Similarly, for changes that occur very slowly from one switching cycle to the

d

next cycle, the derivative— x$ can be approximated with { xsC(k+l)T$] - dt —

> / TS by letting t= kT$ and At = T$ because the period of the lowfrequency perturbation signals is much larger than the switching period T$.

Therefore,

TL J$S,SS = dt-

- JcS(kTs)

r

I IONOD + i

Jil (42)

Since (42) is derived from a difference equation involving the state vectors

at the beginning of successive switching cycles, it is able to predict the

stroboscopic variations of the state vector at the beginning of each switching

cycle, while information about the state vector at time durations in between

the beginning of successive switching cycles is lost. To separate the state

vector observed at stroboscopic time intervals from the continuous state
48

vector x$(t) , the subscript 55

 nas

 &een used ">n (42) to denote the stroboscopic state vector which is valid only at the beginning of each switching

cycle. Therefore, equation (42) is a matrix differential equation describing

the stroboscopic variation of the state variables at the beginning of each

switching cycle. It can be rewritten in the form

d

dt

VC,SS

vc,SS» I'X.SS.

(43)

Equation (43) can be viewed as a differential equation with time-varying

coefficients because the duty ratio ao« which is time-varying from cycle to

cycle under small-signal perturbation, appears as a coefficient associated

with the state vector jc$

 and

 the input vector _uj. Although <XQ does not appear

explicitly as an input variable in the input vector _ui in the equations

derived in Sections 3 and 4, it is an important input variable controlling the

energy flow in the power stage of a converter. Under small-signal perturbation around the equilibrium operating point, (43) can be rewritten as a system

of two linear differential equations with constant coefficients involving the

small-signal variables jcs,ss = C vc,ss ^x.ss 3T and <*d>

 vi > and

 iw

d

Tt-s'

ss =

3fv 3fV

3vc,ss ai'x.ss

3 f I 3 f I

[3VC.SS 3iX,SS J

vc,ss '

. '''x.ss .

+

3fy 3fy 3fy

ill 3fl 3fl

3aD 3VI 31W .

ad

(44)

where the partial derivatives in (44) are evaluated at the equilibrium operating point. After the partial derivatives have been evaluated, (44) is seen

to be a system of linear differential equations with constant coefficients as

shown in (45).
d

dt

c,ss

49

VG.SS

b

!2

 b!3

(45)

L !w J

where

an =

3VC.SS

3fV

ai2 = —3 IX ,SS

- 1^1

3V! I*

3 1W

321 "

322 =

3iX

afi

,ss|*

3fl

b22 = — I 3Vi|*

31W

(46 a)

(46b)

(46c)

(46d)

(46e)

(46 f)

(46g)

(46h)

(461)

(46j)
50

The* symbols in Eqs. (46) indicate evaluation of the partial derivatives at

the equilibrium operating point [ Vc(kT$) Ix(

 kT$) 3T

. Applying the Laplace

transformation to (45), which represents a system of two small-signal linear

differential equations with constant coefficients, gives

s-an

Thus,

s-322

Vc (s)

Ix(s)

bll

022

s)

Vi(s)

'Vc(s) 1

Ix(s)

s- 322

 a!2

321 S"

311

r* TM bll bi2 bi3

[ 21 22 23 J

Ad( s)

Vi(s)

, W s) .

( s- an )( s- 322 ) - ai2 a2i

(47)

For example, the transfer function between the Laplace transformed smallsignal capacitor voltage Vc( s) and the transformation of a small-signal disturbance of the duty ratio Ad( s) is

Yc (s)

s)

( s - 322

a

!2

 b21

(48)

S2 _ s

All six transfer functions relating the two state vsridbles snd the three

input vsriables are of the form indicated in (48) , with one zero and 3 pair of

complex poles. Table 1 lists the constants for these transfer functions.

The transfer function relating the capacitor voltage and the load voltage

is derived in a straightforward manner from Fig. 9(d) . Starting in the time

domain,
51

Table 1. Six Small-Signal Transfer Functions for the Two State Variables

Under Continuous-Mmf-Mode Operation.

FUNCTION FORM

NATURAL RESONAN

G ( 1 + s/Z)

1 + 2? ( s/o)0) + ( s/u>0)2

I KKEQUtNCY wo = V 3H *22 -

 a!2 a21

DAMPING RATIO c = - (an + 322) / 2u>0

FUNCTION

GVc/Ad< S)

GVc/Vi(s)

BVc/lJs)

G

Ix/Ad( s)

GIx/v.(s)

Glx/lw(s)

GAIN CONSTANT G

312 D21 - 322 DU

%

2

31,2 b22 - 322

 b!2

-.

z

312 &23 - 322 bi3

%

2

321 bn - an b2i

V

321 bi2 - an b22

"oZ

321 bis - an b23

»?

ZERO Z

312 D21 - 322 Oil

Oil

312 t>22 - 322 t>12

bi2

312 &23 - 322 D13

b!3

321 DH - 3ii &21

D21

321 &12 -

 all D22

b22

321 ^3 - 3ii t>23

&23
52

dvc

VQ = - ( vc + rc C — )

Transforming to the s domain,

V0(s) = - ( Vc(s) + s rc C Vc(s) )

= - ( 1 + s rC C ) Vc(s) (49)

So far, transfer functions have been derived for the stroboscopic variations of the state variables at the beginning of each switching cycle. However, the information obtained from these stroboscopic variations alone is not

adequate to characterize the power stage in the small-signal frequency domain.

In the discontinuous-mmf mode, the total instantaneous value of the reactor

exciting current is always equal to zero at the beginning of each switching

cycle. That is, ix(kT$) is identically equal to zero, thus making ix,ss equal

to zero since ix,ss is equal to the small-signal variational part of i'x( kT$) •

As a result, using i"x,ss

 to

 derive the input impedance of a power stage of a

dc-to-dc converter operating in the discontinuous-mmf mode indicates an infinite input impedance, which is erroneous. Because of the switching nature

of the power stage and the low-pass characteristics of the output filter, it

is the energy flow over one switching cycle, not the instantaneous power flow,

that is regulated. Therefore, it is more appropriate to use the energy flow

over one switching cycle to derive transfer functions for the two output variables — the supply current and the load voltage.

Going back to the large-signal instantaneous variables, the energy WQ

delivered to load resistance RL in one switching cycle is
f

=

J

53

V0(t) )2

kTs

 RL

The load voltage can be approximated as

VQ( t) = vo,k + VQR( t)

where vos k is the average load voltage over the kth switching cycle and

VQR( t) is the ripple voltage across the load, which has an average value

equal to zero over one switching cycle. Therefore,

v0R(t) )2

MO •/

/

(k+l)Ts i V0 k i f

r dt +

kTs

 RL J

kTS

 RL

)

2

 r(k+l)TS ( v0R(t) )2

kTS L J kTS

 L

dt

Because of the stringent restriction placed on the ripple voltage across the

load of a dc-to-dc converter, the capacitor is usually chosen with sufficient

capacitance and low enough ESR to reduce the ripple voltage across the load to

a level which is very small compared to the nominal load voltage. Therefore,

is very small compared to vo3io and

( v0k )2

. c

R

L

VQ,k )2 TS

dt

Since the ripple voltage VQR( t) is negligible when compared to the average

load voltage vo,k» the load voltage at the beginning of each switching cycle

VQ( kTs) is very close to the average load voltage vos k« Hence,

( VO,K )2 TS ( v0(kTs) )2 TS

Wn = — — - ^^— = - RL RL
54

Therefore, using the stroboscopic load voltage VQ( kT$) to account for the

energy delivered to the load over one switching cycle is a good approximation.

As a result, the transfer function derived in (49), which is valid for the

stroboscopic load voltage v0jSS, is also quite accurate in describing the

small-signal value of the average load voltage v0jk over one switching cycle.

Unlike the load voltage, the supply current usually has a relatively high

amount of ripple content compared to its average value. The energy drawn from

the power supply over one switching cycle is

•J

(fcH)Ts

Ml - | vj i! dt

kTS

Using vi fk and ii^ to represent the average supply voltage and average supply

current over one switching cycle, and viR(t) and iiR(t) to represent the

ripples in the supply voltage and supply current,

Ml = J

(fcH)Ts

) dt

f fcHTs

1ifk + viR(t) iiR(t) dt

J kTs

The ripple viR(t) in the supply voltage is usually negligible compared to

vi,k»

 tne

 supply voltage averaged over one switching cycle. Although the

ripple i'iR(t) .in the supply current is not negligible compared to i'i}k»

 tne

supply current averaged over one switching cycle, it is finite in magnitude.

Therefore, the contribution to MI is mainly from the first integral, i.e.

W-t -

 TS vi, k M,k
55

Since ii k 1S tne supply current averaged over one switching cycle, by

definition

(fcH)Ts

il,k= T I " ix(t) dt

TS J kTS

Since the ripple VIR( t) in the supply voltage over one switching cycle is

small , the value of vj^ is very close to the actual instantaneous voltage

vj(t) . Therefore, the power drawn from the power supply on a per cycle basis

is

P! = Wl / TS

*

 VI

 1I,k

Hence, the modeling of the input port of a power stage should be derived from

the supply current ii^ averaged over one switching cycle. To characterize

the small-signal average supply current ^\ t^, it is convenient to start by

going back to the large-signal instantaneous description of the state variables. Two matrices ^QN and ^QFFl

 are

 introduced such that when the transistor is conducting,

i;

and when the diode is conducting,

" VQ( t)

n(t) = MOFFI jcs(t) = MOFFI

L ixt t)

For both SCVU and TCVU, the supply current is equal to the reflected reactor

exciting current when the transistor is conducting, and is zero otherwise.

Therefore

F vd t) 1

t) = MQN jcs( t) = MON

" L ix( t) J
56

MON = C o Y 3 (so.cvu)

MOFFI = Coo ] (si.cvu)

Using these definitions, the average supply current in the kth switching cycle

i

 is

 defined as

1 fkTsfaDTs 1 /-(k+D

tl.k" T" MONXS(t) dt+ - TS J kTS TSJ ki^

x$(t) dt

(52)

For oo, vi, and iw constant throughout the kth switching cycle, substituting

the expressions for jc$(t) fron> (W aid (16) into (52) gives

JU|ON C kT$+ctDTs r ^ON^t-k^s) ^ON(t-kT$) i

11

 'k * T J kTS I 6 ^ kTs)+-°N"

1( 6 ~^-TM ~l

 I dt

, JlOFFl C

r r e

kTs+aoTs

AQFFl(t-kTs-tON - - i )

(e -DlOFFl Jil

AQN(t-kTs) AQN(t-kTs)

e xsUTsJ+AoN'-Aorr1 e

i

-It)BQNUI

- ~ ~ J

— I _wii ^ ^** ** -«J^'* L ijwW *" .._ *J _\JnTMl I

MOFFI

^OFFl"1 e JcS(kTs<-toN)

(kfl)Ts
57

>k

 TS I -

N e

 -

J

-

s

, JJOFF1 f . AoFFl(l-<*D)TS

i

-_n-l(l-aD)TS)_BoFFl Jill

Since jjoppl is equal to [ 0 0 ] for both SCVU and TCVU, it Is not necessary to

carry any terms premul tip! ied by ^QFFl

 ln tne

 above equations as there is no

contribution to 1it|< from these terms. The above equations, however, are

derived for all four basic converters. In the case of the voltage step- up

converter, deletion of jloFFl

 ancl 1ts

 associated terms is not possible as j^OFFl

is equal to [ 0 1 ] . As a result, ^JOFFI

 and 1ts

 associated terms are not removed from the above equations so that the reader can trace the parallel paths

of derivation of ii^ between the Phase A Report and this Phase B Report.

Substituting t= kT$ + tow into (14),

.

= e jcsUTs) + AoN'

Therefore,

AONODTJ^ON r . _£ON<*D'S ^ONotD'S

I,k = j— I AQN'H6 -_r)_xs(kTS) + A0N"i(AON"i(e -IMctoTsItel

i!!OFFl

-D( e xs( kTs)

, A

+ AON"

1

(

i i

AOFFl"1(AQFFl"1(e -J)-l(l-aD)Ts]iOFFl Jil

(53)
58

_ Expanding the state transition matrices e and e and

retaining only up to second- order terms in T$,

WON I (oQi + - - - ) 2<S,SS

r , _OFFl-oD s

I ( (1-otoU

 +

 - - - + AON(l-aD)aOTS ) 1S.SS

-a 1

+ ( - 2 - * 10N(l-aD)aDTS ) JJI I (54)

where jc$( kT$) has been replaced byjcs^S to follow the notation used earlier.

Equation (54) can be rewritten as

1-I = f ( 2LS,SS» <*D, VL iw ) (55)

Linearizing (54) and applying the Laplace transformation,

Ii(s) = pii Vc (s) + pie Ix(s) + qn Ad(s) + qi2 V^s) + qi3 Iw( s) (56)

where

3

Pll =

3VC.SSI

- + MOFFI C (I-AD)J. + "" " 2 + AON(I-AD)ADTS ) [ i o

(57 a)

A

3f

3iX,SS

f AQNAD^TS — = MQN ( AQI + —-— ] [ i o ]T

.ssr

 z

= _MON ( AQI + ^~— ) [ o i ]T

+ MOFFI ( (i-AD)l + + AONU-AD)ADTS ) [ o i ]T

(57b)
59

a f

= MQN [ ( 1 + AQNAoTs ) XS( kTs) +

+ MOFFI [ ( - 1 - AOFFI(I-AD)TS

+ ( - IOFFI(I-AD)TS +

3f

—3VI

MOFFIIONU-AD)ADTS j C i o o o

3f

—3iw

[ 0 1 0 0 ]T

) 2L

) HI

(57c)

(5?d)

(57 e)

This analysis demonstrates that the small-signal average supply current

I-j(s) depends on the small-signal variables Vc( s) , Ix( s) , ft^( s) , V-j( s) , and

Iv/s) . From Section 5.1, the capacitor voltage vc and the inductor current ix

were found to depend on the duty ratio ct^, the supply voltage v-,-, and the

load-disturbance current iw. Now perturb the power stage with a small-signal

applied to the supply voltage, holding the duty ratio and the load-disturbance

current constant. If the perturbation is performed at a frequency close to

half of the switching frequency, then Vc( s) and Ix( s) will be well attenuated

by the pair of complex poles listed in Table 1. As a result, according to

(56),
60

or

Vj(s ) _ 2_

I-j( s) oj.2

This implies that the input impedance of the power stage is resistive at very

high frequencies because the Glj/V-jls) derived from (57c) is just a numerical

constant independent of frequency. This contradicts the observed phenomenon

that a power stage possesses an inductive input impedance when the perturbation frequency is relatively high. Tracing the path of the modeling process, this discrepancy can be attributed to the assumption that _uj is

piecewise-constant over one switching period.

Although it has been deduced that (56), with the constants calculated

from (57a) to (57e) , is not accurate enough at high frequencies, a dc analysis

shows that the numerical constants from (57a) through (57e) are the correct

gain constants for the various transfer functions from the state variables and

input variables to the average supply current.

Now examine the dependence of the supply current ij on the three input

variables, duty ratio a^t input voltage v-j, and load-disturbance current iw.

Although the duty ratio ad has been treated as a continuous variable for

small-signal perturbation at frequencies below half of the switching frequency, it is actually a discrete time variable. In each switching period,

there is only one single value for the duty ratio OQ» and hence also a single

value for ad• As a result, the dependence of the average supply current ij on

the duty ratio ad» taking both variables in the same switching period, is frequency independent. Therefore, the transfer function from the duty ratio to

the supply current is just a proportionality constant as defined in (57c) .
61

To exactly model how i'i will vary under a small-signal perturbation at

the supply voltage requires the modeling of _ui as a time-varying source over

one switching cycle, which then complicates the modeling excessively. Since

the supply current is identical to the inductor current times the turn ratio

during part of the switching cycle, the frequency dependence of the average

supply current on the supply voltage will be similar to the frequency dependence of the inductor current on the supply voltage. As a result, the frequency dependence of I-j(s) on V-j( s) will be well approximated if it is assumed

that the poles and zeros of

 G I-j/V-j( s

) are equal to the poles and zeros of

G

I X /V^ S^ . As crude an assumption as this may seem to be, it provides an

adequate approximation of

 e I-j/V-j( s^ when compared to experimentally measuread

data. Such an assumption may be justified by the following explanation. Had

G

I-i/V-j(s) been approximated with the numerical constant oj.2 given in (57d),

the input impedance function obtained from such an analysis would still be

quite accurate at dc and very low frequencies. Therefore, using qj.2 given in

(57d) as the dc gain of ^i/V^s) and the poles and zeros of

 GIX/V^ s) as the

poles and zeros of

 G I-j/V^s) g-jves a fair approximation of G Ii/V^s

^ from dc

up to half of the switching frequency. Similarly, the transfer function

G

If/Iy/ s) is approximated by taking the numerical constant qi3 from (57e) as

its dc gain constant and the poles and zeros of

 GIx/*w^ as its poles and

zeros. The six transfer functions describing the load voltage and the supply

current are listed in Table 2.

Substituting the matrices _AQN> .£OFF1>_§ON»

 and

 JOFFl

 from

 U9,CVU) to

(22.CVU) into (42), and comparing with (43),
62

Table 2. Six Transfer Functions Describing the Load Voltage and the Supply

Current for Continuous-Mmf-Mode Operation.

FUNCTION FORM

G ( 1 + S/Z)

(s/o)0)

NATURAL RESONANT FREQUENCY uo = V an 322 - «21

DAMPING RATIO

FUNCTION GAIN CONSTANT G ZERO

22

321 D13 - ail 023

FUNCTION

GVO/VC( s) s r C = P12

= Pll
63

) *

PaiX.SS ( U-OD) - a)g(l-aD)aoTS

iW ( 1 - -T

2 2 ' 2LS

and

f

l — ( {1-

aD) 2 I )

(toga[jrl-a)h(l-aD))2Ts p

- I'X.SS ( aj

( u

a)a 2 T

( oD - u)n(l-ao)

L

LS 2

Computing the derivatives according to (46a) through (46j) gives

ajaTs pRL(l-Ao) 2Ts

ail

 a

 - <oa ( 1 - — + gL ^ (46a,CVU)

(46b,CVU)
64

O

- Ix( kT$)

coe

L

S

 LS

Y(VI-VQ )

- - - ( 1 - u>hd

LS LS

(46c,CVU)

b12 = - (46d,CVU)

bl3 - - RLua ( 1 - ~ --- 1 - ) (46e,CVU)

(46f,CVU)

a22 = -[ wgAo*tahd-Ao) -- = - ; -

 +

 - - - ) (46g,CVU)

b21 = -[• ( 1

(

 1-

 AD) TS

(46h,CVU)

Y o)

b22 = r- ( AD - -=— -- u)hd-Ao)ADTs ) (46i,CVU)

L Z
65

b23 = ( u) e (l-AD) + - 21 --- 2 - ^ (46j ,CVU)

After substituting the matrices J^QN , _AQFF1»_§ON>

 and

 _§OFF1

 from

 (19.CVU) to

(22,CVU) and the matrices JJQN afrt MQFFI from (50.CVU) and (51.CVU) into (57 a)

to (57 e) ,

Pll = 0 (57a,CVU)

P12 = yAo ( 1 - ~ - ) (57b,CVU)

Y

2

(VI-VQ)ADTS

qil = Tlxt^d-AoagTs) + - ; - (57c,CVU)

(57d,CVU)

013 = 0 (57e,CVU)

5.2 Small-Signal Transfer Functions in the Discontinuous-Mmf Mode

In the discontinuous-nunf mode, the inductor current at the beginning of

each switching cycle, ix(kT$), is always equal to zero. As a result, i'x( kT$)

is no longer a dynamic variable. As far as signal flow is concerned, the

power stage can be modeled by the block diagram shown in Fig. 12.

Following the approximation of the derivatives shown in Section 5.1, and

using (33) derived in Section 4.2,
66

T

sl -

_AOFFl2tQFFl2 + _A

-TS ^ ( BoMtO BQNtOM M

> 1S.SS

AQFF2iOFFl( TS- ION-

(58)

Rewriting (58) ,

d

dt

vc.ss

.

 1x ,ss .

3

o

(59)

The derivative for ix,SS is

 zer° because i'x(kTs), the inductor current at

the beginning of a switching cycle, is identically equal to zero for every

switching cycle. Linearizing (59) at the equilibrium operating point, a

small-signal differential equation is obtained.

u • • « •

i. C ,SS C 955 u 1 W

- •• - - 31W

(60)

where the partial derivatives in (60) are evaluated at the equilibrium operating point. Rewriting (60) ,
67

Iw(s)

Yi(s)

Ad(s)

Vc(s) V0(S)

Fig. 12 Interconnection of functional blocks to characterize a converter

power stage in the discontinuous-mmf mode.
^

 v c,ss= an vc,ss

63

b

!2

 vi w (61)

where

3fy

an = (62a)

3fy

= —

3aD *

3JVI

3VT r

(62b)

3fV

b!3 = ~

3i W

(62c)

(62d)

and the *' s in Eqs. (62) indicate that the partial derivatives are evaluated

at the equilibrium operating point. Applying the Laplace transformation on

(61) gives

Thus

s Vc (s) = an Vc (s) + bii Ad(s) + bi2 V^ s) +

Vc (s)

s)

bi2 Vi(s)

( s - an )

(63)

and the three transfer functions relating Vc (s) and the inputs are listed in

Table 3.

As in the case of the continuous-mmf mode operation, the transfer function relating the load voltage and capacitor voltage is

GV0/VC( s)

 = - ( 1 + s rC C ) (64)
69

Table 3. Three Transfer Functions for the Ideal Capacitor Voltage Under

Discontinuous-Mmf-Mode Operation

s- an

GVC/IW(S) = s- an
70

Similar to the modeling of the supply current in the continuous- mmf mode,

the average supply current will be derived over one switching period as a

function of other circuit variables :

1 fkTs+anTs 1 f kT^a

1

I.k= TT- MONXS<ti dt+ —

S J kTs

 TS./ kTs+a

dt

kTs

(65)

Since the inductor current is equal to zero in the time interval between

kTs+aoTs+tQFFl

 and

 (k+l)Ts,

 tne

 contribution to the average supply current

during that time interval is also zero. Hence (65) does not contain the integral in the time interval from kTs+aoTs+tOFFl

 to

 (k+l)T$. Similar to (54)

in the continuous-mmf mode,

T *)Tr. ^fjn~\ITl i~U ^ /lS TSS

lOFFltQFFl2

(66)

r ^ON«O 's IONOD'-IS i

i'i>k = MON [ (aoi

 + —2— ^ -s

»

ss + —i— -1

 J

+ MOFFI | (

• 2T

Equation (66) can be rewritten as

1-I = f ( _*S,SS. <XD» vi, iw ) (67)

Linearizing (66), and then applying the Laplace transformation^

• Ii(s) = Pii Vc (s) + on Ad(s) + qi2 V^ s) + qia lyji s) (68)

where
71

Pll =

3 f

3VC.SS

o

,

 TOFFll _AQFF1TOFF12

 _

+ MOFFI ( -T— + —— + AONTOFFIAD ) [ i o ] i TS 2'S

r 1 AOFFITOFFI

+ MOFFI ( * —— + -

ONA° ^ -s( kT$)

lOFFlTQFFl

( +

TS J \ 3vC */

(69 a)

3f|

= — = HON

3<xD'*

( ( 1 + AQNAoTs ) JCS( kTS) + 1

MOFFI ( AQNTOFFI As(kTs)

 + IONTOFFI M

1 iOFFlTOFFl

AONAD ) x

)-,] (

3tQFFl

(69b)

<U2

3 f I r

= - I =

svir

 L

ilOFFllOFFlTOFFl21

.[ 1 0 0 0 ]T

J

1

7

, _§OFF1TOFF1

+

 ( - 7 T - + lONAD

S

1 /

Ml ( J \

tQFFl| \

(69c)
72

st

3f

L 0 1 0 0

ui

xs( KTS)

(69d)

As in the case of the continuous- ranf mode, the frequency dependence of

the supply current on the supply voltage is approximated by assigning the

poles and zeros of

 GIX/V^S

^ as the poles and zeros of

 GI-|/V^S

^ . However, in

the case of the discontinuous-mmf mode,

 GIX/V^S

^ is identically equal to zero

because ix(kT$) is always equal to zero. That is,

 GIX/V^S

^ is equal to the

numerical constant zero. Hence the transfer function ^I-j/V-j^ 5^ is also modeled as a proportionality constant without any poles or zeros. Similarly, the

transfer function

 G!-J/IW'S

^ is modeled as a numerical constant. Hence the

constants given by (69a) through (69d) need no further modification in the

modeling of the supply current.

In the case of the continuous- mmf mode, the diode conduction time

is simply tQFFl

 = TS ~ ^ON- In tne case of the discontinuous-mmf mode, however, tQFFl not only depends on tQN, but also depends on vj, VQ( kTs) , Vq , VQ,

and I'M. Hence under small-signal perturbation, topFl

 1S al s° a function of

not only od» but also v-j, vc, and iw. As a result, compared to their counterparts in (47) , Eqs. (69) have extra terms involving the partial derivatives of

tOFFl

 wvth

 respect to the other dynamical variables. The five transfer functions relating the output variables are also listed in Table 4.
73

Table 4. Five Transfer Functions Describing the Load Voltage and the Supply

Current for Discontinuous-Mmf-Mode Operation.

FUNCTION

= - ( 1 + s rc C )

Pll
74

Substituting the matrices ^QN> ^OFFl > _£OFF2 » j*ON » j*OFFl > and j*OFF2

(19.CVU) through (24.CVU) into (58) , and comparing with (59) ,

fvV = — o)

^

- *L»ai« ( 1 - - --- ) -

2LSTS LSTS

2L$T$

Unlike operation in the continuous- mmf mode, where tQFFl

 = ( 1 - ao )T$t the

diode conduction time toFFl in the discontinuous- mmf mode depends not only on

the duty ratio <XQ> but also on the supply voltage vj, capacitor voltage

VG( kT$) , and load- disturbance current iy. Although (39,CVU) gives an adequate

approximation on the equilibrium diode conduction time TQFFI as a function of

the other parameters, it is not sufficient to characterize the small- signal

toffi because all parasitic dissipative elements have been deleted in the

approximation process. To find out how toffi will vary under small- signal

perturbation, the large-signal equation (36,CVU) is re-examined.

,,

0 = tQFFl f *

)ffL l

tQFFl (

2LS 2LS 2

Y( vi-V)u ,

( - — - - ) (36.CVU)

Applying partial differentiation on (36,CVU) with respect to VG,SS>
75

P*OFF1

+ 2

2LS

L

s

Defining the expression

0

 -

L

S

2LS

Y( vi-V)u i

L

s

3VC,SS

3tQFFl

J

LS

,cvu)

then solving for the partial derivatives gives

3VC.SS

(72a,CVU)

All parameters in (71.CVU) and (72a,CVU) are obtained from the algorithms

presented in Section 4. Using similar methods,

3 tOFFl I

 TS

L

 s KT ( T°FF1

3aQ I* S

3tQFFl

M^H^H^MV

3VI

- Y{Vi-Vq)(l-UgAoTs) )

- ( 1 - ) )

(72b,CVU)

(72c,CVU)
76

3tQFFl

TOFFI TOFFI

L

S ' ^e

 LS

(72d,CVU)

With the partial derivatives in (72.CVU) derived and evaluated, the partial

derivatives of the function fy defined in (70,CVU) with respect to the dynamic

variables can be found. Defining

p

i = (73.CVU)

then

an =

afy

3vc,ssL3fV

3fV

b12 =

 ~

3fV

bis = —3 1W

p,

/

f

\

/

PI

\3V C

,SS

I \

L)

3tQFFl

(62a,CVU)

(62b,CVU)

(62c,CVU)

(62d,CVU)

After substituting the matrices ^QN> ^OFF1>10N>

 and

 JOFF1

 from

 (19.CVU) to

(22.CVU) and the matrices ^QN and j^OFFl

 from

 (50.CVU) and (51.CVU) into (69a)

to (69d) ,
77

Pll = 0

Y

2

(Vi-VQ)TON

= o

(69a,CVU)

(69b,CVU)

(69c,CVU)

(69d,CVU)
78

6. CLOSED-LOOP FUNCTIONS OF A REGULATED DC-TO-DC CONVERTER

Although the continuous- mmf mode and the discontinuous-mmf mode have been

treated separately in the derivation of the small- signal transfer functions,

they can be treated the same now by working with the functions derived in

various subsections of Section 5. Under small-signal perturbation, a closedloop regulated dc-to-dc converter can be represented in block-diagram form as

shown in Fig. 13. In the continuous-mmf mode, the functions characterizing

the power stage with its load are obtained from the procedures presented in

section 5.1. In the discontinuous-mmf mode, the functions

 GIx/AD^ ,

G

IX/V^S^,

 GI x /Iv/ s

^,and

 GI.j/Ix(s

) are equal to zero, and the remaining

functions characterizing the power stage with its load are obtained from the

procedures presented in section 5.2. Thus, the block diagram in Fig. 13 can

be used to represent a regulated dc-to-dc converter for both continuous-mmf

mode or discontinuous-mmf mode operation.

Sometimes the two transfer functions

 GVf/V0^s^ and GAd/Ve^s^ are combined

into one transfer function

G

VV0

(s) = C ' GVf/V0(s) ][ GA^V^S) ] (75)

where

 GA<j/V0( s) is called the transfer function of the controller.

6.1 Loop Gain of a Regulated DC-to-DC Converter

In the block diagram in Fig. 13, there is only one closed loop, which is

drawn in a heavy line. By tracing around that heavy-line path, the loop gain

T( s) of a regulated converter is found to be :

T(s) - C HVf/Vo(s) ][ GYO/VC(S) ][ GVc/Ad(s) ][
79

• CONVERTER POWER STAGE AND LOAD

FEEDBACK NETWORK

Fig. 13. Block diagram of a regulated dc-to-dc converter operated from a

stiff voltage source.
80

Using (75) ,

T(s) - C - GAd/Vo(s) 1C GVQ/VC(S) ][ GYc/Ad(s) ] {76)

where ^V0/VC^

 s^ and ^VC/A^ ^ are obtained following the procedures explained

in Section 5 .

6.2 Input Impedance of a Regulated DC-to-DC Converter

To find the input impedance Z-j( s) of a closed-loop regulated dc-to-dc

converter, the input admittance function Y-j( s) is derived first.

By taking the reciprocal of Y-j( s) , the input impedance function Zj( s) is

obtained. Using Mason's gain rule on the block diagram shown in Fig. 13, the

input admittance is derived as

(78)

where

) ] (78 a)

T2 * C GIl/Ix(s) ][ GIx/v.(s) ] {78b)

T3 * C SIl/Ix(s) ][ 6Ix/Ad(s) ][ GAd/V0(s) ][ GVo/vc(s) ][ GVC/VI( s) ] {78c)

T4 - C GIi/Ad(s) ][ G^/v^s) ][ GVo/Vc(s) ][ GVC/VI(S) ] (78d)

T5 - C 6Il/Vc(s) ][ GVc/Vi(s) ] {78e)

A * 1 + T(s) (78 f)
81

and T( s) is defined earlier in (76). Taking the reciprocal of Y-j( s) , the

input impedance function is

1 A

=

 T3 + T4 + T5

 (79)

6.3 Audio Susceptibility of a Regulated DC-to-DC Converter

The audio susceptibility S( s) is defined as the ratio of per cent change

in the load voltage to the per cent change in the supply voltage under smallsignal perturbation.

V0(s) / V0 Vx V0(s)

S( S) = Vi(s) / Yj

 = V0 Vi( s)

Using Mason's gain rule on the block diagram shown in Fig. 13

V

I C GVO/VC

(S) ]C

 ST/VT^

 ]

S(s) = - - (80)

YQ A

where A is defined in (78f) .

6.4 Output Impedance of a Regulated DC-to-DC Converter

The output impedance is defined as ZQ( s) = V0(s)/I^(s). Using Mason's

gain rule on the block diagram shown in Fig. 13,

V0(s) t GVo/Vc(s) 1C

- — 2L± - - (si)

where A is defined in (78 f) .
82

7. CONCLUSIONS

The modeling of the power stage of the four most commonly used energystorage dc-to-dc converters — the voltage step-up, current step-up, singlewinding current-or-voltage step-up, and the two-winding current-or-voltage

step-up converters ~ under small-signal 1 ow-frequency perturbation has been

presented in the Phase A Report and this Phase B Report. Starting from the

time domain, large-signal difference equations relating the state variables at

the beginning of each switching cycle are obtained by approximating state

transition matrices by second-order Taylor series. For low-frequency disturbances, the derivatives of the state variables can be approximated by the difference equations. For small-signal perturbation around the equil ibrium

operating point, a set of small-signal differential equations with constant

coefficients is obtained from the large-signal differential equations. Applying the Laplace transformation to the small-signal differential equations then

yields a set of small-signal transfer functions characterizing the power stage

of an energy-storage dc-to-dc converter. Combining these transfer functions

with the small-signal transfer functions of the controller gives valuable

information such as the loop gain, input impedance, output impedance, and

audio susceptibility of the closed-loop regulated dc-to-dc converter.

The procedure for obtaining various small-signal functions of a closedloop regulated dc-to-dc converter is illustrated, with the current step-up

converter on board the Dynamics Explorer Satellite as an example, in Section 8

of the Phase A Report. Following this illustration and the flow charts shown

in Figs. 2 of these two reports, various small-signal functions of. the voltage

step-up, current step-up, single-winding current-or-voltage step-up, and the

two-winding current-or-voltage step-up converters can be obtained.
83

8. REFERENCES

[1] R.C. Wong, H.A. Owen, Jr., and T.G. Wilson, "Derivation of Linearized

Transfer Functions for Switching-Mode Regulators, Phase A — Current

Step-Up and Voltage Step-Up Converters," prepared by Center for SolidState Power Conditioning and Control, Duke University, for NASA Goddard

Space Flight Center under Order No. S-71440B, September 29, 1981.

[2] Y. Yu, R.P. I wens, F.C. Lee, and L.Y. Inouye, "Development of A

Standardized Control Module for DC-DC Converters," NASA Contract Report,

NAS3-18918, prepared by TRW Defense and Space System Groups, August 30,

1977.

[3] R.C. Wong, G.E. Rodriguez, H.A. Owen, Jr., and T.G. Wilson, "Application

of Small-Signal Modeling and Measurement Techniques to the Stability

Analysis of an Integrated Switching-Mode Power System," IEEE PESC Record

- 1980, pp 104-118.

[4] J .J. D'Azzo and C.H. Houpis, "Linear Control System Analysis and Design :

Conventional and Modern," McGraw Hill, 1975, pp 457-460.
84

APPENDIX A

DEFINITION OF SYMBOLS AND ABBREVIATIONS

O.Q Duty ratio of the transistor switch (the ratio of the transistor

conduction time tQN to the switching period T$) , dimension!ess

C Nominal capacitance of the output filter capacitor, farad

CD Equations applying to the current step-up converter only

CVU Equations applying to both the single-winding and the two-winding

current-or-voltage step-up converters

f$ Switching frequency, hertz

I'D Diode current, ampere

ij Supply current, ampere

il,K Average supply current over the kth switching cycle, ampere

ig Transistor current, ampere

iQ.rms Effective value of the transistor current, ampere

iw Load disturbance current, ampere

i'x Current through the ideal inductance in the model of the energystorage reactor, ampere

L Nominal inductance of the single-winding reactor, henry

L$ Nominal secondary inductance of the two-winding reactor, henry

Np Number of turns in the primary winding of the two-winding reactor,

dimensionless

N$ Number of turns in the secondary winding of the two-winding reactor,

dimensionless

PQ Power loss in the transistor switch , watt

PR Power loss in the transistor switch due to an equivalent resistance,

watt

PV Power loss in the transistor switch due to the saturation voltage

drop, watt

re Equivalent series resistance (ESR) of the filter capacitor, ohm
85

rg Equivalent diode resistance, ohm

TO' Dynamic resistance of the diode as measured from a curve tracer, ohm

R|_ Load resistance, ohm

rg Equivalent transistor resistance, ohm

rg' Saturation resistance of the transistor as measured from a curve

tracer, ohm

rp Primary winding resistance of the two-winding reactor, ohm

r$ Secondary winding resistance of the two-winding reactor, ohm

rx Winding resistance of the single-winding reactor, ohm

SCVU Single-winding current-or-voltage step-up converter

TCVU Two-winding current-or-voltage step-up converter

_ui input vector, equal to [ vi i^ Vg Vg ]T

VQ Voltage across the ideal capacitance of the filter capacitor, volt

VD Voltage across the diode switch, volt

VD Break-point voltage of the diode, volt

VE Error voltage of the error comparator, volt

vp Feedback voltage of the feedback network, volt

vi Supply voltage, volt

VQ Load voltage, volt

VQ Voltage across the transistor, volt

VQ Saturation voltage of the transistor, volt

VREF Reference voltage for establishing the desired output voltage, volt

VU Equations applying to the voltage step-up converter only

jc$ state vector, equal to [ VQ i"x ]^

_*S,SS Stroboscopic value of the state vector at the beginning of each

switching cycle

Y Secondary-to-primary turn ratio, equal to N$/Np, dimensionless
86

APPENDIX B

EQUATIONS DEFINING THE CONSTANTS AND THE COEFFICIENTS

IN THE TRANSFER FUNCTIONS

For SCVU, 1 et Y = 1, LS = L, rp = rx, and r$ = r\.

RL

P - 5- (3)

rc+

 RL

0 - - »a VC( kTS)

Rl«a (l-Ao) ( 1 - w

( i

vD

C( re +

NS

- (5)

Np

Y

2 ( rp + rn )

o)a= : — (7)

rs + H)

 + P

«h- r:

(28.CVU)

(30,CVU)
87

Y(VI-YQ)AD TS

2LS

0

 ' ** — --- — ( — • (u^

h)

Iw pR|_aja Y(V

r

 -q

AD (l-»hTs) ( - [ -- coe lw+

(2-cohTs)

^ L

2LS

0 - TOFF1 ( —- - * -_ --- _ J

(32.CVU)

.

TOFFI ( - - --- - --- — * «e iw )

Y(Vi-VQ)ToN(2-WgTON)

H- ( - - - - ) (38.CVU)

Y(VI-VQ)T

-

ON Y(VI-VQ)ADTS

 = - :: - (39,cvu)

v0 + VD v0 + VD

AD= \ — — (4o,cvu)
88

pRLTOFFl2

_ °jls _ .-eW 1

L

 2 2TS

 J 2LSTS 2 2TS J

ail - -«a ( 1 - — + - - ^ (46a,CVU)

ai2 = RLua ( d-Ao) - a)g(l-AD)ADTs -- 5 - ) (46b,CVU)

(46c,CVU)

b12 = . (46d,CVU)

b!3 - - RL«a ( 1 - - --- - ^ (46e,CVU)

p

321 =- ( (1-Ao) -- --- - ^ (46f,CVU)

a22 = -( ugAoWl-AD) - — = - g -

 +

 - 2L - ^ (46g,CVU)
89

pV0

b

21 = — ( - uh(l-Ao)Ts )

p RLOJ a( 1- AQ) TS

( ug - o>h • (ug-a)h)(a)gAg+(oh(l-AQ))Ts -- j - )

— IN ( o»e "

Y(VI-VQ )

VD

) + — ( 1 - u)

(46h,CVU)

(461.CVU)

(46J.CVU)

(57a,CVU)

P12 =

Y

2AD2

TS

~

Y

2

(Vi-VQ)ADTS

(57b,CVU)

(57c,CVU)

(57d,CVU)

(57e,CVU)

an =

3VC.SS 2LsTS

(62a,CVU)

3fV

(62b,CVU)
90

b12 =

3 fV

bi-1 3j = —

 aiw l

PII = o

-

 Y2(Vl'vQ)T°N

LS

= 0

n OT . U = 2TQFF1 ( 2LS

LS

8tOFFl P

=

TS f ^ ( TOFFI

3VI

"eTOFFl2

"2Ts~

2LS

Y(Vl-VQ)a>hTQN

LS

( u) a+u h) TOFF1

(62c,CVU)

(62d,CVU)

(69a,CVU)

(69b,CVU)

(69c,CVU)

(69d,CVU)

(71.CVU)

(72a,CVU)

- Y (Vi-VQ)(l- M gADTS )

(72b,CVU)

(72c,CVU)
9

1

TOFF

1

 TOFF

1

L

S

L

S

(72d,CVU)

Y(VI-VQ)T

ON -

(73.CVU)