chapter 6 soft-switching dc-dc converters outlines types of dc-dc converters classification of...

Chapter 6Soft-Switching dc-dc Converters

Outlines• Types of dc-dc converters• Classification of soft-switching resonant converters• Advantages and disadvantages of ZCS and ZVS• Zero-current switching topologies

– The resonant switch– Steady-state analyses of Quasi-resonant converters

• Zero-voltage switching topologies– Resonant switch arrangements– Steady-state analyses of Quasi-resonant converters

• Generalized analysis– The generalized switching cell– The generalized transformation table– Basic Operation of the ZCS QRC cell– Basic Operation of the ZVS QRC cell

• Zero-Voltage and Zero-Current transition converters– Switching transition– The Boost ZVT PWM Converter

Types of dc-dc Converters• The linear power supplies offer the designer four major advantages:• Simplicity in design, no electrical noise in its output, fast dynamic response

time, and low cost. Applications are limited due to several disadvantages: – The input voltage is at least two or three volts higher than the output voltage

because it can only be used as a step down regulator, – Each regulator is limited to only one output, and – Low efficiency when compared to other switching regulators

• High frequency pulse width modulation (PWM) switching regulator circumvents all the linear regulators’ shortcomings:

– Higher efficiency (68% to 90%)– Power transistors operate at their most efficient points– Multi-output applications are possible– Size and the cost are much lower

• PWM switching converters still have several limitations– Greater circuit complexity compared to the linear power supplies– High Electromagnetic Interference (EMI)– Switching speeds below 100kHz because of high stress levels on power

semiconductor devices

The Resonance Concept• Like switch mode dc-to-dc converters, resonant converters

are used to convert dc-to-dc through an additional conversion stage: the resonant stage.

• Advantages– natural commutation of power switches

– low switching power dissipation

– reduced component stresses, which in turn results in an increased power efficiency and an increased switching frequency

– higher operating frequencies result in reduced size and weight of equipment and results in faster responses; hence, a possible reduction in EMI problems

• From a circuit standpoint, a dc-to-dc resonant converter can be described by three major circuit blocks

• The dc-to-ac input inversion circuit, the resonant energy buffer tank circuit, and the ac-to-dc output rectifying circuit

• The resonant tank serves as an energy buffer between the input and the output is normally synthesized by using a lossless frequency selective network

• The ac-to-dc conversion is achieved by incorporating rectifier circuits at the output section of the converter

Fig 6.1 Typical block diagram of soft-switching dc-to-dc converter

Resonant versus Conventional PWM

• In PWM converters, the switching of semiconductor devices normally occurs at high current levels.

• Therefore, when switching at high frequencies these converters are associated with high power dissipation in their switching devices.

• Furthermore, the PWM converters suffer from EMI caused by high frequency harmonic components associated with their quasi-square switching current and/or voltage waveforms

• In the resonant techniques, the switching losses in the semiconductor devices are avoided due to the fact that current through or voltage across the switching device at the switching point is equal to or near zero.

• Compared to the PWM converters, the resonant converters show a promise of achieving the design of small size and weight converters

• Another advantage of resonant converters over PWM converters is the decrease of the harmonic content in the converter voltage and current waveforms.

Classification of soft-Switching Resonant Converters• Quasi-resonant converters (single-ended)

– Zero-current switching (ZCS)– Zero-voltage switching (ZVS)

• Full-resonance converters (conventional)– Series resonant converter (SRC)– Parallel resonant converter (PRC)

• Quasi-squarewave (QSW) converters– Zero-current switching (ZCS)– Zero-voltage switching (ZVS)

• Zero-clamped topologies– Zero-clamped-voltage (CV)– Zero-clamped-current (CC)

• Class E resonant converters• Dc link resonant inverters• Multi-resonant converters

– Zero-current switching (ZCS)– Zero-voltage switching (ZVS)

• Zero Transition Topologies– Zero-voltage transition (ZVT)– Zero- current transition (ZCT)

Advantages and Disadvantages of ZCS and ZVS

• Power switch is turned ON and OFF at Zero-Voltage and Zero-Current

• In ZCS topologies, the rectifying diode has ZVS

• ZVS topologies, the rectifying diode has ZCS

• Both the ZVS and the ZCS utilize transformer leakage inductances and diode junction capacitors and the output parasitic capacitor of the power switch.

• Major disadvantage of the ZVS and ZCS techniques is that they require variable-frequency control to regulate the output

• In ZCS, the power switch turns-OFF at zero current but at turn-ON, the converter still suffers from the capacitor turn-ON loss caused by the output capacitor of the power switch.

Switching Loci

• Typical switching loci for a hard-switching converter without and with a snubber circuit as shown in Fig. 6.2.

Fig 6.2 Switching loci. (a) Without snubber circuit. (b) With snubber circuit.

(a) (b)

• There are two types of switching losses:

– At turn-off, the power transformer leakage inductance produces high di/dt, which results in a high voltage spike across it

– At turn-on, the switching loss is mainly caused by the dissipation of energy stored in the output parasitic capacitor of the power switch

Switching Losses

(a)

Fig 6.3 (a) ZVS at turn-on. (b) ZCS at turn-off.

(b)

Fig 6.4 Typical switching current, voltage, and power loss waveforms at

(a) turn-off and (b) turn-on.

(a) (b)

Zero-Current Switching TopologiesThe Resonant Switch

• Depending on the inductor-capacitor arrangements, there are two possible types of resonant switch arrangements

• The switch is either an L-type or an M-type and can be implemented as a half-wave or a full-wave, i.e. unidirectional or bi-directional

• LC tank forms the resonant tank that causes ZCS to occur

Fig 6.6 Resonant switch. (a) M-type switch. (b) Half-wave implementation. (c) Full-wave

implementation.

Fig 6.5 Resonant switch. (a) L-type switch. (b) Half-wave implementation. (c)

Full-wave implementation.

(a)

(c)(b)

(a)

(b) (c)

Steady-State Analysis of Quasi-Resonant Converters

• To simplify the steady-state analysis

– The filtering components , Lin, LF and Co are large when

compared to the resonant components L and C

– The output filter - Co - R is treated as a constant current source Io

– The output filter Co - R is treated as a constant voltage source Vo

– Switching devices and diodes are ideal

– Reactive circuit components are ideal

Lo

Lo

Fig 6.7 Conventional converters: (a) buck, (b) boost, and (c) buck-boost

(a) (b) (c)

The Buck Resonant Converter

• Replacing the switch by the resonant-type switch, to obtain a quasi-resonant PWM buck converter

• It can be shown that there are four modes of operation under the steady-state condition

Fig 6.8 (a) Conventional buck converter with L-type resonant switch. (b) Simplified equivalent circuit.

(a) (b)

Fig 6.9 (a) Equivalent circuit for Mode I. (b) Equivalent circuit for mode II. (c) Equivalent circuit for mode III. (d) Equivalent circuit for mode IV.

(a)

(c) (d)

(b)

Buck Converter: Equivalent Modes

Mode I [ 0 t t1]

• Mode I starts at t = 0 when S is turned ON• Assume for t > 0, both S and D are ON• The capacitor voltage, vc , is zero and the input voltage is equal to the inductor

voltage

• The inductor current, iL, is given by

• As long as the inductor current is less than Io, the diode will continue conducting and the capacitor voltage remains at zero.

(6.3)

• Hence, the time interval = t1 is given by

(6.4)

• This is the inductor current charging state.

dt

diLV L

in

tL

Vti in

L )(

1tL

VI in

o

t tLI

Vo

in1 1

Buck Converter: Steady-State Analysis

Mode II [t1 t t2]

• Mode II starts at t1, diode is open resonant stage between L and C

• The first-order differential equations that represent this mode are

(6.5a)

(6.5b)

• Inductor current is given by

(6.6)

• The general solution for iL(t) is given by

(6.7)

• Resonant angular frequency and constants:

• A2 to equal, A3 = Io

oLc Ii

dt

dvC

cinL vV

dt

diL

LC

Ii

LCdt

idL

L 02

2 1

31211 )(cos)(sin AttAttAti ooL

o LC

1A

V

Lin

o1

Steady-State Analysis (cont’d)

The time interval in this mode can be derived at t = t2 by setting iL(t2) = 0,

0

sin 122

ttZ

VIti o

o

inol

therefore,

t t tZ I

Vo

o o

in2 2 1

11

sin

Mode III starts at t = t, when the switch is turned OFF.

(6.11)

(6.10)

)(sin 1ttZ

VIti o

o

inoL

)(cos1)( 1ttVtv oinc

(6.8)

(6.9)

is known as the characteristic impedance ZL

Co

Equations iL and vc are given by


Mode III [t2 t t3]:

At t2, the inductor current becomes zero, and the capacitor linearly discharges from (t2) to zero during t2 to t3.

vc

The capacitor current equals to as given by,Io

oc

C Idt

dvCi

The capacitor voltage vc(t) is obtained from Eq. (6.12) from t2 to t with

as the initial value,

V tc ( )2

(6.12)

22 tVttC

Itv c

oc

(6.13)

The initial value (t2) is obtained from previous mode, cV

V t V t tc in o( ) cos ( )2 2 11 (6.14)

Substituting Eq. (6.14) into Eq. (6.13),

v tI

Ct t V t tc

oin

2 2 11 cos (6.15)

At t = t3, the capacitor voltage becomes zero,

t t tC

IV t t

oin o3 3 2 2 11 cos (6.16)

At this point, the diode turns ON and the circuit enters Mode IV.


Mode IV [t3 t t4]:

At this mode the switch remains OFF, but the diode starts conducting at t = t3. Mode IV continues as long as the switch is OFF, and the output current starts the free-wheeling stage through the diode.

Initial conditionsi t

v tl

c

( )

( )

0

0

By turning ON the switch at t = , the cycle repeats these four modes. The dead time is given by,

Tst4

t T t t ts4 1 2 3 (6.17)


Voltage Gain The expression for the voltage gain,

in

o

V

VM

Fig 6.10 Steady-state current and voltage waveforms of buck L-type

Figure 6.10 shows the steady state waveforms for and for the buck converter with L-type switch.

vc Li

Typical Steady-State Waveforms

Substitute for vc(t) from intervals (t2-t1) and (t3-t2), to yield,

2

1

3

2

]))()(())(cos1([1

221

t

t c

t

t

ooin

so dttVtt

C

IdtttV

TV

The voltage gain ratio is given by,

)])((2

)()(sin)[(

1232

22312

12 tttVtt

CV

Itttt

TV

Vc

in

o

o

o

sin

o

(6.19)

(6.18)

Substitute for (t2-t1), (t3-t2) and VC(t2) from the above modes, a closed form

expression for M in terms of the circuit parameters can be obtained.

Voltage Gain

The total input energy over one switching cycle,

E i V dtin in in

Ts

0

(6.20)

Since is equal to , Eq. (6.20) is rewritten as,iin)(tiL

1 2

10

t

in

t

t

LinLin dtVtidtVtiE(6.21)

Substituting for iL(t) from Eqs. (6.2) and (6.8) into the above integrals, Eq. (6.21)

becomes,

E VV

Lt I t t

V

Zt tin in

ino

in

o oo

2

112

2 1 2 1cos (6.22)

Substituting for in

oo CV

ttItt

)(1cos 23

12

(6.23) E V

tI I t t

V

Z

I t t

CVin in o oin

o o

o

in

1

2 13 2

2

with, ,Eq. (6.23) becomes, C

Z oo

1

E V It

t t t tin in o

12 1 3 2

2 (6.24)

Voltage Gain (cont’d)

The output energy over one switching cycle is:

E I V dt I V To o o

T

o o s

s

0

(6.25)

Equating the input and output energy expressions

I V T V It

t t t to o s in o

12 1 3 2

2(6.26)

From Eq. (6.26) the voltage gain is expressed by,

V

V T

tt t t to

in s

1

21

2 1 3 2

Substituting for t1, (t2-t1) and (t3-t2) from previous equations, the voltage gain

becomes

(6.27)

12

1 cos1sin1

2

1tt

I

CV

V

IZ

V

LI

TV

Vo

o

in

in

oo

oin

o

sin

o

(6.28)

Voltage Gain (cont’d)

To simplify and generalize the gain equation, the following normalized parameters are defined:

normalized output voltage in

o

V

VM

o

o

Z

RQ

o

oo R

VI

o

sns f

ff

normalized load

average output current

normalized switching frequency

(6.29a)

(6.29b)

(6.29c)

(6.29d)

By substituting Eq. (6.29) into Eq. (6.28), the final voltage gain is simplified into

cos1

22 M

Q

Q

MfM ns

(6.30)

where,

sin 1 M

Q(6.31)

Normalization

A plot of the control characteristic curve of M vs. fns under various

normalized loads is given in Fig. 6.11

Fig 6.11 Control characteristic curve of M vs. ns for the ZCS buck converter.

Buck-Control Characteristic Curve

Example 6.1

Consider the following specifications for a ZCS buck converter of Fig. 6.8(a). Assume the parameters are: Vin = 25V, Vo = 12V, Io = 1A, fs = 250kHz

Design for the resonant tank parameters L and C and calculate the peak inductor current, and peak capacitor voltage. Determine the time interval for each mode.

Solution:

The voltage gain is . Select . Determine Q from either the control characteristic curve of Fig. 6.11 or from the gain equation of Eq. (6.30). This results in Q approximately equaling 1. Since , the characteristic impedance is given by,

The second equation in terms of L and C is obtained from fo. From the normalized

switching frequency, fo may be given by,

48.025

12

in

o

V

VM

4.0nsf

o

oo I

VR

12Q

RZ o

o 121

LC

6250.4

so

ns

so

ff

f

ff kHz

In terms of the angular frequency, o,

LCfoo

12

Solving Eqs. (6.32) and (6.33) for L and C, we obtain,

H

rad

ZL

o

o

31006.3

sec/106252

12

6

3

F

ZC

oo

02.0

10625212

113

(6.32)

(6.33)

ZCS Buck Converter

The peak inductor current, is given by,

A

Z

VII

o

inopeakl

3

,

The peak capacitor voltage is:

V50

V2v inpeak,c

The time intervals are calculated from the following expressions:

sV

HA

V

LIt

in

o 122.025

)103()1( 6

1

s

f

V

IZtt

o

in

oo

o

795.0

)25

112(sin

2

1122.0

)(sin1

1

112

o

oin

I

ttVCtt

))(cos1( 1223

s

Ao

79.1

1

67.0cos1102502.0795.0

6

For t’max we have

10

1

110

22

)(

tt

tt

s

ss

522.0

122.04.0

sTst 44

Example 6.1 (cont’d)

The ZCS Boost Converter

The boost-quasi-resonant converter with an M-type switch as shown in Fig. 6.13(a), with its equivalent circuit shown in Fig. 6.13(b).

Fig 6.13 (a) ZCS boost converter with M-type switch. (b) Simplified equivalent circuit.

(a)

(b)

Mode I [ 0 t t1]:

Assume switch and the diode are both ON

The output voltage is given by

dt

diLV L

o

The initial inductor current and capacitor voltage,il ( )0 0 oc Vv )0(

Integrating Eq. (6.34), the inductor current becomes, t

L

Vit

L

Vti o

Lo

L 0

When the resonant inductor current reaches the input current, , the diode turns OFF,

I in

V

Lt Io

in1

with t1 given by,

tI L

Vin

o1

At t = t1, the diode turns OFF since = , and the converter enters Mode II. Li I in

ZCS Boost Converter: Steady-State Analysis

Mode II [t1 t t2]: The switch remains closed, but the diode is OFF at tin Mode II as shown in Fig. 6.14(b). This is a resonant mode during which the capacitor voltage starts decreasing resonantly from its initial value of . When = , the capacitor reaches its negative peak. At t = t2, equals zero, and the switch turns OFF, hence, switching at zero-current.

Vo

LiI inLi

The initial conditions,

v t Vc o( )1 oL Iti )( 1

From Fig. 6.14(b), the first derivatives for iL and vc are,

cL v

dt

diL

Linc iI

dt

dvC

Using the same solution technique used in the buck converter to solve the above differential equations, the expression for iL(t)

)(sin)( 1ttZ

VIti o

o

oinL

v t V t tc o o( ) cos ( ) 1

where oLC

1

At , and the time interval can be obtained from evaluating Eq. (6.37) at to yield,

t t 20)( 2 tiLt t 2

)(sin1

)( 112

o

oin

o V

ZItt

11

[ sin ( )]in o

o o

I Z

V

(6.37)

(6.38)

(6.39)


Fig 6.14 (a) Equivalent circuit for mode I. (b) Equivalent circuit for mode II. (c) Equivalent circuit for mode III. (d) Equivalent circuit for mode IV.

(a)

(c) (d)

(b)

ZCS Boost: Equivalent Circuit Modes

Mode III [t2 t t3]:

Mode III starts at t, and the switch and the diode are both open as shown in Fig. 6.14(c). Since is constant, the capacitor starts charging up by the input current source. The capacitor voltage,

vc

t

t inc dtIC

v2

1

I

Ct t v tin

c( ) ( )2 2 (6.40)

The diode begins conducting at when the capacitor voltage is equal to the output voltage, i.e. vc(t3)=Vo.

t t 3

VI

Ct t v to

inc ( ) ( )3 2 2

Time interval in this period

t tI

CV v tin

o c3 2 2 [ ( )] (6.41)

Mode IV [t3 t t4]:

At t3, the capacitor voltage is clamped to the output voltage, and the diode starts conducting again. The cycle of the mode will repeat again at the time of when S is turned ON again

Ts


Mode IV [t3 t t4]:

At t3, the capacitor voltage is clamped to the output voltage, and the diode starts conducting again. The cycle of the mode will repeat again at the time of when S is turned ON again

Ts


Typical steady state waveforms are shown in Fig. 6.15.

Fig 6.15 Steady-state waveforms of the boost converter with M-type switch.

ZCS Boost-Typical Steady-State Waveforms

Voltage Gain

Conservation of energy per switching cycle to express the voltage gain, M V Vo in /

The input energy is,

(6.42)

The output energy,

(6.43) The output current equals io=Iin-iL and io=Iin for intervals 0tt1 and t3t<Ts,,

(6.44)

The input current is obtained from the conservation of output power as:

E V I Tin in in s

sT

ooo dtViE0

sT

t

oin

t

oLino dtVIdtViIE3

1

0

)(

RV

VI

in

oin

2

ZCS Boost Converter

Substituting for the input current and by evaluating Eq. (6.44), the output energy becomes

(6.45) with and , and use the equations for . and from Eqs. (6.39) and (6.41), Eq. (6.45) becomes,

(6.46)

)()2

1(

)()(

3211

30

1

tTVItL

VtIV

tTVIdttL

VIVE

soino

ino

soin

to

inoo

tI L

Vin

o1 ( )T t T t t t t ts s 3 1 2 1 3 2

t t2 1 t t3 2

)]cos1([2

1 2

oino

sinoino VI

CTIVLIE

The voltage gain expression is given by,

(6.47)M

M

f M

Q

Q

Mns

1

2 21

( cos )

where, , M, Io and fns are given as before.

ZCS Boost-Voltage Gain

Fig 6.16 shows the characteristic curve for M vs. as a function of the normalized load.

f ns

Fig 6.16 Characteristic curve for M vs. ns for the boost ZCS converter.

ZCS Boost-Control Characteristic Curve

Example 6.2

Design a boost ZCS converter for the following parameters: V in = 20V, Vo = 40V. Po

=20W, fs = 250kHz.

Solution:The voltage gain is .Let us select . From the characteristic curve of Fig. 6.16, Q can be approximated to 6.0 The characteristic impedance is given by,

220

40

in

o

V

VM 38.0nsf

Resonant frequency is,

(6.49)

Solve Eqs. (6.48) and (6.49) for L and C

To limit the input ripple current and the output voltage, we select,

33.136

80

Q

RZ o

o

kHzkHz

f

f

ff

o

ns

so

89.65738.0

250

nFZ

C

Hf

ZL

oo

o

o

14.18)1089.6572)(33.13(

11

1022.31089.6572

33.13

2

3

63

FCC

HLL

o

o

6

6

108.1100

10322100

(6.48)

ZCS Boost Converter

Example 6.3

Design a boost converter with ZCS, with the following design parameters: V in=25V,

P0=30W at I0=0.5A, and fs=100kHz. Assume the output voltage ripple % 0.2 isV 0

Solution:

From the characteristic curve of Fig. 6.15, approximate Q to 6 when we assume fn = 0.58.

From Q and Ro, the characteristic impedance is obtained from,

, 4.225

60

120)5.0(

30 ,resistance load The

0

220

0

sV

VM

I

PR

100172.4 .

0.58of kHz

Solving for C and L From obtained from the ripple voltage equation for the conventional

boost converter, which is given by,

00

0 0

1206, and 20

RQ Z

Z Z 20

L

C

3 3

3

2 172 10 1080.7 10 / sec

11080.7 10

o rad

L C

FC 27.46HL 51.18

0 , %2.0 CV

v

o

c

000 CRf

D

V

v

s

o

00 0

3

(10 6.147 0.193) 10 16

100 10 120 0.2 100

s c

DC

f R v V

F

ZCS Boost Converter

The time intervals are given by

sV

ILt ir 370.0

60

2.11051.18 3

01

s

V

IZtt i

o

29.360

2.120sin

107.1080

1

sin1

13

0

0112

s

IZ

Vtt

in

193.0

553.3cos12.120

60

107.1080

1

cos11

3

0

023

s

tttTtt

147.6193.029.3370.010

)()( 2312134

FL

F

DDf

RL

o

scrit

890

8.88

12

20.

Example 6.3 (cont’d)

Fig 6.17 (a) ZCS boost converter with L-type switch. (b) Simplified equivalent circuit. (c) Steady-state waveforms.

(b) (c)(a)

Figure 6.17(a) shows the quasi-resonant boost converter by using the L-type resonant switch, and the simplified circuit and its steady state waveforms are shown in Fig.6. 17(b) and (c), respectively.

Other ZCS Boost Converter

ZCS Buck-boost Converter

Quasi-resonant buck-boost converter by using the L-type switch as shown in Fig. 6.18(a), Fig. 6.18(b) shows the simplified equivalent circuit.

Fig 6.18 (a) ZCS buck-boost converter with L-type switch. (b) Simplified equivalent circuit.

(a) (b)

The Buck-boost converter also leads to four modes of operations. Mode I [ 0 t t1]:

Mode I starts at t = 0, the switch and the diode are both conducting. According to Kirchhoff’s law, the voltage equation can be written as

(6.50)oinL VVdt

tdiL

)(

By integrating both sides of Eq. (6.50) with the initial condition of , is given by,

(6.51) and At t = t1, the inductor current reaches IF, forcing the output diode to stop conducting, so t1

can be express as,

(6.52)

0)0( Li

)(tiL

tL

VVti oin

L

)(

( ) 0cv t

1F

in o

LIt

V V

ZCS Buck-Boost Converter-Steady-State Analysis

Mode II [t1 t t2]:

FL

c

Iti

tv

)(

0)(

1

1

This is a resonant stage between L and C with the initial conditions given by

Applying Kirchhoff’s law, in Fig. 6.19(b), the inductor current and capacitor voltage equations may be given as

coinl vVV

dt

diL

cL F

dvC i I

dt

Solving Eqs. (6.53) for t > t,

)(sin)( 1ttZ

VVIti o

o

oinFl

)(cos1)()( 1ttVVtv ooinc

At ,the inductor current reaches zero, ,and the switch stops conducting. The time interval is given by,

t t 2 0)( 2 tiL

)( 12 tt

12 1

1( ) sin ( )F o

o in o

I Zt t

V V

(6.56)

(6.55)

(6.54)

(6.53b)

(6.53a)


Mode III [t2 t t3]:Mode III starts at t = t2 when the inductor current reaches zero. The switch and the diode are both OFF. The capacitor starts to discharge until it reaches zero, and the diode will start to conduct again at t = t3. During this period, the inductor current is zero.

t

t cF

Fc tvttC

IdtI

Cv

2

)()(1

22

The diode begins to conduct at the end of this mode, , because the capacitor voltage is equal to zero

t t 3

)()(0 223 tvttC

Ic

F

where may be obtained from Eq. (6.55) by evaluating it at t = t2. The expression from

Eq. (6.57) for the time between t2 and t3 is,2( )cv t

)()( 223 tvI

Ctt c

F

Mode IV [t3 t t4]:

Between t3 and t4, the switch remains OFF, but the diode is ON. At the end of the cycle, the switch is closed again when the current is zero. The cycle of the modes will repeat again at T s.


Fig 6.19 (a) Equivalent circuit for mode I. (b) Equivalent circuit for mode II. (c) Equivalent circuit for mode III. (d) Equivalent circuit for mode IV.

(c)

(a) (b)

(d)

ZCS Buck-Boost Converter-Equivalent Circuit Modes

The steady state waveforms shown in Fig. 6.20 are the characteristic waveforms for the switch, , and .vc Li

Fig 6.20 Steady-state waveforms for buck-boost converter with L-type switch.

ZCS Buck-Boost Converter – Typical Steady-State Analysis

Voltage Gain • Conservation of energy per switching cycle will be used as before to obtain the voltage gain,

M V Vo in /

• The buck-boost-ZCS converter gain is given by

M

M

f M

Q

Q

Mns

1 2 21

[ cos( )] (6.59)

Fig 6.21 Characteristic curve for M vs. ns for the ZCS buck-boost converter.

ZCS Buck-Boost Converter

Example 6.4

Consider a buck-boost QRC-ZCS converter with the following specifications:Vin=40V, Po=80W at Io=4A, fs=250kHz, Lo=0.1mH, and Co=6 F. Design values for L and C

and determine the output ripple voltage.

Solution:

The output voltage and load resistance are given by

VVo 204

80 5

4

20oR

The voltage gain is given by 20

0.540

o

in

V

V

With M = 0.5, and fns= 0.17, we select , to yield,3Q kHzfo 6.147017.0

250

From Q, and Z0,

oo

o

ZZ

RQ

5 6.0

5

3oZ 6.0CL

and 3106.1470221 ofCL

3106.147026.01

C

From the above equation C and L are given by, nFC 4.180 23 1.6L C H

The duty cycle D is approximately 33% since the voltage gain for the buck-boost is 0.5. Hence, the voltage ripple is,

%67.6102501065

5.036

fRC

D

V

V

oo

o

ZERO-Voltage Switching Topologies

• The Zero-Voltage-Switching (ZVS) Quasi-resonant converter family. Like the ZCS topologies, M-type or L-type switch arrangements can be used. • The power switch is turned ON at zero-voltage (of course the turn OFF also occurs at zero-voltage). • A flyback diode across the switch (body diode) is used to damp the voltage across the capacitor

Figure 6.23(a) shows a MOSFET switch implementation by including an internal body diode and a parasitic capacitance.We will assume and are too small to be included. If the body diode is not fast enough for the designed application or has limited power capabilities, it is practically possible to block it and use an external, fast flyback diode as shown in Fig. 6.23(b). is used to block , and is the actual diode used to carry the reverse switch current.

Cgd gsC

1D

Ds

D2Ds

Fig 6.23 (a) MOSFET implementation. (b) MOSFET switch with fast flyback diode.

(a) (b)

Switch Implementation

ZVS Resonant Switch Arrangements • Next we investigate the buck, boost and buck-boost ZVS topologies using the L-type and M-type resonant switches. • Figure 6.24(a) shows the two possible switch implementations using L- and M-type resonant switches. • The half-wave L-type and M-type MOSFET implementations are shown in Fig 6.24(b), whereas Fig 6.24(c) shows the full-wave implementations for L- and M-type switches.

(a)

(b)

(c)

Fig 6.24 (a) Resonant switch arrangement types for ZVS operation. (b) Half-wave MOSFET implementation. (c) Full-wave MOSFET implementation.

Steady State Analyses of Quasi-Resonant Converters

The Buck Converter

Replacing the switch in Fig. 6.7(a) by the M-type switch of Fig. 6.24(a), we obtain a new ZVS buck converter as shown in Fig. 6.25(a). The simplified equivalent circuit is given in Fig. 6.25(b).

Fig 6.25 (a) Quasi-resonant buck converter with M-type switch. (b) Simplified equivalent circuit.

(a) (b)

Equivalent circuit modes under the steady state condition.

(e)

(d)(c)

(b)(a)

Fig 6.26 Equivalent circuits for (a) mode I, (b) mode II, (c) mode III (t2 t < t2`), (d) mode III (t2` t < t3), and (e) mode IV.

Under steady-state conditions, there are four modes of operation.

Mode I [0 t t1]

Assume initially the power switch is conducting, and the diode is OFF. Mode I starts at t=0 when the switch is turned OFF.

The initial capacitor voltage, is zero, and the inductor current iscv Io

0)0( cv

oL Ii )0(

Applying KCL to Fig. 6.26(a),

Lc i

dt

dvC

since ,the capacitor starts to charge,oL Ii

1( )c ov t I t

C

The voltage across the output diode,

cinD vVv

As long as vc<Vin, the diode remains OFF.

(6.61)

Steady State Analysis

The capacitor voltage reaches the input voltage, , at ,causing the diode to turn ON. Hence, at ,we have

Vin

t t 1

t t 1

v t Vc in( )1

and can be expressed as t1

tCV

Iin

o1

At , the circuit enters Mode II. t t 1

Mode II [t1 t t2]:

Mode II starts at t1 with diode turns ON, circuit enters the resonant stage. At ,the capacitor voltage tends to go negative, hence, forcing the diode across S to turn ON. The initial condition,

t t 2

v t Vc in( )1 =

oL Iti )( 1

The expressions of the current and the voltage The period between and is given by, t2t1

)(sin1 1

12oo

in

o ZI

Vtt

o

and the inductor current at is,t t 2

cos)( 2 oL Iti 1

0 2 1sin ( ) ( )in

o o

Vt t

I Z =

Steady State Analysis (cont’d)

Mode III [t2 t < t3]:

At t2, the capacitor voltage becomes zero, and the inductor current starts to charge

linearly, and it reaches the output current at t = t3. The body diode of the switch turns ON at t = t2

The initial value of the capacitor voltage in Mode III is zero

v tc ( )2 0

The inductor voltage,

inL V

dt

diL

The inductor current can be expressed as,

cos)()( 2 oin

L IttL

Vti

(6.67)

(6.68)

At , the inductor current reaches the output current ,forcing the diode to turn OFF.

3tt i t Il o( )3

Time interval from - ist3 t2

t tI L

Vo

in3 2 1 ( cos ) (6.69)


Mode IV [t3 t < t4]:

In this mode, the inductor current is trapped and held constant at =Io, with = 0, i.e., Li Cv

0L o

c

i I

v

Mode IV will continue as long as the switch remains on. By turning off the switch at ,the switching cycle repeats. The dead time is given by,t t Ts 4 t t4 3

t t T t t ts4 3 1 2 3 (6.70)


ZVS Buck Converter - Voltage Gain

The input energy is given by,

E i V dtin in in

Ts

0

is the current which is equal to .Hence, we haveiin)(tiL

dtVtidtVtidtVtidtVtiE in

T

tLin

t

tL

t

in

t

tLinLin

s

3

3

2

1 2

10

(6.71)

The inductor current equals the output current in Mode I and IV, and for t1 t t2 and t2 t t3,

223121 )(

2)(sin[ tt

L

VttLCItIVE in

oooinin

)]()(cos 323 tTIttI soo

Substituting for the time intervals t1, (t2-t1), (t3-t2) and (Ts-t3) using the normalized

parameters M, Q, and o

E V IQ

M

ML

RT

ML

R

ML

Rin in oo

so

[ cos cos ]

2 2

2

(6.72)

(6.73)

Fig 6.28 Control characteristic curve of M vs. ns for ZVS buck converter.

The output energy is expressed by,

soo

T

ooo TVIdtVIEs

0

Equating the input and output energy

)]cos1(2

[2

1

Q

M

Q

Mf

V

V ns

in

o

A plot of the control characteristic curve of M vs. is shown in Fig. 6.28. f ns

ZVS Buck Converter - Voltage Gain (cont’d)

ZVS Boost Converter

The quasi-resonant boost converter by using the M-type switch as shown in Fig. 6.29(a) with its simplified circuit shown in Fig. 6.29(b).

(a) (b)

Fig 6.29 (a) Quasi-resonant boost converter with M-type switch. (b) Equivalent circuit.

The four circuit modes of operation are shown in Fig. 6.30.

Fig 6.30 equivalent circuit nodes. (a) Mode I. (b) Mode II. (c) Mode III. (d) Mode IV.

(a)

(c)

(b)

(d)

ZVS Boost Converter – Equivalent Circuit Modes

Mode I [ 0 t t1]: Assume for t < 0, the switch is closed while D is open. At t=0, the switch is turned OFF, allowing the capacitor to charge by the constant current I in,

dt

dvCiiI c

CLin

With the initial capacitor voltage equals zero

v tI

Ctc

in( )

The capacitor voltage reaches the output voltage at , t t 1 v t Vc o( )1

tCV

Io

in1

At ,the diode starts conducting since , and the converter enters Mode II. t t 1 v Vc o

(6.76)

(6.77)

(6.78)

Steady-State Analysis

Mode II [t1 t t2]: At t = t1, the resonant stage begins since D is ON and S is OFF,

The initial conditions are and . v t Vc o( )1 inL Iti )( 1

The expression for is given by )(tvc

)(sin)( 1ttZIVtv oOinoc

Inductor current,

)](cos1[)( 1ttIti oinL

Evaluating Eq. (6.79) at t = t2 with vc(t2)=0, the time interval between to can be found to be,t1 t2

)(sin1

)( 112

oin

o

o ZI

Vtt

(6.79)

(6.80)

(6.81)


Mode III [t2 t t3]:

• Mode III starts at t2 when vc reaches zero and S turns ON at ZVS. The switch and the diode

are both conducting, and the inductor current linearly increases to I in.• At , the diode (anti-parallel diode) turns ON, clamping the voltage across C to zero.

t t 3

The initial conditions at are, t t 2

v tc ( )2 0

))(cos1()( 122 ttIti oinL

Because the capacitor voltage is zero, the inductor voltage is equal to the output voltage.

oL V

dt

diL

The inductor current becomes,

)()()( 22 tittL

Vti L

oL

• To achieve ZVS, the switch can be turned ON anytime after t2 and before t’2. At t = t’2, the

inductor current reverses polarity and the switch picks up the current.

At , reaches zero, resulting in the time interval given in Eq. (6.85) t t 3 Li

)()( 223 tiV

Ltt L

o

(6.84)

(6.83)

(6.82a)

(6.82b)

(6.85)


Fig 6.31 Steady-state waveforms for ZVS boost converter.

Substituting the initial condition into the equation,

))(cos1()( 1223 tttIV

Ltt oin

o

At ,the output diode turns OFF and the entire current flows in the transistor and the inductor.

3tt Iin

(6.86)

Mode IV [t3 t t4]:

At time t3, the inductor current reaches zero, and the output diode turns OFF, but the switch remains closed. The cycle of the mode will repeat again at t = Ts .


Voltage Gain:

• The voltage gain in terms of the normalized parameter:

• A plot of the control characteristic curve of M vs. is shown in Fig. 6.32. f ns

)]cos1(2

[2

1

Q

M

Q

Mf

M

M ns

Fig 6.32 Control characteristic curve of M vs. ns for ZVT boost converter.

(6.87)

Example 6.5Design a ZVS-QRC boost converter for the following design parameters: V in=30V, P0=30W at

V0=38V, fns=0.4, and Ts=4 s. Assume the output voltage ripple is limited to 2% at D = 0.4.

Solution:

The voltage gain is and with fns = 0.4, we obtain Q = 0.2. Using the

switching frequency , and , the resonant frequency is obtained from,

3.1in

o

V

VM

1 1250

4ss

f kHzT s

kHzkHzf

f so 625

4.0

250

4.0

30

1(2 )(625) 10

L C

The second equation in terms of L and C is obtained from,

The load resistance is,

Substituting in the above relation for Q, we obtain

0.2o o

o

R RQ

Z LC

13.4830

382

oR

48.13 240.65

0.2

L

C

Solving the above two equations for C and L, we obtain,

nFC 06.1)65.240)(10)(625)(2(

13

FL 3.61)65.240(1006.1 29

ZVS Boost Converter

To calculate Lo and Co, using the voltage ripple to be 2%, we use the following relation,

0.02s o o

D

f R C

where,

4.0s

n

f

fD

2.0

1004.0

oso Rf

C

F6.1613.48102502.0

403

The critical inductor value is given by,

212

ocrit

s

RL D D

f

H9.13

)4.0(4.01102502

13.48 2

3

To achieve a limited ripple current, it is recommended that Lo be set to be about a 100 times

the critical inductor value. So we select Lo=1.4 mH.

The Buck-Boost Converter

The ZVS buck-boost converter with an M-type switch is shown in Fig. 6.34(a) with its equivalent circuit shown in Fig. 6.34(b).

Fig 6.33 (a) Quasi-resonant boost converter with L-type switch. (b) Simplified equivalent circuit.

Fig 6.34 (a) ZVS buck-boost converter with M-type switch. (b) Simplified equivalent circuit.

(a)

(a)

(b)

(b)

Typical steady state waveforms.

Fig 6.35 Equivalent circuits for (a) mode I, (b) mode II, (c) mode III, and (d) mode IV. (e) Steady-state waveforms for c and iL.

(a) (b)

(c) (d)

(e)

Equivalent Modes

The voltage gain in terms of M, Q, and is given in Eq. (6.88),

1)]cos1(

2[

2

1

Q

M

Q

MfM

ns

fns

Figure 6.36 shows the control characteristic curve for M vs. fns.

Fig 6.36 Control characteristic curve of M vs. ns for ZVS buck-boost converter.

(6.88)

Voltage Gain

Generalized Analysis for ZCS

• Quasi-resonant ZCS and ZVS PWM converters that each dc-dc converter family shares the same switching network

• The switching network representation and analysis, including the switching waveforms, can be generalized for each converter family.

• Only the generalized switching-cell with the generalized and normalized parameters for that family need to be analyzed.

• By using generalized parameters, it is possible to generate a single transformation table from which the voltage converter ratios and other important design parameters for each converter can be obtained directly.

The Generalized Switching-Cell

Figure 6.37(a) and (b) show the generalized switching-cells of the quasi-resonant PWM ZCS and ZVS converters.

Fig 6.37 Switching cells. (a) ZCS quasi-resonant converter cell. (b) ZVS quasi-resonant converter cell.

(a) (b)

The following parameters, will be used

ngV

in

gng V

VV

where is the switching-cell average input voltage as shown in Figure 6.37 gV

The normalized cell output current : nFI

o

FnF I

II

where is the switching-cell average output current.FI

The normalized filter capacitor voltage : nFV

in

FnF V

VV

where is the filter capacitor average voltage.FV

The normalized filter inductor current : nTI

The normalized cell input voltage :

o

TnT I

II

where is the filter inductor average current (in the ZCS-QSW CC family).TI

Generalized Analysis Parameters

The normalized cell output average voltage : nbcV

in

bcnbc V

VV

where is the switching-cell average output voltage. bcV

The normalized current entering node b in the switching cell nbI

o

bnb I

II

where is the average current entering node b. bI

Generalized Analysis Parameters

The Generalized Transformation Table

The Generalized Transformation Table is shown in Table 6.1.

By applying the appropriate cell to the conventional DC/DC converters, the ZCT-QRC family can be formed as shown in Fig. 6.38.

Fig 6.38 The dc-dc ZCS QRC family. (a) Buck. (b) Boost. (c) Buck-boost.

(a)

(b)

(c)

Basic ZVS QRC Topologies

Basic Operation of the ZCS-QRC Cell

There are four modes of operation and their analysis is summarized as follows:

Mode 1 ( ) 1ttto

It is assumed that before , was OFF and was ON to carry . 0tt S DFI

Mode 1 starts when is turned ON while is ON, which causes to charge up linearly until the current through it becomes equal to at causing to turn OFF.

S

D

L

FI 1tt D

Mode 2 ( ) starts when turns OFF while is ON, causing a resonant stage between and to start until the current through drops to zero at causing to turn OFF at zero current (soft-switching).

21 ttt D S

C L L 2tt S

Mode 3 ( ) starts when turns OFF at zero current. The resonant capacitor starts discharging linearly causing the voltage across it to drop to zero again, causing to turn ON at zero voltage at .

32 ttt SD

3tt

Mode 4 ( ) is a steady-state mode and nothing happens in it until is turned ON again to start the next switching cycle.

so Tttt 3 S

Fig 6.38 The dc-dc ZCS QRC family. (d) Cuk. (e) Zeta. (f) SEPIC (continued)

(a)

(b)

(c)

Fig 6.39 Main ZCS QRC switching-cell waveforms.

Typical ZCS QRC Waveform

ZCS – QRC Generalized Steady State Analysis

Mode 1( ) 1ttto 0)( tvc

)()( 0ttL

Vti g

l

0)( 1 tvC

Fl Iti )( 1

Mode 2 ( ) 21 ttt )(cos1)( 1ttVtv ogc

)(sin)( 1ttZ

VIti o

o

gFl

0)( 2 tiL

(6.89)

(6.90)

(6.91)

Mode 3 ( )32 ttt cos1)()( 2 g

Fc Vtt

C

Itv

0)( til

0)( 3 tvc

Mode 4 ( ) so Tttt 3

0)( tvc

0)( tiL

(6.92)(6.93)

(6.94)(6.95)

Generalized Intervals Equations

To simplify the analysis, the following time intervals are defined:

)( 01 tto

)( 12 tto

)( 23 tto

))(( 30 tTt so

Fig 6.40 Equivalent circuits for (a) mode I, (b) mode 2, (c) mode 3, and (d) mode 4.

These intervals can be derived as follows:

From Eqs. (6.89) and (6.90), α is given by,

g

Foo V

IZtt )( 01

By using the normalized parameters, we have

ng

nFo QV

MItt )( 01

From Eqs. (6.92) and (6.93); is given by,

)(sin)( 112

ng

nFo QV

MItt

From Eqs. (6.94) and (6.95), γ is given by,

)cos1()( 23 nF

ngo MI

QVtt

From Fig. 6.39 and the intervals , and , we have given by,

ns

so ftTt

2))(( 30

(6.96)

(6.97)

(6.98)

Generalized Intervals Equations (cont’d)

Generalized Gain Equation

The cell output to input generalized gain can be found using the average output diode D voltage as follows:

aveCaveD VV

sTt

t

Cs

dttvT

0

0

)(1

))(cos1()(

2)

sin)((

123

22312 ttVtt

C

IttV

T gF

og

s

By using the normalized parameters defined previously, we have:

)cossin(

222

ngnFns

nD VQ

MIfV

By substituting for the generalized parameters ( , , and ) from Table 6.1 in Eq. (6.99), we will have the gain equation for each converter in the family.

nDVngV nFI

(6.99)

Generalized Peak Resonant Inductor Current (Peak Switch Current)

• The peak resonant inductor current or peak switch current occurs at when . 2)( 1 tt pLo

pLtt

M

QVII ng

nFpLn ,

• The peak resonant capacitor voltage or peak diode voltage occurs at when . : )( 1tt pCo

pCtt

ngpCn VV 2,

(6.101)

Design Control Curves

Fig. 6.41 shows the control characteristic curves of M vs. fns for the ZCS-QRC family as

an example, Fig. 6.42 shows the average and the rms switch currents as a function of the voltage gain.

Fig 6.41 Control characteristic curves of M vs. ns for (a) ZCS QRC buck, (b) ZCS QRC boost, (c) ZCS QRC buck-boost, Cuk, Zeta, and SEPIC.

(a) (b) (c)

Fig 6.42 Some of the ZCS QRC boost main switch (S) normalized stress. (a) Normalized average current, (b) Normalized rms current.

(a) (b)

ZCS – QRC Boost Switch/Peak Values

Basic Operation of the ZVS-QRC Cell

Mode 1 ( ) 1ttto

By applying this cell to the conventional DC/DC converters, the ZVS-QRC family can be formed as shown in Fig. 6.43.

The typical switching waveforms for the ZVS-QRC cell are shown in Fig. 6.44.

Starts when S is turned ON while D is OFF, which causes C to charge up linearly until its voltage reaches a value equal to Vg at causing D to start conducting. 1tt

Mode 2 ( ) 21 ttt

Starts when D turns ON while S is OFF causing a resonant stage between C and L to start until the voltage across C tends to go negative forcing the switch diode D s to turn ON at t =

t2.

Mode 3 ( ) 32 ttt

Starts when S is turned ON at zero voltage (Zero-Voltage Switching). The resonant inductor current starts charging up linearly until it reaches IF causing D to turn OFF at zero current at . 3tt

Mode 4 ( ) so Tttt 3

is a steady-state mode and nothing happens in it until S is turned OFF again to start the next switching cycle.

Fig 6.43 The dc-dc ZVS QRC family. (a) Buck. (b) Boost. (c) Buck-boost.

(b)

(a)

(c)

ZVS QRC Family

(f)

(g) (e)

Fig 6.43 The dc-dc ZVS QRC family. (d) Cuk. (e) Zeta (f) SEPIC (continued)

ZVS QRC Family (cont’d)

Fig 6.44 Main ZVS QRC switching-cell waveforms.

ZVS QRC Switching Cell

Generalized Steady State Analysis

From the description of the modes of operation, the equivalent circuit for each mode can be drawn as shown in Fig. 6.45 Mode 1[ ] 1ttto

0( ) ( )Fc

Iv t t t

C Fl Iti )( gC Vtv )( 1 Fl Iti )( 1

Mode 2: [ ]21 ttt )(sin)( 1ttIZVtv oFogC

)(cos)( 1ttIti oFL

0)( 2 tvc

2 2 1( ) cos ( ) 0L F oi t I t t

Mode 3: [ ]32 ttt

0)( tvc

)()(cos)( 212 ttL

VttIti g

oFl

3( ) 0cv t

FL Iti )( 3

Mode 4: [ ]so Tttt 3

0)( tvc

FL Iti )(

Generalized Intervals Equations

To simplify the analysis, the following time intervals are defined:

)( 01 tto

)( 12 tto

)( 23 tto

))(( 30 tTt so

These intervals can be derived as follows:

From Equations (6.102) and (6.103):

Fo

go IZ

Vtt )( 01

By Using the normalized parameters defined earlier:

nF

ngo MI

QVtt )( 01


)(sin)( 112

nF

ngo MI

QVtt


)cos1()( 23 ng

nFo QV

MItt

From Fig. 6.44 and the above intervals

ns

so ftTt

2))(( 30

Generalized Gain Equation

The cell output to input generalized gain can be found using the average output diode voltage as follows:

D

gavesaveD VVV

g

Tt

ts

s

VdttvT

s

0

0

)(1

goo

Fog

F

s

VttIZ

ttVttC

I

T

)1)((cos)()(

2

11212

201

By using the normalized parameters,

ngnF

ngnFns

nD VQ

MIV

Q

MIfV

)cos1(

222

(6.108)

Design Control Curves

• By substituting for the generalized parameters from Table 6.1 in the generalized equations, design equations for each topology in the family can be found. • Using computer software, design curves can be plotted. • Fig. 6.46 shows the control characteristic curves for the ZVS-QRC family.

Fig 6.46 Control characteristic curves of M vs. ns for (a) ZVS QRC buck, (b) ZVS QRC boost, (c) ZVS QRC buck-boost, Cuk, Zeta, and SEPIC.

(a) (b) (c)

Zero-voltage and Zero-current Transition Converter

• Traditional converters operate with a sinusoidal current through the power switches, which results in high peak and rms currents for the power transistors and high voltage stresses on the rectifier diodes.

• When the line voltage or load current varies over a wide range, Quasi-Resonant Converters are modulated with a wide switching frequency range, making the circuit design difficult to optimize.

• As a compromise between the PWM and resonant techniques, various soft-switching PWM converter techniques proposed to aim at combining desirable features of both the conventional PWM and Quasi-Resonant techniques without a significant increase in the circulating energy.

Switching Transition

• To overcome the limitations of the quasi-resonant converters, zero-voltage transition (ZVT) or zero-current transition (ZCT) is the solution. Instead of using a series resonant network across the power switch, a shunt resonant network is used across the power switch.

• The features of the ZCT PWM and ZVT PWM soft-switching converters are summarized as follows:– Zero-current/voltage turn-off/on for the power switch– Low voltage/current stresses of the power switch and rectifier diode– Minimal circulating energy– Constant-frequency operation– Soft switching for a wide line and load range

• One disadvantage is that the auxiliary switch does not operate with soft switching; it is hard-switching, but the switching loss is much lower than that of a PWM converter.

Fig 6.48 (a) ZVT PWM switching cell. (b) ZCT PWM switching cell.

(a) (b)

The ZVT and ZCT converters differ from the conventional PWM converters by the introduction of a resonant branch.

Figure 6.48(a) shows the ZVT-PWM switching cell

ZVT & ZCT Switch

The Boost ZVT PWM Converter

• The boost ZVT PWM, shown in Fig. 6.49, by placing the ZVT PWM switching cell shown in Fig. 6.48(a) into the conventional boost converter.

Fig 6.49 (a) Boost ZVT PWM. (b) Simplified equivalent circuit.

(a) (b)

Fig 6.50 Equivalent circuits for the six modes of operation: (a) mode I,

(b) mode II, (c) mode III

(a)

(b)

(c)

The switching cycle is divided into six modes,

Mode I [t0 t t1]

• Mode I starts at t = when the auxiliary switch is turned ON.

• Since the main switch, S, and the auxiliary switch were OFF prior to t = , S1

S1t0

t0

• The capacitor voltage ,is equal to the output voltage and also equal to the inductor voltage as given by,

vc Vo

cl

o vdt

diLV

The inductor current is given by,il

)( 0ttL

Vi ol

The above equation assumes zero initial condition for .il

Steady State Analysis

As long as the inductor current is less than ,the diode will stay conducting and the capacitor voltage remains at .At time t1, the inductor current becomes equal to ,D stops conducting, and the circuit enters Mode II. From the above equation, we have,

I inVo

I in

)( 0ttL

VI o

in

The time interval is given by

o

in

V

ILtt )( 01

This is the inductor current charging state.

Mode II [t1 t t2]Mode II starts at t1 when D is OFF, resulting in a resonant stage between and C .During the time between t1 and t2, the main switch, remains OFF, and is still ON, but both diodes are OFF. The initial capacitor voltage is still ,but the initial has changed to .The first order differential equations that represent this mode are given by,

L

SS1

Vo

il

Iin

( )

cin L

Lc

dvC I i

dtdi

v t Ldt


inLL I

LCi

LCdt

id 112

2

The solution for iL and vc is given by,

The time interval between and is given by

The diode voltage starts to charge up due to the decreasing capacitor voltage.

Substituting for , the diode voltage becomes,

inoo

L IttZ

Vi )(sin 1

))(cos2( 1ttVv ooc

t1 t2

( ) cos ( )t to

2 111

2

v V vd o c

vc

v t V V t td o o o cos ( ) 1

Equation (6.109) is obtained from the above two equations.


Mode III [t2 t t3]:

Mode III starts when the capacitor discharging to zero. In this mode the main switch, S remains OFF, the auxiliary switch, is still ON, and both diodes are OFF.Now,

S1

0)( tvc

Mode IV [t3 t t4]:

Mode IV starts at t = ,when the main switch, S is turned ON and the auxiliary switch, is turned OFF. At ,the initial capacitor voltage is zero, and the inductor starts linearly discharging from (t2) to zero during to .The diode D remain OFF since its voltage is

negative, but turns ON at t = to carry the inductor current.

t3

S1t3

il

t3

t4

D1t3

The input voltage is equal to the inductor voltage, and the output voltage is equal to the negative of inductor voltage, .Lv

LL o

div L V

dt

The inductor current for t > t2, is given by,

2 2( ) ( ) ( )oL L

Vi t t t i t

L


Mode V [t4 t t5]:

In this mode, at t = t4 both switches are OFF, and also both diodes are OFF. The inductor

current is zero, and the input current is only going through the capacitor,

dt

dvCI c

in

The capacitor voltage can be expressed as,

4

1( ) ( )c inv t I t t

C

The capacitor is charging up from zero and will reache the output voltage at t = .The time interval is,

5t

in

o

I

CVtt )( 45

then it enters mode VI at this point.


Mode VI [t5 t t6]

When the capacitor reaches the output voltage, D starts conducting, but in this mode, both switches are still OFF. The diode current will equal the input current immediately. At t = , the capacitor voltage is equal to the output voltage until the auxiliary switch is turned ON again, then the cycle will repeat from mode I. The waveforms for the six modes of operation are shown in Fig.6.51.

t5

Fig 6.51 Steady-state waveforms for the ZVT boost converter of Fig 6.49(b).

Steady State Waveforms

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