Parametric Average Value Modeling of Flyback Converters in ... · A flyback converter is a switching power supply topology widely used in low power applications such as chargers and

PARAMETRIC AVERAGE VALUE MODELING OF FLYBACK CONVER TERS IN

CCM AND DCM INCLUDING PARASITICS AND SNUBBERS

by

Mehmet Sucu

B.A.Sc., Marmara University, 2000 M.A.Sc., Marmara University, 2003

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in

THE FACULTY OF GRADUATE STUDIES

(ELECTRICAL AND COMPUTER ENGINEERING)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

October 2011

© Mehmet Sucu, 2011

ii

Abstract

Modeling of switched-mode DC-DC converters has been receiving significant interest due to

their widespread applications. Averaged modeling is the most common approach (and tool)

that has been used to analyze dynamic performance of converter circuits. Specifically, state-

space averaged models are widely used because of their simplicity and generality. However,

as has been shown in the literature, the challenges of directly applying this approach to

predict the discontinuous variables (states) and include the parasitics and losses have limited

application of this approach to a wider range of converter circuits. The recently introduced

parametric average value models (PAVM) has a potential to overcome this problem.

In this Thesis, first of all a second-order flyback converter has been investigated. An

analytical solution of state-apace averaging and small-signal analysis of the flyback converter

in continuous conduction mode (CCM) and discontinuous conduction mode (DCM) is given

without and with parasitics. The PAVM methodology has been applied to the second-order

model to overcome the problem of discontinuous state during the DCM.

The snubber circuits in flyback converter have also been investigated. Appearance of

snubbers in the model introduces a problem on the output voltage besides improving the

efficiency prediction. It is shown that with the snubbers the conventional state-space

averaging cannot predict the output voltage correctly in CCM and DCM. To solve this

problem the model is partitioned into two different sub-circuits: i) switching sub-circuit

circuit; and ii) non-switching sub-circuit. Thereafter it becomes possible apply the averaging

on the switching sub-circuit only.

Finally, a full-order flyback converter with two RC snubber circuits and all the basic

parasitics is considered. The PAVM methodology has been extended to this class of

switching converter for the first time. It is shown that including the snubbers and parasitics

significantly improves the model accuracy in terms of predicting converter efficiency, which

represents an appreciable improvement over all previously existing average models. The

proposed model has been verified with detailed simulations and hardware measurements.

iii

Table of Contents

Abstract .................................................................................................................................... ii

Table of Contents ................................................................................................................... iii

List of Tables ........................................................................................................................... v

List of Figures ......................................................................................................................... vi

List of Abbreviations ............................................................................................................. ix

Acknowledgements ................................................................................................................. x

Chapter 1 : Introduction ........................................................................................................ 1

1.1 PWM DC-DC Converters ..................................................................................................... 1

1.2 Flyback Converters ............................................................................................................... 1

1.3 Average Value Modeling ...................................................................................................... 2

1.4 Parametric Average-Value Modeling ................................................................................... 4

1.5 Motivations and Objectives .................................................................................................. 4

Chapter 2 : Second Order Flyback Converters ................................................................... 5

2.1 Small-Signal AC Model and State-Space Averaging without Parasitics in CCM ................ 5

2.2 State-Space Averaging in DCM without Parasitics ............................................................ 15

2.3 Small-Signal AC Model and State-Space Averaging with Basic Parasitics in CCM ......... 19

2.4 State-Space Averaging with Parasitics in DCM ................................................................. 32

2.5 Parametric Average Value Modeling in CCM and DCM ................................................... 36

2.5.1 Correction Term ............................................................................................................. 37

2.5.2 Model Implementation .................................................................................................... 38

2.5.3 Case Studies .................................................................................................................... 43

2.5.3.1 Time domain .......................................................................................................... 43

2.5.3.2 Frequency domain .................................................................................................. 45

Chapter 3 : Analysis of Flyback Converter with Snubber Circuits ................................. 47

3.1 Fifth –order Flyback Converter with Snubbers ................................................................... 47

3.2 State-Space Averaging Phenomena with the Snubbers ....................................................... 49

Chapter 4 : Full-order Flyback Converter ......................................................................... 56

4.1 State-Space Averaging in CCM .......................................................................................... 56

4.2 State-Space Averaging in DCM .......................................................................................... 59

iv

4.3 Parametric Average Value Modeling in CCM and DCM ................................................... 62

4.3.1 Model Implementation .................................................................................................... 62

4.4 Case Studies ........................................................................................................................ 67

4.4.1 Time Domain .................................................................................................................. 67

4.4.2 Frequency Domain.......................................................................................................... 69

4.4.3 Efficiency Results ........................................................................................................... 70

Chapter 5 : Conclusion ......................................................................................................... 72

5.1 Future Work ........................................................................................................................ 72

Bibliography .......................................................................................................................... 74

Appendices ............................................................................................................................. 78

Appendix A. The Converters Circuit Parameters .......................................................................... 78

A.1 Second-order Flyback Converter Parameters without Parasitics in CCM ...................... 78

A.2 Second-order Flyback Converter Parameters without Parasitics in DCM ...................... 78

A.3 Second-order Flyback Converter Parameters with Parasitics in CCM ........................... 78

A.4 Second-order Flyback Converter Parameters with Parasitics in DCM ........................... 79

A.5 Fifth-order Flyback Converter Parameters in CCM ....................................................... 79

A.6 Full-order Flyback Converter Parameters in CCM ......................................................... 79

A.7 Full-order Flyback Converter Parameters in DCM ........................................................ 80

Appendix B. Flyback Converter Circuit Diagram ......................................................................... 81

v

List of Tables

Table 4.1 Efficiency comparison of the average-value models .............................................. 71

vi

List of Figures

Figure 2.1 (a) Assumed circuit for the second order Flyback converter without

parasitics; (b) Circuit during subinterval 1; (c) Circuit during

subinterval 2. .......................................................................................................... 6

Figure 2.2 The inductor current ................................................................................................ 6

Figure 2.3 Inductor current and capacitor voltage of second order Flyback

converter without parasitics in CCM. .................................................................. 14

Figure 2.4 Capacitor voltage of second order Flyback converter without parasitics

in CCM. ................................................................................................................ 14

Figure 2.5 Second-order Flyback converter without parasitics during third

subinterval in DCM. ............................................................................................. 15

Figure 2.6 Magnetizing current in DCM for the load 2500R= Ω . ......................................... 16

Figure 2.7 Inductor current and capacitor voltage of second order Flyback

converter without parasitics in DCM. .................................................................. 18

Figure 2.8(a) Second-order Flyback converter with parasitics; (b) Circuit during

subinterval 1 (c) Circuit during subinterval 2. ..................................................... 19

Figure 2.9 Inductor current, capacitor voltage and output voltage of second order

Flyback converter with parasitics in CCM. ......................................................... 31

Figure 2.10 Output voltage of second order Flyback converter with parasitics in

CCM. .................................................................................................................... 31

Figure 2.11 Second-order Flyback converter with parasitics during third

subinterval in DCM. ............................................................................................. 32

Figure 2.12 Inductor current, capacitor voltage and output voltage of second

order Flyback converter with parasitics in DCM. ................................................ 35

Figure 2.13 Variable 3d as a function` of duty-cycle ( )1d and the load ( )R . ....................... 40

Figure 2.14 The correction term 1m as a function of duty-cycle ( )1d and the load

resistance ( )R . ..................................................................................................... 40


resistance ( )R . ..................................................................................................... 41

vii

Figure 2.16 Implementation of the parametric average-value model. .................................... 42

Figure 2.17 Simulated inductor current, capacitor voltage and output voltage of

the second order Flyback converter with parasitics in DCM. .............................. 44

Figure 2.18 Transients in inductor current, capacitor voltage and output voltage

of the second order Flyback converter due to the step change in load. ............... 45

Figure 2.19 Control-to-output transfer function of the second-order Flyback

converter evaluated at 717.05R= Ω and 1 0.381d = . ............................................. 46

Figure 3.1 Fifth-order Flyback converter circuit. ................................................................... 47

Figure 3.2 Measured transformer secondary voltage: (a) without the diode

snubber; and (b) with the diode snubber. ............................................................. 48

Figure 3.3 Simulated output filter capacitor voltage and the output voltage of the

fifth-order Flyback converter with snubbers in CCM. ......................................... 50

Figure 3.4 The predicted secondary current and the diode snubber capacitor

voltage of fifth-order Flyback converter in CCM. ............................................... 51

Figure 3.5 Forth-order Flyback converter without diode snubber. ......................................... 52

Figure 3.6 The simulated output filter capacitor and output voltage of the forth-

order Flyback converter without the diode snubber in CCM. .............................. 53

Figure 3.7 Modified fifth-order Flyback converter circuit. .................................................... 54

Figure 3.8 Proposed state-space averaged model of the fifth-order Flyback

converter using two sub-circuits and sub-models. ............................................... 54

Figure 4.1 Full-order Flyback converter circuit. ..................................................................... 56

Figure 4.2 Predicted state variables of full-order Flyback converter in CCM. ....................... 58

Figure 4.3 Simulated state variables of the full-order Flyback converter in DCM. ............... 60

Figure 4.4 Simulated transformer secondary current of the full-order Flyback

converter in DCM. ............................................................................................... 61

Figure 4.5 The diode current waveform. ................................................................................ 63

viii

Figure 4.6 Variable 3d as a function of the duty-cycle ( )1d and the load

resistance ( )R . ..................................................................................................... 64


resistance ( )R . ..................................................................................................... 65

Figure 4.8 The correction term 3m as a function of the duty-cycle ( )1d and the

load resistance ( )R . ............................................................................................. 65


resistance ( )R . ..................................................................................................... 66


resistance ( )R . ..................................................................................................... 67

Figure 4.11 Measured and simulated output voltage, primary and secondary

current in CCM at constant duty-cycle. ............................................................... 68

Figure 4.12 Simulated output voltage, primary and secondary current during the

transient from DCM to CCM due to the step change in load. ............................. 69

Figure 4.13 Control-to-output transfer function of the full-order Flyback

converter evaluated at 717.05R= Ω and 1 0.381d = ............................................... 70

ix

List of Abbreviations

CCM Continuous Conduction Mode

DCM Discontinuous Conduction Mode

PAVM Parametric Average Value Modeling

PWM Pulse Width Modulation

x

Acknowledgements

I would like to express my appreciation to my supervisor, Dr. Juri Jatskevich, whose strong

academic support have been the most precious assets to my studies and research. I am also

very grateful for the partial financial support that has been made available to me through the

NSERC under the Discovery Grant.

I also like to thank Dr. William Dunford and Dr. Shahriar Mirabbasi who have accepted to be

the committee members and dedicated their time and effort for reading this thesis and

providing their constructive and valuable comments.

My special thanks go to previous and current members of the Power and Energy Systems

Group at UBC who have always supported me and gave their valuable insights into my

research.

1

Chapter 1 : Introduction

1.1 PWM DC-DC Converters

Switch mode dc-dc converters have become an essential element of many commercial and

military applications. Due to their high efficiency, light weight and relatively low cost, the

switching dc-dc converters have generated a significant research interest in the area of their

modeling, analysis, and control. Among various types of dc-dc converters, the Pulse-Width

Modulated (PWM) converters constitute by far the largest group. They have displaced

conventional linear power supplies even at low power levels. Switch-mode dc-dc converters

can be categorized as non-linear periodic time-variant systems due to their inherent switching

operation. The topology depends on instantaneous states of the power switches. This is what

makes their modeling a complex task. Nevertheless, accurate analytical models of PWM dc-

dc converters are essential for the analysis and design in many applications e.g., automobiles,

aeronautics, aerospace, telecommunications, submarines, naval ships, mainframe computers

and medical equipments. Many efforts have been made in the past few decades to model dc-

dc converters and several new models have been proposed. These models are widely used to

study the static and dynamic characteristics of the converters as well as to design their

control systems to achieve specific regulation characteristics [1-8].

1.2 Flyback Converters

A flyback converter is a switching power supply topology widely used in low power

applications such as chargers and PC power supplies. It is basically an implementation of

buck-boost converter and has transformer isolation. The most important advantage is that it

becomes possible to have multiple outputs with a simple modification on the transformer

(adding another secondary winding) and adding few extra components (a diode and a filter

capacitor). Another important advantage is that it has natural isolation between input and

output, which is required by many standards for design of power supplies [9-13].

A detailed model of a flyback converter can be easily implemented using widely available

simulation packages (e.g. Matlab/Simulink, ASMG, PLECS, etc.) [14-16]. Detailed models

are often used during the design process as such models have all the required information to

2

calculate the exact switching transients and component stresses and characteristics. But the

large computation time required for such detailed switching models makes them less

applicable for system-level studies. Instead, the average-value modeling has been used very

effectively for the system-level analysis and studies, wherein the effects of fast switching are

neglected or averaged with respect to the switching interval. A classical state-space averaged

model of a flyback converter [10] considers only the switch losses without any snubber

circuits and has the simplest first-order transformer approximation. A simplified linear circuit

model for obtaining dc and small-signal circuit model is given in [17], which has the basic

parasitics but does not include any snubber circuits and transformer primary and secondary

copper losses. A dc and small-signal circuit model models for a flyback converter operating

in CCM can be found in [12], which has the basic parasitics but again does not include any

snubber circuits and has only a the simplest transformer model without the primary and

secondary losses.

1.3 Average Value Modeling

The averaged-value modeling, wherein the effects of fast switching are “averaged” over a

switching interval, is most frequently applied when investigating power-electronics-based

systems. Continuous large-signal models are typically non-linear and can be linearized

around a desired operating point. Averaged models of dc-dc converters offer several

advantages over the switching models. These advantages are: i) straightforward approach in

determining local transfer-functions; ii) faster simulation of large-signal system-level

transients; and iii) use of general-purpose simulators to linearize converters for designing the

feedback controllers.

A typical switched-inductor dc-dc converter can operate in two modes. One is the

Continuous Conduction Mode (CCM) in which inductor current never falls to zero, and the

second mode is Discontinuous Conduction Mode (DCM) allowing inductor current to

become zero for a portion of switching period. The DCM typically occurs at light loads and

differs from CCM since this mode results into three different switched networks over one

switching cycle (as opposed to two switched networks in the case of CCM operation).

Models for PWM converters operating in CCM based on well-known state-space averaging

3

technique were first introduced in 1970’s [18]. Since then, several circuit-oriented averaging

approaches have also been proposed [3, 19]. Numerous method have been developed for the

average value modeling of PWM dc-dc converters in DCM such as reduced-order state-space

averaging [20], reduced-order averaged-switch modeling [7], equivalent duty ratio models

[3], loss-free resistor model [10], full-order averaged-switch modeling [21], and full-order

state-space averaging [4].

Average value models may be categorized as resulting system of equations (reduced-order

vs. full order); or by derivation methodology (sampled data modeling, circuit averaging,

state-space averaging). The full-order as well as reduced-order models can be obtained by

averaging approaches including sampled data modeling, circuit averaging or state space

averaging. The conventional reduced order models treat the discontinuous variable as a

dependent variable and eliminate its dynamic from the state equations. The elimination of

fast/discontinuous variable is undesirable for application in which this variable is used for

control purposes, which limits the range of applications of such reduced-order models.

State-space averaging is based on the classical averaging theory and involves manipulation of

state-space equations of a converter system. First, a state-space representation of converter is

obtained for each topology and subinterval. Then, the obtained piece-wise linear equations

are weighted by the corresponding time subinterval length and added together. State-space

averaging has been demonstrated to be an effective method to analyze PWM converters.

Analytical averaging, however, is based on so-called small-ripple approximation. Most of the

previous works on averaging methods were derived for a specific ideal topology. In addition,

derivation of state-space average-value model, the equivalent series resistance (ESR) of

circuit components are often neglected and the state variables are considered as linear

segments. Such assumptions result in inaccuracy of the corresponding time constants as well

as the waveforms. If the losses due to the switch and/or active elements are taken into

account, whereby the linear shape of the current waveform would change into exponential

form, the analytically derived models would become significantly more complicated and

challenging. The analytical derivation also becomes more complicated when the number of

energy storage elements (inductors and capacitors) is high.

4

1.4 Parametric Average-Value Modeling

Parametric average-value modeling methodology has been set forth by the UBC researchers.

This methodology has been successfully demonstrated for synchronous machine-converter

systems in [22, 23]. The major point of this approach is to use the detailed simulation for

numerically calculating the key relationships needed for constructing the average-value

model of a certain well-defined form. In doing so, the effect of parasitics included in the

detailed model becomes automatically included in the numerically constructed parametric

functions, which are then used for the state-variable-based average-value models. This

approach also reduces the effort of the model developer and avoids many complicated

analytical derivations. This method has been extended to the PWM dc-dc converters in [24-

27] based on corrected full-order averaged models proposed for circuit averaging [28] and

state-space averaging [4] that very accurately capture the high-frequency dynamics of fast

state variables.

1.5 Motivations and Objectives

The detailed models of PWM dc-dc converters are widely used for design purposes but they

are not desirable for system level studies due to very high computational times, wherein it

has been always required to have more efficient average models. Although there are various

averaged models of the flyback PWM converter available, none of the previously established

models have full order and include all realistic parasitics. Most of the models use the simplest

transformer representation and none of them include the snubber circuits. At the same time,

the snubber circuits are very important components of the flyback PWM converters and have

significant effect on the converter dynamics and efficiency.

This Thesis makes an original contribution and extends the parametric average-value

modeling the flyback converters. The considered converter model includes all the basic

parasitics and high order transformer model with primary and secondary resistances and

leakage inductances. The propose model also includes two RC (resistance and capacitance)

snubber circuits to protect the switch and the diode during the on-off operation. To the best

of our knowledge, this has not been done in any published research on this subject.

5

Chapter 2 : Second Order Flyback Converters

In this Chapter, we consider and approximate (simplified) circuit of the flyback converter,

wherein only two energy storage elements are considered, hence second order converter.

Such approximate converter circuit has been used in the literature for carrying out basic

analysis and averaging methods. A number of modeling techniques have appeared in the

literature, including the current injected approach [19], circuit averaging [7, 21, 29], and

state-space averaging [18] method.

2.1 Small-Signal AC Model and State-Space Averaging without Parasitics in CCM

The state-space description of dynamical systems is a basis of modern control theory. The

state-space averaging method makes use of this description to derive the small-signal

averaged equations of the PWM switching converters. The state-space averaging method is

otherwise identical to the procedure of deriving the small-signal ac model. A benefit of the

state-space averaging procedure is its results: a small-signal averaged model that can always

be obtained, provided that the state equations of the original converter can be written.

Obtaining a small-signal ac model of a basic switched converter circuit, such as buck, boost

without parasitics, can be readily achieved using analytical derivations. But when the

converter circuit has parasitics, it becomes almost impossible and impractical to derive the

higher order state equations. In this case, the state-space equations can be obtained from the

detailed model by using commercially available simulation packages [16, 30], and then used

the state-space description (matrices) to establish the small-signal model [10] (see

Section7.3.2).

In this Section, a small-signal ac model will be derived for a second-order flyback converters

without parasitics. Based on that, the state-space equations will be derived.

A second order flyback converter without parasitics is shown in Figure 2.1(a). Here, n is the

turn ratio of the transformer( )1 2N N .

6

+

mL

n( )i t( )gi t( )ci t

( )v t( )Lv t

C R

D

Mosfet

+

mL


( )v t( )Lv t

C R

+

mL


( )v t( )Lv t

C R

( )a

( )b

( )c

( )gv t

( )gv t

( )gv t

Figure 2.1 (a) Assumed circuit for the second order Flyback converter without parasitics; (b) Circuit

during subinterval 1; (c) Circuit during subinterval 2.

The inductor current ( )i t in CCM is shown in Figure 2.2. During the first interval, Figure

2.1(b), the inductor stores some energy and transfers this energy to the secondary side during

the second interval, Figure 2.1(c).

0.4

0.6

0.8

1

1.2

1.4

1.6

I (A

mp.)

0.999991 0.999993 0.999995 0.999997

Time (s)

1 sd T2 sd T

sT

Figure 2.2 The inductor current

7

During the first subinterval, when the MOSFET conducts and the diode is off, the circuit

reduces to Figure 2.1(b). The inductor voltage ( )Lv t , capacitor current ( )ci t , and converter

input current ( )gi t can be expressed as follows:

( ) ( )L gv t v t= (2.1)

( ) ( )c

v ti t

R= − (2.2)

( ) ( )gi t i t= (2.3)

Applying the small ripple approximation [10] and replacing the voltages and currents with

their respective average values, we obtain

( ) ( )s

L g Tv t v t= (2.4)

( )( )

sT

c

v ti t

R= − (2.5)

( ) ( )s

g Ti t i t= (2.6)

During the second subinterval, MOSFET is off and the diode conducts, which results in

circuit of Figure 2.1(c). The inductor voltage ( )Lv t , capacitor current ( )ci t , and converter

input current ( )gi t are given by

( ) ( )Lv t v t n= (2.7)

( ) ( ) ( )c

v ti t i t n

R

= − +

(2.8)

( ) 0gi t = (2.9)

The small ripple approximation leads to

( ) ( )s

L Tv t v t n= (2.10)

( ) ( )( )

s

s

T

c T

v ti t i t n

R= − − (2.11)

( ) 0gi t = (2.12)

The average inductor voltage now can be found by averaging the subintervals over one

complete switching period. The result is

8

( ) ( ) ( ) ( ) ( )1 2s ss

L gT TTv t v t d t v t nd t= + (2.13)

This leads to the following state equation for the average inductor current

( )

( ) ( ) ( ) ( )1 2s

ss

T

m g TT

d i tL v t d t v t nd t

dt= + (2.14)

The average capacitor current now can be found by averaging the subintervals over one

switching period resulting in the following:

( )( )

( ) ( ) ( )( )

( )1 2 2s s

s s

T T

c T T

v t v ti t d t i t nd t d t

R R= − − − (2.15)

After collecting terms, the average capacitor current becomes

( )( )

( ) ( )( ) ( ) ( )1 2 2

1

s

s s

Tc T T

v ti t d t d t i t nd t

R= − + −

1442443 (2.16)

This leads to the following state equation for the average capacitor voltage

( ) ( )

( ) ( )2s s

s

T T

T

d v t v tC i t nd t

dt R= − − (2.17)

The converter input current now can be found by averaging the subintervals over one

switching period. The result is

( ) ( ) ( )1 0ss

g TTi t i t d t= + (2.18)

The equation (2.14), (2.17), and (2.18) are nonlinear set of differential equations. In order to

construct the converter small-signal ac model, the next step is to perturb and linearize these

equations. Here, we assume that the converter input voltage ( )gv t and duty cycle ( )1d t can

be expressed as quiescent values plus small ac variations, as follows

( ) ( )ˆs

g g gTv t V v t= + (2.19)

( ) ( )1 1 1ˆd t D d t= + (2.20)

In response to these inputs, and after all transients have decayed, the average converter

variables can also be expressed as quiescent values plus small ac variations,

( ) ( )ˆsT

i t I i t= + (2.21)

( ) ( )ˆsT

v t V v t= + (2.22)

9

( ) ( )ˆs

g g gTi t I i t= + (2.23)

With these substitutions, the large-signal averaged inductor equation (2.14) becomes

( )( ) ( )( ) ( )( ) ( )( ) ( )( )1 1 2 1

ˆˆ ˆˆ ˆm g g

d I i tL V v t D d t V v t n D d t

dt

+= + + + + − (2.24)

Upon multiplying this expression out and collecting terms, we obtain

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

1 2 1 1 1 2

1 ( )

1 1

2 ( )

ˆˆ ˆˆ ˆ

ˆ ˆˆ ˆ

m g g g

Dcterms st order acterms linear

g

nd order acterms nonlinear

di tdIL V D VnD V d t v t D Vnd t v t nD

dt dt

v t d t v t nd t

+ = + + + − +

+ −

1442443 14444444244444443

14444244443

(2.25)

As usual, this equation contains three types of terms. The dc term contain no time-varying

quantities. The first order ac terms are linear functions of the ac variations in the circuit,

while the second order ac terms are functions of the products of the ac variations. At this

point, we make an assumption that the ac variations are small in magnitude compared to the

dc quiescent values,

( )( )

( )( )( )

1 1

ˆ

ˆ

ˆ

ˆ

ˆ

g g

g g

v t V

d t D

i t I

v t V

i t I

<<<<<<

<<

<<

(2.26)

If the small signal assumptions (2.26) are satisfied, then the second-order terms are much

smaller in magnitude than the first-order terms and hence can be neglected. The dc terms

must satisfy

1 20 gV D VnD= + (2.27)

The first order ac terms must satisfy

( ) ( ) ( ) ( ) ( )1 1 1 2

ˆˆ ˆˆ ˆm g g

di tL V d t v t D Vnd t v t nD

dt= + − + (2.28)

This result is the linearized equation that describes ac variations in the inductor current.

10

Upon substation of (2.19), (2.20), (2.21), (2.22), and (2.23) into the averaged capacitor

voltage state equation (2.17), we obtain the following

( )( ) ( )( ) ( )( ) ( )( )2 1

ˆ ˆ ˆˆd V v t V v t

C I i t n D d tdt R

+ += − − + − (2.29)


( ) ( ) ( ) ( )

( ) ( )( )

2 1 2

1 ( )

1

2 ( )

ˆ ˆ ˆ ˆ

ˆˆ

Dcterms st order acterms linear


dv t v tdV VC InD Ind t i t nD

dt dt R R

i t nd t

+ = − − + − + −

+

1442443 1444442444443

1442443

(2.30)

Here, we neglect the second-order terms. The dc terms of equation (2.30) must satisfy

20V

InDR

= − − (2.31)

The first-order ac terms of (2.30) lead to the small-signal ac state equation for capacitor

voltage

( ) ( ) ( ) ( )1 2

ˆ ˆ ˆ ˆdv t v tC Ind t i t nD

dt R= − + − (2.32)

Substation of (2.19), (2.20), (2.21), (2.22), and (2.23) into (2.18) results in the following:

( ) ( )( ) ( )( )1 1ˆˆ ˆ

g gI i t I i t D d t+ = + + (2.33)


( ) ( )

( ) ( )( ) ( ) ( )( )1 1 1 1

1 ( ) 2 ( )

ˆ ˆˆ ˆ ˆg g

Dctermsst order acterms linear nd order acterms nonlinear

I i t ID Id t i t D i t d t+ = + + +1442443 14243

(2.34)

The dc term must satisfy

1gI ID= (2.35)

We neglect the second-order terms in (2.34), leaving the following linearized ac equation

( ) ( ) ( )1 1ˆˆ ˆ

gi t Id t i t D= + (2.36)

This result represents the low-frequency ac variations in the converter input current.

The equation of the quiescent values, (2.27), (2.31), and (2.35) are collected below,

11

1 2

2

1

0

0

g

g

V D VnD

VInD

RI ID

= + = − −

=

(2.37)

For given quiescent values of the input voltage gV , and duty cycle 1D , the system of

equations (2.37) can be evaluated to find the quiescent output voltage V , inductor current I ,

and input current dc component gI . The results are then inserted into the small-signal ac

model. The final small signal ac model is summarized below,

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 1 2

1 2

1 1

ˆˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ

ˆˆ ˆ

m g g

g

di tL V d t v t D Vnd t v t nD

dtdv t v t

C Ind t i t nDdt R

i t Id t i t D

= + − +

= − + − = +

(2.38)

The final step is to construct an equivalent circuit of (2.37) and (2.38) using commercially

available simulation tools.

Let us now apply the state-space averaging method to the second order flyback converter

circuit depicted in Figure 2.1(a). The independent state variables of the converter are the

inductor current ( )i t and the capacitor voltage ( )v t , which form the state vector

( ) ( )( )

i tx t

v t

=

(2.39)

The input voltage ( )gv t , is an independent source which should be placed in the input vector,

( ) ( )gu t v t = (2.40)

To model the converter as a system with input and output, we need to find the converter input

current ( )gi t . To calculate this dependent current, it should be included in the output vector

( )y t . Therefore,

( ) ( )gy t i t = (2.41)

Note that it’s not necessary to include the output voltage ( )v t in the output vector since the

voltage is already included in the state vector ( )x t .

12

Next, let us write the state equations for each subinterval. When the switch is on and the

diode is off, the converter circuit of Figure 2.1(b) is obtained. The inductor voltage, capacitor

current, and converter input current are

( ) ( )m g

di tL v t

dt= (2.42)

( ) ( )dv t v t

Cdt R

= − (2.43)

( ) ( )gi t i t= (2.44)

Similar to (2.1), (2.2), and (2.3), after organizing (2.42), (2.43), and (2.44), these equations

can be written in the following standard state-space form:

( )

( )

( )

( )( )( )

( )( )

1 1

10 0

10

0m g

u tx t

A Bdx t

dt

di ti tdt L v tv tdv t

RCdt

= + −

14243123

1424314243

(2.45)

( )( )

[ ]

( )( )( )

[ ]

( )( )1 1

1 0 0g g

C Ey t u tx t

i ti t v t

v t

= +

123 14243123

(2.46)

So, (2.45) and (2.46) define the state-space equation for the first subinterval.

When the MOSFET is off and the diode is on, the converter circuit of Figure 2.1(c) is

obtained. For the second subinterval, the inductor voltage, capacitor current, and converter

input current are given by

( ) ( )m

di tL v t n

dt= (2.47)

( ) ( ) ( )dv t v t

C i t ndt R

= − − (2.48)

( ) 0gi t = (2.49)

After organizing terms, the following state-space equation for the second interval is obtained:

13

( )

( )

( )

( )( )( )

( )( )

2

2

00

01m

g

u tBx t

Adx t

dt

ndi ti tLdt v tv tdv t n

C RCdt

= + − −

14243123

144244314243

(2.50)

( )( )

[ ] ( )( )( )

[ ]

( )( )22

0 0 0g g

ECy t u tx t

i ti t v t

v t

= +

123123 14243123

(2.51)

The next step is to evaluate the averaged state-space model, which is achieved as follows:

2

1 1 2 2 1 2

2

0 00 0

10 1 1

m m

nDn

L LA A D A D D D

n nDRC

C RC C RC

= + = + = − − − − −

(2.52)

1

1 1 2 2 1 2

10

00 0m m

D

L LB B D B D D D

= + = + =

(2.53)

[ ] [ ] [ ]1 1 2 2 1 2 11 0 0 0 0C C D C D D D D= + = + = (2.54)

[ ] [ ] [ ]1 1 2 2 1 20 0 0E E D E D D D= + = + = (2.55)

So the final averaged state-space model of the second order Flyback converter without

parasitics in CCM is

( )

( )( )( ) ( )

21

2

0

10

mm g

nDdi tD

i tLdt L v tv tdv t nD

C RCdt

= + − −

(2.56)

( ) [ ] ( )( ) [ ] ( )1 0 0g g

i ti t D v t

v t

= +

(2.57)

To validate the analytically derived averaged state-space model in CCM, a detailed model of

Figure 2.1(a) has been constructed in PLECS [16]. The averaged state-space equations have

been constructed in Matlab/Simulink [30]. A hardware prototype of the subject converter has

14

been built to validate the results. The parameters of the second-order flyback converter

without parasitics in CCM are given in Appendix A.1. The predicted and measured inductor

current and capacitor voltage are shown in Figure 2.3.

0.4

0.6

0.8

1

1.2

1.4

1.6

I (A

mp.)

0.999991 0.999993 0.999995 0.999997

Time (s)

Detailed Model

Actual Average

Analytical State-Space

Hardware Prototype

Detaied Model

Actual Average


Actual Average andAnalytical State-Space

SeeFigure 2.4

-74

-73

-72

-71

V (

Volt)

Figure 2.3 Inductor current and capacitor voltage of second order Flyback converter without parasitics

in CCM.

0.999991 0.999993 0.999995 0.999997

-73.896

-73.894

-73.892

-73.89

Time (s)

V (

Vo

lt)

Detailed Model

Actual Average


Figure 2.4 Capacitor voltage of second order Flyback converter without parasitics in CCM.

As can be seen in Figure 2.3, the analytically derived state-space averaged model predicts the

averaged inductor current very well. As can be seen in Figure 2.3, there is a difference

between the capacitor voltage waveforms predicted by the detailed model and the

measurements from the hardware prototype, which comes from the simplified detailed model

15

(second-order without parasitics). As can be seen in Figure 2.4, the analytically derived state-

space averaged model predicts the average capacitor voltage very well.

2.2 State-Space Averaging in DCM without Parasitics

When designing a flyback converter, one of the very first challenges is the decision on the

mode of operation. It is known that performance of the flyback converters in CCM and DCM

differs significantly in terms of components stress, output voltage regulation, transient

response, and efficiency. Interested reader can find a comparison of CCM and DCM for the

flyback converters in [31]. If the converter is designed to operate in DCM, it will operate in

DCM for almost all specified loads. If the converter is designed to operate in CCM, it will

operate in CCM at nominal load up to a boundary between CCM and DCM. The value of

magnetizing inductance at the boundary between CCM and DCM is given as [32],

( ) 22

1 1

2

1

2L

m

D R NL

f N

− =

(2.58)

Here, 1D is the duty cycle, LR is the load resistance, f is the switching frequency, and

1 2N N is the turn ratio of the transformer.

In addition to 2 subintervals that occur in CCM, in DCM there is another subinterval that

occurs at light loads. The third interval results in the topological instance of the flyback

converter circuit shown in Figure 2.5.

+

mL


( )v t( )Lv t

C R( )gv t

Figure 2.5 Second-order Flyback converter without parasitics during third subinterval in DCM.

During the first subinterval [see circuit of Figure 2.1(b)], the MOSFET is on and the diode is

off, and the magnetizing inductance stores the energy. During the second subinterval [see

Figure 2.1(c)], the MOSFET is off and the diode is on, and the energy stored in the

16

transformer field is transferred to the secondary side. This energy then flows through the

filter capacitor and is supplied to the load resistor. If the magnetizing current during the

second subinterval goes to zero before the end of the second subinterval, the converter goes

into another stage (DCM, subinterval 3) before it goes back to the first interval again. The

magnetizing current predicted by the detailed model in DCM is shown in Figure 2.6 for the

load 2500R= Ω .

0.123543 0.123546 0.123549

0

0.4

0.8

1.2

Time (s)

I (A

mp.)

1 sd T2 sd T

3 sd T

sT

Figure 2.6 Magnetizing current in DCM for the load 2500R= Ω .

For the flyback converter without parasitics, the first and second subintervals in DCM are the

same as in CCM, as described in Section 2.1. The third subinterval comes into account when

the MOSFET and the diode are both off as shown in Figure 2.5. For this topology, the

inductor voltage ( )Lv t , capacitor current ( )ci t , and converter input current ( )gi t are

( ) 0Lv t = (2.59)

( ) ( )c

v ti t

R= − (2.60)

( ) 0gi t = (2.61)

Let us now apply the state-space averaging method to third sub interval. The state vector

( )x t , the input vector ( )u t , and the output vector( )y t are as define in Section 2.1 in (2.39),

(2.40), and (2.41), respectively. Next, let us write the state equation for third subinterval.

When the switch and the diode are off, the converter circuit of Figure 2.5 is obtained. The

inductor voltage, capacitor current, and converter input current are

17

( )

0m

di tL

dt= (2.62)

( ) ( )dv t v t

Cdt R

= − (2.63)

( ) 0gi t = (2.64)

Similar to (2.59), (2.60), and (2.61), after organizing the terms in (2.62), (2.63), and(2.64),

the following state-space for the third subinterval is formed

( )

( )

( )

( )( )( )

( )( )

3

3

0 00

100 g

u tBx t

A

dx t

dt

di ti tdt v tv tdv t

RCdt

= + −

14243123

1424314243

(2.65)

( )( )

[ ] ( )( )( )

[ ]

( )( )33

0 0 0g g

ECy t u tx t

i ti t v t

v t

= +

123123 14243123

(2.66)

So the next step is to evaluate the state-space averaged model, which goes as following:

1 1 2 2 3 3

2

1 2 3

2

0 00 0 0 0

1 10 01 1

m m

A A D A D A D

nDn

L LD D D

n nDRC RC

C RC C RC

= + +

= + + = − − − − − −

(2.67)

1

1 1 2 2 3 3 1 2 3

10 0

0 00 0m m

D

L LB B D B D B D D D D

= + + = + + =

(2.68)

[ ] [ ] [ ] [ ]1 1 2 2 3 3 1 2 3 11 0 0 0 0 0 0C C D C D C D D D D D= + + = + + = (2.69)

[ ] [ ] [ ] [ ]1 1 2 2 3 3 1 2 30 0 0 0E E D E D E D D D D= + + = + + = (2.70)

So the final state-space averaged equation of the second order flyback converter without

parasitics in DCM becomes:

18

( )

( )( )( ) ( )

21

2

0

10

mm g

nDdi tD

i tLdt L v tv tdv t nD

C RCdt

= + − −

(2.71)

( ) [ ] ( )( ) [ ] ( )1 0 0g g

i ti t D v t

v t

= +

(2.72)

Equations (2.71) and (2.72) are the state-space averaged model for the DCM, which is the

same as (2.56) and (2.57). It only happens when there are no parasitics.

To validate the analytically derived state-space averaged model in DCM, a detailed model of

the converter depicted in Figure 2.1(a) has been constructed in PLECS [16]. The state-space

averaged model has been constructed in Matlab/Simulink [30]. The interval 2d has been

calculated as 0.4409 using (2.58) and 2500LR R= = Ω . The parameters of the second-order

flyback converter without parasitics in CCM are given in Appendix A.2. The resulting

waveforms of inductor current and capacitor voltage are shown in Figure 2.7.

0

0.4

0.8

1.2

I (A

mp

.)

1.699991 1.699993 1.699995 1.699997 1.699999 1.7-102.547

-102.545

-102.543

-102.541

Time (s)

V (

Vo

lt)

Detailed Model

Actual Average


Figure 2.7 Inductor current and capacitor voltage of second order Flyback converter without parasitics

in DCM.

19


magnetizing current with a large error of 18.25%. This error is because the actual inductor

current is discontinuous, which is not properly accounted by the classical state-space

averaging as will be explained in Section 2.5. At the same time, the analytically derived

state-space averaged model predicts the capacitor voltage and the output voltage with a very

small error because these state variables are continuous. The small error in this case comes

from the error in representing the magnetizing current according to (2.71).

2.3 Small-Signal AC Model and State-Space Averaging with Basic Parasitics in CCM

A second order flyback converter with basic parasitics is shown in Figure 2.8(a). Here, n is

the turn ratio of the transformer( )1 2N N .

+

mL

n( )i t( )gi t ( )ci t

( )v t( )Lv t C

R

D

Mosfet

( )a

( )b

( )c

( )gv t

+

cR

swR

dV

( )cv t

+

mL


( )v t( )Lv t C

R( )gv t

cR

swR

( )cv t

+

mL


( )v t( )Lv t C

R( )gv t

+

cR

dV

( )cv t

Figure 2.8(a) Second-order Flyback converter with parasitics; (b) Circuit during subinterval 1 (c) Circuit

during subinterval 2.

20

During the first subinterval, when the MOSFET conducts and the diode is off, the circuit

reduces to Figure 2.8(b). For this interval, the inductor voltage ( )Lv t , capacitor current ( )ci t ,

converter output voltage( )v t , and converter input current ( )gi t are

( ) ( ) ( )L g sw gv t v t R i t= − (2.73)

( ) ( )cc

c

v ti t

R R= −

+ (2.74)

( ) ( )c

c

v t Rv t

R R=

+ (2.75)

( ) ( )gi t i t= (2.76)

We next apply the small ripple approximation and replace the voltages and currents with

their respective average values to obtain the following:

( ) ( ) ( )s s

L g sw gT Tv t v t R i t= − (2.77)

( )( )

sc T

cc

v ti t

R R= −

+ (2.78)

( )( )

sc T

c

v t Rv t

R R=

+ (2.79)

( ) ( )s

g Ti t i t= (2.80)

During the second subinterval, the MOSFET is off and diode conducts, which results in the

circuit of Figure 2.8(c). For this interval, the inductor voltage ( )Lv t , capacitor current ( )ci t ,

converter output voltage ( )v t , and converter input current ( )gi t are given by

( ) ( ) ( )( )L c c c dv t v t i t R V n= − − (2.81)

( ) ( ) ( )c

v ti t i t n

R

= − +

(2.82)

( ) ( ) ( )c c cv t v t i t R= − (2.83)

( ) 0gi t = (2.84)

Applying the small ripple approximation leads to the following:

21

( ) ( ) ( )s s

L c c c dT Tv t v t n i t R n V n= − − (2.85)

( ) ( )( )

s

s

T

c T

v ti t i t n

R= − − (2.86)

( ) ( ) ( )s s

c c cT Tv t v t i t R= − (2.87)

( ) 0gi t = (2.88)

The average inductor voltage now can be found by averaging the subintervals over one

complete switching period. The result is

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )1 1

2 2 2

s s s

s s

L g sw gT T T

c c c dT T

v t v t d t R i t d t

v t nd t i t R nd t V nd t

= −

+ − − (2.89)

This leads to the following equation for the average inductor current

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )1 1

2 2 2

s

s s

s s

Tm g sw gT T

c c c dT T

d i tL v t d t R i t d t

dt

v t nd t i t R nd t V nd t

= −

+ − − (2.90)

The average capacitor current now can be found by averaging the subintervals over one

switching period, which results in the following:

( )( )

( ) ( ) ( )( )

( )1 2 2s s

s s

c T T

c T Tc

v t v ti t d t i t nd t d t

R R R= − − −

+ (2.91)

This leads to the following equation for the average capacitor voltage

( ) ( )

( ) ( ) ( )( )

( )1 2 2s s s

s

c cT T T

Tc

d v t v t v tC d t i t nd t d t

dt R R R= − − −

+ (2.92)

The converter output voltage now can be found by averaging the subintervals over one

switching period, which results in

( )( )

( ) ( ) ( ) ( ) ( )1 2 2s

s s s

c T

c c cT T Tc

v t Rv t d t v t d t i t R d t

R R= + −

+ (2.93)

22

The converter input current can now be found by averaging the subintervals over one

switching period, resulting in the following:

( ) ( ) ( )1 0ss

g TTi t i t d t= + (2.94)

Equations (2.90), (2.92), (2.93), and (2.94) are nonlinear differential equations. Hence, to

construct the converter small-signal ac model, the next step is to perturb and linearize them.

We assume that the converter input voltage ( )gv t and duty cycle ( )1d t can be expressed as

quiescent values plus small ac variations, as follows

( ) ( )ˆs

g g gTv t V v t= + (2.95)

( ) ( )1 1 1ˆd t D d t= + (2.96)

In response to these inputs, and after all transients have decayed, the averaged converter

waveforms can be expressed as quiescent values plus small ac variations as

( ) ( )ˆs

g g gTi t I i t= + (2.97)

( ) ( )ˆsT

i t I i t= + (2.98)

( ) ( )ˆs

c c cTv t V v t= + (2.99)

( ) ( )ˆs

c c cTi t I i t= + (2.100)

( ) ( )ˆsT

v t V v t= + (2.101)

With these substitutions, the large-signal averaged inductor, (2.90), becomes

( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( )

( )( )

1 1 1 1

2 1 2 1

2 1

ˆˆ ˆˆˆ

ˆ ˆˆˆ

ˆ

m g g sw g g

c c c c c

d

d I i tL V v t D d t R I i t D d t

dt

V v t n D d t I i t R n D d t

V n D d t

+= + + − + +

+ + − − + −

− −

(2.102)


23

( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 1 2 2 2

1 1 1 1 1 1

2 1 2 1

1 ( )

ˆ

ˆ ˆ ˆˆˆ

ˆ ˆˆˆ

m g sw g c c c d

Dcterms

g g sw g sw g c

c c c c c d

st order acterms linear

di tdIL V D R I D V nD I R nD V nD

dt dt

V d t v t D R I d t R i t D V nd t

v t nD I R nd t i t R nD V nd t

+ = − + − −

+ − − − + + + − +

1444444442444444443

144444444 2

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

1 1 1 1

1 1

2 ( )

ˆ ˆ ˆˆˆ ˆ

ˆ ˆˆ

g sw g c

c c d


v t d t R i t d t v t nd t

i t R nd t V nd t

− − + + +

444 444444444443

1444444442444444443

(2.103)

As usual, this equation contains three types of terms. The dc term contains no time-varying

quantities. The first order ac terms are linear functions of the ac variations in the circuit,

while the second order ac terms are functions of the products of the ac variations. At this

point, we make an assumption that the ac variations are small in magnitude compared to the

dc quiescent values,

( )( )( )

( )( )( )( )

1 1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

g g

g g

c c

c c

v t V

d t D

i t I

i t I

v t V

i t I

v t V

<<<<<<<<

<<

<<

<<

(2.104)

If the small signal assumptions (2.104) are satisfied, then the second-order terms are much

smaller in magnitude than the first-order terms and hence be neglected. The dc terms must

satisfy

1 1 2 2 20 g sw g c c c dV D R I D V nD I R nD V nD= − + − − (2.105)

The first order ac terms must satisfy

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )1 1 1 1 1 1

2 1 2 1

ˆˆ ˆ ˆˆˆ

ˆ ˆˆˆ

m g g sw g sw g c

c c c c c d

di tL V d t v t D R I d t R i t D V nd t

dt


= + − − −

+ + − + (2.106)

This is the linearized equation that describes ac variations in the inductor current.

24

Upon substation of (2.95)-(2.101) into (2.92), we obtain

( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( )

1 1 2 1

2 1

ˆ ˆ ˆ ˆˆ

ˆ ˆ

c c c c

c

d V v t V v tC D d t I i t n D d t

dt R R

V v tD d t

R

+ += − + − + −

+

+− −

(2.107)


( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 22

1 1 1 21 2

1 ( )

11

ˆ

ˆ ˆˆ ˆˆ ˆ

ˆˆ ˆˆˆ

cc c

c

Dcterms

c c

c c


c

c

dv tdV V D VDC InD

dt dt R R R

V d t v t D Vd t v t DInd t i t nD

R R R R R R

v t d t v ti t nd t

R R

+ = − − − +

+ − − + − + − + +

+ − + ++

14444244443

144444444444424444444444443

( )1

2 ( )

ˆ


d t

R

14444444244444443

(2.108)

Here again we neglect the second-order terms. The dc terms of (2.108) must satisfy

1 220 c

c

V D VDInD

R R R= − − −

+ (2.109)

The first-order ac terms of (2.108) lead to the following small-signal for the ac capacitor

voltage

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 2

1 2

ˆ ˆˆ ˆ ˆˆ ˆc c c

c c

dv t V d t v t D Vd t v t DC Ind t i t nD

dt R R R R R R= − − + − + −

+ + (2.110)

Substation of (2.95)-(2.101) into (2.93) leads to

( )( ) ( )( ) ( )( ) ( )( ) ( )( )

( )( ) ( )( )1 1 2 1

2 1

ˆ ˆ ˆˆ ˆ

ˆˆ

c c

c cc

c c c

V v tV v t R D d t V v t D d t

R R

I i t R D d t

++ = + + + −

+

− + −

(2.111)


25

( )( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

12 2

1 11 2

1 2

1 ( )

11 1

ˆ

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆˆ ˆ ˆˆˆ

cc c c

c

Dcterms

c cc c

c c

c c c c


cc c c

c

V D RV v t V D I R D

R R

V Rd t v t RDV d t v t D

R R R R

I R d t i t R D

v t Rd tv t d t i t R d t

R R

+ = + − +

+ − +

+ ++ + −

+ − + +

14444244443

1444444442444444443

2 ( )nd order acterms nonlinear

1444444442444444443

(2.112)


12 2

cc c c

c

V D RV V D I R D

R R= + −

+ (2.113)


( ) ( ) ( ) ( ) ( ) ( ) ( )1 11 2 1 2

ˆ ˆ ˆ ˆ ˆˆ ˆc cc c c c c c

c c

V Rd t v t RDv t V d t v t D I R d t i t R D

R R R R= + − + + −

+ + (2.114)

Substation of (2.95)-(2.101) into (2.94) leads to

( ) ( )( ) ( )( )1 1ˆˆ ˆ

g gI i t I i t D d t+ = + + (2.115)


( ) ( )

( ) ( )( ) ( ) ( )( )1 1 1 1

1 ( ) 2 ( )

ˆ ˆˆ ˆ ˆg g

Dctermsst order acterms linear nd order acterms nonlinear

I i t ID Id t i t D i t d t+ = + + +1442443 14243

(2.116)


1gI ID= (2.117)


( ) ( ) ( )1 1ˆˆ ˆ

gi t Id t i t D= + (2.118)

This result represents the low-frequency ac variations in the converter input current.

The equations of the quiescent values, (2.105), (2.109), (2.113), and (2.117) are collected

below as

26

1 1 2 2 2

1 22

12 2

1

0

0

g sw g c c c d

c

c

cc c c

c

g

V D R I D V nD I R nD V nD

V D VDInD

R R R

V D RV V D I R D

R R

I ID

= − + − − = − − −+ = + −+

=

(2.119)

For given quiescent values of the input voltage gV , the diode voltage drop dV , and the duty

cycle 1D , the system (2.119) can be evaluated to find the quiescent output voltage V ,

inductor current I , input current gI , capacitor voltage cV , and capacitor current cI .

However, in this problem there are 5 variables but there are only 4 equations. The fifth

equation can be the following

c c cV V I R= − (2.120)

The results are then inserted into the small-signal ac model.

The small signal ac model, (2.106), (2.110), (2.114), and (2.118), is summarized below

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1 1

2 1 2 1

1 1 1 21 2

1 11 2 1

ˆˆ ˆ ˆˆˆ

ˆ ˆˆˆ

ˆ ˆˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ ˆˆ ˆ

m g g sw g sw g c

c c c c c d

c c c

c c

c cc c c c c

c c

di tL V d t v t D R I d t R i t D V nd t

dt


dv t V d t v t D Vd t v t DC Ind t i t nD

dt R R R R R R

V Rd t v t RDv t V d t v t D I R d t i

R R R R

= + − − −

+ + − +

= − − + − + −+ +

= + − + + −+ +

( )

( ) ( ) ( )

2

1 1ˆˆ ˆ

c

g

t R D

i t Id t i t D

= +

(2.121)

The final step is to construct an equivalent circuit of (2.119) and(2.121) using the

commercially available simulation tools.

27

Let us now apply the state-space averaging method to the second order flyback converter of

Figure 2.8(a). The independent state variables as usual are the inductor current ( )i t and the

capacitor voltage ( )cv t , which form the state vector

( ) ( )( )c

i tx t

v t

=

(2.122)

The input voltage ( )gv t , and the diode voltage drop is an independent source, which should

be placed in the input vector as

( ) ( )g

d

v tu t

V

=

(2.123)

To model the converter input port and output port, we need to find the converter input current

( )gi t and output voltage ( )v t . To calculate this dependent current and voltage, it should be

included in the output vector ( )y t as

( ) ( )( )

gi ty t

v t

=

(2.124)

Next, let us write the state equations for each subinterval. When the switch is on and the

diode is off, the converter circuit of Figure 2.8(b) is obtained. The inductor voltage, capacitor

current, output voltage, and converter input current are

( ) ( ) ( )m g sw

di tL v t R i t

dt= − (2.125)

( ) ( )c c

c

dv t v tC

dt R R= −

+ (2.126)

( ) ( )c

c

v t Rv t

R R=

+ (2.127)

( ) ( )gi t i t= (2.128)

Similar to (2.73)-(2.76), after organizing the terms in (2.125)-(2.128), the result can be

written in the following state-space form

28

( )

( )

( )

( )( )( )

( )

( )1

1

0 10

10 0 0

sw

m gm

c dc

u tx tcB

dx t A

dt

Rdi tL i t v tdt L

v t Vdv tRC R Cdt

− = + − +

1424312314243

14243 144424443

(2.129)

( )( )( )

( )( )( )

( )

( )1

1

1 00 0

0 0 0g g

c dc

E u ty t x tC

i t i t v tR

v t v t VR R

= + + 12314243123 123

1442443

(2.130)

So the state-space equations for the first subinterval have been identified.

In the second subinterval, when the MOSFET is off and the diode is on, the converter circuit

of Figure 2.8(c) is obtained. For the second subinterval, the inductor voltage, capacitor

current, output voltage, and converter input current are given by

( ) ( ) ( )2

c cm d

c c

di t i t n R R v t nRL V n

dt R R R R= + −

− − (2.131)

( ) ( ) ( )c c

c c

dv t i t nR v tC

dt R R R R= − −

− − (2.132)

( ) ( ) ( )c c

c c

i t nR R v t Rv t

R R R R= +

− − (2.133)

( ) 0gi t = (2.134)

After organizing them and writing in state-space form, we get

( )

( )

( )

( )( )( )

( )

( )2

2

2

0

10 0

c

gm c m m c mm

c dc

u tx tc cB

dx t Adt

n R R nRdi tn

i t v tRL R L RL R Ldt Lv t VnRdv t

RC R C RC R Cdt

− − − = + − − − −

1424312314243

14243 1444442444443

(2.135)

29

( )( )( )

( )( )( )

( )

( )2

2

0 00 0

0 0g g

cc d

c cE u ty t x t

C

i t i t v tnR R R

v t v t VR R R R

= + − − 12314243123 123

144424443

(2.136)

So the space-space equation of the second interval has also been identified.

The next step is to combine the result and obtain the state-space averaged model as

2

1 1 2 2 1 2

21 2 2

2 1 2

0

1 10

sw c

m m c m m c m

c c c

sw c

m m c m m c m

c c c

R n R R nRL RL R L RL R L

A A D A D D DnR

RC R C RC R C RC R C

R D n R RD nRD

L RL R L RL R L

nRD D D


− − − = + = +

− − − + − −

− + − − =

− − − − + −

(2.137)

1 2

1 1 2 2 1 2

10 0

0 0 0 0 0 0m m m m

D nDn

L L L LB B D B D D D

− − = + = + =

(2.138)

1 1 2 2 1 2

1

2 1 2

0 01 0

0

0

c

c c c

c

c c c

C C D C D D DnR RR R

R R R R R R

D

nR RD RD RD

R R R R R R

= + = +

+ − −

= + − + −

(2.139)

1 1 2 2 1 2

0 0 0 0 0 0

0 0 0 0 0 0E E D E D D D

= + = + =

(2.140)

Therefore, the final state-space averaged model of the second order flyback converter with

parasitics in CCM becomes

30

( )

( )( )( )

( )

21 2 2

2 1 2

1 2

0 0

sw c

m m c m m c m

cc

c c c

gm m

d

R D n R RD nRDdi ti tL RL R L RL R Ldtv tnRD D Ddv t

RC R C RC R C RC R Cdt

D nDv t

L LV

− + − − = − − − − + −

− +

(2.141)

( )( )

( )( )

( )1

2 1 2

00 0

0 0g g

cc d

c c c

Di t i t v t

nR RD RD RDv t v t V

R R R R R R

= + + − + −

(2.142)

To validate the analytically derived state-space averaged model in CCM, a detailed model of

Figure 2.8(a) has been constructed in PLECS [16]. The state-space averaged model has been

constructed in Matlab/Simulink [30]. The same hardware prototype has been used here to

validate the results. The parameters of the second-order flyback converter with basic

parasitics in CCM are given in Appendix A.3. The predicted and measured inductor current,

capacitor voltage, and output voltage are shown in Figure 2.9.


averaged inductor current and capacitor voltage very well. There is a difference between the

detailed model and hardware prototype in terms of the output voltage (see Figure 2.9), which

comes from the simplified detailed model (second-order with parasitics). As seen in Figure

2.10, the analytically derived state-space averaged model predicts the average output voltage

very well.

31

0.5

1

1.5

I (A

mp

.)

-72.496

-72.492

-72.488

Vc (

Vo

lt)

0.999991 0.999993 0.999995 0.999997

-72.6

-72.2

-71.8

-71.4

Time (s)

V (

Vo

lt)

Actual Average andAnalytical State-Space

SeeFigure 2.10

Detailed Model

Actual Average


Detailed Model

Actual Average


Hardware Prototype

Figure 2.9 Inductor current, capacitor voltage and output voltage of second order Flyback converter with

parasitics in CCM.

0.999991 0.999993 0.999995 0.999997

-72.5

-72.495

-72.49

-72.485

-72.48

Time (s)

V (

Volt)

Detailed Model

Actual Average


Figure 2.10 Output voltage of second order Flyback converter with parasitics in CCM.

32

2.4 State-Space Averaging with Parasitics in DCM

In addition to the two subintervals that occur in CCM, here there is another subinterval that

occurs at light loads. The topology of the flyback converter in this subinterval is shown in

Figure 2.11.

+

mL


( )v t( )Lv t C

R( )gv t

cR

( )cv t

Figure 2.11 Second-order Flyback converter with parasitics during third subinterval in DCM.

During the first subinterval depicted in Figure 2.8(b), the MOSFET is on and the diode is off,

and the magnetizing inductance stores some energy. During the second subinterval depicted

in Figure 2.8(c), the MOSFET is off and the diode is on. During this interval, the stored

energy is transferred to secondary side and it flows through the filter capacitor to the load

resistor. If the magnetizing current during the second interval goes to zero before the end of

the second interval, the converter goes to another stage resulting in third subinterval and

DCM, before it goes back to the first interval in the next cycle.

For the flyback converter without parasitics in DCM, the first and second sub intervals

remain of the same as in CCM as described in Section 2.3. The third interval comes into

account when the MOSFET and the diode are off as in Figure 2.11. For this case, the

inductor voltage ( )Lv t , capacitor current ( )ci t , output voltage ( )v t , and converter input

current ( )gi t are

( ) 0Lv t = (2.143)

( ) ( )cc

c

v ti t

R R= −

+ (2.144)

( ) ( )c

c

v t Rv t

R R=

+ (2.145)

33

( ) 0gi t = (2.146)

Let us now apply the state-space averaging method to the third interval of Figure 2.11. The

inductor voltage, capacitor current, output voltage, and converter input current can be written

as

( )

0m

di tL

dt= (2.147)

( ) ( )c c

c

dv t v tC

dt R R= −

+ (2.148)

( ) ( )c

c

v t Rv t

R R=

+ (2.149)

( ) 0gi t = (2.150)

Similar to (2.143)-(2.146), after organizing terms in (2.147)-(2.150), these equations can be

written as

( )

( )

( )

( )( )( )

( )

( )3

3

0 00 0

10 0 0

g

c dcc

B u tx tA

dx t

dt

di ti t v tdtv t Vdv t

RC R Cdt

= + − +

12314243123

14442444314243

(2.151)

( )( )( )

( )( )( )

( )

( )3

3

0 00 0

0 0 0g g

c dc

E u ty t x tC

i t i t v tR

v t v t VR R

= + + 12314243123 123

1442443

(2.152)

which defines the state-space equation for the third subinterval. Hence, the next step is to

evaluate the state-space averaged equations, which goes as following:

34

1 1 2 2 3 3

2

1 2 3

21 2 2

1 32 2

0 0 0

101 1

0

sw c

m m c m m c m

cc c c

sw c

m m c m m c m

c c c

A A D A D A D

R n R R nRL RL R L RL R L

D D DnR

RC R CRC R C RC R C RC R C

R D n R RD nRD

L RL R L RL R L

D DnRD D


= + +

− − − = + + −

− +− − + − −

− + − −= +− − −

− + −

(2.153)

1 2

1 1 2 2 3 3 1 2 3

10 0 0 0

0 00 0 0 0 0 0m m m m

D nDn

L L L LB B D B D B D D D D

− − = + + = + + =

(2.154)

( )

1 1 2 2 3 3 1 2 3

1

1 32 2

0 01 0 0 0

0 0

0

c

c cc c

c

c c c

C C D C D C D D D DnR RR RR

R R R RR R R R

D

R D DnR RD RD

R R R R R R

= + + = + +

+ + − −

= + + − + −

(2.155)

1 1 2 2 3 3 1 2 3

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0E E D E D E D D D D

= + + = + + =

(2.156)

The final state-space averaged model for the second order flyback converter with parasitics

in DCM becomes:

( )

( )( )( )

( )

21 2 2

1 32 2

1 2

0 0

sw c

m m c m m c m

cc

c c c

gm m

d

R D n R RD nRDdi ti tL RL R L RL R Ldtv tD DnRD Ddv t

RC R C RC R C RC R Cdt

D nDv t

L LV

− + − − = + − − − − + −

− +

(2.157)

( )( ) ( ) ( )

( )( )1

1 32 2

00 0

0 0g g

cc d

c c c

Di t i t v t

R D DnR RD RDv t v t V

R R R R R R

= ++ + − + −

(2.158)

35

To validate the analytically derived state-space averaged model in DCM, a detailed model of

the converter circuit depicted Figure 2.8(a) has been constructed in PLECS [16]. The state-

space averaged model has been constructed in Matlab/Simulink [30].The variable 2d has

been calculated using (2.58) and set to 0.4409. The load was assumed as 2500LR R= = Ω .

The parameters of the second-order flyback converter without parasitics in CCM are given in

Appendix A.4. The predicted inductor current and capacitor voltage are shown in Figure

2.12.

0

0.5

1

I (A

mp.)

-102

-101.5

-101

Vc (

Volt)

0.999991 0.999993 0.999995 0.999997-102

-101.5

-101

Time (s)

V (

Volt)

Detailed Model

Actual Average

Analytical State-SpaceDetailed Model and

Actual Average

Detailed Model andActual Average

Figure 2.12 Inductor current, capacitor voltage and output voltage of second order Flyback converter

with parasitics in DCM.

As can be seen in Figure 2.12, the analytically derived state-space averaged model predicts

the magnetizing current with a large error of 17.5%. This error is because the current is

discontinuous, while the conventional state-space averaging fails to correctly take this into

36

account. At the same time, the analytically derived state-space averaged model predicts the

capacitor voltage with a very small error because the capacitor voltage is a continuous state

variable. The small error comes from the error in the magnetizing current defined by (2.157).

2.5 Parametric Average Value Modeling in CCM and DCM

To see the challenges with the conventional state-space averaging in DCM and representation

of parasitics, we take another detailed look at this approach. The state-space averaging is a

well defined approach [18] that has been presented previously in numerous publications, e.g.

[8, 10, 33]. In CCM, the state-space equation is

( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( )1 2 1 21 1x t q t A q t A x t q t B q t B u t•

= + − + + − (2.159)

where ( )q t is the switching function, ,k kA B are the system matrices, and ( )u t is the input

vector. By definition, the so-called fast average of a state variable ( )x t over a switching

interval is

( ) ( )1 st T

s t

x t x t dtT

+

= ∫ (2.160)

where ( )x t is the actual or the true average of ( )x t . Since the averaging is commutative

with respect to differentiation, taking the fast average of (2.159) over a switching interval

yields

( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( )1 2 1 21 1x t d t A d t A x t d t B d t B u t•

= + − + + − (2.161)

This result follows from the fact that the fast average of the switching function over a

switching interval is the duty cycle function

( ) ( )1 st T

s t

d t q t dtT

+

= ∫ (2.162)

It is also assumed that the average of the product is equal to the product of the averages,

especially

Ax A x= ⋅ (2.163)

Bu B u= ⋅ (2.164)

Assumption (2.163) is acceptable if the original switching variables do not deviate

significantly from their average values and also the system matrices 1A and 2A are

37

commutative [18, 34]. But in general, the matrices 1A and 2A are not commutative [35, 36].

So, if we make assumption (2.163), the equation (2.161) is not an exact solution as

mentioned in [18] and Appendix A, but can only be an approximation. Assumption (2.164) is

usually accepted when the source ripple is neglected.

The DCM operation of PWM converters differs from CCM operation by an additional time

interval in each switching cycle during which the inductor current or capacitor voltage is

clamped to zero (or a constant when there are multiple energy storage elements).

Conventional state-space averaging for converters working in DCM has been summarized in

[4, 10, 18]. For this mode, the direct extension of (2.159)-(2.161) results in

( ) ( ) ( ) ( ) ( )1 1 2 2 3 3 1 1 2 2 3 3x t d A d A d A x t d B d B d B u t•

= + + + + + (2.165)

which is no longer accurate. In particular, the local average of the magnetizing current in the

third interval is zero, whereas the state-space averaging implies that this value should be 3d i .

Since 3d and i are not zero, the result of the state-space averaging is not zero. That is why

the discontinuous averaged variable in DCM is higher than the actual averaged value that can

be seen in Figure 2.7 and Figure 2.12.

As shown in Sections 2.2 and 2.4, the conventional state-space averaging is no longer

accurate in DCM. The parametric average-value modeling has been proposed to solve this

problem as documented in numerous publications [22, 24-27].

2.5.1 Correction Term

The state-space averaging involves the weighted sum of the state-space equations

corresponding to different topological instances within a switching interval. In DCM, a

prototypical switching interval is divided into three subintervals as seen in Figure 2.6. To

accurately represent the dynamics of the underlying converter circuit, a corrected full order

state-space averaged model [4] has been proposed for an ideal converter circuit as

( )3 3

1 1k k k k

k k

x d A Mx d B u•

= =

= + ∑ ∑ (2.166)

38

To make state-space averaging work properly in DCM, the so-called correction matrix M is

added in (2.166). The analytical derivation of the correction matrix for an ideal topology

(without parasitics) is given in [4]. Since the circuit of Figure 2.8(a) with parasitics has been

considered in this study, it would be very complicated to extend that analytical solution.

Hence, a numerical solution has been adopted in this Thesis.

The correction term is a diagonal matrix, wherein each state variable has its own correction

coefficient. In this implementation presented in this Thesis, instead of a diagonal matrix, a

column correction vector (each state variable has its own correction term as an element of

column matrix) is proposed as

( ) ( )3 3

1 1

.k k k kk k

x d A x M d B u•

= =

= ∗ + ∑ ∑ (2.167)

where ( ).x M∗ denotes the element wise multiplication. This operation uses less

computational time compare to direct implementation of (2.166).

2.5.2 Model Implementation

A detailed model of the converter depicted in Figure 2.8(a) has been constructed using the

PLECS toolbox. The system matrices ( ), , ,k k k kA B C D in each subinterval can be extracted

numerically using PLECS and Simulink built-in feature for numerical model linearization

and analysis. Since the detailed model includes all the parasitics, the extracted system

matrices will have all the necessary information in them by construction and without any

extra analytical derivations. The elements of correction vector ( )M and ( )3d are obtained as

functions of the duty-cycle ( )1d and the average value of the state variables ( )x . To obtain

the values of ( )3 1,d d x and ( )1,M d x , the detailed model is run in the operation region of

interest (for example; duty-cycle changes between 0.1 and 0.9, and the load ( )R changes

from very low load to very high load) whereas the state variables are averaged numerically

over the prototyping switching interval and saved for the future use. In particular, the

average-value of the state vector ( )x is computed in a steady-state corresponding to a given

operation point. Specifically, in steady-state we have

39

( ) ( )3 3

1 1

0 .k k k kk k

d A x M d B u= =

= ∗ + ∑ ∑ (2.168)

From which an intermediate variable vector p is computed as

( ) ( )13 3

1 1

. k k k kk k

p x M d A d B u−

= =

= ∗ = − ⋅ ∑ ∑ (2.169)

Thereafter the elements of M can be found using the following

j j jM p x= ⋅ ÷ (2.170)

where j denotes number of state variables and ( )j jp x⋅ ÷ denotes element wise division.

To obtain the functions ( )3 1,d d x and ( )1,M d x for the desired operation range, the detailed

simulation is run with different values of control variable ( )1d as well as the load( )R . The

variables resulting from this procedures are 1 2 3, , , , ,cd d d R v i and the correction vector ( )M is

computed using (2.169) and (2.170). These variables are stored for future use in lookup

tables, wherein an interpolation/extrapolation may be used as necessary. The real challenge

here is to calculate the 2d or 3d at any given operation point. If one can calculate either one,

thereafter it becomes easy to calculate the other one since 1d is the control variable which

can be calculated from the magnetizing current. As can be seen in Figure 2.12, the

magnetizing current is zero during the third subinterval whereas the variable 3d can be

calculated.

The final numerical function for 3d is plotted in Figure 2.13, which shows that this function

has a flat surface corresponding to CCM, and varies linearly along the 1d . In DCM, the

surface of 3d becomes non-linear and increases.

40

01000

20003000

40005000

0.10.20.30.40.50.60.70.80.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

R (ohm)

d1

d3

CCM

DCM

Figure 2.13 Variable 3d as a function` of duty-cycle ( )1d and the load ( )R .

The correction term ( )1m of the magnetizing current ( )I is plotted in Figure 2.14 as a

function of duty-cycle ( )1d and the load resistance ( )R . As can be seen in Figure 2.14, on

the one hand, the correction term ( )1m has a flat surface and value of 1 in CCM, which

implies that the state-space averaging captures the correct value of this current in CCM and

no correction is required. On the other hand, it has values higher than 1 corresponding to the

DCM because the conventional state-space averaging does not capture the correct average

values in this region of operation as explained in Section 2.5, and therefore the correction

effort is required.

01000

20003000

40005000

0.10.20.30.40.50.60.70.80.910.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

R (ohm)

d1

m1

CCM

DCM

Figure 2.14 The correction term 1m as a function of duty-cycle ( )1d and the load resistance ( )R .

41

The correction term ( )2m of filter capacitor voltage ( )cv is plotted in Figure 2.15 as a

function of the duty-cycle( )1d and the load resistance ( )R . As can be seen in Figure 2.15,

the calculated value for this correction term is 1 for either CCM or DCM. This value is

consistent with the fact that the capacitor voltage is a continuous state variable with relatively

small ripple, so the conventional state-space averaging predicts the correct average value for

this variable for both modes.

01000

20003000

40005000

0.10.20.30.40.50.60.70.80.91

0.98

0.99

1

1.01

1.02

R (ohm)

d1

m2

CCM and DCM


Once the parametric functions ( )3 1,d d x and ( )1,M d x have been calculated and stored, these

functions become available for the model implementation. Finally, the parametric average-

value model is implemented according to the block diagram shown in Figure 2.16. The

system matrices ( ), , ,k k k kA B C D for each subinterval are calculated numerically. For a given

value of control variable 1d and state vector x , the values of 3d and correction vector ( )M

are acquired through the lookup tables. Total average system matrices ( ), , ,T T T TA B C D are

computed using the system matrices ( ), , ,k k k kA B C D and the variables 1 2,d d and 3d . These

matrices are then used to build the new continuous non-linear state-space average-value

model that replaces the discontinuous detailed model. Thereafter, this parametric average-

value model can be used for large-signal transient studies as well as for numerical

linearization and subsequent small-signal frequency-domain analysis.

42

3d

M

x

1d

u

y

( )

( )

( )

( )

2 1 3

3

1

3

1

3

1

3

1

1

T k k

k

T k k

k

T k k

k

T k k

k

d d d

A d A

B d B

C d C

D d D

=

=

=

=

= - -

=

=

=

=

å

å

å

å

( )T T

T T

x A x M B u

y C x D u

·

= ×* +

= +

, , ,k k k kA B C D

( )

( )3 1

1

,

,

d d x

M d x

Figure 2.16 Implementation of the parametric average-value model.

The previously proposed parametric average-value models [24-27] had problem with the

correcting the output equation. In particular, in the previous formulation the output equation

was defined as

( )3 3

1 1k k k k

k k

y d C Mx d D u= =

= + ∑ ∑ (2.171)

However, my simulation results (which are not shown in this study) have shown that if there

is a correction term ( )M in the output equation, in this case no correction occurs on the

output voltage ( )V . The reason for this is that the state vector ( )x has already been corrected

in the state equation (2.167), and therefore no additional correction is required in the output

equation. The analytical proof of this argument goes as follow:

During the steady-state, we have

1 10 T T T TA Mx B u x M A B u− −= + ⇒ = − (2.172)

When (2.172) is inserted into (2.171), the output equation becomes

( )1 1T T T Ty C M M A B u D u− −= − + (2.173)

After required calculations in (2.173), we get

1T T T Ty C A B u D u−= − + (2.174)

If there is a correction term in (2.171), there is no correction occurs in (2.174). Therefore, the

correct output equation is

43

( )3 3

1 1k k k k

k k

y d C x d D u= =

= + ∑ ∑ (2.175)

2.5.3 Case Studies

The flyback converter shown in Figure 2.8(a) is used here for the case studies. The detailed

model has been implemented using PLECS. The converters parameters are summarized in

the Appendices A.3 and A.4 for CCM and DCM, respectively. The proposed parametric

average-value model has been implemented and compared to the hardware prototype, the

detailed model and the conventional state-space averaging model in both time and frequency

domains.

2.5.3.1 Time domain

To demonstrate the parametric average-value model (PAVM) in DCM, the system is operates

in steady-state defined by 1 0.381d = and 2500R= Ω . The resulting simulated waveforms of

the magnetizing current, filter capacitor voltage and the output voltage are plotted in Figure

2.17. As can be seen in Figure 2.17, the PAVM predicts the average-value of the steady state

waveforms in DCM very well compared to the conventional state-space averaging. Actual

average of the inductor current is 0.46Amp, and the PAVM can predict this value very well.

On the other hand, the analytical state-space averaging can predict this value as

0.55 .Amp which is equal to 18.18% error. Actual average of the capacitor voltage is

101.56V− , and the PAVM can predict this value very well. The analytical state-space

averaging can predict this value as 101.2V− .

44

0

0.5

1

I (A

mp

.)

-102

-101.5

-101

Vc (

Vo

lt)

0.999991 0.999993 0.999995 0.999997-102

-101.5

-101

Time (s)

V (

Vo

lt)

Detailed Model,Actual Average

and PAVM

Detailed Model

Actual Average


PAVM

Actual Averageand PAVM

Detailed Model,Actual Average

and PAVM

Figure 2.17 Simulated inductor current, capacitor voltage and output voltage of the second order Flyback

converter with parasitics in DCM.

The accuracy of the PAVM in predicting the large-signal behaviour in time-domain has also

been verified by studying the effect of sudden change in load. In the following study, the

output of the flyback converter was regulated using a PI controller. The PI controller was

designed to regulate the output voltage at 72V− by adjusting the duty-cycle ( )1d . The

controller parameters are 0.7pK = and 25iK = . In this study, the converter run with a

resistive load of 2000Ω (which results in DCM) until it reaches a steady-state. At 0.2t s= , a

parallel load of 500Ω is added (which results in CCM). The resulting time-domain transients

are shown in Figure 2.18.

45

0

1

2

3

I (A

mp.)

-72

-71.8

-71.6

Vc

(Volt)

0.1996 0.1998 0.2 0.2002 0.2004 0.2006 0.2008 0.201

-72

-71.8

-71.6

Time (s)

V (

Volt)

PAVM

Detailed Model

Detailed Modeland PAVM

Detailed Modeland PAVM

Figure 2.18 Transients in inductor current, capacitor voltage and output voltage of the second order

Flyback converter due to the step change in load.

As it can be observed in Figure 2.17, after the load change at 0.2t s= , the converter switches

operation from DCM to CCM. After load changes the control action brings the output

voltage to about the desired 72V− after about 0.15s which wasn't shown in Figure 2.18 to

demonstrate the response of the PAVM to load change. At the same time, as it can be seen in

Figure 2.17, through all transients the large signal behaviour of the detailed model is

accurately predicted by proposed PAVM.

2.5.3.2 Frequency domain

The control-to-output transfer function is often considered in literature for verifying the

small-signal behaviour of the converter models. The small-signal injection and subsequent

frequency sweep method has been implemented to extract the small-signal transfer function

46

from the detailed simulation and the PAVM corresponding to full load operation condition

defined by 717.05R= Ω and 1 0.381d = . The magnitude and phase of the corresponding

control-to-output transfer function are plotted in Figure 2.19.

The transfer function is evaluated up to150kHz, which is more than one-half of the switching

frequency( )250kHz . Closer to the switching frequency the results become distorted due to the

interaction between the injected perturbations and the converter switching. In general,

considering the frequencies closer to and above the switching frequency has limited use for

the average-value model since the basic assumptions of the averaging are no longer valid.

-20

-10

0

10

20

30

Am

plit

ude (

dB

)

102

103

104

105

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Phase (

°)

Frequency (Hz)

PAVM

Detailed

PAVM

Detailed

Figure 2.19 Control-to-output transfer function of the second-order Flyback converter evaluated at

717.05R= Ω and 1 0.381d = .

47

Chapter 3 : Analysis of Flyback Converter with Snubber Circuits

Real semiconductor devices experience voltage and current stress during the turn-on and

turn-off transitions which can easily damage the circuit elements. If a power electronic

converter stresses a power semiconductor device beyond its ratings, there are two basic ways

of relieving the problem. Either the device can be replaced with one whose ratings exceed the

stresses, or a snubber circuits can be added to the converter in order to reduce the stress to

safe levels. The final choice always involves a trade-off between the cost and availability of

semiconductor devices with required electrical ratings compared to the cost and additional

complexity of using the snubber circuits [9, 37-40]. In some topologies, such as a transformer

isolated Flyback converter, the snubber circuit may be required to protect the switching

transistor due to non-zero transformer leakage inductance.

3.1 Fifth –order Flyback Converter with Snubbers

The circuit with snubbers considered in this Chapter is shown in Figure 3.1. Also, a hardware

prototype that has been built includes the snubbers. The parameters and the detailed circuit

diagram are given in Appendices A.5 and Appendix B, respectively. The circuit is has fifth-

order because it has a high order transformer model with primary and secondary leakage

inductances, and two RCsnubber circuits ( ),ss ss ds dsC R C R− − to protect the MOSFET and

diode during turn-on and turn-off switching transitions.

+

+

sL

mL

R

cR

ssR

dsR

swR

C

dsC

ssC

gV

Mosfet

V

dVDn

pL

Figure 3.1 Fifth-order Flyback converter circuit.

48

To demonstrate the role and effect of the snubber circuit, the measured transformer

secondary voltage waveforms with and without the diode snubber is shown in Figure 3.2.

-100

0

100

200

300

Vs

(Volt)

5.983 5.984 5.985 5.986 5.987 5.988 5.989

x10-3Time (s)

-100

0

100

200

300

Vs

(Volt)

( )a

( )b

Figure 3.2 Measured transformer secondary voltage: (a) without the diode snubber; and (b) with the

diode snubber.

As can be seen in Figure 3.2(a), when the snubber is not used on the secondary side, the

voltage spikes up to 310V and the oscillations die pretty much at the end of the switching

interval. This phenomena causes two problems. First, the spike voltage can cause

breakdowns on the switching device. Secondly, the ringing energy will be radiated creating

noise and electro-magnetic interference (EMI) issues with potential for logic and control

errors. This type of ringing is not acceptable and it is necessary to add the snubber circuit

elements to damp the ringing, or clamp the voltage (with RCD clamps), or active snubbers

[40]. In the hardware prototype, a conventional RC snubber is used because of its simplicity

and low part count. Calculation of the RC elements is not the interest of this Thesis, whereas

the reader can find more information in [9, 37-39].

49

It must be emphasized that the snubbers are not a fundamental part of a power electronic

converter circuit. However, when it comes to averaging of PWM converters, the snubbers

significantly add to the complexity of the problem, and to the best of our knowledge this has

not been considered in the prior literature. At the same time, since the snubber circuits have

an important role in overall system dynamics, it is very desirable to take them into account in

the average-value modeling. On the one hand, the benefit of considering the snubbers will be

the improved accuracy of the average model in terms of capturing and predicting the

converter losses and efficiency, which will be used in Section 4.4.3. On the other hand, their

presence in the circuit has introduced more complexity and theoretical problems which are

almost impossible or impractical to resolve using conventional analytical derivations of the

average-value models as will be explained in more detail in Section 3.2.

3.2 State-Space Averaging Phenomena with the Snubbers

The circuit considered in this Section is shown in Figure 3.1. This circuit is a fifth-order

system which makes it more difficult to derive state equations analytically. A more practical

method to calculate the state-space equation is to built a detailed switching model using any

commercially available software programs and extract the state-space matrices numerically

from the detailed model [16, 30] for each subinterval. These matrices can then be used for the

state-space averaging.

A detailed analysis of the fifth-order Flyback converter will be given in Chapter 4. In this

Section, we will focus on only the problem caused by the snubbers, not the whole converter

circuit. The problem occurs on the output filter capacitor voltage, which also translates to the

output voltage which has been defined as an output in state-space output equation. To

demonstrate this problem caused by the snubbers, a detailed model has been built using

PLECS [16]. The corresponding converter parameters are summarized in Appendix A.5. The

state matrices , , ,k k k kA B C D have been extracted for each subinterval in CCM. Finally, the

state-space averaged model has been calculated using these matrices and given duty ratio.

The simulated output filter capacitor voltage and the output voltage are shown Figure 3.3.

50

0.0499 0.04991 0.04992 0.04993

-80

-60

-40

-20

0

Time (s)

V (

Vo

lt)

Detailed Model

Actual Average

State-Space Averaging

c

-80

-60

-40

-20

0

V (

Vo

lt)

Detailed Modeland Actual Average

Detailed Modeland Actual Average

Figure 3.3 Simulated output filter capacitor voltage and the output voltage of the fifth-order Flyback

converter with snubbers in CCM.

As can be seen in Figure 3.3, the output voltage and capacitor voltage is about 72V− , which

corresponds to the considered operating point in CCM. However, the state-space averaged

predicts the very low value of about1.25V− . Since the output voltage and the capacitor

voltage are continuous and slow variables, the conventional state-space averaging should

predict the average value correctly but in this can it does not! The reason why this is

happening can be understood by examining the Figure 3.1. The output filter capacitor voltage

is created by the charging current that flows through the diode. On the one hand, if the

average current is predicted correctly, the capacitor voltage will also be correct. But on the

other hand, if we look at the final state-space averaged equation (3.1), the extracted from the

detailed model diode current is not a state variable. As a result, the output capacitor voltage

does not depend on the diode current in the averaged state-space model. Instead, the output

filter capacitor voltage depends on the secondary side current ( )sI , the diode snubber

capacitor voltage( )dsCV , and the diode voltage drop ( )dV . The diode voltage drop is a small

constant that is not significant in this discussion. Instead, let us look at the averaged

secondary current ( )sI , and the diode snubber capacitor voltage ( )dsCV shown in Figure 3.4.

51

7 9

5 6 7 7 5

5 6 6

7 9 7

205.33 0 0 28393 141.96

0 7.946 10 1.333 10 0 0

2.593 10 1.572 10 1.572 10 3.115 10 1.557 10

43538 2.6 10 2.6 10 5.231 10 26157

3.123 10 0 0 3.752 10 3.123 10

ss

ds

c

C

p

s

C

v

v

i

i

v

•

•

•

•

•

− − × ×

= × − × − × × × − × × − × −

× × − ×

6 5

5

7

0 141.96

0 0

2.509 10 2.593 10

4.151 10 43543

0 3.123 10

ss

ds

c

C

p

s

C

g

d

v

v

i

i

v

V

V

− + × × − × − ×

(3.1)

-0.4

-0.2

0

0.2

0.4

I(A

mp.)

s

0.01232 0.012321 0.012322 0.012323 0.012324 0.012325 0.012326-100

0

100

200

Time (s)

VC

ds(V

olt)

Detailed Model

Actual Average


Actual Average andState- Space Averaging

Figure 3.4 The predicted secondary current and the diode snubber capacitor voltage of fifth-order

Flyback converter in CCM.

As can be seen in Figure 3.4, the averaged state-space model predicts the secondary current

very low at 0.01− , but its actual value must be about 0.1− . The reason being is that the

secondary current is a discontinuous variable even though the converter operates in CCM.

Also, the diode snubber capacitor voltage represents another problem. Here, he averaged

state-space model predicts this value as zero which is also the actual averaged because of the

52

characteristic of the snubber as it releases the energy stored during the ringing. As a result,

the output filter capacitor voltage has a very large error which translates into the large error

in predicted output voltage.

For the purpose of further investigation, let us look at a circuit without a diode snubber,

shown in Figure 3.5. This circuit is a forth-order system.

+

+sL

mL

R

cR

ssRswR

C

ssC

gV

Mosfet

V

dVDn

pL

Figure 3.5 Forth-order Flyback converter without diode snubber.

A detailed model of this simplifies fourth-order converter has been built in PLECS [16]. The

corresponding parameters are summarized in Appendix A.5. The state-space matrices

, , ,k k k kA B C D have been extracted for the CCM in each subinterval. Finally, the state-space

averaged model has been implemented using these matrices and the given duty ratio. The

corresponding simulated output filter capacitor voltage and the output voltage are shown in

Figure 3.6. As can be seen in Figure 3.6, the state-space averaged model predicts the output

filter capacitor cV and output voltage V with a very small error. The reason for this small

error is the discontinuous secondary current, which the state-space averaging cannot predict

this current correctly.

53

-72.256

-72.252

-72.248

-72.244

-72.24

Vc (

Volt)

0.199991 0.199993 0.199995 0.199997 0.199999-72.28

-72.27

-72.26

-72.25

-72.24

-72.23

Time (s)

V (

Volt)

Detailed Model

Actual Average


Figure 3.6 The simulated output filter capacitor and output voltage of the forth-order Flyback converter

without the diode snubber in CCM.

Let us now look at the averaged state-space equation (3.2) of the fort-order flyback converter.

The output filter capacitor voltage depends on only the secondary current. Since the state-

space averaging predicts the secondary current with an error, because of the discontinuity,

the output filter capacitor has the error but much less than the fifth-order model.

7 9

5 6 7

5 6

6 5

5

63.383 0 0 28042

0 8.116 10 1.316 10 0

2.561 10 1.548 10 1.548 10 23049

43000 2.561 10 2.561 10 3870

0 0

0 0

1.562 10 2.561 10

2.561 10

ss ss

cc

C C

pp

s

s

vv

v v

ii

ii

•

•

•

•

− − × × = × − × − × − × × −

+× ×

− × 43005

g

d

V

V

−

(3.2)

As it is shown in this Section, the state-space averaging has a problem when the diode

snubber is present. The way to fix this problem is to introduce the diode current as a variable

54

in the averaged state-space model. To achieve that, the circuit has to be separated into two

circuits as shown in Figure 3.7.

+

+

sL

mL

R

cR

ssR

dsR

swR

C

dsC

ssC

gV

Mosfet

V

dVDn

pL

1Circuit 2Circuit

Figure 3.7 Modified fifth-order Flyback converter circuit.

The Circuit 1 is a switched circuit, and on the other Circuit 2 is non switched and can be

removed from the state-space averaging. After such partitioning, it is easy to extract the state

matrices of the Circuit 1 in CCM or DCM for each subinterval using detailed model. Here,

the diode current must be defined as an output in the state-space model, which will be the

input of the Circuit 2. Then, the state-space equations of the Circuit 2 can either be extracted

from the detailed model or calculated analytically. After extracting state-space model for

these circuits, the overall system can be modelled in CCM as shown in Figure 3.8.

x Ax Bu

y Cx Eu

·

= +

= +

( ) ( )

( ) ( )1 1 2 2 1 1 2 2

1 1 2 2 1 1 2 2

x A d A d x B d B d u

y C d C d x E d E d u

·

= + + +

= + + +

State Variables

gV

dVdi

cv

v

ssCvdsCv

pi si

uu

State Variable

1Circuit 2Circuit Figure 3.8 Proposed state-space averaged model of the fifth-order Flyback converter using two sub-

circuits and sub-models.

If the proposed state-space averaged model is implemented, it will be seen that the predicted

output filter capacitor voltage cv and output voltage v is same as Figure 3.3, which is still

55

incorrect. The reason is that the diode current is a discontinuous variable, and so the state-

space averaging of the Circuit 1 does not predict the diode current correctly, which translates

into the corresponding error in the predicted output voltage.

56

Chapter 4 : Full-order Flyback Converter

4.1 State-Space Averaging in CCM

The full circuit considered in this Chapter is shown in Figure 4.1. This circuit corresponds to

the full-order Flyback converter with parasitics and two snubber circuits (one snubber in the

MOSFET and another snubber is on the secondary side with the diode). As mentioned

earlier, when the circuit gets complicated, it is no longer practical to derive the state

equations analytically. There are well defined algorithms [41] and software tools [14-16] that

automatically generate and dynamically update the state-space model for each new

topological state of the system being used. Regardless of the approach or tool used, it is

assumed that inside each subinterval ( )k the system state model may be expressed by the

system matrices , , ,k k k kA B C D .

+

+

sL

mL

R

cR

ssR

dsR

swR

C

dsC

ssC

gV

Mosfet

V

dVDn

pLpR

sR

Figure 4.1 Full-order Flyback converter circuit.

The detailed model has been implemented using PLECS [16]. The converter parameters are

summarized in Appendix A.6. The state matrices , , ,k k k kA B C D have been extracted for each

subinterval in CCM. Finally, the state-space model has been implemented using these

matrices and the given duty ratio. The simulated state variables are shown in Figure 4.2.

57

As can be seen in Figure 4.2, the state-space averaging predicts the switch snubber capacitor

voltage ( )ssCV , and the diode snubber capacitor voltage ( )

dsCV correctly as expected because

these are continuous state variables. However, the state-space averaging predicts the output

filter capacitor voltage ( )cV with a large error. Moreover, the transformers primary ( )pI and

secondary current( )sI , are also predicted with an error. First, the currents deviate from their

average values because of the oscillations seen during the topology changes and as a result

the assumption (2.163) is no longer valid as explained in Section 2.5. Second, both currents

are discontinuous variables although the converter operates in CCM. Regardless of the mode

of operation, CCM or DCM, these currents as individual state variables are discontinuous

and (2.161) is no longer accurate. In particular, the local average of the primary current in the

second interval ( )2d T is zero (see Figure 4.2), whereas the state-space averaging implies that

this value should be 2 pd i . Since 2d and pi are not zero, the result of the state-space

averaging is not zero. Similar scenario is applied to secondary current.

As a result, when it comes to state-space averaging of Flyback converters (if the transformer

model has leakage inductances, which means primary and secondary currents are state

variables), it needs special consideration in CCM as well as in DCM which will be explained

in Section4.2.

58

-80

-60

-40

-20

0

Vc

(Volt)

0

10

20

30

40

V Cs

s(V

olt)

-1

0

1

2

3

I p(V

olt)

-0.4

-0.2

0

0.2

0.4

I s(V

olt)

0.071399 0.071401 0.071403 0.071405 0.071407-100

0

100

200

Time (s)

VC

ds

(Volt)

Detailed Model

Actual Average


Detailed ModelActual Average

Actual Average andState-Space Averaging


Figure 4.2 Predicted state variables of full-order Flyback converter in CCM.

59

4.2 State-Space Averaging in DCM

The DCM operation of PWM converters differs from CCM operation by an additional time

interval in each switching cycle during which the inductor current or capacitor voltage is

clamped to zero. In Flyback converters, there is an exception in this case if there is a diode

snubber in the circuit. To demonstrate this point, a detailed model of the converter circuit

depicted in Figure 4.1 has been implemented using PLECS [16]. The corresponding

parameters are summarized in Appendix A.7. The state matrices , , ,k k k kA B C D have been

extracted for each subinterval in DCM. Finally, the state-space averaged model has been

implemented using these matrices and the given duty ratio. The resulting simulated state

variables are shown in Figure 4.3.

60

-100

-50

0

Vc

(Volt)

0

20

40

VC

ss

(Volt)

-0.2

0

0.2

I s(A

mp.)

0.099959 0.099961 0.099963 0.099965 0.099967-100

0

100

200

Time (s)

VC

ds

(Volt)

-0.5

0

0.5

1

1.5

I p(A

mp.)

Detailed Model

Actual Average


Detailed ModelActual Average



1 sd T2 sd T 3 sd T

sT

see Fig 4.4

Figure 4.3 Simulated state variables of the full-order Flyback converter in DCM.

61

0.099963 0.099964 0.099965

-0.04

0

0.04

0.08

Time (s)

I s(A

mp

.)

3 sd T

Figure 4.4 Simulated transformer secondary current of the full-order Flyback converter in DCM.

During the first interval ( )1 sd T , when the MOSFET is on and the diode is off, the primary

side of the transformer stores the energy in the field. Right after the switch change its state,

the stored energy on primary side is transferred to the secondary side, and this energy is spent

during the second interval ( )2 sd T . In CCM, the switches go back to their original position

before this energy being spent. But in DCM this energy is spent before the switches change

their positions. That is where there is an exception. If there is no snubber circuit, after the

energy is spent, the secondary current will stay at zero until switches go back to their original

state. If there is a snubber circuit, after the energy is spent, the energy stored in the diode

snubber capacitor will flow in the other direction until it reaches zero or switches go back

their original state. This special case makes it difficult to identify 2d and 3d in the model

with the presence of parasitics, which will be explained is Section 4.3.1.

As can be seen in Figure 4.3, the state-space averaging predicts the switch snubber capacitor

voltage ( )ssCV , and the diode snubber capacitor voltage ( )

dsCV correctly as expected because

these are continuous state variables. However, the state-space averaging predicts the output

filter capacitor voltage ( )cV with a large error. The reason has already been explained in

Section 3.2. Also the transformer primary ( )pI and secondary current ( )sI are predicted with

an error. As explained in Section 4.1, these currents are discontinuous in either CCM or

DCM. As a result, the conventional state-space averaging method also does not produce the

correct results in DCM.

62

4.3 Parametric Average Value Modeling in CCM and DCM

In this Section, the parametric average-value modeling is extended to the full-order converter

operation in CCM and DCM, which to the best of our knowledge has not been done for the

transformer isolated topologies.

4.3.1 Model Implementation

A detailed model of the converter depicted in Figure 4.1 has been implemented using

PLECS. The system matrices ( ), , ,k k k kA B C D for each subinterval have been extracted

numerically using PLECS and Simulink. Since the detailed model has all the parasitics, the

extracted system matrices have all information required. The element of correction vector

( )M and ( )3d are obtained as parametric functions of the duty-cycle ( )1d , the diode current

( )di and the average value of the state variables ( )x . To obtain the values of ( )3 1,d d x and

( )1,M d x , the detailed model has been run in the operation region of interest (for example;

duty-cycle changes between 0.1 and 0.9, and the load ( )R changes from very low load to

very high load) whereas the state variables are averaged numerically over the prototyping

switching interval. In particular, the average-value of the state vector ( )x is computed in a

steady-state corresponding to given operation point. Specifically, in the steady-state from

equation(2.168) an intermediate vector p is computed using (2.169). Thereafter, the

elements of M are found using (2.170).

To obtain the functions ( )3 1,d d x and ( )1,M d x for the desired operation range, the detailed

simulation is run with different values of control variable ( )1d as well as the load resistance

( )R . The variables resulting from this procedure are 1 2 3 ,, , , , , , , ,ss dsd c C p s Cd d d R i v v i i v . Then, the

correction vector ( )M is computed using (2.169) and (2.170). These variables are stored for

future use in lookup tables. The real challenge here is to calculate the 2d or 3d at any given

operation point. As can be seen in Figure 4.3 and Figure 4.4, there is no easy way to read the

intervals from the state variables' waveforms because of the ringing waveforms caused by

snubbers. To achieve that, the diode current is used, as seen in Figure 4.5. As can be seen in

63

Figure 4.5 , the diode current is zero during the first and third subinterval. Since we know the

first interval, it becomes easy to calculate the third and then second subintervals.

0.105993 0.105995 0.105997 0.105999-0.16

-0.12

-0.08

-0.04

0

Time (s)

I d(A

mp.)

2 sd T ( )1 3 sd d T+

sT

Figure 4.5 The diode current waveform.

The final numerical function for 3d is plotted in Figure 4.6. The variable 3d has a flat surface

corresponding to CCM and varies linearly along the 1d . In DCM, the surface of 3d becomes

non-linear and increases.

As explained in Section 3.2, the filter capacitor voltage ( )cV has a large error even though it

is a continuous voltage. To fix this problem a model proposed in Section 3.2 (see Figure 3.8)

has been used here. If the secondary current is corrected, the diode current will be corrected

as well, and as a result the output capacitor voltage and the output voltage will be fixed

without a correction term. In the PAVM, the correction term 1m of filter capacitor voltage

( )cV can be set to 1.

64

0

1000

2000

3000

0.10.20.30.40.50.60.70.80.910

0.1

0.2

0.3

0.4

0.5

R (ohm)

d1

d3 CCM

DCM

Figure 4.6 Variable 3d as a function of the duty-cycle ( )1d and the load resistance ( )R .

The correction term ( )2m of the primary side snubber capacitor voltage ( )ssCV is plotted in

Figure 4.7 as a function of the duty-cycle ( )1d and the load resistance ( )R . The correction

term ( )3m of the primary current ( )pI is plotted in Figure 4.8 as a function of the duty-cycle

( )1d and the load resistance ( )R . The correction term of the primary current has no flat

surface because some correction is needed in CCM and DCM as explained in Section 4.1.

The correction term ( )4m of the secondary current ( )sI is plotted in Figure 4.9 as a function

of the duty-cycle ( )1d and the load resistance ( )R . Similar to correction term the secondary

current ( )3m , the current ( )sI is always discontinuous and needs correction in CCM and

DCM.

65

01000

20003000

0.10.20.30.40.50.60.70.80.96

0.98

1

1.02

1.04

R (ohm)

d1

m2

CCM and DCM


01000

20003000

0.10.20.30.40.50.60.70.80

2

4

6

8

10

12

14

R (ohm)

d1

m3

CCM and DCM

Figure 4.8 The correction term 3m as a function of the duty-cycle ( )1d and the load resistance ( )R .

66

0

1000

2000

3000

0.10.20.30.40.50.60.70.80.910

0.2

0.4

0.6

0.8

1

1.2

1.4

R (ohm)

d1

m4

CCM and DCM


The correction term ( )5m of the diode’s snubber capacitor voltage ( )dsCV is plotted in Figure

4.10. This correction term has a problem. Since the diode’s snubber capacitor voltage ( )dsCV

is a continuous state variable, the correction term must be equal to 1. In theory, the average

value of the diode snubber capacitor voltage is zero because it stores energy during one

interval and discharge this energy during another interval. As a result, the average value is

should be equal to zero. But when the value of p calculated using (2.170), it doesn’t give

exactly zero but gives very small numbers such as 132 10−⋅ due to the numerical precision of

calculations. Then this is used to calculate the correction term ( )5m . In theory, dsCv must be

equal to zero but in the detailed simulation again we get very small numbers because of the

ringing waveforms. Since the capacitor voltage is a continuous state variable, in the model

implementation the correction term ( )5m is set to 1.

67

0

10002000

3000

0.10.20.30.40.50.60.70.8-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x10

-5

R (ohm)

d1

m5

CCM and DCM


Once the functions ( )3 1,d d x and ( )1,M d x are available and stored, the parametric average-

value model can be implemented as explained in section 2.5.2.

4.4 Case Studies

The flyback converter shown in Figure 4.1 is used for case studies. The detailed model is

implemented in PLECS [16]. The converters parameters are summarized in the AppendixA.6

and A.7 in CCM and DCM, respectively, correspond to the hardware prototype that has been

built, Appendix B. The proposed model, Figure 2.16, has been implemented and compared to

the hardware prototype, the detailed model and conventional state-space averaging model in

both time and frequency domains.

4.4.1 Time Domain

To demonstrate the parametric average-value model in CCM, the following study has been

implemented in the detailed model, the hardware prototype, the conventional stat-space

averaged model and the proposed PAVM. The system is assumed to initially operate in a

steady state defined by 1 0.381d = and 717.05R= Ω . The resulting measured and simulated

output voltage, the primary current and the secondary current shown in Figure 4.11.

As can be seen in Figure 4.11, the developed PAVM predicts the average-value of the

waveforms well in CCM as compared to conventional state-space averaging which does not.

68

There is a difference between measured output voltage and detailed model’s output voltage.

The reason is that the detailed model doesn’t have all the parameters the hardware model has

such as stray capacitances.

-71.7

-71.6

-71.5

-71.4

V (

Vo

lt)

0

1

2

I p(A

mp

.)

0.009992 0.009993 0.009994 0.009995 0.009996-0.4

-0.2

0

0.2

0.4

Time (s)

I s(A

mp

.)

Detailed Model

Hardware Prototype

Actual Average

PAVM

Detailed Model


Actual Average

PAVM

Detailed Model




Figure 4.11 Measured and simulated output voltage, primary and secondary current in CCM at constant

duty-cycle.

The accuracy of the PAVM in predicting the large-signal behaviour in time-domain has also

been verified by studying the effect of sudden change in load. In the following study, the

output of the flyback converter was regulated using the same PI controller designed to

regulate the output voltage at 72V− . In the study being considered, the converter initially

operate in DCM with a 2000Ω load. At 0.01t s= , a parallel load of 500Ω is added which

changes the mode to CCM. The resulting time-domain transients are shown in Figure 4.12.

After load changes the control action brings the output voltage to the desired72V− . Through

69

all transients, the large signal behaviour of the detailed model is accurately predicted by

developed PAVM.

-72

-71.9

-71.8

-71.7V

(V

olt)

-2

-1

0

1

2

3

4

I p(A

mp.)

0.0098 0.0102 0.0106 0.011 0.0114-1

-0.5

0

0.5

Time (s)

I s(A

mp.)

Detailed Model

PAVM

Detailed Model

PAVM

Detailed Model PAVM

Figure 4.12 Simulated output voltage, primary and secondary current during the transient from DCM to

CCM due to the step change in load.

4.4.2 Frequency Domain

The control-to-output transfer function of the full-order converter predicted by the detailed

and the developed PAVM models have been extracted again for the full load operating point

defined by 717.05R= Ω and 1 0.381d = . The magnitude and phase of the corresponding

function are plotted in Figure 4.13. As can be seen in this figure, the developed PAVM

predicts the small-signal characteristic with good agreement with the detailed switching

model.

70

-40

-20

0

20

40

60

Am

plit

ude (

dB

)

102

103

104

105

0

20

40

60

80

100

120

140

160

Phase (

°)

Frequency (Hz)

PAVM

DetailedModel

DetailedModel

PAVM

Figure 4.13 Control-to-output transfer function of the full-order Flyback converter evaluated at

717.05R= Ω and 1 0.381d = .

4.4.3 Efficiency Results

One of the most common use of averaged models is system level modeling, wherein the

models only appear as a black box in the system level modeling with input and output ports.

It is therefore very important that the developed model accurately predicts the terminal

characteristics of the converter module. Among such terminal characteristics the total

converter efficiency is of significant importance. The comparisons of efficiencies predicted

by various averaged models are shown in Table 4.1 assuming the converter operates at full

load in CCM, ( )1717.05 , 0.381R d= Ω = .

71

Table 4.1 Efficiency comparison of the average-value models

Model Type Efficiency (%)

2nd Order Model without Parasitics

Detailed Model 99.88

State-space Averaging 99.77

2nd Order Model with Parasitics


State-space Averaging 98.14

PAVM 98.11

5nd Full-order


State-space Averaging cannot predict

PAVM 85.41

Measured from the base hardware prototype 83.45

As can be seen in Table 4.1, the ability of the model to account for the losses and predict the

efficiency improves with increasing the model order from 2 to 5, wherein the traditional

second-order models are significantly off and over estimate the efficiency to the 98%.

Moreover, the classical state-space averaging cannot be simply extended to the 5th order

model and therefore cannot be used for calculating the efficiency correctly. At the same time,

the proposed PAVM of full-order model predicts the efficiency with only a small error

compared to the hardware prototype and its detailed switching model from which it was

established.

72

Chapter 5 : Conclusion

In this Thesis, the recently established parametric average-value modeling methodology has

been extended to the transformer-isolated Flyback converter topology which includes

parasitics and snubbers. It is shown that the developed model captures includes the effect

parasitics and losses, and is therefore capable of accurately predicting the terminal

characteristics of the converter such as efficiency with the accuracy that has not been

attainable by any previously established average-value models. The developed method

overcomes the complexity and challenges common to many previously developed models

when the parasitics of the circuit elements and snubbers are considered. The numerically

constructed model can function in both CCM and DCM. It has been shown that obtaining an

accurate full order average-value model requires extracting the duty-ratio constraint and the

correction term. The functions of the duty-ratio constraint and correction terms were obtained

numerically by running the detailed simulation at desired operation range. Once the model

established, the resulting model is continuous and valid for large-signal time-domain

transient studies as well as for linearization and subsequent small-signal characterization of

the overall system over a wide range.

It has also been shown that direct application of conventional state-space averaging method

is no longer accurate for flyback converters in CCM when the transformer leakage

inductances are taken into account. A detailed analysis of flyback converters working in

CCM has been done to present the problems in CCM. Regardless of the converter operating

mode (CCM or DCM) the primary and secondary currents are discontinuous. As a result, the

conventional assumptions used for DC-DC converters are no longer acceptable for flyback

converters.

5.1 Future Work

As a next step, the parametric average-value modeling can be extended to other DC-DC

topologies such as forward converters (H Bridge) and Push Pull converters for high voltage

applications to have a full set of models for DC-DC converters. Another potential topic for

73

future research could be to investigate the converters with source ripple and input filters

using PAVM. Since it is very easy to include parasitics in PAVM, one could also include in

the model stray capacitances and core losses in the magnetic components.

There are other kinds of snubber circuits in the practical applications such as RCD

(resistance, capacitance and diode) and active snubbers. They also could be included in

PAVM.

74

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78

Appendices

Appendix A. The Converters Circuit Parameters

A.1 Second-order Flyback Converter Parameters without Parasitics in CCM

1

20.009

:3.8 ,55 , , Re 2705

250

0.381

: 0635

27

1/ 6

:1 ,400 , cov , 1 04

2

g

s

m

v V

Mosfet A V N chanel International ctifier IRLL

f kHz

d

Transformer ICEComponents ICA

L H

n

Diode A V ultra fast re ery diode Central Semiconductor CorpCMR U

C

µ

=

−==

−=

=− −

= 2 ,100 , min , 100 22

717.05

F V alu umelectrolyticcapacitor Sanyo MV AX

R

µ= Ω

A.2 Second-order Flyback Converter Parameters without Parasitics in DCM

1 220.009 , 250 , 0.381, 0.4409

27 , 1/ 6, 22 , 2500g s

m

v V f kHz d d

L H n C F Rµ µ= = = =

= = = = Ω

A.3 Second-order Flyback Converter Parameters with Parasitics in CCM

1

20.009

:3.8 ,55 , , Re 2705

0.04

250

0.381

: 0635

27

1/ 6

:1 ,400 , cov ,

g

sw

s

m

v V


R

f kHz

d


L H

n

Diode A V ultra fast re ery diode Central Semiconductor CorpCM

µ

=

−= Ω

==

−=

=− 1 04

1.25

22 ,100 , min , 100 22

0.09

717.05

d

c

R U

V V

C F V alu umelectrolyticcapacitor Sanyo MV AX

R

R

µ

−=

== Ω

= Ω

79

A.4 Second-order Flyback Converter Parameters with Parasitics in DCM

1 220.009 , 250 , 0.381, 0.4409

27 , 1/ 6, 22 , 2500

0.04 , 1.25 , 0.09

g s

m

sw d c

v V f kHz d d

L H n C F R

R V V R

µ µ= = = =

= = = = Ω= Ω = = Ω

A.5 Fifth-order Flyback Converter Parameters in CCM

1

20.009

:3.8 ,55 , , Re 2705

0.04 , 250

0.381

470 , 10

: 0635

27 , 0.2 , 0.8

1/ 6

:1 ,400 , co

g

sw s

ss ss

m p s

v V


R f kHz

d

C pF R


L H L H L H

n

Diode A V ultra fast re

µ µ µ

=

−= Ω =

== = Ω

−= = =

=− v , 1 04

1.25

100 , 200

22 ,100 , min , 100 22 , 0.09

717.05

d

ds ds

c

ery diode Central Semiconductor CorpCMR U

V V

C pF R

C F V alu umelectrolyticcapacitor Sanyo MV AX R

R

µ

−== = Ω

= = Ω= Ω

A.6 Full-order Flyback Converter Parameters in CCM

1

20.009

:3.8 ,55 , , Re 2705

0.04 , 250

0.381

470 , 10

: 0635

27 , 0.2 , 0.8

210 , 1.35

1/ 6

:1 ,40

g

sw s

ss ss

m p s

p s

v V


R f kHz

d

C pF R


L H L H L H

R m R

n

Diode A

µ µ µ

=

−= Ω =

== = Ω

−= = =

= Ω = Ω

=0 , cov , 1 04

1.25

100 , 200

22 ,100 , min , 100 22 , 0.09

717.05

d

ds ds

c

V ultra fast re ery diode Central Semiconductor CorpCMR U

V V

C pF R


R

µ

− −== = Ω

= = Ω= Ω

80

A.7 Full-order Flyback Converter Parameters in DCM

1

20.009

:3.8 ,55 , , Re 2705

0.04 , 250

0.381

470 , 10

: 0635

27 , 0.2 , 0.8

210 , 1.35

1/ 6

:1 ,40

g

sw s

ss ss

m p s

p s

v V


R f kHz

d

C pF R


L H L H L H

R m R

n

Diode A

µ µ µ

=

−= Ω =

== = Ω

−= = =

= Ω = Ω

=0 , cov , 1 04

1.25

100 , 200

22 ,100 , min , 100 22 , 0.09

2500

d

ds ds

c

V ultra fast re ery diode Central Semiconductor CorpCMR U

V V

C pF R


R

µ

− −== = Ω

= = Ω= Ω

81

Appendix B. Flyback Converter Circuit Diagram

Documents

Parametric Average Value Modeling of Flyback Converters in ... · A flyback converter is a switching power supply topology widely used in low power applications such as chargers and